No title

Author: Jonathan Wahl (Academic Editor)

27 downloads 547 Views 3MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

Vol. 104, No. 1

DUKE MATHEMATICAL JOURNAL

© 2000

ON THE SET-THEORETICAL YANG-BAXTER EQUATION JIANG-HUA LU, MIN YAN, and YONG-CHANG ZHU 1. Introduction. Let V be a vector space. Let R : V ⊗V → V ⊗V be an invertible linear transformation. The Yang-Baxter equation is the equality R12 R13 R23 = R23 R13 R12

(1)

of linear transformations on V ⊗ V ⊗ V . Denote τ (w ⊗ v) = v ⊗ w : V ⊗ V → V ⊗ V and σ = τ ◦ R. Then (1) is equivalent to the braid relation σ12 σ23 σ12 = σ23 σ12 σ23 .

(2)

Because of this, a solution of (1) gives rise to a linear representation of the braid group Bn on V ⊗n for every n. In [D], Drinfel’d raised the question of ﬁnding set-theoretical solutions of the YangBaxter equation. Speciﬁcally, we consider a set S and an invertible map R : S × S → S ×S. We think of the Yang-Baxter equation (1) as an equality of maps from S ×S ×S to S × S × S. As in the linear case, a solution of (1) on a set S gives rise to an action of Bn on the set S n . By studying Poisson groups, Weinstein and Xu [WX] found a way of constructing set-theoretical solutions of the Yang-Baxter equation. Later on, Etingof, Schedler, and Soloviev [ESS] gave a complete classiﬁcation of the nondegenerate set-theoretical solutions R of the Yang-Baxter equation satisfying (τ ◦ R)2 = id (where τ (w, v) = (v, w)). In this paper, we present the following construction of set-theoretical solutions of the Yang-Baxter equation. Theorem 1. Let G be a group. Let ξ and η be left and right actions of G on itself, denoted by (u, v) → ξ(u) v and (u, v) → uη(v) , respectively. If the two actions satisfy the compatibility condition uv = ξ(u) v uη(v) , (3) Received 24 August 1999. 2000 Mathematics Subject Classiﬁcation. Primary 16W30, 81R50; Secondary 57M25. Lu’s research partially supported by a National Science Foundation postdoctoral fellowship and by National Science Foundation grant number DMS-9803624. Yan’s research supported by Research Grant Council earmark grant number HKUST 6071/98P. Zhu’s research supported by Research Grant Council earmark grant number HKUST 629/95P. 1

2

LU, YAN, AND ZHU

then

R(u, v) = uη(v) , ξ(u) v

is invertible and is a solution of the Yang-Baxter equation on the set G. In Section 2, we give two conceptual reformulations of the conditions in the theorem. The second reformulation immediately indicates that our construction generalizes the one in [ESS]. In Section 3, we further show that the construction in [WX] is also a special case of our construction. General set-theoretical solutions of the Yang-Baxter equation are related to our construction in the following way. Given a nondegenerate solution R of the YangBaxter equation on a set S, we may consider triples (G, ξ, η) satisfying the conditions of Theorem 1 together with maps S → G commuting with the solutions of the Yang-Baxter equation. We can construct a universal triple and the associated map in the sense that any other triple factors through the map by a homomorphism of the triples. The group in the universal triple is generated by S under the relation uv = yx whenever R(u, v) = (x, y). This group was ﬁrst introduced in [ESS]. The conceptual reason behind the construction in Theorem 1 is the following. It is well known that representations of quasi-triangular Hopf algebras give solutions of the Yang-Baxter equation. Therefore we establish a set-theoretical theory of Hopf algebras in [LYZ1]. Then we establish the corresponding theory of quasi-triangular structures in [LYZ2]. Just as in the usual theory of Hopf algebras, such set-theoretical quasi-triangular structures lead us to the set-theoretical solutions of the Yang-Baxter equation. Acknowledgments. We would like to thank the referee for pointing out the reformulation of our construction in terms of bijective 1-cocycles, which simpliﬁes the presentation of the paper. 2. Braiding operators and bijective 1-cocycles. We start with the proof of our main theorem. Proof of Theorem 1. We verify the braid relation (2) for σ (u, v) = τ R(u, v) = (ξ(u) v, uη(v) ). Denote σ12 σ23 σ12 (u, v, w) = (u1 , v1 , w1 ),

σ23 σ12 σ23 (u, v, w) = (u2 , v2 , w2 ).

The compatibility condition (3) implies that u1 v1 w1 = u2 v2 w2 . Thus it sufﬁces to ξ(u) η(v) prove u1 = u2 and w1 = w2 . By direct computation, we ﬁnd that u1 = ξ( v)ξ(u ) w and u2 = ξ(u)ξ(v) w. Then we have ξ(ξ(u) v)ξ(uη(v) )

ξ(u) v)(uη(v) ))

w = ξ((

(3) ξ(uv)

w=

w = ξ(u)ξ(v) w,

where the ﬁrst and third equalities are due to the fact that ξ is a left action. The equality w1 = w2 can be proved in a similar way.

3

ON THE SET-THEORETICAL YANG-BAXTER EQUATION

To see that R is invertible, let (x, y) = R(u, v), that is, x = uη(v) and y = ξ(u) v. Then ξ(x) −1 −1 (3) (3) (v ) u = ξ(x) (v −1 ) x η(v ) = xv −1 = y −1 u, −1

−1

which implies v −1 = ξ(x ) (y −1 ). Similarly, we have u−1 = (x −1 )η(y ) . Thus, we conclude that R(x −1 , y −1 ) = (u−1 , v −1 ). In other words, if we denote ı(u, v) = (u−1 , v −1 ), then (ıR)2 = id. In particular, this implies that R is invertible. Next we give two alternative descriptions of our construction. Deﬁnition. Let G be a group with multiplication m. A braiding operator on G is a bijective map σ : G × G → G × G satisfying (1) for any u, v, w ∈ G, σ (uv, w) = (id ×m)σ12 σ23 (u, v, w),

(4)

σ (u, vw) = (m × id)σ23 σ12 (u, v, w);

(5)

(2) for any u ∈ G, σ (e, u) = (u, e),

σ (u, e) = (e, u);

(6)

(3) for any u, v ∈ G, mσ (u, v) = uv.

(7)

Let G be any group. Then σ (u, v) = (v, v −1 uv) is a braiding operator, called the conjugate braiding. The corresponding solution R(u, v) = (v −1 uv, v) of the settheoretical Yang-Baxter equation is the conjugate solution. The conjugate solution appeared at the end of [D], and it was attributed to Venkov. Deﬁnition. Let G and A be groups. Let G act on the left of A as an automorphism, denoted as (u, a) → u · a. A bijective 1-cocycle of G with coefﬁcients in A is a bijection π : G → A such that π(uv) = π(u) u · π(v) , for any u, v ∈ G. (8) The deﬁnition is consistent with the one in [EG] and differs from the one in [ESS] by the inverse. We also remark that if (u, a) → u · a is only assumed to be an action, then the bijection and the 1-cocycle condition (8) imply that the action must be an automorphism of A. Theorem 2. Over any group G, the following data are equivalent: (1) (ξ, η): compatible left and right actions of G on itself; (2) σ : braiding operator; (3) (A, π ): a group acted upon by G as an automorphism and a bijective 1-cocycle of G with coefﬁcients in the group.

4

LU, YAN, AND ZHU

Proof. Case 1: (ξ, η) ⇔ σ . The equivalence between (ξ, η) and σ is simply given by σ (u, v) = (ξ(u) v, uη(v) ). First, we assume that ξ and η are compatible and verify that σ is a braiding operator. Denote σ (uv, w) = (u1 , v1 ),

(id ×m)σ12 σ23 (u, v, w) = (u2 , v2 ).

A direct computation gives u1 = ξ(uv) w and u2 = ξ(u)ξ(v) w, which are equal because ξ is a left action. Moreover, the compatibility condition (3) implies that u1 v1 = uvw = u2 v2 . Therefore, we also have v1 = v2 , and the equality (4) is veriﬁed. The equality (5) can be veriﬁed in a similar way. Since ξ is an action, we have ξ(e) v = v. Taking u = e in (3), we have v = ξ(e) ( v)(eη(v) ) = v(eη(v) ), which implies eη(v) = e. This proves the ﬁrst equality in (6). The second equality can be proved similarly. Finally, the equality (7) is exactly the compatibility condition (3). This completes the proof that σ is a braiding operator. Conversely, we assume that σ is a braiding operator. Then by comparing the ﬁrst coordinates of (4), we see that ξ(uv) w = ξ(u)ξ(v) w. Moreover, the ﬁrst equality in (6) implies ξ(e) u = u. This proves ξ is a left action. Similarly, η is a right action. The compatibility condition (3) then follows directly from (7). Case 2: (ξ, η) ⇔ (A, π ). Given compatible actions ξ and η, we take A = G with the following product −1 u v = u ξ(u ) v .

(9)

By replacing v in the formula with ξ(u) v, we ﬁnd that uv = u ξ(u) v, which means exactly that π = id : G → A is a bijective 1-cocycle. It remains to show that (9) is indeed a group structure and that ξ acts as an automorphism of A. Clearly, e is a left unit, and ξ(u) (u−1 ) is a right inverse of u with regard to . By the compatibility condition, we have ξ(u−1 ) −1 v = v (u−1 )η(v) .

uv = u

(10)

−1

Then it is easy to see that e is also a right unit, and (uη(u ) )−1 is a left inverse of u. To prove the associativity of and that ξ acts as an automorphism, we ﬁrst use the compatibility condition (3) to get ξ(u)

(vx) uη(vx) = uvx = ξ(u) v uη(v) x η(x) η(v) . = ξ(u) v ξ(u ) x uη(v)

(11)

5

ON THE SET-THEORETICAL YANG-BAXTER EQUATION

Since η is a right action, the right factors on both sides are the same. Thus by canceling −1 the factor and taking x = ξ(v ) w, we conclude that ξ(u)

(11) ξ(u) ξ(uη(v) )

(v w) = ξ(u) (vx) = (3) ξ(u)

=

v

ξ((ξ(u) v)−1 u)

v

(a) ξ(u)

w=

(a) ξ(u) ξ(uη(v) v −1 )

x=

v

w

v ξ(u) w,

(12)

where we use (a) to denote the use of the fact that ξ is a left action. Moreover, ξ(u−1 )

(12) −1 ξ(u−1 ) v)−1 u−1 ) (v w) = u ξ(u ) v ξ(( w ξ((uv)−1 ) = (u v) w = (u v) w.

u (v w) = u

Conversely, we use π to identify A with G, and denote by the pullback of the product of A to G. Moreover, we take ξ to be the pullback of the action of G on A. Then the 1-cocycle condition means that (9) holds. Finally, we deﬁne η according to the compatibility condition, so that (10) also holds. It remains to show that η is indeed a right action. Since ξ is a left action, we have −1 e e = e(ξ(e ) e) = e. This implies that e is also a unit with respect to . Then by taking u = e in (10), we ﬁnd that v η(e) = v. On the other hand, by the compatibility condition, we still have the equality (11). Moreover, since ξ acts as an automorphism of (G, ), we have ξ(u)

η(v) )

(vx) = ξ(u) (v w) = ξ(u) v ξ(u) w = ξ(u) v ξ(u

x,

where we use part of the computations in the last equality of (12). Then we conclude from the equality (11) that η(x) uη(vx) = uη(v) . This completes the proof that η is a right action. Corollary 3. Any braiding operator is invertible and satisﬁes the braid relation (2). We ﬁnish the section with two useful properties of braiding operators. Proposition 4. Let σ be a braiding operator on a group G. Then σ 2 = id if and only if the star product (9) is commutative. −1

−1

Proof. We have σ −1 (u, v) = (((v −1 )η(u ) )−1 , (ξ(v ) (u−1 ))−1 ) from the proof of the invertibility of R in Theorem 1. Therefore σ 2 = id holds if and only if −1 −1 (ξ(u) v)((v −1 )η(u ) ) = e. On the other hand, we have u v = u(ξ(u ) v) and v −1 u = u((v −1 )η(u) )−1 from (10). Therefore is commutative if and only if (ξ(u ) v) ((v −1 )η(u) ) = e. This condition is clearly equivalent to the condition for σ 2 = id.

6

LU, YAN, AND ZHU

Proposition 5. Let σ be a braiding operator on a group G. If σ (u, v) = (y, x), then σ x −1 , y −1 = v −1 , u−1 ,

σ u−1 , y = v, x −1 ,

σ x, v −1 = y −1 , u .

Proof. The equality σ (x −1 , y −1 ) = (v −1 , u−1 ) can be found in the proof of the −1 invertibility of R in Theorem 1. To prove σ (u−1 , y) = (v, x −1 ), we note that ξ(u ) y = −1 v implies σ (u , y) = (v, z) for some z. Then from the compatibility condition (7) we have u−1 y = vz. On the other hand, by applying the compatibility condition to σ (u, v) = (y, x), we have uv = yx. Thus we conclude that z = x −1 . The proof of the equality σ (x, v −1 ) = (y −1 , u) is similar. 3. Comparison with known solutions. Besides the conjugate solution, there have been two general methods for constructing set-theoretical solutions of the YangBaxter equation, given by Weinstein and Xu [WX] and by Etingof, Schedler, and Soloviev [ESS]. The reformulation in terms of bijective 1-cocycles indicates that our construction generalizes the one in [ESS]. In this section, we compare our construction with the conjugate solution and the solution in [WX]. Denote the conjugate braiding operator on (G, ) by σ (u, v) = v, v −1 u v , where

−1 −1 v −1 = ξ(v) v −1 = v η(v )

is the inverse with respect to the new group structure. Theorem 6. Let σ be a braiding operator on a group G. Let σ be the conjugate braiding operator induced by σ . Then the transformation Tn (u1 , u2 , . . . , un−1 , un ) = u1 , ξ(u1 ) u2 , . . . , ξ(u1 ···un−2 ) un−1 , ξ(u1 ···un−1 ) un is an equivalence between the two actions of the braid group Bn on Gn induced by σ and σ . In [ESS, Proposition 1.7], Etingof, Schedler, and Soloviev showed that the action of the braid group Bn on Gn induced by their solution is equivalent to the usual action via the symmetric group Sn . Note that the action via the symmetric group Sn is induced by the conjugate solution in an abelian group structure. Thus the theorem generalizes the result in [ESS]. We emphasize that the conjugation is with regard to a different group structure on G. Moreover, although the braid group actions are equivalent, the underlying solutions of the Yang-Baxter equation are not necessarily equivalent.

ON THE SET-THEORETICAL YANG-BAXTER EQUATION

7

Proof. It is quite easy to see that Tn is invertible. We mainly need to show that Tn intertwines the two actions. First, we consider the beginning case n = 2. The equality T2 σ = σ T2 means ξ(ξ(u) v)

−1 u ξ(u) v . uη(v) = ξ(u) v

(13)

We have u

ξ(u) −1 v = uξ(u ) ξ(u) v = uv.

(14)

If we substitute u and v by ξ(u) v and uη(v) , respectively, in (14), then we have ξ(u)

v

ξ(ξ(u) v)

uη(v)

=

ξ(u) η(v) = uv. v u

(15)

ξ(u)

Comparing (14) and (15), we have u (ξ(u) v) = ξ(u) v (ξ( v) (uη(v) )), which proves equation (13). In general, let σi,i+1 and σi,i+1 be the actions of the braid group Bn on Gn induced from σ and σ . Then we need to show that Tn , Tn σi,i+1 = σi,i+1

i = 1, 2, . . . , n − 1.

(16)

By induction, we assume that (16) holds for n − 1. Denote Un (u1 , u2 , . . . , un−1 , un ) = u1 , ξ(u1 ) u2 , . . . , ξ(u1 ) un−1 , ξ(u1 ) un . Since ξ is a left action, we have Tn = Un (id × Tn−1 ).

(17)

Since ξ is an automorphism of (G, ), we have ξ(w)

−1 ξ(w) ξ(w) u v. v −1 u v = ξ(w) v

It is easy to see that this implies = σi,i+1 Un Un σi,i+1

(18)

for i ≥ 2. Combining (17), (18), and the inductive hypothesis, we conclude that (16) T is the same as T σ = σ T , holds for i ≥ 2. Finally, the equality Tn σ1,2 = σ1,2 n 2 2 which has already been proved. Now we turn to the solution given by Weinstein and Xu in [WX]. For two subgroups G+ and G− of G, we say that G = G+ G− is a unique factorization if for any g ∈ G, there are unique g+ ∈ G+ and g− ∈ G− such that g = g+ g− . −1 −1 and g− , respectively. The We denote (g+ )−1 ∈ G+ and (g− )−1 ∈ G− simply by g+

8

LU, YAN, AND ZHU

unique factorization induces a left action of the product group G+ × G− on G by −1 (g+ , g− ) · a = g− ag− .

Moreover, it is easy to verify that −1 π(g+ , g− ) = g+ g−

is a bijective 1-cocycle of G+ × G− with coefﬁcients in G. From these data, we may construct a set-theoretical solution of the braid relation over G+ ×G− . Then we may use the bijection π to translate the solution to get a solution over G. A detailed computation shows that this solution is −1 −1 −1 σ (g, h) = h−1 − gh− , h− gh− + h h− gh− + . This solution appeared in [WX, Theorem 9.2], under the assumption that G is a factorizable Poisson-Lie group. (We also note that g− is deﬁned in [WX] by the −1 equality g = g+ g− , so that in their formula, the “negative components” differ from ours by the inverse.) 4. A universal construction. Let a bijective map σ : S ×S → S ×S be a solution of the braid relation (2) on a set S. Denote σ (u, v) = (y, x).

(19)

We call σ nondegenerate if (i) for any ﬁxed u, the map v → y : S → S is bijective; (ii) for any ﬁxed v, the map u → x : S → S is bijective. A nondegenerate solution has the following property. Lemma 7. Suppose σ : S × S → S × S is nondegenerate. Then (1) for any (u, y), there is unique (v, x) such that (19) holds; (2) for any (v, x), there is unique (u, y) such that (19) holds. Proof. Suppose u, y are given. Then we ﬁx u and ﬁnd the unique v corresponding to y via (i). Once v is found, we may ﬁx v and further ﬁnd the unique x corresponding to u via (ii). This proves the ﬁrst statement. The second statement can be proved similarly. The best way to understand and utilize the conclusion of Lemma 7 is through the graphical interpretation. It is well known that solution (19) of the braid relation ﬁts into the braiding scheme in Figure 1 (which we call a σ -square). The bijectivity of σ means that any row determines the whole σ -square. The conclusion of Lemma 7 means that any column determines the whole σ -square. Moreover, the braid relation means that if we start with the same triple (u, v, w), then the two

ON THE SET-THEORETICAL YANG-BAXTER EQUATION

9

y x ..... ... . . ..... . .... ....... .... σ −1 .... ...... . . . ..... ... ... .... u v

u v ..... ... . . ..... . ..... ...... .... .. .........σ . . . ..... ... ... .... y x

Figure 1. σ -square and σ −1 -square

u v w ..... ... .... ..... ... . . . ..... ..... ... ..... ... . . ... .... ........ . ... . . . . . . . . . ..... ..... . . . . . . ..... ... ..... ........ ... ..... ... .. .. .... ........ . . .. . ..... . . . .... . ... ..... .... . . .. . . ..... .... ... ..... . ... . . ........ . ... . . . . . . . ..... . .. .. ..... y z x

u v w ..... ... .... ..... ... . . . ..... ..... ... ..... ... . ... .... ......... . ... . . ..... . . . ... ... ..... .... ... ..... ........ ... ..... .. ... ..... . ... .... ....... . ... . . ..... . . ..... ....... . . . . ..... .. .. . . . . . . ... ..... .. ... ... .. ......... . ... . . . ..... . . .. .. ..... y z x

Figure 2. Braid relation

chains of three σ -squares in Figure 2 will give the same results (z, y, x) at the bottoms (note that the two braids are topologically equivalent). The purpose of this section is to construct a universal group G = G(S, σ ) with a braiding operator σ G from a set S and from a nondegenerate set-theoretical solution σ of the braid relation on S. As in [ESS], we take the group G to be generated by the set S, subject to the relation uv = yx whenever σ (u, v) = (y, x), which is necessary by the compatibility condition on σ . Then we need to extend σ to a braiding operator σ G on G. To construct σ G , we need to know in detail how G is constructed. Let S be another copy of S, with x ∈ S denoting the element corresponding to x ∈ S. Let S¯ be the disjoint union of S and S . Let U (S) =

∞ i=0

S¯ i = {e}

S¯ S¯ × S¯ S¯ × S¯ × S¯ ···

10

LU, YAN, AND ZHU

¯ Let ∼ be the equivalence relation on U (S) be the free monoid generated by S. generated by guu h ∼ gh, gu uh ∼ gh, and guvh ∼ gyxh whenever σ (u, v) = (y, x), where g, h ∈ U (S) and u, v, x, y ∈ S. Then G = U (S)/ ∼. ¯ Note that x ∈ S becomes We start the construction of σ G by extending σ to S. −1 x in G. Therefore, by Proposition 5, the extension σ¯ : S¯ × S¯ → S¯ × S¯ should be given as follows: If σ (u, v) = (y, x), then σ¯ (u, v) = (y, x),

σ¯ (x , y ) = (v , u ),

σ¯ (u , y) = (v, x ),

σ¯ (x, v ) = (y , u).

By the discussion after the proof of Lemma 7, our assumption implies that σ¯ is well deﬁned. The deﬁnition of σ¯ has the graphical interpretation that the σ -square in Figure 1 induces the σ¯ -squares in Figure 3. u v ..... .... ..... . . . ..... ..... ..... . σ¯ .... ......... . . . ..... . . . . ... y x

y x. . ..... . ..... ..... .. ..... ... ..... σ¯ .... ......... . . . ..... . . ..... u v

y u. . ..... ..... ........ ..... ... ..... σ¯ .... ......... . . . ..... . . ..... v x

x v. ..... . ..... . .. ..... ....... ..... . σ¯ .... ......... . . . ..... . . . . ... u y

Figure 3. Deﬁnition of σ¯

Proposition 8. σ¯ satisﬁes the braid relation. Proof. S¯ × S¯ × S¯ is a disjoint union of eight subsets of the form S1 ×S2 ×S3 , where each Si is either S or S . We show that σ¯ satisﬁes the braid relation on S × S × S, S × S × S , and S × S × S. The other cases can be similarly proved. Case 1: Braid relation on S × S × S. This follows from the fact that σ satisﬁes the braid relation. Case 2: Braid relation on S × S × S . For (u, v, w ) ∈ S × S × S , the triple σ¯ 12 σ¯ 23 σ¯ 12 (u, v, w ) is computed as in the left graph in Figure 4. According to Figure 3, the left graph is equivalent to the right graph. By Lemma 7, we see that u, v, w in the right graph in Figure 4 successively determine the other elements u1 , v1 , w1 , v2 , v3 , u2 in the graph. On the other hand, the triple σ¯ 23 σ¯ 12 σ¯ 23 (u, v, w ) is computed as in the left graph in Figure 5. According to Figure 3, the left graph is equivalent to the middle graph. Furthermore, by the braid relation (see Figure 2) and the bijectivity of σ , the middle graph gives rise to the right graph. Note that the right graphs in Figures 4 and 5 have the same structures and the same u, v, w. As we have argued before, the other elements in the graphs must be the same.

ON THE SET-THEORETICAL YANG-BAXTER EQUATION u v w ..... ... .. . . ..... . ... ..... ....... ... ..... ... . . ... . . ... ........ ... . . . ..... ......u1 .. w ....v. 1 . . . ... . ..... ... ..... ....... ... ... . ......... ... . . . ..... . ... .... .... w ....u1 .... v2 . ... 1 ..... . . ..... ...... .... ..... . .. ..... ... . ... ....... . . ... . ..... .. .. . v3 w1 u1

11

v3 w1 w ..... ... . ..... .... ... . . . ..... .... ... .... ... ... . ........ . . . ..... . ... .... v .... .... v3 .... v2 ... 1 ..... . . . ... ..... ...... ... ..... . ... ..... . ..... ... . . . . . . ... . . ..... .....u1 .......u2 . v1 . . . . . ..... .. .... . . . . . . . ... .. .. ... .... ...... . ... . . ..... . . . ... ... .... u2 u v

Figure 4. σ¯ 12 σ¯ 23 σ¯ 12 on S × S × S u v w ..... . ... . . . ..... . ... ..... ....... ... ..... ... ... .. .... ... .... ......... . . . .... w¯ .... u .. ..... .... v¯1 ... 1 ..... ....... ... ..... . ... ..... ... ... ....... ... . . . . . . . ... . ..... ..... u¯ . . v ¯ w¯ 1 . . 2 . ..... . . ... 1 . ..... .... ... ..... . . .... .. ..... ... .... ........ . . . ... . ... u¯ 1 v¯3 w¯ 2

v3 w¯ 2 w¯ ..... ... .... ..... . . . . ... .... ...... ... ... ... . . . . . . . ... . . ..... ... ... . . . . . ..... . ..... v¯2 ...w ..¯...1 ....... w ... . . . ..... .. .. .... ... ... .. ......... . . . ... . ..... . ... v .... ....v¯2 .... v¯1 .. ..... . . . ... ..... ..... ..... ... . . . ... ... ........ . . ... . . . . . . .... . ... u v u¯ 1

w v¯ w¯ 2 ..... ... 3 .... ..... . . ... . ..... ..... .... ..... ... . ... ... ........ . . . ..... . ... . . ... ∗ .. .....v¯3 .... ∗ ... ..... . . . ... ..... ..... ... ..... . ... . ... ........ ... . . . . . . . .. . ..... . ..... u¯ 1 . ∗ ∗ . . . . ..... . . ... . ..... .... ... . . . .... .... .... ........ . .. . . ..... .. .. . .... u v u¯ 1

Figure 5. σ¯ 23 σ¯ 12 σ¯ 23 on S × S × S

Thus, we conclude that σ¯ 12 σ¯ 23 σ¯ 12 (u, v, w ) = (u2 , v3 , w1 ) = (u¯ 1 , v¯3 , w¯ 2 ) = σ¯ 23 σ¯ 12 σ¯ 23 (u, v, w ). Case 3: Braid relation on S × S × S. For (u, v , w) ∈ S × S × S, the triples σ¯ 12 σ¯ 23 σ¯ 12 (u, v , w) and σ¯ 23 σ¯ 12 σ¯ 23 (u, v , w) are computed as in the graphs in Figure 6. According to Figure 3, the graphs are equivalent to six basic σ squares, four of which ﬁt into a braid relation (see Figure 7). Since the end results of the braid relation in Figure 7 must be the same, we conclude that the result is of the form (v2 , z, v¯2 ). Thus, we get two σ -squares in Figure 8.

12

LU, YAN, AND ZHU

u w v ..... ... .. . . ..... . ... . . . ..... .... ... ..... ... . . ... . . ... ........ ... . . . ..... ......u .. w ....v. 1 . . . ... 1 . ..... ... ..... ....... ... ... . ......... ... . . . ..... . ... .... .... w ....u .... v2 . ... 1 .....1 . . ..... ...... .... ..... . .. ..... ... . ... ....... . . ... . ..... .. .. . w1 u2 v3

u w .v.... ... .. . ... ..... ....... ..... ... ... ..... ... . ... .... ......... ... . . . ..... . . ... u . ... v¯ ... w¯ 1 ..... ..... ........ 1 ... ..... .. ... ..... ... . ...... ... . . . . . . ... . . ..... .....v¯2 .......u¯ 1 . w¯ 1 . . . . . ..... . . .... . ..... .... ... ..... ..... ... . . . ..... . ... .. ..... .. .. ... v¯3 u¯ 1 w¯ 2

Figure 6. Braid relation on S × S × S

u1 u2 ..... . . . . . ..... ..... ....... .... . ........ . . . ..... ... .... ..... v3 v2 w¯ 2 w¯ 1 ..... . ..... .... . . . ..... .... .... . ......... . . . ..... ... ... ..... v¯3 v¯2

v1 v v¯ ... 1 ..... . .... ..... . . . . . ... ..... .... ... ..... . ... . . . ..... . . ... . ..... ... .. ..... u ......u . . ..... .. v¯1 ... 1 . . . . . ..... ... ... ..... . ... . ......... ... . . . ..... . ... .... .... ....u1 .... u¯ 1 . ... v¯2 ..... . . . . . ..... ... .... ..... . ... ..... . .... ... ....... . . . ... ..... ..... ∗ ∗ v¯2

v1 v v¯ ..... .1 ... .... ..... ... . . . ..... .... ... .... ... . ........ ... . . . ..... . ... .... .... .... v1 .... w ... w¯ 1 . ..... . . ... ..... ...... ... ..... . ..... ... . .... ... . . . . . . . . ... . ..... ....w .......v¯2 . . w¯ 1 1 . . . . . ..... ... ..... ....... ... ..... .. ... .... ........ ... . . . ..... . . . . .. v2 ∗ ∗

Figure 7. Interpreting braid relation on S × S × S

Comparing these σ -squares in Figure 8 with the σ -squares in Figure 7, we have u2 = u¯ 1 ,

v3 = z = v¯3 ,

w1 = w¯ 2

from the bijectivity of σ . This further implies σ¯ 12 σ¯ 23 σ¯ 12 (u, v , w) = (u2 , v3 , w1 ) = (u¯ 1 , v¯3 , w¯ 2 ) = σ¯ 23 σ¯ 12 σ¯ 23 (u, v , w). Next we extend σ¯ to σ U over U (S). The extension can be better understood through the graphical meaning of the braid relation. The idea for the deﬁnition of σ U comes from (4) and (5). Note that the product of u, v ∈ S¯ in U (S) is simply the pair (u, v) ∈ S¯ 2 . Therefore, if σ U satisﬁes (4) and (5),

ON THE SET-THEORETICAL YANG-BAXTER EQUATION

u1 u¯ 1 ..... . .... ..... . . . ..... .... .... . ........σ . . . ..... ... .... ..... v2 z

13

w2 w¯ 1 ..... . .... ..... . . . ..... .... .... . ........σ . . . ..... ... .... ..... z v¯2

Figure 8. Two σ -squares from S × S × S

then we should have σ U (u, v), w = σ¯ 12 σ¯ 23 (u, v, w),

σ U u, (v, w) = σ¯ 23 σ¯ 12 (u, v, w),

where the triples on the right sides are considered as in S¯ × S¯ 2 and S¯ 2 × S¯ because (1 × m)(u, v, w) = (u, (v, w)) and (m × 1)(u, v, w) = ((u, v), w) in U (S). Figure 9 provides the graphical interpretation of the deﬁnitions. w (u v) ..... ... .. . . . . . ..... ... ..... ...... ... ... ¯ ... ... .. ......σ . .....23 . . ... ..... ... . . .... . ... ... ..... . . ... . ..... ..... ... . ..... ... ..... σ¯ 12 ... . . . ... ........ ... . . . .... .. .... z (y x)

u (v w) ..... ... ... . . ..... . ... ..... ...... ... ..... ... .....σ¯ 12 ... . . ... ....... ... . . . ..... ..... ..... .... . ... . . ..... .... ... ..... .. ... .. ... . .... σ¯ 23 .... ......... .. . . . .... .. .... x y) (z

Figure 9. σ U on S¯ × S¯ 2 and S¯ 2 × S¯

This leads to the natural generalization that deﬁnes σ U ((u1 , . . . , um ), (v1 , . . . , vn )). For example, the graph in Figure 10 suggests that we deﬁne σ U (u1 , u2 , u3 ), (v1 , v2 ) = σ¯ 23 σ¯ 12 σ¯ 34 σ¯ 23 σ¯ 45 σ¯ 34 (u1 , u2 , u3 ), (v1 , v2 ) . In general, we deﬁne σ U : U (S) × U (S) → U (S) × U (S) by (1) σ U (e, u) = (u, e), σ U (u, e) = (e, u); (2) on S¯ m × S¯ n , m, n ≥ 1, σ U = σ¯ n,n+1 · · · σ¯ 23 σ¯ 12 σ¯ n+1,n+2 · · · σ¯ 34 σ¯ 23 · · · σ¯ n+m−1,n+m · · · σ¯ m+1,m+2 σ¯ m,m+1 , and the image is considered to be in S¯ n × S¯ m .

14

LU, YAN, AND ZHU

(u1 u2 u3 ) (v1 v2 ) ..... . . ... ... .... ....... ..... . . . ... ..... . . . . . . . . .. . ..... . .... ..... σ¯ 34 ......... ........ ..... ..... .... ..... ..... ..... σ¯ 23 ....... ... .......σ¯ 45 ..... .... ..... .. . .. . . ¯ 34 ......... ..... σ¯ 12........... .... .......σ .. .... .... ......... σ¯ 23 ......... . . .... . . . . . . . . . . . . . ..... .. .... ... ...... . . . . . . . ... .. . . .. (y1 y2 ) (x1 x2 x3 ) Figure 10. σ U on S¯ 3 × S¯ 2

It is easy to see (e.g., from the graphical interpretation) that the deﬁnition above is equivalent to the requirement that the ﬁrst two conditions for σ U being a braiding operator on U (S) (see the conditions (4), (5), and (6)) are satisﬁed. Moreover, σ U still satisﬁes the braid relation. This can be seen graphically by “thickening” each of the three threads in Figure 2 into parallel threads. The result is a pair of graphs that are related by a sequence of the usual braid relations (i.e., Figure 2) with some additional vertical lines (i.e., identity maps) added on both sides. For example, the graphs in Figure 11 show that σ U satisﬁes the braid relation over S¯ 3 × S¯ 2 × S¯ 2 . ............... ... ... ............... .............. ... ... ............... ....... ... ... ............... ... ... . . . . . . . ... ... . . . . . ......... ...................... . ... ... . . . . . . . . . . . . . . . . ............... .. .... .... . . . . . . . . . . . ............... ...... ... ... ... ............... ........ ... ... ................. ... ... ... ... ................... . . ... ... ......... .......................... . . . . ......... ........ .. .. .. .. ........ .......... ... ... ... .......... ................ .......... ..... ... ... ... ... ... ... .................. . . . . ... ... ... ......... .............. . . . .......... ... ... ... .. .. ........

.......... ... ... ... .. .......... .............. ... ... ... .......... ........ .. ... ... ...... . .... ... ... ........................ ... ... ... . . . ... ... ... ........ ................ ............ ... ... ............. . . ............... . ... ... ............... ............... ... ... ............... .. ... ... . . . . . . . . . ... ... . . ........ ...................... . ... ... . . . ............ . . . . . . . .. . . ............... .... .... . . . . . ............... ...... ... .. .. ............... .......... ... ... ................ ... ... ................... ... ... . . .......... .......................... . . .... .... . ............... .. .. .. . ........

Figure 11. Braid relation for σ U on S¯ 3 × S¯ 2 × S¯ 2

It remains to reduce the operator σ U over U (S) to an operator σ G over G. We ﬁrst consider the reduction with respect to the relations guu h ∼ gh and gu uh ∼ gh in U (S). Note that in the case σ (u, v) = (y, x), it is easy to show that σ U ((u, u ), y) = (y, (x, x )) by the graph in Figure 12. Now for g, h, k ∈ U (S), we embed several copies (the number equals the length of k) of the graph in Figure 12 (with various y) into the graph for the deﬁnition of

ON THE SET-THEORETICAL YANG-BAXTER EQUATION

15

y (u u ) ..... ... ... . . ..... ... . ..... ...... ... .... ... ... . ........ . . . ..... . ... .. .... ... u ....v . . ... x ..... . ..... ....... ... ..... .. ... ..... ... . . . ... . ... ........ . ... . . .... . .... y (x x) Figure 12. A special case of σ U consistent with uu ∼ e

σ U (gh, k). The result, as shown in Figure 13 (the region bounded by the dotted square is where the embedding happens), is the graph for the deﬁnition of σ U (guu h, k). (g (g u u h) h) k k ... ... ... . . . . .. . . ... ... ..... ..... ..... ....... ... ...... ... ... ... ...... ... ...... ... ... ... ... . . . ..... ... ... ........ .. .......... ............... .... .... . . . . . ... . . . . . . ... .. ... ........ .. .... ... . ...................... ... .... . ... ... ......... ... ... ... . . . . . ... .... ... ... ......... ......... ...... ... ...... ... . .. ... ... . . . . . .... ... ... .... . . ... . . . . . . . . . . . . . . . . . . . . . ... ... .... ... ... ... ... ... .. .. . . . . .... .... r r (s x x t) (s t) Figure 13. A general case of σ U consistent with uu ∼ e

It is easy to see from the graphs in Figure 13 that σ U (guu h, k) is obtained from by inserting some xx in an appropriate place. Similarly, σ U (g, huu k), U σ (guu h, k), and σ U (g, hu uk) are obtained from σ U (g, hk) and/or σ U (gh, k) by inserting some xx and/or x x in some appropriate places. This implies that σ U is consistent with the relations guu h ∼ gh and gu uh ∼ gh. Next, we consider the reduction with respect to the relation guvh ∼ gyxh whenever σ (u, v) = (y, x). Again, we ﬁrst consider the simplest case. Suppose u, v, w ∈ S, σ (u, v) = (y, x), and σ U (u, v), w = (r1 , s1 , t1 ), σ U (y, x), w = (r2 , s2 , t2 ). σ U (gh, k)

By the braid relation, the two graphs in Figure 14 should have the same outcome at the bottoms. In particular, we conclude that r1 = r2 and σ (s1 , t1 ) = (s2 , t2 ). Moreover,

16

LU, YAN, AND ZHU

w (u v) ..... . ... .... ..... . . . . . ... ..... .... ... ..... ... . . . . . . . . . ... . ..... ... ... . . . . . ..... y) .... (x . ..... .. w . ... . . . . ..... ... ... ..... . ... . ........ ... . . . ..... . ... .... .... .... .... ... . ..... . . . . ... . ..... ... ..... . ... . ... . ....... . . . ..... ... ... . . . . .. . .. .. r2 (s2 t2 )

w (u v) ..... . ... .... ..... ... . . . ..... .... ... ... ... ... . ......... . . . ..... . ... .... .... ... .... ... ..... . . . ..... ..... ... ..... .. ... ..... ... . ..... ... . . . . . . . ... . . ..... ..... (s1 ....... r1 . . t ) . . . . ..... . 1 .... ..... ...... ... ..... . ... .... ......... ... . . . ..... . . . . .. r2 (s2 t2 )

Figure 14. A special case of σ U consistent with uv ∼ yx

the argument above also applies to the case that some of u, v, w are assumed to be in S , because of Proposition 8. As in the proof of the consistency with the relations guu h ∼ gh and gu uh ∼ gh, we may embed Figure 14 (and its generalization over ¯ into the graphs for the deﬁnitions of σ U (guvh, k) and σ U (gyxh, k) and ﬁnd that S) σ U (gyxh, k) is obtained from σ U (guvh, k) by applying σ at an appropriate adjacent pair of coordinates. The similar statements can be proved regarding σ U (g, huvk) and σ U (g, hyxk). This implies that σ U is consistent with the relation guvh ∼ gyxh. Thus, we have shown that σ U can be reduced to an operator σ G on G. Since σ U satisﬁes the conditions (4), (5), and (6), σ G satisﬁes the same conditions. To conclude that σ G is a braiding operator, it remains to explain that σ G satisﬁes the compatibility condition (7). But this is quite obvious because, by the deﬁnition, (g, h) ∈ U (S) × U (S) and σ U (g, h) differ by the successive applications of σ on some adjacent pairs of coordinates in the words (i.e., ignore the parentheses). Such applications of σ are considered as identity in G. Therefore, we conclude that gh ∈ G is equal to the multiplication of the two coordinates in σ G (g, h) ∈ G × G. Now we are ready to state the main result of this section. Theorem 9. Let a bijective map σ : S × S → S × S be a nondegenerate solution of the braid relation. Let G(S, σ ) be the group generated by S and be subject to the relation uv ∼ yx whenever σ (u, v) = (y, x). Let i : S → G(S, σ ) be the canonical map. Then (1) there is a unique braiding operator σ G on G(S, σ ) such that σ G (i × i) = (i × i)σ ; (2) the group G(S, σ ) and the braiding operator σ G have the following universal property: If σ is a braiding operator on a group G and if f : S → G is a braiding-preserving map, then there is a unique braiding-preserving group homomorphism φ : G → G such that f = φi.

ON THE SET-THEORETICAL YANG-BAXTER EQUATION

17

We have shown the existence of σ G by constructing it. The uniqueness follows from the conditions (4), (5), and (6), as we explained when we deﬁned σ U . The universal property is also tautological from the construction. In the special case σ 2 = id studied in [ESS], it is not difﬁcult to use the graphical interpretation to show that (σ G )2 = id. By Proposition 4, the star product (9) on the universal group G(S, σ ) is commutative. Then it is easy to show that (G(S, σ ), ) is in fact free abelian with S as a basis. In particular, i is an embedding, and we recover the classiﬁcation theorem in [ESS]. In general, a set-theoretical solution of the Yang-Baxter equation is embedded in a group with a braiding operator if and only if the canonical map i : S → G(S, σ ) into the universal construction is an embedding. However, the following example from P. Etingof indicates that the canonical map is not always an embedding. For two commuting automorphisms f, g of S, σ (u, v) = (g(v), f (u)) is a nondegenerate solution of the braid relation. In this case, we have uv = f g(u)f g(v) in G(S, σ ). So if fg has at least one ﬁxed point u in S, then f g(v) = v in G(S, σ ) for all v, although it may not be so in S. We end the paper with a remark on the universal group G(S, σ ). Proposition 10. If S is a ﬁnite set, then G(S, σ ) has a ﬁnitely generated abelian normal subgroup of ﬁnite index. ¯ be the permutation group of i(S) ¯ = i(S) ∪ i(S)−1 , the image Proof. Let P (i(S)) U of S, and the inverses in G(S, σ ). Since σ exchanges S¯ i × S¯ and S¯ × S¯ i , the actions ξ and η (associated to σ G ) induce a homomorphism and an antihomomorphism of groups ¯ . ξS¯ , ηS¯ : G(S, σ ) −→ P i(S) ¯ is ﬁnite, A = ker ξ ¯ ∩ ker η ¯ is a normal subgroup of ﬁnite index. We Since P (i(S)) S S claim that A is abelian. ¯ By the compatibility Suppose g ∈ ker ηS¯ . Then i(u)η(g) = i(u) for any u ∈ S. ξ(i(u)) η(g) g)(i(u) ) = (ξ(i(u)) g)i(u). Therefore condition (3), we also have i(u)g = ( we conclude that ξ(i(u))

g = i(u)gi(u)−1 .

(20)

Suppose we have shown that hη(g) = h for all elements h ∈ G(S, σ ) of length less ¯ Then we write an element than or equal to n−1 (with regard to the generating set S). of length less than or equal to n as h i(u), where h has length less than or equal to ¯ By the condition (4), we have n − 1 and u ∈ S.

η(g)

h i(u)

= hη(

ξ(i(u)) g)

i(u)η(g) = hη(i(u)gi(u)

−1 )

i(u),

(21)

where we use g ∈ ker ηS¯ and (20) in the last equality. From the deﬁnition of σ U , we see that hη(i(u)) still has length less than or equal to n − 1. Thus, by the inductive

18

LU, YAN, AND ZHU

assumption, we have hη(i(u)gi(u)

−1 )

=

η(i(u)) η(g) η(i(u)−1 ) η(i(u)) η(i(u)−1 ) h = h = h.

(22)

Combining (21) and (22) together, we conclude that η(g) ﬁxes hi(u). Thus, we have shown that any g ∈ ker ηS¯ actually ﬁxes the whole group G(S, σ ) under η. Similarly, any g ∈ ker ξS¯ ﬁxes the whole group G(S, σ ) under ξ . Therefore, if g, h ∈ A, then we have σ G (g, h) = ξ(g) h, g η(h) = (h, g). By the compatibility condition (3), this further implies gh = hg. This proves that A is abelian. Finally, A is ﬁnitely generated because a subgroup A of a ﬁnitely generated group G(S, σ ) of ﬁnite index is always ﬁnitely generated. This fact can be proved, for example, by considering the pullback to the ﬁnitely generated free group with a homomorphism onto G(S, σ ). The subgroup A constructed in the last proposition contains all the “trivial” actions of G(S, σ ), so that it does not essentially contribute to the construction of our solution. In the special case σ 2 = id, we already know (G(S, σ ), ) is a free abelian group of rank |S|. Then the triviality of A acting on G(S, σ ) implies that the original product and the new star product coincide on A. Therefore, A is also a subgroup of (G(S, σ ), ) of ﬁnite index, and A itself must be a free abelian group of rank |S|. This leaves the interesting question of the rank of A in general. We suspect that the rank is also |S|. But we do not know how to show it. References [D] [EG] [ESS] [LYZ1] [LYZ2]

[WX]

V. Drinfel’d, “On some unsolved problems in quantum group theory” in Quantum Groups (Leningrad, 1990), Lecture Notes in Math. 1510, Springer, Berlin, 1992, 1–8. P. Etingof and S. Gelaki, A method of construction of ﬁnite-dimensional triangular semisimple Hopf algebras, Math. Res. Lett. 5 (1998), 551–561. P. Etingof, T. Schedler, and A. Soloviev, Set-theoretical solutions to the quantum YangBaxter equation, Duke Math. J. 100 (1999), 169–209. J.-H. Lu, M. Yan, and Y.-C. Zhu, On Hopf algebras with positive bases, preprint, 1998. , Quasi-triangular structures on Hopf algebras with positive bases, to appear in Coloquio de Grupos Cuánticos y Álgebras de Hopf (La Falda, Argentina, 1999), ed. N. Andruskiewitsch et al., Contemp. Math. A. Weinstein and P. Xu, Classical solutions of the quantum Yang-Baxter equation, Comm. Math. Phys. 148 (1992), 309–343.

Lu: Department of Mathematics, University of Arizona, Tucson, Arizona 85721, USA Yan: Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Special Administrative Region, China Zhu: Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Special Administrative Region, China

Vol. 104, No. 1

DUKE MATHEMATICAL JOURNAL

© 2000

KIRILLOV THEORY FOR GL2 (Ᏸ) WHERE Ᏸ IS A DIVISION ALGEBRA OVER A NON-ARCHIMEDEAN LOCAL FIELD DIPENDRA PRASAD and A. RAGHURAM

Contents 1. Introduction and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Basic structure theory of GL2 (Ᏸ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Jacquet modules of principal series representations . . . . . . . . . . . . . . . . . . . . . . . 3. Kirillov theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. New forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. New forms for principal series representations . . . . . . . . . . . . . . . . . . . . . . . 4.2. New forms for spherical representations in the Kirillov model . . . . . . . . . 5. Supercuspidal representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Shalika model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 19 22 23 24 25 30 32 34 34 42

1. Introduction and notation 1.1. Introduction. The aim of this work is to develop Kirillov theory for irreducible admissible representations of GL2 (Ᏸ) for a division algebra Ᏸ over a nonArchimedean local ﬁeld F . We apply this theory to develop a theory of new forms for such representations. The Kirillov theory developed here is in close analogy with the case of GL2 (F ). We recall (see Jacquet and Langlands [12]) that the Kirillov model K(π) of an irreducible admissible representation π of GL2 (F ) consists of a certain space of locally constant functions on F ∗ , which vanish outside compact subsets of F , and contains Cc∞ (F ∗ ) with codimension at most 2. The action of B, the standard Borel subgroup consisting of upper triangular matrices in GL2 (F ), on K(π) is given by a b f (x) = ωπ (d)ψF d −1 xb f d −1 xa , 0 d where ψF is a ﬁxed nontrivial additive 0 1 character of F , and ωπ is the central quasi character of π. The action of w = −1 0 , the Weyl group element, is given in terms of Fourier transforms. The explicit formula for this involves the -factors (actually γ -factors) of π twisted by characters of F ∗ . Received 8 March 1999. Revision received 11 November 1999. 2000 Mathematics Subject Classiﬁcation. Primary 22E50; Secondary 22E35, 11S37, 11S45. 19

20

PRASAD AND RAGHURAM

The theory of the Kirillov model depends on the existence and uniqueness of the Whittaker model for π , or equivalently on the existence and uniqueness of Whittaker functionals. Deﬁnition 1.1. A linear functional l : π → C is called a Whittaker functional if 1 X v = ψF (X)l(v) 0 1

l

for all v in π and for all X in F . The basic theorem here is that every inﬁnite-dimensional irreducible admissible representation π of GL2 (F ) admits a nonzero Whittaker functional that is unique up to scalars. For the representation theory of GL2 (Ᏸ), we introduce a concept called degenerate Whittaker functional, which is deﬁned as follows. Deﬁnition 1.2. Let π be a representation of GL2 (Ᏸ). A linear form : π → C is called a degenerate Whittaker functional if

1 X v = (X)(v) 0 1

for all v in π and for all X in Ᏸ, where (X) = ψF (TᏰ/F (X)). The reduced trace map from Ᏸ to F is denoted by TᏰ/F . We need a concept called the twisted Jacquet module to analyze the space of degenerate Whittaker functionals. Deﬁnition 1.3. Let (π, V ) be a representation of GL2 (Ᏸ). Deﬁne V (N, ) to be the C-span of vectors of the form π 01 X1 v − (X)v for v ∈ V and for X ∈ Ᏸ. Deﬁne the twisted Jacquet module as VN, = V /V (N, ). Since TᏰ/F (ABA−1 ) = TᏰ/F (B), VN, is a module over Ᏸ∗ =

A 0 : A ∈ Ᏸ∗ . 0 A

The twisted Jacquet module is the maximal quotient of V on which N acts via the character . The space of degenerate Whittaker functionals is just HomC (VN, , C). By a theorem of Moeglin and Waldspurger [17], the twisted Jacquet module of any irreducible admissible representation is ﬁnite-dimensional. We refer to the second author’s thesis [23] for details on this. In Section 2 the twisted Jacquet module is explicitly computed for parabolically induced representations from which ﬁnite dimensionality falls out (see Theorem 2.1).

KIRILLOV THEORY FOR GL2 (Ᏸ)

21

In Section 3 we begin our analysis of Kirillov theory. The Kirillov model that we consider is realized on a space of functions on Ᏸ∗ with values in VN, . We denote this space of functions by ᏷(π). It turns out that ᏷(π) contains Cc∞ (Ᏸ∗ , VN, ) as a subspace with ﬁnite codimension and the minimal parabolic subgroup P consisting of upper triangular matrices acts on ᏷(π) by the formula D A B f (X) = ψ D −1 XB π 0 0 D

0 f D −1 XA . D

(1)

Furthermore, the representation π is supercuspidal if and only if ᏷(π) = Cc∞ (Ᏸ∗ , VN, ). The main theorem of this section is recorded in Theorem 3.1. As a corollary to this theorem, we get that taking duals commutes with taking the twisted Jacquet module (see Proposition 3.1). To have a complete picture as to how G acts on the Kirillov space, we need to describe how the Weyl group element acts on ᏷(π). Since w A0 D0 w −1 = D0 A0 , w gives rise to an intertwining on the Kirillov model ᏷(π) for the corresponding two actions of Ᏸ∗ × Ᏸ∗ . It should be possible to make this action more explicit, but we have not been able to do that. We apply Kirillov theory as developed in Section 3 to develop a theory of new forms considered in Section 4. The statements are akin to Casselman [7], although our proofs are modeled on Deligne [8]. Given an irreducible admissible representation (π, V ) of G, we consider vectors ﬁxed under a certain congruence subgroup #01 (m).

Let Vm denote V #0 (m) , the space of vectors in V that are ﬁxed by #01 (m). We prove in Section 4 that for m large enough, there are vectors ﬁxed under #01 (m), and the space of ﬁxed vectors is intimately connected with the twisted Jacquet module of π. The main theorem proved about new forms is that if C(π) (called the conductor of π in the sense of new forms) is the smallest nonnegative integer m for which Vm = (0), then VC(π) is isomorphic to VN, as Ᏸ∗ -modules. (Actually, we are able to prove this only for irreducible principal series and those supercuspidal representations that are obtained by compact induction.) Every time we increase the level by 1, one more copy of VN, gets added to the space of ﬁxed vectors. We also derive an explicit formula connecting the conductor of π in the sense of new forms and the exponent of the epsilon factor attached to π as in [10]. These results are considered in Section 4.1 for principal series, and Section 5 for supercuspidal representations (see Theorems 4.1, 5.1, 5.2, and 5.3). In Section 4.2, we also give an explicit form of the spherical vector in the Kirillov model for unramiﬁed principal series representations of GL2 (Ᏸ) (see Theorem 4.2). In Section 5 we construct a family of supercuspidal representations of G. This is done by identifying what are called very cuspidal representations of maximal open compact mod center subgroups of G. Compactly inducing them to G gives supercuspidal representations. We then take up these representations and compute their twisted Jacquet modules, conductors in the sense of epsilon factors and also in the 1

22

PRASAD AND RAGHURAM

sense of new forms, and ﬁnally identify the space of new forms as the twisted Jacquet module (see Propositions 5.2, 5.3, and 5.4). In Section 6 we take up the concept of the Shalika model (see Deﬁnition 6.1), which is closely related to the Kirillov model. We prove that if π admits a Shalika model, then it admits a unique one. We also prove that if π admits a Shalika model, then π is self-contragredient (see Theorem 6.2). The present work highlights the importance of the space of degenerate Whittaker models for the representation theory of GL2 (Ᏸ). It would be very nice if there was a way of predicting the structure of the space of degenerate Whittaker model as a representation space of Ᏸ∗ . A conjecture of B. H. Gross and the ﬁrst author gives an answer to this question in terms of certain local root numbers when Ᏸ is a quaternion division algebra (which is the only case when representations of Ᏸ∗ occurring in the space of degenerate Whittaker models appear with multiplicity 1). This conjecture and some of its consequences are elaborated upon in [22]. Acknowledgments. The authors would like to thank Professor M. S. Raghunathan for helpful conversations. The second author would like to thank the Mehta Research Institute, Allahabad, where most of this work was done. The authors also thank the referee for a careful reading of the paper and some pertinent remarks. 1.2. Notation. Let F be a non-Archimedean local ﬁeld. Let Ᏸ be a division algebra with center F and of index d over F , that is, of dimension d 2 over F . Let ᏻF be the ring of integers in F , and let ᏻ be the ring of integers in Ᏸ. Let %F be a uniformizer in F , and let % be a uniformizer in D such that % d = %F . Let PF be the unique maximal ideal in ᏻF , and let P be the unique maximal ideal in ᏻ. Let vF be the valuation on F with vF (%F ) = 1, and let vᏰ be the valuation on Ᏸ with vᏰ (% ) = 1. These valuations uniquely determine the normalized multiplicative valuations | · |F and | · | on F and Ᏸ, respectively, by the formulae |%F |F = q −1 and |% | = q −d , where q is the cardinality of the residue ﬁeld of F . Let TᏰ/F be the reduced trace map from Ᏸ to F . Let NᏰ/F denote the reduced norm map from Ᏸ to F. Let ψF be a nontrivial additive character on F so chosen that the maximal fractional ideal in F on which ψF is trivial is ᏻF . Let be the character on Ᏸ obtained by composing the reduced trace and the character ψF . Let ᏹ = M2 (Ᏸ) be the matrix algebra of 2 × 2 matrices with entries in Ᏸ. It is a central simple algebra over F of dimension 4d 2 . Let G stand for the group GL2 (Ᏸ) = ᏹ× . So G may be regarded as the F -points of a linear algebraic group deﬁned over F. Let P be the minimal parabolic subgroup of upper triangular matrices in G. Let P = M · N, where M is the Levi part of P consisting of diagonal matrices and N is the unipotent radical of P consisting of upper triangular matrices with 1’s on the diagonal. Let P denote the parabolic subgroup opposed to P consisting of lower triangular matrices. The character of Ᏸ is also thought of as a character of N. Let S be the “Shalika subgroup” of P consisting of all matrices of the form A0 B A in P . Then S is the subgroup of P consisting of all elements that leave invariant.

KIRILLOV THEORY FOR GL2 (Ᏸ)

23

0 1 Let D ∗ be M ∩ S. Let w = −1 0 be the Weyl group element in G. Let K denote the maximal open compact subgroup GL2 (ᏻ). Let H be an l-group (see [1] for the deﬁnition of an l-group), and let B be a closed subgroup of H. Let (σ, W ) be a smooth representation of B. The notation IndH B (σ ) stands for unnormalized induction from B to H of the representation σ of B. The notation indH B (σ ) stands for the unnormalized compact induction from B to H of the representation σ of B. 1.3. Basic structure theory of GL2 (Ᏸ). In this section we collect some theorems on the structure of GL2 (Ᏸ), which are used in the rest of this article. No proofs are given as these are all well known and easy to prove. Proposition 1.1 (Bruhat decomposition). With the notation as above, we have G = P P wP . Note that the “big Bruhat cell” P wP can also be written as NwMN = NMwN. Proposition 1.2 (Iwasawa decomposition). With the notation as above, we have G = K · P = P · K. Proposition 1.3 (Cartan decomposition). With the notation as above, we have G = K · A · K, where A is the submonoid of G generated by the elements

% −1 0 0 1

and

%

0 0 %

.

Proposition 1.4 (Maximal compact mod center subgroups) . Let K1 = A B open GL2 (ᏻ), and let K2 = C be the cyclic ( ᏻ ) : C ≡ 0 (mod P ) . Let Z ∈ GL 2 1 D % 0 , and let Z be the cyclic group generated by group generated by z = 1 2 0 % z2 = %0 01 . Note that Z1 normalizes K1 , and Z2 normalizes K2 . Let H1 = Z1 K1 , and let H2 = Z2 K2 . Then H1 and H2 are maximal, open, compact modulo center subgroups of G, and any subgroup that is open and compact modulo center can be conjugated inside H1 or H2 . Furthermore, H1 and H2 are not conjugate to each other. For a proof of Proposition 1.4, refer to [4]. Proposition 1.5 (Iwasawa decomposition II). With the notation as above, we have (1) G = H1 · P = P · H1 , (2) G = H2 · P = P · H2 . Proposition 1.6 (Cartan decomposition II). With the notation as above, we have (1) G = H1 · A · H1 , (2) G = H2 · A · H2 , (3) G = H2 · A · H1 .

24

PRASAD AND RAGHURAM

We need to consider certain congruence subgroups of K = GL2 (ᏻ). Let A B ∈ GL2 (ᏻ) : C ≡ 0 mod P m C D

#0 (m) := and

A B ∈ GL2 (ᏻ) : C ≡ 0, D ≡ 1 mod P m . C D

#01 (m) :=

We also use the following notation. If X1 , . . . , X4 are subsets of Ᏸ, then

X1 X3

X2 A1 := X4 A3

A2 : A i ∈ Xi . A4

Proposition 1.7 (Iwahori With the notation as above, we have × factorization). (1) #0 (m) = N(P m ) ᏻ0 ᏻ0× N(ᏻ), × (2) #01 (m) = N(P m ) ᏻ0 1+0Pm N(ᏻ). Here N(Pm ) = N (m) = N ∩ #0 (m) and N(ᏻ) = N ∩ K. Proposition 1.8. For all m ≥ 1, the subgroup of G generated by #01 (m) and N(m − 1) is #01 (m − 1). Proposition 1.9. The reduced trace map TᏰ/F has the property [(m+d−1)/d]

TᏰ/F (P m ) = PF

.

The conductor of the character of Ᏸ is P1−d . 2. Jacquet modules of principal series representations. In this section we explicitly calculate the Jacquet module and the twisted Jacquet module for a parabolically induced representation of G. Let π1 and π2 be irreducible (necessarily ﬁnite-dimensional) representations of Ᏸ∗ . Consider the representation π1 ⊗π2 ⊗1/2 of M where is the character of Ᏸ∗ × Ᏸ∗ deﬁned by (X, Y ) = |XY −1 |. We think of this as a representation of P by extending it trivially across N. Denote by V (π1 , π2 ) the representation of G induced from this representation of P . So V (π1 , π2 ) is the space of functions

f ∈ C ∞ G, π1 ⊗ π2 : f

A B g = (A, D)1/2 π1 (A) ⊗ π2 (D) f (g) . 0 D

We call V (π1 , π2 ) a principal series representation. In general it is not irreducible, and when it is irreducible, we explicitly mention it.

KIRILLOV THEORY FOR GL2 (Ᏸ)

25

The following theorem computes the twisted Jacquet module for such a representation V (π1 , π2 ). Theorem 2.1. For irreducible representations π1 and π2 of Ᏸ∗ , let V (π1 , π2 ) be the representation of GL2 (Ᏸ) as deﬁned above. Then there is a natural isomorphism of Ᏸ∗ -modules (V (π1 , π2 ))N, π1 ⊗ π2 . Proof. The proof is a simple consequence of the Bruhat decomposition, which gives rise to the following exact sequence of P -modules: 0 −→ Cc∞ (N ) ⊗ π1 ⊗ π2 −→ V (π1 , π2 ) −→ π1 ⊗ π2 ⊗ 1/2 −→ 0 and the elementary fact that the twisted Jacquet functor of Cc∞ (N) is C. We refer to [21, Proposition 7] for more details. The Jacquet module of V (π1 , π2 ), namely, the maximal quotient of V (π1 , π2 ) on which N acts trivially, can also be computed as in the proof of the above theorem. We state this as the following theorem and leave the reader to ﬁll in the details. Theorem 2.2. For irreducible representations π1 and π2 of Ᏸ∗ , let V (π1 , π2 ) denote the corresponding principal series representation of G. The semisimpliﬁcation of the Jacquet module of V (π1 , π2 ) is given by 1/2 · [(π1 ⊗ π2 ) ⊕ (π2 ⊗ π1 )] as M-modules. 3. Kirillov theory. In this section we develop Kirillov theory for irreducible representations of G. We prove that an irreducible admissible representation (π, V ) of GL2 (Ᏸ) can be realized on a certain space of functions on Ᏸ∗ with values in a ﬁnitedimensional vector space, namely, VN, , on which the parabolic subgroup P acts in a very explicit way. This space of functions contains Cc∞ (Ᏸ∗ , VN, ) as a subspace of ﬁnite codimension. By Proposition 1.1, we just need to know how the Weyl group element acts to get a complete understanding of π, but which we have not been able to achieve here. Let L : V → VN, = V /V (N, ) be the canonical projection. For any v ∈ V , let ϕv denote the VN, valued function on Ᏸ∗ given by A 0 v . ϕv (A) = L π 0 1 We state some preliminary lemmas that give some properties of the functions ϕv ’s on Ᏸ∗ with values in VN, . These lemmas are almost identical to the case of GL2 (F ) as written in [9]. We omit the proofs of these lemmas, as the proofs given for the corresponding statements in [9] go through mutatis mutandis to our case. B v, then Lemma 3.1. If v = A0 D D 0 ϕv D −1 XA . ϕv (X) = D −1 XB π 0 D

26

PRASAD AND RAGHURAM

Lemma 3.2. The function ϕv is a locally constant function on Ᏸ∗ that vanishes outside a compact subset of Ᏸ. Lemma 3.3. The map v → ϕv from V to C ∞ (Ᏸ∗ , VN, ) is an injective linear map. Let ᏷(π ) denote {ϕv : v ∈ V }. From the previous lemmas, ᏷(π) is a C-vector space of locally constant functions on Ᏸ∗ with values in VN, , vanishing outside compact subsets of Ᏸ, on which P acts by a simple formula. By Lemma 3.3, since the map v → ϕv is injective, we can give a G action to ᏷(π) by borrowing the G action on V . The map v → ϕv has a fairly natural interpretation as follows. If (ρ, W ) is any representation of S, then by Frobenius reciprocity, we have HomP π, IndPS (ρ) HomS (π|S , ρ). If N acts via on ρ, then since VN, is the largest quotient of V on which N acts via , we have HomS (π|S , ρ) HomS (πN, , ρ). Taking ρ as πN, , we get HomP π, IndPS (πN, ) HomS (πN, , πN, ). It is easy to see that the map v → ϕv is the pullback 4 of the identity map on VN, in the above isomorphism. We have implicitly identiﬁed functions in IndPS (ρ) as functions on Ᏸ∗ with values in ρ on which P acts exactly as in Lemma 3.1. So we have 4(v) = ϕv for all v ∈ V , and the image of 4 is ᏷(π) and, in particular, 4(v)(1) = L(v). The next lemma can be thought of as a p-adic analogue of the method of little groups of Mackey and Wigner (see [24, Section 8.2]) of describing irreducible representations of a group that is an extension of some group by an abelian group. Lemma 3.4. Given an irreducible representation (σ, W ) of S, the Shalika subgroup of G, on which N acts via , the representation indPS (σ ) of P , obtained by compactly inducing σ to P , is irreducible. Proof. The proof is based on the proof of [3, Proposition 4.7.3]. We can consider the representation space of indPS (σ ) as Cc∞ (Ᏸ∗ , W ) on which P acts via the formula D A B f (X) = D −1 XB σ 0 0 D

0 f D −1 XA . D

If v ∈ W and U is an open compact set in Ᏸ∗ , let fU,v be the function that takes the value v on U and 0 elsewhere. Clearly, such functions fU,v span V = Cc∞ (Ᏸ∗ , W ) as v and U vary.

KIRILLOV THEORY FOR GL2 (Ᏸ)

27

Let V1 be a nonzero, P -stable subspace of V , and let W1 = f (X) : X ∈ Ᏸ∗ , f ∈ V1 . Let 0 = f ∈ V1 . Therefore, there exists A ∈ Ᏸ∗ such that f (A) = v = 0. For φ ∈ Cc∞ (Ᏸ), let

1 X f dX, fφ = φ(X) 0 1 Ᏸ which gives that

fφ (Y ) =

Ᏸ

(Y )f (Y ). φ(X)(XY )f (Y ) dX = φ

is the characteristic function of U , Note that fφ is in V1 . Now choose φ such that φ ∗ an open compact subset of Ᏸ containing A on which f is constant, and hence the constant value is v, to get that fφ = fU,v . The formula D 0 f =f DU D −1 , σ D 0 v 0 D U,v 0 D implies that W1 = W. So now choose any w ∈ W and choose a function g ∈ V1 such that g(C) = w for some C ∈ Ᏸ∗ . Looking at gφ and choosing φ appropriately, we get that for arbitrarily small neighbourhoods U of C, fU,w ∈ V1 . The formula A 0 f = fU ·A−1 ,w 0 1 U,w implies that given any w ∈ W and any A ∈ Ᏸ∗ for all small enough neighbourhoods U of A, we have fU,w ∈ V1 . Hence, V1 = V , which is what we wanted to prove. Theorem 3.1. (1) For all n ∈ N and for all v ∈ V , n·ϕv −ϕv has compact support in Ᏸ∗ . (2) The Kirillov space ᏷(π ) contains all functions in Cc∞ (Ᏸ∗ , VN, ). (3) The Jacquet module of π, namely, the maximal quotient of π on which N acts trivially, denoted by πN , is ᏷(π)/Cc∞ (Ᏸ∗ , VN, ). (4) The representation π is supercuspidal if and only if Cc∞ (Ᏸ∗ , VN, ) = ᏷(π). (5) The Jacquet module is ﬁnite-dimensional, that is, Cc∞ (Ᏸ∗ , VN, ) has ﬁnite codimension in ᏷(π ). Proof. Let f = n · ϕv − ϕv , where n = 01 A1 . Then f (X) = ((XA) − 1)ϕv (X). So for all X ∈ A−1 ᏻ, we get f (X) = 0 since the conductor of is P1−d . By Lemma 3.2, the support of the function f lies in a compact subset of Ᏸ, and by the above argument, this compact subset does not contain 0; hence, it is a compact subset of Ᏸ∗ . This proves (1).

28

PRASAD AND RAGHURAM

Note that Cc∞ (Ᏸ∗ , VN, ) = indPS (VN, ). The twisted Jacquet module VN, is a ﬁnite-dimensional S = Ᏸ∗ · N-module on which N acts via . Since Ᏸ∗ is compact modulo center and the center acts via ωπ , the central character of π, VN, is a semisimple module over S. So let VN, = m1 π1 ⊕ m2 π2 ⊕ · · · ⊕ mk πk , where πi ’s are mutually inequivalent irreducible S-modules. Induction being an exact functor, we have IndPS (VN, ) = ⊕ki=1 mi IndPS (πi ), indPS (VN, ) = ⊕ki=1 mi indPS (πi ). For any nonzero vector α ∈ πi and any 1 ≤ j ≤ mi , let v be a vector in V such that L(v) is α in the j th copy of πi and 0 everywhere else. (Here L is the canonical map from V to VN, ). Let f be the corresponding function in ᏷(π). Let f = frs with j frs ∈ IndPS (πr ) as the sth copy. So frs (1) = 0 unless r = i and s = j , and fi (1) = α. 1−d Take any n ∈ / N(P ) such that (n) = 1, and let g = n · f − f. Then by (1), g is in Cc∞ (Ᏸ∗ , VN. ). If we write g as grs , then grs (1) = 0 unless r = i and s = j , in which case it is ((n) − 1)α in the = 0.As proof of the previous lemma, take any ∞ 1 X φ ∈ Cc (Ᏸ), and let gφ = Ᏸ φ(X) 0 1 g dX, which is essentially a ﬁnite sum as g s (Y )g(Y ) = φ gr . Now choose has compact support in Ᏸ∗ , and so we get gφ (Y ) = φ is the characteristic function of U , an open compact neighbourhood of φ such that φ 1, such that all the functions grs are constant on U. Therefore, gφ is a function that j takes the constant value α on U and is 0 elsewhere. Hence, gφ ∈ indPS (πi ), where j πi is πi sitting as the j th copy in VN, . So for all 1 ≤ i ≤ k and all 1 ≤ j ≤ mi , j j P ᏷(π ) ∩ indP S (πi ) is nonempty and so by Lemma 3.4, indS (πi ) ⊂ ᏷(π). Hence, indPS (VN, ) ⊂ ᏷(π ). This proves (2). To prove (3) using (1), we just need to show that any f ∈ Cc∞ (Ᏸ∗ , VN, ) is a ﬁnite sum of functions as in (1). So for any such f , consider

1 X · f dX. π g= 0 1 P−m So for all A in Ᏸ∗ , we have

1 X · f (A) dX = π (AX)f (A) dX. g(A) = 0 1 P−m P−m Since f has compact support, there exists a ≤ b such that supp(f ) ⊂ P a and f (P b ) = 0. Choose any m ≥ b + d. Then X → (AX) is a nontrivial character on P−m for any A ∈ supp(f ). This gives that g is identically zero, from which it follows that f is a ﬁnite sum of functions of the form π(n)φ − φ. This proves (3). Statement (4) follows from (3) and the deﬁnition of supercuspidality. Statement (5) follows from (4) if π is a supercuspidal representation, and if π is a subrepresentation of a parabolically induced representation, then it follows from Theorem 2.2.

KIRILLOV THEORY FOR GL2 (Ᏸ)

29

Proposition 3.1. If (π, V ) is an irreducible admissible representation of G, let (π ∨ , V ∨ ) be its contragredient representation. Then there is a natural isomorphism of Ᏸ∗ modules (πN, )∨ (π ∨ )N, . Proof. By Theorem 3.1(3), we have the exact sequence of P -modules 0 −→ indPS (πN, ) −→ π −→ πN −→ 0, where πN is the usual Jacquet module of π. Let (π|P )∨ be the smooth dual of π considered as a P -module. So π ∨ ⊂ (π|P )∨ . Dualizing the above exact sequence of P -modules, we get 0 −→ (πN )∨ −→ (π|P )∨ −→ IndPS (πN, )∨ −→ 0. Since N acts trivially on πN , ((πN )∨ )N, = (0). Therefore, (π |P )∨ N, IndPS (πN, )∨ N, . If (σ, U ) is a ﬁnite-dimensional representation of the Shalika subgroup S on which N acts via a ﬁxed nontrivial character , then identifying IndPS (σ ) with functions on Ᏸ∗ with values in U , we get exactly as in statements (1) and (3) of Theorem 3.1 that N operates trivially on IndPS (σ )/ indPS (σ ). Hence, (IndPS (σ ))N, (indPS (σ ))N, σ. Taking σ = (πN, )∨ , we ﬁnd that (π|P )∨ N, (πN, )∨ . Thus, under the natural inclusion of π ∨ into (π|P )∨ , (π ∨ )N, becomes a submodule of (πN, )∨ . Interchanging the roles of π and π ∨ and noting that (π ∨ )N, is ﬁnitedimensional completes the proof of the proposition. We omit the proof of the following easy corollary in which we give a P -equivariant pairing between an irreducible supercuspidal representation of G and its contragredient in terms of their Kirillov models. Corollary 3.1. Let π be an irreducible admissible supercuspidal representation of G = GL2 (Ᏸ). Let ᏷ (π ) and ᏷ (π ∨ ) be the Kirillov models of π and its contragredient π ∨ with respect to the additive characters and , respectively. Using the identiﬁcation in Proposition 3.1, the map ᏷ (π) × ᏷ (π ∨ ) → C given by sending the pair (f, g) for f ∈ ᏷ (π ) and g ∈ ᏷ (π ∨ ) to the number f, g deﬁned by

f (X), g(X) dX, f, g = Ᏸ∗

gives a P -invariant duality between ᏷ (π) and ᏷ (π ∨ ).

30

PRASAD AND RAGHURAM

Remark 3.1. If Ᏸ is a ﬁeld, we know that an irreducible supercuspidal representation of GL2 (Ᏸ) remains irreducible when restricted to P (see [1, Section 5]). It follows that the natural GL2 (Ᏸ)-invariant pairing on π ×π ∨ is the unique P -invariant bilinear pairing on π × π ∨ . Therefore, the pairing in Corollary 3.1 is automatically GL2 (Ᏸ)-invariant. When Ᏸ is not a ﬁeld, an irreducible supercuspidal representation of GL2 (Ᏸ) may not be irreducible when restricted to P (equivalently, the space of degenerate Whittaker models may not be irreducible as a Ᏸ∗ -module), and therefore it is not clear if the pairing deﬁned in Corollary 3.1 is GL2 (Ᏸ) invariant in general. 4. New forms. In this section we investigate the space of ﬁxed vectors under a certain type of congruence subgroup for any irreducible admissible representation of G. Deﬁne for m ≥ 1, A B 1 m #0 (m) := ∈ K : C ≡ 0, D ≡ 1 mod P C D and #01 (0) = K = GL2 (ᏻ). By a new form for (π, V ) we mean a vector in V ﬁxed under #01 (m) for some m but that is not ﬁxed under #01 (m−1) if m ≥ 1, or a nonzero vector ﬁxed under #01 (0). For all m ≥ 0, we use the notation Vm := V #0 (m) . 1

Also, for convenience, let V−1 := (0). We start with a proposition that states that new forms exist. Proposition 4.1. If (π, V ) is an irreducible admissible inﬁnite-dimensional representation of G, then there exists an integer m ≥ 0 such that π admits a #01 (m)ﬁxed vector. Proof. Since V is inﬁnite-dimensional, VN, is not (0). Let α be any vector in VN, , and let fα be the function that is 0 outside ᏻ× and takes the value α on ᏻ× . From Theorem 3.1, since fα is in Cc∞ (Ᏸ∗ , VN, ), it belongs to ᏷(π).For all A ∈ ᏻ× and for all B ∈ ᏻ, it is easily checked that fα is left-invariant by A0 B1 . Since π is a smooth representation, there is an m such that fα is left-invariant by K(m) where K(m) is the principal congruence subgroup of level m. The proof follows using Proposition 1.7. Deﬁnition 4.1. For an irreducible admissible representation π of G, let C(π) denote the least nonnegative integer k such that π admits a nonzero vector ﬁxed under #01 (k). This integer C(π) is called the conductor of π in the sense of new forms. Fix an additive character ψF of F such that the largest fractional ideal of F on which ψF is trivial is ᏻF . Let Ce (π) be the integer c such that the epsilon factor (π, s, ψF ) associated to π as in [10] is up to a constant q −cs . This integer Ce (π) is called the conductor of π in the sense of epsilon factors.

KIRILLOV THEORY FOR GL2 (Ᏸ)

31

Lemma 4.1. Let π be any irreducible admissible representation of G. Let (π, V ) be realized in its Kirillov model. Let m ≥ C(π). If f ∈ Vm , then (1) f (xu) = f (x) for all x ∈ Ᏸ∗ and all u ∈ ᏻ× , (2) supp(f ) ⊂ P1−d . Proof. Since for any unit u, u0 01 ∈ #01 (m), we get (1) using the formula in 1 X Lemma 3.1. Since 0 1 ∈ #01 (m) for any X ∈ ᏻ, by Lemma 3.1, we have

(XY ) − 1 f (Y ) = 0

for all Y ∈ Ᏸ∗ . So if Y ∈ / P1−d , then it is possible to choose an X ∈ ᏻ such that (XY ) = 1, which implies that f vanishes on any such Y. This proves (2). Lemma 4.2. Let (π, V ) be any irreducible admissible representation of G. Let (π, V ) be realized in its Kirillov model. Let c = C(π) be the conductor of π in the sense of new forms. Then we have the following. (1) If c = 0 and if 0 = f ∈ V0 , then f (% 1−d ᏻ× ) = 0. (2) If m ≥ max{1, C(π )} and if f ∈ Vm is such that f (% 1−d ᏻ× ) = 0, that is, supp(f ) ⊂ P2−d , then 1 0 · f ∈ Vm−1 . 0 % −1 Proof. To prove (1), if possible let 0 = f ∈ V0 and f (% 1−d ᏻ× ) = 0. Observe that such an f is ﬁxed by N(P−1 ). Let H be the subgroup of G generated by GL2 (ᏻ) and N(P−1 ). The matrix identity

1 % −1 0 1

1 −%

0 0 = 1 −%

% 0 1 % −1 = −1 % 0 1

0 % −1

−n gives that for all n ∈ Z, the matrix %0 %0n is in H. Now choose any A ∈ P 2−d −n such that f (A) = 0. Since %0 %0n f = f , we get f (% −n A% −n ) = 0. Choose n large enough to get a contradiction. This proves(1). For the sake of brevity, let x denote the matrix 01 %0 . Let g = x −1 ·f. So supp(g) ⊂ 1−d P , which together with Lemma 3.1 implies that g is ﬁxed by N(ᏻ). Since f ∈ Vm , we get that g is ﬁxed by x −1 #01 (m)x

=

ᏻ×

P m−1

P . 1 + Pm

So g is ﬁxed by the subgroup of G generated by x −1 #01 (m)x and N(ᏻ), which by Proposition 1.7 is the same as the subgroup of G generated by #01 (m) and N(m − 1), which by Proposition 1.8 is #01 (m − 1). This proves (2).

32

PRASAD AND RAGHURAM

Lemma 4.3. Let (π, V ) be an irreducible admissible representation of G. For all m ≥ C(π ), we have dimC (Vm ) − dimC (Vm−1 ) ≤ dimC (VN, ). Proof. As before, let x denote the matrix 01 %0 . Clearly, x · Vm−1 is ﬁxed by x#01 (m−1)x −1 , which contains #01 (m). Hence, x ·Vm−1 ⊂ Vm . So to prove the lemma, it is enough to show that dimC (Vm /x · Vm−1 ) ≤ dimC (VN, ). Let dimC (VN, ) = r. Let (π, V ) be realized in its Kirillov model. Let f1 , f2 , . . . , fr+1 be r +1 vectors in Vm . By Lemma 4.1, we can choose a1 , a2 , . . . , ar+1 ∈ C such that f = r+1 i=1 ai fi vanishes on % 1−d ᏻ× . By Lemma 4.2, we get that f ∈ x · Vm−1 . The following corollary of the proof of the previous lemma is improved later. Corollary 4.1. For any irreducible admissible representation of GL2 (Ᏸ), the space of new forms is a Ᏸ∗ -submodule of the twisted Jacquet module. 4.1. New forms for principal series representations. In this section we study new forms and conductors for principal series representations. Lemma 4.4. Let K = GL2 (ᏻ), and let P (ᏻ) consist of all upper triangular matrices in K. For i = 0, . . . , m − 1, let γi = %1 i 01 , and let γm = 01 01 . Then 1 K = m i=0 #0 (m)γi P (ᏻ).

A B i / Pi+1 for 1 ≤ i ≤ m, then noting Proof. If k = C D ∈ K and if C ∈ P and C ∈ × −i × that A ∈ ᏻ and C% ∈ ᏻ , we have −1 i 1 0 % −i C % −i CA−1 B AC % 0 A B = , C D 0 1 %i 1 0 D − CA−1 B which gives that k ∈ #01 (m)γi P (ᏻ). If C ∈ ᏻ× , multiplying k on the right by we may assume that C = 1. In this case, we have 1 0 A B · p, =γ· 1 1 1 D m A(1+% m )−1 m )(B−AD) where γ = 1−A% and p = 01 D+(1+% . AD−B −% m 1+% m

C −1 0

0 C −1

,

Lemma 4.5. For 0 ≤ i ≤ m, we have γi−1 #01 (m)γi ∩ P (ᏻ) is equal to (i) A − 1, B − (1 − D)/% i ≡ 0 mod P m−i A B ∈K : . 0 D (ii) D ≡ 1 mod Pi B Proof. Let x = A0 D ∈ P (ᏻ) be such that γi xγi−1 ∈ #01 (m). Multiplying the matrices and using the deﬁning congruences for #01 (m) gives that D + % i B − 1 and % i A − D% i − % i B% i are in P m , which simpliﬁes to what we want.

KIRILLOV THEORY FOR GL2 (Ᏸ)

33

Theorem 4.1. Let π1 and π2 be irreducible (necessarily ﬁnite-dimensional) representations of Ᏸ∗ . Let ni be the smallest nonnegative integer such that πi is trivial on Ᏸ∗ (ni ), for i = 1, 2. Here Ᏸ∗ (r) denotes the subgroup 1 + P r if r ≥ 1 and ᏻ× if r = 0. Then we have the following. (1) The smallest integer m such that V (π1 , π2 ) admits a #01 (m)-ﬁxed vector is m = n1 + n2 ; that is, the conductor in the sense of new forms is given by C V (π1 , π2 ) = n1 + n2 . (2) Furthermore, V (π1 , π2 )#0 (n1 +n2 ) π1 ⊗ π2 as Ᏸ∗ -modules. 1 (3) For k ≥ n1 + n2 , V #0 (k) (k − n1 − n2 + 1)(π1 ⊗ π2 ) as Ᏸ∗ -modules. (4) If V (π1 , π2 ) is irreducible, then the conductor in the sense of epsilon factors is given by Ce V (π1 , π2 ) = n1 + n2 + 2(d − 1). 1

Proof. Using Iwasawa decomposition (see Proposition 1.2), we have V (π1 , π2 ) = IndK P (ᏻ) (π1 ⊗ π2 ). By Mackey’s theorem on restriction of an induced repesentation, the restriction of V (π1 , π2 ) to #01 (m) is m # 1 (m) IndH0i (π1 ⊗ π2 ), i=0

where Hi = P (ᏻ)∩γi−1 #01 (m)γi is considered to be a subgroup of #01 (m) in a natural way. Therefore, m 1 V (π1 , π2 )#0 (m) (π1 ⊗ π2 )Hi . i=0

Since M ∩ Hi is by the previous lemma Ᏸ∗ (m − i) × Ᏸ∗ (i), which is a normal subgroup of M, if (π1 ⊗ π2 )Hi = (0), then m − i ≥ n1 and i ≥ n2 . This is possible if and only if m ≥ n1 + n2 , and if so, there are exactly m − (n1 + n2 ) + 1 many i’s such that (π1 ⊗ π2 ) has an Hi -ﬁxed vector. 1 Note that Ᏸ∗ leaves #01 (m) invariant, and hence V (π1 , π2 )#0 (m) has a Ᏸ∗ -module structure. It is easy to see that the above isomorphisms are Ᏸ∗ -equivariant. We have proved statements (1), (2), and (3). To prove statement (4), we note that by [10] we have V (π1 , π2 ), s, ψF = (π1 , s, ψF ) · (π2 , s, ψF ). It can be seen as in [14] that the exponent of q occurring in (πi , s, ψF ) is ni + d − 1 for i = 1, 2.

34

PRASAD AND RAGHURAM

4.2. New forms for spherical representations in the Kirillov model. An irreducible admissible representation π of G = GL2 (Ᏸ) is said to be spherical if π admits a nonzero vector ﬁxed by the maximal compact subgroup K = GL2 (ᏻ). So in other words, spherical representations are exactly those whose conductor is 0 in the sense of new forms. It can be seen that an inﬁnite-dimensional representation π is spherical if and only if π is an irreducible unramiﬁed principal series representation. See [23] for a proof of this statement. Let π1 and π2 be 1-dimensional representations of Ᏸ∗ that are trivial on ᏻ× . The representations πi are therefore of the form πi (X) = αi (NᏰ/F (X)) for unramiﬁed characters αi of F ∗ . The representation V (π1 , π2 ) is called an unramiﬁed principal series representation of G. By Theorem 4.1, the principal series representation V (π1 , π2 ) has a GL2 (ᏻ)-ﬁxed vector that is unique up to scalars. The aim of this section is to describe this vector, also called the spherical vector, in the Kirillov model. Theorem 4.2. Let πi (X) = αi (NᏰ/F (X)) for unramiﬁed characters αi of F ∗ for i = 1, 2. The principal series representation V (π1 , π2 ) has a GL2 (ᏻ)-ﬁxed vector that is unique up to scalars described in the Kirillov model as j f (X) = |X|1/2 α1i (%F )α2 (%F ) i+j =vᏰ (X)+d−1 i≥0, j ≥0

if vᏰ (x) ≥ 1 − d, and f (X) = 0 otherwise. Proof. We only sketch the argument that is exactly as in the ﬁeld case as given, for instance, in [9]. Let f ∈ ᏷(V (π1 , π2 )) be the spherical vector in the Kirillov model. By Lemma 4.1, we know that f vanishes outside P1−d , and the value of f (X), for any X ∈ Ᏸ∗ , depends only on |X|. Hence, f is completely determined by the sequence of numbers n {fn }n≥1−d , where % 0 fn = f (% ). Let y = 0 1 . The charactersitic function χ of the double coset KyK is an element of the spherical Hecke algebra Cc∞ (K\G/K). Hence, we get π(χ)f = cf for some constant c. Unraveling this while using the decomposition % ∗ 1 0 K, K∪ KyK = 0 % 0 1 where ∗ runs over representatives of ᏻ/P gives a recurrence relation among the numbers fn . Solving this recurrence relation gives the required formula. 5. Supercuspidal representations. Let H be one of the two maximal compact modulo center subgroups deﬁned in Proposition 1.4. We deﬁne a class of representations of H , called very cuspidal, that when induced to GL2 (Ᏸ) produce irreducible, supercuspidal representations. Both the deﬁnition and the proofs are exactly as in the ﬁeld case. We would expect that all supercuspidal representations of GL2 (Ᏸ) can

KIRILLOV THEORY FOR GL2 (Ᏸ)

35

be obtained in this way, but we have not been successful in showing this. We begin with the deﬁnition of a very cuspidal representation. For this we must ﬁrst deﬁne a ﬁltration on H . Deﬁne a ﬁltration on ᏹ indexed by Z as A1 (m) := P m M2 (ᏻ). A decreasing ﬁltration H1 (m) on H1 is now deﬁned by H1 (m) := 1 + A1 (m) for all m ≥ 1, and H1 (0) := GL2 (ᏻ) = K1 (see Proposition 1.4). Similarly deﬁne two more ﬁltrations on ᏹ as

A2 (m) := P m

and

0 B2 (m) := %

ᏻ ᏻ P ᏻ

1 A2 (m). 0

A decreasing ﬁltration H2 (m) is now deﬁned by H2 (m) := 1 + A2 (m) for all m ≥ 1; deﬁne H2 (0) := K2 (see Proposition 1.4). Deﬁnition 5.1. Let H be either H1 or H2 . A ﬁnite-dimensional irreducible representation (σ, W ) of H is called very cuspidal of level m if it is trivial on H (m) and admits no nonzero vector ﬁxed by the subgroup N(P m−1 ) ⊂ H /H (m). With this deﬁnition, the following lemma is clear. Lemma 5.1. Let (σ, W ) be a very cuspidal representation of level m of H . Then σ restricted to N(ᏻ) breaks up into eigencharacters of ᏻ, all of which have conductor P m , that is, trivial on P m and nontrivial on P m−1 . Furthermore, any such character occurs with the same multiplicity in σ. If we denote this common multiplicity as r(σ ), then the dimension of σ is given by dim(σ ) = r(σ )q d(m−1) q d − 1 . Lemma 5.2. Let U denote the subgroup of G given by u 0 U := : u ∈ ᏻ× . 0 1 Let (σ, W ) be a very cuspidal representation of level m of H . Then dim W U = r(σ ).

36

PRASAD AND RAGHURAM

Proof. Decompose σ as a sum of various characters of N(ᏻ). Since the inner conjugation action of the image of U in H on N(ᏻ) permutes the characters of N(ᏻ) of conductor P m simply transitively, the lemma follows. Proposition 5.1. Let H denote either H1 or H2 . Let (σ, W ) and (σ , W ) be two irreducible very cuspidal representations of level m of H. Let (π, V ) = indG H (σ ), and ). Then let (π , V ) = indG (σ H dim HomG (π, π ) = dim HomH (σ, σ ) . In particular, (π, V ) is an irreducible admissible supercuspidal representation of G. Proof. We use Kutzko’s version of Mackey’s theorem (see [15]), which describes the space of intertwining operators of two induced representations. For this we need a set of representatives % a 0 of (H, H ) double cosets in G, which by Proposition 1.3 can be taken to be 0 1 a≥0 . We have G HomG (π, π ) ⊂ HomG indG HomH ∩g −1 ·H ·ga (σ, ga σ ), H (σ ), IndH (σ ) = a≥0

where ga denotes the matrix ﬁeld case.

a

%a 0 0 1

. The following claim is proved exactly as in the

Claim. If HomH ∩g −1 ·H ·ga (σ, ga σ ) = 0, then a = 0. a

The claim implies the proposition because then the only double coset that can support any nonzero intertwining operator is the identity double coset. The following easy lemma, whose proof is omitted, is used in the next proposition. Lemma 5.3. With respect to the nondegenerate symmetric pairing on ᏹ, given by (X, Y ) → ψ = (Tᏹ/F (XY )), we have (1) A1 (m)⊥ = A1 (−m + 1 − d), (2) A2 (m)⊥ = B2 (−m − d). Proposition 5.2. For i = 1 or 2, let (σ, W ) be a very cuspidal representation of Hi of level m, and let (π, V ) = indG Hi (σ ). Then we have the following. (1) The conductor of π in the sense of epsilon factors is Ce (π) = 2m + i − 1 + 2(d − 1). (2) The representation π is minimal, that is, Ce (π) ≤ Ce (π ⊗ χ) for all quasi characters χ of F ∗ . Proof. We start by computing the epsilon factor of π. We use the notation f1 ∼ f2 for two functions of one complex variable s if there is a constant c such that f1 (s) = cf2 (s) for all s. We refer the reader to [10] for notation and also a way of associating

KIRILLOV THEORY FOR GL2 (Ᏸ)

37

epsilon factors. It is easily checked that if π is an irreducible admissible supercuspidal representation of G, then so is its contragredient π ∨ , and for any such representation, the L-function associated to it as in [10] is the constant function 1. If we can ﬁnd a function φ ∈ Cc∞ (ᏹ) and a matrix coefﬁcient f of π such that the associated zeta integral

s φ(x)f (x)Nᏹ/F (x) dx Z(φ, s, f ) = F

G

is a constant, then by the functional equation, we would get that , −s, f ∨ (π, s, ψF ) ∼ Z φ is the Fourier transform of φ with respect to the character ψ and f ∨ is the where φ function given by f ∨ (g) = f (g −1 ) for all g ∈ G. We give the details in the case i = 1 and leave the other case to the reader since the argument is similar. We choose φ to be the characteristic function of H1 (m) thought of as a (compact open) subset of ᏹ. Choose a vector w ∈ W and a linear form w ∗ on W such that w, w∗ = 1. (Later on we choose this vector w more carefully.) Let f be the function on G that is 0 outside H1 and is given by f (h) = σ (h−1 )w, w ∗ on H1 ; then f is a matrix coefﬁcient of π. It is easily seen that for this choice of f and φ, the zeta integral Z(φ, s, f ) is a constant by using the facts that reduced norm maps H1 (m) into the group of units in F ∗ and that σ is trivial on H1 (m). , −s, f ∨ ). An easy computation using Now it sufﬁces to compute the integral Z(φ Lemma 5.3 gives that x∈ / A1 (−m + 1 − d), (x) = 0, φ ψF Tᏹ/F (x) , x ∈ A1 (−m + 1 − d). So, abbreviating T (x) for Tᏹ/F (x), we have

−s ∨ Z φ , −s, f = ψF T (x) f ∨ (x)Nᏹ/F (x)F dx. A1 (−m+1−d)∩G

Using the fact that supp(f ) = H1 , this integral can be split up as ∞ i=−m+1−d

% i H1 (0)

−s ψF T (x) f ∨ (x)Nᏹ/F (x)F dx.

By putting x = % i y, we get ∞ i=−m+1−d

q

2is H1 (0)

ψF T % i y f ∨ % i y dy.

Let Ᏽi denote the integral occurring above.

38

PRASAD AND RAGHURAM

Claim. For all i ≥ −m + 2 − d, Ᏽi = 0. Assuming the claim for the time being, we get that (π, s, ψF ) ∼ q 2(−m+1−d)s since, after all, the epsilon factor is some exponential and hence nonzero. From Deﬁnition 4.1 it follows that Ce (π) = 2m+2(d −1). So it sufﬁces to prove the claim. An easy veriﬁcation gives that

Ᏽi =

ψF T % i y f ∨ % i y .

y∈H1 (0)/H1 (m)

This can be further split up as

ψF T % i tu f u−1 t −1 % −i .

t∈H1 (0)/H1 (m−1) u∈H1 (m−1)/H1 (m)

If m ≥ 2, then choose w such that it is an eigenvector for H1 (m − 1)/H1 (m) with a nontrivial eigencharacter χ, and if m = 1, then we can take f restricted to H1 (0)/H1 (1) = GL2 (Fq d ) as a nontrivial matrix coefﬁcient (of a cuspidal representation of GL2 (Fq d )). We give further details for the case when m ≥ 2 and leave the case of m = 1 to the reader. The previous expression for Ᏽi can now be written by putting u = 1 + a where a ∈ A1 (m − 1)/A1 (m) as

f t −1 % −i ψF T % i t

t∈H1 (0)/H1 (m−1)

ψF T % i ta χ (a)

a∈A1 (m−1)/A1 (m)

for an appropriate nontrivial character χ on A1 (m−1)/A1 (m). Now since i ≥ −m+ 2 − d, we get by Lemma 5.3 that ψF (T (% i ta)) = 1. This gives that Ᏽi = 0 since the inner summation is 0 as χ is a nontrivial character on a ﬁnite abelian group A1 (m − 1)/A1 (m). This proves the claim. Proposition 5.3. Let H denote either H1 or H2 . Let (σ, W ) be an irreducible very cuspidal representation of level m ofH , and let (π, V ) = indG H (σ ). Let m be −m+1−d 1 A A). Then the natural the character on N0 = N(ᏻ) given by m 0 1 = (% map from W to V factors to give an isomorphism WN0 ,m VN, . In particular, dim(VN, ) = r(σ ). a Proof. It follows from Proposition 1.5 that %0 01 a∈Z is a complete set of (N, H ) double coset representatives. Using [15] and Frobenius reciprocity, we have Hom(πN, , C) HomN (π, ) G HomG indG H (σ ), IndN () HomH ∩g −1 Ng (σ, g ). g∈N\G/H

KIRILLOV THEORY FOR GL2 (Ᏸ)

39

a Any g = %0 01 normalizes N, and hence H ∩ g −1 Ng = N(ᏻ). Furthermore, the character g on N(ᏻ) is given by 1 A −1 1 A g g = g = (% a A). 0 1 0 1 This gives that the conductor of g is equal to −a +1−d. From Lemma 5.1 we know that σ restricted to N(ᏻ) breaks up into eigencharacters, all of which have conductor P m . This gives that the only double coset that can support a nonzero intertwining operator is when a = −m + 1 − d, and for this value of a we have g = m . We therefore get that Hom(πN, , C) HomN(ᏻ) (σ, m ) Hom(σN0 ,m , C), or by dualizing we get πN, σN0 ,m . By Lemma 5.1 we also get dim(πN, ) = r(σ ). Proposition 5.4. For i = 1 or 2, let (σ, W ) be an irreducible very cuspidal representation of level m of Hi , and let (π, V ) = indG Hi (σ ). Then (1) V #0 (2m+i−2) = (0), 1 (2) dim(V #0 (2m+i−1) ) = dim(VN, ), (3) C(π ) = 2m + i − 1. 1

Proof. We onlyH = H1 , the other case being similar. Since G = H AH consider n where A = %0 01 , n ≥ 0 , it follows from Mackey theory that the restriction of indG H (σ ) to H is a sum of representations Wn = indH #0 (n) (σ ). From the deﬁnition of very cuspidality, it is easy to see that Wn is a representation of H of level m+n, and that the possible conductors of characters of N(ᏻ) appearing in Wn lie between max{0, m−n} and m+n. Since #(2m−1) ⊂ #01 (2m−1), this proves m (1). Now observe that conjugation by %0 01 takes #01 (2m) into the subgroup U of H , whose image in H /H (m) consists of diagonal entries with 1 at the right-hand 1 bottom corner. Lemma 5.2 therefore implies that dim(V #0 (2m+i−1) ) ≥ dim(VN, ), and hence (2) follows by Lemma 4.3. We summarise our analysis of new forms and conductors in the following two theorems. These theorems have been proved only for those representations that are either principal series or are supercuspidal representations which are induced from a very cuspidal representation of a maximal compact mod center subgroup. We expect these theorems to be true for all irreducible inﬁnite-dimensional representations.

40

PRASAD AND RAGHURAM

Theorem 5.1. Let (π, V ) be an irreducible admissible inﬁnite-dimensional representation of G. Then the space of new forms VC(π) is isomorphic as Ᏸ∗ -modules to the twisted Jacquet module VN, . Theorem 5.2. Let π be an irreducible admissible representation of GL2 (Ᏸ). Then the conductor in the sense of new forms is related to the conductor in the sense of epsilon factors by the formula Ce (π) − C(π) = 2(d − 1), where d is the index of the division algebra Ᏸ over F. Theorem 5.1 can be strengthened as follows. Theorem 5.3. Let (π, V ) be an irreducible admissible representation that is either a principal series representation or a supercuspidal representation that is compactly induced from a very cuspidal representation of a maximal open compact mod center subgroup. Let x denote the element 01 %0 . Then for all m ≥ 0, Vc+m Vc ⊕ x · Vc ⊕ · · · ⊕ x m · Vc , dimC (Vc+m ) = (m + 1) dimC (VN, ). Proof. The proof is by induction on m. The case m = 0 follows from Theorem 5.1. We prove that Vc+m = Vc+m−1 ⊕ x m · Vc , which completes the proof of the theorem. Since x m #01 (c)x −m contains #01 (c + m), we have that x m · Vc is contained in Vc+m . So by Lemma 4.3 and Theorem 5.1, the theorem follows if we show that any v ∈ Vc for which x m ·v ∈ Vc+m−1 is necessarily trivial. But this follows from Proposition 1.8 since if Gv is the stabilizer in G of such a vector v, then Gv contains #01 (c) and also x −m #01 (c + m − 1)x m , which in turn contains N(c − 1). Our ﬁnal result in this section identiﬁes new forms of supercuspidal representations explicitly in the Kirillov model. Proposition 5.5. For i = 1 or 2, let (σ, W ) be an irreducible very cuspidal representation of level m of Hi , and let (π, V ) = indG Hi (σ ). Let c = C(π) = 2m − 1 + i be the conductor of π in the sense of new forms. Then (i) f (Xu) = f (X), ∀u ∈ ᏻ× , ∀X ∈ Ᏸ∗ Vc = f ∈ ᏷(π ) : , (ii) supp(f ) ⊂ % 1−d ᏻ× where we recall that Vc = V #0 (c) . 1

Proof. We give the details for the case i = 1 and leave the other case, which is very similar, to the reader. For w ∈ W U , let gw ∈ V be given by 0, x∈ / H1 , gw (x) = σ (x)w, x ∈ H1 .

KIRILLOV THEORY FOR GL2 (Ᏸ)

Also, let

% −m fw := 0

41

0 g . 1 w

So fw is a new form for π , and by Lemma 5.2 and Proposition 5.4, these are all the new forms of π . By Lemma 4.1 and Theorem 5.3, it is enough to show that the vector φw in the Kirillov space ᏷(π) corresponding to fw vanishes on P2−d . Let X ∈ P 2−d , so we need to show that φw (X) = 0. From Section 3 we need to show that π X0 01 fw ∈ V (N, ). Hence, it is enough to show that there exists a k ∈ Z such that

1 Y X 0 (Y ) f (x) dY = 0, ∀x ∈ G. 0 1 0 1 w Pk We may write X = u% r where r ≥ 2 − d and u ∈ ᏻ× . Using the deﬁnitions of fw and gw , it is enough to show that there exists a k ∈ Z such that r−m

% Y dY = 0, ∀x ∈ G, (Y )gw xu 0 1 Pk −1 u 0 where xu = u0 01 x 0 1 . It is clear that as x varies over G, so does xu . Using a Proposition 1.5, we may write G = H1 · P1 , where P1 = %0 Z1 : a ∈ Z, Z ∈ Ᏸ . Now by the deﬁnition of gw and this decomposition for G, it is enough to show that there exists a k ∈ Z such that r−m a

% % Z Y dY = 0, ∀a ∈ Z, ∀Z ∈ Ᏸ. (Y ) gw 0 1 0 1 Pk t Since supp(gw ) = H1 and since %0 T1 ∈ H1 if and only if t = 0 and T ∈ ᏻ, it is enough to show that there exists a k ∈ Z such that

1 % m−r Y + Z dY = 0, ∀Z ∈ Pm−r+k . (Y ) gw 0 1 Pk Making the substitution T = % m−r Y + Z, we need to show that there exists k ∈ Z such that

1 T dT = 0. (% r−m T ) gw 0 1 Pm−r+k Now choose k = r −1 and since r ≥ 2−d and the conductor of is P1−d , the above integral boils down to

1 T 1 T w dT = 0, dT = gw σ 0 1 0 1 Pm−1 Pm−1 by deﬁnition of very cuspidality of σ.

42

PRASAD AND RAGHURAM

Remark 5.1. In this article we deal with a family of supercuspidal representations of G which are obtained by compactly inducing very cuspidal representations of maximal open compact subgroups. We follow [16] for this. We may also follow [6] and get another notion of very cuspidal representation of these subgroups, which also compactly induce to give irreducible admissible supercuspidal representations of G. This approach is taken in [23], to which the reader may refer for details. This other approach also lends itself to all the computations made here, namely, determining the twisted Jacquet module, computing the epsilon factors (really the exponent occuring in these factors), and determining the space of new forms and also the conductor in the sense of new forms. Remark 5.2. Following Howe’s construction of supercuspidal representations of GLn (F ) (see [11]), we may in the tame case (i.e., when p does not divide 2d where p is the characteristic of the residue ﬁeld of F ) attach supercuspidal representations of G to “admissible characters” of degree 2d ﬁeld extensions of F. (This is expected; as in the tame case, via the local Langlands correspondence, an irreducible representation of the Weil group WF of F is monomial. See [25].) See especially [18] for a lucid discussion of this theme.The second author is ﬁlling in the details of this approach and the results will appear elsewhere. Remark 5.3. There is an elaborate theory by C. J. Bushnell and P. C. Kutzko in [5] that describes the theory of types for GLn (F ). P. Broussous has been adapting their formalism in the context of forms of GL(n). The reader is referred to his papers, particularly [2], which applies to our context and gives another approach to handle supercuspidal representations of G. 6. Shalika model. There is a notion called the Shalika model that is closely related to the Kirillov model. Note that in general we do not have a multiplicity-1 theorem for the Kirillov model, but if we add a further condition on the model we are interested in, then we do get a multiplicity-1 theorem, but the price we pay is that only a small number of representations admit such a model. In this section we consider the notion of Shalika models and prove that an irreducible representation π admits at most one Shalika model and if it does admit a Shalika model, then π is self-contragredient. The reader is referred to the article of Jacquet and Rallis [13] where analogous notions are considered for GL(2n, F ). Deﬁnition 6.1. A linear functional : V → C for an irreducible admissible representation (π, V ) of G is called a Shalika functional if A X v = A−1 X (v) π 0 A for all A ∈ Ᏸ∗ , X ∈ Ᏸ, and v ∈ V . The space of all Shalika functionals for π is denoted by ᏿π .

KIRILLOV THEORY FOR GL2 (Ᏸ)

43

Before we come to the statement of the main result of this section, we recall the following theorem of the ﬁrst author (see [20] and [21]). Theorem 6.1. Let π be an irreducible admissible representation of G = GL2 (Ᏸ), and let M be the subgroup of diagonal matrices. Then dimC HomM (π, C) ≤ 1. Furthermore, if dimC HomM (π, C) = 1, then π π ∨ . The following is the main theorem of this section. Theorem 6.2. If (π, V ) is an irreducible admissible representation of GL2 (Ᏸ), then the dimension of the space of Shalika functionals is at most 1, that is, dimC (᏿π ) ≤ 1. Furthermore, if dimC (᏿π ) = 1, then π π ∨ . Proof. By Theorem 2.1 we know explicitly the twisted Jacquet module of any parabolically induced representation, and the above theorem falls out as an easy corollary for any subquotient of such a representation. We can therefore assume in the rest of the proof that π is a supercuspidal representation for which we construct an injective linear map from ᏿π to HomM (π, C), which by the earlier theorem, completes the proof of this theorem. To construct an injective map from ᏿π to HomM (π, C), we note that a Shalika functional is in particular a degenerate Whittaker functional. Therefore, we have an S-invariant linear map πN,ψ → C, which when composed with the natural map from π to πN, gives rise to . We therefore have the following sequence of homomorphisms: π −→ Cc∞ Ᏸ∗ , πN,ψ −→ Cc∞ (Ᏸ∗ ) −→ C, where the map Cc∞ (Ᏸ∗ ) → C, is given by integrating over Ᏸ∗ . The ﬁrst arrow above is an isomorphism, and the second is a surjection. The composite of all the three maps gives rise to a nonzero M-invariant linear map on π, completing the proof of the theorem. References [1] [2] [3] [4]

I. N. Bernshtein and A. V. Zelevinsky, Representation theory of GL(n, F ) where F is a non-Archimedean local ﬁeld, Russian Math. Surveys 31 (1976), 1–68. P. Broussous, Minimal strata for GL(m, Ᏸ), J. Reine Angew. Math. 514 (1999), 199–236. D. Bump, Automorphic Forms and Representations, Cambridge Stud. Adv. Math. 55, Cambridge Univ. Press, Cambridge, 1997. C. J. Bushnell and A. Frohlich, Nonabelian congruence Gauss sums and p-adic simple algebras, Proc. London Math. Soc. (3) 50 (1985), 207–264.

44 [5] [6] [7] [8]

[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

[24] [25]

PRASAD AND RAGHURAM C. J. Bushnell and P. C. Kutzko, The Admissible Dual of GL(N) via Compact Open Subgroups, Ann. of Math. Stud. 129, Princeton Univ. Press, Princeton, 1993. H. Carayol, Représentations cuspidales du groupe linéaire, Ann. Sci. École Norm. Sup. (4) 17 (1984), 191–225. W. Casselman, On some results of Atkin and Lehner, Math. Ann. 201 (1973), 301–314. P. Deligne, “Formes modulaires et représentations de GL(2)” in Modular Functions of One Variable, II (Antwerp, 1972), Lecture Notes in Math. 349, Springer, Berlin, 1973, 55–105. R. Godement, Notes on Jacquet-Langlands theory, unpublished notes, Institute for Advanced Study, Princeton, 1970. R. Godement and H. Jacquet, Zeta Functions of Simple Algebras, Lecture Notes in Math. 260, Springer, Berlin, 1972. R. E. Howe, Tamely ramiﬁed supercuspidal representations of GLn , Paciﬁc J. Math. 73 (1977), 437–460. H. Jacquet and R. Langlands, Automorphic Forms on GL2 , Lecture Notes in Math. 114, Springer, Berlin, 1970. H. Jacquet and S. Rallis, Uniqueness of linear periods, Compositio Math. 102 (1996), 65–123. H. Koch and E.-W. Zink, Zur Korrespondenz von Darstellungen der Galoisgruppen und der zentralen Divisionsalgebren über lokalen Körpern, Math. Nachr. 98 (1980), 83–119. P. C. Kutzko, Mackey’s theorem for non-unitary representations, Proc. Amer. Math. Soc. 64 (1977), 173–175. , On the supercuspidal representations of GL2 , Amer. J. Math. 100 (1978), 43–60. C. Moeglin and J.-L. Waldspurger, Modèles de Whittaker dégénérés pour des groups padiques, Math. Z. 196 (1987), 427–452. A. Moy, Local constants and the tame Langlands correspondence, Amer. J. Math. 108 (1986), 863–930. D. Prasad, Trilinear forms for representations of GL(2) and local epsilon factors, Compositio Math. 75 (1990), 1–46. , On the self-dual representations of a p-adic group, Internat. Math. Res. Notices 1999, 443–452. , Some remarks on representations of a division algebra and of the Galois groups of local ﬁelds, J. Number Theory 74 (1999), 73–97. , The space of degenerate Whittaker model, preprint, 1998. A. Raghuram, Some topics in algebraic groups: Representation theory of GL2 (Ᏸ) where Ᏸ is a division algebra over a non-Archimedean local ﬁeld, thesis, Tata Institute of Fundamental Research, University of Mumbai, 1999. J.-P. Serre, Linear Representations of Finite Groups, Grad. Texts in Math. 42, Springer, New York, 1977. J. Tate, “Number theoretic background” in Automorphic Forms, Representations and Lfunctions (Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math. 33, Amer. Math. Soc., Providence, 1979, 3–26.

Prasad: Mehta Research Institute, Chhatnag Road, Jhusi, Allahabad-211019, India; [email protected] Raghuram: School of Mathematics, Tata Institute of Fundamental Research, Dr. Homi Bhabha Road, Colaba, Mumbai 400005, India; [email protected]; Current: Department of Mathematics, University of Toronto, 100 Saint George Street, Toronto, Ontario M5S 3G3, Canada; [email protected]

Vol. 104, No. 1

DUKE MATHEMATICAL JOURNAL

© 2000

ON THE NUMBER OF NONREAL ZEROS OF REAL ENTIRE FUNCTIONS AND THE FOURIER-PÓLYA CONJECTURE HASEO KI and YOUNG-ONE KIM

This paper is concerned with a general theorem on the number of nonreal zeros of transcendental functions. J. Fourier formulated the theorem in his work Analyse des équations déterminées in 1831, but he did not give a proof. Roughly speaking, the theorem states that if a real entire function f (x) can be expressed as a product of linear factors, then we can count the nonreal zeros of f (x) by observing the behavior of the derivatives of f (x) on the real axis alone. As we shall see in the sequel, this theorem completely justiﬁes his former argument, by which he tried to prove that the √ function J0 (2 x ) has only real zeros. It seems that no complete proof of the theorem is known, and no general theorem has been published that justiﬁes the argument. Later, in 1930, G. Pólya published a paper entitled Some problems connected with Fourier’s work on transcendental equations [P3]. In this paper, Pólya conjectured two hypothetical theorems that are closely related to Fourier’s unproved theorem. In fact, he conjectured three, but he proved that two of them are equivalent to each other. The ﬁrst hypothetical theorem is a modernized formulation of the theorem, and it justiﬁes Fourier’s argument completely. The second conjecture was proved in 1990, but it is impossible to justify the argument using the conjecture alone. In the present paper, we prove Pólya’s formulation of the theorem (his ﬁrst conjecture) as well as its extensions, give a very simple and direct proof of the second conjecture mentioned above, and exhibit some applications of the results. In particular, we completely justify Fourier’s argument by our general theorems. Acknowledgments. Professor Fefferman has encouraged and helped us to publish this paper. The authors truly thank him for this. The authors also thank Professor Csordas for his kind interest in the results and his valuable suggestions on the proof of the Pólya-Wiman conjecture. 1. Historical introduction. In this section, we brieﬂy explain our results as well as their background. A real entire function is an entire function that assumes only real values on the real axis. Fourier’s unproven theorem asserts that we can know the number of nonreal zeros of such real entire functions by counting their critical points, which are deﬁned as follows: Let f (x) be a real analytic function deﬁned in an open Received 11 November 1999. 2000 Mathematics Subject Classiﬁcation. Primary 30D15, 30D20. Kim’s work supported by the Korea Science and Engineering Foundation (KOSEF) through the Global Analysis Research Center (GARC) at Seoul National University. 45

46

KI AND KIM

interval I of the real axis, and let l be a positive integer. Suppose that c ∈ I is a zero of f (l) (x) of multiplicity m ≥ 1; that is, f (l) (c) = · · · = f (l+m−1) (c) = 0

and

f (l+m) (c) = 0.

Put k = 0 if f (l−1) (c) = 0. Otherwise, put m  , if m is even,   2  m+1 , if m is odd and f (l−1) (c)f (l+m) (c) > 0, (1.1) k=  2      m − 1 , if m is odd and f (l−1) (c)f (l+m) (c) < 0. 2 Then we say that f (l) (x) has k critical zeros and m − k noncritical zeros at x = c. A point in I is said to be a critical point of f (x) if some derivative of f (x) has a critical zero at the point. For example, cosh x has inﬁnitely many critical points at the origin with no other critical points, and the polynomial 1 − x 2 + x 8 has four critical points in the whole real axis. Observe that a real analytic function f (x) has a critical point if and only if some derivative of f (x) has more real zeros than guaranteed by Rolle’s theorem, or equivalently, f (x) has fewer real zeros than the upper bound given by the Budan-Fourier rule, which is generalized and precisely stated by A. Hurwitz [H2, pp. 589–590]. Let f (x) be a real polynomial of degree d > 1. For n = 0, 1, . . . , d − 1, let 2Jn denote the number of nonreal zeros of f (n) (x). Then it is easy to see that f (n) (x) has exactly Jn−1 − Jn critical zeros for n = 1, . . . , d − 1. Since f (d−1) (x) certainly has only real zeros, we obtain the following rule of de Gua: a real polynomial has just as many critical points as couples of nonreal zeros. Throughout this paper, we say that de Gua’s rule is applicable to a real entire function f (x) whenever it has just as many critical points as couples of nonreal zeros. This paragraph is extracted from Pólya’s paper [P3, pp. 21–23]. In his work √ Théorie de la chaleur published in 1822, Fourier proved that the function J0 (2 x ) has no critical points. (For a proof of this, see Section 5.2 of the present paper.) Then he applied de Gua’s algebraical rule to the transcendental function without further inquiry and concluded that it has only real zeros. After the publication of the work, Cauchy and Poisson expressed doubt about the validity of Fourier’s reasoning. In particular, Poisson remarked that in order to apply de Gua’s rule to a function f (x), it must be supposed that the derivatives f (n) (x) of high enough order have only real zeros. Later Fourier tried to justify his argument by establishing a general theorem; he formulated the following theorem in his work Analyse des équations déterminées, but he did not give a proof. Fourier’s unproved theorem. Let f (x) be a real entire function and suppose that f (x) can be expressed as the product of a ﬁnite or an inﬁnite number of linear

THE FOURIER-PÓLYA CONJECTURE

47

factors of the form

1−

x , α

1−

x , β

1−

x , . . .. γ

Then f (x) has just as many critical points as couples of nonreal zeros. Note that if an entire function is of order less than 1, then Hadamard’s theorem implies that it is of genus 0, that is, it can be expressed as an absolutely convergent √ product of linear factors, and that the function J0 (2 x ) is of order 21 . Also note that some functions, such as cos x, cosh x, or Riemann’s ξ -function, can be expressed as an inﬁnite product of linear factors though they are not of genus 0. In 1841, M. A. Stern studied the roots of transcendental equations that were known at the time and asserted the same theorem, but he also did not give a proof [St, pp. 47–50]. It seems that mathematicians at that time thought the theorem was an obvious one, but the authors do not know the reason. Later, in 1930, Pólya conjectured the following three hypothetical theorems that are closely related to the theorem [P3]. Theorem A. A real integral function of genus 0 has just as many critical points as couples of imaginary zeros. Theorem B. If a real integral function of genus 1∗ has only a ﬁnite number of imaginary zeros, it has just as many critical points as couples of imaginary zeros. Theorem C. If a real integral function f (z) of genus 1∗ has only a ﬁnite number of imaginary zeros, its derivatives from a certain one onward, let us say f (n) (z), f (n+1) (z), . . . , have only real zeros. A real entire function f (x) is of genus 1∗ if it can be expressed in the form 2 f (x) = e−αx g(x), where α ≥ 0 and g(x) is a real polynomial or a real entire function of genus 0 or 1. The hypothetical Theorem A is narrower than Fourier’s unproven theorem; nevertheless, it is general enough to justify Fourier’s argument. The assertions of Theorems A and B are the same, but their assumptions are quite different. Just because of this difference, it is impossible to justify Fourier’s argument using√Theorem B alone. All we can conclude from Theorem B is that the function J0 (2 x ) has either only real zeros or inﬁnitely many nonreal zeros. In the same paper, Pólya proved that Theorems B and C are equivalent: He proved that if a real entire function f (x) of genus 1∗ has only a ﬁnite number of nonreal zeros, then its derivative f (x) is also of genus 1∗ and has ﬁnitely many nonreal zeros. He also proved that if the function f (x) has 2J nonreal zeros and f (x) has 2J nonreal zeros, then f (x) has exactly J − J critical zeros. The hypothetical Theorem C is known as the Pólya-Wiman conjecture, because A. Wiman also conjectured it [Wi1]. We call the hypothetical Theorem A the FourierPólya conjecture, and the main purpose of this paper is to prove it completely (see Theorem 4.1). In fact, we obtain the following more general theorems.

48

KI AND KIM

Theorem 4.2. Let f (x) be a real entire function that is at most of order 1 and minimum type, let b1 , b2 , . . . denote the real zeros of f (x) that are different from zero, and suppose that bj−1 −→ β (r → ∞) |bj |
for some real number β. Then for each real constant δ, the function eδx f (x) has just as many critical points as couples of nonreal zeros. Theorem 4.3. Let f (x) be an even or odd real entire function that is at most of order 2 and minimum type, let b1 , b2 , . . . denote the real zeros of f (x) that are different from zero, and suppose that bj−2 < ∞. j

Then for each nonnegative real constant α, the function e−αx f (x) has just as many critical points as couples of nonreal zeros. 2

In the papers [P3], [P4], [Wi1], [Wi2], Pólya and Wiman proved various partial results of the Pólya-Wiman conjecture. In particular, Pólya proved the conjecture for functions of order less than 43 in [P4]. Therefore, he also proved the Fourier-Pólya conjecture for the functions that have ﬁnitely many nonreal zeros. After the proof of his 43 theorem, Pólya published various results about the inﬂuence of sign changes of derivatives of a function on its analytic character [BP1], [BP2], [P5], [P6], and [PW]. In our proof of the Fourier-Pólya conjecture, we use a theorem of S. Bernstein that is closely related to these results. In 1987, T. Craven, G. Csordas, and W. Smith proved the Pólya-Wiman conjecture in the case where f (z) is of genus 0 or 1 [CCS2], [CCS1]. In 1990, Kim proved the Pólya-Wiman conjecture in the general case by completing their proof [K1]. Later he gave a very simple and direct proof of the Pólya-Wiman conjecture [K2], but the proof is incompletely described. Recently, Kim proved the Fourier-Pólya conjecture under the additional assumption that the zeros are distributed in the inﬁnite strip | Im z| ≤ A for some A > 0 [K3], [K4]. All of these results, however, are unable to justify Fourier’s argument, because they assume some additional conditions on the number or distribution of nonreal zeros. In 1997, Kim proved a very weak form of the Fourier-Pólya conjecture that assumes no conditions on the distribution of nonreal zeros [K5]. Based on the ideas of this result, the authors completed a proof of the Fourier-Pólya conjecture [KK]. These two results will not be published, because the essential parts of them are contained in this paper. This paper is organized as follows. In Section 2, we give a very simple and direct proof of the Pólya-Wiman conjecture. In Section 3, we obtain some technical results on the number of critical points of real analytic functions. These results are used in Section 4 to prove the Fourier-Pólya conjecture (see Theorem 4.1) as well as its

THE FOURIER-PÓLYA CONJECTURE

49

extensions (see Theorems 4.2 and 4.3). Finally, in Section 5, we apply these theorems to obtain very simple proofs of some classical results of Hurwitz, Laguerre, and Pólya. 2. The Pólya-Wiman conjecture. As it is mentioned in the introduction, the Pólya-Wiman conjecture was completely proved in 1990 (see [CCS2], [CCS1], and [K1]), but the proof is very complicated. Since the Pólya-Wiman conjecture plays a crucial role in our proof of the Fourier-Pólya conjecture (and its extensions), and since there is a very simple and direct proof of the Pólya-Wiman conjecture, we give the proof here for completeness as well as for the reader’s convenience. Let us introduce the following notation of R. P. Boas: Let 0 < ρ < ∞ and 0 ≤ τ ≤ ∞. An entire function is said to be of growth (ρ, τ ) if it is of order less than ρ or is of order ρ with type not exceeding τ . Thus, for each ﬁxed (ρ, τ ), the class of all entire functions of growth (ρ, τ ) is closed under addition and differentiation; if τ = 0 or ∞, then it is closed under multiplication, too. The general properties of entire functions that are needed in this paper can be found in [B]. We need some preliminaries. The class of all real entire functions of genus 1∗ that have ﬁnitely many nonreal zeros is denoted by ᏸ. Then ᏸ is closed under differentiation, and differentiation does not increase the number of nonreal zeros of a function in the class ᏸ. This fact is an easy consequence of Rolle’s theorem, Hurwitz’s theorem, and the following theorem of Laguerre and Pólya. The Laguerre-Pólya theorem. A real entire function is of genus 1∗ and has only real zeros if and only if it can be uniformly approximated on compact sets in the complex plane by a sequence of real polynomials with only real zeros. Proof. See [L, Chapter 8] or [PS]. Let f (z) be in the class ᏸ, and let c1 , c¯1 , . . . , cJ , c¯J denote the nonreal zeros of f (z). Then f (z) can be expressed in the form (2.1)

n −αz2 +βz

f (z) = cz e

k

J z 1− ez/ak (z − cj )(z − c¯j ), ak j =1

where c and β are real constants, n is a nonnegative integer, α ≥ 0, and ak ’s are the zeros of f (z) that are different from 0. Of course, it must be supposed that real −2 |a | < ∞. The logarithmic derivative of f (z) is given by k k

J 1 1 1 1 f (z) n + . = − 2αz + β + + + f (z) z z − a k ak z − cj z − c¯j k

j =1

By a direct calculation, we have

1 1 −2 Im z Im + = |z − Re c|2 − (Im c)2 . 2 2 z − c z − c¯ |z − c| |z − c| ¯

50

KI AND KIM

Therefore, if z ∈ / R and if |z−Re cj | > | Im cj | for all j = 1, . . . , J , then Im f (z)/f (z) = 0, in particular f (z) = 0. In this way, we obtain the following theorem, which was originally formulated by J. L. W. V. Jensen [J, p. 190]. Jensen’s theorem. Let f (z) be a nonconstant real entire function in the class

ᏸ. If z1 is a nonreal zero of f (z), then there is a nonreal zero z0 of f (z) such that

|z1 − Re z0 | ≤ Im z0 . For an entire function f (z) and r > 0, let M(r; f ) denote the maximum modulus of f (z) on the circle |z| = r. Lemma 2.1. Let A and B be arbitrary positive real numbers, and let C be the positive real number satisfying 2AC(C − B) = 1, √ that is, C = (B + B 2 + 2A−1 )/2. If f (z) is an entire function of growth (2, A), then we have √

1/n √ M B n; f (n) 2 lim n ≤ 2ACeAC . n→∞ n! Proof. Let a be an arbitrary real number greater than A, and let c be the positive real number satisfying 2ac(c − B) = 1. Since f (z) is of growth (2, A), there is a positive real number r1 such that M(r; f ) < ear

2

(r ≥ r1 ).

Then Cauchy’s integral formula implies that for each n = 1, 2, . . . ,

1/n √ √ M B n; f (n) ar 2 − log r − B n (2.2) log ≤ n! n

√ r > max r1 , B n .

It is easy to see that for each n = 1, 2, . . . , the right-hand side of (2.2) attains its minimum value 2 2aceac log √ n √ at r = c n, so that lim

n→∞

√

√

1/n M B n; f (n) 2 n ≤ 2aceac . n!

Since a > A is arbitrary, this proves our assertion.

51

THE FOURIER-PÓLYA CONJECTURE

Let f (z) be an entire function, let {zn }∞ n=0 be a sequence of complex numbers, and suppose that f (n) (zn ) = 0 for n = 0, 1, 2, . . . . Then for each positive integer n, we have z ζ1 ζn−1 f (z) = ··· f (n) (ζn ) dζn · · · dζ2 dζ1 (z ∈ C), z0

z1

zn−1

so that (2.3)

|f (z)| ≤

n Mn |z − z0 | + |z0 − z1 | + · · · + |zn−2 − zn−1 | n!

(z ∈ C),

where Mn = M(|z| + |z − z0 | + |z0 − z1 | + · · · + |zn−2 − zn−1 |; f (n) ). For a detailed proof of this inequality, see [G, pp. 11–13]. Lemma 2.2. Let A, B, and C be as in Lemma 2.1, let f (z) be an entire function of growth (2, A), and suppose that 2ABCeAC < 1. 2

If there is a sequence {zn }∞ n=0 of complex numbers such that (1) f (n) (zn ) = 0 for n = 0, 1, 2, . . . , and √ (2) lim n→∞ (|z0 − z1 | + |z1 − z2 | + · · · + |zn−1 − zn |)/ n < B, then f (z) = 0 for all z ∈ C. Proof. From Lemma 2.1, it follows that √

1/n √ M B n; f (n) 2 lim B n ≤ 2ABCeAC , n→∞ n! and hence we have

√

M B n; f (n) √ n lim B n = 0, n→∞ n!

(2.4)

because we have assumed that 2ABCeAC < 1. Let {zn }∞ n=0 be a sequence of complex numbers satisfying (1) and (2), and let z be an arbitrary complex number. Since {zn }∞ n=0 satisﬁes (2), there is a positive integer n1 such that √ |z| + |z − z0 | + |z0 − z1 | + · · · + |zn−2 − zn−1 | ≤ B n (n ≥ n1 ). 2

Then (2.3) implies that √

M B n; f (n) √ n |f (z)| ≤ B n n!

(n ≥ n1 ),

because f (n) (zn ) = 0 for n = 0, 1, 2, . . . . Finally, (2.4) gives the desired result.

52

KI AND KIM

Remark 2.1. Lemma 2.2 is a special case of Theorem III of W. Gontcharoff [G]. Now we can prove the Pólya-Wiman conjecture. Theorem 2.1 (The Pólya-Wiman conjecture). For each real entire function f (z) in the class ᏸ, there is a positive integer n such that the functions f (n) (z), f (n+1) (z), . . . have only real zeros. Proof. Let f (z) be a nonconstant, real entire function in the class ᏸ. Since the class ᏸ is closed under differentiation, and differentiation does not increase the number of nonreal zeros of a function in the class ᏸ, it is enough to show that there is a positive

integer n such that f (n) (z) has only real zeros. To obtain a contradiction, assume that for each nonnegative integer n, f (n) (z) has a nonreal zero. For n = 0, 1, 2, . . . , let Xn denote the set of nonreal zeros of f (n) (z) in the upper half plane Im z > 0. Then each Xn is ﬁnite by Jensen’s theorem andan inductive argument. Since we have assumed that each Xn is nonempty, X = ∞ n=0 Xn is a nonempty compact space with respect to the product topology. For n = 1, 2, . . . , let En be deﬁned as follows: En = (ζ0 , ζ1 , ζ2 , . . . ) ∈ X : |ζj +1 − Re ζj | ≤ Im ζj , j = 0, 1, . . . , n .

Then each En is a closed subset of the compact space X and E1 ⊃ E2 ⊃ · · · . Moreover, Jensen’s theorem implies that En = ∅ for each n = 1, 2, . . . . Therefore, we have ∞ ∞ n=1 En = ∅. This means that there is an inﬁnite sequence {zn }n=0 of complex numbers such that Im zn > 0,

f (n) (zn ) = 0

and

|zn+1 − Re zn | ≤ Im zn

(n = 0, 1, 2, . . . ).

For n = 0, 1, 2, . . . , let zn = αn + iβn with αn ∈ R and βn > 0. Then {βn } is a decreasing sequence of positive real numbers. Moreover, by an induction or the Cauchy-Schwarz inequality, we have |zm − zm+1 | + |zm+1 − zm+2 | + · · · + |zm+n−1 − zm+n | ≤ βm − βm+n + n(βm 2 − βm+n 2 ), for m = 0, 1, 2, . . . , and for n = 1, 2, . . . . Let β = limn→∞ βn . Then we have lim

(2.5)

n→∞

|zm − zm+1 | + |zm+1 − zm+2 | + · · · + |zm+n−1 − zm+n | √ n ≤ βm 2 − β 2 (m = 0, 1, 2, . . . ).

THE FOURIER-PÓLYA CONJECTURE

53

Since f (z) is in the class ᏸ, it is of the form (2.1). Thus f (z) is of growth (2, A) for 2 some positive real√number A. Choose a positive real number B so that 2ABCeAC < 1 where C = (B + B 2 + 2A−1 )/2. From (2.5), there is a positive integer m such that lim

n→∞

|zm − zm+1 | + |zm+1 − zm+2 | + · · · + |zm+n−1 − zm+n | < B. √ n

Since f (m) (z) is also of growth (2, A), Lemma 2.2 implies that f (m) (z) = 0 for all z ∈ C; that is, f (z) is a real polynomial. This is the desired contradiction: if f (z) is a real polynomial, then there must be a positive integer n such that f (n) (z) has only real zeros. This proves the Pólya-Wiman conjecture. Remarks 2.2. (a) This proof of the Pólya-Wiman conjecture is a revised version of Kim’s proof given in [K2]. (b) Using the same argument as above, we can prove the following generalization of the Pólya-Wiman conjecture: If a real entire function f (z) of genus 1∗ has only a ﬁnite number of zeros outside the strip | Im z| ≤ A for some A ≥ 0, then there is a positive integer n such that the functions f (n) (z), f (n+1) (z), . . . have no zeros outside the strip | Im z| ≤ A. In the proof of this theorem, we must use the obvious strip version of the Laguerre-Pólya theorem, which is a direct consequence of [L, Theorem 8.3]. For completeness, we restate Pólya’s hypothetical Theorem B and sketch a proof of it. Theorem 2.2. Every real entire function in the class ᏸ has just as many critical points as couples of nonreal zeros. Proof. Let f (z) be a real entire function in the class ᏸ, and let 2Jn denote the number of nonreal zeros of f (n) (z) for each n = 0, 1, . . . . Then [P3, Theorem I] states that {Jn } is a decreasing sequence of nonnegative integers and that f (z) has exactly J0 − lim Jn n→∞

critical points. (For another proof of this fact, see [K3, Section 2].) Now the PólyaWiman conjecture implies that limn→∞ Jn = 0, and this proves our assertion. Remark 2.3. It may be remarked that if a real entire function f (z) is not of genus 1∗ , then lim Jn > 0,

n→∞

where 2Jn denotes the number of nonreal zeros of f (n) (z) for n = 0, 1, 2, . . . (see [HSW], [HW1], [HW2], [LO], and [Sh]).

54

KI AND KIM

3. Critical points of real analytic functions. Let f (x) be a real analytic function deﬁned in an open interval I of the real axis, and let a, b ∈ I, a ≤ b. For λ = 0, 1, 2, . . . , let N[a,b] (f (λ) ) denote the number of zeros of f (λ) (x) in the closed interval [a, b]. Similarly, let K[a,b] (f (λ) ) denote the number of critical zeros of f (λ) (x) in [a, b] for λ = 1, 2, . . . . If λ ≥ 1 and if f (λ−1) (a)f (λ) (a)f (λ−1) (b)f (λ) (b) = 0, then (1.1) implies that

(3.1)

2K[a,b] f (λ) = N[a,b] f (λ) − N[a,b] f (λ−1) 1 − sg f (λ−1) (a)f (λ) (a) − sg f (λ−1) (b)f (λ) (b) , 2

here sg r = r/|r|, r = 0. For a detailed proof of this equation, see [H2, pp. 585–587]. The following is an easy consequence of Hurwitz’s theorem. Theorem 3.1. Let {fn (x)} be a sequence of real analytic functions in an open interval I of the real axis, let c ∈ I , let l be a positive integer, and suppose that {fn (x)} converges to a real analytic function f (x) uniformly in a neighborhood of c in the complex plane. If f (l) (x) has k (≥ 1) critical zeros at x = c, and if (3.2) m = 2k +

1 lim sg f (l−1) (c − +)f (l) (c − +) − sg f (l−1) (c + +)f (l) (c + +) , 2 +→0+

then there is a positive real number +1 such that for each positive real number + less than +1 , there is a positive integer N such that for each n = N, N + 1, . . . , we have l+m λ=l

K[c−+,c++] fn(λ) = k.

Proof. From (1.1) and (3.2), it follows that f (l−1) (c) = 0,

f (l) (c) = · · · = f (l+m−1) (c) = 0,

f (l+m) (c) = 0.

Let +1 be so small that the functions f (l−1) (x), . . . , f (l+m) (x) have no zeros in the deleted +1 -neighborhood of the point c in the complex plane, and also that {fn (x)} converges uniformly to f (x) in the same neighborhood. Let + be a positive real number less than +1 . Then there is a positive integer N such that for n = N, N +1, . . . , (λ) and for λ = l −1, . . . , l +m, the functions fn (x) and f (λ) (x) have the same number of zeros in the +-neighborhood of c in the complex plane, and sg fn(λ) (c − +) = sg f (λ) (c − +),

sg fn(λ) (c + +) = sg f (λ) (c + +).

Since f (l−1) (x) and f (l+m) (x) have no zeros in the +-neighborhood of c in the complex plane, (3.1) implies the desired result.

55

THE FOURIER-PÓLYA CONJECTURE

Corollary. Let {fn (x)} be a sequence of real entire functions converging to a real entire function f (x) uniformly on compact sets in the complex plane, and let K be a positive integer. If f (x) has at least K critical points, then fn (x) also has at least K critical points whenever n becomes sufﬁciently large. λ+µ

If f (λ) (a)f (λ+1) (a) · · · f (λ+µ) (a) = 0, let Vλ (f, a) denote the number of sign λ+µ changes in the sequence f (λ)(a), f (λ+1)(a), . . . , f (λ+µ)(a). For example, Vλ (e−x , 0) = µ. Then (3.1) is equivalent to

λ λ 2K[a,b] f (λ) = N[a,b] f (λ) − N[a,b] f (λ−1) + Vλ−1 (f, a) − Vλ−1 (f, b).

(3.3)

The following is a consequence of Rolle’s theorem. Theorem 3.2. Let a be a nonnegative real number, let l be a positive integer, let f (x) be a real analytic function deﬁned in an open interval containing [0, a], and let g(x) = xf (x). Then l

K[0,a] f

λ=1

(λ)

≤

l+1

K[0,a] g (λ) .

λ=1

Proof. There is a positive real number + such that

N[−+,a++] f (λ) = N[0,a] f (λ) ,

K[−+,a++] f (λ) = K[0,a] f (λ) ,

N[−+,a++] g (λ) = N[0,a] g (λ) (λ= 0, 1, . . . , l+1), K[−+,a++] g (λ) = K[0,a] g (λ) (λ = 1, . . . , l + 1),

and g(−+)g (−+) < 0,

(3.4)

sg f (λ) (−+) = sg g (λ+1) (−+)

(λ = 0, 1, . . . , l).

For simplicity, write

N[0,a] f (λ) = Nλ ,

N[0,a] g (λ) = N˜ λ

(λ = 0, 1, . . . , l + 1),

K[0,a] f (λ) = Kλ ,

K[0,a] g (λ) = K˜ λ

(λ = 1, . . . , l + 1).

and

Then (3.3) implies that

(3.5)

2

l+1 λ=1

K˜ λ − 2

l λ=1

Kλ = [N˜ l+1 − Nl ] + [N0 − N˜ 0 ] + V0l+1 (g, −+) − V0l (f, −+) + V0l+1 (f, a + +) − V0l+1 (g, a + +) − Vll+1 (f, a + +).

56

KI AND KIM

Since g(x) = xf (x) and x l g (l+1) (x) = (d/dx)x l+1 f (l) (x), we have (3.6)

N0 − N˜ 0 = −1

N˜ l+1 − Nl ≥ 0.

and

From (3.4), it follows that V0l+1 (g, −+) − V0l (f, −+) = 1.

(3.7)

Since g(a ++) = (a ++)f (a ++) and g (λ) (a ++) = (a ++)f (λ) (a ++)+λf (λ−1) (a + +), λ = 1, 2, . . . , l + 1, we have sg g(a + +) = sg f (a + +), and sg g (λ) (a + +) = sg f (λ) (a + +) or sg g (λ) (a + +) = sg f (λ−1) (a + +), for λ = 1, . . . , l + 1. Therefore, V0l+1 (f, a + +) − V0l+1 (g, a + +) ≥ 0.

(3.8)

From (3.5), (3.6), (3.7), and (3.8), we obtain 2

l+1

K˜ λ − 2

λ=1

l λ=1

Kλ ≥ −Vll+1 (f, a + +),

and this proves our assertion. Corollary. Let f (x) be a real entire function, and let K be a positive integer. If f (x) has at least K critical points, then for each a ∈ R, (x − a)f (x) also has at least K critical points. Let f (x) be a real entire function, and let P (x) be a nonconstant, real polynomial that has only real zeros. Then the above corollary implies that the number of critical points of P (x)f (x) is at least that of f (x). Moreover, if f (x) is in the class ᏸ, then by Theorem 2.2, the number of critical points of P (x)f (x) is the same as that of f (x). The following example, however, shows that this is not the case in general. Example. The real entire function e2x + 1 = 2ex cosh x obviously has no critical points. Let K be an arbitrary positive integer. Then the function ex

K−1 n=0

1+

x2

2 n + (π/2)

has exactly K critical points by Theorem 2.2, because it is in the class ᏸ. Since ∞ x2 1+

2 −→ 1 n + (π/2) n=N uniformly on compact sets in the complex plane as N → ∞, there is a positive integer

THE FOURIER-PÓLYA CONJECTURE

57

N1 > K such that for each N ≥ N1 , the real entire function ∞ K−1 x2 x2 x e 1+ 1+

2

2 n + (π/2) n + (π/2) n=N n=0 has at least K critical points by the corollary to Theorem 3.1. Since ∞ x2 cosh x = 1+

2 , n + (π/2) n=0 we see that the real entire function e2x + 1 is the product of a real polynomial and a real entire function that has at least K critical points. Therefore, the number of critical points of P (x)f (x) can be smaller than that of f (x) if P (x) has nonreal zeros. Since 2 cosh x has inﬁnitely many critical points and

x n 2x 2 cosh x = e−x e2x + 1 = lim 1 − e +1 , n→∞ n the corollary to Theorem 3.1 implies that there is a positive integer N such that

x N 2x e +1 1− N has critical points. Therefore, the number of critical points of P (x)f (x) can be strictly greater than that of f (x), if f (x) is not in the class ᏸ. 4. The Fourier-Pólya conjecture. Our starting point is a class of transcendental functions deﬁned as follows. A real entire function f (x) is in the class ᏹ if (M1) f (x) = eβx g(x), where β is a real number and g(x) is a real entire function of growth (1, 0), (M2) f (x) has inﬁnitely many nonreal zeros, and (M3) f (x) has only a ﬁnite number of real zeros. The most important feature of the class ᏹ is that every function in the class has inﬁnitely many critical points. In order to prove this, we need some lemmas. Lemma 4.1. Let f (x) be in the class ᏹ. Then f (x) is unbounded on the real axis and f (x) has inﬁnitely many zeros. Proof. Let f (x) be given by f (x) = eβx g(x) with β ∈ R, and let g(x) be a real entire function of growth (1, 0). By a Phragmén-Lindelöf theorem [B, Theorem 1.4.3], an entire function of growth (1, 0) that is bounded on a line is a constant. In particular, f (x) is unbounded on the real axis when β = 0. Now suppose that β = 0. For r > 0, let M(r) and m(r) denote the maximum modulus and the minimum modulus of g(x) on the circle |x| = r, respectively. By the argument given in [B, pp. 43–44], we have (4.1)

lim m(r)M(r) = ∞.

r→∞

58

KI AND KIM

Since g(x) is of growth (1, 0), M(r) = o(e+r ) for each positive real number +. Hence (4.1) implies that there is a sequence {rj } of positive real numbers such that rj → ∞ as j → ∞ and such that

|β| m(rj ) ≥ exp − rj (j = 1, 2, . . . ). 2 Therefore, lim supx→∞ |f (x)| = ∞ when β > 0, and lim supx→−∞ |f (x)| = ∞ when β < 0. This proves the ﬁrst assertion. Suppose that f (x) has only a ﬁnite number of zeros. Since every transcendental entire function of growth (1, 0) has inﬁnitely many zeros, we must have β = 0 and d/dx(eβx g(x)) = f (x) = eβx P (x) for some polynomial P (x) of degree m ≥ 0. Then we have g (x) + βg(x) = P (x), so that

g (m+2) (x) + βg (m+1) (x) = 0.

Therefore, we have g (m+1) (x) = Ce−βx for some constant C. This, however, is impossible because g(x) is a transcendental function of growth (1, 0). This contradiction shows that f (x) has inﬁnitely many zeros. Lemma 4.2. Let f (x) be in the class ᏹ. If f (x) is not in the class ᏹ, then f (x) has inﬁnitely many critical zeros. Proof. Let f (x) be as in the proof of Lemma 4.1, and suppose that f (x) is not in the class ᏹ. In order to prove that f (x) has inﬁnitely many critical zeros, it is enough to show that f (x) has inﬁnitely many real zeros, because f (x) has only a ﬁnite number of real zeros. Since f (x) = eβx (g (x) + βg(x)), the condition (M1) is satisﬁed. Since we have assumed that f (x) is not in the class ᏹ, it follows that either f (x) has a ﬁnite number of nonreal zeros or f (x) has inﬁnitely many real zeros. From Lemma 4.1, f (x) has inﬁnitely many zeros. Hence, f (x) has inﬁnitely many real zeros in the ﬁrst case, too. Next, we show that every function in the class ᏹ has at least one critical point. In the proof of this fact, the following theorem plays a crucial role. The Bernstein-Widder theorem. A function f (x) is absolutely monotone in (−∞, 0) if and only if there is an increasing function F (t) deﬁned in [0, ∞) such that ∞ f (x) = etx dF (t) (x < 0). 0

Proof. See [Tim, p. 144]. Recall that a real valued C ∞ -function f (x) deﬁned in an interval I of the real axis is absolutely monotone if f (n) (x) > 0 for n = 0, 1, 2, . . . and x ∈ I .

THE FOURIER-PÓLYA CONJECTURE

59

Lemma 4.3. Every function in the class ᏹ has a critical point. Proof. We ﬁrst show that if a function in the class ᏹ does not vanish on the real axis, then it has a critical point. Let f (x) be in the class ᏹ, and suppose that f (x) has no real zeros. To obtain a contradiction, assume that f (x) has no critical points. Then Lemma 4.2 implies that f (n) ∈ ᏹ for all n = 0, 1, 2, . . . . Suppose for a moment that f (a) = 0 for some real number a. Then (3.1) implies that x = a is the only real zero of f (x) and that

sg f (x)f (x) =

1, if x < a, −1, if x > a.

This implies that f (x) is bounded on the real axis, which is impossible by Lemma 4.1. Therefore, we must have f (x) = 0 for all real x. Since f (n) ∈ ᏹ for all n = 0, 1, 2, . . . , an inductive argument shows that f (n) (x) does not vanish on the real axis for each nonnegative integer n. Hence, we can assume, without loss of generality, that f (n) (x) > 0

(4.2)

(n = 0, 1, 2, . . . , x ∈ R).

In particular, f (x) is absolutely monotone in (−∞, 0). Since f (x) is an entire function, there is a bounded increasing function F (t) deﬁned in the interval [0, ∞) such that ∞ f (x) = (4.3) etx dF (t) (x ≤ 0), 0

by the Bernstein-Widder theorem. Since f (x) is in the class ᏹ, (4.2) implies that there are a nonnegative real number β and a real entire function g(x) of growth (1, 0) such that f (x) = eβx g(x).

(4.4) From (4.3) and (4.4), we obtain g

(n)

∞

(0) =

(t − β)n dF (t)

(n = 0, 1, 2, . . . ).

0

Since g(x) is not a constant function, there is a subinterval [a, b] of [0, ∞) such that β∈ / [a, b] and F (b) − F (a) > 0. Then we have g (2n) (0) ≥

b a

(t − β)2n dF (t)

60

KI AND KIM

≥

2n min |t − β|

a≤t≤b

F (b) − F (a)

(n = 0, 1, 2, . . . ),

which is impossible, because g(x) is an entire function of growth (1, 0). This contradiction shows that f (x) must have a critical point. We have shown that every function in the class ᏹ that does not vanish on the real axis has a critical point. In general, every function h(x) in the class ᏹ can be written in the form h(x) = (x − a1 ) · · · (x − aN )f (x), where a1 , . . . , aN are the real zeros of h(x) and f (x) is a function in the class ᏹ that does not vanish on the real axis. Therefore, our assertion follows from the corollary to Theorem 3.2. Proposition. Every function in the class ᏹ has inﬁnitely many critical points. / Proof. Let f (x) be in the class ᏹ. If there is a positive integer n such that f (n) ∈ ᏹ, then Lemma 4.2 gives the desired result. So assume that f (n) ∈ ᏹ for all n = 0, 1, 2, . . . . From Lemma 4.3, f (x) has a critical point so that there is a positive integer n1 such that f (n1 ) (x) has a critical zero. Since f (n1 ) ∈ ᏹ, the same argument shows that there is an integer n2 > n1 such that f (n2 ) (x) has a critical zero, and so on. Therefore, f (x) has inﬁnitely many critical points in this case, too. The Fourier-Pólya conjecture is an immediate consequence of this proposition. Theorem 4.1 (The Fourier-Pólya conjecture). Every real entire function of genus 0 has just as many critical points as couples of nonreal zeros. Proof. Let f (x) be a real entire function of genus 0. Then f (x) is of growth (1, 0). If f (x) has only a ﬁnite number of nonreal zeros, then Theorem 2.2 would imply the desired result. Hence, we may assume that f (x) has inﬁnitely many nonreal zeros. Then we must show that f (x) has inﬁnitely many critical points. If f (x) has only a ﬁnite number of real zeros, then f (x) is in the class ᏹ, and therefore the proposition gives the desired result. Now suppose that f (x) has inﬁnitely many real zeros. Then we can write f (x) = g(x)

∞

1−

j =1

x aj

,

where a1 , a2 , . . . are the real zeros of f (x) that are different from zero and g(x) is a function in the class ᏹ. From here on, we show that for arbitrary positive integer K, f (x) has at least K critical points. Let K be an arbitrary positive integer. Since g(x) is in the class ᏹ, the proposition implies that g(x) has at least K critical points. Then the corollary to Theorem 3.1

THE FOURIER-PÓLYA CONJECTURE

61

implies that there is a positive integer J such that

∞ x g(x) 1− aj j =J +1

has at least K critical points, because

∞ x g(x) 1− −→ g(x) (J → ∞), aj j =J +1

uniformly on compact sets in the complex plane. Finally, the corollary to Theorem 3.2 implies that

∞ 1 x f (x) = (a1 − x) · · · (aJ − x)g(x) 1− a1 · · · a J aj j =J +1

has at least K critical points. Since K is arbitrary, this proves the Fourier-Pólya conjecture. In order to extend this result, we need a general theorem on the zeros of entire functions. Let f (z) be an entire function of ﬁnite order, and let p be a positive integer that is greater than or equal to the order of f (z). Let a1 , a2 , . . . denote the zeros of f (z) that are different from zero. Then we have j |aj |−p−1 < ∞, and hence f (z) can be expressed in the form

z p p f (z) = czn eα1 z+···+αp z (4.5) 1− e(z/aj )+···+(1/p)(z/aj ) , aj j

where c, α1 , . . . , αp are constants, n is a nonnegative integer, and the product converges absolutely. The following well-known theorem of E. Lindelöf gives a necessary and sufﬁcient condition for f (z) to be of growth (p, 0). Lindelöf’s theorem. The function f (z) given by (4.5) is of growth (p, 0) if and only if #{j : |aj | < r} = o(r p ) as r → ∞, and −p aj −→ −pαp (r → ∞). |aj |
Proof. See [B, pp. 27–30]. Now we can extend the Fourier-Pólya conjecture as follows. Theorem 4.2. Let f (x) be a real entire function of growth (1, 0), let b1 , b2 , . . . denote the real zeros of f (x) that are different from zero, and suppose that (4.6) bj−1 −→ β (r → ∞) |bj |
62

KI AND KIM

for some real number β. Then for each real constant δ, the function eδx f (x) has just as many critical points as couples of nonreal zeros. Proof. From Theorem 2.2 and the proposition, we may assume that f (x) has inﬁnitely many real zeros as well as inﬁnitely many nonreal zeros. Let c1 , c2 , . . . denote the nonreal zeros of f (x). Since f (x) is of growth (1, 0), there are a nonnegative integer n and real constants c and α such that n αx

f (x) = cx e

∞ j =1

x 1− bj

e

∞

x/bj

k=1

x 1− ex/ck , ck

and Lindelöf’s theorem implies that #{j : |bj | < r} + #{k : |ck | < r} = o(r) (r → ∞)

(4.7) and

 α + lim 

(4.8)

r→∞

|bj |
bj−1 +

|ck |
 ck−1  = 0.

Then (4.6) and (4.8) imply that

(4.9)

|ck |
ck−1 −→ γ

(r → ∞)

for some real number γ , and we have α + β + γ = 0. Let δ be an arbitrary real constant. From (4.7), (4.9), and Lindelöf’s theorem, the function

∞ x g(x) = cx n e−γ x 1− ex/ck ck k=1

is of growth (1, 0). Hence the function h(x) = eδx g(x) is in the class ᏹ, so that it has inﬁnitely many critical points by the proposition. Since 



eδx f (x) = expα + γ +

|bj |
 



x x −1  x/bj  1− e 1− h(x) x bj bj bj |bj |≥r

|bj |
63

THE FOURIER-PÓLYA CONJECTURE

for each positive real number r, and since   

x exp α + γ + bj−1  x  1− ex/bj −→ 1 bj |bj |
(r → ∞)

|bj |≥r

uniformly on compact sets in the complex plane, the same argument as in the proof of the Fourier-Pólya conjecture gives the desired result. Remark 4.1. Theorem 4.2 implies that if f (x) is a real entire function of genus 0, then for each real constant δ, the function eδx f (x) has just as many critical points as couples of nonreal zeros. However, the example given in Section 3 shows that this is not the case in general if we merely assume that the function f (x) can be expressed as a product of linear factors. Remark 4.2. Lindelöf’s theorem implies that every entire function of growth (1, 0) can be expressed as a product of linear factors, but the product may not converge absolutely. Theorem 4.2 implies that de Gua’s rule is applicable to some real entire functions of growth (1, 0) that are not of genus zero. The authors, however, do not know whether or not de Gua’s rule can be applied to every real entire function of growth (1, 0). As we mentioned in the introduction, the Fourier-Pólya conjecture cannot be applied to functions such as cos x, cosh x, or 5(x), because they are not of genus 0. Theorem 4.2 is also not applicable to such functions. The following theorem, however, states that de Gua’s rule can be applied to such functions. Theorem 4.3. Let f (x) be an even or odd real entire function of growth (2, 0), let b1 , b2 , . . . denote the real zeros of f (x) that are different from zero, and suppose that (4.10) bj−2 < ∞. j

Then for each nonnegative real constant α, the function e−αx f (x) has just as many critical points as couples of nonreal zeros. 2

Proof. Let α be a nonnegative real constant. If f (x) has only a ﬁnite number of 2 nonreal zeros, then (4.10) implies that the function e−αx f (x) is in the class ᏸ, so that Theorem 2.2 gives the desired result. Therefore, we need only to consider those functions that have inﬁnitely many nonreal zeros. In order to deal with such functions, we introduce the following notation. A real entire function g(x) is in the class ᏺ if 2 (N1) g(x) = eβx h(x), where β is a real number and h(x) is a real entire function of growth (2, 0), (N2) g(x) is an even function or an odd function, (N3) g(x) has inﬁnitely many nonreal zeros, (N4) g(x) has only a ﬁnite number of real zeros, and (N5) g(x) has only a ﬁnite number of critical points at the origin x = 0.

64

KI AND KIM

First of all, we show that every function in the class ᏺ has at least one critical point. Let g(x) be a real entire function in the class ᏺ, and suppose that g(x) is an even function. Then there are a real constant β and an even real entire function h(x) of growth (2, 0) such that g(x) = eβx h(x). 2

(4.11) Since g(x) is even, we can write

∞

g(x) =

am x 2m .

m=0

From (N5), g(x) has only a ﬁnite number of critical points at the origin x = 0. Hence there is a positive integer m1 such that (4.12)

am am+1 < 0

Let G(x) =

(m ≥ m1 ).

∞

am x m ,

m=0

so that g(x) = G(x 2 ). Then (4.12) implies that for each m = m1 , m1 +1, . . . , G(m) (x) has no zeros in the interval (−∞, 0], and hence for each m = 0, 1, 2, . . . , G(m) (x) has only a ﬁnite number of zeros in (−∞, 0]. In particular, G(x) has only a ﬁnite number of critical points in the interval (−∞, 0]. On the other hand, G(x) has only a ﬁnite number of positive real zeros, because g(x) = G(x 2 ) has only a ﬁnite number of real zeros by (N4). Consequently, G(x) has only a ﬁnite number of real zeros. Since g(x) is in the class ᏺ, it has inﬁnitely many nonreal zeros by (N3), so that either G(x) has inﬁnitely many negative real zeros or G(x) has inﬁnitely many nonreal zeros. But G(x) has only a ﬁnite number of (negative) real zeros, and hence we conclude that G(x) has inﬁnitely many nonreal zeros. Moreover, (4.11) implies that G(x) = eβx H (x), where H (x) is deﬁned by H (x 2 ) = h(x). Since h(x) is an even, real entire function of growth (2, 0), it follows that H (x) is a real entire function of growth (1, 0). Therefore, G(x) is in the class ᏹ. Now the proposition implies that G(x) has inﬁnitely many critical points. Since G(x) has only a ﬁnite number of critical points in the interval (−∞, 0], we conclude that G(x) has inﬁnitely many critical points in the interval (0, ∞). In particular, there are a positive integer N and a positive real number c such that G(N) (x) has a critical zero at x = c. For n = 0, 1, . . . , N − 1, let gn (x) = G(n) (x 2 ). Since √ √ √

c > 0, gN−1 ( c) = G(N−1) (c) = 0, gN−1 ( c) = 2 c G(N) (c) = 0, and

√

( c) = 2G(N) (c) + 4c G(N+1) (c) = 4c G(N+1) (c), gN−1 √

it follows that gN−1 (x) has a critical zero at x = c. Hence, the function gN−1 (x)

THE FOURIER-PÓLYA CONJECTURE

65

has a critical point. Since d (N−2) 2

gN−2 G x = 2xG(N−1) x 2 = 2xgN−1 (x), (x) = dx

(x) has a critical point, so gN−2 (x) has a critical point. Theorem 3.2 implies that gN−2 Then the same argument shows that gN−3 (x) has a critical point, and so on. In this way, we have proved that g(x) = g0 (x) has a critical point. Consequently, every even function in the class ᏺ has a critical point. If g(x) is an odd function in the class ᏺ, then x −1 g(x) is an even function in the class ᏺ. Hence Theorem 3.2 implies that every odd function in the class ᏺ also has a critical point. If g(x) is in the class ᏺ, then it is obvious that g (x) satisﬁes the conditions (N1) and (N2). Moreover, it is not hard to see that if g(x) is in the class ᏺ, then g (x) has inﬁnitely many zeros. Then the same argument as in the proof of the proposition shows that every real entire function in the class ᏺ has inﬁnitely many critical points. Finally, the same argument as in the proof of Theorem 4.2 gives the desired result. Remark 4.3. Theorem 4.3 implies that every even or odd real entire function of genus 1∗ has just as many critical points as couples of nonreal zeros. However, the example given in Section 3 shows that there is a real entire function of genus 1∗ that has inﬁnitely many nonreal zeros but no critical points. Remark 4.4. Let f (x) be a real entire function that is not of genus 1∗ , and suppose that f (x) has only real zeros. Then either f (x) has a critical point or f (x) has no critical points. In the ﬁrst case, de Gua’s rule does not hold for f (x). In the second case, no derivative of f (x) has critical points, but as it is remarked in Remark 2.3, there is a positive integer n such that f (n) (x) has a nonreal zero, so that de Gua’s rule does not hold for the function f (n) (x). From this observation, it seems that de Gua’s rule is meaningless for functions that are not of genus 1∗ . Remark 4.5. Recall that Riemann’s ξ -function 5(x) is an even real entire function of order 1, and the Riemann hypothesis states that 5(x) has only real zeros (see [Tit, p. 29]). Therefore, if we could show that 5(x) has no critical points, then the Riemann hypothesis would follow. 5. Applications. In this section, we apply the results obtained in the preceding section to obtain very simple proofs of some classical results of Hurwitz, Laguerre, and Pólya. 5.1. A multiplier sequence of Laguerre. A sequence {γn }∞ n=0 of real numbers is said to be a multiplier sequence (of the ﬁrst kind) if it has the following special property: If a0 + a1 x + · · · + ad x d is a real polynomial with only real zeros, then the polynomial γ0 a0 + γ1 a1 x + · · · + γd ad x d also has only real zeros.

66

KI AND KIM

According to Pólya and Schur [PS], E. Laguerre proved that if −1 ≤ q ≤ 1, then the sequence (5.1)

1, q, q 4 , q 9 , . . . , q n , . . . 2

is a multiplier sequence. This fact can be proved very easily if we apply the FourierPólya conjecture and the following well-known theorem of Pólya and Schur. The Pólya-Schur theorem. A sequence {γn } of real numbers is a multiplier sequence if and only if the function 9(x) =

∞

γn

n=0

xn n!

can be uniformly approximated on compact sets in the complex plane by a sequence of real polynomials all of whose zeros are real and of the same sign. Proof. See [L, Chapter 8] or [PS, p. 104]. Let a real number q with |q| ≤ 1 be given. In order to show that (5.1) is a multiplier sequence, we need only to consider the case where 0 < q < 1. If q = 0 or q = ±1, then there is nothing to prove. The case −1 < q < 0 follows from the case 0 < q < 1. Deﬁne f (x) by ∞ n 2x f (x) = , qn n! n=0

and suppose that 0 < q < 1. Then f (x) is a real entire function of order 0, is positive in the interval [0, ∞), and satisﬁes

(5.2) f (x) = qf q 2 x (x ∈ C). In particular, f (x) has no (critical) zeros in the interval [0, ∞). Let a be an arbitrary negative real number that satisﬁes f (a)f (a) = 0. Then (3.3) implies that (5.3)

2K[a,0] (f ) = N[a,0] (f ) − N[a,0] (f ) + V01 (f, a),

because f (0), f (0) > 0. On the other hand, (5.2) implies that N[a,0] (f ) = N[q 2 a,0] (f ) ≤ N[a,0] (f ), because 0 < q < 1. Therefore, (5.3) implies that 2K[a,0] (f ) ≤ V01 (f, a) ≤ 1, and hence we must have K[a,0] (f ) = 0. Since we can take a arbitrarily close to −∞,

THE FOURIER-PÓLYA CONJECTURE

67

f (x) has no critical zeros. Since f

(x) = q 3 f (q 2 x), the same argument as above shows that f

(x) also has no critical zeros, and so on. In this way, we have shown that f (x) has no critical points. Since f (x) is of order 0, it is of genus 0. Hence, the Fourier-Pólya conjecture implies that f (x) has only real zeros. Since f (x) = 0 for 0 ≤ x < ∞ and since f (x) is of genus 0, it can be uniformly approximated on compact sets in the complex plane by a sequence of real polynomials with only negative real zeros. Therefore, the Pólya-Schur theorem gives the desired result. Moreover, all (real) zeros of f (x) are simple. To see this, suppose that f (x) has a multiple zero at x = a. Then a < 0, and there is a positive real number + > 0 such that f (a − +)f (a − +) < 0 and a < q 2 (a − +), so that N[a−+,0] (f ) ≥ N[q 2 (a−+),0] (f ) + 2. Since f (x) has no critical zeros, we have 0 = 2K[a−+,0] (f ) = N[a−+,0] (f ) − N[a−+,0] (f ) + 1 = N[q 2 (a−+),0] (f ) − N[a−+,0] (f ) + 1 ≤ −1, which is a contradiction. 5.2. Hurwitz’s theorem on the zeros of Bessel functions. Let ν be a real constant. The Bessel function Jν (z) of order ν (of the ﬁrst kind) is given by Jν (z) =

∞ n=0

z 2n+ν (−1)n . n! ;(n + ν + 1) 2

A well-known theorem of Hurwitz states that (i) when ν > −1, the zeros of Jν (z) are all real, (ii) if s is a nonnegative integer and −2s − 2 < ν < −2s − 1, then Jν (z) has 4s + 2 nonreal zeros, of which two are purely imaginary, and (iii) if s is a positive integer and −2s − 1 < ν < −2s, Jν (z) has 4s nonreal zeros, of which none are purely imaginary (see [H1], [P2], and [Wa]). Note that we do not need to consider the functions J−1 (z), J−2 (z), . . . , because J−m (z) = (−1)m Jm (z) for m = 1, 2, . . . . Deﬁne gν (x) by (5.4)

gν (x) =

∞ n=0

so that Jν (z) =

xn 1 , ;(n + ν + 1) n!

z ν 2

z2 . − 4

gν

Then gν (x) is a real entire function of order 21 , so that it is of genus 0.

68

KI AND KIM

Lemma. Hurwitz’s theorem follows once we prove the following. (I) When ν > −1, gν (x) has no critical points and no positive real zeros. (II) If s is a nonnegative integer and −2s − 2 < ν < −2s − 1, gν (x) has exactly s critical points and one positive real zero. (III) If s is a positive integer and −2s − 1 < ν < −2s, gν (x) has exactly s critical points and no positive real zeros. Proof. From (5.4), we have the following: (5.5) (5.6) (5.7)

lim gν (x) = ∞,

x→∞

gν(l) (x) = gν+l (x)

(l = 0, 1, 2, . . . ),

gν(l−1) (x) − (ν + l)gν(l) (x) − xgν(l+1) (x) = 0

(l = 1, 2, . . . ).

First of all, from (5.4), it is clear that gν (x) has no critical points at the origin x = 0. Moreover, if a = 0, then x = a cannot be a multiple zero of the functions (k) gν (x), k = 0, 1, 2, . . . , because of (5.7). Let k be a positive integer, let a < 0, and (k) (k−1) (k+1) assume that gν (a) = 0. Then (5.7) implies that gν (a) − agν (a) = 0. Since (k) (k−1) (k+1) x = a is a simple zero of gν (x) and a < 0, it follows that gν (a)gν (a) < 0; (k) that is, x = a is a noncritical zero of gν (x). Therefore, we conclude that gν (x) has no critical points in the interval (−∞, 0]. Next, we look at the behavior of gν (x) in the interval (0, ∞). If ν > −1, then all (k) coefﬁcients of (5.4) are positive, so that the functions gν (x), k = 0, 1, 2, . . . , are strictly positive on the positive real axis. In particular, gν (x) has no critical points and no positive real zeros in the case where ν > −1. This proves (I). To proceed further, we need a lemma. Lemma. Let s be a nonnegative integer. Then we have the following: (5.8) (5.9)

If −2s − 1 < ν < −2s, then gν (x) > 0 for x ≥ 0. If −2s − 2 < ν < −2s − 1, then gν (x) has exactly one positive real zero.

Proof. We prove (5.8) and (5.9) by an induction on s. We have shown that if −1 < ν < 0, then gν (x) > 0 for x ≥ 0. If −2 < ν < −1, then gν (0) < 0, limx→∞ gν (x) = ∞, and gν (x) = gν+1 (x) > 0 for x ≥ 0. Hence, gν (x) has exactly one positive real zero. Now suppose that s > 0, and assume that the induction hypothesis holds. Let −2s − 1 < ν < −2s. Then gν (0) > 0. Moreover, the induction hypothesis implies that gν (x) = gν+1 (x) has exactly one positive real zero, say, at x = c, and that gν

(x) = gν+2 (x) > 0 for x ≥ 0. From (5.7), it follows that gν (c) = cgν

(c) > 0. Therefore, gν (x) > 0 for x ≥ 0. Next assume that −2s − 2 < ν < −2s − 1. Then gν (0) < 0 and limx→∞ gν (x) = ∞. Since −2s − 1 < ν + 1 < −2s, it follows that gν (x) = gν+1 (x) > 0 for x ≥ 0. Hence, gν (x) has exactly one positive real zero. This completes the induction.

THE FOURIER-PÓLYA CONJECTURE

69

Now we can count the critical points of gν (x) in the case where ν < −1. Let s be a nonnegative integer, and suppose that −2s − 2 < ν < −2s − 1. Then (5.9) implies (l) that gν (x) has exactly one positive real zero. Since gν (x) = gν+l (x) > 0 for x ≥ 0 whenever l ≥ 2s +1, (3.3), (5.5), and (5.6) imply that for all sufﬁciently large positive real numbers b, we have 2

∞ l=1

2s+1 K[0,b] gν(l) = 2 K[0,b] gν(l) l=1

= N[0,b] gν(2s+1) − N[0,b] (gν ) + V02s+1 (gν , 0) − V02s+1 (gν , b) = 2s.

Therefore, gν (x) has exactly s critical points and one positive real zero. This proves (II). Finally, suppose that s is a positive integer and that −2s −1 < ν < −2s. Then (5.8) implies that gν (x) > 0 for x ≥ 0, and the same argument as above shows that gν (x) has exactly s critical points. This proves (III). 5.3. Some results of Pólya on the zeros of trigonometric integrals. Let f (t) be a nonnegative increasing function deﬁned in the unit interval [0, 1], and suppose that #1 0 < 0 f (t) dt < ∞. Let U (x) and V (x) be deﬁned by 1 1 U (x) = f (t) cos xt dt and V (x) = f (t) sin xt dt. 0

0

Obviously, U (x) is an even function and V (x) is an odd function. Moreover, it is easy to see that these functions are real entire functions of growth (1, 1). In [P1], Pólya proved in two different ways that all the zeros of U (x) and V (x) are real: His ﬁrst method is based on a theorem of S. Kakeya on the location of zeros of polynomials, and the second one is based on a method of Hurwitz that uses partial fraction development. In fact, he proved more precise statements. To do this, he distinguished the exceptional case from the general case as follows: The function f (t) is in the exceptional case if it is piecewise constant, has only a ﬁnite number of jumping points, and all of its jumping points are rational. If f (t) is not in the exceptional case, it is said to be in the general case. Then he proved the following theorems using his second method [P1, pp. 368–375]. Theorem 5.1. If f (0) > 0, f (t) > 0 for 0 < t < 1, or f (t) is in the general case, then all zeros of U (x) are real and simple, U (x) = 0 for 0 ≤ x ≤ π/2, and each of the intervals

π 3π 3π 5π 5π 7π , , , , , ,... 2 2 2 2 2 2 contains exactly one zero of U (x).

70

KI AND KIM

Theorem 5.2. If f (t) is in the general case, then all zeros of V (x) are real and simple, V (x) = 0 for 0 < x ≤ π, and each of the intervals (π, 2π ),

(2π, 3π),

(3π, 4π), . . .

contains exactly one zero of V (x). Remark 5.1. His ﬁrst method also shows that if f (t) is in the general case, then all zeros of U (x) and V (x) are real and simple, but it does not give the precise information on the location of zeros as above. Remark 5.2. Since every nonnegative increasing function can be approximated by a sequence of nonnegative increasing functions that are in the general case, these two theorems imply that U (x) and V (x) also have real zeros only in the exceptional case. In the remainder of this paper, we give a new proof of these theorems. Proof of Theorems 5.1 and 5.2. First of all, it is clear that U (x) > 0 for 0 ≤ x ≤ #1 π/2 and V (x) > 0 for 0 < x ≤ π, because f (t) is nonnegative and 0 f (t) dt > 0. #1 Since f (t) is increasing and 0 f (t) dt < ∞, it follows that limt→1 (1 − t)f (t) = 0. Hence, by partial integration, we obtain

(5.10)

(cos x + x sin x)U (x) + x cos xU (x) 1 1 − (t cos x cos xt + sin x sin xt) df (t), = f (0) + 0

(5.11)

(sin x − x cos x)V (x) + x sin xV (x) 1 = (1 − cos x)f (0) + 1 − (t sin x sin xt + cos x cos xt) df (t). 0

If f (0) > 0, or f (t) > 0 for 0 < t < 1, then the right-hand side of (5.10) is strictly positive for x ≥ 0. On the other hand, if f (t) is in the general case, then the right-hand sides of (5.10) and (5.11) are strictly positive for x > 0. Hence, from the assumptions of the theorems, we have (5.12)

(cos x + x sin x)U (x) + x cos xU (x) > 0

(x > 0),

(sin x − x cos x)V (x) + x sin xV (x) > 0

(x > 0).

and (5.13)

The following are direct consequences of (5.12) and (5.13): (5.14)

x>0

and

U (x) = 0 ⇒ cos xU (x) > 0,

(5.15)

x>0

and

V (x) = 0 ⇒ sin xV (x) > 0,

THE FOURIER-PÓLYA CONJECTURE

(5.16) (5.17)

(−1)n U

π 2

+ nπ > 0

(−1)n+1 V (nπ) > 0

71

(n = 0, 1, 2, . . . ), (n = 1, 2, . . . ).

From (5.14) and (5.16), it follows that each of the intervals

3π 5π 5π 7π π 3π , , , , , ,... 2 2 2 2 2 2 contain exactly one (simple) zero of U (x). Similarly, each of the intervals (π, 2π ),

(2π, 3π),

(3π, 4π), . . .

contain exactly one (simple) zero of V (x). Therefore, it remains only to show that U (x) and V (x) have only real zeros. Now we show that U (x) and V (x) have no critical points. For n = 1, 2, . . . , let an denote the zero of U (x) that lies in the interval (nπ − π/2, nπ + π/2). Since U (x) > 0 for −π/2 ≤ x ≤ π/2, we have N[0,an ] (U ) = n for n = 1, 2, . . . . Since #1 U (x) = − 0 tf (t) sin xt dt and the function tf (t) is in the general case, it follows that N[0,nπ ] (U ) = n for n = 1, 2, . . . . Let n be an arbitrary positive integer. Then we can ﬁnd a positive real number + such that + < π/2, an + + < nπ + π/2, and U (an + +)U (an + +) > 0. Then we have 2K[−+,an ++] (U ) = N[−+,an ++] (U ) − N[−+,an ++] (U ) + V01 (U, −+) − V01 (U, an + +) ≤ N[−+,(n+1)π] (U ) − n = 1, so that

K[−+,an ++] (U ) = 0.

Since n is arbitrary, we conclude that U (x) has no critical zeros in the interval [0, ∞). Since U (x) is an even function, the same is true of the interval (−∞, 0]. Thus we have shown that U (x) has no critical zeros. Similarly, V (x) has no critical zeros. #1 #1 Since U

(x) = − 0 t 2 f (t) cos xt dt, V

(x) = − 0 t 2 f (t) sin xt dt, and the function t 2 f (t) is in the general case, the same argument shows that U

(x) and V

(x) have no critical zeros, and so on. Consequently, the functions U (x) and V (x) have no critical points. Finally, Theorem 4.3 gives the desired result. References [B] [BP1] [BP2] [CCS1]

R. P. Boas, Entire Functions, Academic Press, New York, 1954. R. P. Boas and G. Pólya, Generalizations of completely convex functions, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 323–325. , Inﬂuence of the signs of the derivatives of a function on its analytic character, Duke Math. J. 9 (1942), 406–424. T. Craven, G. Csordas, and W. Smith, Zeros of derivatives of entire functions, Proc.

72 [CCS2] [G] [HSW] [HW1] [HW2] [H1] [H2] [J] [KK] [K1] [K2] [K3] [K4] [K5] [L] [LO]

[P1] [P2] [P3] [P4] [P5] [P6] [PS] [PW] [Sh] [St] [Tim]

KI AND KIM Amer. Math. Soc. 101 (1987), 323–326. , The zeros of derivatives of entire functions and the Pólya-Wiman conjecture, Ann. of Math. (2) 125 (1987), 405–431. W. Gontcharoff, Recherches sur les dériveés des fonctions analytiques, Ann. École Norm. 47 (1930), 1–78. S. Hellerstein, L. C. Shen, and J. Williamson, Reality of the zeros of an entire function and its derivatives, Trans. Amer. Math. Soc. 275 (1983), 319–331. S. Hellerstein and J. Williamson, Derivatives of entire functions and a question of Pólya, Trans. Amer. Math. Soc. 227 (1977), 227–249. , Derivatives of entire functions and a question of Pólya, II, Trans. Amer. Math. Soc. 234 (1977), 497–503. A. Hurwitz, Über die Nullstellen der Bessel’schen Funktion, Math. Ann. 33 (1889), 246–266. , Über den Satz von Budan-Fourier, Math. Ann. 71 (1912), 584–591. J. L. W. V. Jensen, Recherches sur la théorie des équations, Acta Math. 36 (1913), 181–195. H. Ki and Y. O. Kim, Proof of the Fourier-Pólya conjecture, preprint, 1997. Y. O. Kim, A proof of the Pólya-Wiman conjecture, Proc. Amer. Math. Soc. 109 (1990), 1045–1052. , On a theorem of Craven, Csordas and Smith, Complex Variables Theory Appl. 22 (1993), 207–209. , Critical points of real entire functions and a conjecture of Pólya, Proc. Amer. Math. Soc. 124 (1996), 819–830. , Critical points of real entire functions whose zeros are distributed in an inﬁnite strip, J. Math. Anal. Appl. 204 (1996), 472–481. , On the number of critical points of real analytic functions, preprint, 1997. B. Ja. Levin, Distribution of Zeros of Entire Functions, Transl. Math. Monogr. 5, Amer. Math. Soc., Providence, 1980. B. Ja. Levin and I. V. Ostrovskii, The dependence of the growth of an entire function on the distribution of the zeros of its derivatives, Amer. Math. Soc. Transl. Ser. 2 32 (1963), 323–357. G. Pólya, Über die Nullstellen gewisser ganzer Funktionen, Math. Z. 2 (1918), 352–383. , Über einen Satz von Laguerre, Jber. Deutsch. Math.-Verein. 38 (1929), 161–168. , Some problems connected with Fourier’s work on transcendental equations, Quart. J. Math. Oxford Ser. (2) 1 (1930), 21–34. , Über die Realität der Nullstellen fast aller Ableitungen gewisser ganzer Funktionen, Math. Ann. 114 (1937), 622–634. , On functions whose derivatives do not vanish in a given interval, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 216–218. , On the zeros of the derivatives of a function and its analytic character, Bull. Amer. Math. Soc. 49 (1943), 178–191. G. Pólya and J. Schur, Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, J. Reine Angew. Math. 144 (1914), 89–113. G. Pólya and N. Wiener, On the oscillation of the derivatives of a periodic function, Trans. Amer. Math. Soc. 52 (1942), 249–256. T. Sheil-Small, On the zeros of the derivatives of real entire functions and Wiman’s conjecture, Ann. of Math. (2) 129 (1989), 179–193. M. A. Stern, Über die Auﬂösung der transcendenten Gleichungen, J. Reine Angew. Math. 22 (1841), 1–62. A. F. Timan, Theory of Approximation of Functions of a Real Variable, Dover, New York, 1994.

THE FOURIER-PÓLYA CONJECTURE [Tit] [Wa] [Wi1]

[Wi2]

73

E. C. Titchmarsh, The Theory of the Riemann Zeta-function, 2d. ed., Oxford Univ. Press, New York, 1986. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2d ed., Cambridge Univ. Press, Cambridge, 1944. A. Wiman, Über eine asymptotische Eigenschaft der Ableitungen der ganzen Funktionen von den Geschlechtern 1 und 2 mit einer endlichen Anzahl von Nullstellen, Math. Ann. 104 (1930), 169–181. , Über die Realität der Nullstellen fast aller Ableitungen gewisser ganzer Funktionen, Math. Ann. 114 (1937), 617–621.

Ki: Department of Mathematics, Yonsei University, Seoul 120-749, Korea; haseo@bubble. yonsei.ac.kr Kim: Department of Mathematics, Sejong University, Seoul 143-747, Korea; kimyo@kunja. sejong.ac.kr

Vol. 104, No. 1

DUKE MATHEMATICAL JOURNAL

© 2000

ABEL-JACOBI MAPPINGS AND FINITENESS OF MOTIVIC COHOMOLOGY GROUPS KANETOMO SATO

Contents 1. Notation and generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2. Varieties over local ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3. Varieties over p-adic ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4. Number ﬁeld case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5. Image of secondary regulator maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6. Kernel of Abel-Jacobi mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Appendix: Interpretation as a motivic cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Introduction. Let X be a projective smooth variety over a global ﬁeld k. Let k denote a separable closure of k, and let X denote the scalar extension of X to k. Let i (X, Q(n)) denote the motivic cohomology groups of X with Q-coefﬁcients in the Hᏹ sense of Bloch [B3] or Voevodsky [Vo]. For example, we have 2n Hᏹ X, Q(n) CHn (X)Q , where the right-hand side is the Chow group of codimension n cycles on X tensored with Q. By combining cycle classes from motivic cohomology to continuous étale cohomology of X over k with the Hochschild-Serre spectral sequence r+s r Hcont Gk , Hes´ t X, Ql (n) ⇒ Hcont X, Ql (n) , we get a homomorphism i−1 i 1 X, Ql (n) , aji,n l : Hᏹ X, Q(n) hom ⊗Q Ql −→ HGal Gk , Hét

(∗)

which is called the l-adic Abel-Jacobi mapping (cf. §5, (5.1)). The meaning of the subscript hom in the group on the left-hand side is described in the main body of the paper below (cf. appendix, (A.3)). In the case of Chow groups, it means cycles homologically equivalent to zero. Otherwise, it means the full group (cf. §1, Lemma 1.1). The general philosophy of mixed motives of Bloch-Beilinson and the Beilinson-Deligne conjecture on the triviality of motivic 2-extensions imply that these mappings should Received 7 June 1999. Revision received 1 December 1999. 2000 Mathematics Subject Classiﬁcation. Primary 19F27; Secondary 11G25, 11G35. Author’s work supported by the Japan Society for the Promotion of Science Research Fellowships for Young Scientists. 75

76

KANETOMO SATO

be injective. In this paper, we are mainly concerned with the injectivity of l-adic AbelJacobi mappings. We prove in some cases that it is a consequence of the conjecture that higher Chow groups are ﬁnitely generated. We also construct a projective smooth surface over a global function ﬁeld whose Abel-Jacobi mappings are injective. (0.1) First of all, we consider the following situation. Let Fq be a ﬁnite ﬁeld. Let X be a projective smooth and geometrically connected threefold over Fq , which belongs to the class A(Fq ) of Soulé [So1, §3.3]. This class is closed under product and contains curves, unirational varieties of dimension at most 3, abelian varieties, and Fermat hypersurfaces (loc. cit. §3.3.3). We ﬁx a closed embedding X ⊂ PN := PN Fq . Replacing Fq by a ﬁnite ﬁeld extension if necessary, we take a linear subvariety A ⊂ PN of codimension two satisfying the following (see, e.g., [SGA7, Exp. XVII]): (1) A intersects X transversally; that is, Z := A ∩ X is smooth and of codimension two in X. (2) The ﬁbration π : XZ → P1 given by the pencil of A is generically smooth. Here XZ denotes the blow-up of X along Z. (We do not assume anything about singular ﬁbers.) We write k for the function ﬁeld of P1 , and X for the generic ﬁber of the ﬁbration π. For this projective smooth surface X over k, we prove the following theorem. Theorem 0.1. For any prime number l prime to ch(Fq ), the l-adic Abel-Jacobi mapping 3 1 2 aj3,2 l : Hᏹ X, Q(2) ⊗Q Ql −→ HGal Gk , He´ t X, Ql (2) is injective. Here we note that the subscript hom in (∗) means the full group in this case. This result follows from the ﬁniteness of the higher Chow group CH2 (XZ , 1) (see [So1]) and the main result, Theorem 0.2 stated below (see §6 (6.2) and Remark 6.4 for details and results on other l-adic Abel-Jacobi mappings; see also [Ras1, §2] for the case dim X = 2). (0.2) We consider the following more general setting. Let k be an algebraic function ﬁeld in one variable over a ﬁnite ﬁeld Fq . Let X be a proper smooth geometrically connected k-variety. We write C for the projective smooth curve over Fq whose function ﬁeld is k. We take a nonempty open subset U0 of C and a proper smooth ﬁbration π0 : X0 → U0 whose generic ﬁber is X. We ﬁx a prime number l invertible in k and an arbitrary integer n at least 2, and consider the following l-adic Abel-Jacobi mapping: n+1 1 ajln+1,n : Hᏹ X, Q(n) ⊗Q Ql −→ HGal Gk , Hen´ t X, Ql (n) , where the subscript hom in (∗) means the full group also in this case. In §5 and §6, we prove the following result.

77

ABEL-JACOBI MAPPINGS

Theorem 0.2 (Theorem 6.2). We assume the Bloch-Kato conjecture in degree n (see §6, Conjecture 6.1) for all ﬁnitely generated ﬁelds F over Fq . Then we have Ker ajln+1,n lim CHn (XU , n − 1) ⊗ Zl l- Div ⊗ Q . − → U ⊂U0 open

Here XU denotes π0−1 (U ), and for an abelian group M, Ml- Div denotes the maximal l-divisible subgroup. Consequently, if the higher Chow groups CHn (XU , n − 1) are is injective. ﬁnitely generated abelian groups for any U ⊂ U0 , then the map ajn+1,n l Note that if n = 2 or l = 2, Theorem 0.2 is unconditional by the Merkur’evSuslin theorem [MS] and by the Voevodsky theorem [Vo]. Theorem 0.2 is proved in the following way. First, by the recent results on motivic complexes (see [SuVo], [GeLe]), we show, assuming the Bloch-Kato conjecture, that the kernel of the l-adic cycle class map n+1 n+1 Hᏹ X, Q(n) Q −→ Hind (∗2) X, Ql (n) l

is isomorphic to the right-hand side in Theorem 0.2 (cf. §6 (6.1) for details; see also i (X)). Here Hi (X) is endowed with the Hochschild(1.8) for the deﬁnition of Hind ind Serre ﬁltration F• associated with the covering X → X, and for an integer p, the pth p n−p graded piece is isomorphic to Hind (k, He´ t (X, Ql (n))) (cf. §1, Lemma 1.1). Since the l-adic Abel-Jacobi mapping ajln+1,n is the composite map by deﬁnition (cf. (5.1)), (∗2) n+1 n+1 n+1 Hᏹ X, Q(n) Q −−−→ Hind X, Ql (n) F1 Hind X, Ql (n) l ⊂ 1 1 −→ Hind k, Hen´ t X, Ql (n) −→ HGal Gk , Hen´ t X, Ql (n) , our essential task is to prove that the secondary class map n+1 X, Ql (n) Ker ajln+1,n −→ F2 Hind

(∗3)

has trivial image. This triviality is a higher-Chow-theoretic analogue of a recent work of Raskind (see [Ras2, Theorem 0.1]). We follow an analogue of his local-toglobal argument (cf. [Ras2, Proposition 3.6]) to prove that the map (∗3) is trivial (see Theorem 5.1(2), Corollary 5.4 below). The following technical result plays a key role. Theorem 0.3 (Corollary 2.2). Let X be a proper smooth geometrically connected variety over a classical local ﬁeld K. Let l be a prime number prime to ch(K). Then the composite map n+1 n+1 2 vln : HGal GK , Hen−1 X, Q (n) −→ H (n) −→ lim H (n) X, Q V , Q l l l ´t e´ t e´ t − → V ⊂X open

is injective for any n ≥ 2. Here the ﬁrst map is obtained from the Hochschild-Serre spectral sequence.

78

KANETOMO SATO

(0.3) Finally, we explain the outline of the proof of Theorem 0.3. We consider the Ql /Zl -coefﬁcient variant of Theorem 0.3, namely, we study the kernel of the following composite map (see [Sal, 2.5, 3.8] for the case ch(K) = 0 and n = 2): 2 X, Q /Z (n) GK , Hen−1 αln : HGal l l ´t −→ Hen+1 X, Ql /Zl (n) ´t −→ Hen+1 /Z (n) . Spec K(X), Q l l ´t Theorem 0.3 follows from the result that this map αln has ﬁnite kernel for any prime number l = ch(K), and is injective for almost all l = ch(K) (cf. Theorem 2.1). Essentially, we have to deal with the case where X does not have potentially good reduction. Using the alteration theorem of de Jong [dJ], we reduce the problem to the case where X has a projective regular model with strictly semistable reduction. We write p for the residual exponential characteristic of K. In the case l = p, our problem is reduced to calculating the weight ﬁltration on the étale cohomology group Hen−1 (X, Ql /Zl (n)) by results on vanishing cycles (see [RZ, Satz 2.21, 2.23]) and to ´t extract quotients of weight −2 by the key commutative diagram (2.2.2). The injectivity of αln for almost all l is a delicate problem and seems to be related to the degeneration of the vanishing cycle spectral sequence with integral coefﬁcients (2.5.2). However, actually, we do not need this degeneration for our proof (cf. (2.6)). In the case where K is a p-adic ﬁeld and l = p, we use the p-adic Hodge theory (see [HK], [K3], [Ts]) and the weight spectral sequence for log crystalline cohomology (see [Mo, §3]). Acknowledgments. The author heartily thanks Professor Shuji Saito. The technical main result, Theorem 0.3, was inspired by a discussion with him. Thanks are also due to the referee, whose many detailed comments much improved the presentation of this paper. The author also expresses gratitude to Professor Takeshi Saito, Atsushi Shiho, Takao Yamazaki, and Noriyuki Otsubo, who carefully read the earlier version of this manuscript and gave him helpful comments. Finally, the author heartily thanks Professor Uwe Jannsen, who gave him much valuable advice on arguments in §2. 1. Notation and generalities. In this section, we ﬁx some notation and recall some generalities. (1.1) For an abelian group M and a positive integer n, n M and M/n, denote n → M, respectively. For a prime number the kernel and the cokernel of the map M − l, M{l} denotes the subgroup of l-primary torsion elements in M, Ml- div denotes the subgroup of l-divisible elements Ml- div := ∩ν∈N l ν M, and Ml- Div denotes the maximal l-divisible subgroup of M. In general, we have

ABEL-JACOBI MAPPINGS

79

Ml- div = Ker M −→ lim M/l ν , ← − ν −1 Ml- Div = Im Hom Z l , M −→ M ,

(1.1.1) (1.1.2)

and Ml- Div ⊂ Ml- div ⊂ M, but we do not necessarily have Ml- Div = Ml- div . If the projective system {l ν M}ν∈N satisﬁes the Mittag-Lefﬂer condition, then we have Ml- Div = Ml- div (cf. [J1, §4]). If M is ﬁnitely generated over Z or Zl , then we have Ml- Div = Ml- div = 0. (1.2) For a ﬁeld k, k × denotes the multiplicative group, k denotes a ﬁxed separable closure, and Gk denotes the absolute Galois group Gal(k/k). For a geometrically connected k-variety X, X denotes the scalar extension X ⊗k k, and k(X) denotes the ﬁeld of rational functions. For a scalar extension X ⊗k L, we often write L(X) for the function ﬁeld. For a commutative ring R with unity and an étale sheaf Ᏺ on Spec R, we write Hei´ t (R, Ᏺ) for Hei´ t (Spec R, Ᏺ). If R is a ﬁeld, we identify the étale i (G , Ᏺ| cohomology group Hei´ t (R, Ᏺ) with the Galois cohomology group HGal R Spec R ) by the standard comparison fact. In Sections 2–4, unless indicated otherwise, all cohomology groups of schemes are taken over the étale topology. (1.3) For a scheme X and a nonnegative integer q, Xq denotes the set of the points on X of codimension q. For a point x ∈ X, κ(x) denotes the residue ﬁeld. For a Zariski presheaf P on X, we deﬁne the coniveau ﬁltration N• on P (X) by Ker P (X) −→ P (X \ Z) . Nq P (X) := Z⊂X: closed of codim. ≥q

For a positive integer n invertible on X, µn denotes the étale sheaf of nth roots of unity. For a nonnegative integer i, Z/nZ(i) denotes µ⊗i n . For a prime number l invertible on X, Ql /Zl (i) denotes lim ν Z/ l ν Z(i). − → (1.4) In this paragraph, (1.5), and (1.6), we ﬁx the notation in §2 and §3. Let k be a local ﬁeld, that is, a complete discrete valuation ﬁeld with ﬁnite residue ﬁeld. Let ᏻ denote the ring of integers, and let F denote the residue ﬁeld. Let knr denote the maximal unramiﬁed ﬁeld extension in k, let Ik denote the Galois group Gal(k/knr ), and let ᏻnr denote the normalization of ᏻ in knr . We call a scheme X of ﬁnite type and ﬂat over ᏻ semistable if X is, everywhere, étale locally isomorphic to Spec ᏻ[t0 , . . . , td ]/(t0 · · · tr − π) for some integer r such that 0 ≤ r ≤ d. Here π denotes a prime of ᏻ, and d denotes the relative dimension. Then X is regular. Further, we call X strict if any irreducible component of the special ﬁber X ⊗ᏻ F is smooth over F. (1.5) Let X be a regular scheme of ﬁnite type, ﬂat and strictly semistable over ᏻ. In §2 and §3, we are mainly concerned with the case X not smooth over ᏻ. Let Y

80

KANETOMO SATO

denote the special ﬁber X ⊗ᏻ F. For an integer i ≥ 1, we deﬁne the reduced closed subscheme Y i ⊂ X of pure codimension i by Y i :=

∪

{η1 ,...,ηi }⊂Y 0

Yη1 ∩ Yη2 ∩ · · · ∩ Yηi .

Here for an η ∈ Y 0 , Yη denotes the corresponding irreducible component of Y . Clearly Y 1 coincides with Y . The complement X \ Y 2 is smooth over ᏻ, and Y i \ Y i+1 is smooth over F. We write Y (i) for the disjoint union of the irreducible components of Y i , which is smooth over F. For a positive integer i, J i denotes the set of the generic points of Y i . For example, J 1 coincides with Y 0 . For the scalar extension Xnr := X ⊗ᏻ ᏻnr , we deﬁne Y i , and J i is deﬁned similarly. (1.6) In §3, we use the following notation. Let i be a nonnegative integer. For a smooth variety Y over a ﬁnite ﬁeld F, Wr *iY,log denotes the étale subsheaf of the logarithmic part of the Hodge-Witt sheaf Wr *iY deﬁned by Illusie (see [I1]). We have an exact sequence of prosheaves (see [I1, Chap. I, 5.7.2]) 1−F

0 −→ W· *iY,log −→ W· *iY −−−→ W· *iY −→ 0.

(1.6.1)

Here F denotes the Frobenius operator (see [I1, Chap. I]). i denotes the For the special ﬁber Y of a ﬂat semistable family X over ᏻ, Wr ωY,log étale subsheaf of the logarithmic part of the modiﬁed Hodge-Witt sheaf Wr ωYi deﬁned i by Hyodo (see [H1], [H2]). We recall the deﬁnition of Wr ωY,log given in [H1, (1.5)]. × We write X for the generic ﬁber X ⊗ᏻ k. First we deﬁne the étale sheaf ᏻ∼ Y on Y by × ∗ ! ᏻ∼ Y := Coker (1 + π · Ga ,X ) ⊕ f (j )∗ Gm ,k −→ j∗ Gm ,X Y . Here j (resp., j ! ) denotes the open immersion X .→ X (resp., Spec k .→ Spec ᏻ), f denotes the structure morphism X → Spec ᏻ, and π denotes a ﬁxed prime of ᏻ. × By deﬁnition, there is a natural map Gm ,Y → ᏻ∼ Y . The kernel is Gm ,F |Y , and the cokernel is diag. ∗ ! Coker Gm ,X ⊕f (j )∗ Gm ,k −→ j∗ Gm ,X Y Coker ZY −−−→ ⊕ (iy )∗ Z . y∈Y 0

Here we have used a standard purity for Gm ,X (see [G, §6, (6.4)]). For a point y ∈ Y , × is the quotient of G , by we wrote iy for the canonical map y → Y . Hence, ᏻ∼ m Y Y Gm ,F |Y outside of the singular locus Y 2 , and there exists a differential symbol × ⊗i −→ ⊕ (iy )∗ Wr *iy,log dlog : ᏻ∼ Y y∈Y 0

i (cf. [I1, p. 580, (3.23.1)]). The sheaf Wr ωY,log is deﬁned by the image of the map ∼ × ⊗i i i (1.6.2) −→ ⊕ (iy )∗ Wr *y,log . Wr ωY,log := Im dlog : ᏻY y∈Y 0

1 r × For example, Wr ωY,log is isomorphic to ᏻ∼ Y /p .

81

ABEL-JACOBI MAPPINGS

The modiﬁed Hodge-Witt sheaf Wr ωYi is equipped with the Frobenius operator F and the monodromy operator ᏺ (see [H2]). The same exact sequence as (1.6.1) holds i i for Wr ωY,log and Wr ωYi (cf. [H1, (2.5)], the second deﬁnition of Wr ωY,log , and [I2, (2.2)]): 1−F

i −→ W· ωYi −−−→ W· ωYi −→ 0. 0 −→ W· ωY,log

(1.6.3)

(1.7) In this paragraph and the next, we ﬁx some notation in §5 and §6. We do not omit the indication of the topology of cohomology groups. For a scheme V and a prime number l invertible on V , we put Hei´ t V , Zl (n) := limHei´ t V , Z/ l ν Z(n) . ← − ν If V is a scheme of ﬁnite type over a local ﬁeld or Spec Z[l −1 ], then this group i (V , Z (n)) (cf. [J1, §3 coincides with the continuous étale cohomology group Hcont l (3.1)]). Similarly, for a proﬁnite group G and a projective system {Fν }ν∈N of discrete G-modules, we put

i i (G, Fν ). G, limFν := limHGal HGal ← − ← − ν ν If Fν is ﬁnite for any ν ∈ N, we have

1 1 HGal G, limFν Hcont G, limFν . ← − ← − ν ν

(1.7.1)

Here the right-hand side is the continuous Galois cohomology of Tate (see [T2]). (1.8) In §5 and §6, we work in the following setting. Let k be a global ﬁeld, that is, an algebraic number ﬁeld (case (N)) or a function ﬁeld in one variable over a ﬁnite ﬁeld (case (F)). For a ﬁnite set of places of k, we write kS for the maximal algebraic galois ﬁeld extension of k unramiﬁed outside of S, and write GS for the Galois group Gal(kS /k). In the case (N), we put C := Spec ᏻk , where ᏻk denotes the integral closure of Z in k. In the case (F), we write C for the proper smooth model of k over the algebraic closure of the prime ﬁeld in k. Let X be a proper smooth geometrically connected variety over k. We take an afﬁne open subset U0 ⊂ C, and a proper smooth U0 -scheme X0 whose generic ﬁber is X. We ﬁx a prime number l. Replacing U0 and X0 by suitable nonempty open subschemes if necessary, we may assume that l is invertible on U0 . For nonnegative integers i and n, we put i Hind (1.8.1) X, Zl (n) := lim Hei´ t XU , Zl (n) . − → U ⊂U0 open

Here XU denotes X0 ×U0 U . This group does not depend on the model X0 by [EGA4, 8.8.2.5].

82

KANETOMO SATO

We review the Hochschild-Serre ﬁltration F• ⊂ H∗i (X, Zl (n)) (∗ ∈ {cont, ind}) associated with the covering X = X ⊗k k → X. As for the continuous étale cohomology, we have the Hochschild-Serre spectral sequence [J1, (3.3)]: p,q p q p+q E2 = Hcont Gk , He´ t X, Zl (n) ⇒ Hcont X, Zl (n) . (1.8.2) i (X, Z (n)) is deﬁned by that resulting from this spectral The ﬁltration F• on Hcont l sequence. On the other hand, for an open U ⊂ U0 , we put SU := (C \ U ) ∪ {inﬁnite places}. For each open U of U0 , the Leray spectral sequence for the structure morphism fU : XU → U may be written as p,q p q p+q E2 = HGal GSU , He´ t X, Zl (n) ⇒ He´ t XU , Zl (n) (1.8.3)

by the proper and smooth base change theorem and [Mi, II, 2.9]. Since this spectral sequence is contravariantly functorial in U , we have the following spectral sequence by passing to the inductive limit: p,q p q p+q E2 = Hind k, He´ t X, Zl (n) ⇒ Hind X, Zl (n) , (1.8.4) where we put p q Hind k, He´ t X, Zl (n) :=

lim − →

U ⊂U0 open

p q HGal GSU , He´ t X, Zl (n) .

i (X, Z (n)) by that resulting from (1.8.4). Since the We deﬁne the ﬁltration F• on Hind l canonical maps Gk → GSU and X → XU induce a homomorphism of spectral sei (X, Z (n)) and Hi (X, Z (n)) quences from (1.8.3) to (1.8.2), the ﬁltration F• on Hind l l cont i (X, Z (n)) → Hi (X, Z (n)). is compatible with the canonical map Hind l l cont

Lemma 1.1. For the spectral sequences (1.8.2), (1.8.3), and (1.8.4), we have = 0 if p ≥ 3, or if p = 0 and q = 2n. Consequently, we have

p,q E2 ⊗ Q

p,q

E2

p,q

⊗ Q E∞ ⊗ Q,

if (p, q) = (0, 2n), (2, 2n − 1). p,q

Proof. First, we prove that E2 ⊗Q = 0 for p ≥ 3. In the case (N), by a standard p,q norm argument, it sufﬁces to prove that E2 = 0 for p ≥ 3 if k is totally imaginary. We assume that k is the case (F) or a totally imaginary number ﬁeld. By the global duality theorem of Poitou and Tate [T1, Theorem 3.1], we have cd(Gk ) = cd(GSU ) = q p,q 2. Hence, E2 of (1.8.3) and (1.8.4) are trivial for p ≥ 3. We put T q := He´ t (X, Zl (n)), q and Mν := T q / l ν T q . By the exact sequence p−1 p p 0 −→ lim 1 HGal Gk , Mνq ν −→ Hcont Gk , T q −→ lim HGal Gk , Mνq ν −→ 0 ← − ← −

83

ABEL-JACOBI MAPPINGS p,q

(cf. [T2, 2.2]), the E2 -terms of (1.8.2) are trivial for p ≥ 3, because the projective m (G , M q )} system {HGal k ν ν∈N satisﬁes the Mittag-Lefﬂer condition for m = 2 and is trivial for m > 2. 0,q Finally, we prove that E2 is torsion if q = 2n. We ﬁx a place p on U , and write Yp for the reduction of X at p, which is proper and smooth over the residue ﬁeld κ(p). By the proper and smooth base change theorem, we have Gal(κ(p)/κ(p)) q , (T q )Gk (T q )GSU ⊂ He´ t Yp ⊗κ(p) κ(p), Zl (n) and the last group is ﬁnite if q = 2n by Deligne’s proof of the Weil conjecture (see [D3, 3.3.9]). This completes the proof. Remark 1.2. In the above situation, the spectral sequences (1.8.2), (1.8.3), and (1.8.4) degenerate at E2 up to torsion. If X is projective and smooth, this follows from Deligne’s criterion [D1] and the hard Lefschetz theorem (see also [D3, (1.1.2)], [Ras2, Theorem 1.1]). If X is proper and smooth, then the problem is reduced to the projective smooth case by [dJ, Theorem 4.1], [EGA4, Théorème 8.8.2], and a standard norm argument. 2. Varieties over local ﬁelds. In this section and §3, we prove Theorem 0.3. More precisely, we prove the following stronger result. Theorem 2.1. Let X be a proper smooth geometrically connected variety over a local ﬁeld k. Let n be an arbitrary integer at least 2. Then the composite map αln : H2 k, Hn−1 X, Ql /Zl (n) −→ Hn+1 X, Ql /Zl (n) −→ Hn+1 k(X), Ql /Zl (n) has ﬁnite kernel for any prime number l = ch(k), and is injective for almost all l = ch(k). First, admitting Theorem 2.1, we prove Theorem 0.3, which we use in §5. Corollary 2.2 (Theorem 0.3). Let X, k, and n be as in Theorem 2.1. Then the composite map vln : H2 k, Hn−1 X, Ql (n) −→ Hn+1 X, Ql (n) −→ lim Hn+1 V , Ql (n) − → V ⊂X open

is injective for any prime number l different from ch(k). Proof. We write tln for the composite map

tln : H2 k, Hn−1 X, Zl (n) −→ Hn+1 X, Zl (n) −→

lim − →

V ⊂X open

! Hn+1 V , Zl (n) .

Here ! means the quotient by the torsion part. Then we have the following commutative

84

KANETOMO SATO

diagram with exact columns (up to ﬁnite groups): H2 k, Hn−1 X, Zl (n) H2 k, H

n−1

tln

/

lim − →

V ⊂X open

! Hn+1 V , Zl (n) ∩

X, Ql (n)

H2 k, Hn−1 X, Ql /Zl (n)

vln

αln

/

lim − →

V ⊂X open

H

n+1

V , Ql (n)

/ Hn+1 k(X), Ql /Zl (n) .

Applying the snake lemma to this diagram, we have an exact sequence (up to ﬁnite groups) Ker tln −→ Ker vln −→ Ker αln . Since Ker(tln ) is ﬁnitely generated over Zl and Ker(αln ) is ﬁnite by Theorem 2.1, the Ql -vector space Ker(vln ) is trivial. This completes the proof. Remark 2.3. The paper [Sz] contains a proposition (attributed to Colliot-Thélène, see [Sz, Proposition 5.5]; assuming the Bloch-Kato conjecture in degree 3, cf. Conjecture 6.1): for a projective smooth surface X over a p-adic ﬁeld, the group CH3 (X, 1){l} is ﬁnite for l = p (resp., 1 = p), if the Albanese variety has potentially good reduction (resp., if H1 (X, ᏻX ) = 0). In the proof, Theorem 2.1 plays a key role. (2.1) We begin the proof of Theorem 2.1. We write p for the characteristic of the residue ﬁeld F of k. First, we assume that X has potentially good reduction over k. Then the group H2 (k, Hn−1 (X, Ql /Zl (n))) is ﬁnite for any l = ch(k) and trivial for almost all l = ch(k) by a result of Gabber [Ga] and the following well-known fact. Lemma 2.4. Let i and n be positive integers with i = 2n, and let l be a prime number different from ch(k). We assume that X has potentially good reduction over k. Then the group H2 (k, Hi−2 (X, Ql /Zl (n))) is ﬁnite. Proof. By the Tate duality (see [T1, Theorem 2.1]) and the Poincaré duality, we only have to show that the group H2d−i+2 (X, Zl (d − n + 1))Gk is ﬁnite, where d := dim X. Henceforth, we may assume that X has good reduction over k. Since i = 2n, we have 2d − i + 2 = 2(d − n + 1). Hence, if l = p, this ﬁniteness follows from the proper and smooth base change theorem and Deligne’s proof of the Weil conjecture (see [D3, 3.3.9]). If k is a p-adic ﬁeld and l = p, this ﬁniteness is a consequence of the crystalline conjecture (see [FoMe], [Fa], and [KoM]) and the Katz-Messing theorem [KzM] (see also [B1, Proposition 2.4] for the case i = 3 and n = 2). We write Y for the reduction variety of X over F and write q for the cardinality of F. By the crystalline conjecture,

85

ABEL-JACOBI MAPPINGS

we have G φ =q d−n+1 2d−i+2 He2d−i+2 X, Ql (d − n + 1) k ⊂ Hcrys Y/W (F) QF ´t (see [J2, p. 344] for more details), where φF denotes the crystalline Frobenius operator with respect to F. Furthermore, the right-hand side is trivial by the Katz-Messing theorem, and hence the left-hand side is also trivial. We remark that the Katz-Messing theorem holds for any proper smooth variety over a ﬁnite ﬁeld by the alteration theorem of de Jong (see [dJ, Theorem 4.1]). In the following, we assume that X does not have potentially good reduction. We reduce Theorem 2.1 to the semistable reduction case. By the alteration theorem of de Jong (see [dJ, Theorem 6.5]), we take a proper, surjective, and generically ﬁnite morphism f : X ! → X such that X! is projective smooth over k and has a regular model X! that is projective, ﬂat, and strictly semistable over ᏻL . Here L denotes 6(X ! , ᏻX! ), and ᏻL denotes the ring of integers in L. Then we have a commutative diagram H2 k, Hn−1 X ⊗k k, Ql /Zl (n)

αln

f∗

H2 L, Hn−1 X ! ⊗L L, Ql /Zl (n)

/ Hn+1 k(X), Ql /Zl (n) f∗

αln

/ Hn+1 L(X ! ), Ql /Zl (n) .

The lower horizontal arrow is the map αln for X! . The left vertical map f ∗ has ﬁnite kernel, which is annihilated by an integer independent of l: [L : k] L(X ! ) : k(X) = L(X ! ) : k(X) . Therefore Theorem 2.1 follows from the following result. Theorem 2.5. Let n be an integer at least 2. We assume that X has a regular model X that is projective, ﬂat, and strictly semistable over ᏻ. Then (1) αLn := ⊕all l=p αln has ﬁnite kernel. (2) If k is a p-adic ﬁeld, then αpn has ﬁnite kernel. We prove Theorem 2.5(1) in this section. The proof of Theorem 2.5(2) is given in §3. (2.2) We put

We begin the proof of Theorem 2.5(1). Let the notation be as in (1.1)–(1.5). Q/Z! := ⊕ Ql /Zl . all l=p

We always assume that Y n is not empty and, hence, that dim Y ≥ n − 1. Otherwise, our problem is obvious by Lemma 2.8 below.

86

KANETOMO SATO

For a positive integer m prime to p, let R • 7Z/mZ denote the sheaf of vanishing cycles on Y e´ t with respect to the canonical map j : X → Xnr , that is, R i 7Z/mZ := R i j ∗ Z/mZ Y . (2.2.1) In the following, we ﬁx an ordering Y1 , Y2 , . . . , YN of the irreducible components of Y . To control Ker(αLn ), we construct the following commutative diagram of natural homomorphisms using this ordering: H2 k, Hn−1 X, Q/Z! (n) γLn

H2 k, ⊕ R n−1 7Q/Z! (n)z z∈J

n αL

n

/ Hn+1 k(X), Q/Z! (n) dLn

βLn

/ ⊕ H1 (x, Q/Z! ).

(2.2.2)

x∈J n

The maps dLn and γLn are deﬁned below. We deﬁne βLn in (2.3)–(2.4), which is injective by the construction. Then our task is to prove the following lemma. Lemma 2.6. γLn has ﬁnite kernel. We prove Lemma 2.6 in (2.5)–(2.6). Deﬁnition of dLn . For an x ∈ J n , the map dLn is deﬁned as follows. Let Yν1 , Yν2 , . . . , Yνn (ν1 < ν2 < · · · < νn ) be the irreducible components of Y that contain x. Then we have a sequence of specializations Spec k(X) −→ y1 −→ y2 −→ · · · −→ yn = x yi ∈ J i (2.2.3) determined by the law that yi is the generic point of the connected component of Yν1 ∩Yν2 ∩· · ·∩Yνi that contains x. We deﬁne dLn to be the composite of the boundary maps of the étale cohomology groups [K2, §1 (1.3)(i)]: Hn+1 k(X), Q/Z! (n) −→ Hn y1 , Q/Z! (n − 1) (2.2.4) −→ Hn−1 y2 , Q/Z! (n − 2) −→ · · · −→ H1 (x, Q/Z! ). This map essentially depends on the sequence (2.2.3). Deﬁnition of γLn . The map γLn is deﬁned by the composite of natural maps, γLn : H2 k, Hn−1 X, Q/Z! (n) −→ H2 k, H0 Y , R n−1 7Q/Z! (n) 2 n−1 ! −→ H k, ⊕ R 7Q/Z (n)z . z∈J n

Here the ﬁrst map is the canonical map attached to the canonical morphism j : X → Xnr (cf. (2.2.1), (2.5.2)).

87

ABEL-JACOBI MAPPINGS

(2.3)

We put < :=

all l=p Zl .

Fix a point z ∈ J n . We put Vz := Spec ᏻsh Xnr ,z ,

that is, the strict henselization of Xnr at z, and put Dz := Vz ⊗ᏻnr F. We write Uz for the complement Vz \ Dz . Clearly, Dz is a simple normal crossing divisor on Vz . Let x ∈ J n be the image of z, and let Yν1 , Yν2 , . . . , Yνn (ν1 < ν2 < · · · < νn ) be the irreducible components of Y that contain x. We write Di (i = 1, 2, . . . , n) for the irreducible component of Dz lying over Yνi . By [RZ, Satz 2.21], we have, for any ν, the cohomological purity H1 Vz \ Dν , <(1) < (2.3.1) and

H q Vz \ D ν , < = 0

for q = 1.

We write cν for the inverse image of 1 ∈ < under the isomorphism (2.3.1). By this fact and the smooth purity, we ﬁx an isomorphism δz : Hn Uz , Q/Z! (n) Q/Z! , which sends (c1 ∪ c2 ∪ · · · ∪ cn ) ⊗ a to a (a ∈ Q/Z! ) (cf. [SGA7, Exp. I, (3.1.2)]). (2.4) map

For a point z ∈ J n , we have a homomorphism ιz deﬁned by the composite

H1 Ik , R n−1 7Q/Z! (n)z −→ H1 Ik , Hn−1 Uz ⊗ k, Q/Z! (n) knr

δz −→ H Uz , Q/Z! (n) Q/Z! . ⊂

n

(2.4.1)

Here the second map is injective by the fact that cd(Ik ) = 1 (cf. [S, Chap. II, §3.3]). This map ιz is injective by deﬁnition, and further the GF -coinvariant H1 (F, ⊕z∈J n ιz ) is also injective. In fact, by a theorem of Rapoport and Zink [RZ, Satz 2.23], (see (2.5) below), the second map in (2.4.1) is bijective. We deﬁne the desired map βLn by the composite of H1 (F, ⊕z∈J n ιz ) and the natural injective map ⊂ 1 0 ! H F, ⊕ H (z, Q/Z ) −→ ⊕ H1 (x, Q/Z! ), z∈J n

x∈J n

which comes from the Hochschild-Serre spectral sequence. By the construction, βLn is injective. Lemma 2.7. The diagram (2.2.2) is commutative. Proof. For each z ∈ J n , we ﬁx a sequence of specializations Spec knr (X) −→ w1 −→ w2 −→ · · · −→ wn = z wi ∈ J i

(2.4.2)

88

KANETOMO SATO

in the following way. Let x ∈ J n be the image of z, and let Yν1 , Yν2 , . . . , Yνn (ν1 < ν2 < · · · < νn ) be the irreducible components of Y that contain x. We write Wi (i = 1, 2, . . . , n) for the irreducible component of Y that lies over Yνi and contains z. Then we deﬁne wi to be the generic point of the connected component of W1 ∩W2 ∩· · ·∩Wi , which contains z. By the naturality of the functor H1 (F, •) and inﬂation maps, our task is to prove the commutativity of the following diagram: H1 Ik , Hn−1 X, Q/Z! (n) 1

γ nL

⊕ H Ik , R n−1 7Q/Z! (n)z

z∈J n

α nL

/ Hn knr (X), Q/Z! (n) d nL

⊕z∈J n ιz

! / ⊕ Q/Z .

(2.4.3)

z∈J n

n

Here the maps α nL, γ nL, and d L are deﬁned in the same way as αLn , γLn , and dLn , n respectively, and the sequence (2.4.2) is used in the deﬁnition of d L. Since we have not used the properness of X in the construction of the diagram (2.4.3) or (2.2.2), we only have to check the commutativity by localizing Xnr around z for each z ∈ J n . We ﬁx a z ∈ J n , and use the notation in (3.3). We write Kz for the fraction ﬁeld of sh ᏻXnr ,z . Let d z denote the successive boundary map deﬁned in the same way as d nL: d z : Hn Kz , Q/Z! (n) −→ Q/Z! . By the deﬁnitions of ιz and α nL, it sufﬁces to show that the following diagram is commutative: / Hn Kz , Q/Z! (n) Hn Uz , Q/Z! (n) HH HH vv HH vv v H v v δz HH$ zvv d z ! Q/Z . We consider the ﬁrst step of d z , Hn Kz , <(n) −→ Hn−1 w1 , <(n − 1) .

(2.4.4)

Here w1 denotes the generic point of D1 . For each ν (1 ≤ ν ≤ n), we ﬁx a uniformizer πν of the divisor Dν in Uz (recall that Dν ’s and Uz are all regular (cf. (2.3)). The image of πν under the natural map Kz× → H1 (Kz , <(1)) coincides with cν . Because Dz is a simple normal crossing divisor in Uz , the elements π2 , . . . , πn ∈ Kz are units with respect to the discrete valuation given by π1 . Hence, the map (2.4.4) sends the element c1 ∪ c2 ∪ · · · ∪ cn to c2 ∪ · · · ∪ cn (see [K2, Lemma 1.4(1)]). Here for each ν ≥ 2, cν denotes the image of the residue class of πν (in κ(w1 )) under the natural map κ(w1 )× → H 1 (w1 , <(1)). By the same arguments for the other steps, (c1 ∪ c2 ∪ · · · ∪ cn ) ⊗ a (a ∈ Q/Z! ) maps to a under d z . This completes the proof.

89

ABEL-JACOBI MAPPINGS

(2.5) We prove Lemma 2.6. We recall a theorem of Rapoport and Zink (see [RZ, Satz 2.23]); for an integer i ≥ 0, the sheaf R i 7Q/Z! is (noncanonically) quasi-isomorphic to a complex of étale sheaves on Y : ⊕ (ix )∗ Q/Z! (−i) −→

x∈J i+1

⊕ (ix )∗ Q/Z! (−i)

x∈J i+2

−→ · · · −→

⊕ (ix )∗ Q/Z! (−i)

(2.5.1)

x∈J d+1

if i ≤ d := dim Y , and otherwise R i 7Q/Z! = 0. Here for a point x ∈ Y , ix denotes the canonical map x → Y , and the maps in (2.5.1) are given by the alternate sum determined by the ordering Y1 , Y2 , . . . , YN . By this fact, the support of any nonzero section of R n−1 7Q/Z! is of pure codimension zero in Y n , and hence the Leray spectral sequence p,q E2 = Hp Y , R q 7Q/Z! (n) ⇒ Hp+q X, Q/Z! (n) (2.5.2) gives us an exact sequence of Gk -modules 0 −→ L1 Hn−1 X, Q/Z! (n) −→ Hn−1 X, Q/Z! (n) −→ ⊕ R z∈J

n

n−1

(2.5.3)

!

7Q/Z (n)z .

Here L• denotes the resulting ﬁltration from the spectral sequence (2.5.2). Lemma 2.6 is reduced to two claims. Lemma 2.8. (1) The group H2 (k, L1 Hn−1 (X, Q/Z! (n))) is ﬁnite. (2) Assume that any irreducible component of Y n is geometrically connected over F. Then the complex associated with the sequence (2.5.3), γLn 2 1 n−1 2 n−1 2 n−1 H k, L H (X) −→ H k, H (X) −−→ H k, ⊕ R 7z , z∈J

n

is exact. The assumption in (2) is satisﬁed after base change by some ﬁnite unramiﬁed extension of ᏻ. Hence, by a norm argument, Lemma 2.8 implies Lemma 2.6. We prove Lemma 2.8(2). In that situation, for any i ≤ n, the action of GF on the set J i is trivial; that is, J i is naturally identiﬁed with J i . Our claim follows from the exactness of ﬁnite coefﬁcient variant of (2.5.3) and the following Lemma 2.9(2). Lemma 2.9. (1) Let 0 → M1 → M2 → M3 be an exact sequence of ﬁnite Gk modules. Assume that Gk acts on M3 (−1) trivially. Then the associated complex H2 (k, M1 ) −→ H2 (k, M2 ) −→ H2 (k, M3 ) is exact.

90

KANETOMO SATO

(2) Let {0 → M1,ν → M2,ν → M3,ν }ν∈N be an inductive system of exact sequences of ﬁnite Gk -modules, and write 0 → M1 → M2 → M3 for the inductive limit. Assume that Gk acts on M3,ν (−1) trivially for any ν. Then the associated complex H2 (k, M1 ) −→ H2 (k, M2 ) −→ H2 (k, M3 ) is exact. Proof. Lemma 2.9(2) immediately follows from (1). We prove Lemma 2.9(1). We put M4 := Im(M2 → M3 ). Then we have a long exact sequence of Galois cohomology groups · · · −→ H2 (k, M1 ) −→ H2 (k, M2 ) −→ H2 (k, M4 ) −→ 0. Hence, it sufﬁces to show that the map H2 (k, M4 ) → H2 (k, M3 ) is injective, which follows from the Tate duality (see [T1, Theorem 2.1]). (2.6) We prove Lemma 2.8(1). By the result of Rapoport and Zink, which we recalled in (2.5), the inertia group Ik acts trivially on the E∞ -terms of the Leray spectral sequence (2.5.2). Hence it sufﬁces to show that for any i with 1 ≤ i ≤ n − 1, the group i,n−1−i H1 F, E∞ (−1) = H1 F, gr iL Hn−1 X, Q/Z! (n − 1) is ﬁnite. The following lemma immediately follows from the fact that cd(GF ) = 1. Lemma 2.10. Let f : M1 → M2 be a homomorphism of torsion discrete GF modules. We assume that the groups H0 (F, M1 ), H1 (F, M1 ), H0 (F, M2 ), and H1 (F, M2 ) are ﬁnite. Then the groups H0 (F, Ker(f )), H1 (F, Ker(f )), H0 (F, Coker(f )), and H1 (F, Coker(f )) are ﬁnite. We apply this lemma to the Leray spectral sequence (2.5.2). We consider the following statement (Lr ) with respect to an integer r ≥ 2: p,q (Lr ) : Hi F, Er (−1) is ﬁnite for any i ∈ {0, 1}and any pair (p, q) satisfying (r − 1)p + rq < r(n − 1). p,q

Since E2 = 0 if p < 0 or q < 0, the statement (Lr−1 ) implies (Lr ) by Lemma 2.10, and hence it sufﬁces to show (L2 ). We prove (L2 ). Fix a q < n − 1. By the Rapoport-Zink theorem (2.5.1), we have a spectral sequence of GF -modules ! s,t E1

= Ht Y (q+s+1) , Q/Z! (n − 1 − q) ⇒ Hs+t Y , R q 7Q/Z! (n − 1) .

Here !E1s,t = 0 for t < 0 or s < 0, and Y (q+s+1) is a projective smooth scheme over F for any s. Hence, for any t satisfying t < 2(n − 1 − q), any i and s, the group

91

ABEL-JACOBI MAPPINGS

Hi (F, !E1s,t ) is ﬁnite by Deligne’s proof of the Weil conjecture (see [D3, 3.3.9]), and a theorem of Gabber (see [Ga]; see [CTSS, §2.1] for details). By applying Lemma 2.10 s,t to this spectral sequence, we obtain the ﬁniteness of Hi (F, !E∞ ) for any i ∈ {0, 1} and any (s, t) satisfying s + t < 2(n − 1 − q). This implies (L2 ). Remark 2.11. For any integer i with i ≤ n, the group H2 (k, Hi−2 (X, Q/Z! (n))) is ﬁnite. This follows from a similar argument to (2.6). Hence F2 Hi (X, Q/Z! (n)) is ﬁnite for any i ≤ n and any proper smooth X over k by the alteration argument in (2.1). 3. Varieties over p-adic ﬁelds. In this section, we prove Theorem 2.5(2). (3.1) Let k be a p-adic ﬁeld. Let the notation be as in (1.1)–(1.6). First, we reduce our problem to the case dim X = n − 1. If dim X < n − 1, the problem is clear, because the group H2 (k, Hn−1 (X, Qp /Zp (n))) is ﬁnite by Lemma 3.5 below. If dim X > n−1, we apply the Lefschetz argument of Salberger (see [Sal, §1]) to the generic ﬁber X. We review this reduction argument brieﬂy. We take a smooth hyperplane section ı : X1 → X by the Bertini theorem. Let R denote the local ring of X at the generic point of X1 . The restriction map Hn+1 (X) → Hn+1 (k(X)) factors through Hn+1 (R): Hn+1 X, Qp /Zp (n) −→ Hn+1 R, Qp /Zp (n) −→ Hn+1 k(X), Qp /Zp (n) , and the last map is injective by the Bloch-Ogus theorem (see [BO, Theorem 4.2]). We have a commutative diagram H2 k, Hn−1 X, Qp /Zp (n)

αpn

ı∗

H2 k, Hn−1 X 1 , Qp /Zp (n)

/ Hn+1 R, Qp /Zp (n) ı∗

αpn

/ Hn+1 k(X1 ), Qp /Zp (n) .

If dim X > n, the left vertical arrow is bijective by the weak Lefschetz theorem. If dim X = n, this map has ﬁnite kernel by the hard Lefschetz theorem for X(C)an . Thus our problem is reduced to the case dim X = n − 1. After this argument, we apply the reduction argument in (2.1) and assume that X has a model X over ᏻ as in Theorem 2.5. Remark 3.1. We follow this procedure in order to simplify the argument on the ﬁniteness of the group of global sections of the modiﬁed logarithmic Hodge-Witt sheaf Wr ωn−1 on Y (see (3.4) and (3.5)). This ﬁniteness is established in the case Y ,log where Y is smooth of arbitrary dimension (see [I1, p. 534, (2.5.7)]). (3.2)

In this paragraph and the next, we construct a commutative diagram (3.3.1),

92

KANETOMO SATO

which is analogous to (2.2.2). The map αpn factors through the injection: αp! n H2 k, Hn−1 X, Qp /Zp (n) −−−→ H1 F, Hn knr (X), Qp /Zp (n) ⊂ −→ Hn+1 k(X), Qp /Zp (n) . Hence, it sufﬁces to control the kernel of αp! n . By a theorem of Hyodo (see [H1, Theorem 1.6]), for any positive integer r, there exists a natural Gk -homomorphism

hrn−1 : Hn−1 X, Z/p r Z(n − 1) −→ H0 Y , Wr ωYn−1 , (3.2.1) ,log which ﬁts into a commutative diagram H2 k, Hn−1 X, Qp /Zp (n)

αp! n

/ H1 F, Hn knr (X), Qp /Zp (n)

γ np

1 0 lim H F, H Y , Wr ωYn−1 ,log − → r

d1

/ lim H1 F, ⊕ H0 w, Wr *n−1 w,log − → w∈J 1 r

(3.2.2)

.

Here the map γ np is the Gk -coinvariant of the map lim r hrn−1 , and the bottom horizontal − → map is the pullback by the canonical map w → Y (cf. (1.6.2)). The map d1 is deﬁned as follows. Fix a w ∈ J 1 . Let Kw denote the fraction ﬁeld of the henselian local ring ᏻhXnr ,w . We deﬁne the natural map δ1,w by the composite map δ1,w : Hn knr (X), Qp /Zp (n) (∗) −→ Hn Kw , Qp /Zp (n) KnM (Kw ) ⊗ Qp /Zp (3.2.3)

∂ n−1 M 0 −→ Kn−1 (κ(w)) ⊗ Qp /Zp −→ lim H w, Wr *w,log . − → r The isomorphism (∗) is due to a theorem of Bloch and Kato (see [BK1, Theorem 5.12]) and is the inverse of the Galois symbol mapping (see [T2]; see also Conjecture 6.1 below). The map ∂ is the tame symbol of the Milnor K-groups (see [K2, §1, (1.1)]). The last map is the differential symbol and known to be bijective (see [BK1, Theorem 2.1]). The map d1 is deﬁned to be that induced by the direct sum of δ1,w ’s. Remark 3.2. In fact, since dim Y = n − 1, we have a natural boundary map

n−1 Hn+1 k(X), Z/p r Z(n) −→ H1 y, Wr *y,log for any generic point y ∈ Y 0 = J 1 by a theorem of Kato (see [K1, §0, Theorem 3]; cf. [K2, §1, (1.3)(ii)]). We do not use this fact here.

93

ABEL-JACOBI MAPPINGS

(3.3)

We put

W∞ ωn−1 := limWr ωn−1 Y ,log Y ,log − → r

in the following. By an analogy to (2.2.2), we ﬁx an ordering Y1 , Y2 , . . . , YN of the irreducible components of Y and construct a commutative diagram H2 k, Hn−1 X, Qp /Zp (n)

αp! n

/ H1 F, Hn knr (X), Qp /Zp (n)

γ np

1 0 H F, H Y , W∞ ωn−1

dpn

(3.3.1)

/ ⊕ n Qp /Zp .

βpn

x∈J

Y ,log

The maps dpn and βpn are deﬁned in the rest of this paragraph. Theorem 2.5(2) immediately follows from this diagram and the following two lemmas. Lemma 3.3. The map γ np has ﬁnite kernel and cokernel. Lemma 3.4. The map βpn has ﬁnite kernel. We prove Lemma 3.3 in the next paragraph (3.4). The proof of Lemma 3.4 is given in (3.5)–(3.6). Deﬁnitions of βpn and dpn . Fix a positive integer r. For a point x ∈ Y , we write ix for the canonical map x → Y . First we deﬁne the homomorphism n−1 ιr : Wr ωY,log −→ ⊕ (ix )∗ Z/p r Z x∈J n

of étale sheaves as follows. Fix a point x ∈ J n . Then by the ordering Y1 , Y2 , . . . , YN , we have a sequence of specializations (3.3.2) y1 −→ y2 −→ · · · −→ yn = x yi ∈ J i (cf. §2 (2.2.3)). We deﬁne ιr to be the composite of the natural map (cf. (1.6.2)) and the boundary maps arising from the purity for logarithmic Hodge-Witt sheaves (see [Gr]): n−1 Wr ωY,log −→ (iy1 )∗ Wr *yn−1 −→ (iy2 )∗ Wr *yn−2 −→ · · · −→ (ix )∗ Z/p r Z. (3.3.3) 1 ,log 2 ,log

We deﬁne the map βpn by the composite map

1 Q /Z −→ H F, ⊕ H1 (F, ι∞ ) : H1 F, H 0 Y , W∞ ωYn−1 p p ⊕ Qp /Zp . ,log x∈J n

x∈J n

For a point x ∈ J n , the map dpn is deﬁned as follows. Let z ∈ J n be a point lying above x. We have a sequence of specializations w1 −→ w2 −→ · · · −→ wn = z wi ∈ J i

94

KANETOMO SATO

determined in the same way as in §2 (2.4.2). Then we have a GF -homomorphism δz deﬁned by the composite of the boundary map δ1,w1 (3.2.3) and the boundary map deﬁned in the same way as in (3.3.3):

δ1,w1 n−1 −→ Qp /Zp . δz : Hn knr (X), Qp /Zp (n) −−−→ H0 w1 , W∞ *w 1 ,log We deﬁne the map dpn for an x ∈ J n to be that induced by the direct sum of δz ’s for all z’s lying above x. Clearly the diagram (3.3.1) is commutative by the construction and the commutativity of (3.2.2). (3.4)

and

We prove Lemma 3.3. We put

, Tp := lim H0 Y , Wr ωYn−1 ,log ← − r

Vp := Tp ⊗Zp Qp

n−1 0 H Y , W ω = lim Ap := H0 Y , W∞ ωYn−1 r Y ,log . ,log − → r

We write hˆ n−1 for the map induced by (3.2.1): hˆ n−1 := lim hrn−1 ⊗Zp Qp : Hn−1 X, Qp (n − 1) −→ Vp . ← − r We use the following lemma, which is a consequence of the p-adic Hodge theory of Hyodo, Kato, and Tsuji (see [HK], [K3], and [Ts]). Lemma 3.5. The map hˆ n−1 induces an isomorphism of GF -modules Hn−1 X, Qp (n − 1) I Vp . k

Proof. See [Sat, Lemma 3.3] for the proof. This lemma also holds for the case dim Y < n − 1; the left-hand side is zero in this case. This can be checked by the same way as in [Sat, Lemma 3.3]. Proof of Lemma 3.3. Since dim Y = n−1, the cohomology group Hi Y , Wr ωn−1 Y ,log is ﬁnite for any r ≥ 1 and i ≥ 0 by the exact sequences 0 −→ Wr−1 ωn−1 −→ Wr ωn−1 −→ ωn−1 −→ 0 Y ,log Y ,log Y ,log (a variant of [H1, (1.5.2)]) and C−1

0 −→ ωn−1 −→ ωn−1 −−−→ ωn−1 −→ 0 Y ,log Y Y (a variant of [H1, (1.5.1)]). Hence we have an exact sequence

n−1 1 0 −→ Tp −→ Vp −→ Ap −→ lim H Y , Wr ωY ,log . ← − r tor

95

ABEL-JACOBI MAPPINGS

Tp is a ﬁnitely generated Zp -module, and the last group is ﬁnite. Therefore, γ np is isogenous if and only if the Gk -coinvariant (hˆ n−1 )Gk is bijective. In particular, Lemma 3.3 follows from Lemma 3.5. (3.5) We prove Lemma 3.4. We deﬁne the étale sheaf Mr on Y by the exact sequence ιr

n−1 0 −→ Mr −→ Wr ωY,log −→ ⊕x∈J n (ix )∗ Z/p r Z.

(3.5.1)

Then we have an exact sequence of GF -modules

ι ∞ 0 −→ H0 Y , M∞ −→ H0 Y , W∞ ωn−1 −−→ ⊕ Qp /Zp . Y ,log x∈J n

By an easier analogue of Lemma 2.9, the induced complex

H F, H Y , M∞ 1

0

−→ H F, H 1

0

Y , W∞ ωYn−1 ,log

βpn

−−→ H

1

F, ⊕ Qp /Zp x∈J n

is exact modulo ﬁnite groups. To prove the ﬁniteness of Ker(βpn ), it sufﬁces to show that the group H1 (F, H0 (Y , M∞ )) is ﬁnite. We put N ∞ := lim H0 Y , Mr ⊗ Qp . ← − Zp r Once we have proved the following lemma, Lemma 3.4 is proved. Lemma 3.6. (1) H1 (F, H0 (Y , M∞ )) is ﬁnite. ⇔ we have H1 (F, N ∞ ) = 0. (2) We have H1 (F, N ∞ ) = 0. We prove (2) in (3.6), and we prove (1) here. Since dim Y = n−1, any point x ∈ J n is a closed point of Y , and the intersection of the closures of any two distinct points x, x ! ∈ J n is empty. Hence, the map ιr is surjective as a homomorphism of sheaves by n−1 the deﬁnition of Wr ωY,log (cf. (1.6.2), (3.3)). Therefore, we have an exact sequence 0 −→ Mr−1 −→ Mr −→ M1 −→ 0 by the deﬁnition of Mr (3.5.1). The group H i (Y , Mr ) is ﬁnite for any r ≥ 1 and i ≥ 0 by the ﬁniteness of H i (Y , Wr ωn−1 ) (see (3.4)). Hence we have an exact sequence Y ,log

0 −→ lim H Y , Mr −→ N ← − r 0

∞

−→ H Y , M∞ −→ lim H Y , Mr ← − r 0

1

. tor

The group limr H0 (Y , Mr ) is ﬁnitely generated as a Zp -module, and the last group is ← − ﬁnite. These facts show Lemma 3.6(1).

96

KANETOMO SATO

(3.6) We prove Lemma 3.6(2). By a result of Mokrane (see [Mo, 3.23]), we have the weight spectral sequence E1−s,i+s =

⊕

j ≥0,j ≥−s

i−2j −s (2j +s+1)

Hcrys

Y

/W (F)

i ⇒ P• Hlog-crys Y /W (F) Q .

Q

(−j − s)

Here P• denotes the resulting ﬁltration, and for an F -isocrystal (H, G) and a positive integer r, H (−r) denotes (H, p r G). To be precise, we have used the ordering Y1 , Y2 , . . . , YN in the construction of the above spectral sequence. Taking the Qp subspaces where the Frobenius operator acts by p n−1 , we have a spectral sequence of GF -modules E1−s,i+s =

⊕

j ≥0,j ≥−s

n−1−j −s F =1 Hi−n+1−j Y (2j +s+1) , W*Y (2j +s+1) Q

F =1 ⇒ P• Hi−n+1 Y , WωYn−1 .

(3.6.1)

Q

Here we have used the fact that the functor F -isocrystals/F −→ (Qp -vector spaces);

(H, G) & −→ H G=p

r

is exact for any r ∈ Z, the fact that

F =1 G=pn−1 i Hlog-crys Y /W (F) Q Hi−n+1 Y , WωYn−1 , Q

and the similar fact for the strata (cf. [I2, II, §3, p. 598]). In the left-hand side of the spectral sequence (3.6.1), if i = n−1, then j is necessarily equal to zero, and for any s outside of [0, n−1], we have E1−s,n−1+s = 0. If i ≤ n−2, we have E1−s,i+s = 0 for any s. Hence, we have an exact sequence

δ n−1 0 0 −→ P2−n H0 Y , WωYn−1 −→ H Y , Wω −→ H0 Y (n) , Qp ,log Y ,log Q

Q

(cf. §1, (1.6.3)). The map δ arises from a residue mapping of Poincaré type; that is, it coincides with the map coming from ιr . Hence, we have

0 lim H Y , M ⊗Zp Qp = N ∞ . P2−n H0 Y , WωYn−1 r ,log Q ← − r The E1−s,n−1+s -terms that contribute to this Qp -vector space are

F =1 (1.6.1)

n−1−s 0 (s+1) H Y , W* H0 Y (s+1) , W*n−1−s (s+1) (s+1) Y Y ,log Q

Q

(0 ≤ s < n − 1).

97

ABEL-JACOBI MAPPINGS

By the Katz-Messing theorem (see [KzM]) and an argument of Colliot-Thélène, Sansuc, and Soulé, the eigenvalues of the arithmetic Frobenius operator on these vector spaces are different from 1 (cf. [CTSS, p. 784]). Hence we have

H1 (F, N ∞ ) H1 F, P2−n H0 Y , Wωn−1 = 0. Y ,log Q

(3.6.2)

This completes the proof. 4. Number ﬁeld case. The general notation is as we ﬁxed in §1 (1.1)–(1.3). (4.1) Let X be a proper smooth geometrically connected variety over a number ﬁeld k, and let l be an arbitrary prime number. For a positive integer n, let Hn+1 (X, l) denote the (n + 1)-st étale cohomology group of X with Ql /Zl (n)-coefﬁcients. This group has two canonical ﬁltrations: one is the coniveau ﬁltration N• (cf. §1 (1.3)), and the other is the Hochschild-Serre ﬁltration F• associated with the covering X := X ⊗k k sep → X (see (4.2.1) below). We deﬁne the subgroup H n (X, l) ⊂ Hn+1 (X, l) to be the intersection of N1 Hn+1 (X, l) and the maximal divisible subgroup of F2 Hn+1 (X, l). In the following, n is an arbitrary integer at least 2. In this section, we prove the following result, using §2, Theorem 2.1. Theorem 4.1. Let k be a number ﬁeld, and let X be a proper smooth geometrically connected k-variety. Then the group H n (X, l) is ﬁnite for any l and trivial for almost all l. See Remark 4.3 below for the case n = 1, and see also Proposition A.5(1) below for a consequence of Theorem 4.1. The case n = 2 was originally proved by Salberger (see [Sal, main lemma 3.9]). The outline of the proof of “Theorem 2.1⇒Theorem 4.1” is the same as his argument. However, because the details are slightly different (see the p-adic part of Lemma 2.4), we include the proof here. Remark 4.2. We do not know, a priori, whether the group H n (X, l) is ﬁnite or not. For example, for a Tate curve E over Qp , we have F2 H3 (E, l) Ql /Zl , and N1 H3 (E, l) Qp× ⊗Ql /Zl (cf. [LS, (2-9)], [Sai]). Therefore, neither N1 H3 (E, l) nor F2 H3 (E, l) is ﬁnite. Nevertheless, it turns out that the group H 2 (E, l) is trivial for any l in this case (see [Sal, Lemma 2.5]; see also [Ras1, p. 189, Claim]). Furthermore, for a projective smooth curve C over a totally imaginary number ﬁeld that has a rational point, the group H 2 (C, l) is trivial for any prime number l (see [Ras1, proof of Theorem 3.7]). (4.2) We begin the proof of Theorem 4.1. First, we assume that k is totally imaginary. Then the Galois cohomological dimension of k is two (see [T1, Theorem 3.1]). For a prime number l, we let αln denote the composite map f g αln : H2 k, Hn−1 X, Ql /Zl (n) −→ Hn+1 X, Ql /Zl (n) −→ Hn+1 k(X), Ql /Zl (n) .

98

KANETOMO SATO

Here Ker(g) N1 Hn+1 (X, Ql /Zl (n)) (cf. (1.3)), and the map f comes from the Hochschild-Serre spectral sequence p,q (4.2.1) E2 = Hp k, Hq X, Ql /Zl (n) ⇒ Hp+q X, Ql /Zl (n) and the fact that cdl (k) = 2. When we write F• for the resulting ﬁltration from this spectral sequence, F2 Hn+1 (X, Ql /Zl (n)) is nothing other than the image of f . Further the quotient of H2 (k, Hn−1 (X, Ql /Zl (n))) by the maximal l-divisible subgroup has a ﬁnite exponent since cd(k) = 2. Therefore the group H n (X, l) is written as

H n (X, l) Im Ker αln |H2 (k,Hn−1 (X,Ql /Zl (n)))l- Div −→ Hn+1 X, Ql /Zl (n) . (4.2.2) We prove that Ker(αln |H2 (k,Hn−1 (X,Ql /Zl (n)))l- Div ) is ﬁnite for any l and trivial for almost all l. (4.3) For a ﬁnite place p of k, we write kp for the completion of k at p, and αpn,l for the composite H2 kp, Hn−1 X, Ql /Zl (n) −→Hn+1 Xkp , Ql /Zl (n) −→ Hn+1 kp(X), Ql /Zl (n) . Here we have identiﬁed the étale cohomology groups of X ⊗k k with that of X ⊗k kp by the proper base change theorem. We deduce the ﬁniteness of Ker(αln |H2 (k,Hn−1 (X,Ql /Zl (n)))l- Div ) from that of Ker(αpn,l ) by a Hasse principle of Jannsen (see [J2, Theorem 3]). We consider a commutative diagram H2 k, Hn−1 (X)

/ Hn+1 k(X)

αln

res

2 n−1 H kp, H (X)

p :ﬁnite

/

res

H

p :ﬁnite

n+1

kp(X) .

(4.3.1)

Here we have omitted the coefﬁcient Ql /Zl (n). The bottom horizontal map is the composite map

⊕αpn ,l n+1 ⊂ n+1 H2 kp, Hn−1 (X) −−−−→ H H kp(X) −→ kp(X) .

p :ﬁnite

p :ﬁnite

p :ﬁnite

The left vertical arrow in (4.3.1) has ﬁnite kernel and cokernel, and further, if Hn−1 (X) is divisible, this map is bijective (see [J2, Theorem 3]). The group Hn−1 (X, Ql /Zl (n)) is divisible for almost all primes l, because the cotorsion is isomorphic to the l-primary n (X(C)an , Z), and this group is torsion part of the singular cohomology group Hsing

ABEL-JACOBI MAPPINGS

99

ﬁnitely generated. Therefore, Theorem 4.1 is a consequence of the following two claims. (1) For any place p where X has potentially good reduction and any prime l, we have H2 kp, Hn−1 X, Ql /Zl (n) l- Div = 0.

(2) For any place p, αpn,l has ﬁnite kernel for any l and is injective for almost all l. The claim (1) follows from Lemma 2.4, and (2) follows from Theorem 2.1. Finally, in the case where k has a real place, we also reduce Theorem 4.1 to (1) and (2). We apply the same argument as in (2.2) and (2.3) if l = 2. As for the case l = 2, F2 Hn+1 (X) is isomorphic to H2 k, Hn−1 X, Q2 /Z2 (n)

modulo ﬁnite 2-primary torsion (see [T1, Theorem 3.1]). Remark 4.3. For a smooth variety X over a ﬁeld k, H 1 (X, l) is trivial for any prime number l = ch(k) if X has a k-rational point x0 . One can show this by a standard purity for Brauer groups (see [G, §6]) and the Lefschetz argument in (3.1) (replace X1 by the closed point x0 ). Hence in general, H 1 (X, l) has a ﬁnite exponent for any prime l = ch(k) and is trivial for almost all primes by a standard norm argument. 5. Image of secondary regulator maps. The aim of this section is to control the image of secondary regulator maps of varieties over a global ﬁeld. The main results of this section are stated in (5.2) below. We use the notation in §1 (1.1)–(1.3), (1.7), and (1.8). (5.1) In this paragraph, we deﬁne l-adic regulator maps, l-adic Abel-Jacobi mappings (5.1.2), and secondary regulator maps (5.1.3) (see also [Ras2, §2]). Let k be a global ﬁeld, and let X be a proper smooth geometrically connected variety over k. For these X and k, we ﬁx U0 and X0 as in (1.8). We ﬁx a prime number l different from ch(k). Replacing U0 by a suitable nonempty open subset if necessary, we may assume that l is invertible on U0 . Let i and n be positive integers with i < 2n. For both ∗ ∈ {ind, cont}, we deﬁne the l-adic regulator map: i i regi,n l,∗ : Hᏹ X, Z(n) Q −→ H∗ X, Ql (n) . l

i (X, Z(n)) denotes the higher Chow group CHn (X, 2n − i) deﬁned by Bloch Here Hᏹ (see [B3]). For ∗ = cont, we deﬁne regi,n l,cont by the higher cycle class map (see [B2, i,n §4]). For ∗ = ind, we deﬁne regl,ind by the limit of the higher cycle class maps of XU := X0 ×U0 U (U ⊂ U0 open): i i Hᏹ X, Z(n) Q lim CHn (XU , 2n − i)Ql −→ Hind (5.1.1) X, Ql (n) . − → l U ⊂U0 open

100

KANETOMO SATO

Here, in the case (N), we put CHn (XU , 2n − i) := CHdim XU −n (XU , 2n − i), the higher Chow homology group deﬁned by Levine (see [Le1, §0.6]), and dim XU denotes the Krull dimension of XU (see [Le2, §12.3] for the deﬁnition of higher cycle class maps). We deﬁne l-adic Abel-Jacobi mappings. We write F• ⊂ H∗i (X, Ql (n)) (∗ ∈ {ind, cont}) for the Hochschild-Serre ﬁltration associated with the covering X → X. By Lemma 1.1 in (1.8), for each ∗ ∈ {ind, cont}, we have F3 H∗i X, Ql (n) = 0, gr 0F H∗i X, Ql (n) = 0, and

p p i−p gr F H∗i X, Ql (n) H∗ k, He´ t X, Ql (n)

i,n for p = 1, 2. We write aji,n l,∗ for the following map induced by regl,∗ :

i−1 i 1 X, Ql (n) , aji,n l,∗ : Hᏹ X, Z(n) Q −→ H∗ k, He´ t l

(5.1.2)

which we call the l-adic Abel-Jacobi mapping. We put Tli,n (X) := Ker(aji,n l,ind ).

Then we have Tli,n (X) Ker(aji,n l,cont ), because we have the following commutative diagram: i X, Z(n) Hᏹ PQPlP i,n n n aji,n PPPajl,cont n l,ind nn PPP nn n PPP n nn n w ' inf 1 k, Hi−1 X, Q (n) / H1 k, Hi−1 X, Ql (n) Hind l cont e´ t e´ t and the bottom horizontal arrow is injective. Finally, we have the secondary class map for both ∗ ∈ {ind, cont}, i,n i,n i−2 2 X, Ql (n) , (5.1.3) 2 regl,∗ : Tl (X) −→ H∗ k, He´ t which we call the secondary l-adic regulator map in the following. (5.2) We put S0 := C \ U0 ∪ {inﬁnite places}. We ﬁx a positive integer n, and for an integer q, we put q V q := He´ t X, Ql (n) . The main result of this section is the following, which we prove in (5.3) below. Theorem 5.1. (1) If i < 2n, then the canonical map 2 2 HGal GS0 , V i−2 −→ Hind k, V i−2

ABEL-JACOBI MAPPINGS

101

is surjective. Consequently, the image of the secondary map 2 regi,n l,ind is ﬁnite-dimensional over Ql . Further in the case (F), if X has good reduction everywhere, the map i,n 2 regl,ind is trivial. (2) Assume n ≥ 2. Then Im(2 regn+1,n l,ind ) is contained in the image of the following 2 n−1 Ql -vector space in Hind (k, V ): 2 2 n−1 2 n−1 −→ ⊕ HGal kp, V . Ker aS0 : HGal GS0 , V p∈S0

Here for a place p of k, kp denotes the completion at p, and the map aS20 is the restriction map. i,n Because the map 2 regi,n l,cont factors through 2 regl,ind , these claims on the image of

i,n 2 regl,ind

also hold for 2 regi,n l,cont .

Remark 5.2. Theorem 5.1(1) is standard and is a consequence of [D3, 3.2.3, 3.3.9] (see the proof below). In the case i = 2n, it is known that Im(reg2n,n l,∗ ) is ﬁnitedimensional over Ql (cf. [Ras2, 2.5]). When we replace V n−1 by V i−2 , the statement corresponding to Theorem 5.1(2) also holds for 2 regi,n l,∗ if i ≤ n. In fact, in this case, 2 (k , V i−2 ) = 0 for any place p (see Remark 2.11). we have HGal p Corollary 5.3. Let k be a global ﬁeld, and let X be a proper smooth variety over k. If 2n − i > 1, then the image of the regulator map regi,n l,∗ is ﬁnite-dimensional over Ql for both ∗ = ind and cont. Corollary 5.4. Let k be the case (F), and let X be a proper smooth variety over is zero for both ∗ = ind and cont. k, and n ≥ 2. Then the secondary map 2 regn+1,n l,∗ n+1,n n+1,n (X). In other words, we have Ker(regn+1,n l,ind ) Ker(regl,cont ) Tl

These corollaries immediately follow from Theorem 5.1 and results of Jannsen (see [J2, §2 Lemma 4, §6 Theorem 4]). (5.3) We prove Theorem 5.1 (see also [Ras2, Proposition 3.6]). First, we prove Theorem 5.1(1). For any open subset U ⊂ U0 , we have the localization exact sequence σ 2 2 2 · · · −→ HGal GS0 , V i−2 −→ HGal GSU , V i−2 −→ ⊕ HGal kp, V i−2 −→· · · , p∈SU \S0

and the inﬂation map σ is surjective by Deligne’s proof of the Weil conjecture (cf. 2 (k , V i−2 ) = 0 for any p ∈ S \ S , since i < 2n. Therefore, the Lemma 2.4): HGal p U 0 2 (G , V i−2 ) → H2 (k, V i−2 ) is surjective. Further in the case canonical map HGal S0 ind 2 (G , V i−2 ) = 0 by theorems (F), if X has good reduction everywhere, we have HGal S0 of Deligne (see [D3, 3.2.3, 3.3.9]). We prove Theorem 5.1(2). For a place p of k, we deﬁne the regulator map regn+1,n l,´et n+1,n of Xkp := X ⊗k kp by regl,cont (cf. (5.1)). By Theorem 5.1(1), it sufﬁces to show that

102

KANETOMO SATO

n+1,n the right vertical map 2 regl,´ in the following commutative diagram is zero for any et p ∈ S0 .

n+1,n Xkp / ⊕ Tl

Tln+1,n (X)

p∈S0

⊕2 regn+1,n l,´et

n+1,n 2 regl,ind

2 HGal

GS0

, V n−1

surj.

2 kp, V n−1 / ⊕ HGal

/ H2 k, V n−1 ind

p∈S0

n+1,n Here Tln+1,n (Xkp ) denotes the kernel of the following map induced by regl,´ : et

n+1,n n+1 regl,´ : Hᏹ Xkp , Z(n) Q −→ Hen+1 Xkp , Ql (n) /F2 Hei´ t Xkp , Ql (n) . ´t et l

n+1,n is the secondary regulator map of Xkp , and we have used The right vertical 2 regl,´ et the fact that 2 HGal kp, V n−1 F2 Hen+1 (5.3.1) Xkp , Ql (n) ´t

for any p ∈ S0 (cf. Corollary 2.2; see also [D1] and [D3, (1.1.2)]). n+1,n of Xkp is trivial for any p ∈ S0 . Fix a p ∈ S0 . If We prove that the map 2 regl,´ et 2 (k , V n−1 ) = 0 by X has potentially good reduction at p, our problem is clear: HGal p n+1,n (Xkp ) is contained in the Lemma 2.4. Otherwise, we claim that the image of Tl kernel of the composite map ! 2 lim Hen+1 kp, V n−1 −→ Hen+1 Xkp , Ql (n) −→ X , Ql (n) , HGal ´t ´t − → ! X ⊂Xkp open

which is injective by Corollary 2.2. We prove this claim. In fact, the composite map regn+1,n l,´et n+1 Hᏹ Xkp , Z(n) Q −−−−−→ Hen+1 Xkp , Ql (n) −→ ´t l

lim − →

X! ⊂Xkp open

! Hen+1 X , Ql (n) ´t

vanishes by the fact that regulator maps are compatible with restriction maps and by the deﬁnition of higher Chow groups: n+1 ! lim Hᏹ X , Z(n) CHn Spec kp(X), n − 1 = 0. − → ! X ⊂Xkp open

This completes the proof. 6. Kernel of Abel-Jacobi mappings. In this section, we prove Theorems 0.1 and 0.2.

ABEL-JACOBI MAPPINGS

103

(6.1) Let k be a function ﬁeld in one variable over a ﬁnite ﬁeld (case (F)), let l be a prime number invertible in k, and let X be a proper smooth geometrically connected k-variety. The notation remains as in §5. For a smooth variety V over a ﬁeld F , we i (V , Z(n)) for the higher Chow group CHn (V , 2n − i). First, let us recall the write Hᏹ Bloch-Kato conjecture (see [BK1, §5]). Conjecture 6.1 (Bloch-Kato). Let F be a ﬁeld, and let n be a positive integer. Then for any prime number l different from ch(F ), the Galois symbol mapping n F, µ⊗n KnM (F )/ l −→ HGal l is bijective. The deﬁnition of Galois symbol mappings is given in [T2]. The case n = 0 is clear, the case n = 1 follows from Theorem 90 of Hilbert, and the case n = 2 is proved by Merkur’ev and Suslin (see [MS]). In his recent work [Vo], Voevodsky proved the case where l = 2 and n is arbitrary. Now we prove Theorem 0.2. Theorem 6.2 (Theorem 0.2). Let n be an arbitrary integer at least 2. We assume Conjecture 6.1 in degree n for all ﬁnitely generated ﬁelds F over Fq . Then we have n+1 Hᏹ XU , Z(n) ⊗ Zl l- Div ⊗ Q . Tln+1,n (X) lim − → U ⊂U0 open

n+1 (XU , Z(n))⊗Zl are ﬁnitely generated over Zl , then Consequently, if the groups Hᏹ n+1,n the l-adic Abel-Jacobi mapping ajl,ind is injective.

Proof. By Corollary 5.4 for ∗ = ind, it sufﬁces to show that for any open U ⊂ U0 , the kernel of the cycle class map n+1 Hᏹ XU , Z(n) ⊗ Zl −→ Hen+1 XU , Zl (n) ´t n+1 (XU , Z(n)) ⊗ Zl )l- Div . is isomorphic to (Hᏹ Fix a nonempty open subset U ⊂ U0 . By Conjecture 6.1 and [GeLe, Theorem 1.1] (see also [BK1, Lemma 5.13]), we have an exact sequence n+1 · · · −→ Hen´ t XU , Z/ l ν Z(n) −→ Hᏹ XU , Z(n) (6.1.1) ×l ν n+1 ν −−→ Hᏹ XU , Z(n) −→ Hen+1 X , Z/ l Z(n) U ´t

for any ν ∈ N, where the last map is the cycle class map (cf. (A.2) below). Therefore we have n+1 Hᏹ XU , Z(n) ⊗ Zl l- div Ker regn+1,n l,XU by (1.1.1). Further, we have n+1 n+1 Hᏹ XU , Z(n) ⊗ Zl l- div = Hᏹ XU , Z(n) ⊗ Zl l- Div ,

104

KANETOMO SATO

n+1 because the projective system {l ν Hᏹ (XU , Z(n))}ν consists of ﬁnite groups by the exact sequence (6.1.1). This completes the proof.

(6.2) We prove Theorem 0.1. Let us recall the situation (0.1). Since Z and X belong to A(Fq ), XZ also belongs to A(Fq ) (cf. [So1, §3.3.1]). Hence, the higher 3 (X , Z(2)) is ﬁnite by [So1, Théorème 4], [GrSu, 4.19], and [La, Chow group Hᏹ Z (2.8)]. We claim that the Abel-Jacobi mapping aj3,2 l,ind (cf. (5.1)) of X/k is injective for any prime number l invertible in k. In fact, by the Merkur’ev-Suslin theorem (see [MS]) and Theorem 6.2, it sufﬁces to show the following. 3 (π −1 (U ), Proposition 6.3. For an arbitrary open subset U of P1 , the group Hᏹ Z(2)) is a ﬁnitely generated abelian group modulo ﬁnite exponent ch(Fq )-primary 3 (π −1 (U ), Z(2)) ⊗ Z is ﬁnitely generated over torsion. Consequently, the group Hᏹ l Zl for any open U of P1 and any prime number l different from ch(Fq ).

Proof. We ﬁx a nonempty open subset U ⊂ P1 . Proposition 6.3 follows from the localization sequence of higher Chow groups (see [B3, Theorem 3.1], [B4]): −1 3 3 π (U ), Z(2) XZ , Z(2) −→ Hᏹ · · · −→ Hᏹ CH1 π −1 (p) −→ · · · , −→

(6.2.1)

p∈P1 \U 3 (X , Z(2)), and [CTSS, p. 788, Corollaire 7]. the ﬁniteness of Hᏹ Z

Remark 6.4. (1) If the ﬁbration π in (0.1) comes from a Lefschetz pencil, the proof of Theorem 0.1 is simpler. In fact, the triviality of secondary regulator maps over local ﬁelds follows from the local Lefschetz theory (cf. [D2, (4.3.2)]) and Deligne’s proof of the Weil conjecture (see [D3, 3.3.1]) instead of Corollary 2.2. (2) In the situation (0.1), the map 5 1 4 aj5,3 l,ind : Hᏹ X, Z(3) Q −→ Hind k, He´ t X, Ql (3) l

is, via trace maps, isomorphic to the following map coming from the Kummer theory: 1 k × ⊗ Ql −→ Hind k, Ql (1) , and hence bijective (see also (A.3) below). Here the trace map for the left-hand sides means the composite map f∗ 5 1 Hᏹ X, Z(3) −−→ Hᏹ Spec k, Z(1) −→ k ×

(6.2.2)

(we wrote f for the structure morphism X → Spec k; see [B3, §6] for the last iso5 (X , Z(3)) has ﬁnite exponent by [La, (2.8)] and morphism). In fact, the group Hᏹ Z

ABEL-JACOBI MAPPINGS

105

[So1, Théorème 4]. Since the ﬁbers of π : XZ → P1 are connected (cf. [Ha, Chap. III, 7.9]), the Chow homology group CH0 (π −1 (p)) is a ﬁnitely generated abelian group of rank 1 for any closed point p ∈ C (cf. [CTSS, p. 788, Corollaire 7], and [J3, (12.10.2), Theorem 12.12]). Hence, the kernel of the trace map (6.2.2) is torsion by the localization sequence for higher Chow groups (see [B3, §3], [B4]) and by the argument in [Ras1, Lemma 2.2]. Consequently, the map (6.2.2) is injective modulo ! torsion. Furthermore, when we ﬁx an embedding X ⊂ PN k and take a hyperplane section H ⊂ X, the composite map f∗ ?.H 2 f∗ 1 1 5 1 Hᏹ Spec k, Z(1) −−→ Hᏹ X, Z(1) −−−→ Hᏹ X, Z(3) −−→ Hᏹ Spec k, Z(1) coincides with multiplication by the integer degX (H 2 ), by the projection formula (see [B3, (5.8)]). Thus the trace map (6.2.2) is also surjective modulo torsion, and the map aj5,3 l,ind is bijective by the functoriality of cycle class maps. In particular, we have the hard Lefschetz isomorphism 1 5 ?.H 2 : Hᏹ X, Z(1) Q −→ Hᏹ X, Z(3) Q . Appendix: Interpretation as a motivic cohomology (A.1) In this appendix, we observe the relation between the ﬁniteness of the group H n (cf. §4) and the injectivity of the l-adic Abel-Jacobi mapping ajln+1,n by the motivic complex Zᏹ (n). We use the notation in §1 (1.1)–(1.3), (1.7), and (1.8). (A.2) Let X be a smooth variety over a ﬁeld k, and let n be a positive integer. We write Zᏹ (n) for the Zariski sheaﬁﬁcation of the complex of presheaves U & → Zn (U, •)[−2n]. Here Zn (U, •) denotes the complex of algebraic cycles computing Bloch’s higher Chow groups of U (see [B3]). The complex Zᏹ (n) is a candidate for the motivic complex of Beilinson and Lichtenbaum (see [Li1, §5]). By results of Bloch, for any integers i and n, we have the isomorphisms (n) K2n−i (X)Q CHn (X, 2n − i)Q HiZar X, Zᏹ (n) Q . (A.2.1) Here the ﬁrst map is the Riemann-Roch mapping (see [B3, §7, §9]), and the second isomorphism is due to [B3, Theorem 3.2] and [B4]. The superscript (n) in the lefthand side denotes the nth Adams eigenspace: (n) Km (X)Q := x ∈ Km (X)Q ; ψ s (x) = s n · x for any s ∈ Z \ {0} , and ψ s (s ∈ Z\{0}) denotes the Adams operator (see [So2]). For any positive integer r invertible in k, the complex Zᏹ (n) is expected to ﬁt into the following distinguished triangle in the derived category of Zariski sheaves on X: ×r

Zᏹ (n) −−→ Zᏹ (n) −→ τ≤n Rα∗ µ⊗n r −→ Zᏹ (n)[1].

(A.2.2)

106

KANETOMO SATO

Here α : Xe´ t → XZar denotes the natural continuous map of small sites, and the second map is the cycle class map constructed in [GeLe, §3.9]. i (X, Z(n)) := CHn (X, 2n − i) Hi (X, Z (n)). We put Hᏹ ᏹ Zar Remark A.1. By recent works of Suslin and Voevodsky [SuVo] and Geisser and Levine [GeLe], the complex Zᏹ (n) ﬁts into the distinguished triangle (A.2.2), when we assume the Bloch-Kato conjecture in degree n (see Conjecture 6.1) for all ﬁnitely generated ﬁelds over k. In particular, Zᏹ (n) ﬁts into (A.2.2) if n = 2, by the Merkur’evSuslin theorem (see [MS]; see also [Li2], [Li3]), or if ch(k) = 2 and r = 2ν for some ν > 0 by the Voevodsky theorem (see [Vo]). (A.3) Let k be a global ﬁeld, and let l be a prime number invertible in k. We assume that X is proper smooth over k. In the following, we are concerned with the l-adic Abel-Jacobi map i−1 i 1 aji,n X, Ql (n) (A.3.1) l,ind : Hᏹ X, Z(n) hom ⊗ Ql −→ Hind k, He´ t i (X, Z(n)) into Hi (X, Z (n)) (cf. (5.1)). induced by the higher cycle class map of Hᏹ l ind Here we put i G i Hᏹ X, Z(n) hom := Ker Hᏹ X, Z(n) −→ Hei´ t X, Zl (n) k .

In the case where k is an algebraic number ﬁeld (case (N)), we expect that the map aji,n l should be injective for any i and n by the philosophy of mixed motives and Deligne’s conjecture (cf. [J3, §11]); the realization functor for regular schemes of ﬁnite type over Spec Z is faithful (cf. [J3, 11.6]), and there are no nontrivial motivic 2-extensions over k. After Beilinson [Be], Jannsen conjectures that aji,n l,ind is bijective if i ≤ n (see [J2], [J3, §13]). Bloch and Kato conjecture that the image of aji,n l,ind (for 1 1 general i and n) coincides with the subspace Hg ⊂ Hind deﬁned by using the ring Bcrys (for Ql ) of Fontaine (cf. [BK2, §3, §5]). On the other hand, in the case where k is a function ﬁeld in one variable over a ﬁnite ﬁeld (case (F)), Jannsen conjectures that the map aji,n l,ind is bijective (see [J3, 12.18]). (A.4)

In the following, we observe the case i = n + 1.

Lemma A.2. We have a canonical isomorphism n+1 X, τ≤n Rα∗ Ql /Zl (n) −→ N1 Hen+1 X, Ql /Zl (n) . HZar ´t Here N• denotes the coniveau ﬁltration. Proof. By the deﬁnition of the truncation τ≤n , we have an exact sequence n+1 X, τ≤n Rα∗ Ql /Zl (n) 0 −→ HZar −→ Hen+1 X, Ql /Zl (n) ´t 0 X, R n+1 α∗ Ql /Zl (n) . −→ HZar

107

ABEL-JACOBI MAPPINGS

The last group is a subgroup of Hen+1 (k(X), Ql /Zl (n)) by the Bloch-Ogus theorem ´t (see [BO, Theorem 4.2]). To advance our observation, we introduce an analogue of Bass’s conjecture (see [Ba, §9]) on algebraic K-groups. Conjecture A.3. For a smooth variety Y over a ﬁnite ﬁeld and any integers i i (Y, Z(n)) is a ﬁnitely generated abelian group. and n, the group Hᏹ We use the following consequence of Conjecture A.3. Lemma A.4 (assuming Conjecture A.3). Let k be the case (F), and let X be a i (X, Z(n)) contains no nontrivial l-divisible smooth variety over k. Then the group Hᏹ subgroup for any integers i and n and for any prime number l. Proof. Let C be the model of k (cf. (1.8)). We take a nonempty open subset U0 ⊂ C, and a smooth U0 -scheme X0 whose generic ﬁber is X. For a closed point p ∈ (U0 )1 , we write Yp for the ﬁber X0 ×U0 Spec κ(p). Then we have the localization exact sequence of higher Chow groups (see [B3, Theorem 3.1], [B4]): a i−1 i i X0 , Z(n) −→ Hᏹ Hᏹ X, Z(n) −→ Yp, Z(n − 1) −→ · · · , · · · −→ Hᏹ p∈U0

and hence an exact sequence i−1 i 0 −→ Im(a) −→ Hᏹ Hᏹ X, Z(n) −→ Yp, Z(n − 1) . p∈U0

It sufﬁces to check −1 −1 i−1 Hom Z l , Im(a) = Hom Z l , Hᏹ Yp, Z(n − 1) = 0 p∈U0

assuming Conjecture A.3 (cf. (1.1)). In fact, by Conjecture A.3, Im(a) is ﬁnitely i−1 generated, and we have Hom(Z[l −1 ], Im(a)) = 0. Further, since Hᏹ (Yp, Z(n − 1)) 1 is also ﬁnitely generated for any p ∈ (U0 ) by Conjecture A.3, we have i−1 i−1 Hom Z l −1 , Hᏹ Yp, Z(n − 1) ⊂ Hom Z l −1 , Hᏹ Yp, Z(n − 1) p∈U0

p∈U0

= 0. This completes the proof.

p∈U0

i−1 Hom Z l −1 , Hᏹ Yp, Z(n − 1)

108

KANETOMO SATO

We observe what Theorem 4.1 implies, using the motivic complex Zᏹ (n). We also prove that Theorem 0.2 implies that H n (X, l)l- Div = 0, assuming the exact triangle (A.2.2) and Conjecture A.3 (see §4 for the deﬁnition of the group H n (X, l)). Proposition A.5. Let X be a proper smooth geometrically connected variety over a global ﬁeld k, and let l be a prime number invertible in k. We assume that the complex Zᏹ (n) ﬁts into the distinguished triangle (A.2.2). (1) Let k be the case (N ), and n ≥ 2. Then we have ! n+1,n n+1 Hᏹ X, Z(n) ⊗ Zl l- Div ⊗ Q. Ker ajl,ind Here ! means the quotient by the torsion part. (2) Let k be the case (F ), and n ≥ 3. We assume Conjecture A.3. Then we have H n (X, l)l- Div = 0. Proof. For simplicity, we assume that the group Hen´ t (X, Ql /Zl (n)) is divisible. The general case follows from the same proof as below considered in the category of the abelian groups modulo ﬁnite exponent groups. We put ! n+1 X, Z(n) , T := Hᏹ V := T ⊗ Q, A := T ⊗ Q/Z. Here ! means the quotient by the torsion part. By (A.2.2) and Lemma A.2, there is a natural inclusion A{l} ⊂ N1 Hen+1 X, Ql /Zl (n) l- Div . (A.4.1) ´t Further we put ! ! 1 Gk , Hen´ t X, Zl (n) , Tl := HGal 1 Gk , Hen´ t X, Ql (n) , Vl := Tl ⊗ Q HGal 1 Gk , Hen´ t X, Ql /Zl (n) . Al := HGal We prove the following claim. Claim. (a) There is a commutative diagram of exact sequences 0

/ T ⊗ Zl f

0

/ Tl

/ V ⊗ Q Ql ajn+1,n l

/ Vl

/ A{l} g

/ Al .

(b) There is a natural inclusion Ker(f ) ⊗ Q/Z ⊂ H n (X, l).

/0

(A.4.2)

ABEL-JACOBI MAPPINGS

109

Proof of Claim ⇒ Proposition A.5(1). By Theorem 4.1, we have H n (X, l)l- Div = 0 and Claim (b) implies that Ker(f ) ⊗ Q/Z = 0. Since Ker(f ) is torsion-free, Ker(f ) is l-divisible. Further, Tl contains no nontrivial l-divisible subgroup (see 1 (G , Hn (X, Z (n))! ) is ﬁnite. [T2, 2.1]) since the torsion part of the group HGal k l e´ t n+1,n Therefore, we have Ker(f ) (T ⊗ Zl )l- Div and Ker(ajl ) Ker(f ) ⊗ Q n+1 (Hᏹ (X, Z(n))! ⊗ Zl )l- Div ⊗ Q. Proof of Claim and Proposition A.5(2). (a) Clearly, the horizontal rows in (A.4.2) are exact. First, we deﬁne the map g. Under the inclusion (A.4.1), the group A{l} is contained in the subgroup F1 Hen+1 (X, Ql /Zl (n)), since A{l} is divisible and the ´t groups G gr 0F Hen+1 X, Ql /Zl (n) k X, Ql /Zl (n) ⊂ Hen+1 ´t ´t are ﬁnite by Lemma 1.1. We deﬁne the map g by the composite of the natural inclusion and the projection map: X, Ql /Zl (n) −→ gr 1F Hen+1 A{l} .→ F1 Hen+1 X, Ql /Zl (n) ⊂ Al . ´t ´t f is deﬁned by the map induced by the right square in (A.4.2). We prove Claim (b). We put F2 := F2 Hen+1 X, Ql /Zl (n) , X, Ql /Zl (n) N1 := N1 Hen+1 ´t ´t for simplicity. We make the following two points: • Ker(g) is written as (A.4.1) 1 N l- Div ∩ F2 .

Ker(g) A{l} ∩ F2 ⊂

(A.4.3)

• We have

H n (X, l)l- Div = N1 ∩ F2 l- Div l- Div N1 l- Div ∩ F2 l- Div .

Therefore, there is a natural inclusion Ker(g)l- Div ⊂ H n (X, l)l- Div . By applying the snake lemma to the diagram (A.4.2), we have an exact sequence 0 −→ Ker(f ) −→ Ker(f ) ⊗ Q −→ Ker(g) −→ Coker(f ) , (A.4.4) and

Ker(f ) ⊗ Q/Z ⊂ Ker(g)l- Div ⊂ H n (X, l)l- Div .

This shows Claim (b). n+2 We prove Proposition A.5(2). By Lemma A.4, we have HZar (X, Zᏹ (n))l- Div = 0. Hence, by (A.2.2) and Lemma A.2, we have A{l} = T ⊗ Ql /Zl N1 l- Div ,

110 and by (A.4.3),

KANETOMO SATO

H n (X, l)l- Div Ker(g)l- Div .

Further, the group Tl is ﬁnitely generated as a Zl -module if n ≥ 3 (see [J2, §2, Lemma 4]). Therefore we have H n (X, l)l- Div Ker(f ) ⊗ Q/Z by the exact sequence (A.4.4). Since ajln+1,n is injective by (A.2.2), Theorem 0.2 (cf. §6), and Conjecture A.3, we have H n (X, l)l- Div = 0. References H. Bass, “Some problems in ‘classical’ algebraic K-theory” in Algebraic K-Theory, Vol. II: “Classical” Algebraic K-Theory and Connections with Arithmetic (Seattle, Wash., 1972), ed. H. Bass, Lecture Notes in Math. 342, Springer, Berlin, 1973, 3–73. [Be] A. Beilinson, Higher regulators and values of L-functions, J. Soviet Math. 30 (1985), 2036–2070. [B1] S. Bloch, Algebraic K-theory and class ﬁeld theory for arithmetic surfaces, Ann. of Math. (2) 114 (1981), 229–265. , “Algebraic cycles and the Beilinson conjectures” in The Lefschetz Centennial [B2] Conference (Mexico City, 1984), Part I, Contemp. Math. 58, Amer. Math. Soc., Providence, 1986, 65–79. , Algebraic cycles and higher K-theory, Adv. Math. 61 (1986), 267–304. [B3] [B4] , The moving lemma for higher Chow groups, J. Algebraic Geom. 3 (1994), 537– 568. [BK1] S. Bloch and K. Kato, p-adic étale cohomology, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 107–152. [BK2] , “L-functions and Tamagawa numbers of motives” in The Grothendieck Festschrift, Vol. I, Progr. Math. 86, Birkhäuser, Boston, 1990, 333–400. [BO] S. Bloch and A. Ogus, Gersten’s conjecture and the homology of schemes, Ann. Sci. École Norm. Sup. (4) 7 (1974), 181–201. [CTSS] J.-L. Colliot-Thélène, J.-J. Sansuc, and C. Soulé, Torsion dans le groupe de Chow de codimension deux, Duke Math. J. 50 (1983), 763–801. [D1] P. Deligne, Théorème de Lefschetz et critères de dégénéresence de suites spectrales, Inst. Hautes Études Sci. Publ. Math. 35 (1968), 259–278. , La conjecture de Weil, I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–308. [D2] [D3] , La conjecture de Weil, II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252. [SGA4 21 ] P. Deligne, J.-F. Boutot, A. Grothendieck, L. Illusie, and J.-L. Verdier, Cohomologie étale, Séminaire de Géométrie Algébrique du Bois-Marie (SGA 4 21 ), Lecture Notes in Math. 569, Springer, Berlin, 1977. [Fa] G. Faltings, “Crystalline cohomology and p-adic Galois-representations” in Algebraic Analysis, Geometry, and Number Theory (Baltimore, 1988), ed. J.-I. Igusa, Johns Hopkins Univ. Press, Baltimore, 1989, 25–80. [FoMe] J.-M. Fontaine and W. Messing, “p-adic periods and p-adic étale cohomology” in Current Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985), ed. K. A. Ribet, Contemp. Math. 67, Amer. Math. Soc., Providence, 1987, 179–207. [Ga] O. Gabber, Sur la torsion dans la cohomologie l-adique d’une variété, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983), 179–182. [GeLe] T. Geisser and M. Levine, The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky, [Ba]

ABEL-JACOBI MAPPINGS

[Gr] [GrSu] [G] [SGA7]

[EGA4]

[Ha] [H1] [H2] [HK]

[I1] [I2]

[J1] [J2]

[J3] [dJ] [K1]

[K2] [K3]

[KoM] [KzM] [La] [LS]

111

to appear in J. Reine Angew. Math. M. Gros, Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique, Mém. Soc. Math. France (N.S.) (1985), no. 21. M. Gros and N. Suwa, Application d’Abel-Jacobi p-adique et cycles algébriques, Duke Math. J. 57 (1988), 579–613. A. Grothendieck, “Le groupe de Brauer, III: Exemples et compléments” in Dix exposés sur la cohomologie des schémas, North-Holland, Amsterdam, 1968, 88–188. A. Grothendieck, P. Deligne, and N. Katz, Groupes de monodromie en géométrie algébrique, Séminaire de Géométrie Algébrique du Bois-Marie (SGA 7), Lecture Notes in Math. 288, 340, Springer, Berlin, 1972–73. A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas, I, Inst. Hautes Études Sci. Publ. Math. 20 (1964); II, 24 (1965); III, 28 (1966); IV, 32 (1967). R. Hartshorne, Algebraic Geometry, Graduate Texts in Math. 52, Springer, New York, 1977. O. Hyodo, A note on p-adic etale cohomology in the semistable reduction case, Invent. Math. 91 (1988), 543–557. , On the de Rham-Witt complex attached to a semi-stable family, Compositio Math. 78 (1991), 241–260. O. Hyodo and K. Kato, “Semi-stable reduction and crystalline cohomology with logarithmic poles” in Périodes p-adiques (Bures-sur-Yvette, 1988), Astérisque 223, Soc. Math. France, Montrouge, 1994, 221–268. L. Illusie, Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. École Norm. Sup. (4) 12 (1979), 501–661. , “Appendix: Réduction semi-stable ordinaire, cohomologie étale p-adique et cohomologie de de Rham d’après Bloch-Kato et Hyodo” in Périodes p-adiques (Buressur-Yvette, 1988), Astérisque 223, Soc. Math. France, Montrouge, 1994, 209–220. U. Jannsen, Continuous étale cohomology, Math. Ann. 280 (1988), 207–245. , “On the l-adic cohomology of varieties over number ﬁelds and its Galois cohomology” in Galois Group over Q, Math. Sci. Res. Inst. Publ. 16, Springer, New York, 1989. , Mixed Motives and Algebraic K-Theory, Lecture Notes in Math. 1400, Springer, Berlin, 1990. A. J. de Jong, Smoothness, semi-stability, and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93. K. Kato, “Galois cohomology of complete discrete valuation ﬁelds” in Algebraic KTheory (Oberwolfach, 1980), Part II, ed. R. K. Dennis, Lecture Notes in Math. 967, Springer, Berlin, 1982, 215–238. , A Hasse principle for two-dimensional global ﬁelds (with an appendix by J.-L. Colliot-Thélène), J. Reine Angew. Math. 366 (1986), 142–183. , “Semi-stable reduction and p-adic étale cohomology” in Périodes p-adiques (Bures-sur-Yvette, 1988), Astérisque 223, Soc. Math. France, Montrouge, 1994, 269–293. K. Kato and W. Messing, Syntomic cohomology and p-adic étale cohomology, Tôhoku Math. J. (2) 44 (1992), 1–9. N. Katz and W. Messing, Some consequences of the Riemann hypothesis for varieties over ﬁnite ﬁelds, Invent. Math. 23 (1974), 73–77. S. E. Landsburg, Relative Chow groups, Illinois J. Math. 35 (1991), 618–641. A. Langer and S. Saito, Torsion zero-cycles on the self-product of a modular elliptic curve, Duke Math. J. 85 (1996), 315–357.

112 [Le1] [Le2] [Li1]

[Li2] [Li3] [MS] [Mi] [Mo] [Q]

[RZ]

[Ras1] [Ras2] [Sai] [Sal]

[Sat] [S] [So1] [So2] [SuVo] [Sz] [T1]

[T2] [Ts] [Vo]

KANETOMO SATO M. Levine, Techniques of localization in the theory of algebraic cycles, preprint, 1998. , K-theory and motivic cohomology of schemes, preprint, 1999. S. Lichtenbaum, “Values of zeta functions at nonnegative integers” in Number Theory (Noordwijkerhout, 1983), ed. H. Jager, Lecture Notes in Math. 1068, Springer, Berlin, 1984, 127–138. , The construction of weight-two arithmetic cohomology, Invent. Math. 88 (1987), 183–215. , “New results on weight-two motivic cohomology” in The Grothendieck Festschrift, Vol. III, Progr. Math. 88, Birkhäuser, Boston, 1990, 35–55. A. S. Merkur’ev and A. A. Suslin, K-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Math. USSR-Izv. 21 (1983), 307–340. J. S. Milne, Arithmetic Duality Theorems, Perspect. Math. 1, Academic Press, Boston, 1986. A. Mokrane, La suite spectrale des poids en cohomologie de Hyodo-Kato, Duke Math. J. 72 (1993), 301–337. D. Quillen, “Higher algebraic K-theory, I” in Algebraic K-Theory, Vol. I: Higher KTheories (Seattle, Wash., 1972), ed. H. Bass, Lecture Notes in Math. 341, Springer, Berlin, 1973, 85–147. M. Rapoport and T. Zink, Über die lokale Zetafunktion von Shimuravarietäten: Monodromieﬁltration und verschwindende Zyklen in ungleicher Charakteristik, Invent. Math. 68 (1982), 21–101. W. Raskind, On K1 of curves over global ﬁelds, Math. Ann. 288 (1990), 179–193. , Higher l-adic Abel-Jacobi mappings and ﬁltrations on Chow groups, Duke Math. J. 78 (1995), 33–57. S. Saito, Class ﬁeld theory for curves over local ﬁelds, J. Number Theory 21 (1985), 44–80. P. Salberger, “Torsion cycles of codimension 2 and l-adic realizations of motivic cohomology” in Séminaire de Théorie des Nombres (Paris, 1991/92), ed. S. David, Progr. Math. 116, Birkhäuser, Boston, 1993, 247–277. K. Sato, Injectivity of the torsion cycle map of codimension two of varieties over p-adic ﬁelds with semi-stable reduction, J. Reine Angew. Math. 501 (1998), 221–235. J.-P. Serre, Cohomologie galoisienne, 5th ed., Lecture Notes in Math. 5, Springer, Berlin, 1994. C. Soulé, Groupes de Chow et K-théorie de variétés sur un corps ﬁni, Math. Ann. 268 (1984), 317–345. , Opérations en K-théorie algébrique, Canad. J. Math. 37 (1985), 488–550. A. Suslin and V. Voevodsky, Bloch-Kato conjecture and motivic cohomology with ﬁnite coefﬁcients, preprint, 1999. T. Szamuely, Sur la théorie des corps de classes pour les variétés sur les corps p-adiques, preprint, 1999. J. Tate, “Duality theorems in Galois cohomology over number ﬁelds” in Proceedings of the International Congress of Mathematicians (Stockholm, 1962), Inst. Mittag-Lefﬂer, Djursholm, 1963, 288–295. , Relations between K2 and Galois cohomology, Invent. Math. 36 (1976), 257–274. T. Tsuji, p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math. 137 (1999), 233–411. V. Voevodsky, The Milnor conjecture, preprint, 1996.

Department of Mathematics, Faculty of Science, Tokyo Institute of Technology, 12-1 Oh-okayama 2-chome, Meguro-ku, Tokyo 152-8552, Japan; [email protected]

Vol. 104, No. 1

DUKE MATHEMATICAL JOURNAL

© 2000

CARLEMAN INEQUALITIES AND THE HEAT OPERATOR LUIS ESCAURIAZA

1. Introduction. The unique continuation property is best understood for secondorder elliptic operators. The classic paper by Carleman [8] established the strong unique continuation theorem for second-order elliptic operators that need not have analytic coefﬁcients. The powerful technique he used, the so-called “Carleman weighted inequality,” has played a central role in later developments. In the 1950s, Aronszajn [3] and Cordes [11] generalized Carleman’s result to higher dimensions. In recent years, this subject attracted attention from a great number of people. Many efforts have been made to relax the smoothness hypothesis on the coefﬁcients (see [7], [15], [31], [2], and [16]). Using this method, Jerison and Kenig [19] obtained the n/2 strong unique continuation property for operators of the form + V with V ∈ Lloc , n ≥ 3. Further improvements have been made in considering other classes of coefﬁcients (Fefferman-Phong class and Kato class) (see [30], [9], [12], and [24]), in extending the result to operators with ﬁrst derivative terms and variable coefﬁcients (see [32], [33], [39], and [40]), and so on (see [18], [13], [14], [23], [5], and [22] and the references therein). For second-order linear parabolic operators with time-independent coefﬁcients, the strong unique continuation property was reduced in [25] to the previously established elliptic counterparts. In particular, it is shown in [25] that if u is a solution of u + ∂t u + V (x)u = 0 (x0 , t0 ) ∈ ST , and

Br (x0 )

in ST = × (0, T ), V ∈ L(n+1)/2 ( ),

u2 (x, t0 ) dx ≤ CN r N

for any integer N . Then u(x, t0 ) ≡ 0 for x ∈ , and in addition, assuming that u = 0 on ∂ × (0, T ), then u ≡ 0 in ST . The reduction from time-independent parabolic equations to elliptic equations, a basic technique used in [25], relies on a representation formula for solutions of parabolic equations in terms of eigenfunctions of the corresponding elliptic operator, and therefore cannot be applied to more general equations with time-dependent coefﬁcients (for the weak unique continuation, see also [17], [41]). Received 4 May 1999. 2000 Mathematics Subject Classiﬁcation. Primary 35K10; Secondary 35B05. Author’s work supported by Basque Government grant number P-I997/22. 113

114

LUIS ESCAURIAZA

Time-dependent parabolic equations with variable coefﬁcients are treated in [29], [34], where a weak unique continuation theorem is proven using a Carleman inequality. In [29] this was established for variable C 2 second-order coefﬁcients and bounded ﬁrst- and zero-order terms, while in [34] unbounded time-dependent potentials were treated. In particular, it is shown that if u veriﬁes |u + ∂t u| ≤ |V (x, t)u|

in ST ,

(n+2)/2

where V ∈ Lloc (dx dt) and u ≡ 0 in an open set W ⊂ ST , then u(·, s) ≡ 0 for all times s ∈ (0, T ) such that the hyperplane t = s has nonempty intersection with W . Until now, there have been only two works dealing with the strong unique continuation property for parabolic equations with time-dependent coefﬁcients [27], [10]. In [27] it was shown that if u veriﬁes |u + ∂t u| ≤ C[|∇u| + |u|]

in Rn × (0, T )

√ for some C > 0 and |u(x, t)| ≤ CN (|x| + t)N for any integer N, then u ≡ 0 in Rn × (0, T ). The technique used is based on a frequency function that measures the space-time vanishing rate of a solution. In particular, if u solves u + ∂t u = 0 in 2 Rn × (0, T ), setting Gt (x) = t −n/2 e−|x| /4t , Poon deﬁnes tD(t) 2 H (t) = u Gt dx, , D(t) = |∇u|2 Gt dx, N(t) = H (t) and using the divergence theorem shows that for 0 < t < T , x H (t) = 2D(t), D(t) = u ∂t u + · ∇u Gt dx, 2t 2 d x t log H (t) = 2N(t), (tD) (t) = 2 ∂t u + · ∇u Gt dx, dt 2t 2t x x 2 2 N (t) = 2 ∂t u + · ∇u Gt dx u Gt dx − u ∂t u + · ∇u Gt dx 2 . 2t 2t H (t) This identity shows that the frequency function N is nondecreasing and t H (t) ≤ 4N(T ) H for 0 < t < T . 2 Thus, H (t) veriﬁes a doubling property and u cannot vanish to inﬁnity order at zero unless u ≡ 0 in Rn × (0, T ). In this work, we prove the following Carleman inequality associated to this frequency function and derive the strong unique continuation property stated in Theorem 2 for the heat operator.

115

CARLEMAN INEQUALITIES AND THE HEAT OPERATOR

Theorem 1. Let n ≥ 2, r > n/2, 1 ≤ s ≤ ∞, and 1 ≤ p ≤ p¯ ≤ ∞ verify (1.1)

1 1 1 − = , p p¯ r

1 1 + = 1, p p¯

n 1 + ≤ 1. 2r s

Then there are numbers 1 ≤ q ≤ q¯ ≤ ∞ verifying 1/q −1/q¯ = 1/s such that if α ∈ R and β = 2α − n/p¯ − 2/q¯ > 0 is not an integer, then there is a constant N(n, r, s, µ) depending on n, r, s, and the distance µ from β to the set of positive integers, such that the inequality −α −|x|2 /8t t e f Lq¯ Lp¯ t x (1.2) 2 ≤ N (n, r, s, µ)t −α+1−(n/2r)−(1/s) e−|x| /8t (f + ∂t f ) Lq Lp t

x

holds for every f ∈ C0∞ (Rn+1 \ {(0, 0)}). Here the integration with respect to t takes place on R+ and the integration with respect to x takes place on Rn . In particular, for p, p, ¯ and r as above, the inequality holds when either (1.3)

n < r < ∞, 2

1 < q ≤ q¯ < ∞,

0≤

1 1 n − ≤ 1− q q¯ 2r

or r = ∞,

(1.4)

1 ≤ q ≤ q¯ ≤ ∞.

When n = 1, the same holds provided that 1 ≤ r, s ≤ ∞, and 1 ≤ p ≤ p¯ ≤ ∞ verify (1.1), and either (1.5) and

1 ≤ r < ∞,

1 ≤ q ≤ q¯ ≤ ∞,

1 1 , q q¯

1 ≡ 1 − , 0 , 2r

1 1 1 − ≤ 1− q q¯ 2r

1 1, 2r

or (1.6)

r = ∞,

1 ≤ q ≤ q¯ ≤ ∞.

Theorem 2. Suppose that n ≥ 2, 0 < T < +∞, and 1−(n/2r)−(1/s) s V t L ((0,T ), dt; Lr (Rn )) is ﬁnite for some r and s verifying n/2 < r ≤ ∞, 1 ≤ s < ∞, and n/2r + 1/s ≤ 1. Assume that u veriﬁes |u + ∂t u| ≤ |V (x, t)u|

for (x, t) ∈ Rn × (0, T ),

116

LUIS ESCAURIAZA

and for some constant b > 0 and all k ∈ N, √ k 2 |u(x, t)| ≤ Ck |x| + t eb|x| . (1.7) Then u ≡ 0 in Rn × (0, T ). When s = ∞, the same conclusion holds provided that 1−(n/2r) ∞ V t ≤ ε, L ((0,T ), dt; Lr (Rn )) where ε depends on n and r. When n = 1, the same holds but also when 1 ≤ r ≤ ∞. We believe that Theorem 1 is the ﬁrst Carleman inequality in the literature implying the strong unique continuation property for parabolic equations. Also observe that in Theorem 2, we need to ask the solution u to be globally deﬁned in the space-variable in Rn × (0, T ) for some T > 0, and verify a certain growth condition at inﬁnity to obtain the fact that u is identically zero. On the contrary, in the elliptic case, the known Carleman inequalities imply the unique continuation property for solutions that are locally deﬁned. This may seem a pitfall of this Carleman inequality, but the classic counterexample by Tychonoff of nonuniqueness in the Dirichlet problem for the heat 2 operator when the solutions grow faster than eN|x| at inﬁnity can be modiﬁed to −α show that the conditions in Theorem 2 are natural. In fact, if α > 1 and g(t) = e−t , then the function [20] u(x, t) =

+∞ x 2k (−1)k g (k) (t) (2k)! k=0

veriﬁes u + ∂t u = 0 in R × [0, 1), u ∈ C ∞ (R × [0, 1)), and for some θ ∈ (0, 1) depending on α 2 x t −α |u(x, t)| ≤ exp − , θt 2 showing that u vanishes to inﬁnite order in the space-time variables at any point (x, 0), x ∈ R, but u is not identically zero in R × [0, 1). We reduce the proof of the Carleman inequality by means of changes of variables to the following equivalent estimates for the Hermite subgroup. Theorem 3. Let n, p, p, ¯ q, q, ¯ r, and s be as in Theorem 1. Then if β > 0 is not an integer, there is a constant N > 0 depending on n, r, s, and the distance from β to the set of nonnegative integers such that the inequality

u q¯ p¯ ≤ N − |x|2 + ∂t + (2β + n) u q p Lt Lx

Lt Lx

holds for all u ∈ C0∞ (Rn+1 ). Here the integration with respect to t takes place on R and the integration with respect to x takes place on Rn .

117

CARLEMAN INEQUALITIES AND THE HEAT OPERATOR

A simple scaling argument shows that the above inequality is false when n/2r + 1/s > 1. Moreover, observe that when n ≥ 2, n/2 < r ≤ ∞, and p, p¯ are as in Theorem 1, we have that if α ∈ R, β = 2α − n/p¯ − 1 > 0 is not an integer, and f ∈ C0∞ (Rn+1 \ {(0, 0)}), then −α −|x|2 /8t t e f

p¯

L2t Lx

2 ≤ N t −α+1−(n/2r) e−|x| /8t (f + ∂t f ) L2 Lp , x

t

and since p ≤ 2 ≤ p, ¯ from Minkowsky’s inequality it follows that −α −|x|2 /8t 2 t e f Lp¯ L2 ≤ N t −α+1−(n/2r) e−|x| /8t (f + ∂t f ) Lp L2 . x

x

t

t

Using truncations, it is simple to verify that this inequality still holds when f (x, t) = ϕ(x) and ϕ ∈ C0∞ (Rn \ {0}). From Stirling’s formula [1], it follows that for n/2 < r ≤ ∞ and ϕ as before, ϕ|x|−t p¯ ≤ Nt −1+(n/2r) ϕ|x|−t+2−(n/r) p L L x

x

when t ∈ R and where N depends on n, r and the distance from t − n/p¯ to N. We believe that (1.2) also holds in the end-point case r = n/2, but we have not succeeded in obtaining this result. The previous argument shows that a proof of (1.2) with r = n/2 would imply the Carleman inequality proved by Jerison and Kenig in [19]. In Section 2, we prove Theorems 1 and 3, and in Section 3, we apply Theorem 1 to obtain the strong unique continuation property. Acknowledgments. The author would like to thank Luis Vega for the useful conversations we had related to this work. 2. Proof of the estimates. To reduce the proof of Theorem 1 to proving Theo2 rem 2, deﬁne for f ∈ C0∞ (Rn+1 \ {(0, 0)}), g(x, t) = t −α e−|x| /8t f (x, t). A simple calculation shows that (1.2) is equivalent to the inequality (2.1) gLq¯ Lp¯ t

x

1−(n/2r)−(1/s) n + 4α |x|2 x g− g ≤ N t g + ∂t g + · ∇g + 2t 4t 16t

q

p

Lt Lx

,

n+1 for any g ∈ C0∞ (R+ ) and where the integration with respect to t takes place in R+ . ¯ q)) ¯ , again a calculation and a similar Deﬁning u(x, t) = g(2xe2t , e4t )e4t ((n/2p)+(1/ change of variables on the right-hand side of (2.1) show that the last inequality is equivalent to Theorem 2 with β = 2α − n/p¯ − 2/q. ¯ Recall that the Hermite functions hk on R [37, Chapter 1] are given by

√ −1/2 dk 2 2 (−1)k k e−x e−x /2 . hk (x) = 2k k! π dx

118

LUIS ESCAURIAZA

The Hermite functions +α , α ∈ Nn , on Rn are deﬁned by taking the product of the 1-dimensional Hermite functions hαj : +α (x) =

n

hαj (xj ).

j =1

Then they form a complete orthonormal system for L2 (Rn ), and if H = − |x|2 is the Hermite operator on Rn , then H +α = −(2|α| + n)+α . Given f ∈ C0∞ (Rn ), the Fourier-Hermite coefﬁcients of f are given by f (x)+α (x) dx cα = Rn

and f (x) =

1/2

cα +α (x),

f L2 (Rn ) =

α

|cα |2

.

α

Deﬁning Pk to be the projection onto the kth eigenspace +α (x)+α (y)f (y) dy, Pk f = Rn |α|=k

β∈ / N, β > 0, we have that for u ∈ C0∞ (Rn+1 ), it is u = Sβ (f ), where Sβ (f ) =

t

e−2(β−k)(t−s) Pk f (·, s) (x) ds

−∞ k<β +∞

−

t

e−2(k−β)(s−t) Pk f (·, s) (x) ds

k>β

and

f = − |x|2 + ∂t + (2β + n) u. ¯ q, Hence, to prove (2.1), it sufﬁces to show that for any f ∈ C0∞ (Rn+1 ) and p, p, and q¯ as in Theorem 1, we have (2.2)

Sβ (f )Lq¯ Lp¯ ≤ Cf Lqt Lpx . t

x

119

CARLEMAN INEQUALITIES AND THE HEAT OPERATOR

To obtain these inequalities, we ﬁrst prove that they hold when r = ∞ in (1.4) and (1.6). Then, using an analytic family of operators [35], we obtain (2.2) when n/2 < r ≤ n in (1.3), and the other cases in (1.3) follow from the complex method of interpolation [6, Theorem 5.1.2] between the cases r = n and r = ∞. When n = 1, we prove (2.2) when r = 1, and the other cases follow by the complex method of interpolation between the cases r = 1 and r = ∞. The main tool used in one of the end-points of the analytic interpolation is the following restriction theorem for the projections associated to Hermite functions that was ﬁrst proved by Karadzhov [21] (see also [38, the remark to Proposition 1]). Theorem 4. Assume that n ≥ 2. Then there is a constant N depending only on n such that for any f ∈ C0∞ (Rn ) and k ≥ 0, Pk f L2n/n−2 (Rn ) ≤ N f L2n/n+2 (Rn ) . Proof of (2.2) when r = ∞ and n ≥ 1. Writing Sβ (f ) = Iβ (f ) − I Iβ (f ) where Iβ (f ) = I Iβ (f ) =

t

e−2(β−k)(t−s) cα (s) ds+α (x),

k<β |α|=k −∞ +∞

cα (t) =

t

Rn

e−2(k−β)(s−t) cα (s) ds+α (x),

k>β |α|=k

f (y, t)+α (y) dy,

and setting gβ (t) = Iβ (f )(·, t)L2x , the orthogonality of the Hermite functions and differentiation implies that  gβ (t) = 

k<β |α|=k

gβ (t)gβ (t) + 2

t −∞

e−2(β−k)(t−s) cα (s) ds

(β − k)

k<β |α|=k

=

k<β |α|=k

t −∞

cα (t)

e

2

−2(β−k)(t−s)

t −∞

1/2 

,

2 cα (s) ds

e−2(β−k)(t−s) cα (s) ds.

Letting µ denote the distance from β to N, from the Cauchy-Schwarz inequality, we have that gβ (t)gβ (t) + 2µgβ2 (t) ≤ gβ (t)f (·, t)L2x ,

120

LUIS ESCAURIAZA

and dividing this inequality by gβ (t), gβ (t) + 2µgβ (t) ≤ f (·, t)L2x . Then, integrating this inequality in (−∞, t), t e−2µ(t−s) f (·, s)L2x ds. (2.3) gβ (t) ≤ −∞

Proceeding in the same way with fβ (t) = I Iβ (f )L2x , we obtain −fβ (t) + 2µfβ (t) ≤ f (·, t)L2x , and integrating this inequality over (t, +∞), +∞ fβ (t) ≤ (2.4) e−2µ(s−t) f (·, s)L2x ds. t

Thus, from (2.3), (2.4), and the Hausdorff-Young inequality, Iβ (f )Lq¯ L2 + I Iβ (f )Lq¯ L2 ≤ N f Lqt L2 t

t

x

x

x

when 1 ≤ q ≤ q¯ ≤ ∞ and with N depending on s and µ, showing that (2.2) holds when r = ∞. Proof of (2.2) when n/2 < r ≤ n. For f ∈ L1 (R), denote the Fourier transform of f as ˆ e−2πitτ f (t) dt. f (τ ) = R

Setting

+∞

3(z) =

t z−1 e−t dt

0

and Tk,z (t) =

1 e−kt t z−1 ℵ(0,+∞) (t), 3(z)

Sk,z (t) =

1 e−k|t| |t|z−1 ℵ(−∞,0) (t) 3(z)

for k > 0 and z > 0, it is simple to verify that (2.5)

Tˆk,z (τ ) =

1 , (2π iτ + k)z

where ξ z = ez log ξ and −π < arg ξ < π .

Sˆk,z (τ ) =

1 , (−2πiτ + k)z

CARLEMAN INEQUALITIES AND THE HEAT OPERATOR

121

Now for n/2 < r ≤ n and 0 < z < 1, consider the family of operators t

1 Iβz (f )(x, t) = e−2(β−k)(t−s) (t − s)(2rz/n)−1 Pk f (·, s) (x) ds, 3(2rz/n) −∞ k<β

and from (2.5), we have z (f )(x, τ ) = Iˆβ,r

k<β

1

2πiτ + 2(β − k)

2rz/n Pk fˆ(·, τ ) (x).

Clearly, Iβz is an analytic family of operators for 0 < z < 1 and continuous for 0 ≤ z ≤ 1 and from the orthogonality of the Hermite functions and Plancherel z I (f ) 2 2 ≤ eπ|y| µ−2rx/n f 2 2 , β L L L L t

t

x

x

for any z = x + iy verifying x ≥ 0. In particular, if y ∈ R, then iy I (f ) 2 2 ≤ eπ|y| f 2 2 . (2.6) β L L L L t

We can also write

t

x

x

iy Iβ

as a vector-valued singular integral operator +∞

iy iy Kβ (t − s) f (·, s) (x) ds, Iβ (f )(x, t) = −∞

where if ϕ

∈ C0∞ (Rn ),

iy

Kβ (t)(ϕ)(x) =

then

1 ℵ(0,+∞) (t) e−2(β−k)t t (2riy/n)−1 Pk (ϕ)(x). 3(2riy/n) k<β

Moreover, it is simple to verify that for some constant N depending on n, r, and µ, we have N|y| iy K (t) 2 2 ≤ Ne , β Lx Lx |t| (2.7) iy K (t − s) − K iy (t) 2 2 dt ≤ NeN|y| β β L L x

|t|>2|s|

x

for all t, s ∈ R. Then from (2.6), (2.7), and [28, Theorem 1.3], it follows that iy

I (f ) q0 2 ≤ NeN|y| f q0 2 when f ∈ C ∞ Rn+1 , 1 < q0 < ∞. + 0 β L L L L t

t

x

x

On the other hand, from Theorem 4 and Minkowski’s inequality, 1+iy Iβ (f )(·, t) 2n/(n−2) Lx

≤ NeN|y|

t

−∞ k<β

e−2(β−k)(t−s) (t − s)(2r/n)−1 f (·, s)L2n/(n+2) ds, x

122

LUIS ESCAURIAZA

and for t ≥ 0,

e−2(β−k)t t (2r/n)−1 ≤ Nt (2r/n)−2 e−µt ,

k<β

where N is as above. The last two inequalities, the Hausdorff-Young inequality, and fractional integration [36] imply that 1+iy I (f )Lq¯1 L2n/(n−2) ≤ NeN|y| f Lq1 L2n/(n+2) β x

t

t

x

when either n/2 < r < n and (1/q1 , 1/q¯1 ) is in the closed polygon with vertices 2r 2r (0, 0), (1, 1), 1, 2 − , − 1, 0 n n and

1 1 , q1 q¯1

2r ≡ 1, 2 − , n

2r − 1, 0 , n

or when r = n and (1/q1 , 1/q¯1 ) is in the closed triangle with vertices (0, 0), (1, 1), (1, 0). Clearly, the family of operators Iβz is admissible, and since for 1 < q0 < ∞, 1 ≤ q1 , q¯1 ≤ ∞, it is

q

q¯

2n/(n−2) n/2r

Lt 0 L2x , Lt 1 Lx

q¯

q0 2 q1 2n/(n+2) q p L t Lx , L t Lx = L t Lx , n/2r

p¯

= L t Lx ,

where p, p¯ are as in (1.1) and 1 n 1 n 1 = 1− + , q 2r q0 2r q1

n 1 1 n 1 = 1− + q¯ 2r q0 2r q¯1

[6], using Stein’s theorem of analytic interpolation [35] when z = n/2r, we obtain Iβ (f )Lq¯ Lp¯ ≤ N f Lqt Lpx t

x

when n/2 < r ≤ n and p, p, ¯ q, and q¯ are as in (1.1) and (1.3). A similar argument, but considering the analytic family of operators

I Iβz (f )(x, t) =

1 3(2rz/n)

+∞ t

e−2(k−β)(s−t) (s − t)(2rz/n)−1 Pk f (·, s) (x) ds,

k>β

shows that the above estimates hold with Iβ replaced by I Iβ , thus proving (2.2).

CARLEMAN INEQUALITIES AND THE HEAT OPERATOR

123

Proof of (2.2) when n = 1 and r = 1. Writing Pk f (x) = φk (x, y)f (y) dy, R

it is well known [37, Chapter 1] that the following generating identity holds for |z| < 1, z ∈ C: +∞ k=0

−1/2 −(1/2)(1+z2 )/(1−z2 )(|x|2 +|y|2 )+2zxy/(1−z2 ) 1 zk φk (x, y) = √ 1 − z2 e . π

Setting g(z, x, y) =

2 2 2 2 2 √1 e−(1/2)(1+z )/(1−z )(|x| +|y| )+2zxy/(1−z ) , π

it is easy to verify that (2.8)

1 |g(z, x, y)| ≤ √ π

for z ∈ C, |z| ≤ 1 and all x, y ∈ R.

Using Cauchy’s integral formula, we have that for any f ∈ C0∞ (R), Iβ (f )(x, t) = I Iβ (f )(x, t) =

t

−∞ R

Iβ x, y, e2(t−s) f (y, s) dy ds,

+∞

t

R

I Iβ x, y, e−2(s−t) f (y, s) dy ds,

where if m ∈ N and m < β < m + 1,

g(z, x, y) zm+1 − r m+1 dz for r > 1, √ 1 − z2 (z − r) z1+m |z|=1 r 1+m−β g(z, x, y) dz for r < 1. I Iβ (x, y, r) = √ 2π i |z|=1 1 − z2 (z − r) z1+m r −β Iβ (x, y, r) = 2π i

Now from (2.8) and for r < 1, 1+m−β g(z, x, y) r 1+m−β ≤ r |dz| dz √ √ |z|=1 1 − z2 (z − r) z1+m |z−1|≤2(1−r) 1 − z2 (1 − r) r 1+m−β + |dz| √ |z−1|≥2(1−r) 1 − z2 (z − 1) ≤ C (1 − r)−1/2 r 1+m−β .

124 Thus,

LUIS ESCAURIAZA

I Iβ x, y, e−2(s−t) ≤ C (s − t)−1/2 e−µ(s−t)

A similar argument shows that

Iβ x, y, e2(t−s) ≤ C (t − s)−1/2 e−µ(t−s)

for s > t. for s < t,

and fractional integration and Minkowsky’s inequality imply Iβ (f )Lq¯ L∞ + I Iβ (f )Lq¯ L∞ ≤ N f Lqt L1 t

t

x

x

x

when (1/q, 1/q) ¯ is in the closed polygon with vertices (0, 0), (1, 1), (1/2, 0), and (1, 1/2), and (1/q, 1/q) ¯ is different from the last two points, proving (1.5) when r = 1. The remaining cases follow by the complex method of interpolation between the cases r = ∞ and r = 1. 3. The strong unique continuation property. The argument to obtain Theorem 2 from Theorem 1 is well known, but for completeness we include it here. Assume that u veriﬁes the conditions in Theorem 2. If C = N(n, r, s, µ) is the constant appearing in the inequality (1.1), we choose T sufﬁciently small so that 32b < 1/T and 1−(n/2r)−(1/s) V t s (3.1) ≤ 1/2C. L ((0,T ), dt; Lr (Rn )) For ε > 0 and R > 1, let ϕε ∈ C0∞ (R) and ϕR ∈ C0∞ (Rn ) be such that ϕε (t) = 0 for t ≤ ε or t ≥ 3/4T , ϕε (t) = 1 for 2ε ≤ t ≤ T /2, and ϕR (x) = 0 for |x| ≥ 2R, ϕR (x) = 1 when |x| ≤ R. Applying the Carleman inequality (1.1) to uϕε ϕR , we obtain −α −|x|2 /8t t e ϕε ϕR u q¯ p¯ Lt Lx

2 ≤ C t −α+1−(n/2r)−(1/s) e−|x| /8t V ϕε ϕR uLq Lp t x −α+1−(n/2r)−(1/s) −|x|2 /8t +C t e ϕε ϕR u Lq Lp t x −α+1−(n/2r)−(1/s) −|x|2 /8t +C t e ϕε uϕR Lq Lp t x −α+1−(n/2r)−(1/s) −|x|2 /8t + 2C t e ϕε ∇u∇ϕR q

p

Lt Lx

= I (ε, R) + I I (ε, R) + I I I (ε, R).

We show that for ﬁxed α and ε, I I I (ε, R) → 0 when R → +∞. To this end, we use the interior estimate 

2 q/p 1/q q/p 1/q 0 −δ /2 |∇u|p dx dt ≤ N δ |u + ∂t u|p dx dt −δ 2

Bδ

−δ 2

B2δ

+ δ −3−n+(2/q)+(n/p)

0

−δ 2 B2δ

 |u| dx dt,

CARLEMAN INEQUALITIES AND THE HEAT OPERATOR

125

and since p ≤ r and q ≤ s, the growth condition (1.7) on u with k = 0 implies that there is a constant N depending on n, α, p, q, r, s, T , and ε such that I I I (ε, R) ≤ Ne(4b−(1/8T ))R . 2

On the other hand, from Hölder’s inequality, (3.1), and (1.7), 1 2 2 C t −α+1−(n/2r)−(1/s) e−|x| /8t V ϕε uLq Lp ≤ t −α e−|x| /8t ϕε uLq¯ Lp¯ , t x t x 2 2 p. lim sup I I (ε, R) ≤ (T /2)−α+1−(n/2r)−(1/s) e−|x| /8t u q Lt (0,T )Lx

ε→0

Thus, −|x|2 /8t e u

q¯

p¯

Lt (0,T /4)Lx

2 ≤ N 2−α e−|x| /8t uLq (0,T )Lp , t

x

and if we now choose α → +∞ with d(2α −n/p¯ −2/q, ¯ N) = 1/2, we see that u ≡ 0 on Rn × (0, T /4). References [1] [2] [3] [4] [5]

[6] [7] [8] [9] [10] [11]

[12] [13]

L. V. Ahlfors, Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, McGraw-Hill, New York, 1953. W. O. Amerin, A. M. Berthier, and V. Georgescu, Lp -inequalities for the Laplacian and unique continuation, Ann. Inst. Fourier (Grenoble) 31 (1981), 153–168. N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl. (9) 36 (1957), 235–249. N. Aronszajn, A. Krzywicki, and J. Szarski, A unique continuation theorem for exterior differential forms on Riemannian manifolds, Ark. Mat. 4 (1962), 417–453. B. Barcelo, C. E. Kenig, A. Ruiz, and C. D. Sogge, Weighted Sobolev inequalities and unique continuation for the Laplacian plus lower order terms, Illinois J. Math. 32 (1988), 230–245. J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Grundlehren Math. Wiss. 223, Springer, Berlin, 1976. A.-M. Berthier, Sur le spectre ponctuel de l’opérateur de Schrödinger, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), A393–A395. T. Carleman, Sur un problème d’unicité pur les systèmes d’équations aux dérivées partielles à deux variables indépendantes, Ark. Mat., Astr. Fys. 26 (1939), no. 17. S. Chanillo and E. Sawyer, Unique continuation for + V and the C. Fefferman-Phong class, Trans. Amer. Math. Soc. 318 (1990), 275–300. X. Y. Chen, A strong unique continuation theorem for parabolic equations, Math. Ann. 311 (1998), 603–630. H. O. Cordes, Über die eindeutige Bestimmtheit der Lösungen elliptischer Differentialgleichungen durch Anfangsvorgaben, Nachr. Akad. Wiss. Göttingen. Math-Phys. Kl. IIa 1956, 239–258. E. B. Fabes, N. Garofalo, and F. H. Lin, A partial answer to a conjecture of B. Simon concerning unique continuation, J. Funct. Anal. 88 (1990), 194–210. N. Garofalo and F. H. Lin, Monotonicity properties of variational integrals, Ap -weights and unique continuation, Indiana Univ. Math. J. 35 (1986), 245–268.

126 [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

[24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]

LUIS ESCAURIAZA , Unique continuation for elliptic operators: A geometric-variational approach, Comm. Pure Appl. Math. 40 (1987), 347–366. V. Georgescu, On the unique continuation property for Schrödinger Hamiltonians, Helv. Phys. Acta 52 (1979), 655–670. L. Hörmander, Uniqueness theorems for second order elliptic differential equations, Comm. Partial Differential Equations 8 (1983), 21–64. S. Ito and H. Yamabe, A unique continuation theorem for solutions of a parabolic differential equation, J. Math. Soc. Japan 10 (1958), 314–321. D. Jerison, Carleman inequalities for the Dirac and Laplace operators and unique continuation, Adv. Math. 62 (1986), 118–134. D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math. (2) 121 (1985), 463–494. F. John, Partial Differential Equations, 4th ed., Appl. Math. Sci. 1, Springer, New York, 1982. G. Karadzhov, Riesz summability of multiple Hermite series in Lp spaces, C. R. Acad. Bulgare Sci. 47 (1994), 5–8. J. L. Kazdan, Unique continuation in geometry, Comm. Pure Appl. Math. 41 (1988), 667–681. C. E. Kenig, A. Ruiz, and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefﬁcient differential operators, Duke Math. J. 55 (1987), 329–347. K. Kurata, A unique continuation theorem for uniformly elliptic equations with strongly singular potentials, Comm. Partial Differential Equations 18 (1993), 1161–1189. F.-H. Lin, A uniqueness theorem for parabolic equations, Comm. Pure Appl. Math. 43 (1990), 127–136. , Nodal sets of solutions of elliptic and parabolic equations, Comm. Pure Appl. Math. 44 (1991), 287–308. C.-C. Poon, Unique continuation for parabolic equations, Comm. Partial Differential Equations 21 (1996), 521–539. J. L. Rubio de Francia, F. J. Ruiz, and J. L. Torrea, Calderón-Zygmund theory for operatorvalued kernels, Adv. Math. 62 (1988), 7–48. J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations 66 (1987), 118–139. E. T. Sawyer, Unique continuation for Schrödinger operators in dimensions three or less, Ann. Inst. Fourier (Grenoble) 34 (1984), 189–200. M. Schechter and B. Simon, Unique continuation for Schrödinger operators with unbounded potentials, J. Math. Anal. Appl. 77 (1980), 482–492. C. D. Sogge, Oscillatory integrals and unique continuation for second order elliptic differential equations, J. Amer. Math. Soc. 2 (1989), 491–515. , Strong uniqueness theorems for second order elliptic differential equations, Amer. J. Math. 112 (1990), 943–984. , A unique continuation theorem for second order parabolic differential operators, Ark. Mat. 28 (1990), 159–182. E. M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482–492. , Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton Univ. Press, Princeton, 1970. S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Math. Notes 42, Princeton Univ. Press, Princeton, 1993. , Hermite and special Hermite expansions revisited, Duke Math. J. 94 (1998), 257–278. T. Wolff, Unique continuation for |u| ≤ V |∇u| and related problems, Rev. Mat. Iberoamericana 6 (1990), 155–200. , A property of measures in Rn and an application to unique continuation, Geom. Funct.

CARLEMAN INEQUALITIES AND THE HEAT OPERATOR

[41]

127

Anal. 2 (1992), 225–284. H. Yamabe, A unique continuation theorem of a diffusion equation, Ann. of Math. (2) 69 (1959), 462–466.

Departamento de Matemáticas, Universidad del País Vasco, 48080 Bilbao, Spain; [email protected]

Vol. 104, No. 1

DUKE MATHEMATICAL JOURNAL

© 2000

PANEITZ-TYPE OPERATORS AND APPLICATIONS ZINDINE DJADLI, EMMANUEL HEBEY, and MICHEL LEDOUX

To the memory of André Lichnerowicz Given (M, g) a smooth 4-dimensional Riemannian manifold, let Sg be the scalar curvature of g, and let Rcg be the Ricci curvature of g. The Paneitz operator, discovered in [21], is the fourth-order operator deﬁned by 2 Sg g − 2 Rcg du, Pg4 u = 2g u − divg 3 where g u = − divg ∇u is the Laplacian of u with respect to g. When (M, g) is the 4-dimensional standard unit sphere (S 4 , h), we get that Ph4 u = 2h u + 2 h u. The Paneitz operator is conformally invariant in the sense that if g˜ = e2ϕ g is a conformal metric to g, then for all u ∈ C ∞ (M), Pg˜4 u = e−4ϕ Pg4 (u). The 2-dimensional analogue of this relation is g˜ u = e−2ϕ g u. When the dimension is 2, it is well known that the scalar curvatures of g and g˜ are related by the equation 1 1 g ϕ + Sg = Sg˜ e2ϕ . 2 2 When the dimension is 4, we get that Pg4 ϕ + Q4g = Q4g˜ e4ϕ , where Q4g =

1 g Sg + Sg2 − 3| Rcg |2 . 6

Received 10 March 1999. 2000 Mathematics Subject Classiﬁcation. Primary 58E35; Secondary 35A15. 129

130

DJADLI, HEBEY, AND LEDOUX

With respect to the Euler-Poincaré characteristic, in dimension 2, we have that 1 Sg dvg , χ(M) = 4π M while in dimension 4, χ (M) =

1 8π 2

M

1 |Wg |2 + Q4g dvg , 4

where Wg stands for the Weyl tensor of g. Beautiful works on Pg4 have been developed recently by Beckner [2], Branson-Chang-Yang [4], Chang-Yang [7], Chang-GurskyYang [6], and Gursky [12]. Also see the survey Chang [5]. The Paneitz operator was generalized to higher dimensions by Branson [3]. Given (M, g) a smooth compact Riemannian n-manifold, n ≥ 5, let Pgn be the operator deﬁned by n−4 n Pgn u = 2g u − divg an Sg g + bn Rcg du + Qg u, 2 where Qng = and

1 n3 − 4n2 + 16n − 16 2 2 g Sg + Sg − | Rcg |2 2 2 2(n − 1) 8(n − 1) (n − 2) (n − 2)2   an =

(n − 2)2 + 4 , 2(n − 1)(n − 2)  bn = − 4 . n−2

If g˜ = ϕ 4/(n−4) g is a conformal metric to g, then for all u ∈ C ∞ (M), Pgn (uϕ) = ϕ (n+4)/(n−4) Pg˜n u. In particular,

n − 4 n (n+4)/(n−4) Qg˜ ϕ . 2 These two relations have well-known analogues when dealing with the conformal Laplacian Lng , the second order operator whose expression is given by Pgn ϕ =

Lng u = g u +

n−2 Sg u. 4(n − 1)

If g˜ = ϕ 4/(n−2) g is a conformal metric to g, for all u ∈ C ∞ (M), we get that Lng (uϕ) = ϕ (n+2)/(n−2) Lng˜ (u).

PANEITZ-TYPE OPERATORS AND APPLICATIONS

In particular, Lng ϕ =

131

n−2 Sg˜ ϕ (n+2)/(n−2) . 4(n − 1)

On the standard unit sphere (S n , h), n ≥ 5, the expression of Phn is Phn u = 2h u + cn h u + dn u, where

 1  cn = (n2 − 2n − 4), 2  dn = n − 4 n(n2 − 4), 16

(0.1)

(0.2)

that is, cn = n(n−1)an +(n−1)bn , dn = (n−4)Qnh /2. Note that Phn = (Lnh )2 −2Lnh . Given (M, g) a smooth n-dimensional compact Riemannian manifold, n ≥ 5, and α > 0 real, we let Pg be the fourth-order operator deﬁned by Pg u = 2g u + α g u.

(0.3)

Keeping in mind the expression of Pg4 on the standard sphere, we refer to Pg as a Paneitz-type operator. Results and remarks often shift between operators like Pg , and Paneitz-Branson-type operators like P˜g u = 2g u + α g u + βu, where α and β are real numbers. Here we should regard Pg as the essential part of P˜g , like the Laplacian is the essential part of the conformal Laplacian, and note that the Paneitz-Branson operator Pgn reduces to P˜g when g is Einstein. A natural space when studying Pg is the Sobolev space H22 (M) deﬁned as the completion of C ∞ (M) with respect to the norm

u 2 = ∇ 2 u 22 + ∇u 22 + u 22 . Following standard notations, · p in the above expression stands for the Lp -norm (with respect to the Riemannian measure dvg ). As is well known and easy to see, for all u ∈ C ∞ (M), ( g u)2 ≤ n|∇ 2 u|2 . Conversely, by the Bochner-Lichnerowicz-Weitzenböck formula we get that |∇ 2 u|2 dvg = | g u|2 dvg − Rcg (∇u, ∇u) dvg M M M ≤ | g u|2 dvg + k |∇u|2 dvg , M

M

132

DJADLI, HEBEY, AND LEDOUX

where k is such that Rcg ≥ −k. Hence, · H 2 deﬁned by 2

u 2H 2 =

M

2

H22 (M)

is a norm on in what follows,

(Pg u)u dvg +

M

that is equivalent to the above more classical one · . Here and

M

u2 dvg

(Pg u)u dvg =

( g u) dvg + α 2

M

M

|∇u|2 dvg .

By the Sobolev embedding theorem (see, e.g., [14] and recall that n ≥ 5), we get an ! embedding of H22 (M) in L2 (M), where 2! =

2n . n−4

This embedding is critical. It is also continuous, so that there exist A ∈ R and B ∈ R such that for all u ∈ H22 (M), (S1)

u 22! ≤ A (Pg u)u dvg + B u 22 . M

Another possible inequality is that for all u ∈ H22 (M),

u 22! ≤ A (Pg u)u dvg + B u 2H 2 , M

(S2)

1

where · H 2 is the usual norm of H12 (M) given by 1

u 2H 2 = ∇u 22 + u 22 . 1

Given i = 1, 2, we deﬁne

(i) Aopt (M)

as the best constant A in (Si). Formally,

(i) Aopt (M) = inf A ∈ R s.t. ∃B ∈ R with the property that (Si) is valid , where by “(Si) is valid”, we mean that (Si) holds with A and B for all u ∈ H22 (M). (i) Similarly, we deﬁne Bopt (M) as the best constant B in (Si), that is,

(i) Bopt (M) = inf B ∈ R s.t. ∃A ∈ R with the property that (Si) is valid . Section 1 of this paper is devoted to the study of these best constants. We determine their exact values and investigate the corresponding optimal inequalities. Applications to a fourth-order partial differential equation of critical growth are presented in Section 2, with special attention devoted to the standard sphere and the Paneitz-Branson

PANEITZ-TYPE OPERATORS AND APPLICATIONS

133

operator Phn given by (0.1). The results we formulate do involve at some point conditions on the parameter α > 0 in the deﬁnition (0.3) of Pg . It appears that α plays a central role. Results may be expressed equivalently in terms of conditions on α and β for the operator P˜g u = 2g u + α g u + βu (and we do so for the operator Phn on the sphere). For the sake of comparison with the classical Sobolev inequalities, and in order to preserve the energy as (Pg u)u dvg , we deal with deﬁnition (0.3) for Paneitz-type operators. 1. Optimal inequalities. As a starting point, let us consider the Euclidean inequality ∀u ∈ Ᏸ(Rn ), n ≥ 5,

2/2!

2!

Rn

|u| dx

≤K

Rn

( u)2 dx,

where stands for the Laplacian with respect to the Euclidean metric. The best constant K in this inequality was studied by Edmunds, Fortunato, and Janelli [10], Lieb [18], and Lions [19]. If K0 stands for this best constant, it was shown that

n 4/n '(n)−4/n . K0−1 = π 2 n(n − 4)(n2 − 4)' 2 The sharp Euclidean Sobolev inequality is then

!

Rn

2/2!

|u|2 dx

≤ K0

Rn

( u)2 dx.

(1.1)

Its extremal functions are −(n−4)/2 , u(,x0 (x) = |x − x0 |2 + ( 2 and we have that

!

2 −1 . 2 u1,x0 = n(n − 4)(n2 − 4)u1,x 0

Given α > 0 real, in what follows, we let P be as in (0.3), with g = δ being the Euclidean metric. In other words, P u = 2 u + α u. The ﬁrst result we prove is the following. Lemma 1.1. For all u ∈ Ᏸ(Rn ), n ≥ 5,

!

Rn

|u|2 dx

2/2!

≤ K0

and K0 is the best constant in this inequality.

Rn

(P u)u dx,

(1.2)

134

DJADLI, HEBEY, AND LEDOUX

Proof. Suppose that A > 0 is such that for all u ∈ Ᏸ(Rn ),

2/2!

2!

Rn

|u| dx

≤A

Rn

(P u)u dx.

(1.3)

Let Ꮾ be the unit ball in Rn . Then for all u ∈ Ᏸ(Ꮾ),

2/2!

!

Ꮾ

|u|2 dx

≤A

Ꮾ

(P u)u dx.

Also let 2 (Ꮾ) = completion of Ᏸ(Ꮾ) w.r.t. · H 2 , H0,2 2

2 H0,1 (Ꮾ) = completion of Ᏸ(Ꮾ) w.r.t. · H 2 , 1

where, as above,

u 2H 2 = 2

Ꮾ

(P u)u dx +

Ꮾ

u2 dx.

2 (Ꮾ) in H 2 (Ꮾ) is compact, while the embedding of H 2 (Ꮾ) The embedding of H0,2 0,1 0,1 2 in L (Ꮾ) is continuous. It easily follows that for any ( > 0, there exists B( > 0 such that for all u ∈ Ᏸ(Ꮾ),

u 2H 2 ≤ ( u 2H 2 + B( u 22 . 1

In particular,

∇u 22 ≤

2

( B( − 1 + (

u 22 +

u 22 , 1 − α( 1 − α(

and we get that for any ( > 0, there exists B( > 0 such that for all u ∈ Ᏸ(Ꮾ),

2/2!

!

Ꮾ

|u|2 dx

≤ (A + ()

Ꮾ

( u)2 dx + B(

Ꮾ

u2 dx.

Let Ꮾδ be the ball of center 0 and radius δ > 0. For any δ < 1, any ( > 0, and all u ∈ Ᏸ(Ꮾδ ), Ꮾδ

2!

2/2!

|u| dx

≤ (A + ()

( u) dx + B( 2

Ꮾδ

Ꮾδ

By Hölder’s inequality, u dx ≤ |Ꮾδ | 2

Ꮾδ

1−2/2!

Ꮾδ

2!

|u| dx

2/2! ,

u2 dx.

PANEITZ-TYPE OPERATORS AND APPLICATIONS

135

where |Ꮾδ | stands for the Euclidean volume of Ꮾδ . Choosing δ small, we are led to the following: for any ( > 0, there exists δ > 0 such that for all u ∈ Ᏸ(Ꮾδ ), 2/2! 2! |u| dx ≤ (A + 2() ( u)2 dx. Ꮾδ

Ꮾδ

Given u ∈ Ᏸ(Rn ) and λ > 0, let uλ (x) = u(λx). For λ large, uλ

2!

Ꮾδ

while

2/2!

|uλ | dx

=λ

2!

4−n Ꮾδ

∈ Ᏸ(Ꮾδ ). In addition,

2/2!

|u| dx

,

( uλ ) dx = λ 2

Ꮾδ

4−n Ꮾδ

( u)2 dx.

As a consequence, for all u ∈ Ᏸ(Rn ), 2/2! ! |u|2 dx ≤ (A + 2() Rn

Rn

( u)2 dx,

so that if A is as in (1.3), then A + 2( ≥ K0 . Since ( is arbitrary, A ≥ K0 . Independently, we clearly get from (1.1) that for all u ∈ Ᏸ(Rn ), 2/2! 2! |u| dx ≤ K0 (P u)u dx, Rn

Rn

so that A = K0 in (1.3) is an admissible value. Lemma 1.1 is established. The following lemma easily follows from Lemma 1.1. Lemma 1.2. Let (M, g) be a smooth, compact, n-dimensional Riemannian manifold, n ≥ 5. Any constant A in (S1) or (S2), whatever the constant B, has to be greater than or equal to K0 . Proof. Given x0 ∈ M, we consider a geodesic, normal coordinates system at x0 . We let Bx0 (δ) be the ball on which this coordinates system is deﬁned. Given ( > 0, choosing δ sufﬁciently small, and by local comparison of the Riemannian metric with the Euclidean metric in the above coordinates system, we easily get from (S1) and (S2) that for all u ∈ Ᏸ(Ꮾδ ), 2 ˜ (P u)u dx + B u

(1.4)

u 22! ≤ (A + () 2, n R 2 2 ˜ (P u)u dx + B u

, (1.5)

u 2! ≤ (A + () H2 Rn

1

where the norms and P in the above expressions are understood with respect to the Euclidean metric. Passing in the coordinates system, we indeed have that ( g u)2 ≤ ( u)2 + (ˆ (δ)|∇ 2 u|2 + (ˆ (δ)|∇u|2 ,

136

DJADLI, HEBEY, AND LEDOUX

where (ˆ (δ) → 0 as δ → 0, while by the Bochner-Lichnerowicz-Weitzenböck formula, 2 2 |∇ u| dx = ( u)2 dx. Rn

Rn

As in the proof of Lemma 1.1, it is enough to deal with (1.4). Applying Hölder’s inequality to the L2 -norm of u, with (1.4) as in the proof of Lemma 1.1, we get that for all ( > 0, there exists δ > 0, such that for all u ∈ Ᏸ(Ꮾδ ), 2 (P u)u dx.

u 2! ≤ (A + 2() Rn

For homogeneity reasons, see once more the proof of Lemma 1.1; such an inequality, if valid for functions with small support, has to be valid for all functions with compact support. By Lemma 1.1, this implies that A+2( ≥ K0 . Since ( is arbitrary, this proves the result. (i)

The answer to the ﬁrst question we asked, dealing with the exact value of Aopt (M), now follows from Lemma 1.2. Theorem 1.1. Let (M, g) be a smooth, compact, n-dimensional Riemannian manifold, n ≥ 5, and let Pg be as in (0.3), that is, Pg u = 2g u + α g u. For any ( > 0, there exists B( ∈ R such that for all u ∈ H22 (M), 2

u 2! ≤ (1 + ()K0 (Pg u)u dvg + B( u 22 , M

u 22! ≤ (1 + ()K0 (Pg u)u dvg + B( u 2H 2 . M

(1)

(1.6) (1.7)

1

(2)

In particular, Aopt (M) = Aopt (M) = K0 . Moreover, we can choose B( depending only on n, (, α, a bound for the Ricci curvature Rcg of g, and a positive lower bound for the injectivity radius ig of g. Proof. It sufﬁces to prove the result for (1.6), since

u 22 ≤ u 2H 2 . 1

Let , > 0 be such that | Rcg | ≤ ,, and let i > 0 be such that ig ≥ i. By a well-known result of Anderson [1], the C 1,1/2 -harmonic radius of g is bounded from below by a positive constant depending mainly on n, ,, and i. More precisely, given δ > 0, there exists rH = rH (n, δ, ,, i), depending only on these constants, such that the following holds: for any x ∈ M, there exists a harmonic chart ϕ on the ball Bx (rH ) with the

PANEITZ-TYPE OPERATORS AND APPLICATIONS

137

property that 1 δij ≤ gij ≤ (1 + δ)δij as bilinear forms, 1+δ n n ∂k gij (y) − ∂k gij (z) 3/2 rH sup ∂k gij (x) + rH sup ≤ δ, dg (y, z)1/2 x y=z k=1

k=1

where dg is the distance with respect to g, and the gij ’s stand for the components of g in the harmonic coordinates system. Without loss of generality, we may assume that rH ≤ 1/2. Let ( > 0 be given. The expression of the Laplacian in harmonic coordinates is g u = −g ij ∂ij u. As we easily check, if δ = δ(() is small enough, for all x ∈ M, and all u ∈

Ᏸ(Bx (rH )), then

P (u ◦ ϕ −1 ) (u ◦ ϕ −1 ) dx ≤ (1 + ()

Rn

M

(Pg u)u dvg .

Similarly, for all x ∈ M, and all u ∈ Ᏸ(Bx (rH )), ! ! (u ◦ ϕ −1 )2 dx. |u|2 dvg ≤ (1 + () Rn

M

It follows from these relations and Lemma 1.1 that for any ( > 0, there exists δ > 0 such that for all u ∈ Ᏸ(Bx (rH )),

2!

M

|u| dvg

2/2!

( K0 (Pg u)u dvg . ≤ 1+ 2 M

(Sloc )

Since M is compact, it can be covered by a ﬁnite number of balls Bxi (rH /2), i = 1, . . . , N . As is by now classical, we may choose these balls such that any point in M has a neighborhood that intersects at most S0 of the Bxi (rH )’s, where the integer S0 = S0 (n, (, ,, i) depends only on n, (, ,, and i. Let αi ∈ Ᏸ(Bxi (rH )) be such that 0 ≤ αi ≤ 1 and αi = 1 in Bxi (rH /2). We set α2 ηi = i 2 . k αk Clearly, (ηi ) is a partition of unity subordinated to the covering (Bxi (rH )) such that √ the ηi ’s are smooth. As we easily check, we may assume that  √ ∇ ηi ≤ H1 ,    g √ηi ≤ H2 ,   √  ∇ g ηi ≤ H3 ,

138

DJADLI, HEBEY, AND LEDOUX

where H1 , H2 , and H3 only depend on n, (, ,, and i. Given u ∈ C ∞ (M), we write that √ 2 ηi u2 ! = ηi u ! .

u 22! = u2 2! /2 ≤ 2 /2 2 i

With (Sloc ), it follows that for all

2!

M

2/2!

|u| dvg

i

u ∈ C ∞ (M),

( √ 2 K0 ≤ 1+ g ηi u dvg 2 M i

√ 2 ( ∇ ηi u dvg . K0 +α 1+ 2 M

i

Hence, M

!

|u|2 dvg

2/2!

( ≤ 1+ ηi ( g u)2 dvg K0 2 M i

√ 2 2 ( K0 + 1+ g ηi u dvg 2 M i

2 √ ( K0 +4 1+ ∇ ηi , ∇u g dvg 2 M i

√ ( √ K0 +2 1+ ηi u( g u) g ηi dvg 2 M i

√ ( √ K0 −4 1+ ηi ( g u) ∇ ηi , ∇u g dvg 2 M i

√ √ ( K0 −4 1+ g ηi u ∇ ηi , ∇u g dvg 2 M i

( K0 +α 1+ ηi |∇u|2 dvg 2 M i

√ 2 2 ( ∇ ηi u dvg +α 1+ K0 2 M i

( √ √ K0 + 2α 1 + u ηi ∇ ηi , ∇u g dvg . 2 M

(1.8)

i

Since rH ≤ 1/2, and if δ is sufﬁciently small, we get that for every x ∈ M, ϕ(Bx (rH )) ⊂ Ꮾ, where Ꮾ stands for the unit ball in Rn with center 0. As in the proof of Lemma 1.1, given (0 > 0, there exists C1 , depending only on (0 , such that for all u ∈ Ᏸ(Ꮾ), (0 |∇u|2 dx ≤ ( u)2 dx + C1 u2 dx. 2 Ꮾ Ꮾ Ꮾ

139

PANEITZ-TYPE OPERATORS AND APPLICATIONS

As before, this implies that for all x ∈ M, and all u ∈ Ᏸ(Bx (rH )), 2 2 |∇u| dvg ≤ (0 ( g u) dvg + C2 u2 dvg , M

M

M

where C2 depends only on (0 . Let us now deal with the various terms in (1.8). As a starting point, ηi ( g u)2 dvg = ( g u)2 dvg M

i

M

and

√ 2 g ηi u2 dvg ≤ S0 H22

M

i

M

u2 dvg .

To ease the notations, set Ꮾi = Bxi (rH ). Then, as is easily checked, √ √ 2 2 ∇ ηi |∇u|2 dvg ∇ ηi , ∇u g dvg ≤ i

M

i

≤ H12

M

i

≤ H12 (0

|∇u|2 dvg

Ꮾi

i

≤ S0 H12 (0

Ꮾi

M

( g u)

2

dvg + H12 C2

Ꮾi

u2 dvg

( g u)

2

dvg + S0 H12 C2

M

u2 dvg .

Similarly, √ √ √ √ ∇ ηi , ∇u g u g ηi dvg ηi u( g u) g ηi dvg = M M i i √ √ 2 + ηi |∇u|2 g ηi dvg i

+

M

i

√ M

Hence, √ √ ηi u( g u) g ηi dvg M i ≤ (H1 H2 + H3 ) |u||∇u| dvg + H22 Ꮾi

i

1 ≤ (H1 H2 + H3 ) 2 i

i

√ ηi u ∇u, ∇ g ηi g dvg .

Ꮾi

|∇u|2 dvg

1 (H1 H2 + H3 ) + H22 u2 dvg + |∇u|2 dvg . 2 Ꮾi Ꮾi

i

140

DJADLI, HEBEY, AND LEDOUX

As above, this leads to √ √ ηi u( g u) g ηi dvg M i S0 2 ≤ (H1 H2 + H3 ) + S0 H2 (0 ( g u)2 dvg 2 M S0 S0 (H1 H2 + H3 ) + (H1 H2 + H3 ) + S0 H22 C2 + u2 dvg . 2 2 M On the other hand, √ √ 1 ηi ( g u) ∇ ηi , ∇u g dvg = ( g u)∇ηi , ∇ug dvg = 0, 2 M M i

i

while √ √ g ηi u ∇ ηi , ∇u g dvg M i ≤ H 1 H2 |u||∇u| dvg ≤

H1 H 2 2

Ꮾi

i

i

Ꮾi

u2 dvg +

H1 H2 2

i

Ꮾi

|∇u|2 dvg .

Here again, it follows that √ √ g ηi u ∇ ηi , ∇u g dvg M i H 1 H2 H1 H2 2 ≤ S0 (0 ( g u) dvg + S0 (1 + C2 ) u2 dvg . 2 2 M M Moreover, ηi |∇u|2 dvg ≤ i

while

M

i

Ꮾi

|∇u|2 dvg ≤ S0 (0

M

( g u)2 dvg + S0 C2

√ 2 ∇ ηi u2 dvg ≤ S0 H 2 u2 dvg 2 i

M

M

M

u2 dvg ,

141

PANEITZ-TYPE OPERATORS AND APPLICATIONS

and

√ √ u ηi ∇ ηi , ∇u g dvg M i H2 H2 ≤ H2 |u||∇u| dvg ≤ u2 dvg + |∇u|2 dvg 2 2 Ꮾi Ꮾi Ꮾi i i i H2 H2 S0 (0 ( g u)2 dvg + S0 (1 + C2 ) ≤ u2 dvg . 2 2 M M

Summarizing our various estimates, we get that for all u ∈ C ∞ (M), M

!

2/2!

|u|2 dvg

≤

1+

( K0 + C3 (0 2

M

( g u)2 dvg + C4

M

u2 dvg ,

where the constant C3 , explicitly known, depends only on n, (, α, ,, and i, and where the constant C4 , explicitly known, depends only on n, (, α, (0 , ,, and i. Choosing (0 such that ( C3 (0 = K0 2 proves the theorem. (1)

(2)

By Theorem 1.1, Aopt (M) = Aopt (M) = K0 . It is natural to ask whether these constants are attained in (S1) and (S2), a question to which we now turn. We do believe that the answer should be positive for what concerns (S2), and a partial result in that direction is given by Theorem 1.2. The answer for (S1) might be more delicate as shown by Proposition 1.2. As a starting point, we state the following elementary lemma that is used in the proof of Theorem 1.2. Lemma 1.3. Given M a smooth, compact, n-dimensional manifold, n ≥ 5, and g1 , g2 = ϕ 4/(n−4) g1 two conformal metrics on M, Pg2 u u dvg2 = Pg1 (uϕ) (uϕ) dvg1 + ᏾ M

M

u ∈ C ∞ (M),

for all where Pg1 and Pg2 are as in (0.3) with g = g1 , g2 , and where ᏾, explicitly known, is made of lower-order terms in the sense that for all i = 1, 2, and all u ∈ C ∞ (M), |∇u|2 dvgi + Bi u2 dvgi , |᏾| ≤ Ai M

M

the constants Ai and Bi being independent of u. Proof. The proof is simple. We may see the result as an easy consequence of the relation Pgn2 (u) = ϕ −(n+4)/(n−4) Pgn1 (uϕ). We may also proceed by direct computation.

142

DJADLI, HEBEY, AND LEDOUX

We now prove the following theorem, which yields a partial answer to the question we raised above. As already mentioned, the case where (M, g) is not conformally ﬂat remains open. Theorem 1.2. Let (M, g) be a smooth, compact, n-dimensional Riemannian man(2) ifold, n ≥ 5. If g is conformally ﬂat, then Aopt (M) = K0 is attained in (S2). In other words, there exists B ∈ R such that

u 22!

≤ K0

M

(Pg u)u dvg + B u 2H 2 1

for all u ∈ H22 (M). Proof. Since (M, g) is compact and conformally ﬂat, it can be covered by a ﬁnite number of open subsets 4i , i = 1, . . . , N , such that (1) for any i, there exists ϕi ∈ C ∞ (M), ϕi > 0, with the property that the metric 4/(n−4) g is ﬂat on 4i ; gi = ϕi (2) for any i, and if gi is as above, (4i , gi ) is isometric to an open subset of Rn equipped with the Euclidean metric. Let (ηi ) be a partition of unity subordinated to the covering (4i ). Without loss of gen√ erality, we may assume that for any i, ηi ∈ C ∞ (M). Given u ∈ C ∞ (M), write that

2/2!

2!

M

|u| dvg

2/2! √ 2! ≤ ηi u dvg . M

i

For any i, let ψi = ϕi−1 so that g = ψi

gi . Then for all u ∈ C ∞ (M),

4/(n−4)

M

√ 2! ηi u dvg =

M

√ ! ηi uψi 2 dvg . i

It follows from Lemma 1.1 that M

√ ! ηi uψi 2 dvg

2/2!

≤ K0

i

M

Pgi

√ √ ηi uψi ηi uψi dvgi .

By Lemma 1.3, for any i, and for all u ∈ C ∞ (M), M

√ 2! ηi u dvg

2/2!

≤ K0

M

Pg

√ √ ηi u ηi u dvg

+ Ai

M

|∇u|2 dvg + Bi

M

u2 dvg .

143

PANEITZ-TYPE OPERATORS AND APPLICATIONS

Hence, for all u ∈ C ∞ (M),

2/2!

!

M

|u|2 dvg

≤ K0

M

i

+

Pg

√ √ ηi u ηi u dvg

Ai

i

|∇u| dvg + 2

M

Bi

M

i

u2 dvg .

Now, as in the proof of Theorem 1.1, √ √ Pg ηi u ηi u dvg i

M

≤

i

≤

M

M

i

√ 2 g ηi u dvg + α i

ηi ( g u)2 dvg +

M

i

M

√ 2 ∇ ηi u dvg

√ 2 g ηi u2 dvg

√ √ 2 √ +4 ηi u( g u) g ηi dvg ∇ ηi , ∇u g dvg + 2 −4 −4 +α

i

M

i

M

i

M

√

i

M

√ ηi ( g u) ∇ ηi , ∇u g dvg

+ 2α

√ 2 ∇ ηi u2 dvg ηi |∇u| dvg + α 2

M

i

√ √ g ηi u ∇ ηi , ∇u g dvg

i

M

M

i

√ √ u ηi ∇ ηi , ∇u g dvg .

We again analyze the various terms in this expression. First, ηi ( g u)2 dvg = ( g u)2 dvg , i

while

M

M

i

M

√ 2 √ 2 g ηi u2 dvg ≤ N max max g ηi i

M

M

u2 dvg .

Similarly,

√ √ 2 2 |∇u|2 dvg . ∇ ηi , ∇u g dvg ≤ N max max ∇ ηi i

M

i

M

M

144

DJADLI, HEBEY, AND LEDOUX

On the other hand, √ √ √ √ ηi u( g u) g ηi dvg = ∇ ηi , ∇u g u g ηi dvg i

M

+ + so that

M

i

√ i

M

i

M

√

√ 2 ηi |∇u|2 g ηi dvg √ ηi u ∇u, ∇ g ηi g dvg ,

√ √ ηi u( g u) g ηi dvg M i √ √ N ≤ max max ∇ ηi max max g ηi u2 dvg i M i M 2 M √ √ N max max ∇ ηi + |∇u|2 dvg max max g ηi i M i M 2 M

√ 2 + N max max g ηi |∇u|2 dvg i

M

M

N √ max max ∇ g ηi + u2 dvg i M 2 M N √ max max ∇ g ηi + |∇u|2 dvg . i M 2 M Furthermore, √ √ 1 ηi ( g u) ∇ ηi , ∇u g dvg = ( g u)∇ηi , ∇ug dvg = 0, 2 M M i

while

i

√ √ g ηi u ∇ ηi , ∇u g dvg M i

√ √ N ≤ max max ∇ ηi max max g ηi u2 dvg i M i M 2 M √ √ N + max max g ηi max max ∇ ηi |∇u|2 dvg . i M i M 2 M

For the next terms, note that i

M

ηi |∇u|2 dvg =

M

|∇u|2 dvg

145

PANEITZ-TYPE OPERATORS AND APPLICATIONS

√ 2 2 ∇ ηi u2 dvg ≤ N max max ∇ √ηi u2 dvg .

and

i

i

M

M

M

Finally, √ √ N √ max u η η , ∇u dv max η u2 dvg ∇ ∇ ≤ i i g i g i M 2 M M i

N √ max max ∇ ηi + |∇u|2 dvg . i M 2 M As a conclusion, for all u ∈ C ∞ (M), 2/2! 2! |u| dvg ≤ K0 ( g u)2 dvg + A |∇u|2 dvg + B u2 dvg , M

M

M

M

where A and B are constants that do not depend on u. In particular, and for all u ∈ C ∞ (M), 2

u 2! ≤ K0 (Pg u)u dvg + C u 2H 2 , M

1

where C does not depend on u. Theorem 1.2 is thus proved. (2)

When dealing with conformally ﬂat manifolds, Theorem 1.2 indicates that Aopt (M) = K0 is attained in (S2). The situation for (S1) is more complicated. In particular, α in the deﬁnition (0.3) of Pg has to play a role. This is what we prove in Proposition 1.2 below. As a ﬁrst result, we establish the following. Proposition 1.1. Let (S n , h) be the standard unit sphere of Rn+1 , n ≥ 5, and let be the Paneitz-Branson operator given by (0.1), that is,

Phn

Phn u = 2h u + cn h u + dn u. Then

Phn u u dvh 1 = infn ! ∞ 2/2 u∈C (S )\{0} K0 2! S n |u| dvh Sn

with the additional property that for any β > 1, and any x0 ∈ S n , if uβ =

(β 2 − 1)(n−4)/4 , (β − cos r)(n−4)/2

where r is the distance on S n to x0 , then !

Phn uβ = dn uβ2 −1 , and uβ realizes the inﬁmum in the left-hand side of (1.9).

(1.9)

146

DJADLI, HEBEY, AND LEDOUX

Proof. We ﬁrst prove (1.9). For that purpose, let x0 be some point on S n , and let 6 : S n \{x0 } → Rn be the stereographic projection of pole x0 . If δ stands for the Euclidean metric of Rn , then (6−1 )7 h = ϕ 4/(n−4) δ, where

ϕ(x) =

4 (1 + |x|2 )2

(n−4)/4

.

By conformal invariance of the Paneitz-Branson operator Pgn , we get that for all u ∈ Ᏸ(Rn ), 2 2 2 2 n ( (uϕ)) dx Rn ( h˜ u) + cn |∇u| + dn u dvh˜ = R (1.10) ! !, 2! dv 2/2 2! dx 2/2 |u| |uϕ| h˜ Rn Rn where h˜ = (6−1 )7 h. Suppose now that 2 2 2 1 S n ( h u) + cn |∇u| + dn u dvh inf < , 2/2! ! u∈C ∞ (S n )\{0} K 2 0 |u| dv h Sn

(1.11)

and let u0 ∈ C ∞ (S n ), u0 ≡ 0, be such that 2 2 2 1 S n ( h u0 ) + cn |∇u0 | + dn u0 dvh < . 2/2! ! K 2 0 S n |u0 | dvh We let (ηs ), s > 0 small, be a family of smooth functions on S n having the property that 0 ≤ ηs ≤ 1, ηs = 0 on BP (s), ηs = 1 on S n \BP (2s), and  C1   , |∇ηs | ≤ s  C  | h ηs | ≤ 2 , s2 where C1 , C2 are positive constants that do not depend on s. In order to get such a family, we might ﬁx some ηs0 as above, for instance, radially symmetric, and set then, for s ≤ s0 , ηs = ηs0 (r/s). As we easily check, 2 2 2 2 2 2 S n ( h us ) + cn |∇us | + dn us dvh S n ( h u0 ) + cn |∇u0 | + dn u0 dvh = , lim ! ! s→0 2! dv 2/2 2! dv 2/2 |u | |u | n n s h 0 h S S where us = ηs u0 . Just note here that lim

1

s→0 s 2

Vh BP (2s)\BP (s) = 0,

PANEITZ-TYPE OPERATORS AND APPLICATIONS

147

where Vh (4) stands for the volume of 4 with respect to h. Choosing s sufﬁciently small, it follows from (1.10) and (1.11) that there exists u˜ s ∈ Ᏸ(Rn ) of the form u˜ s = us ◦ 6−1 ϕ, such that

u˜ s

2

dx 1 2/2! < K . ! 0 ˜ s |2 dx Rn |u Rn

This contradicts (1.1) so that (1.9) follows. Now let uβ be as in the statement of the proposition. As is well known, see, for instance, [15], there exists a conformal diffeomorphism ϕβ of (S n , h) such that ϕβ7 h = uβ

4/(n−4)

h.

By conformal invariance of the Paneitz-Branson operator Pgn , this implies that !

Phn uβ = dn uβ2 −1 . On the other hand, it is easily seen that −4/n

dn K0 = ωn

,

where ωn stands for the volume of S n with respect to h. This follows from the relations 4n/2 ωn−1 = and ωn−1 ' Noting that

Sn

Phn uβ

n 2

= 2π n/2 .

!

Sn

we get that

2'(n) ωn−1 '(n/2)2

u2β dvh = ωn ,

1 uβ dvh = K0

Sn

! u2β dvh

2/2! .

This ends the proof of the proposition. As an ending result in the study of the best ﬁrst constant, we now prove the following. Since (S n , h) is conformally ﬂat, this result has to be compared to Theorem 1.2. Due to the lack of concentration, the approach we use does not allow us to conclude when n = 5.

148

DJADLI, HEBEY, AND LEDOUX

Proposition 1.2. Let (S n , h) be the standard unit sphere of Rn+1 , n ≥ 6, and let Ph be as in (0.3) with g = h, that is, Ph u = 2h u + α h u. There exists B ∈ R such that for all u ∈ H22 (S n ),

u 22! ≤ K0

Sn

(Ph u)u dvh + B u 22

if and only if α ≥ cn , where cn is as in (0.2). Proof. If α ≥ cn , the result follows from (1.9), and we may take B = K0 dn . Suppose, on the contrary, that α < cn . For β > 1 real, and r the distance on S n to a given point, we let uβ be as in Proposition 1.1, that is, uβ = Then, for any β > 1,

(β 2 − 1)(n−4)/4 . (β − cos r)(n−4)/2

Phn uβ uβ dvh 1

2/2! = K . ! 0 2 S n uβ dvh Sn

Let B be given. Writing that Ph u + Bu = Phn u + (α − cn ) h u + (B − dn )u, and since

!

Sn

u2β dvh = ωn ,

it follows that for any β > 1, 2 S n Ph uβ uβ dvh + B uβ 2

uβ 22! 1 1 = + K0 ωn2/2!

(α − cn )

Sn

∇uβ 2 dvh + (B − dn )

Performing the changes of variables x = tan(r/2) and y = easily seen that Sn

∇uβ 2 dvh = C1 (β, n)(β − 1)

+∞ 0

√

Sn

u2β dvh

(1.12) .

(β + 1)/(β − 1)x, it is

y n+1 dy

, 4 1 + ((β − 1)/(β + 1))y 2 (1 + y 2 )n−2

149

PANEITZ-TYPE OPERATORS AND APPLICATIONS

and that u2β dvh = C2 (β, n)(β − 1) Sn

+∞

×

1+((β−1)/(β+1))y 2

0

y n−1 dy

, 1/(β−1)+(1/(β+1))y 2 (1 + y 2 )n−4

3

where

2n (n − 4)2 ωn−1 2n ωn−1 , C (β, n) = . 2 (β + 1)3 (β + 1)2 By the dominated convergence theorem, if n > 6, +∞ y n−1 dy = 0, lim 3 β→1+ 0 1 + ((β − 1)/(β + 1))y 2 1/(β − 1) + (1/(β + 1))y 2 (1 + y 2 )n−4 C1 (β, n) =

while

lim

β→1+ 0

+∞

y n+1 dy

4

1 + ((β − 1)/(β + 1))y 2 (1 + y 2 )n−2

+∞

= 0

y n+1 dy . (1 + y 2 )n−2

The latter integral is a ﬁnite positive constant. It follows that if α < cn and n > 6, then for β > 1 sufﬁciently close to 1, 2 u2β dvh < 0. ∇uβ dvh + (B − dn ) (α − cn ) Sn

Sn

We then get with (1.12) that for β > 1 sufﬁciently close to 1, 1 (Ph uβ )uβ dvh + B uβ 22 <

uβ 22! . n K 0 S This proves the proposition when n > 6. If n =√6, we decompose our integrals into +∞ 1 1/ β−1 +∞ three pieces by writing that 0 = 0+ 1 + 1/√β−1 . Easy computations then give us that 1 1 A1 (β − 1) ln ≤ |∇uβ |2 dvh ≤ A2 (β − 1) ln β −1 β −1 S6 for some positive constants A1 < A2 independent of β, while 2 S 6 uβ dvh lim = 0. β→1+ (β − 1) ln(1/(β − 1)) As above, this gives that if α < c6 , then for β > 1 sufﬁciently close to 1, 2 (α − c6 ) u2β dvh < 0. ∇uβ dvh + (B − d6 ) S6

The proposition is proved.

S6

150

DJADLI, HEBEY, AND LEDOUX

Parallel with the study of the best ﬁrst constant, we can ask similar questions on the best second constant. We state our results on this parallel program without any proofs. Details on these proofs can be found in Djadli-Hebey-Ledoux [9]. As a starting point, it is easily seen that whatever (M, g) smooth and compact of dimension n ≥ 5 is, (1)

(2)

−4/n

Bopt (M) = Bopt (M) = Vg

,

where Vg is the volume of (M, g). Moreover, we can prove that these constants are attained in the sense that there always exists A ∈ R such that for all u ∈ H22 (M), −4/n

u 22! ≤ A (Pg u)u dvg + Vg

u 22 , (1.13) M

and such that for all u ∈ H22 (M), −4/n 2

u 2! ≤ A (Pg u)u dvg + Vg

u 2H 2 . M

1

There A can be chosen such that it depends only on n, α, a lower bound on the Ricci curvature of g, a lower bound on the volume of M with respect to g, and an upper bound on the diameter of M with respect to g. Looking for more precise information on the remaining constant A, an easy statement is that A in (1.13) has to be such that A≥

2! − 2 −4/n Vg , λ1 (λ1 + α)

(1.14)

where λ1 is the ﬁrst nonzero eigenvalue of g . In the speciﬁc case of the standard unit sphere (S n , h), as proved by Beckner [2], the Sobolev inequality

u 2p ≤

p − 2 2/p−1 2/p−1 ωn

∇u 22 + ωn

u 22 n

holds for all p ∈ [2, 27 ]. By the variational characterization of the ﬁrst nonzero eigenvalue λ1 of h , and the Bochner-Lichnerowicz-Weitzenböck formula, it follows that for every u ∈ H22 (S n ), p − 2 2/p−1 2/p−1

u 2p ≤ ωn (Ph u)u dvh + ωn

u 22 . n n(n + α) S It is natural to question whether or not this Beckner’s type inequality extends to real numbers p such that p > 27 . Assuming that this is the case, and, in particular, that the inequality holds for all p ∈ [2, 2! ], we would get that 8 −4/n −4/n

u 22! ≤ ωn (Ph u)u dvh + ωn u 22 . (1.15) n n(n − 4)(n + α) S

151

PANEITZ-TYPE OPERATORS AND APPLICATIONS

Observe that the ﬁrst constant in this inequality is the constant given by (1.14) when the manifold considered is the standard sphere. Let cn be as in (0.2). We can prove (see [9] for details) that if α ≤ cn , then for all u ∈ H22 (S n ),

u 22! ≤

8 −4/n ωn n(n − 4)(n + α)

−4/n

Sn

(Ph u)u dvh + ωn

u 22 ,

and the two constants in this inequality cannot be lowered. In a similar way, we can prove that if α > cn , then for all u ∈ H22 (S n ),

u 22! ≤

16 −4/n ωn n(n − 4)(n2 − 4)

−4/n

Sn

(Ph u)u dvh + ωn

u 22 ,

and again the two constants in this inequality cannot be lowered. In particular, (1.15) is true if α ≤ cn , but false if α > cn . It follows that Beckner’s inequality does not extend to p = 2! . As an ending remark, coming back to an arbitrary, smooth, compact, Riemannian manifold of dimension n ≥ 5, we mention that it is possible to prove that if g is such that Rcg ≥ n − 1, then for all u ∈ H22 (M), −4/n

u 22! ≤ AVg

M

−4/n

(Pg u)u dvg + Vg

u 22 ,

(1.16)

ˆ α) be the where A = A(n, α), explicitly known, depends only on n and α. Let A(n, constant involved in the above inequalities on the sphere

ˆ α) = A(n,

 8    n(n − 4)(n + α)   

if α ≤ cn ,

16 n(n − 4)(n2 − 4)

if α > cn .

With respect to what was proved by Ilias [16] when dealing with the Sobolev space ˆ α) H12 , an open question we are left with is whether or not (1.16) holds with A = A(n, when g is such that Rcg ≥ n − 1. 2. On a fourth-order partial differential equation. Let (M, g) be a smooth, compact, n-dimensional Riemannian manifold, n ≥ 5, and let α, a be two positive real numbers. Let f be a smooth real-valued function on M. We are here concerned with the fourth-order partial differential equation Pg u + au = f u2

! −1

,

where, as in (0.3), Pg u = 2g u + α g u.

(E)

152

DJADLI, HEBEY, AND LEDOUX

When referring to a solution of (E), we assume that the solution is positive and smooth. Multiplying (E) by u, and integrating over M, a necessary condition for (E) to have a solution is that f is positive somewhere on M. For u in H22 (M), we let Ig (u) = We also let

M

(Pg u)u dvg + a

u2 dvg .

M

2 2! Ᏼf = u ∈ H2 (M) f |u| dvg = 1 . M

Our ﬁrst result here is the following theorem. Theorem 2.1. Let (M, g) be a smooth, compact, n-dimensional Riemannian manifold, n ≥ 5, let Pg be the operator given by (0.3), let a be some positive real number, and let f be a smooth positive function on M. Then the inequality inf Ig (u) ≤

u∈Ᏼf

1

(2.1)

!

(maxM f )2/2 K0

always holds, with the additional property that if the inequality in (2.1) is strict, and if a ≤ α 2 /4, then the inﬁmum in the left-hand side of (2.1) is attained by a smooth positive function. In particular, if the inequality in (2.1) is strict and a ≤ α 2 /4, then (E) possesses a smooth positive solution. Proof. We start by proving (2.1). Suppose on the contrary that inf Ig (u) >

u∈Ᏼf

1 !

(maxM f )2/2 K0

.

Then there exists ( > 0 such that for all u ∈ H22 (M),

1 (maxM f )2/2

!

2!

M

f |u| dvg

2/2!

≤ K0 (1 − ()

M

(Pg u)u dvg + B u 22 ,

where, for instance, B = aK0 . If x0 is a point where f is maximum, for r > 0 sufﬁciently small, and all x ∈ Bx0 (r),

( 2! /2 . f (x) ≥ f (x0 ) 1 − 2 Let (ˆ = (/(2 − () and Bˆ = B(1 − ((/2))−1 . Then for all u ∈ Ᏸ(Bx0 (r)), 2 ˆ

u 22! ≤ K0 (1 − (ˆ ) (Pg u)u dvg + B u

2. M

PANEITZ-TYPE OPERATORS AND APPLICATIONS

153

The same arguments as the ones used in the proof of Lemma 1.2 then lead to a contradiction. It follows that (2.1) holds. Let us now prove the second part of the theorem. Let q ∈ (2, 2! ). Set µq = infq Ig (u), u∈Ᏼf

where q

Ᏼf = u ∈ H22 (M)

M

f |u|q dvg = 1 .

Since the embedding of H22 (M) in Lq (M) is compact, we know from classical variq ational arguments that µq is attained. In other words, there exists uq ∈ Ᏼf such that Ig (uq ) = µq . In particular, uq is a weak solution of

Pg uq + auq = µq f |uq |q−2 uq . By classical bootstrap, uq ∈ Ls (M) for all s. It easily follows that uq is in fact C 3 . Mimicking what is done in Van der Vorst [23], let u˜ q be the solution of α α g u˜ q + u˜ q = g uq + uq . 2 2 Clearly, u˜ q is C 2 , and α g u˜ q ± uq + u˜ q ± uq ≥ 0. 2 It follows from the maximum principle that u˜ q ≥ |uq |, and that u˜ q > 0. Noting that α 2 α 2 g u˜ q + u˜ q dvg = g uq + uq dvg , 2 2 M M it follows from the assumption a ≤ α 2 /4 that α 2 α2 Ig (u˜ q ) = u˜ 2q dvg g u˜ q + u˜ q dvg + a − 2 4 M M α2 2 2 = µq + a − u˜ q dvg − uq dvg ≤ µq . 4 M M

On the other hand,

q

M

f u˜ q dvg ≥

M

f |uq |q dvg .

Hence, uˆ q =

1 q

˜ q dvg Mfu

1/q u˜ q

154

DJADLI, HEBEY, AND LEDOUX

realises µq . Here again, uˆ q is a solution of q−1

Pg uˆ q + a uˆ q = µq f uˆ q

.

By classical regularity, uˆ q is in fact C ∞ . The family (uˆ q ) is obviously bounded in H22 (M). Up to the extraction of a subsequence, and for q → 2! , it converges weakly to some nonnegative u in H22 (M). The embedding H22 (M) ⊂ H12 (M) being compact, we may also assume that it converges strongly to u in H12 (M). It follows from classical arguments that u is a weak solution of Pg u + au = µf u2 where µ is given by

! −1

,

µ = lim sup µq . q→2!

By Lemma 2.1 below, u ∈ Ls (M) for all s. It easily follows that u is C 4 . From the maximum principle, and noting that

α 2 g + u ≥ 0, 2 we get that u is either positive or the zero function. In both cases, it is actually C ∞ . Let µ0 = inf Ig (u). u∈Ᏼf

It is easily seen that µ ≤ µ0 . Coming back to the family (uˆ q ), we have that 2/q q f uˆ q dvg 1=

M

≤ max f

2/q

M

≤ max f M

2/q

q

M

2/q

uˆ q dvg

2(1/q−1/2! )

Vg

2/2!

!

M

uˆ 2q dvg

,

where Vg stands for the volume of M with respect to g. Under the assumption that inequality (2.1) is strict, there exists ( > 0 such that µ0 (K0 + () ≤

1−( (maxM f )2/2

!

.

We ﬁx such an (. It follows from Theorem 1.1 that there exists a constant B, independent of q, such that Pg uˆ q uˆ q dvg + B uˆ q 22 .

uˆ q 22! ≤ (K0 + () M

155

PANEITZ-TYPE OPERATORS AND APPLICATIONS

Therefore,

2/q 2(1/q−1/2! ) 1 ≤ max f Vg (K0 + () µq + C uˆ q 22 , M

where C is independent of q. As q → 2! , and since µ ≤ µ0 , we get that ( ≤ C u 22 . In particular, u ≡ 0. It thus follows that u is a smooth positive solution of (E). We are left with the proof that µ0 is attained. As we easily check, ! f u2 dvg ≤ 1. M

Besides,

M

(Pg u)u dvg + a

M

u2 dvg = µ

!

M

f u2 dvg ,

while according to the deﬁnition of µ0 ,

M

(Pg u)u dvg + a

It follows that

M

µ

f u dvg

M

!

2/2!

f u2 dvg

.

1−2/2!

2!

M

Hence, µ = µ0 and

u2 dvg ≥ µ0

≥ µ0 .

!

M

f u2 dvg = 1.

In particular, u achieves the inﬁmum of the deﬁnition of µ0 . This ends the proof of the theorem. The following lemma, based on ideas developed in Van der Vorst [23], has been used in the proof of Theorem 2.1. Lemma 2.1. Let (M, g) be a smooth, compact, n-dimensional Riemannian manifold, let α be a positive real number, let b be a real-valued function deﬁned on M, and let u ∈ H22 (M) be a weak solution of 2g u + α g u +

α2 u = bu. 4

If b ∈ Ln/4 (M), then u ∈ Ls (M) for all s ≥ 1. Proof. We proceed as in Van der Vorst [23]. As a starting point, we claim that for any ( > 0, there exists q( ∈ Ln/4 (M), f( ∈ L∞ (M), and a constant K( > 0 such that bu = q( u + f( ,

q( n/4 < (,

f( ∞ ≤ K( .

156

DJADLI, HEBEY, AND LEDOUX

Here we may assume that b ≡ 0, and we let

4k = x ∈ M/|b| < k ,

4l = x ∈ M/|u| < l , where if (ˆ is such that (2(ˆ )4/n = (/2, k and l are chosen such that

b Ln/4 (M\4k ) < (ˆ

b Ln/4 (M\4l ) < (ˆ ,

and

4k ∩ 4l = ∅

b ≡ 0

and

on 4k ∩ 4l .

Given p ≥ 1 an integer that we ﬁx below, let 1  b in 4k ∩ 4l , q( = p  b in (M\4k ) ∪ (M\4l ), and

f( = b − q( u.

Clearly, f( = 0 on M\(4k ∩ 4l ). On the other hand, n/4

q( n/4 = |q( |n/4 dvg + |q( |n/4 dvg ≤

4k ∩4l

4k ∩4l

|q( |n/4 dvg +

M\(4k ∩4l )

M\4k

|q( |n/4 dvg +

n/4 1 ≤ |b|n/4 dvg + 2(ˆ p 4k ∩4l so that

q( n/4 ≤

1 1

b n/4 + (. p 2

Choosing p such that p > 2 b n/4 /(, we get that

q( n/4 < (. Now, since f( = 0 on M\(4k ∩ 4l ),

1

f( ∞ ≤ 1 − kl, p

and this proves the above claim. The equation 2g u + α g u +

α2 u = bu 4

M\4l

|q( |n/4 dvg

PANEITZ-TYPE OPERATORS AND APPLICATIONS

157

may now be written as L2g u = q( u + f( , where

α Lg u = g u + u. 2

For any s > 1 and any f ∈ Ls (M), there exists one and only one u ∈ H4s (M) such that L2g u = f . We let Ᏼ( be the operator Ᏼ( u = (Lg )−2 (q( u).

The preceding equation becomes u − Ᏼ( u = (Lg )−2 (f( ). Let v ∈ Ls (M), let s ≥ 2! , and let u( be such that L2g u( = q( v. Set sˆ = ns/(n + 4s). Clearly, q( v ∈ Lsˆ (M), and it follows from elliptic-type arguments that

u( s ≤ C q( v sˆ . By Hölder’s inequality,

q( v sˆ ≤ q( n/4 v s so that

u( s ≤ C( v s . In other words, for all s ≥ 2! , Ᏼ( acts from Ls (M) into Ls (M), and its norm is less than or equal to C(. Let s ≥ 2! be given. For ( > 0 sufﬁciently small, 1

Ᏼ( Ls →Ls < , 2 and the operator

I − Ᏼ( : Ls (M) −→ Ls (M)

has an inverse. Since !

I − Ᏼ( u = (Lg )−2 (f( )

and since u ∈ L2 (M) and f( ∈ L∞ (M), we get that u ∈ Ls (M). The lemma is proved.

158

DJADLI, HEBEY, AND LEDOUX

On what concerns Theorem 2.1, the equality in (2.1) holds when (M, g) is the standard unit sphere (S n , h), Pg + a = Phn , and f is a positive constant. This is just equation (1.9) of Proposition 1.1: n 1 S n Ph u u dvh (2.2) inf 2/2! = K . ! u∈C ∞ (S n )\{0} 2 0 S n |u| dvh Independently, let

u0 ≡

M

f dvg

Clearly, u0 ∈ Ᏼf , and Ig (u0 ) =

−1/2!

aVg

Mf

dvg

.

2/2! .

We then get the following result from Theorem 2.1. Corollary 2.1. Let (M, g) be a smooth, compact, n-dimensional Riemannian manifold, n ≥ 5, let Pg be the operator given by (0.3), let a > 0 be real, and let f be a smooth, positive function deﬁned on M. If a ≤ α 2 /4 and if ! 2! /2−1 M f dvg , (2.3) > (aK0 )2 /2 Vg Vg maxM f where Vg stands for the volume of M with respect to g, then (E) possesses a smooth positive solution. Here again, the standard unit sphere plays a particular role in this result. As already mentioned in the proof of Proposition 1.1, −4/n

dn K0 = ωn

,

(2.4)

where ωn is the volume of S n with respect to h. It follows that if (M, g) = (S n , h) and Pg +a = Phn , then the right-hand side in (2.3) is 1. On the contrary, the left-hand side is always less than or equal to 1. The strict inequality (2.3) is therefore never satisﬁed when (M, g) = (S n , h) and Pg + a = Phn . On the other hand, the condition a ≤ α 2 /4 does hold for Phn , and we indeed do have that dn ≤ cn2 /4. As we easily check, the difﬁculty mentioned above disapears when considering quotients of S n . The volume there becomes smaller, and the following result holds. In the particular case n ≤ 7, see also Theorem 2.2. Corollary 2.2. Let (S n , h) be the standard n-dimensional unit sphere, n ≥ 5. For any ( ∈ (0, 1), there exists an integer k( with the following property: if f smooth on S n is invariant under the action of a subgroup G of O(n + 1) acting freely on S n and of order k ≥ k( , and if f is such that f − 1 C 0 < (, then the equation 2h u + cn h u + dn u = f u2

! −1

159

PANEITZ-TYPE OPERATORS AND APPLICATIONS

possesses a smooth, positive, G-invariant solution. In particular, there exists a metric g in the conformal class of h for which Qng = f , where Qng =

1 n3 − 4n2 + 16n − 16 2 2 g Sg + Sg − | Rcg |2 , 2 2 2(n − 1) 8(n − 1) (n − 2) (n − 2)2

and for which G ⊂ Isomg (S n ), where Isomg (S n ) stands for the isometry group of S n with respect to g. Proof. Let M be the quotient manifold S n /G, and let g0 be its standard metric induced by h. We still denote by f the quotient of f on M. The Paneitz-Branson operator on M is given by Pgn0 u = 2g0 u + cn g0 u + dn u, and as already mentioned, dn ≤ cn2 /4. If f is such that f − 1 C 0 < (, then

Mf

dvg0 1−( > . Vg0 maxM f 1+( Now, (dn K0 )2 Then set

! /2

k( =

2! /2−1

Vg0

1+( 1−(

=

1 ! k 2 /2−1

2/(2! −2)

(dn K0 )2

! /2

2! /2−1

ωn

.

(dn K0 )

2! /(2! −2)

ωn + 1,

where [x] stands for the largest integer not exceeding x. If k ≥ k( , then ! 2! /2−1 M f dvg0 > (dn K0 )2 /2 Vg0 . Vg0 maxM f Noticing that the existence of a solution to the equation on M gives the existence of a G-invariant solution of the equation on S n , the result follows from Theorem 2.1. We now concentrate on the study of (E) when (M, g) is the standard sphere and Pg + a = Phn . The following result, together with Theorem 2.3, shows that cn in the deﬁnition of Phn is critical. In the study of (E) on the standard sphere, we do get obstructions by Theorem 2.3 when α = cn and a = dn . These obstructions disappear according to Corollary 2.3 if α < cn . When studying (E) on the standard sphere, both the medium term α = cn and the nonlinear growth p = 2! − 1 are critical. For more details on such assertions, we refer to the remark after the proof of Theorem 2.3.

160

DJADLI, HEBEY, AND LEDOUX

Corollary 2.3. Let (S n , h) be the standard n-dimensional unit sphere, n ≥ 6, let α and a be two positive real numbers, and let f be a smooth, positive function on S n . If a ≤ α 2 /4, α < cn , and if (n − 6)

h f (x0 ) 8n(n − 1) < (cn − α) f (x0 ) (n − 4)(n + 2)

for at least one x0 where f is maximum, then the equation 2h u + α h u + au = f u2

! −1

possesses a smooth, positive solution. Proof. Let x0 be a point where f is maximum, and r be the distance on S n to x0 . For β > 1, we let uβ be the function uβ =

(β 2 − 1)(n−4)/4 . (β − cos r)(n−4)/2 4/(n−4)

As already mentioned in the proof of Proposition 1.1, h and uβ It follows that ! Phn uβ = dn uβ2 −1 ,

h are isometric.

where Phn is the Paneitz-Branson operator on the sphere, as deﬁned in (0.1). According to the developments made in the proof of Proposition 1.2, (Ph uβ )uβ dvh + a u2β dvh Sn Sn 1 1 + o (β − 1) ln if n = 6 = d6 ω6 + A (α − c6 ) (β − 1) ln β −1 β −1 = dn ωn + 2n−3 (n − 4)2 (α − cn )(β − 1)ωn−1 I + o(β − 1) where A is some positive constant and

+∞

I= 0

We now write

y n+1 dy . (1 + y 2 )n−2

f = f (x0 ) + (1 − cos r)fˆ.

It is easily seen that lim

t→0+

ˆ dσ ωn−1 =− h f (x0 ), n Vh ∂Bx0 (t) ∂Bx0 (t) f

if n > 6,

PANEITZ-TYPE OPERATORS AND APPLICATIONS

161

where Vh (∂Bx0 (t)) stands for the area of ∂Bx0 (t) with respect√to the metric induced by h. By the changes of variables x = tan(r/2), and then y = (β + 1)/(β − 1)x, we get that ! −1 (β − 1) (1 − cos r)u2β dvh Bx0 (t)

2n+1 ωn−1 = β +1

√

(β+1)/(β−1)T

y n+1 dy 1 + ((β − 1)/(β + 1))y 2 (1 + y 2 )n

0

for all t ∈ (0, π ), where T = tan(t/2). It easily follows that 2n ωn−1 ! lim (β − 1) J h f (x0 ), (1 − cos r)fˆu2β dvh = − n β→1+ Sn where

+∞

J= 0

y n+1 dy . (1 + y 2 )n

As a consequence, 2n ωn−1 h f (x0 ) 2! + o(β − 1) . f uβ dvh = f (x0 )ωn 1 − (β − 1)J nωn f (x0 ) Sn −4/n

Since dn K0 = ωn , and since x0 is a point where f is maximum, we get that 2 S n (Ph uβ )uβ dvh + a S n uβ dvh 2/2! 2! S n f uβ dvh 1 = 1 + B (α − c6 ) εβ + o(εβ ) if n = 6 ! K0 (maxM f )2/2 1 1 + C(β − 1) + o(β − 1) if n > 6, = ! K0 (maxM f )2/2 where εβ = (β − 1) ln (1/(β − 1)), B > 0 does not depend on β, and 2n+1 ωn−1 2nI h f (x0 ) (α − cn ) . C= + 2 J 2! nωn f (x0 ) (n − 4)J As we easily check, see, for instance, Demengel-Hebey [8], I=

1 '((n + 2)/2)'((n − 6)/2) 2 '(n − 2)

Hence,

and

J=

1 '((n + 2)/2)'((n − 2)/2) . 2 '(n)

8n(n − 1) 2nI = . (n − 6)(n − 4)(n + 2) (n2 − 4)J

162

DJADLI, HEBEY, AND LEDOUX

Under our assumptions C < 0, so that for β > 1 sufﬁciently close to 1, 2 1 S n (Ph uβ )uβ dvh + a S n uβ dvh < . ! ! ! 2/2 2 dv K0 (maxM f )2/2 f u n h β S The result now follows from Theorem 2.1. The equation involved in the study of the prescribed scalar curvature problem on the sphere, also referred to as the Kazdan-Warner problem or the Nirenberg problem, is the equation n(n − 2) 7 u = f u2 −1 , h u + 4 where 27 = 2n/(n − 2). In the study of this equation, a celebrated result of Escobar and Schoen [11] states that if n = 3 and if f is invariant under the action of a nontrivial subgroup G of O(4) acting freely on S 3 , then the above equation possesses a smooth, positive G-invariant solution. In particular, under these assumptions, f is the scalar curvature of a G-invariant conformal metric to h. The same result was proved by Moser [20] when n = 2 and f is assumed to be invariant under the action of the antipodal group G = {I d, −I d}, the only group acting freely on S n when the dimension n is even. A natural question is whether such types of results do hold for the equation ! Phn u = f u2 −1 . This is the subject of the following theorem. As a ﬁrst remark, note that by EdmundsFortunato-Janelli [10] and Pucci-Serrin [22], low dimensions for the Euclidean biharmonic operator are n = 5, 6, 7. As another remark, we mention that there should be an analogue of our result when G acts without ﬁxed points (i.e., for any x, the G-orbit of x has at least two elements). Concerning the above mentioned scalar curvature problem on the sphere, this was proved by Hebey [13]. Theorem 2.2. Let (S n , h) be the standard n-dimensional unit sphere, n = 5, 6, or 7, and let f be a smooth positive function on S n . We assume that f is invariant under the action of a nontrivial subgroup G of O(n + 1) acting freely on S n , and if n = 6 or 7, we assume that h f (x) = 0 for at least one x where f is maximum. Then the equation ! Phn u = f u2 −1 possesses a smooth, positive G-invariant solution. In particular, there exists a metric g in the conformal class of h for which Qng = f , where Qng =

1 n3 − 4n2 + 16n − 16 2 2 g Sg + Sg − | Rcg |2 , 2 2 2(n − 1) 8(n − 1) (n − 2) (n − 2)2

and for which G ⊂ Isomg (S n ), where Isomg (S n ) stands for the isometry group of S n with respect to g.

163

PANEITZ-TYPE OPERATORS AND APPLICATIONS

Proof. Let M be the quotient manifold S n /G, and let g0 be its standard metric induced by h. Also let u ∈ H22 (M), u ≡ 0, and let u˜ be the function on S n induced by u. As we easily check, n n ˜ u˜ dvg0 ˜ u˜ dvh M Pg0 u S n Ph u 1−2/2! 2/2! , 2/2! = k ! ! 2 dv 2 dv f | u| ˜ f | u| ˜ g n h 0 M S where k is the number of elements in G, and f in the right-hand side of this relation stands for the quotient of f on M. The existence of a solution to the equation on M leads to the existence of a G-invariant solution to the equation on S n . Therefore, as a consequence of Theorem 2.1, it sufﬁces to show that

! Phn u u dvh k 1−2/2 inf 2/2! < (max n f )2/2! K , u∈, 2! S 0 S n f |u| dvh Sn

(2.5)

where , stands for the subset of H22 (S n ) consisting of nonzero G-invariant functions. Now let x1 be a point where f is maximum, and denote by OG (x1 ) = {x1 , . . . , xk } the G-orbit of x1 . If ri stands for the distance on S n to xi , let ui,β , β > 1 be the functions on S n deﬁned by ui,β =

(β 2 − 1)(n−4)/4 . (β − cos ri )(n−4)/2

4/(n−4)

As already mentioned, h and ui,β

h are isometric. In particular, !

2 −1 Phn ui,β = dn ui,β

and

!

Sn

u2i,β dvh = ωn .

Then let uβ =

k

ui,β .

i=1

On the one hand, uβ is G-invariant. On the other hand, Sn

Phn uβ

uβ dvh = kdn ωn + kdn

S

2! −1 u1,β n

k i=2

ui,β

dvh .

164

DJADLI, HEBEY, AND LEDOUX

Set

A(n) = 2

1+3n/4

ωn−1

+∞ 0

y n−1 dy . (1 + y 2 )(n+4)/2

We claim that for all t in (0, π), lim (β − 1)

!

1−n/4

β→1+

Bx1 (t)

2 −1 u1,β dvh = A(n).

Indeed, by the change of variables x = tan(r1 /2) and y = get that

√

(β + 1)/(β − 1)x, we

!

Bx1 (t)

2 −1 u1,β dvh

= C1 (n, β)(β − 1)n/4−1 √(β+1)/(β−1)T y n−1 dy × , (n−4)/2 0 (1 + y 2 )(n+4)/2 1 + ((β − 1)/(β + 1))y 2 where C1 (n, β) = 2n (β + 1)1−n/4 ωn−1 and T = tan(t/2). The above claim then easily follows. With easier arguments, for all h ∈ C 0 (S n ), and all t in (0, π), lim (β − 1)

2−n/2

β→1+

S n \Bx1 (t)

2! −1 hu1,β

k

ui,β dvh = 0.

i=2

It follows from these two relations that for all h ∈ C 0 (S n ), all t in (0, π), and all open subset 4 of S n that contains x1 , lim (β − 1)

β→1+

2−n/2 4

2! −1 hu1,β

k

ui,β dvh = A(n)

i=2

k

u˜ i,1 (x1 ) h(x1 ),

(2.6)

i=2

where u˜ i,1 = (1 − cos ri )2−n/2 . Then let t0 > 0 be such that Bxi (t0 ) ∩ Bxj (t0 ) = ∅ for i = j . Since !

!

(a + b)2 ≥ a 2 + 2! a 2

! −1

b,

165

PANEITZ-TYPE OPERATORS AND APPLICATIONS

we may write that Sn

!

f u2β dvh ≥ ≥

k

!

i=1 Bxi (t0 )

k

!

i=1 Bxi (t0 )

=k

Bx1 (t0 )

f u2β dvh f u2i,β dvh + 2!

! f u21,β dvh + 2! k

It follows that 2! f uβ dvh ≥ kf (x1 )ωn − kf (x1 ) Sn

+k

Bx1 (t0 )

k

!

i=1 j =i Bxi (t0 )

Bx1 (t0 )

2! −1 f u1,β

2 −1 f ui,β uj,β dvh

k

ui,β dvh .

i=2

!

S n \Bx1 (t0 )

2!

f − f (x1 ) u1,β dvh + 2! k

u21,β dvh

2! −1

Bx1 (t0 )

f u1,β

k

ui,β dvh .

i=2

It is easily seen that lim (β − 1)2−n/2

β→1+

On the other hand, for any 5 ≤ n ≤ 7, 2−n/2 lim (β − 1) β→1+

!

S n \Bx1 (t0 )

Bx1 (t0 )

u21,β dvh = 0.

! f − f (x1 ) u21,β dvh = 0.

(2.7)

(2.8)

Indeed, suppose that n = 5. Since x1 is a critical point for f , there exists a constant C > 0 such that for all x ∈ Bx1 (t0 ), f (x) − f (x1 ) ≤ C(1 − cos r1 ). √ With the change of variables x = tan(r1 /2) and y = (β + 1)/(β − 1)x, we get that 2−n/2 2! (β − 1) (1 − cos r1 )u1,β dvh = O (β − 1)3−n/2 , Bx1 (t0 )

from which (2.8) follows. Suppose then that n = 6 or 7 and that x1 is such that h f (x1 ) = 0. We may write that there exists a constant C > 0 such that for all x ∈ Bx1 (t0 ), f (x) − f (x1 ) ≤ C(1 − cos r1 )2 .

166

DJADLI, HEBEY, AND LEDOUX

As above, we get that (β − 1)

2−n/2 Bx1 (t0 )

! (1 − cos r1 )2 u21,β dvh = O (β − 1)4−n/2 ,

from which (2.8) also follows. Now, by (2.4), (2.6), (2.7), and (2.8), n ! 1 + (β − 1)n/2−2 Ak (n) + o (β − 1)n/2−2 k 1−2/2 S n Ph uβ uβ dvh , × ! = ! 2! dv 2/2 f (x1 )2/2 K0 1 + 2(β − 1)n/2−2 Ak (n) + o (β − 1)n/2−2 f u n h β S where Ak (n) > 0 is given by k

Ak (n) =

A(n) u˜ i,1 (x1 ). ωn i=2

Hence, for every β > 1 sufﬁciently close to 1, n ! k 1−2/2 S n Ph uβ uβ dvh . 2/2! < ! 2! f (x1 )2/2 K0 S n f uβ dvh In particular, since uβ is G-invariant, and f (x1 ) = maxS n f , n ! k 1−2/2 S n Ph u u dvh inf ! < (max n f )2/2! K . u∈, 2! dv 2/2 S 0 f |u| n h S This is exactly inequality (2.5). The theorem is thus proved. A celebrated result of Kazdan and Warner [17] states that the scalar curvature equation on the sphere (S n , h) possesses obstructions. We prove here that such obstructions hold similarly for the equation Phn u = f u2

! −1

.

In the statement of Theorem 2.3, (∇f ∇ϕ) stands for the pointwise scalar product with respect to h of ∇f and ∇ϕ. Theorem 2.3. Let (S n , h) be the standard n-dimensional unit sphere, n ≥ 5, and let f be a smooth function on S n , positive somewhere on S n . If u is a smooth positive solution of the equation Phn u = f u2

! −1

,

(2.9)

where Phn is as in (0.1), then for any eigenfunction ϕ of h associated to the ﬁrst nonzero eigenvalue λ1 = n, ! (∇f ∇ϕ)u2 dvh = 0. Sn

PANEITZ-TYPE OPERATORS AND APPLICATIONS

167

In particular, for any ( > 0 and any eigenfunction ϕ ≡ 0 of h associated to the ﬁrst nonzero eigenvalue λ1 = n, (2.9) with f = 1+(ϕ does not possess a smooth, positive solution. Proof. The proof mainly follows what was done in Kazdan and Warner [17]. Let ϕ be an eigenfunction of h associated to the ﬁrst nonzero eigenvalue λ1 = n, and let u be a smooth function on S n . As it is easy to see, 2 h u (∇u∇ϕ) # ( h u)(∇ h u∇ϕ) − (n − 2)( h u)(∇u∇ϕ) − 2( h u)2 ϕ, where the sign “#” means that the relation holds modulo terms in divergence form. Clearly, 1 ( h u)(∇ h u∇ϕ) # nϕ( h u)2 2 so that n−4 ( h u)2 ϕ − (n − 2)( h u)(∇u∇ϕ). ( h u)2 (∇u∇ϕ) # 2 Suppose now that u is a solution of (2.9). Then !

( h u)2 ϕ # 2( h u)(∇u∇ϕ) − nuϕ( h u) + f ϕu2 − dn u2 ϕ − cn uϕ( h u) so that n−4 (n + cn )uϕ( h u) + dn u(∇u∇ϕ) 2 n−4 n−4 ! ! + f ϕu2 − dn u2 ϕ # f u2 −1 (∇u∇ϕ). 2 2

(cn − 2)( h u)(∇u∇ϕ) −

Since

n−2 ( h u)(∇u∇ϕ) # − 2

1 2 nu ϕ − uϕ( h u) 2

and

n−4 n−2 (cn − 2) = (n + cn ), 2 2 the terms uϕ( h u) disappear. We then get that −

n−4 n−4 n−2 ! ! (cn − 2)nu2 ϕ + dn u(∇u∇ϕ) + f ϕu2 − dn u2 ϕ # f u2 −1 (∇u∇ϕ). 4 2 2

Now it is easily seen that u(∇u∇ϕ) # and that f u2

! −1

(∇u∇ϕ) # −

n 2 u ϕ 2

1 2! n ! u (∇f ∇ϕ) + ! f u2 ϕ. ! 2 2

168

DJADLI, HEBEY, AND LEDOUX

Therefore,

n(n − 2) 1 ! (cn − 2) u2 ϕ + ! u2 (∇f ∇ϕ) # 0. 2dn − 4 2

Since 2dn = we ﬁnd that

n(n − 2) (cn − 2), 4

!

u2 (∇f ∇ϕ) # 0. This ends the proof of the theorem. To conclude, we collect a few remarks on Corollary 2.3 and Theorem 2.3. First, let ϕ be an eigenfunction of h associated to the ﬁrst nonzero eigenvalue λ1 = n, and, for ε > 0, set fε = 1 + εϕ. Given α > 0, consider the equation !

2h u + α h u + dn u = fε u2 −1 . (Eαε ) √ According to Corollary 2.3, if n > 6 and α ∈ [2 dn , cn ), then there exists εα > 0 such that if ε ≤ εα , (Eαε ) possesses a smooth positive solution. On the contrary, by Theorem 2.3, for all n ≥ 5 and all ε > 0, (Ecεn ) does not possess any smooth positive solution. This is one of the possible illustrations of the criticality of cn we mentioned before stating Corollary 2.3. As another remark, note that Theorem 2.3, together with Theorem 2.1, gives another proof of (1.9). Indeed, suppose by contradiction that (1.9) is false. In other words, assume that n 1 S n Ph u u dvh infn < . ! ∞ 2/2 ! u∈C (S )\{0} K0 |u|2 dv Sn

h

Then for any f sufﬁciently close to 1 in the C 0 -topology, n 1 S n Ph u u dvh inf 2/2! < (max n f )2/2! K . ! u∈C ∞ (S n )\{0} 2 S 0 S n f |u| dvh It follows from Theorem 2.1 that for such an f , the equation Phn u = f u2

! −1

has a smooth, positive solution. This is in contradiction to the last part of Theorem 2.3 and thus proves (1.9). References [1] [2]

M. T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102 (1990), 429–445. W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. (2) 138 (1993), 213–242.

PANEITZ-TYPE OPERATORS AND APPLICATIONS [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

169

T. P. Branson, Group representations arising from Lorentz conformal geometry, J. Funct. Anal. 74 (1987), 199–291. T. P. Branson, S.-Y. A. Chang, and P. C. Yang, Estimates and extremals for zeta function determinants on four-manifolds, Comm. Math. Phys. 149 (1992), 241–262. S.-Y. A. Chang, On a fourth order PDE in conformal geometry, preprint, 1997. S.-Y. A. Chang, M. J. Gursky, and P. C. Yang, Regularity of a fourth order nonlinear PDE with critical exponent, Amer J. Math. 121 (1999), 215–257. S.-Y. A. Chang and P. C. Yang, Extremal metrics of zeta function determinants on 4-manifolds, Ann. of Math. (2) 142 (1995), 171–212. F. Demengel and E. Hebey, On some nonlinear equations involving the p-Laplacian with critical Sobolev growth, Adv. Differential Equations 3 (1998), 533–574. Z. Djadli, E. Hebey, and M. Ledoux, Sharp inequalities involving Paneitz-type operators, preprint, 1999. D. E. Edmunds, F. Fortunato, and E. Janelli, Critical exponents, critical dimensions, and the biharmonic operator, Arch. Rational Mech. Anal. 112 (1990), 269–289. F. Escobar and R. Schoen, Conformal metrics with prescribed scalar curvature, Invent. Math. 86 (1986), 243–254. M. J. Gursky, The Weyl functional, de Rham cohomology, and Kähler-Einstein metrics, Ann. of Math. (2) 148 (1998), 315–337. E. Hebey, Changements de métriques conformes sur la sphère: Le problème de Nirenberg, Bull. Sci. Math. (2) 114 (1990), 215–242. , Sobolev Spaces on Riemannian Manifolds, Lecture Notes in Math. 1635, Springer, Berlin, 1996. , Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lect. Notes Math. 5, Courant Institute of Mathematical Sciences, New York, 1999. S. Ilias, Constantes explicites pour les inégalités de Sobolev sur les variétés riemanniennes compactes, Ann. Inst. Fourier (Grenoble) 33 (1983), 151–165. J. L. Kazdan and F. W. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geom. 10 (1975), 113–134. E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), 349–374. P.-L. Lions, The concentration-compactness principle in the calculus of variations: The limit case, I, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145–201; II, (1985), no. 2, 45–121. J. Moser, “On a nonlinear problem in differential geometry” in Dynamical Systems (Salvador, Brazil, 1971), Academic Press, New York, 1973, 273–280. S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudoRiemannian manifolds, preprint, 1983. P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl. (9) 69 (1990), 55–83. R. C. A. M. van der Vorst, Best constants for the embedding of the space H 2 ∩ H01 (4) into L2N/(N −4) (4), Differential Integral Equations 6 (1993), 259–276.

Djadli: Université de Cergy-Pontoise, Département de Mathématiques, Site de SaintMartin, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France; Zindine.Djadli @math.u-cergy.fr Hebey: Université de Cergy-Pontoise, Département de Mathématiques, Site de SaintMartin, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France; Emmanuel.Hebey @math.u-cergy.fr Ledoux: Université Paul Sabatier, Département de Mathématiques, 118 Route de Barbonne, 31062 Toulouse, France; [email protected]

Vol. 104, No. 1

DUKE MATHEMATICAL JOURNAL

© 2000

THE GAUSS MAP AND A NONCOMPACT RIEMANN-ROCH FORMULA FOR CONSTRUCTIBLE SHEAVES ON SEMIABELIAN VARIETIES J. FRANECKI and M. KAPRANOV 1. Introduction and statement of results. Let X be a smooth algebraic variety over C, and let F be a constructible sheaf of C-vector spaces on X. As in other situations, we have the Riemann-Roch problem: express χ(X, F) = i (−1)i dim H i (X, F) in terms of some intrinsic geometric invariants of X and F. One such invariant is the characteristic cycle CC(F), which is a formal Z-linear combination ν nν [ ν ] of irreducible conic Lagrangian subvarieties ν in the cotangent bundle T ∗ X; see [14] for background. When X is compact, the Riemann-Roch problem has a nice solution, namely (see [13]), χ (X, F) = CC(F), [X] T ∗ X , (1.1) where the right-hand side is the intersection index, in T ∗ X, of CC(F) and the zero section X ⊂ T ∗ X. It can be calculated, for instance, by ﬁrst deforming X to the graph of a C ∞ 1-form so that the intersection becomes transverse, and then counting intersection points (with multiplicities and signs). Both the deﬁnition of CC(F) and the formula (1.1) (for compact X) extend to the case when F is a bounded constructible complex (i.e., a complex of sheaves with constructible cohomology). When X is not compact, χ(X, F) still makes sense, but (1.1) is not applicable. We face, therefore, an interesting noncompact Riemann-Roch problem of ﬁnding χ (X, F) in terms of invariants intrinsic to X (in particular, not involving the choice of compactiﬁcation). The purpose of this paper is to exhibit such a “noncompact Riemann-Roch formula” in a particular class of situations, namely, suppose that X = G is an algebraic group with Lie algebra g. For γ ∈ g∗ let ωγ be the corresponding left-invariant 1-form on G, and let γ ⊂ T ∗ G be its graph. The γ then form a natural family of deformations of X, and we can use them to make sense of the intersection index in (1.1) even when G is not compact. More precisely, if ⊂ T ∗ G is an irreducible conic Lagrangian subvariety and if γ ∈ g∗ is generic, then ∩ γ consists of ﬁnitely many transversal intersection points; their number is denoted gdeg( ) and called the Gaussian degree of . To explain the name, recall that has the form TZ∗ X for an irreducible subvariety Received 10 December 1999. Revision received 21 December 1999. 2000 Mathematics Subject Classiﬁcation. Primary 32S60; Secondary 20G20. Kapranov partially supported by National Science Foundation grant number 9623044. 171

172

FRANECKI AND KAPRANOV

Z ⊂ X (notation: TZ∗ X always means the closure of the conormal bundle to the smooth locus Zsm of Z). Denoting k = dim(Z), we have the left Gauss map Z : Z −→ G(k, g),

z −→ z−1 (Tz Z) ⊂ Te G = g,

which is a rational map, regular on Zsm . The number gdeg( ) is the degree of Z in an appropriate sense; see Section 2. For example, if Z is a hypersurface, then the source and the target of Z have the same dimension, and gdeg( ) is the degree of Z in the usual sense. Note that gdeg( ) ≥ 0 by construction. We now formulate the main result of this note. Recall (see [2]) that a semiabelian variety is an algebraic group G, which is an extension (1.2)

1 −→ T −→ G −→ A −→ 1,

where A is an abelian variety and T ∼ = (C∗ )n is an algebraic torus. Theorem 1.3. Let G be asemiabelian variety, let F be a bounded constructible complex on G and CC(F) = ν nν [ ν ]. Then nν gdeg( ν ). χ (G, F) = ν

Corollary 1.4. If, in the situation of Theorem 1.3, F is a perverse sheaf, then χ (G, F)≥ 0. Indeed, for F a perverse sheaf, all nν ≥ 0. Here we use the conventions of [1] for the deﬁnition of (middle) perversity. Corollary 1.5. If G is semiabelian and if Z ⊂ G is a smooth closed subvariety, then the number (−1)dim(Z) χ (Z, C) is nonnegative and coincides with gdeg(Z). Indeed, CZ [dim(Z)] is perverse and its characteristic cycle is [TZ∗ G] taken with multiplicity 1. Corollary 1.4 for G = (C∗ )n was proven by Loeser-Sabbah [15] and given a different proof (applicable to étale sheaves) by Gabber-Loeser [4]. In the case when G is an abelian variety, Corollary 1.4 seems to be new, even though Theorem 1.3 in this case does not need a special proof, being a consequence of (1.1). Moreover, since the general proof of Theorem 1.3 given below is extremely simple and transparent, we believe that our approach exhibits the true reason behind the Loeser-Sabbah observation. Further, another result of [4] identiﬁes irreducible perverse sheaves F on (C∗ )n such that χ ((C∗ )n , F) = 1, with complexes of solutions of hypergeometric systems (essentially, of the A-hypergeometric systems; see [5], [11]; see [6] for a comparison of the two points of view). On the other hand, in [12], the second author classiﬁed irreducible hypersurfaces in (C∗ )n for which the Gauss map has degree 1 and identiﬁed them with (reduced) A-discriminantal hypersurfaces. The latter describe the characteristic varieties of the A-hypergeometric system. Thus our approach explains the analogy between these two results.

A NONCOMPACT RIEMANN-ROCH FORMULA

173

Unfortunately, Theorem 1.3 cannot be straightforwardly generalized to more general algebraic groups. For example, if G is nonabelian reductive, then it contains afﬁne spaces Am for m ranging from zero to the number of positive roots of G. The Euler characteristic of Am being 1, Corollary 1.5 (and thus Theorem 1.3) cannot hold. The same applies when G, while commutative, contains Ga , the additive group. Nevertheless, we believe that there should exist generalizations of Theorem 1.3 that involve some particular classes of constructible sheaves and complexes. We are grateful to V. Ginzburg for several useful remarks on the ﬁrst version of the text. A part of the results of this paper were included in the thesis of the ﬁrst author [3]. 2. The Gauss map and the Gaussian degree. Let G be a complex algebraic group, let g be its Lie algebra, and let G(k, g) be the Grassmannian of k-dimensional linear subspaces in g. If Z ⊂ G is an irreducible k-dimensional subvariety, we have the rational map Z : Z −→ G(k, g) called the (left) Gauss map and deﬁned as follows. For x ∈ G, let lx : G −→ G,

lx (y) = xy

be the left translation by x. Then, for a smooth point z ∈ Z, the value Z (z) is the image of dx lx −1 : Tz Z −→ Te G = g. We want to associate to Z a nonnegative integer called its degree. To this end, consider ﬁrst a more general situation. Let M be an irreducible k-dimensional variety, let V be an n-dimensional vector space, and let f : M → G(k, V ) be a rational map. Replacing, if necessary, M with its Zariski open subset, we can assume that f is regular. Consider the ﬂag variety F (k, n − 1, V ) and its projections p

q

G(k, V ) ←− F (k, n − 1, V ) −→ G(n − 1, V ) = P (V ∗ ). = G(k, V )×M F (k, n − 1, V ) be the ﬁber product with respect to f and p. Let M is an irreducible Since p is a smooth map with (n − k − 1)-dimensional ﬁbers, M → P(V ∗ ), variety of dimension n − 1. The map q induces a regular map q : M whose source and target have the same dimension. We deﬁne deg(f ) to be the degree of the map q . The following is then clear. Proposition 2.1. If W ⊂ V is a generic hyperplane, then deg(f ) is equal to the number of x ∈ M such that f (x) ∈ G(k, W ) ⊂ G(k, V ). For any such x, the map f is locally (in the analytic or étale topology) an embedding near x and the intersection of f (M) and G(k, W ) is transversal at x.

174

FRANECKI AND KAPRANOV

= M, and deg(f ) is the degree of f in the usual When k = n − 1, we have M sense. We now specialize to the case where M = Z, V = g, n = dim(G), and f = Z . The number deg(Z ) is denoted by gdeg(Z) and called the Gaussian degree of Z. Let = TZ∗ G be the conic Lagrangian variety associated to Z. We write gdeg( ) for gdeg(Z). As in Section 1, for γ ∈ g∗ , let γ ⊂ T ∗ X be the graph of the leftinvariant 1-form ωγ on G associated to γ . Proposition 2.1 easily implies the following proposition. Proposition 2.2. Let γ ∈ g∗ be a generic linear functional. Then ∩ γ consists of ﬁnitely many points that are smooth on and in which the intersection is transverse. The number of these points is equal to gdeg( ). 3. Characteristic cycle of an open embedding. A nonintrinsic way to ﬁnd χ(U, F), where U is a noncompact manifold, is to apply (1.1) to Rj∗ F, where j : U *→ X is a smooth compactiﬁcation. We do indeed use this approach in the proof of Theorem 1.3, so we recall the (now well-known) procedure of ﬁnding CC(Rj∗ F) from CC(F); see [7], [17]. Let X be a not necessarily compact smooth variety, let f ∈ C[X] be a regular function, let U ⊂ X be the open set {f = 0}, and let j : U *→ X be the embedding. Let ⊂ T ∗ U be an irreducible conic Lagrangian variety. For s ∈ C∗ , let

#s = + sd log f = ξ + s(d log f )(x), x | (x, ξ ) ∈ . (3.1) This is a closed (no longer conic) Lagrangian subvariety in T ∗ X. The total space of the family of #s is a subvariety # ⊂ T ∗ X × C∗ . The limit lims→0 #s (also called the specialization of # in [7]) is an effective Lagrangian cycle in T ∗ X deﬁned as follows. We ﬁrst take the closure # in T ∗ X × C and then form the schemetheoretic intersection # ∩ (T ∗ X × {0}). The cycle lims→0 #s is obtained by taking the irreducible components of this intersection with the multiplicities given by the scheme structure. We extend this construction by Z-linearity to conic Lagrangian cycles in T ∗ U . Thus, if , is such a cycle, we have the family of nonconic cycles ,s# , s ∈ C∗ and the conic cycle lims→0 ,s# in T ∗ X. Now, the fact we need is as follows. Theorem 3.2. If F is a bounded constructible complex on U , then CC(Rj∗ F) = lim CC(F)#s . s→0

This statement can be obtained from [7, Theorem 3.2] by applying the RiemannHilbert correspondence, or from [17, Theorem 3.1], which is applicable to the more general case of R-constructible sheaves. (To be precise, the concepts of the characteristic cycle used in [7]–[9] and [13], [14], and [17] refer to different contexts: holonomic D-modules vs. constructible complexes. The compatibility of these two

A NONCOMPACT RIEMANN-ROCH FORMULA

175

deﬁnitions of the characteristic cycle under the Riemann-Hilbert correspondence follows from the results of [7].) Consider now a nominally more general situation (cf. [9, Appendix A]): let X be as before but suppose that we have n regular functions f1 , . . . , fn ∈ C[X]. Let U be the intersection of the n open sets {fi = 0}, and let j : U *→ X be the embedding. Of course, this situation can be analyzed by applying Theorem 3.2 to f = f1 · · · fn , but it is convenient for us to have a more ﬂexible formulation. For a point s = (s1 , . . . , sn ) ∈ (C∗ )n and a conic Lagrangian variety ⊂ T ∗ U , we form, similarly to (3.1), a nonconic Lagrangian variety

#s = +

(3.3)

n

si d log fi ⊂ T ∗ X.

i=1

The total space of this family lies in T ∗ X × (C∗ )n . Taking the closure in T ∗ X × Cn and then intersecting with T ∗ X × {(0, . . . , 0)} deﬁnes, similarly to the above, a conic Lagrangian cycle lims→(0,...,0) #s . Of course, this “limit” could be taken along any curve in Cn passing through zero and generically lying in (C∗ )n . As before, we extend this construction by linearity to conic Lagrangian cycles in T ∗ U . The next statement follows by iterated application of Theorem 3.2. Theorem 3.4. If F is a bounded constructible complex on U , then CC(Rj∗ F) =

lim

s→(0,...,0)

CC(F)#s ,

s = (s1 , . . . , sn ) ∈ (C∗ )n .

Taking the limit along different curves approaching (0, . . . , 0) corresponds, roughly, to different choices of our equation for the reducible hypersurface X\U . For example, mi restricting tomithe curve with parametric equation si = t , mi > 0 corresponds to taking i fi as an equation. We now need a slight globalization of Theorem 3.4. First of all, let (L, ∇) be a line bundle on X with an algebraic ﬂat connection. If f is a regular section of L, then f −1 ∇f is a scalar 1-form regular over the open set {f = 0}. We denote this form by ∇ log f . Suppose now that we have n line bundles with ﬂat connections (Li , ∇i ) on X, i = 1, . . . , n. Suppose fi ∈ (X, Li ), i = 1, . . . , n and U ⊂ X is the intersection of the open sets {fi = 0}. As before, let j : U *→ X be the embedding. Theorem 3.5. For a bounded constructible complex F on U , we have

n CC(Rj∗ F) = lim CC(F) + si ∇i log fi . s→(0,...,0)

i=1

Proof. As before, it is enough to consider the case n = 1, as the general case can be obtained by iteration, as in [9, Appendix A]. The statement for n = 1 is a consequence of [8, Theorem 6.3], which deals with the more general case of the zero

176

FRANECKI AND KAPRANOV

locus of a section f of an arbitrary line bundle L, not necessarily with connection. The recipe in this case is to consider the “twisted cotangent bundles” (T ∗ X)(s) , s ∈ C, deﬁned as the symplectic quotients of T ∗ L by the hamiltonian action of C∗ induced by dilations of L. Now, if L is equipped with a ﬂat connection ∇, then all the (T ∗ X)(s) become identiﬁed with T ∗ X and the formulation of [8, Theorem 6.3] reduces to our statement. 4. Proof of Theorem 1.3 for G = (C∗ )n . We ﬁrst consider the case when G = (C∗ )n is an algebraic torus. Let z1 , . . . , zn , zi = 0, be the standard coordinates in (C∗ )n . We compactify G by the projective space Pn with homogeneous coordinates (t0 : · · · : tn ) by j : (C∗ )n *→ Pn ,

(z1 , . . . , zn ) −→ (1 : z1 : · · · : zn ).

For ν = 0, . . . , n let Anν ⊂ Pn be the afﬁne chart given by tν = 0. This is an afﬁne (ν) (ν) space with coordinates zi , i ∈ {0, . . . , n} \ {ν} given by zi = ti /tν . Denote by jν

kν

(C∗ )n *→ Anν *→ Pn (0)

the embeddings. For ν = 0, we have zi = zi . We now apply Theorem 3.4 to U = (C∗ )n , X = An0 , fi = zi , and our constructible complex F. The recipe of the theorem requires us to introduce the family of 1-forms ωs =

n

si d log zi ,

s = (s1 , . . . , sn ) ∈ (C∗ )n .

i=1

These forms are precisely the invariant 1-forms on G = (C∗ )n . We can view s as an element of g∗ , where g = Cn is the Lie algebra of G. Theorem 3.4 then gives us CC(Rj0∗ F) = CC(Rj∗ F)|An0 = lim (CC(F) + ωs ),

(4.1)

s→0

the limit being taken in T ∗ An0 . Next, we apply Theorem 3.4 to U = (C∗ )n and X = Anν with arbitrary ν ∈ (ν) {0, . . . , n}. Then we should take fi = zi , i ∈ {0, . . . , n} \ {ν} and consider the 1forms (ν) (ν) ωs = si d log zi , s ∈ (C∗ ){0,...,n}\{ν} . i=ν

(ν)

(ν)

Now, each zi is a Laurent monomial in the z1 , . . . , zn , so d log zi is an (integer) (ν) linear combination of d log z1 , . . . , d log zn . Therefore, ωs = ωs , where s = φν (s ) is an image of s under a linear transformation φν : C{0,...,n}\{ν} → Cn . It is clear that φν is invertible; in particular, s → 0 if and only if s → 0. This means that the answers

A NONCOMPACT RIEMANN-ROCH FORMULA

177

given by Theorem 3.4 for the CC(Rjν∗ F) glue together into one global answer, as follows: CC(Rj∗ F) = lim CC(F) + ωs . (4.2) s→0 s∈E

Here the limit is taken in T ∗ Pn and s runs over the set E = (s1 , . . . , sn ) ∈ (C∗ )n : si = sj , i = j . (The condition s ∈ E is equivalent to φν−1 (s) ∈ (C∗ ){0,...,n}\{ν} for any ν.) Now, Theorem 1.3 for G = (C∗ )n would follow from (4.2) and the next lemma. Lemma 4.3. Let ⊂ T ∗ (C∗ )n be an irreducible conic Lagrangian variety. Then    lim ( + ωs ), [Pn ] s→0 s∈E

= gdeg( ).

T ∗ Pn

Proof. Let s ⊂ T ∗ (C∗ )n be the graph of ωs . If s ∈ E, then s is closed in T ∗ Pn as well. By translation, intersecting + ωs with [Pn ] is equivalent to intersecting

with −s . By Proposition 2.2, there exists a Zariski open, nonempty set F ⊂ Cn such that for s (0) ∈ F , the intersection ∩ −s (0) consists of gdeg( ) smooth transverse points. Since E is also Zariski open, F ∩ E meets any polydisk Pε = (z1 , . . . , zn ) ∈ (C∗ )n : 0 < |zi | < ε in an open dense set. By the “continuity of intersection,” it follows that for |ε| 1 and s (0) ∈ Pε ∩ F ∩ E,    lim ( + ωs ), [Pn ] s→0 s∈E

= |−s (0) ∩ | = gdeg( );

T ∗ Pn

this completes the proof. 5. Proof of Theorem 1.3 in general. Let A be an abelian variety. We recall some elementary properties of line bundles on A (see [10], [16]). If L is such a bundle, by La , a ∈ A we denote its ﬁber at a. By L0 we denote the total space of L with the zero section deleted, so L0 is a principal C∗ -bundle on A. As for any base variety, the correspondence L → L0 is an equivalence between the category of line bundles and isomorphisms and the category of principal C∗ -bundles. Assume that L has degree zero. Then the theorem of the square (see [16]) provides identiﬁcations La ⊗ Lb −→L L0 ∼ a+b = C,

178

FRANECKI AND KAPRANOV

that make L0 into a group, namely, a semiabelian variety ﬁtting into an extension (5.1)

p

0 −→ C∗ −→ L0 −→ A −→ 0.

Next, any bundle L of degree zero has a ﬂat connection. All such connections form an afﬁne space Conn(L) over the vector space H 0 (A, 1 ) = a∗ . Here a is the Lie algebra of A. Let L be the Lie algebra of the group L0 , so that we have the exact sequence (5.2)

π

0 −→ a∗ −→ L∗ −→ C −→ 0.

= p∗ L the pullback of L to L0 . For any ∇ ∈ Conn(L), let ∇ be its Denote by L over pullback to a connection in L. Denote also by f the tautological section of L 0 L (given by the identity map). log f , ∇ ∈ Conn(L) are invariant (with respect Proposition 5.3. The 1-forms ∇ to the group structure on L0 ). Their images in L∗ under the evaluation at 0 form the subspace π −1 (1), where π is as in (5.2). log f is just the Lie algebraProof. Let us ﬁrst show the invariance. Note that ∇ valued 1-form on the total space of the principal C∗ -bundle L0 , describing the connection ∇ in L0 in the standard approach of differential geometry. So its invariance follows from the fact that the line bundle (L, ∇) (or, what is equivalent, the C∗ -bundle (L0 , ∇)) satisﬁes the theorem on the square as a bundle with connection. More precisely, if m, q1 , q2 : A × A → A are the group structure and the two projections, then the isomorphism µ : q1∗ L ⊗ q2∗ L −→ m∗ L given by the theorem on the square is an isomorphism of bundles with connection. In particular, the induced isomorphism µa : La ⊗L → la∗ L is an isomorphism of bundles with connection on A. But the translation l(a,λ) , λ ∈ La \ {0} on L0 is just given by µa (λ⊗−). This shows the invariance. As for the second assertion of Proposition 5.3, it is again obvious from the interpretation of λ log f as the Lie algebra-valued 1-form describing the connection and the identiﬁcation L00 = C∗ . log f , s ∈ C∗ , ∇ ∈ Conn(L) form a nonempty Corollary 5.4. The 1-forms s ∇ Zariski open set in the space of all invariant 1-forms on L0 . Consider now several line bundles of degree 0, say L1 , . . . , Ln , on A, and let (5.5)

p

0 −→ (C∗ )n −→ G = L01 ×A · · · ×A L0n −→ A −→ 0

be the associated semiabelian variety. i ) be the tautological section. As before, i = p ∗ Li and let fi ∈ H 0 (G, L We denote L i . Corollary 5.4 easily implies for ∇i ∈ Conn(Li ) we denote by ∇i its pullback to L the following proposition.

A NONCOMPACT RIEMANN-ROCH FORMULA

179

i log fi for s1 , . . . , sn Proposition 5.6. Suppose n ≥ 0. Then the 1-forms ni=1 si ∇ ∗ ∈ C , ∇i ∈ Conn(Li ) form a Zariski open dense set in the space of all invariant 1-forms on G. We now turn to the proof of Theorem 1.3. Our approach is similar to [18, §2]. First of all, it is known that any semiabelian variety has the form (5.5), which we assume. Next, we assume n > 0, since for n = 0, the group G = A is compact, and the theorem follows from (1.1) and from Proposition 2.2. We set L0 = ᏻA and compactify G by embedding it into the relative projectivization ρ

j : G *→ P := P(L0 ⊕ · · · ⊕ Ln ) −→ A. We can think of P as having homogeneous coordinates (t0 : · · · ; tn ) with ti being not a function any more but rather a section of ρ ∗ Li . Note that ρ ∗ Li has a ﬂat connection i . The variety P is the union of the induced from ∇i , whose restriction to G is ∇ relative afﬁne charts Aν = i=ν Li . Inside A0 , the complement of G is given by the condition that one of the tautological sections (still denoted by fi ) of the pullback of Li vanishes. ∗ n Take generic ∇i ∈ Conn(Li ) so that for a Zariski open, dense set of s ∈ (C ) , the 1-form ωs = si ∇i log fi satisﬁes Proposition 2.2. Then, we mimic the arguments of Section 4 but in the relative situation, using Theorem 3.5. We ﬁnd that i log fi , si ∇ CC(Rj∗ F) = lim CC(F) + s→0 s∈E

i

the limit being taken in T ∗ P. After this, the proof is identical to the argument at the end of Section 4, and the theorem is proven. References [1]

[2] [3] [4] [5] [6]

[7] [8]

A. A. Beilinson, J. Bernstein, and P. Deligne, “Faisceaux Pervers” in Analysis and Topology on Singular Spaces (Luminy, 1981), I, Astérisque 100, Soc. Math. France, Montrouge, 1982, 5–171. C. Chai and G. Faltings, Degeneration of Abelian Varieties, Ergeb. Math. Grenzgeb (3) 22, Springer, Berlin, 1990. J. Franecki, The Gauss map and Euler characteristic on algebraic groups, thesis, Northwestern Univ., Evanston, Ill., 1998. O. Gabber and F. Loeser, Faisceaux pervers l-adiques sur un tore, Duke Math. J. 83 (1996), 501–606. I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Generalized Euler integrals and A-hypergeometric functions, Adv. Math. 84 (1990), 255–271. , “Hypergeometric functions, toric varieties and Newton polyhedra” in Special Functions (Okayama, 1990), ed. M. Kashiwara and T. Miwa, ICM-90 Satell. Conf. Proc., Springer, Tokyo, 1991, 104-121. V. Ginsburg, Characteristic varieties and vanishing cycles, Invent. Math. 84 (1986), 327–402. , G-modules, Springer’s representations and bivariant Chern classes, Adv. Math. 61 (1986), 1–48.

180 [9] [10] [11] [12] [13] [14] [15]

[16] [17]

[18]

FRANECKI AND KAPRANOV , “Admissible modules on a symmetric space” in Orbites unipotentes et représentations, III, Astérisque 173–174, Soc. Math. France, Montrouge, 1989, 9–10, 199–255. P. Grifﬁths and J. Harris, Principles of Algebraic Geometry, Pure Appl. Math., Wiley, New York, 1978. J. Horn, Über die Konvergenz hypergeometrischer Reihen zweier und dreier Veränderlichen, Math. Ann. 34 (1889), 544–600. M. M. Kapranov, A characterization of A-discriminantal hypersurfaces in terms of the logarithmic Gauss map, Math. Ann. 290 (1991), 277–285. M. Kashiwara, “Index theorem for constructible sheaves” in Differential Systems and Singularities (Luminy, 1983), Astérisque 130, Soc. Math. France, Montrouge, 1985, 193–209. M. Kashiwara and P. Schapira, Sheaves on Manifolds, Grundlehren Math. Wiss. 292, Springer, Berlin, 1990. F. Loeser and C. Sabbah, Caractérisation des Ᏸ-modules hypergéométriques irréductibles sur le tore, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), 735–738; II, 315 (1992), 1263–1264. D. Mumford, Abelian Varieties, Oxford Univ. Press, London, 1970. W. Schmid and K. Vilonen, “Characters, characteristic cycles, and nilpotent orbits” in Geometry, Topology, and Physics, ed. S.-T. Yau, Conf. Proc. Lecture Notes Geom. Topology 6, International Press, Cambridge, Mass., 1995, 329–340. P. Vojta, Integral points on subvarieties of semiabelian varieties, I, Invent. Math. 126 (1996), 133–181.

Franecki: Department of Mathematics, Loyola University, 6525 North Sheridan Road, Chicago, Illinois 60620, USA; [email protected] Kapranov: Department of Mathematics, Northwestern University, Evanston, Illinois 60208, USA; [email protected]

Vol. 104, No. 2

DUKE MATHEMATICAL JOURNAL

© 2000

EXTREMAL HERMITIAN METRICS ON RIEMANN SURFACES WITH SINGULARITIES GUOFANG WANG and XIAOHUA ZHU

0. Introduction. It is a well-known consequence of the classical uniformization theorem that there is a metric with constant Gaussian curvature in each conformal class of any compact Riemann surface. It is natural to ask how to generalize this classical uniformization theory to compact surfaces with conical singularities and with nonempty boundary, or to ﬁnd a “best metric” on such surfaces. However, there are surfaces with conical singularities that do not admit a metric with constant curvature. For example, a football with two different singular angles does not admit a metric with constant curvature. (For the existence or nonexistence results of constant curvature metric in a surface with conical singularities, see [T], [M], [CL2], [CY], [LT], and [UY].) Recently, instead of using metrics of constant curvature, in [Ch5], [Ch4], X. X. Chen started to use the extremal Hermitian metrics to generalize the classical uniformization theory to Riemann surfaces with ﬁnite conical singularities. On any football there is at least an extremal Hermitian metric. It was claimed in [Ch4], [Ch2] that there is at least an extremal Hermitian metric on any surface with boundary. This problem may be regarded as the simplest nontrivial case of Calabi’s extremal metrics on Kähler manifolds (cf. [Ca1] and [Ca2]). Let M be a compact Riemann surface with nonempty boundary ∂M. For any Hermitian metric g0 on M, consider the set Ᏻ(M) of metrics with the same area that are pointwise conformal to g0 and agree with g0 in a small neighborhood of ∂M. In the closure of this set Ᏻ(M) under some suitable H 2,2 -norm, we deﬁne the energy functional E(g) = Kg2 dg, (0.1) M

where Kg is the Gaussian curvature of g. A critical point of this functional is called an extremal Hermitian metric. It is easy to see that the Euler-Lagrange equation of the energy functional is g Kg + Kg2 = c

(0.2)

for some constant c. Let g = e2ψ |dz|2 be written in local coordinates. Chen [Ch5] observed that (0.2) is equivalent to K,zzz = 0, Received 9 July 1999. Revision received 29 November 1999. 2000 Mathematics Subject Classiﬁcation. Primary 58E11; Secondary 53C55. 181

(0.3)

182

WANG AND ZHU

where ∂ψ ∂K ∂ 2K −2 , ∂z∂z ∂z ∂z and K,zz¯z denotes the third covariant derivative of K with respect to the metric g. A special case of (0.3) is K = Const. The question when a surface with singularities admits a metric of constant curvature is still open, though there are many interesting results about this problem. In [LT] and [UY], sufﬁcient and necessary conditions were given for special cases. A metric with curvature satisfying K,zz =

K,zz = 0

(0.4)

is called HCMU (this means that the Hessian of the Curvature of the Metric is Umbilical). Another special case of (0.3) is the case where K is an HCMU metric but not a metric with constant curvature. On any football, there is an HCMU metric. In [Ch5], there are more examples of HCMU metrics that are not metrics of constant curvature. In this paper, we discuss the extremal Hermitian metrics with ﬁnite energy and area on Riemann surfaces with conical singularities. In [Ch4], Chen classiﬁed all extremal Hermitian metrics with ﬁnite energy and area for compact Riemann surfaces with ﬁnite cusp singularities. He proved the following theorem. Theorem A. Let M be a compact Riemann surface, and let g be an extremal Hermitian metric with ﬁnite energy and area on M \ {pj }j =1,...,n . Suppose that all singularities pj are weak cusps. Then the following classiﬁcation holds. (i) If genus(M) ≥ 1, then K ≡ Const. (ii) If M = S 2 and n ≥ 3, then K ≡ Const. (iii) If M = S 2 and n = 2, then there is no extremal Hermitian metric. (iv) If M = S 2 and n = 1, then g is a rotationally symmetric metric that is determined uniquely by the total area. In particular, g is a metric with cusps. For the deﬁnitions of cusp and weak cusp singularity, see Section 1. In this paper, we ﬁrst show in Proposition 2.2 that any singular point of each extremal metric with ﬁnite area and energy on a singular surface is either a weak cusp or a conical singularity. Then we generalize Theorem A as follows. Main theorem. Let M be a compact Riemann surface, let g be an extremal Hermitian metric with ﬁnite energy and area on M \ {pj }j =1,...,n , and let K be the Gaussian curvature of g. Then g is a conical metric with singular angle αj (j = 1, . . . , n) (which may include some weak cusps). Furthermore, if all singular angles satisfy 2παj ≤ π, then the following classiﬁcation holds.

(0.5)

EXTREMAL METRICS ON SURFACES WITH SINGULARITIES

183

(i) If genus(M) ≥ 1, then K ≡ Const. (ii) If M = S 2 and n ≥ 3, then K ≡ Const. (iii) If M = S 2 and n = 2, there are two cases. (a) If both singular points are cusps, then there is no extremal Hermitian metric. (b) If one of the singular points is not a cusp, then g is a rotationally symmetric, extremal Hermitian metric that is determined uniquely by the total area and two angles 2π αj . (iv) If M = S 2 and n = 1, then g is a rotationally symmetric metric that is determined uniquely by the total area and angle 2πα. In view of a result of [LT] (see also Remark 5.2), we believe that our main theorem is optimal. In other words, there may be an extremal Hermitian metric with ﬁnite energy and area on a compact Riemann surface that is not HCMU if some singular conical angle is larger than π . To establish a general theory for the existence of the extremal metrics, we have to consider compactness of the energy functional E or the convergence of extremal metrics. In [Ch3], Chen analyzed the convergence of sequences of metrics with ﬁnite area and energy in an interesting, somewhat geometric way. We consider the convergence of the sequences of extremal Hermitian metrics with ﬁnite area and energy using a more analytic method in a forthcoming paper. In the limit, we may obtain an extremal metric on R2 . This leads to classiﬁcation of all extremal metrics with ﬁnite area and energy on R2 . The classiﬁcation of such metrics on R2 will be discussed in that paper. The main difﬁculty for obtaining the classiﬁcation is that the Gaussian curvature K may become unbounded near singular points. We study the asymptotic behavior of extremal Hermitian metrics with ﬁnite energy and area near the singular point and sharpen some of Chen’s estimates. Then we use the Kelvin transform and a simple cut-off function to transfer our problem to an entire solution of an almost extremal metric equation on R2 , on which we can apply the global Poisson potential integral technique. By a careful analysis, we show that K can be extended continuously to the singular points, provided that singular angles are less than or equal to 2π. Finally, we show that any extremal Hermitian metric of ﬁnite area and energy is HCMU, provided that the singular angles are less than or equal to π. The organization of this paper is as follows. In Section 1, by using the rescaling method, we study the asymptotic behavior of an extremal Hermitian metric with ﬁnite energy and area near the singular point. Then in Section 2, by using the global Poisson integral technique on R 2 , we prove that the metric is in fact weak cuspidal or conical. In Section 3, we study the asymptotic behavior of Gaussian curvature as well as its second covariant derivative near a singular point. The proof is more technical, based on the global Poisson potential integral analysis. In Section 4, we improve the estimate of the second covariant derivative of Gaussian curvature. After establishing the above preliminary results, we give the proof of the main theorem in Section 5.

184

WANG AND ZHU

Acknowledgments. We would like to thank Professor Weiyue Ding for bringing to our attention the problem of extremal Hermitian metrics and X. X. Chen for sending us his preprints. Part of this work was done while the ﬁrst author was visiting the Max-Planck-Institut für Mathematik in Leipzig. He would also like to thank Professor J. Jost and the Institut for their hospitality. 1. Asymptotic behavior of metric. Let D be a disk centered at the origin. Suppose that g = e2ψ |dz|2 is an extremal metric with ﬁnite energy and area on D \ {0}. Let K be the Gaussian curvature of g. Then ψ and K satisfy the following system on D \ {0}: K = −K 2 e2ψ + ce2ψ , (1.1) ψ = −Ke2ψ , for some constant c. Lemma 1.1. Let g = e2ψ |dz|2 be an extremal metric with ﬁnite energy and area on D \ {0}. Then lim

max

r→0 |x|=|x |=r

ψ(x) + ln |x| = 1. ψ(x ) + ln |x |

Before we give the proof of Lemma 1.1, we state a useful lemma for positive harmonic functions, which can be found in [Ch4]. For any ﬁxed a > 0, let Ta = {x|e−a < |x| < ea }. Denote the best Harnack constant by βa for positive harmonic functions in the domain T1 ⊂ Ta ; that is, max f f is a positive harmonic function in Ta . βa = sup min f We have the following lemma. Lemma 1.2. We have lima→+∞ βa = 1. Proof of Lemma 1.1. It is sufﬁcient to prove that for any sequence {xi } → 0, ψ(x) + ln |x| = 1. i→+∞ e−1 |x |≤|x|, |x |≤e1 |x | ψ(x ) + ln |x | i i lim

sup

Let ψi (y) = ψ(|xi |y) + ln |xi |

and

Ki (y) = K(|xi |y).

(1.2)

By (1.1), ψi satisﬁes ψi = −Ki e2ψi . From Lemma 1.2, for any ε > 0, we can choose a sufﬁciently large number a such that βa < 1 + ε. Let vi be a smooth function on Ta satisfying

EXTREMAL METRICS ON SURFACES WITH SINGULARITIES

vi = −Ki e2ψi

185

on Ta ,

vi |∂Ta = 0. Then from a result in [Ch3], sup ψi −→ −∞

as i −→ +∞,

Ta

(1.3)

we have Ki (y)e2ψi (y) 2 dy ≤ Ki (y)eψi (y) 2 dy Ta

Ta

=

2 K(x)eψ(x) dx −→ 0

e−a |xi |≤|x|≤ea |xi |

as i −→ +∞.

By the standard L2 -estimate, we get sup |vi | −→ 0 Ta

as i −→ +∞.

(1.4)

Let ui = ψi − vi . From (1.3) and (1.4), we have sup |ui | −→ −∞

as i −→ +∞.

Ta

It is clear that ui = 0

on Ta .

Applying Lemma 1.1 to {ui }, we get lim

ui (x) < 1 + ε. sup u (x )

lim

ψi (x) < 1 + ε. sup ψ (x )

i→+∞ x,x ∈T1

It follows that

i→+∞ x,x ∈T1

Consequently, lim

sup

i→+∞ e−1 |x |≤|x|, |x |≤e1 |x | i i

i

i

ψ(x) + ln |x| ψ(x ) + ln |x | < 1 + ε.

(1.5)

This proves the lemma. Deﬁnition 1.1. (1) Let g = e2ψ |dz|2 be an extremal metric on D \{0}. The singular point x = 0 is called a weak conical point with singular angle 2πα > 0 if and only if ψ satisﬁes 2π ∂ψ(r, θ) 1 lim r + 1 − α dθ = 0. (1.6) r→0 2π 0 ∂r

186

WANG AND ZHU

If α = 0 in (1.6), then the singular point x = 0 is called a weak cusp. (2) If ψ can be locally expressed as ψ(x) = (α − 1) ln |x| + ρ(x)

(1.7)

with ρ(x) a smooth function on D and α > 0, then the singular point x = 0 is called a conical point with singular angle 2πα. (3) If ψ can be locally expressed as ψ(x) = − ln |x| + ln ρ(x)

(1.8)

with ρ(x) a smooth positive function on D, then the singular point x = 0 is called a cusp point. Remark. In Deﬁnition 1.1, if 0 is a cusp point of the metric g = e2ψ |dz|2 of ﬁnite area and energy, then ρ in (1.8) has the form (− ln |z|)−γ h with γ ∈ (1/2, 3/2) and a smooth function h > 0. We ﬁrst have the following lemma. Lemma 1.3 [Ch4]. Let g = e2ψ |dz|2 be an extremal Hermitian metric with ﬁnite energy and area on D \ {0}. Then x = 0 is a weak cusp or weak conical singular point with angle 2π α > 0; that is, 2π 1 ∂ψ(r, θ) lim + 1 − α dθ = 0. r r→0 2π 0 ∂r We can show that a weak cusp point of an extremal Hermitian metric with ﬁnite area and energy is actually a cusp (see [Ch4]). In the present paper, we generalize this result for any weak conical point (cf. Proposition 2.2). In this section, we ﬁrst prove the following proposition. Proposition 1.1. Let g = e2ψ |dz|2 be an extremal metric with ﬁnite energy and area on D \ {0}. Then there is a nonnegative number α ∈ [0, ∞) such that ψ(x) = α − 1. |x|→0 ln |x| lim

(1.9)

Proof. By Lemma 1.3, there is a number α such that 2π 1 ∂ψ(r, θ) + 1 − α dθ −→ 0 as r −→ 0. r 2π 0 ∂r It follows that there is a point (r, θr ) for each r such that ψ(r, θr ) + (1 − α) ln r −→ 0 ln r

as r −→ 0.

(1.10)

EXTREMAL METRICS ON SURFACES WITH SINGULARITIES

Thus from Lemma 1.1, for any ε > 0, we have ψ(r, θ ) + (1 − α) ln r ψ(r, θ) + ln r − α ln r = ln r ln r (1 ± ε)ψ(r, θ ) + ln r − α ln r r ≤ ln r ψ(r, θr ) + (1 − α) ln r + ε ψ(r, θr ) + ln r ≤ ln r ln r as r → 0. Combining (1.10) and (1.11), we get ψ(r, θ ) + (1 − α) ln r −→ εα ln r

187

(1.11)

as r −→ 0.

Since ε can be taken arbitrarily small, we get lim

Moreover, since

|x|→0 De

2ψ(x) dx

ψ(x) = α − 1. ln |x|

(1.12)

is bounded, it is clear that α is nonnegative.

2. Conical structure of metrics. In this section, we consider the conical structure of extremal Hermitian metrics. The following estimate was proved in [Ch4], which is generalized in Proposition 2.1. For completeness, we still include the proof. Lemma 2.1. Let g = e2ψ be an extremal metric with ﬁnite energy and area on D \ {0}, and let K be the Gaussian curvature. Then lim |x|2 |K(x)|e2ψ(x) = 0.

|x|→0

Proof. It sufﬁces to prove that for any sequence {|xi |} → 0, lim |xi |2 |K(xi )|e2ψ(xi ) = 0.

|xi |→0

Let ψi and Ki be the two smooth functions on T1 deﬁned by (1.2). By (1.1), it is clear that Ki = −Ki2 e2ψi + Ce2ψi

on T1 .

Set ai = inf ψi (y) y∈T1

and

K˜ i = Ki eai .

We have K˜ i = −K˜ i2 e2ψi + ce2ψi eai = fi

on T1 .

(2.1)

188 Since

WANG AND ZHU

T1

K˜ i2 dy ≤

T1

Ki eψi

2

dy

=

and

Keψ

e−1 |xi |≤|x|≤e1 |xi |

2

dx −→ 0

as i −→ +∞,

2 2ψi fi dy ≤ Ki e dy + |c| e2ψ dy T1

T1

T1

=

e−1 |xi |≤|x|≤e1 |xi |

Keψ

2

dx

+ |c|

e−1 |xi |≤|x|≤e1 |xi |

e2ψ dx −→ 0

as i −→ +∞,

the standard elliptic estimate implies that K˜ i Lp (T1/2 ) −→ 0

for any p ≥ 1.

(2.2)

By the Hölder inequality, it follows that fi Ls (T1/2 ) −→ 0

for any s < 2.

Thus, from the standard Lp -estimate, we get K˜ i H 2,s (T1/4 ) −→ 0. In particular, Ki (1)eai ≤ sup K˜ i −→ 0 T1/4

as i −→ +∞.

(2.3)

This, together with Lemma 1.1, implies that |xi |2 |Ki (xi )|e2ψ(xi ) = Ki (1)e2ψi (1) ≤ e2 Ki (1)eai −→ 0

as i −→ +∞.

Lemma 2.2 [Ch4]. Let D ⊂ R 2 be a disk, and let f ∈ L1 (D). Suppose that lim|x|→0 |x|2 f (x) = 0. Let v(x) be the Poisson potential of f ; that is, ln |x − y|f (y) dy. v(x) = D

Then the following two statements hold: (1) lim|x|→0 v(x)/ ln |x| = 0, (2) limr→0 sup|x|=|x |=r (v(x) − v(x )) = 0.

189

EXTREMAL METRICS ON SURFACES WITH SINGULARITIES

Lemma 2.3. Let g = e2ψ |dz|2 be an extremal metric with ﬁnite energy and area on D \ {0}. Then lim sup ψ(x) − ψ(x ) = 0. (2.4) r→0 |x|=|x |=r

Proof. First from (1.1) and the Hölder inequality, we notice that 2ψ = ψ dx Ke dx D\{0}

D\{0}

≤

1/2 D\{0}

K 2 e2ψ dx

1/2

D\{0}

e2ψ dx

.

Then we deﬁne a Lp -integral function on D: ln |x − y|ψ(y) dy. v(x) = D

From Lemma 2.1, we have |x|2 ψ(x) = −|x|2 K(x)e2ψ(x) −→ 0

as |x| −→ 0.

Applying Lemma 2.2, we obtain v(x) =0 |x|→0 ln |x| lim

and lim

sup

v(x) − v(x ) = 0.

r→0 |x|=|x |=r

(2.5)

(2.6)

Let u(x) = ψ(x) − v(x). It is clear that u = 0

on D \ {0}.

From (2.5) and Proposition 1.1, we have lim

|x|→0

u(x) ψ(x) = lim = α − 1. ln |x| |x|→0 ln |x|

Thus there is a smooth function u (x) on D such that u(x) = (α − 1) ln |x| + u (x). In particular, we have lim

sup

u(x) − u(x ) = 0.

r→0 |x|=|x |=r

Combining (2.6) and (2.7), we get lim

sup

r→0 |x|=|x |=r

ψ(x) − ψ(x ) = 0.

(2.7)

190

WANG AND ZHU

Now we improve Lemma 2.1 to the following proposition, which is crucial to us. Proposition 2.1. Let g = e2ψ |dz|2 be an extremal metric with ﬁnite energy and area on D \ {0}, and let K be the Gaussian curvature of g. Then lim |x||K(x)|eψ(x) = 0.

(2.8)

|x|→0

Proof. As in Lemma 2.1, it sufﬁces to prove that for any sequence {xi } → 0, lim |xi ||K(xi )|eψ(xi ) = 0.

|xi |→0

From (2.3), we have lim |K(xi )|eai = 0,

(2.9)

i

where ai = inf y∈T1 ψi (y) = inf y∈T1 {ψ(|xi |y) + ln |xi |}. Let ψ˜ i (y) = ψ(|xi |y) + (1 − α) ln(|xi ||y|) on T1 . By Lemma 1.3, it is easy to see that 2π 2π ∂ ψ˜ i (r, θ ) 1 1 ∂ψ(r , θ) = lim r + 1 − α dθ = 0, (2.10) r lim i→+∞ 2π 0 ∂r ∂r r →0 2π 0 where r = |xi |r and e−1 ≤ r ≤ e1 . It follows that 2ψ ∂ ψ˜ i (r, θ) dθ = 0 lim i→+∞ ∂r 0 uniformly for e−1 ≤ r ≤ e1 . In particular, lim Osce−1 ≤r≤e1 i→+∞

0

2ψ

ψ˜ i (r, θ) dθ

= 0.

(2.11)

On the other hand, by Lemma 2.3,

lim Oscθ ψ˜ i (r, θ) = 0

i→+∞

(2.12)

uniformly for e−1 ≤ r ≤ e1 . Combining (2.11) and (2.12), we get

lim Osc(r,θ)∈T1 ψ˜ i (r, θ) = 0. i→+∞

Consequently,

Oscy∈T1 ψi (y) = Oscy∈T1 ψ˜ i (y) + α ln(|xi ||y|) − ln |y| ≤ 3(α + 1), when i is sufﬁciently large.

(2.13)

EXTREMAL METRICS ON SURFACES WITH SINGULARITIES

191

In view of (2.9) and (2.13), we get lim |xi ||K(xi )|eψ(xi ) ≤ lim |K(xi )| sup eψi (y)

|xi |→0

|xi |→0

y∈T1

≤ lim e |xi |→0

3(α+1)

|K(xi )|eai = 0.

Remark. We can also show an --regularity-type theorem for extremal Hermitian metrics (see such a result for harmonic maps in [Sc]), from which Proposition 2.1 can be deduced. We believe that Proposition 2.1 is optimal. Lemma 2.4. Let u(x) and f (x) be two smooth functions on R 2 satisfying u(x) = f (x) on R 2 . Suppose that for some positive number ε, |u(x)| ≤ o(|x|)

|f (x)| ≤ O |x|−(2+ε)

and

(2.14)

as |x| → +∞. Then there is a uniform C > 0 such that as |x| → +∞, β ln |x| − C ≤ u(x) ≤ β ln |x| + C, where β = (1/2π )

R 2 f (y) dy.

Proof. Let 1 w(x) = 2π

R2

ln |x − y| − ln(|y| + 1) f (y) dy.

It is clear that w(x) = f (x)

on R 2 .

We claim that there is a uniform constant C > 0 such that (2.15) β ln |x| − C ≤ w(x) ≤ β ln |x| + C

as |x| → +∞, where β = (1/2π) R 2 f (y) dy. To see this, we need only to verify that 1 |I | = ln |x − y| − ln(|y| + 1) − ln |x| f (y) dy 2π R 2 is uniformly bounded as |x| → +∞. First, from (2.14) we notice that ln(|y| + 1)|f (y)| dy ≤ C1 (|y| + 1)−(2+ε ) dy ≤ C2 , R2

for another small constant ε > 0.

R2

(2.16)

192

WANG AND ZHU

Decompose I as follows: I = I1 + I2 + I3 , where I1 , I2 , and I3 are integrals on the regions D1 = {y; |y| ≤ R}, D2 = {y; |x −y| ≤ 1 and y ≥ R}, and D3 = {y; |x −y| ≥ 1 and y ≥ R}, respectively. We estimate |x| + 1 |I1 | ≤ ln(|y| + 1)|f (y)| dy + ln |f (y)| dy |x| − R R 2 R2 |x| + 1 −→ C2 as |x| −→ +∞, ≤ C2 + C3 ln |x| − R ln |y| dy + 2 ln(|y| + 2)f (y) dy ≤ C5 , |I2 | ≤ C4 |y|≤1

and

|I3 | ≤ ≤

|x−y|≥1,|y|>R

|y|≥R

|y|>R

ln |x − y| − ln(|y| + 1) − ln(|x| + 1)|f (y)| dy

2 ln(|y| + 1)|f (y)| dy ≤ C6 .

It follows that there is a uniform constant C > 0 such that |I | ≤ |I1 | + |I2 | + |I3 | ≤ C

as |x| −→ +∞;

that is, β ln |x| − C ≤ w(x) ≤ β ln |x| + C

as |x| −→ +∞.

(2.17)

Let v(x) = u(x)−w(x). It is clear that v is a harmonic function on R 2 . From (2.17) and (2.14), we have v(x) ≤ o(|x|). It follows that v(x) ≡ Const. Again from (2.17), we see that there is a uniform constant C such that β ln |x| − C ≤ u(x) ≤ β ln |x| + C as |x| → +∞. Proposition 2.2. Let g = e2ψ(z) |dz|2 be an extremal Hermitian metric with ﬁnite energy and area on D \ {0}. Then g is either a weak cusp metric or a conical metric with singular angle 2π α > 0. Proof. By Lemma 1.3, x = 0 is a weak cusp or a weak conical singular point with angle 2π α > 0. The case where x = 0 is a weak conical singular point was already considered in [Ch2]. Thus we assume that x = 0 is a weak conical singular point with angle 2π α. It is convenient to make a change of coordinates. Let w = 1/z, k(w) = K(1/w), and e2φ(w) |dw|2 = e2ψ(z) |dz|2 . (When there is no confusion, we also denote k by

193

EXTREMAL METRICS ON SURFACES WITH SINGULARITIES

K.) Then

1 φ(w) = ψ − 2 ln |w| w

on R 2 \ B(R/2) for some sufﬁciently large R. Clearly, φ(w) satisﬁes R φ = −Ke2φ on R 2 \ B . 2

(2.18)

(2.19)

Moreover, from (2.18) and Proposition 1.1, we have φ(y) = −(1 + α). |y|→+∞ ln |y| lim

(2.20)

Let η ≥ 0 be a smooth cut-off function on R 2 → [0, 1] such that   0 on B R , 2 η(y) =  1 on R 2 \ B(R). Let φ ∗ = ηφ. Then by (2.19), φ ∗ is a smooth function on R 2 and satisﬁes φ ∗ = −ηKe2φ + 2∇η∇φ + φη.

(2.21)

We denote 6φ ∗ by h. Moreover, by (2.20), (2.21), and Proposition 2.1, we have φ ∗ (w) = −(1 + α) |w|→+∞ ln |w|

(2.22)

|h(w)| = − ηK(w)e2φ(w) ≤ K(w)e2φ(w) ≤ o |w|−(2+α) for large |w|.

(2.23)

lim

and

Thus we can apply Lemma 2.4 to (2.21) to show that there is a uniform constant C > 0 such that as |y| → +∞,

β ln |w| − C ≤ φ(w) ≤ β ln |w| + C,

(2.24)

where β = (1/2π ) R 2 h dy. Equations (2.20) and (2.24) imply that β = −(1 + α). Combining (2.18) and (2.24), we get −(1 − α) ln |x| − C ≤ ψ(x) ≤ −(1 − α) ln |x| + C as |x| → 0. Thus we see that g is a conical metric with the singular angle 2πα > 0 at x = 0.

194

WANG AND ZHU

3. Asymptotic behavior of the Gaussian curvature. We need the following lemma. Lemma 3.1. Let u(x) and f (x) be two smooth functions on R 2 satisfying u(x) = f (x) on R 2 . Suppose that f (x) ∈ L1 (R 2 ) and |u(x)| ≤ o(|x|) as |x| −→ +∞. Then lim

where f¯ = (1/2π )

|y|→+∞

u(y) = f¯, ln |y|

R 2 f (y) dy.

Proof. Let w(x) =

1 2π

R2

ln |x − y| − ln(|y| + 1) f (y) dy.

It is easy to see that w = f

on R 2 .

We claim that

where f¯ = (1/2π )

I=

w(x) −→ f¯ as |x| −→ +∞, ln |x| R 2 f (y) dy.

R2

(3.1)

To see this, we need only to verify that

ln |x − y| − ln(|y| + 1) − ln |x| f (y) dy −→ 0 ln |x|

as |x| → +∞. Decompose R 2 into three domains: D1 = {y; |x − y| ≤ 1}, D2 = {y; |x − y| ≥ 1 and y ≤ R}, and D3 = {y; |x − y| ≥ 1 and y ≥ R}; and write I = I1 + I2 + I3 , where I1 , I2 , I3 are integrals over these domains, respectively. We estimate 1 |f (y)| dy + ln |x − y||f (y)| dy |I1 | ≤ 3 ln |x| |x−y|≤1 |x−y|≤1 C1 |f (y)| dy + ln |y| dy −→ 0 as |x| −→ +∞, ≤3 ln |x| y|≤1 |y|≤|x−1| ln |x + R| + ln(R + 1) − ln |x| |f (y)| dy −→ 0 as |x| −→ +∞, |I2 | ≤ ln |x| R2

EXTREMAL METRICS ON SURFACES WITH SINGULARITIES

and |I3 | ≤ ≤

ln |x − y| − ln(|y| + 1) − ln |x| 3 ln(|x| + 1) ln |x|

ln |x|

|y|≥R

|y|≥R

|f (y)| dy −→ 0

195

|f (y)| dy

as |R| −→ +∞.

It follows that |I | ≤ |I1 | + |I2 | + |I3 | −→ 0

as |x| −→ +∞.

Let v(x) = u(x) − w(x). We have v(x) = 0

on R 2 .

Moreover, from (3.1) and the assumption of the lemma, we have |v(x)| ≤ o(|x|). It follows that v(x) ≡ Const. Again from (3.1), we get u(x) w(x) + Const = −→ f¯ as |x| −→ +∞. ln |x| ln |x| Now we want to show that the Gaussian curvature K can be extended continuously to singular points with angles 2παi , provided that αi ≤ 1 for any i. Proposition 3.1. Let M be a compact Riemann surface, and let g be an extremal Hermitian metric with ﬁnite energy and area on M \ {pj }j =1,...,n . Assume that all conical singular angles satisfy 2πα ≤ 2π. Then the Gaussian curvature function K(x) of g can be extended continuously to the singular points {pj }j =1,...,n . Proof. We divide the proof into several steps. When a singular point is a weak cusp, such a result was shown in [Ch4]. Hence, in the sequel, we only consider conical singularities. Step 1. We ﬁrst obtain an asymptotic behavior of the Gaussian curvature K. Lemma 3.2. Let g = e2ψ(z) |dz|2 be an extremal Hermitian metric on D \ {0}, and let K(x) be the Gaussian curvature of g. Suppose that x = 0 is a conical singular point with angle 2π α ≤ 2π . Then K(x) = −β, |x|→0 ln |x| lim

for some constant β. Furthermore, we have −β ln |x| − C ≤ K(x) ≤ −β ln |x| + C.

(3.2)

Proof. As in the proof of Proposition 2.2, we make a change of coordinates w = 1/z and e2φ(w) |dw|2 = e2ψ(z) |dz|2 . Then φ(w) is an extremal Hermitian metric

196

WANG AND ZHU

with curvature K(w) = K(z) on R 2 \ B(R/2) for some sufﬁciently large R, and the Gaussian curvature K(w) satisﬁes 2φ R 2 2 . (3.3) on R \ B K = − K + c e 2 Let η ≥ 0 be a smooth cut-off function on R 2 as in Proposition 2.2 and K ∗ (y) = η(y)K(y). By (3.3), K ∗ (y) is a smooth function on R 2 and satisﬁes K ∗ = η − K 2 + c e2φ + 2∇η∇K + Kη on R 2 . (3.4) We also denote 6K ∗ by f . From Proposition 2.1, we have |w|K ∗ (w)eφ(w) = K(z)eψ(z) |z| −→ 0

as |w| −→ ∞,

which implies that K ∗ (w) K(w) = lim = 0. |w|→+∞ |w| |w|→+∞ |w| lim

(Here we have used the assumption that α ≤ 1.) Since 2φ(w) 2 ≤ f (w) dw η(w) − K (w) + c e dw R2

R2

+ K(w)η(w) dw B(R)\B(R/2) − K 2 (w) + ce2φ(w) dw + C ≤ ≤

R 2 \B(R/2)

D\{0}

− K 2 (x) + ce2ψ(x) dx + C < ∞,

applying Lemma 3.1 to (3.4), we obtain K(w) K ∗ (w) = lim = β, |w|→∞ ln |w| |w|→∞ ln |w| lim

where β = (1/2π )

R 2 f (y) dy.

Therefore, we have K(x) −→ −β. |x|→0 ln |x| lim

(3.5)

In view of (3.5), we can improve the above estimates for f to |f (x)| ≤ C|x|−2−α . Now applying Lemma 2.4 to (3.4), we see that there is a uniform constant C > 0 such that −β ln |x| − C ≤ K(x) ≤ −β ln |x| + C.

EXTREMAL METRICS ON SURFACES WITH SINGULARITIES

From Lemma 3.2 and Proposition 2.2, we see that 1 φ ∗ (x) = ln |x − y| − ln(|y| + 1) h dy + c1 2π R 2

197

(3.6)

and 1 K (x) = 2π ∗

R2

ln |x − y| − ln(|y| + 1) f dy + c2

(3.7)

for some constants c1 and c2 , where h and f are determined by (2.21) and (3.4), respectively. Furthermore, we have φ = φ ∗ and K = K ∗ on R 2 \ B(R). Step 2. We study the asymptotic behavior of the ﬁrst derivatives of K. Lemma 3.3. Let (r, θ ) be the polar coordinate system on R 2 . Then we have rKr − β ≤ C(ln r)2 r −2α , (3.8) |Kθ | ≤ C(ln r)2 r −2α , rφr + α + 1 ≤ C ln rr −2α , |φθ | ≤ C ln rr −2α ,

(3.9) (3.10) (3.11)

for large r = |x| > R. Moreover,

K = β ln r + b0 + O (ln r)2 r −2α , φ = −(α + 1) ln r + a0 + O ln rr −2α ,

(3.12) (3.13)

for some constants a0 and b0 . Proof. Since rKr = x1 K1 + x2 K2 , from (3.7) we have 1 y · (x − y) rKr − β = f dy 2π R 2 |x − y|2 1 1 y · (x − y) y · (x − y) = f dy + − K 2 + c e2φ dy 2π |y|≤R |x − y|2 2π |y|≥R |x − y|2 y · (x − y) 1 + + − K 2 + c e2φ dy, = 2 2π |x − y| R≤|y|≤-r -r≤|y|≤-kr kr≤|y|≤+∞ (3.14) where k is a large constant and - is a small one. From Proposition 2.2 and Lemma 3.2, we estimate that 1 C y · (x − y) ≤ 1 (3.15) f dy |y||f | dy ≤ , 2 2π πr r |x − y| |y|≤R

|y|≤R

198

WANG AND ZHU

1 2π

R≤|y|≤-r

2 ≤ r

-r R

2φ y · (x − y) 2 − K + c e dy |x − y|2 2 −2α

2β (ln s) s 2

ds ≤ C-

1−2α

(3.16) 2 −2α

(ln r) r

,

2φ y · (x − y) 2 − K + c e dy 2 -r≤|y|≤kr |x − y| 1 1 2 −1−2α 2 −1−2α ≤ (ln r) r 2β dy 2π |y|≤(k+1)r |y|

1 2π

(3.17)

≤ Ck- −1−2α (ln r)2 r −2α , and

2φ(x−y) y · (x − y) 2 dy −K +c e 2 kr≤|y| |x − y| 1 ≤ 2β 2 (ln r)2 r −1−2α dy 2π kr≤|y|

1 2π

(3.18)

≤ Ck −2α (ln r)2 r −2α . Then (3.8) follows from these estimates and from (3.14). The proofs of (3.9), (3.10), and (3.11) are similar to that of (3.8), so we omit them. From (3.8), we have Kr − β ≤ C(ln r)2 r −1−2α . r This implies that for any θ ∈ [0, 2π], there are numbers b(θ) such that lim (K − β ln r) = b(θ).

r→+∞

In view of (3.9), we have b(θ) ≡ b0 , and as a consequence (3.12) holds. Similarly, from (3.10) and (3.11), we can get (3.13). This proves Lemma 3.3. Step 3. Now we improve the previous estimates. Lemma 3.4. For large r, we have rKr = β + e2a0

β2 (ln r)2 r −2α + O ln rr −2α 2α

(3.19)

β ln rr −2α + O r −2α . 2α

(3.20)

and rφr = −α − 1 + e2a0

EXTREMAL METRICS ON SURFACES WITH SINGULARITIES

199

Proof. Let x = (r0 cos θ0 , r0 sin θ0 ) and y = (r cos θ, r sin θ). It is easy to see that r 2 − r02 y · (x − y) −1 1+ . (3.21) = 2 |x − y|2 r 2 + r02 − 2 cos(θ − θ0 )rr0 Moreover, we have 2π 0

r 2 − r02 2 r 2 + r0 − 2 cos(θ − θ0 )rr0

dθ = 2π

as r > r0

(3.22)

as r < r0 .

(3.23)

and 0

r 2 − r02

2π

r 2 + r02 − 2 cos(θ − θ0 )rr0

dθ = −2π

On the other hand, from (3.12) and (3.13), we have − K 2 + c e2φ = e2a0 β 2 (ln r)2 r −2−2α + O ln rr −2−2α . Moreover, as in (3.16), (3.17), and (3.18), we estimate that y · (x − y) 1 2 −1−2α (ln r) r dy ≤ C ln r0 r0−2α . 2 2π |y|≥R |x − y| Applying (3.22) and (3.23) to (3.14), we obtain +∞ 2a0 2 r0 Kr0 = β + e β (ln r)2 r −1−2α dr r0 1 y · (x − y) 2 −1−2α +O (ln r) r dy +O 2 r0 |y|≥R |x − y| = β + e2a0

β2 (ln r0 )2 r0−2α + O ln r0 r0−2α . 2α

This proves (3.19). Similarly, we can prove (3.20). Step 4. In this step, we estimate the asymptotic behavior of a second derivative of K, which is crucial to the proof of Proposition 3.1. Lemma 3.5. For large r, we have

r(rKr )r − Kθ θ = −e2a0 β 2 (ln r)2 r −2α + O ln rr −2α .

(3.24)

Proof. Here we use the same notation as in the proof of Lemma 3.4. Recall that for |r| ≥ R, 1 (x − y) · y r0 Kr0 = β + f (x − y) dy. 2π R2 |y|2

200

WANG AND ZHU

We compute r0 (r0 Kr0 )r0 x ·y (x − y) · y = f (x − y) dy + x1 ∇1 f (x − y) + x2 ∇2 f (x − y) dy 2 2 |y| R2 |y| R2 y · (x − y) y · (x − y) =β+ f (y) dy + x1 ∇1 f (y) + x2 ∇2 f (y) dy 2 2 2 2 |x − y| |x − y| R R 1 y · (x − y) =β+ − K 2 (y) + c e2φ(y) dy 2 2π |y|≥R |x − y| y · (x − y) 1 − 2K(rKr ) + 2rφr − K 2 + c e2φ dy + 2 2π |y|≥R |x − y| + g1 (x) + g2 (x), (3.25) where g1 (x) =

1 2π

g2 (x) =

1 2π

and

R2

y · (x − y) (x1 − y1 )f1 + (x2 − y2 )f2 dy 2 |x − y|

|y|≤R

1 y · (x − y) f dy + 2π |x − y|2

Similarly, for |r| ≥ R, by using 1 Kθ0 = 2π we have Kθ0 θ0

1 =− 2π

R2

R2

|y|≤R

y · (x − y) rfr dy. |x − y|2

(x − y) · y f (x − y) dy + β, |y|2

1 x ·y f (x − y) dy + 2π |y|2

R2

(x − y) · y − x2 ∇1 f + x1 ∇1 f (x − y) dy |y|2

1 x · (x − y) y · (x − y) f (y) dy + − x2 ∇1 f + x1 ∇1 f (y) dy 2 2 2π R2 |x − y| R2 |x − y| 2φ(y) y · (x − y) 1 2 − K + c e dy = −β − 2π |y|≥R |x − y|2 1 y · (x − y) + − 2KKθ0 + 2φθ0 − K 2 + c e2φ dy + h1 (x) + h2 (x), 2 2π |y|≥R |x − y| (3.26)

=−

1 2π

where y = (−y2 , y1 ) and

EXTREMAL METRICS ON SURFACES WITH SINGULARITIES

h1 (x) =

1 2π

and 1 h2 (x) = 2π

R2

201

y · (x − y) (y2 − x2 )f1 + (x1 − y1 )f2 dy 2 |x − y|

|y|≤R

1 y · (x − y) fθ dy − 2 2π |x − y|

|y|≤R

y · (x − y) f dy. |x − y|2

The following identity is important for us: 1 1 y · ∇y f dy = −2 · f dy = −2β. g1 (x) − h1 (x) = 2π R 2 2π R 2 It is easy to check that |g2 (x)| ≤

C |x|

(3.27)

|h2 (x)| ≤

C . |x|

(3.28)

and

Combining (3.25) and (3.26) and applying the previous identity, we get r0 (r0 Kr0 )r0 − Kθ0 θ0 1 y · (x − y) = − 2K(rKr ) + 2rφr − K 2 + c e2φ dy 2 2π |y|≥R |x − y| y · (x − y) 2 − K(y)2 + c e2φ(y) dy + 2 2π |y|≥R |x − y| 2φ y · (x − y) 1 2 + 2φ + c e dy − 2KK − K − θ θ 2π |y|≥R |x − y|2 + g2 (x) − h2 (x). Let 1 p(x) = 2π

|y|≥R

1 +2 2π

y · (x − y) − 2K(rKr ) + 2rφr − K 2 + c e2φ dy |x − y|2

|y|≥R

y · (x − y) − K(y)2 + c e2φ(y) dy. |x − y|2

Then by Lemmas 3.3 and 3.4, we have y · (x − y) 1 − 2(α + 1) − K 2 + c e2φ + O ln rr −2α dy p(x) = 2 2π |y|≥R |x − y| 1 y · (x − y) + − K(y)2 + c e2φ(y) dy 2 π |y|≥R |x − y|

(3.29)

202

WANG AND ZHU

=

−2α 2π

|y|≥R

y · (x − y) 2a0 2 − e β (ln r)2 r −2−2α + O ln rr −2−2α dy. 2 |x − y|

By using the same argument as in the proof of Lemma 3.4, we have the following estimate: +∞ p(x) = 2αe2a0 β 2 (ln s)2 s −1−2α + O ln ss −1−2α ds r (3.30) 2a0 2 2 −2α −2α = e β (ln r) r + O ln rr . From (3.9) and (3.11), we also have 2φ y · (x − y) 1 2 − 2KKθ + 2φθ − K + c e dy ≤ Cr −3α . 2π |y|≥R |x − y|2

(3.31)

By inserting (3.27), (3.28), (3.30), and (3.31) into (3.29), we get (3.24) immediately. Step 5. In this step, we show that β = 0 by using the holomorphicity of K,zz in D \ {0}. Lemma 3.6. Let g = e2ψ(z) |dz|2 be an extremal Hermitian metric with ﬁnite energy and area on D \ {0}, and let K(x) be the Gaussian curvature of g. Suppose that x = 0 is a conical singular point with angle 2πα ≤ 2π. Then K(x) = 0; |x|→0 ln |x| lim

that is, β = 0. Proof. Keep the notation in Lemma 3.2. It is easy to check that √ 1 rKr − −1Kθ , w √ −1 Kww = 2 rKr − r(rKr )r − Kθθ − −1(Kθ − 2rKrθ ) , w Kw =

and 1 1 K = − 2K φ K ,ww ww w w z4 z4 √ 1 = − 2 rKr − r(rKr )r − Kθθ − −1(Kθ − 2rKrθ ) z √ 2 − 2 rKr · rφr − Kθ φθ − −1(Kθ · rφr + φθ · rKθ ) . z

K,zz =

(3.32)

EXTREMAL METRICS ON SURFACES WITH SINGULARITIES

203

By Lemmas 3.4 and 3.5, we have rKr · rφr = −β(α + 1) −

α + 1 2a0 2 e β (ln r)2 r −2α + O ln rr −2α 2α

and

1 rKr − r(rKr )r − Kθ θ = β + 1 + e2a0 β 2 (ln r)2 r −2α + O ln rr −2α . 2α

Thus from (3.31), we get 1 1 1 2a0 2 K,zz = − 2 − β − 2αβ + + 1 − (1 + α) e β (ln r)2 r −2α 2α α z −2α + O ln rr + an imaginary part . Since K,zz is a holomorphic function on D \ {0}, we conclude that 1 2a0 2 −1 2a0 2 1 + 1 − (1 + α) e β = e β = 0, 2α α α which yields β = 0. Proposition 3.1 follows from Lemmas 3.3 and 3.6. 4. Estimate of the second covariant derivative of curvature Lemma 4.1. Let g = e2ψ(z) |dz|2 be an extremal Hermitian metric with ﬁnite energy and area on D \ {0}, and let K(x) be the Gaussian curvature of g. If x = 0 is a weak cusp or a conical singular point with angle 2πα ≤ 2π, then 1 |K,zz | ≤ O 2 as z −→ 0. (4.1) z Proof. Let Ki (y) and ψi (y) be the two smooth functions on T1 deﬁned by (1.2). Let K˜ i (y) = |xi |ε Ki (y) for the small positive number ε. Then by Proposition 3.1, we have K˜ i (y) −→ 0

uniformly on T1

as i −→ +∞.

Moreover, by Propositions 2.1 and 2.2 and by the assumption that α ≤ 1, we have K˜ i = −Ki2 e2ψi |xi |ε + Ce2ψi |xi |ε −→ 0 The standard L2 -estimate implies that K˜ i (y)H 2,2 (T3/4 ) −→ 0.

uniformly on T1 .

(4.2)

204

WANG AND ZHU

In particular, 2 ∇ K˜ i (y) 2 L (T

3/4

−→ 0 )

∇ K˜ i (y)

and

L4 (T3/4 )

−→ 0.

(4.3)

On the other hand, by Proposition 2.2 and (2.13), we have ψi = −Ki e2ψi −→ 0 uniformly on T1 , Oscy∈T1 ψi (y) ≤ 3(α + 1). It follows by the standard L2 -estimate that ψi (y)H 2,2 (T3/4 ) ≤ C1 for some uniform C1 . In particular, there is a uniform constant C2 > 0 such that ∇ψi (y) 4 ≤ C2 . (4.4) L (T ) 3/4

Since ∂ 2 K˜ i ∂ψi ∂ K˜ i −2 , K˜ i,zz = ∂z∂z ∂z ∂z from (4.3) and (4.4), we have K˜ i,zz L2 (T3/4 ) ≤ ∇ 2 K˜ i (y)L2 (T ) 3/4 2 2 + 4∇ K˜ i (y)L4 (T ) ∇ψi (y)L4 (T

3/4 )

3/4

It follows that for any δ > 0, K,zz (x)|xi |1+ε+δ 2 dx = e−3/4 |zi |≤z≤e3/4 |zi |

Therefore, we get B(|xi |)\B((1/2)|xi |)

T3/4

−→ 0

as i −→ +∞.

K˜ i,zz (y)|xi |δ 2 dy ≤ C3 |xi |2δ .

K,zz (x)|x|1+ε+δ 2 dx ≤ C3 |xi |2δ .

(4.5)

Using the standard iteration of (4.5), we see that there is a number C4 such that for any small positive number ε , K,zz (x)|x|1+ε 2 dx ≤ C4 . (4.6) D

EXTREMAL METRICS ON SURFACES WITH SINGULARITIES

205

In particular, D

K,zz (x)|x|2 2 dx ≤ C4 .

It follows that K,zz z2 is a holomorphic function on D because K,zz z2 is a holomorphic function on D \ {0}. Thus there is a uniform constant C such that 1 |K,zz | ≤ C 2 as z −→ 0. z We improve Lemma 4.1 in the following proposition. Proposition 4.1. Let g = e2ψ(z) |dz|2 be an extremal Hermitian metric with ﬁnite energy and area on D \ {0}, and let K(x) be the Gaussian curvature of g. Suppose that x = 0 is a weak cusp or a conical singular point with angle 2πα ≤ 2π. Then 1 |K,zz | ≤ O as z −→ 0. (4.7) z √ Proof. Let u = ln |z| = ln r and w = u + −1θ, where θ ∈ [0, 2π]. Let φ(u, θ) = ψ(r, θ) + u. It is clear that g = e2φ du2 + dθ 2 = e2ψ(z) |dz|2 . In view of Lemma 3.3, we see that there is a number β such that 2π 1 ∂K lim (u, θ) dθ = β, u→−∞ 2π 0 ∂u where K(u, θ ) is the Gaussian curvature of g. From (4.8), we have

2π (1/2π) 0 K(u, θ) dθ lim = β. u→−∞ u

(4.8)

(4.9)

Comparing (4.9) with (3.2) in Lemma 3.2, together with Proposition 3.1, we get β = 0. Let {ui } → −∞ and Ti (2) = {(u, θ) | ui < u < ui + 2}. Similar to (2.13), we have

OscTi (2) φ(u, θ ) ≤ OscTi (2) φ(u, θ) − αu + OscTi (2) {αu} −→ 2α (4.10) uniformly on Ti (2) when i is sufﬁciently large. From (4.7) and Lemma 3.3, we have lim OscTi (2) K(u, θ) = 0.

i→−∞

(4.11)

206

WANG AND ZHU

Proposition 2.1 tells us that

K(u, θ ) = − K 2 (u, θ) + c e2φ(u,θ) −→ 0

as u → −∞. It follows from this, together with the standard L2 -theory and (4.11), that ∇ 2 KL2 (Ti (1)) −→ 0

∇KL4 (Ti (1)) −→ 0

and

(4.12)

as i → +∞. Similarly, from (2.24) and φ(u, θ) = −K(u, θ)e2φ(u,θ) −→ 0 as u → −∞, we have ∇φL4 (Ti (1)) ≤ C

(4.13)

for some uniform constant C. Since ∂φ ∂K ∂ 2K −2 , ∂w∂w ∂w ∂w

K,ww = we have

K,ww 2L2 (T (1)) ≤ ∇ 2 K2L2 (T (1)) + 4∇φ2L4 (T (1)) ∇K2L4 (T (1)) . i

i

i

i

It follows that K,ww 2L2 (T (1)) −→ 0 i

as i −→ +∞.

(4.14)

On the other hand, by Lemma 4.1, we may assume that 1 1 + c1 + η(z), z z2 where c2 and c1 are two constants and η(z) is a holomorphic function on D. Then K,zz = c2

K,ww = z2 K,zz = c2 + c1 z + z2 η(z). It follows that

K,ww 2L2 (T (1)) ≥ |c2 |2 i

Ti (1)

dy

dy − 2eui |c2 | |c1 | + |η(z)| Ti (1) dy −→ 2π|c2 |2 − 2e2ui |c1 |2 + |η(z)|2 Ti (1)

as i −→ +∞. (4.15)

Comparing (4.14) with (4.15), we get c2 = 0, and consequently 1 K,zz = c1 + η(z). z This proves the proposition.

(4.16)

EXTREMAL METRICS ON SURFACES WITH SINGULARITIES

207

5. Extremal metrics on Riemann surfaces Lemma 5.1. Let g = e2ψ(z) |dz|2 be an extremal Hermitian metric with ﬁnite energy and area on D \ {0}, and let K(x) be the Gaussian curvature of g. Suppose that x = 0 is a weak cusp or a conical singular point with angle 2πα ≤ π. Then e−2ψ(z) |K,z | ≤ O(|z|).

(5.1)

Proof. Since −2ψ(z) e K,z z = e−2ψ(z) K,zz , by Proposition 4.1, we have −2ψ(z) e K,z z ≤ O |z|1−2α . Let v(z) be the ∂-potential of (e−2ψ(z) K,z )z . From the standard elliptic estimate, we have v(z) ∈ C 2−2α (D). On the other hand, as in the proof of Proposition 4.1, it is easy to see that K(u, θ )H 2,p (Ti (1)) −→ 0

as i −→ +∞

for any p ≥ 1. In particular, |K,w | = |zK,z | −→ 0

as z −→ 0,

where w = 1/z. Thus by the assumption α ≤ 1/2, we get e−2ψ(z) |K,z | −→ 0

as z −→ 0.

(5.2)

Let η(z) = v(z) − e−2ψ(z) K,z . Then η(z) is a holomorphic function on D, and consequently |η(z)| = O(|z|) by (5.2). This shows that e−2ψ(z) |K,z | ≤ |η(z)| + |v(x)| ≤ O(|z|). Proof of the main theorem. The ﬁrst statement has been proven in Proposition 2.2. We need to prove the second statement. Proposition 4.1 and Lemma 5.1 imply that we can integrate (0.3) by parts to get (0.4), that is, the extremal Hermitian metric g is HCMU (cf. [Ch4]). Now (i) follows from the fact that there is no nontrivial holomorphic vector ﬁeld on a compact Riemann surface with genus(M) ≥ 1; (ii) follows from the fact that there are only two zero points of a nontrivial holomorphic vector on S 2 . The proof of (iii) follows from the fact that if M = S 2 and n = 2, there are two cases: (a) if both singular points are cusp, then there is no extremal Hermitian metric (cf. [Ch4]);

208

WANG AND ZHU

(b) if at least one of the singular points is not a cusp, then there is a unique, rotationally symmetric extremal Hermitian metric that can be constructed by solving an ODE system and determined by the total area and two angles 2παj . Finally, (iv) follows from a result of [Ch5], namely, that ψ is radially symmetric if g is HCMU, M = S 2 , n = 1, and in addition, if α is not an integer. Now we can assume that an HCMU metric is of form g = du2 + f (u)2 dθ 2 for some smooth positive function f (u), where u ∈ [0, +∞) is the conformal parameter and θ ∈ S 1 . It is easy to see that f and the Gaussian curvature K = f¨/f satisfy the ODE system (cf. [Ch4]),  ˙ f (u) = b1 K, (5.3) K˙ 2 = −1 K 3 − 3cK + b2 3 with f˙(0) = α1 and f˙(∞) = α2 , where c > 0, b1 , b2 can be determined uniquely by the singular angle 2π α and area 2πA. In fact, let K 3 − 3cK + b2 = (K − β1 )(K − β2 )(K − β3 )

(5.4)

with β1 , β2 ≥ β3 . Then by the identities 2f˙ = b1 − K 2 + c and

∞

A=

f (u) du = b1 K(∞) − K(0) = b1 (β2 − β1 ),

0

we get ﬁve algebraic equations:   β 3 − 3cβ1 + b2 = 0,   1     β 3 − 3cβ2 + b2 = 0,   2 2α1 = b1 (−β12 + c),     2α2 = b1 (−β22 + c),      A = b1 (β2 − β1 ).

(5.5)

From the above equations, we can determine c, b1 , b2 uniquely by the two angles 2π αj and the area A unless α1 = α2 = 0. Therefore, we can solve the system (5.3) and get the unique solution f (u), and consequently we can construct the rotationally symmetric extremal Hermitian metric, which is determined uniquely by αj and A (cf. [La]). This completes the proof of our main theorem. Remark 5.2. For the case of M = S 2 and n = 3, Luo and Tian [LT] proved the following result: Let the singular angles 2παi ∈ (0, 2π), i = 1, 2, 3. Then there

EXTREMAL METRICS ON SURFACES WITH SINGULARITIES

209

2 exists a metric with positive constant Gaussian curvature on S \ {pj } if and only if α1 + α2 + α3 > 1 and i (αi − αj ) < 1, j = 1, 2, 3. Thus if we choose α1 = α2 > 1/2 and α3 > 0 such that α1 + α2 − 2α3 > 1, then there is no metric on S 2 \ {pj } with positive constant Gaussian curvature. See also [CL4]. On the other hand, it is conjectured that extremal metrics exist on any surface with conical singularities (see [Ch5]). If the conjecture is true, the above result shows that the classiﬁcation in our main theorem is optimal.

Note added in proof. After submitting this paper, we learnt from X. X. Chen that he proved that any surface (with or without boundary) with conical singularities of angles strictly less than 2π admits at least one extremal Hermitian metric. See [Ch1] for his proof. From this result, our classiﬁcation is optimal as discussed in Remark 5.2. References [Ca1] [Ca2] [CY1] [CY2] [CL1] [CL2] [CL3] [CL4] [Ch1] [Ch2]

[Ch3] [Ch4] [Ch5] [DT] [KW] [La] [LT]

E. Calabi, “Extremal Kähler metrics” in Seminar on Differential Geometry, Ann. of Math. Stud. 102, Princeton Univ. Press, Princeton, 1982, 259–290. , “Extremal Kähler metrics, II” in Differential Geometry and Complex Analysis, Springer, Berlin, 1985, 95–114. S. Chang and P. Yang, Conformal deformation of metrics on S 2 , J. Differential Geom. 27 (1988), 259–296. , Extremal metrics of zeta function determinants on 4-manifolds, Ann. of Math. (2) 142 (1995), 171–212. W. X. Chen and C. Li, Classiﬁcation of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), 615–622. , Prescribing Gaussian curvatures on surfaces with conical singularities, J. Geom. Anal. 1 (1991), 359–372. , Qualitative properties of solutions to some nonlinear elliptic equations in R 2 , Duke Math. J. 71 (1993), 427–439. , What kinds of singular surfaces can admit constant curvature?, Duke Math. J. 78 (1995), 437–451. X. X. Chen, Extremal Hermitian metrics on Riemannian surfaces, Internat. Math. Res. Notices 1998, 781–797. , “Extremal metrics in Riemann surfaces and the uniformization theorem” in Chern Symposium (Berkeley, Calif., 1998), Sel. MSRI Video Arch. 2, Math. Sci. Res. Inst., Berkeley, Calif., 1998 (CD-ROM). , Weak limits of Riemannian metrics in surfaces with integral curvature bound, Calc. Var. Partial Differential Equations 6 (1998), 189–226. , Extremal Hermitian metrics on Riemann surfaces, Calc. Var. Partial Differential Equations 8 (1999), 191–232. , Obstruction to existence of metric where curvature has umbilical Hessian in a surface with conical singularities, preprint, 1996. W. Ding and G. Tian, Energy identity for a class of approximate harmonic maps from surfaces, Comm. Anal. Geom. 3 (1995), 543–554. J. L. Kazdan and F. W. Warner, Curvature functions for compact 2-manifolds, Ann. of Math. (2) 99 (1974), 14–47. S. Lang, Elliptic Functions, 2d ed., Grad. Texts in Math. 112, Springer, New York, 1987. F. Luo and G. Tian, Liouville equation and spherical convex polytopes, Proc. Amer. Math. Soc. 116 (1992), 1119–1129.

210 [M] [Sc]

[SU] [T] [UY]

WANG AND ZHU R. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988), 222–224. R. Schoen, “Analytic aspects of the harmonic map problem” in Seminar on Nonlinear Partial Diffrential Equations (Berkeley, Calif., 1983), Math. Sci. Res. Inst. Publ. 2, Springer, New York, 1984, 321–358. J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. (2) 113 (1981), 1–24. M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991), 793–821. M. Umehara and K. Yamada, Metrics of constant curvature 1 with three conical singularities on the 2-sphere, Illinois J. Math. 44 (2000), 72–94.

Wang: Institute of Mathematics, Academia Sinica, Beijing 100080, China; Current: MaxPlanck-Institut für Mathematik, Inselstrasse 22-26, D-04103 Leipzig, Germany; gwang@ mis.mpg.de Zhu: Institute of Mathematics, Peking University, Beijing, China; [email protected]. edu.cn

Vol. 104, No. 2

DUKE MATHEMATICAL JOURNAL

© 2000

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS DENIS BENOIS

Contents §0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 §1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 1.1. Rings of p-adic periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 1.2. Classiﬁcation of p-adic representations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 1.3. Computation of Galois cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 §2. The explicit reciprocity law of Coleman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 2.1. The Kummer map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 2.2. The isomorphism h2 : H 2 (C . (ᏻKn (1))) H 2 (GKn , Zp (1)) . . . . . . . . . . . 228 2.3. The explicit reciprocity law for Qp (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 §3. The functions Ek,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 3.1. Construction of Ek,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 3.2. The residue formula for Ek,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 §4. Construction of families of points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 4.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 4.2. The homomorphisms T ,k,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 §5. Explicit reciprocity law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 5.1. Cohomological pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 5.2. Interpolation of exponential maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 5.3. The explicit reciprocity law of Perrin-Riou . . . . . . . . . . . . . . . . . . . . . . . . . . 264 §0. Introduction 0.1. In his paper [F2], J.-M. Fontaine worked out a general approach to the classiﬁcation of p-adic representations of local ﬁelds. His method is based on the relation between local ﬁelds of characteristic zero and functional local ﬁelds, given by the ﬁeld of norms functor (see [Win]). More precisely, Fontaine established an equivalence between the category of Zp -representations of a local ﬁeld of characteristic zero and the category Mét ᏻK of étale modules over a 2-dimensional local ring ᏻK , equipped with a Frobenius and a continuous action of the cyclotomic Galois group. Received 12 May 1998. Revision received 29 November 1999. 2000 Mathematics Subject Classiﬁcation. Primary 11S15; Secondary 11R23. Author partially supported by Russian Foundation for Fundamental Investigations grant number 97-01-00058-a and by Volkswagen Forschung. 211

212

DENIS BENOIS

As an application of this idea he showed that the reciprocity law of Bloch and Kato can be deduced from the explicit formula for Witt pairing in characteristic p (see [F3]; see also [C, §8], for a reﬁned version of this result). In this paper we apply the theory of Fontaine to Iwasawa theory of crystalline representations. In particular, for crystalline representations of ﬁnite height we prove the explicit reciprocity law conjectured by B. Perrin-Riou [PR2]. Our approach differs from the methods of P. Colmez [Cz1] and K. Kato, M. Kurihara, and T. Tsuji [KKT] and essentially uses the following results: (a) computation of Galois cohomology of p-adic representations using complexes of -modules (see L. Herr [H1], [H2]), (b) classiﬁcation of crystalline representations of ﬁnite height in terms of modules (see N. Wach [W]). Since we are restricted to representations of ﬁnite height, this result is weaker than the mentioned results of Colmez and Kato, Kurihara, and Tsuji, where the conjecture of Perrin-Riou is proved for all crystalline representations. On the other hand, Fontaine conjectured that any crystalline representation of an unramiﬁed local ﬁeld is of ﬁnite height, and our result therefore covers the general case modulo this conjecture. In [W], Wach showed that the conjecture of Fontaine holds for representations whose Hodge ﬁltration has length less than p. Recently Colmez [Cz2] proved this in full generality, but his approach uses essentially the results of [Cz1] concerning the explicit reciprocity law. The search of a more direct proof seems to be an interesting problem. Finally note that, using the results of Herr, one can extend the construction of a generalized Coleman map, given in [PR2], to all p-adic representations (from Fontaine, unpublished). F. Cherbonnier and P. Colmez [CCz1], [CCz2] showed that for de Rham representations this map is closely related to the logarithmic map, constructed in [Cz1] and, therefore, closely related to the dual exponential map. Acknowledgments. Some part of this work was done during my stay at the Institut für Experimentelle Mathematik (Essen), Université de Limoges, Université de Paris VI, and at the Institut Henri Poincaré in 1996–1997. I am very grateful to these institutions for their hospitality. Finally, I would like to thank the referee for several suggestions for the improvement of the ﬁrst version of this paper. 0.2. Notation. Throughout this paper p is a ﬁxed odd prime number, K a ﬁnite unramiﬁed extension of Qp with residue ﬁeld k, W = W (k) its ring of integers, and σ ¯ the absolute Frobenius of K. Let K¯ be an algebraic closure of K, GK = Gal(K/K), n ¯ and C the completion of K. As usual, µpn denote the group of p th roots of unity. p We ﬁx a system of generators ε = (ζpn )n0 , ζpn ∈ µpn , such that ζpn = ζpn−1 for all n. Then ε can be viewed as a generator of Zp (1). Let Kn be the cyclotomic extension of K obtained by adjoining ζpn , and let K∞ = ∞ n=0 Kn . Put Γ = Gal(K∞ /K), Γn = Gal(K∞ /Kn ), and Λ = Zp [[Γ ]]. Denote by χ : Γ → Z∗p the cyclotomic character. The ring ᏾ = W [[X]] of formal power series has a natural Λ-module structure given by g(X) = (1 + X)χ(g) − 1. Moreover, it is equipped with the ring

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

homomorphisms

σ : ᏾ −→ ᏾,

ai X i

σ

=

213

aiσ X i

and ϕ : ᏾ −→ ᏾,

ai X i

ϕ

=

aiσ ϕ(X)i ,

ϕ(X) = (1 + X)p − 1

which commute with Γ . Denote by ᏾ψ the W -submodule of ᏾ consisting of all power series f (X) such that ζ ∈µp f (ζ (1+X)−1) = 0. Then ᏾ψ is a free W [[Γ ]]module of rank 1 generated by 1 + X. The differential operator D = (1 + X)d/dX is invertible on ᏾ψ . For a topological GK -module N we denote by N(k) its twisting by χ k , and we denote by T wkε : N → N(k) the map n → n⊗ε k . Further we denote by H ∗ (GK , N ) the continuous cohomology of N (see [T]). In particular, if N is discrete, then H ∗ (GK , N ) coincides with usual Galois cohomology. 0.3. Interpolation of exponential maps. In this section we review the theory of Perrin-Riou [PR2]. We start with a classical result on p-adic interpolation. Let f : N → Qp be an arbitrary function, deﬁned on the set of positive integers, and let f (i) = ai . The theorem of Mahler states that f can be interpolated by a continuous p-adic function if and only if j j −i j lim ai = 0. (−1) j →∞ i i=0

In particular, applying this result to the function (1 − p k−1 )ζ (1 − k) and using well-known congruences for Bernoulli numbers, one can construct a Kubota-Leopoldt p-adic zeta function. Now let V be a crystalline representation of GK , and let Dcris (V ) be the ﬁltered module of Fontaine associated to V . Then Dcris (V ) is a K-vector space equipped with an exhaustive separated decreasing ﬁltration Fili Dcris (V ) and with an injective σ -semilinear endomorphism ϕ : Dcris (V ) → Dcris (V ). In particular, Dcris (V (−j )) = Dcris (V ) ⊗ ej , where ϕ(ej ) = p j ej and Filk Dcris (V (−j )) = Filk−j Dcris (V ) ⊗ ej . Perrin-Riou interpolated a family of Bloch-Kato exponential maps expV (k),Kn : Dcris (V ) ⊗K Kn −→ H 1 GKn , V (k) . Namely, let ᏴK be the subset of K[[X]] consisting of all power series that converge ¯ Put Ᏸ(V ) = ᏾ψ ⊗W Dcris (V ) and on the maximal ideal of K.

Ᏼ(V ) = α ∈ ᏴK ⊗K Dcris (V ) | (1 − ϕ)α ∈ Ᏸ(V ) . Assume ﬁrst that all Hodge weights of V are 0. Fix a lattice T of V stable under the action of GK , and denote by M a W -lattice of Dcris (V ), associated to T (this

214

DENIS BENOIS

choice is not unique). Then the set     Ᏼ(T ) = α ∈ W [[X]] ⊗W M α ζ (1 + X) − 1 = pα ϕ (X)   ζ ∈µ p

is a W -lattice of Ᏼ(V ). Perrin-Riou constructed a system of homomorphisms T ,k,n : Ᏼ(T ) −→ H 1 GKn , T (k) satisfying the following properties. (m) (i) Interpolation property. Let T ,r+k,n denote the map T ,k,n modulo p m . Let j 2 and m = j (n − 1) + 1. Then for any r ∈ Zp one has j (m) k j ε (−1) T w−k ◦ resKm /Kn T ,r+k,n = 0. k k=0

(ii) Twisting. Let α ∈ Ᏼ(T ). Then T (−1),k+1,n (D ⊗ e1 )α = −kp n T ,k,n (α). (iii) For any α ∈ Ᏼ(T ) and n 1 one has cor Kn+1 /Kn T ,k,n+1 (α) = T ,k,n (σ ⊗ ϕ)α . (iv) Relation to exponential maps. Let V ,k,n be the maps obtained from T ,k,n by ⊗Qp . Let k 1 and α(X) ∈ Ᏼ(V (k)). Then V ,k,n D k ⊗ ek α = (−1)k (k − 1)!p (k−1)n expV (k),Kn α ζpn − 1 for any n ∈ N. Now let V be an arbitrary crystalline representation. Using property (ii) we deﬁne the maps V ,k,n for almost all k putting V ,k,n : Ᏼ(V ) −→ H 1 GKn , V (k) , V ,k,n (α) =

(−1)h p −hn V (−h),h+k,n D h ⊗ eh α , k · (k + 1) × · · · × (k + h − 1)

where h is sufﬁciently big. 0.4. Main result. We conserve the previous notation. Consider the dual representation V ∗ = HomQp (V , Qp ). The natural duality Dcris (V ) × Dcris (V ∗ ) −→ Qp

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

215

can be extended by linearity to the map [ , ] : Ᏸ(V ) × Ᏸ(V ∗ ) −→ Qp ⊗ ᏾. The main result of this paper gives an explicit description of the cohomological pairing ∪ ( , )k,n : H 1 GKn , V (k) × H 1 GKn , V ∗ (1 − k) −→ Qp in terms of homomorphisms V ,k,n . Theorem. Let V be a crystalline representation of ﬁnite height. For α ∈ Ᏼ(V ) and β ∈ Ᏼ(V ∗ ) put f = (1 − ϕ)α, g = (1 − ϕ)β. Then for almost all k ∈ Z one has (−1)k V ,k,n (α), V ∗ ,1−k,n (β) k,n = Tr D −k f, D k−1 g (ζ − 1). K/Q p n p ζ ∈µpn

This result is equivalent to the explicit reciprocity law conjectured in [PR2]. For V = Qp (1) we essentially obtain the reciprocity law of Coleman [Cn2]. 0.5. The outline of the proof. The proof of this theorem reduces, roughly speaking, to an explicit construction of V ,k,n for all k ∈ Z. In [KKT], Kato, Kurihara, and Tsuji constructed so-called syntomic complexes that allow one to compute H ∗ (GKn , T (k)) in terms of differential forms and Frobenius. (Note that the approach of Colmez is based on other ideas and gives also some generalization of the maps V ,k,n to de Rham representations). In this paper we work with complexes of -modules introduced in [H1]. Namely, for any n we can consider T as a representation of GKn . Let ᏹn denote the -module associated to T (see [F2]). By the mentioned result of Herr, H ∗ (GKn , T (k)) is isomorphic to the cohomology of the complex f g C . ᏹn (k) : 0 −→ ᏹn (k) −→ ᏹn (k) ⊕ ᏹn (k) −→ ᏹn (k) −→ 0, where f (x) = ((ϕ − 1)x, (γn − 1)x) and g(y, z) = (γn − 1)y + (1 − ϕ)z. Using this isomorphism we construct the maps T ,k,n in terms of C . (ᏹn (k)). To link ᏹn and Dcris (V ), we use the result of Wach [W] concerning crystalline representations of ﬁnite height. The pairing H 1 (GKn , V (k))×H 1 (GKn , V (1−k)) → Qp can be computed explicitly in terms of Cn. (ᏹn (k)) and Cn. (ᏹn∗ (1−k)) (see [H2]). Applying Herr’s formula to the maps T ,k,n we obtain the explicit reciprocity law. In fact, the main difference between this approach and the method of [KKT] is the use of complexes C . (ᏹn (k)) instead of syntomic complexes. The organization of the paper is as follows. In §1 we review the theory of p-adic periods, in particular the computation of Galois cohomology using -modules. In §2 the simplest case V = Qp (1) is considered. We describe the exponential map ∗ Kn → H 1 (GKn , Qp (1)) in terms of the complex C . (ᏻKn (1)) and give a short proof of Coleman’s explicit reciprocity law.

216

DENIS BENOIS

In §3 we deﬁne functions Ek,n , which play a role of “negative exponents” if k 0, and we study their formal properties. In the last two sections, we assume that V is a crystalline representation of ﬁnite height. In §4 we construct the homomorphisms T ,k,n via -modules and prove properties (i)–(iv). In §5 we prove the main result and deduce from it the explicit reciprocity law of Perrin-Riou. §1. Preliminaries 1.1. Rings of p-adic periods 1.1.1. The rings W (R), Sn , ᏻKn , and Sˆ PD (see [F1], [F2]). Let K be a ﬁnite unramiﬁed extension of Qp with residue ﬁeld k and valuation ring W = W (k). Let ¯ and OC its ring of integers. K¯ be an algebraic closure of K, C the completion of K, Denote by v : C → R ∪ {∞} the valuation of C normalized so that v(p) = 1. Let R = lim n OC /pOC be the inverse limit of copies of OC /pOC under the pth power ← − homomorphism. An element x ∈ R can be viewed as a sequence (xn )n∈N such that p pn xn+1 = xn for all n. Let xˆn ∈ OC be a lifting of xn . Then the sequence xˆm+n converges to some x (m) ∈ OC which does not depend on the choice of liftings. The ring R equipped with the valuation vR (x) = v(x (0) ) is a complete local ring of characteristic ¯ Moreover, it is integrally closed in the ﬁeld of fractions Fr(R). p with residue ﬁeld k. We ﬁx a generator ε = (ζpn )n∈N of Zp (1), and we denote by the same letter the unique element ε ∈ R such that ε (n) = ζpn . The rings of Witt vectors W (R) and W (Fr(R)) are complete rings of characteristic ¯ Moreover, they are endowed with natural actions of the Galois zero containing W (k). group GK and Frobenius ϕ. Let [ ] : R → W (R) be the Teichmüller map. Then any u = (u0 , u1 , . . . ) ∈ W (R) can be written in the form u=

∞

p−m m

um

p .

m=0 n

For n 0 put πn = ϕ −n (π ) = [ε]1/p −1, and let Sn = W [[πn ]] be the W -subring of W (R), generated by πn . To simplify notation we put π = π0 and S = S0 . The rings Sn are stable under the actions of GK and ϕ; moreover, g(πn ) = (1 + πn )χ(g) − 1, ϕ(πn ) = (1 + πn )p − 1, where χ denotes the cyclotomic character. It is not difﬁcult to see that πn is invertible in W (Fr(R)). Then Sn [πn−1 ] can be considered as a subring of W (Fr(R)) and we denote by ᏻKn its p-adic completion. Thus, ᏻKn = W {{πn }} is a 2-dimensional local ring with residue ﬁeld k((εn − 1)),

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

217

where εn = ϕ −n (ε). One has ϕ(ᏻKn ) = ᏻKn−1 . We denote by ᏻnr Kn the maximal unramiﬁed extension of ᏻKn , contained in W (Fr(R)), and we denote by ᏻˆ nr Kn its is isomorphic to the separable closure of p-adic completion. The residue ﬁeld of ᏻˆ nr Kn k((εn − 1)) in W (Fr(R)). The sequence 1−ϕ ˆ nr 0 −→ Zp −→ ᏻˆ nr Kn −−−→ ᏻKn −→ 0

is exact. Finally we review the deﬁnition of the ring Sˆ PD which plays an important role in the classiﬁcation of crystalline representations in terms of -modules. Consider the PD-envelope of S: 2 3 π π S PD = S , ,... . 2! 3! Denote by I i the ideal of S PD generated by π m /m!, m i, and put S Sˆ PD = lim i . ← −I PD

i

1.1.2. The rings BdR and Bcris (see [F1], [F4]). Let θ : W (R) → OC be the map given by the formula ∞ ∞ m m [um ]p u(0) = θ m p . m=0

m=0

Then θ is a surjective ring homomorphism and ker(θ) is generated by the element v=

p−1

[ε]i/p .

i=0

Let WK (R) = W (R)⊗W K. By linearity, θ can be extended to a map θK : WK (R) → C. Put WK1 (R) = ker(θK ), WKi (R) = (WK1 (R))i , and deﬁne + = lim WK (R)/WKi (R). BdR ← − i

+ BdR

It may be shown that is a complete discrete valuation ring with residue ﬁeld C. + Moreover, it is equipped with a GK -action. There is a natural embedding K¯ ⊂ BdR . Since θ (π ) = θ ([ε] − 1) = 0, the series log(1 + π) =

∞ m=1

(−1)m+1

πm m

+ + . The element t generates the maximal ideal of BdR . The converges to some t ∈ BdR Galois group GK acts on t by the formula g(t) = χ(g)t.

218

DENIS BENOIS

+ −1 + Let BdR = BdR [t ] be the ﬁeld of fractions of BdR . This is a complete discrete valuation ﬁeld, equipped with a GK -action and an exhaustive, separated decreasing + . As a GK -module, Fili BdR / Fili+1 BdR C(i). ﬁltration Fili BdR = t i BdR Consider the PD-envelope of W (R) with respect to the map θ : W (R) → OC : 2 3 v v PD W (R) = W (R) , ,... . 2! 3! + + Denote by Acris its p-adic completion. Let Bcris = Acris ⊗Zp Qp and Bcris = Bcris [t −1 ]. i The ring Bcris is a subring of BdR endowed with induced ﬁltration Fil Bcris and GK action. Moreover, it has a Frobenius operator ϕ, induced by ϕ on W (R). In particular, one has ϕ(t) = pt. The rings BdR and Bcris are related to each other via the fundamental exact sequence f

+ −→ 0, 0 −→ Qp −→ Bcris −→ Bcris ⊕ BdR /BdR + where f (x) = ((1 − ϕ)x, x (mod BdR )) (see [F4], [BK], and [dSh]). Finally, note PD ˆ that S can be viewed as a subring of Acris . To prove it, write π k /k! in the form p v k π1k /k!. Suppose that k p m . From the congruence π1 ≡ π (mod p) one has m−1 m−1 pm π1 ≡ π p (mod p m ). Then π k /k! = p m αv k /k! + p m−1 βv k+p /(k + p m−1 )! m with α, β ∈ S. Hence I

⊂ p m−1 Acris and Sˆ PD ⊂ Acris .

1.2. Classiﬁcation of p-adic representations 1.2.1. Crystalline representations and de Rham representations (see [F1], [F5]). Let L be a ﬁnite totally ramiﬁed extension of K. Denote by RepQp (GL ) the category ¯ For a p-adic representation V , deﬁne of p-adic representations of GL = Gal(K/L). G DdR (V ) = V ⊗Qp BdR L . Then DdR (V ) is an L-vector space with a decreasing ﬁltration given by Fili DdR (V ) = (V ⊗Qp Fili BdR )GL . The representation V is said to be de Rham if dimL DdR (V ) = dimQp (V ). (In general, one has dimL DdR (V ) dimQp (V ).) For a de Rham representation we have the Hodge-Tate decomposition V ⊗Qp C ⊕ Gr −i DdR (V ) ⊗L C(i), where Gr k (DdR (V )) = Filk DdR (V )/ Filk+1 DdR (V ). Analogously one deﬁnes G Dcris (V ) = V ⊗Qp Bcris L . Then Dcris (V ) is a K-vector space equipped with a Frobenius operator and a ﬁltration, induced by the natural inclusion Dcris (V )⊗K L ⊂ DdR (V ). One has dimK Dcris (V ) dimL DdR (V ) dimQp (V ), and V is said to be crystalline if the equality holds here.

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

219

For the resulting categories we have Repcris (GL ) ⊂ RepdR (GL ) ⊂ RepQp (GL ). The functor DdR (resp., Dcris ) restricted on RepdR (GL ) (resp., Repcris (GL )) commutes with tensor products, direct sums, and duals. Moreover, Dcris is an equivalence between Repcris (GL ) and some subcategory of the category of ﬁltered K-modules. 1.2.2. Classiﬁcation of Zp -adic representations in terms of -modules (see [F2]). Fontaine [F2] found a classiﬁcation of all p-adic representations in terms of étale modules over a 2-dimensional local ring endowed with semilinear actions of Frobenius and Γ commuting to each other. In particular, his theory allows us to compute the Galois cohomology of p-adic representations using complexes of such modules (see §1.3 below). We review these results only for cyclotomic ground ﬁelds. Let A be a commutative Noetherian ring with a ﬂat map ϕ : A → A. An A-module ᏹ is said to be a ϕ-module if it is equipped with a ϕ- semilinear map ϕᏹ : ᏹ → ᏹ. (We often write ϕ instead ϕᏹ .) Let ᏹϕ = Aϕ ⊗A ᏹ be the module obtained from ᏹ by extending scalars ϕ : A → A, and let ᏹ : ᏹϕ → ᏹϕ be the induced linear map; that is, ᏹ (a ⊗ m) = a ⊗ ϕᏹ (m). ᏹ is called an étale module over A if it is ﬁnitely generated and if ᏹ is a bijection. Fix n ∈ N and consider the ring ᏻKn , with Frobenius operator ϕ and continuous action of Γn . An étale -module over ᏻKn is an étale ᏻKn -module equipped with a semilinear action of Γn , commuting with ϕ. The category Mét ᏻKn of étale modules over ᏻKn is a ⊗-category. Let RepZp (GKn ) be the category of Zp -representations, that is, the category of ﬁnite Zp -modules, equipped with a continuous linear action of GKn . Theorem (J.-M. Fontaine). (i) The functor DᏻKn : RepZp GKn −→ Mét ᏻK , n

GK∞ DᏻKn (T ) = T ⊗Zp ᏻˆ nr Kn is an equivalence of categories. (ii) The functor VᏻKn : Mét ᏻK −→ RepZp GKn , n

VᏻKn (ᏹ) = ᏹ ⊗ᏻKn ᏻˆ nr Kn

ϕ=1

is quasi inverse to DᏻKn . Remarks. (1) Let T be a representation of GKn , and let m n. Then DᏻKm (T ) = ᏻKm ⊗ᏻKn DᏻKn (T ).

220

DENIS BENOIS

(2) Denote by Reptor(GKn ) the category of p-torsion Galois representations, and denote by Mét ᏻKn ,tor the category of p-torsion -modules. Then DᏻKn and VᏻKn induce equivalences of these categories. Passing to inductive limits one obtains an equivalence between the category MGKn ,tor of p-torsion GKn -modules and the cate− ét gory Mind ᏻK ,tor . n

1.2.3. Classiﬁcation of crystalline representations in terms of -modules (see [F2], [W]). In this section we review the results of Fontaine and Wach. ˜ ˜ Let S˜n = ᏻˆ nr Kn ∩ W (R), and let, for simplicity, S = S0 . For a free Zp -module T of ﬁnite rank endowed with a continuous action of GKn , deﬁne G DSn (T ) = T ⊗Zp S˜n K∞ . Then DSn (T ) is a free Sn -module with natural actions of ϕ and Γn . One has rangSn DSn (T ) rangᏻKn DᏻKn (T ) = rangZp T . T is called a representation of ﬁnite height if rangSn DSn (T ) = rangZp T . Theorem (N. Wach). Let T be a representation of ﬁnite height. Then the following two conditions are equivalent: (i) V = T ⊗Zp Qp is crystalline. (ii) There exists a free Sn -submodule N ⊂ DSn (T ) of rank d such that Γn acts trivially on (N/πn N)(−h) for some h ∈ Z. One can also set

G DS (T ) = T ⊗Zp S˜ K∞ .

It is easy to see that DS (T ) = ϕ n DSn (T ) and that (ii) is equivalent to the existence of a free submodule NS ⊂ DS (T ) of rank d such that Γn acts trivially on (NS /πNS )(−h). Recall brieﬂy the implication (ii) ⇒ (i). Taking T (−h) instead of T we can suppose that h = 0, that is, that Γn acts trivially on NS /πNS . Let π π2 πm PD S(n) = S n , ,..., ,... . p 2!p 2n m!p mn i

PD generated by the elements π m /p mn m!, m i, and Denote by I(n) the ideal of S(n) deﬁne i PD PD Sˆ(n) = lim S(n) I(n) . ← − i

PD = Sˆ PD . The Frobenius operator on S can be extended to the Note that Sˆ(0) PD → Sˆ PD , and we have, therefore, a homomorphism ϕ n : Sˆ PD → A ϕ : Sˆ(n) cris . (n−1) (n)

map

By successive approximations it is not difﬁcult to show that there exists a unique PD )Γn of the natural projection N ⊗ PD ˆ Sˆ(n) section NS /π NS (NS ⊗ S ˆ Sˆ(n) → NS /πNS .

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

221

Consider a sequence of maps n n PD Γn ϕ ⊗ϕ ˆ S Sˆ(n) ˆ S Sˆ PD Γn ⊂ T ⊗ Acris GKn . NS /π NS NS ⊗ −−−−→ NS ⊗

Since rangZp NS /π NS = d, it implies that (T ⊗Acris )GKn contains a W -lattice of rank d. Thus dimQp Dcris (V ) = d, and V is crystalline. ˆ S Sˆ PD )Γn ⊂ (T ⊗ Acris )GKn , the W ˆ S Sˆ PD )Γn ⊂ (DS (T )⊗ Note that since (NS ⊗ module ˆ S Sˆ PD Γn M = DS (T )⊗ is a lattice of Dcris (V ) stable under the Frobenius operator ϕ. 1.3. Computation of Galois cohomology 1.3.1. Construction of complexes (see [H1]). Denote by MGKn ,pm the category of pm -torsion GKn -modules. The functor DᏻKn gives an equivalence between this − ét category and the category Mind ᏻKn ,p m whose objects are inductive limits of étale m -modules of p -torsion. Note that these categories have enough injective objects. Put ᏻKn ,pm = ᏻKn /p m ᏻKn . For any T ∈ MGKn ,pm there is a natural isomorphism HomGKn Z/p m Z, T Hom ᏻKn ,pm , DᏻKn (T ) , and therefore we obtain isomorphisms of derived functors: Ext iGK Z/p m Z, T Ext i ᏻKn ,pm , DᏻKn (T ) . n

The uniqueness of a derived functor implies that Ext iGK (Z/p m Z, T ) is isomorphic to n

H i (GKn , T ) (see [CE, Chap. 6, Prop. 4.1.3]). Fix a generator γn ∈ Γn , and consider the complex β

α

P . : 0 ←− ᏻKn ,pm [ϕ, γn ] ←− ᏻKn ,pm [ϕ, γn ]⊕ ᏻKn ,pm [ϕ, γn ] ←− ᏻKn ,pm [ϕ, γn ] ←− 0, where α(y, z) = (ϕ − 1)y + (γn − 1)z and β(x) = ((γn − 1)x, (1 − ϕ)x). Then P . is a − ét projective resolution of ᏻKn ,pm . To any ᏹ ∈ Mind ᏻK ,tor we associate the complex n

f

g

C . (ᏹ) : 0 −→ ᏹ −→ ᏹ ⊕ ᏹ −→ ᏹ −→ 0, where f (m1 ) = ((ϕ − 1)m1 , (γn − 1)m1 ) and g(m2 , m3 ) = (γn − 1)m2 + (1 − ϕ)m3 . An easy computation shows that Hom (P . , ᏹ) is isomorphic to C . (ᏹ). Thus, if ᏹ = DᏻKn (T ), then H i (GKn , T ) is isomorphic to H i (C . (ᏹ)). Passing to direct limits we obtain a system of natural isomorphisms hi : H i (C . (ᏹ)) H i GKn , T for all T ∈ MGKn ,tor .

222

DENIS BENOIS

In this paper it is more convenient to use continuous Galois cohomology. Let T be a Zp -representation of GKn , and let ᏹ = DᏻKn (T ). Then, repeating the argument of Tate [T, Prop. 2.2], we obtain an isomorphism between H i (GKn , T ) and H i (C . (ᏹ)) which also is denoted by hi . In particular, h0 : ᏹϕ=1,γn =1 H 0 (GKn , T ). The isomorphism h1 also has a rather simple description in terms of cocycles. Tensoring the exact sequence 1−ϕ

ˆ nr 0 −→ Zp −→ ᏻˆ nr Kn −−−→ ᏻKn −→ 0 with T , one obtains an exact sequence 1−ϕ

ˆ nr 0 −→ T −→ ᏹ ⊗ᏻKn ᏻˆ nr Kn −−−→ ᏹ ⊗ᏻKn ᏻKn −→ 0. Proposition 1.3.2. Let m1 , m2 ∈ ᏹ, and assume that (γn − 1)m1 = (ϕ − 1)m2 . 1 Let u ∈ ᏹ ⊗ᏻKn ᏻˆ nr Kn be a solution of the equation (1 − ϕ)u = m1 . Then h sends cl(m1 , m2 ) to the class of the cocycle k(g)−1 g −→ ug − u + 1 + γn + · · · + γn m2 , k(g)

where γn

= g|K∞ .

Proof. Let ᏺm1 ,m2 = ᏹ ⊕ ᏻKn e, where the action of ϕ and γn is given by ϕ(e) = e + m1 and γn (e) = e + m2 . Then the long exact cohomology sequence associated to the short exact sequence (1)

0 −→ ᏹ −→ ᏺm1 ,m2 −→ ᏻKn −→ 0

1 : H 0 (C . (ᏻ )) → H 1 (C . (ᏹ)). An easy gives the connecting homomorphism δ Kn 1 (1). Applying V diagram search shows that cl(m1 , m2 ) = δ ᏻKn to the sequence (1) one obtains an exact sequence α

0 −→ T −→ Tm1 ,m2 −→ Zp −→ 0 1 : Z → H 1 (G , T ). Let (1 − ϕ)u = m . and the connecting homomorphism δGal p Kn 1 nr and (1 − ϕ)(u + e) = 0; that is, u + e ∈ T Then u + e ∈ ᏺm1 ,m2 ⊗ᏻKn Oˆ K m ,m 1 2. n 1 (1) can be represented by the cocycle g −→ (u + e)g − (u + e) = ug − u + Thus δGal k(g)−1 (1 + γn + · · · + γn )m2 . The proposition follows now from commutativity of the diagram

H 0 (C . (ᏹ))

1 δ

h1

h0

Zp

/ H 1 (C . (ᏹ))

1 δGal

/ H 1 GK , T . n

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

223

The following proposition, which can be easily proved by the same method, plays an important role in this paper. Proposition 1.3.3. Let T and U be Zp -representations of GKn . Put ᏹ = DᏻKn (T ) and ᏺ = DᏻKn (U ). Then the cup products ∪ H i GKn , T × H j (GKn , U ) −→ H i+j (GKn , T ⊗ U ) have the following explicit description in terms of complexes C . (ᏹ) and C . (ᏺ): (i) h0 (cl(m)) ∪ h0 (cl(n)) = h0 (cl(m ⊗ n)), (ii) h0 (cl(m)) ∪ h1 (cl(n1 , n2 )) = h1 (cl(m ⊗ n1 , m ⊗ n2 )), (iii) h1 (cl(m1 , m2 )) ∪ h1 (cl(n1 , n2 )) = h2 (cl(m2 ⊗ γn (n1 ) − m1 ⊗ ϕ(n2 ))). Proof. See [H2, Prop. 4.3]. Proposition 1.3.4. (i) DᏻKn (Zp (1)) = ᏻKn (1). (ii) The map TRn : H 2 C . ᏻKn (1) −→ Zp given by the formula TRn (α ⊗ ε) = −

pn αdπn Tr K/Qp ◦ resπn log χ(γn ) 1 + πn

is an isomorphism. Proof. See [H2, Th. 4.4]. Remark. In §2 we show that TRn coincides with the canonical isomorphism H 2 (GKn , Zp (1)) Zp . §2. The explicit reciprocity law of Coleman 2.1. The Kummer map. In this section, K is a ﬁnite unramiﬁed extension of Qp (p ! = 2) with the ring of integers W = W (k), σ the absolute Frobenius of K, ᏾ = W [[X]] the ring of formal power series equipped with Frobenius endomorphisms σ and ϕ, and D = (1 + X)d/dX. There is a natural isomorphism between ᏾ and Sn = W [[πn ]] which sends X to πn . Lemma 2.1.1. (i) For any f (X) ∈ ᏾ and γ ∈ Γn , one has γf (πn ) ≡ f (πn ) +

χ(γ ) − 1 Df (πn )π pn

In particular, γf (πn ) ≡ f (πn ) (mod π). (ii) We have ϕ(π ) ≡ 0 (mod (pπ, π p )). (iii) Dϕ = pϕD.

mod π 2 .

224

DENIS BENOIS

Proof. To simplify notation, put κn (γ ) =

(χ(γ ) − 1) . pn

Taking into account that χ (γ ) ≡ 1 (mod p n ), one can write γ (πn ) = (1 + πn )(1 + π )κn (γ ) − 1 ≡ πn +

χ(γ ) − 1 (1 + πn )π pn

mod π 2 .

Then, expanding γf (πn ) = f (γ (πn )) in powers of π, we obtain γf (πn ) ≡ f (πn ) +

χ(γ ) − 1 Df (πn )π pn

mod π 2 .

The proofs of (ii) and (iii) are straightforward, and they are omitted here. 2.1.2. The function D. Let E0 : ᏾ → W/(σ − 1)W be the map given by mod (σ − 1)W . E0 (f (X)) = Df (0) Denote by m = (p, X) the maximal ideal of ᏾, and put Ꮽ = 1+ m. In [Cn1], Coleman constructed an exact sequence D

0 −→ (1 + X)Zp −→ Ꮽ −→ ᏾E0 =0 −→ 0, where the homomorphism D is given by the formula ϕ log F (X). D(F (X)) = 1 − p Put τ = 1/π + 1/2. Lemma 2.1.3. Let F (X) ∈ Ꮽ and f (X) = D(F (X)). Then for any γ ∈ Γn there exists a unique aF,γ (πn ) ∈ Sn such that (i) (ϕ − 1)(aF,γ (πn ) ⊗ ε) = (γ − 1)(f (πn )τ ⊗ ε), (ii) aF,γ (πn ) ≡ p −n (1 − χ(γ ))D log F (πn ) (mod π). Proof. By Lemma 2.1.1 we can write γf (πn ) = f (πn ) +

χ(γ ) − 1 Df (πn )π pn

mod π 2 .

Similarly one has γ (π ) ≡ χ (γ )π +

χ(γ )(χ(γ ) − 1) 2 π 2

These congruences imply that (γ − 1) f (πn )τ ⊗ ε ≡ −a˜ F,γ (πn ) ⊗ ε

mod π 3 .

(mod π),

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

225

where a˜ F,γ (πn ) = p −n (1−χ (γ ))D log F (πn ). Using the identity D ◦ϕ = pϕ ◦D we can write ϕ −n (ϕ − 1)a˜ F,γ (πn ) = p (χ (g) − 1)D 1 − log F (πn ) = p −n (χ(γ ) − 1)Df (πn ). p Hence (ϕ − 1)a˜ F,γ (πn ) ⊗ ε ≡ (γ − 1)(f (πn )τ ⊗ ε) (mod π). Since ϕ − 1 is invertible on πSn , it implies that there exists a unique aF,γ (πn ) ≡ a˜ F,γ (πn ) (mod π ) such that (ϕ − 1)(aF,γ (πn ) ⊗ ε) = (γ − 1)(f (πn )τ ⊗ ε), and the lemma is proved. 2.1.4.

Fix a generator γn of Γn = Gal(K∞ /Kn ), and consider the complex Cn. =

C . (ᏻKn (1)):

f

g

0 −→ ᏻKn (1) −→ ᏻKn (1) ⊕ ᏻKn (1) −→ ᏻKn (1) −→ 0, where f (α) = ((ϕ − 1)α, (γn − 1)α) and g(α1 , α2 ) = (γn − 1)α1 + (1 − ϕ)α2 . The short exact sequence m

p 0 −→ µpm −→ K¯ ∗ −−→ K¯ ∗ −→ 0

gives rise to the connecting homomorphism δn,m : Kn∗ → H 1 (GKn , µpm ). Passing to the inverse limit over m, we obtain the Kummer map δn : Kn∗ −→ H 1 GKn , Zp (1) . The main result of this section is an explicit description of this homomorphism in terms of C . (ᏻKn (1)). Proposition 2.1.5. Let

ιn : Ꮽ −→ H 1 Cn.

be the homomorphism F (X) → cl(f (πn )τ ⊗ε, aF,γn (πn )⊗ε) with f (X) = D(F (X)), and let ρn (F ) = F (ζpn − 1). Then the diagram Ꮽ

−ιn

ρn

Kn∗

δn

/ H 1 C. n

h1

/ H 1 GK , Zp (1) n

is commutative. 2.1.6. Proof of Proposition 2.1.5. In fact, this proposition is the main lemma of [A] reformulated in cohomological terms. In his paper, V. A. Abrashkin uses another choice of Frobenius, given by ϕ(X) = X p , but all of his arguments work in our case, and we repeat them below with some modiﬁcations.

226

DENIS BENOIS

2.1.6.1. It follows from Proposition 1.3.2 that h1 (ιn (F (πn ))) coincides with the class of the cocycle ψF (g) ⊗ ε given by k(g)−1 aF,γn (πn ) ⊗ ε , ψF (g) ⊗ ε = g(u ⊗ ε) − (u ⊗ ε) + 1 + γn + · · · + γn k(g)

where (1 − ϕ)u = f (πn )τ and γn = g|K∞ . Since γn (πn ) ≡ πn (mod π), one has k(g)−1 aF,γn (πn ) ⊗ ε ≡ p −n (1 − χ(g))D log F (πn ) ⊗ ε (mod π). 1 + γn + · · · + γn This congruence implies that ψF (g) ≡ χ (g)ug − u + p −n (1 − χ(g))D log F (πn ) (mod π). We now interpret ψF (g) in terms of Acris . Denote by I the ideal of Acris generated by π 2 and π p−1 /p. Lemma 2.1.6.2. There exists a unique x ∈ Fil1 Acris such that x ≡ u(π − π 2 /2) (mod I ) and ϕ x = f (πn ). 1− p p−1 Proof. Let v = i=0 [ε]i/p . Then u1 = uπ1 is a solution of the equation π (v − ϕ)u1 = f (πn ) 1 + . 2 The reduction of (v − ϕ)X modulo p is vX ¯ − Xp . Since R is integrally closed in Fr(R), it follows, by successive approximation modulo p m , that u1 ∈ W (R). Put x˜ = u(π − π 2 /2) = u1 v(1 − π/2). Then x˜ ∈ W 1 (R) and one has ϕ(x) ˜ π2 (ϕ(π) − π) x˜ − = f (πn ) 1 − + ϕ(u1 ) . ϕ(v) 4 2 Consider this equation in Acris . An easy computation shows that ϕ(x) ˜ ϕ(v) ϕ(π) ϕ(x) ˜ − = ϕ(u1 ) 1 − 1− . ϕ(v) p 2 p Summing these two equations we obtain that ϕ x˜ = f (πn ) + α, 1− p where α = −f (πn )π 2 /4 + ϕ(u1 )β and β = (ϕ(π) − π)/2 + (1 − ϕ(π)/2)(1 − ϕ(v)/p). Using the identity = π −1 ((1 + π)p − 1) it is easy to check that β ∈ I . ϕ(v) ∞ Then α ∈ I and the series m=1 (ϕ/p)m (α) converges to some element of I because ϕ/p is topologically nilpotent on I . Thus, there exists a unique x ∈ Fil1 Acris such that x ≡ x˜ (mod I ) and ϕ x = f (πn ). 1− p The lemma is proved.

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

227

2.1.6.3. From the deﬁnition of the ring Acris it is easy to see that the maps log : 1 + Fil1 Acris → Fil1 Acris and exp : Fil1 Acris → 1 + Fil1 Acris , given by usual power series log(1 + X) = X − X 2 /2 + · · · and exp(1 + X) = 1 + X + X 2 /2! + · · · , respectively, are isomorphisms inverse to each other. Since θ(F (πn )g /F (πn )) = 1, one has F (πn )g /F (πn ) ∈ 1 + Fil1 Acris . Hence the element F (πn )g g µF (g) = x − x − log F (πn ) belongs to Fil1 Acris for any g ∈ GKn . Since g2 g F (πn )g1 g2 g2 F (πn )g2 1 + x − x − log µF (g1 g2 ) = x − x − log F (πn ) F (πn ) = µF (g1 )g2 + µF (g2 ), the map µF : GKn → Fil1 Acris is a cocycle. We show that µF (g) = ψF (g)t. Indeed (1 − ϕ/p)µF (g) = 0, and hence µF (g) has a form µF (g) = c(g)t with c(g) ∈ Qp . On the other hand, from the congruences F (πn )g ≡ F (πn )+p −n (χ(g)− 1)D log F (πn )π (mod π 2 ) and x g − x ≡ (χ(g)ug − u)π (mod I ), it follows that µF (g) ≡ χ (g)ug − u π + p −n (1 − χ(g))D log F (πn )π ≡ ψF (g)t (mod I ). Hence µF (g) = ψF (g)t. In particular, one has [ε]ψF (g) = exp(µF (g)) =

exp(x)g F (πn ) . exp(x) F (πn )g

2.1.6.4. Let y = exp(x). Then the equation (1 − ϕ/p)x = f (πn ) can be written in the form yp = exp(pf (πn )). yϕ Consider the short exact sequence ν

1 −→ [ε]Zp −→ 1 + W 1 (R) −→ 1 + pW (R) −→ 1, where ν(a) = a p /a ϕ (see [dSh]). It shows that the inclusion W (R) ⊂ Acris gives a one-to-one correspondence between solutions Y of Y p /Y ϕ = exp(pf (πn )) and solutions X = log Y of (1 − ϕ/p)X = f (πn ). Hence, in fact, y ∈ 1 + W 1 (R). Furthermore, by induction on m, it is easy to see that yp ϕ m−1 2 ϕ m−2 m = exp pf (π ) + p f (π ) + · · · + p f (π ) n n n . m yϕ m

On the other hand, from the deﬁnition of D it follows that m m m−1 m−2 F (πn )p = F (πn )ϕ exp pf ϕ (πn ) + p 2 f ϕ (πn ) + · · · + pm f (πn ) .

228

DENIS BENOIS m

m

m

m

Hence y p /y ϕ = F (πn )p /F (πn )ϕ . Let z = ϕ −m (yF (πn )−1 ). Applying the map θ : W (R) → OC to both sides of this equation, we obtain that −1 m θ(z)p = F ζpn − 1 . Hence the connecting map δn,m sends F (ζpn − 1) to the class of the cocycle g → θ (z/zg ). On the other hand, one has yF (πn )g z −ψ (g) = θ ◦ ϕ −m [ε]−ψF (g) = ζpm F , θ g = θ ◦ ϕ −m g z y F (πn ) and the proposition is proved. Remark. The function D and its inverse E were ﬁrst deﬁned for the Frobenius ϕ(X) = X p in connection with explicit reciprocity laws (see [AH], [Sh], [Br], [V], and [Hen]). 2.2. The isomorphism h2 : H 2 (C . (ᏻKn (1))) H 2 (GKn , Zp (1)). In this section we describe the canonical isomorphism H 2 (GKn , Zp (1)) Zp using -modules and the map TRn , deﬁned in Proposition 1.3.4. We start with a purely formal proposition that allows us to rewrite the integral of Coleman [Cn2] in terms of residues. Proposition 2.2.1. For any f (X) ∈ ᏾ one has dπn = p −n f (ζ − 1). res π −1 f (πn ) 1 + πn ζ ∈µpn

2.2.2. Proof of Proposition 2.2.1

Lemma 2.2.2.1. Let ᏾ψ = {f (X) ∈ ᏾ | ζ ∈µp f (ζ (1 + X) − 1) = 0}. Let m 1 and 0 i < n. Then for any f (X) ∈ ᏾ψ one has dπn −m = 0. res πi f (πn ) 1 + πn Proof. Recall that ᏾ψ is topologically generated by (1 + X)a , (a, p) = 1 and that D is invertible on ᏾ψ . From the identity −(m+1) D πi−m D −1 f (πn ) = πi−m f (πn ) − mp n−i πi (1 + πi )D −1 f (πn ), it follows that res

πi−m f (πn )

dπn 1 + πn

=p

n−i

res

−(m+1) πi f1 (πn )

dπn , 1 + πn

where f1 (πn ) = m(1 + πi )D −1 f (πn ). If i < n, then f1 (X) ∈ ᏾ψ . Hence we can continue this process and obtain the congruence dπn dπn −(m+k) −m k(n−i) res πi fk (πn ) =p ≡ 0 mod p k(n−i) res πi f (πn ) 1 + πn 1 + πn for any k ∈ N. The residue is therefore equal to zero and the lemma is proved.

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

229

2.2.2.2. Now we can prove the proposition. Since the ring W [[X]] is topologically i generated by the series (1 + X)m , m ∈ Z, we may assume that f (X) = (1 + X)p a with (a, p) = 1. First suppose i < n. Taking into account that ϕ ◦ res = res ◦ϕ we obtain ϕ i −1 −1 pi a dπn −i a dπn = p res πi (1 + πn ) =0 res π (1 + πn ) 1 + πn 1 + πn by the lemma just proved. On the other hand, in this case, i f (ζ − 1) = ζ p a = 0. ζ ∈µpn

ζ ∈µpn

Suppose now that i n. Then −1 pi a dπn −1 dπn = res π =1 res π (1 + πn ) 1 + πn 1 + πn and ζ ∈µpn f (ζ − 1) = p n . The proposition is proved. 2.2.3. Denote by invn the canonical isomorphism H 2 (GKn , Zp (1)) Zp , and consider the cup product H 1 GKn , Zp (1) × H 1 GKn , Zp −→ H 2 GKn , Zp (1) . Let κn : Γn → Zp be the additive character given by κn (γ ) = p−n log χ(γ ). Then for any α ∈ U1 (Kn ) one has 1 invn δn (α) ∪ κn = − n Tr Kn /Qp log α, p where Tr Kn /Qp denotes the trace map Kn → Qp (see, e.g., [Se, Chap. 14, Prop. 3]). Proposition 2.2.4. Let f (X) ∈ ᏾.Then the composite map invn ◦h2 : H 2 Cn. −→ Zp sends cl(π −1 f (πn ) ⊗ ε) to



− log−1 χ(γn ) Tr K/Qp 

 f (ζ − 1) .

ζ ∈µpn

2.2.5. Proof of Proposition 2.2.4 Lemma 2.2.5.1. Let F (X) ∈ Ꮽ. Put f (X) = D(F (X)) and α = F (ζpn − 1). Then   f (ζ − 1) . Tr Kn /Qp log α = Tr K/Qp  ζ ∈µpn

230

DENIS BENOIS

Proof. Note that log α is well deﬁned because α ∈ U1 (Kn ). By multiplicativity of both sides of the formula it is sufﬁcient to consider the two following cases. (a) F (X) = a ∈ 1 + pW (k). Then f (ζ − 1) = (1 − ϕ/p) log a and one has   1 Tr K/Qp  Tr K/Qp log α = Tr Kn /Qp log α. f (ζ − 1) = p n 1 − p ζ ∈µpn

(b) F (X) ≡ 1 (mod X). Denote by Pm the set of p m -primitive roots of unity. Taking m into account that f ϕ (X)|X=ζpn −1 = 0 for m > n, we obtain Tr Kn /Qp log α = Tr Kn /Qp = Tr Kn /Qp

m=0 n

 = Tr K/Qp   = Tr K/Qp   = Tr K/Qp 

∞ m ϕ

f (X)|X=ζpn −1

pm p

−m σ m

f

m=0 n

p −m

m=0

ϕm

ζ ∈Pn

n

ζ

pm



−1 



f (ζ − 1)

m=0 ζ ∈Pn−m

f

pm ζpn − 1



f (ζ − 1) .

ζ ∈µpn

The lemma is proved. 2.2.5.2.

We pass to the proof of the proposition. At ﬁrst, consider the case f (X) ∈

᏾E0 =0 . Then there exists F (X) ∈ Ꮽ such that f (X) = D(F (X)). Put α = F (ζpn −1).

By Proposition 2.1.5, h1 sends the class of the pair −(f (πn )τ ⊗ ε, aF,γn (πn ) ⊗ ε) to δn (α). Similarly, by Proposition 1.3.2, κn corresponds to the class cl(0, κn (γn )). From Proposition 1.3.3 it follows that the cup product of these classes is equal to cl(π −1 f (πn )κn (γn ) ⊗ ε). On the other hand, by Lemma 2.2.5.1, one has   1 f (ζ − 1) . invn δn (α) ∪ κn = − n Tr K/Qp  p ζ ∈µpn

Hence invn ◦h2 sends cl(π −1 f (πn )κn ⊗ ε) to −p−n Tr K/Qp ( f (X) ∈ ᏾E0 =0 .

ζ ∈µpn

f (ζ − 1)), and

The general case can be reduced to this the proposition is proved for case in the following way. For f (X) = am X m ∈ ᏾, put f1 (X) = f (X)−a1 (1+X).

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

231

Then f1 (X) ∈ ᏾E0 =0 and one has dπn dπn −1 −1 − resπn π f1 (πn ) = a1 resπn π −1 dπn = 0. resπn π f (πn ) 1 + πn 1 + πn It now follows from Proposition 1.3.4 that cl(π −1 f (πn ) ⊗ ε) = cl(π −1 f1 (πn ) ⊗ ε). On the other hand, f1 (ζ − 1) = f (ζ − 1) − a1 ζ= f (ζ − 1). ζ ∈µpn

ζ ∈µpn

ζ ∈µpn

ζ ∈µpn

This completes the proof of the proposition. Now we can state the main result of this section. Theorem 2.2.6. The isomorphism invn ◦h2 : H 2 Cn. −→ Zp coincides with the map TRn ; that is, invn ◦h2 cl(α ⊗ ε) = −

αdπn pn Tr K/Qp res . log χ(γn ) 1 + πn

Proof. Since K/Qp is unramiﬁed, there exists a ∈ W such that Tr K/Qp a = 1. Put z = a/π ⊗ ε. From Propositions 2.2.1 and 2.2.4 it follows that invn ◦h2 (cl(z)) = TRn (cl(z)) = −

pn . log χ(γn )

Since pn / log χ (γn ) is a p-adic unit, the class cl(z) generates H 2 (Cn. ). We have shown that invn ◦h2 coincides with TRn on cl(z). Hence, they coincide everywhere. 2.3. The explicit reciprocity law for Qp (1). In [Ka1], Kato showed that explicit reciprocity laws can be viewed as an explicit computation of cup products in syntomic cohomology theory (see also [Ku]). In this section we use the complex Cn. to give a new proof of the explicit reciprocity law of Coleman [Cn2]. Suppose that n 1, and consider the cup product inv × id ∪ 2 n H 1 GKn , µpn × H 1 GKn , µpn −→ H 2 GKn , µ⊗ pn −−−−−→ µpn , where, to simplify notation, we write invn for invn (mod p n ). Let δn,n : Kn∗ → H 1 (GKn , µpn ) be the connecting map associated to the Kummer exact sequence. The Hilbert symbol ( , )n : Kn∗ × Kn∗ −→ µpn is a bilinear pairing given by (α, β)n = (invn × id)(δn,n (α), δn,n (β)).

232

DENIS BENOIS

Proposition 2.3.1. Let α, β ∈ U (Kn ), and let F (X), G(X) ∈ Ꮽ be such that F (ζpn − 1) = α, G(ζpn − 1) = β. Then n (α, β)n = ζp[F,G] , n

where [F, G]n = Tr K/Qp res

1 1 D(G(πn )) d log F (πn ) − D(F (πn )) d log G(πn )ϕ . π p

Proof. Let f (X) = D(F (X)) and g(X) = D(G(X)). Consider the commutative diagram H 1 Cn. /p n Cn. × H 1 Cn. /p n Cn.

∪

h1 ×h1

H GKn , µpn × H 1 GKn , µpn 1

∪

/ H 2 C . /p n C . n n h2

/ H GK , µ p n ⊗ µ p n n

2

invn × id

µp n . By Proposition 2.1.5 we have δn,n (α) = −h1 cl f (πn )τ ⊗ ε, aF,γn (πn ) ⊗ ε and

δn,n (β) = −h1 cl g(πn )τ ⊗ ε, aG,γn (πn ) ⊗ ε .

Then from Proposition 1.3.3 it follows that δn,n (α) ∪ δn,n (β) = h2 (cl(Hα,β ⊗ ε2 )), where Hα,β = aF,γn (πn )χ(γn )γn (g(πn )τ ) − f (πn )ϕ aG,γn (πn ) τ. Using the congruence 2.1.3(ii) and the congruence γn (g(πn )τ ) ≡ χ −1 (γn )g(πn )τ (mod Sn ), we can write Hα,β ≡

ϕ 1 − χ (γn ) D log F (πn )D(G(πn )) − D log G(πn ) D(F (πn )) n p π

(mod Sn ).

Note that (D log G(πn ))ϕ = (1/p)D(log G(πn )ϕ ). Applying Theorem 2.2.6 and taking into account that p−n log χ(γn ) ≡ p −n (χ(γn ) − 1) (mod p n ), we obtain that invn ◦h2 (cl(Hα,β ⊗ ε)) is congruent modulo pn to 1 1 ϕ D(G(πn )) d log F (πn ) − D(F (πn ))d log G(πn ) . Tr K/Qp res π p The proposition is proved.

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

233

Remark. In this proposition we considered only the case α, β ∈ U (Kn ). The general case is not much more difﬁcult but demands additional computations and is omitted here. Corollary 2.3.2. Using Proposition 2.2.1 we can write the explicit reciprocity law in the form of Coleman: [F, G]n =

1 Tr D log F D(G) − (D log G)ϕ D(F ) X=ζ −1 . K/Q p n p ζ ∈µpn

Corollary 2.3.3. Suppose that α and β are universal norms of K∞ /Kn . Then one can take F and G such that f = D(F ), g = D(G) ∈ ᏾ψ , and   1 [F, G]n = n Tr K/Qp  Df (ζ − 1)g(ζ − 1) . p ζ ∈µpn

Proof. It follows from the interpolation theorem of Coleman [Cn1] that one can choose F such that F ζ (1 + X) − 1 = F ϕ (X). ζ ∈µp

An easy computation shows that D(F ) ∈ ᏾ψ . Analogously we can choose G such that D(G) ∈ ᏾ψ . Furthermore, for any u(X) ∈ ᏾ψ and v(X) ∈ ᏾ one has ζ ∈µpn v ϕ (ζ − 1)u(ζ − 1) = 0. Using this formula we obtain that

(D log F )ϕ (ζ − 1)g(ζ − 1) = 0

ζ ∈µpn

and

(D log G)ϕ (ζ − 1)f (ζ − 1) = 0.

ζ ∈µpn

Hence ζ ∈µpn

D log F D(G) − (D log G)ϕ D(F ) X=ζ −1 = Df (ζ − 1)g(ζ − 1), ζ ∈µpn

and the corollary is proved. §3. The functions Ek,n 3.1. Construction of Ek,n . In this section we deﬁne a family of functions Ek,n , k ∈ Zp , which play an important role in our construction of cohomology classes.

234

DENIS BENOIS

Lemma 3.1.1. Let m 1, and let t −m = (i) for all i −m one has

∞

i=−m am,i π

i.

Then

i +m vp (am,i ) − ; p−1

+ (ii) pm−1 t −m ∈ π −m S + π Bcris .

Proof. (i) Since t = log(1 + π) = t

−1

=π

−1

∞

j =1 (−1)

j −1 π j /j ,

one has

2 π π2 π π2 1+ − +··· + − +··· +··· . 2 3 2 3

The series t −m consists, therefore, of terms of the form ±

π j1 +···+js −m , (j1 + 1) × · · · × (js + 1)

jk , s 1.

Put i = j1 + · · · + js − m. If we write jk in the form jk = p lk uk − 1,

lk 0, (uk , p) = 1,

then, taking into account the estimate x , p−1

logp (1 + x)

x p − 1,

we obtain that lk [jk /(p − 1)]. Hence

i +m vp (j1 + 1) × · · · × (js + 1) = l1 + · · · + ls , p−1

and the ﬁrst assertion is proved. (ii) From (i) it immediately follows that if i 0, then m vp p m−1 am,i m − 1 − 0. p−1 Hence to prove (ii) it is sufﬁcient to show that the sequence am,i+1 π i converges to + . We check that zero in Bcris am,i+1 π i ∈ p l W (R)PD , where (2)

l

i − Am , p2

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

and Am does not depend on i. One has π = vπ1 , where v = W 1 (R) and π1 = [ε]1/p − 1. Since p−1 p−1 vp θ π1 = v p ζp − 1 = 1, p−1

one can write π1

p−1

k=0 [ε]

k/p

235 generates

in the form p−1

π1

= pα + vβ,

α, β ∈ W (R).

Let i = (p − 1)k + r, where 0 r p − 1. Then πi = πr

k

p s cs v kp−s ,

cs ∈ W (R).

s=0

Since v kp−s /(kp − s)! ∈ W (R)PD , the inequality (2) follows from an easy estimate kp − s kp − s r +m+1 + + vp am,i+1 p s (kp − s)! s − k − p−1 p p2 k(p − 1) m+1 −1+ − p−1 p2 i − Am , p2 where Am = [(m + 1)/(p − 1)] + 2. The lemma is proved.

i 3.1.2. The ring An . Let K[[πn−1 , πn ]] the set of all power series ∞ i=−∞ ai πn , ai ∈ −1 K. The rings ᏻKn = W {{πn }} and K[[πn ]] can be viewed as subsets of K[[πn , πn ]]. For k, l 0 deﬁne p k t l−k . fk,l (π) = l! (i) The series fk,l (π ) has a form i l−k ak,l π i with ai ∈ Qp . If k > l, then Lemma 3.1.1 implies p−2 i (i) k− . vp ak,l p−1 p−1 From this estimate it follows that for any set of gk,l (πn ) ∈ Sn the series gk,l (πn )fk,l (π)

k,l 0

converges to some g(πn ) = bm πnm ∈ K[[πn−1 , πn ]]. In addition, vp (bm ) 1 for m 0 and limm→−∞ bm = 0. Denote by An the subset of K[[πn−1 , πn ]] consisting of all such series. Then An ⊂ p ᏻKn + πn K[[πn ]]. Since l 1 + l2 fk1 +k2 ,l1 +l2 (π), fk1 ,l1 (π )fk2 ,l2 (π) = l1

236

DENIS BENOIS

An has a natural ring structure. Sn and Sˆ PD can be viewed as subrings of An . The action of Γ on these rings extends to An . Let k ∈ Zp and n ∈ N. For f (X) ∈ ᏾ψ deﬁne Ek,n : ᏾ψ −→

1 An p

as follows: Ek,n (f ) =

∞ (1 − k)(2 − k) × · · · × (i − k − 1)

ti

i=1

p n(i−1) D −i (f (πn )).

Since D −i (f (πn )γ ) = χ (γ )−i (D −i f (πn ))γ , the map Ek,n is a W [[Γ ]]-homomorphism. The main properties of Ek,n are collected in the following proposition. Proposition 3.1.3. (i) For any k, l ∈ Zp one has Ek,n (f ) ≡ El,n (f ) mod πn , p n−1 . (ii) Let r ∈ Zp and j 2. Then j k=0

j (−1) Er+k,n (f ) ≡ 0 k k

mod πn , p j (n−1)+1 .

(iii) For any k ∈ Zp one has tEk,n (Df ) = f (πn ) + p n (1 − k)Ek−1,n (f ). (iv) Let γ ∈ Γn and k ! = 0. Then γ Ek,n (f ) ≡ χ (γ )−k Ek,n (f ) +

1 − χ(γ )−k f (πn ) (mod πn ). kp n

For k = 0 this congruence takes the form γ E0,n (f ) ≡ E0,n (f ) +

log χ(γ ) f (πn ) (mod πn ). pn

(v) Let n 1, and let γn be a generator of Γn . Then for any f ∈ ᏾ψ one has   p−1 j ϕ ◦ γn  Ek,n+1 (f ) ⊗ ε k ≡ Ek,n (f σ ) + A(γn )f σ (πn ) ⊗ ε k (mod πn ), j =0

where

1 χ(γn )pk − 1 −1 . A(γn ) = n kp p χ(γn )k − 1

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

237

For k = 0 this congruence takes the form   p−1 j ϕ ◦ γn  E0,n+1 (f ) ⊗ ε k ≡ E0,n (f σ ) ⊗ ε k j =0

+

p − 1 log χ(γn ) σ f (πn ) ⊗ ε k 2 pn

(mod πn ).

3.1.4. Proof of Proposition 3.1.3 3.1.4.1.

Let i 2. By Lemma 3.1.1 one has

pn(i−1) t −i = p (n−1)(i−1) p i−1 t −i ≡ 0

mod p (n−1)(i−1) , π ,

and (i) immediately follows from this congruence. 3.1.4.2. We repeat the arguments of Perrin-Riou [PR2, Prop. 2.4.4]. Suppose at ﬁrst that r ∈ Z. Differentiating i times the identity X r−1 (1 − X)j =

j j (−1)k X r+k−1 k k=0

and putting X = 1, one obtains j i j (−1)k (r + k − m) = 0 k k=0

for

0 i j − 1.

m=1

By continuity, this formula holds for all r ∈ Zp . One has j ∞ i in D −i−1 (f ) j j p Er+k,n (f ) = (−1)k (−1)k (m − r − k) k k t i+1 k=0 k=0 i=0 m=1   j −1 −i−1 j i D (f ) in  j ·p (−1)k (m − r − k) = k t i+1

j

i=0

k=0

m=1

  j ∞ i j D −i−1 (f ) in  + ·p (−1)k (m − r − k) . k t i+1 i=j

k=0

m=1

The ﬁrst term in this sum is equal to zero. From §3.1.4.1 it follows that p i t −(i+1) ∈ p ᏻKn + K[[πn ]] if i 2. Hence pin t −(i+1) ≡ 0 (mod (p j (n−1)+1 , πn )) for i j , and (ii) is proved.

238 3.1.4.3.

DENIS BENOIS

The third property follows from an easy computation

t · Ek,n (Df ) = t ·

∞ (1 − k)(2 − k) × · · · × (i − k − 1)

ti

i=1

= f (πn ) +

p (i−1)n D 1−i f (πn )

∞ (1 − k)(2 − k) × · · · × (i − k − 1)

t i−1

i=2 n

p (i−1)n D 1−i f (πn )

= f (πn ) + p (1 − k)Ek−1,n (f ). 3.1.4.4. Proof of (iv). To simplify notation put κn (γ ) = (χ(γ )−1)/p n . Let f (x) ∈ ᏾ψ . At ﬁrst we show that ∞

γ Ek,n (f ) =

(3)

ck,i D i f (πn )t i ,

i=−∞

where

ck,i =

p n(m−1)

j −m=i j 0, m1

(1 − k)(2 − k) × · · · × (m − k − 1)κn (γ )j . j !χ(γ )m

Since both sides of (3) are W -linear and the series (1 + X)a , (a, p) = 1 generate ᏾ψ topologically, it is sufﬁcient to check this formula for f (X) = (1 + X)a . Taking into j account that 1 + π = exp(t) = ∞ j =0 t /j !, one has, in this case, ∞ (1 − k) × · · · × (m − k − 1) n(m−1) γ Ek,n (f ) = (1 + πn ) p (1 + π)aκn (γ ) a m χ(γ )m t m a

= (1 + πn )a = (1 + πn )a

m=1 ∞

∞

(1 − k) × · · · × (m − k − 1) n(m−1) a j κn (γ )j j t p · a m χ(γ )m t m j!

i=1 ∞

j =0

ck,i a i t i =

i=−∞

∞

ck,i D i f (πn )t i ,

i=−∞

and (3) is checked. For i 1 one has ck,−i =

∞

p

n(i+j −1) κn (γ )

j =0

=p

n(i−1)

χ (γ )

−i

i−1 m=1

j χ(γ )−i−j i+j −1

j!

(m − k) ·

(m − k)

m=1

j 0

j k −i χ(γ )−1 − 1 j

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

= pn(i−1) χ (γ )−i

i−1

239

k−i (m − k) · 1 + (χ(γ )−1 − 1)

m=1

= p n(i−1) χ (γ )−k

i−1

(m − k).

m=1

Hence γ Ek,n (f ) ≡ χ (γ )−k Ek,n (f ) + ck,0 (mod πn ). Similarly it is not difﬁcult to compute ck,0 . For k ! = 0 we have 1 1−k n χ (γ )−1 κn (γ ) + p χ(γ )−2 κn (γ )2 + · · · 1! 2! −k i 1 1 − χ(γ )−k = n 1− . 1 − χ(γ )−1 = kp kp n i

ck,0 =

i 0

Now let k = 0. Then c0,0 = p

−n

1 − χ(γ )−1 i−1 = p −n log χ(γ ). i i 1

3.1.4.5. We now prove the congruence (v). Let Dπi = (1+πi )d/dπi . Recall that κn (γ ) = (χ (γ ) − 1)/p n . Put Fa (π1 ) =

p−1

j

(1 + π1 )aκn (γn ) .

j =0

We need the following auxiliary result. Lemma 3.1.4.6. Let fa = (1 + X)a , where (a, p) = 1. Then p−1 j =0

j γn Ek,n+1 (fa ) ⊗ ε k m ∞ (−1)m p nm m Fa (π1 ) −m−1 k = D (i − k)Dπ1 fa (πn+1 ) ⊗ ε . m! t m=0

i=1

Proof. Since p−1 j =0

j

κn (γn )j s (1 + π1 )aκn (γn ) = a −s Dπs 1 Fa (π1 ),

240

DENIS BENOIS

one has p−1

j χ (γn )j k γn

j =0

D −i fa (πn+1 ) ti

= fa (πn+1 )

p−1 j =0

= fa (πn−1 )

s

j k−i j 1 + p n κn γ n (1 + π1 )aκn (γn ) i i at

p ns

k − i Dπs 1 Fa (π1 ) . a s+i t i s

Let m = s + i − 1. Since

(1 − k)(2 − k) × · · · × (i − k − 1) k − i (i − 1)! s m! (1 − k)(2 − k) × · · · × (m − k) = (−1)s , m! s!(i − 1)!

the summing over i 1 gives p−1

j χ (γn )j k γn Ek,n+1 (fa )

j =0

= fa

∞ (−1)m p nm (1 − k) × · · · × (m − k) a m+1 m! m=0 m (−1)i−1 (i − 1)!p i−1 × Dπs 1 Fa (π1 ) ti s i+s=m+1

=

∞ m=0

(−1)m p nm (1 − k) × · · · × (m − k) m Fa (π1 ) Dπ1 D −m−1 fa . m! t

The lemma is proved. 3.1.4.7. We pass to the proof of (v). From the congruence 1/t ≡ 1/π(mod K[[π]]) it follows that (m − 1)! m−1 1 Dπ ≡ (−1)m−1 mod K[[π]] . π tm It is easy to see that Fa (π1 ) ≡

p−1 j =0

(1 + π1 )aj =

(1 + π)a − 1 (1 + π1 )a − 1

(mod π).

It follows that Fa (π1 )/t ≡ 1/π1 (mod K[[π1 ]]), and one has Fa (π1 ) (m − 1)! ϕ Dπm−1 ≡ (−1)m−1 mod K[[π]] . 1 m t t

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

241

Hence  (4)

ϕ ◦

p−1 j =0



j γn  Ek,n+1 (fa ) ⊗ ε k ≡ Ek,n faσ ⊗ ε k

mod K[[π]] .

Since fa generates ᏾ψ , it holds for all f ∈ ᏾ψ . Let k ! = 0. Applying the operator p−1 j ϕ◦ j =0 γn to both sides of tEk+1,n+1 (Df ) ⊗ ε k = f (πn+1 ) ⊗ ε k − kp n+1 Ek,n+1 (f ) ⊗ ε k and using (4) for Ek+1,n+1 , we can write that p−1

j k ptEk+1,n Df σ ⊗ ε k ≡ χ γn f σ (πn ) ⊗ ε k j =0



− kp n+1 ϕ ◦

p−1



j γn  Ek,n+1 (f ) ⊗ ε k

j =0

modulo πn . Taking into account that tEk+1,n (Df σ ) = f σ (πn ) − kp n Ek,n (f σ ), we obtain (v) for k ! = 0. But by continuity it holds also for k = 0 since p(p − 1) 1 χ (γn )pk − 1 − p = log χ(γn ). k→0 k χ (γn )k − 1 2 lim

The proposition is proved. Remarks 3.1.5. (i) From the proof of property (iv) it follows that ∞ ck,i D i f (πn )t i χ (γ )k γ − 1 En,k (f ) = i=0

with ck,i = p

−(i+1)n

i+1 ∞ (1 − k) × · · · × (j − k) j χ (γ ) − 1 1 − χ(γ )−1 . 1−k (i + j + 1)! χ (γ ) j =0

This formula is used in §3.2.3 below. (ii) Estimating the coefﬁcients of Ek,n more accurately, it is not difﬁcult to prove the following reﬁned version of congruence (i): mod p n , πn if p 5. Ek,n (f ) ≡ El,n (f )

242

DENIS BENOIS

3.2. The residue formula for Ek,n . The main result of this section is the following formula, which plays an important role in our proof of the explicit reciprocity law (see §5). Proposition 3.2.1. Let γ ∈ Γn and f, g ∈ ᏾ψ . Then for any k ∈ Z one has dπn −k res Ek+1,n (f ) · χ (γ ) γ − 1 E−k,n (g) 1 + πn −k−1 D f (πn )D k g(πn ) dπn k log χ(γ ) res = (−1) . pn π 1 + πn Proof. The rest of the section is devoted to the proof of this proposition. Lemma 3.2.2. Let f ∈ ᏾ψ , g ∈ ᏾, and k 0. If 0 m k, then dπn m res t Ek+1,n (f )g(πn ) 1 + πn −k−1 nm D f (πn )D k−m g(πn ) dπn k p k! res . = (−1) (k − m)! π 1 + πn If m > k, the residue is equal to zero. Proof. By deﬁnition one has 1 kp n k!p kn Ek+1,n (f ) = D −1 f (πn ) − 2 D −2 f (πn ) + · · · + (−1)k k+1 D −k−1 f (πn ). t t t Suppose that m > k. Then t m Ek+1,n (f ) ∈ K[[πn ]] and hence dπn = 0. res t m Ek+1,n (f )g(πn ) 1 + πn On the other hand, t k Ek+1,n (f ) ≡ (−1)k k!p kn D −k−1 f (πn )/π (mod K[[πn ]]). Hence dπn k res t Ek+1,n (f )g(πn ) 1 + πn −k−1 D f (πn )D k−m g(πn ) dπn k kn . = (−1) p k! res π 1 + πn Thus, the lemma is proved for all m k. The case m < k can be reduced to m = k in the following way. It is easy to see that D(t k+1 Ek+1,n (f )) = t k f (πn ). Using this formula we obtain D t m Ek+1,n (f )g(πn ) = D t m−k−1 t k+1 Ek+1,n (f ) g(πn ) = p n (m − k − 1)t m−1 Ek+1,n (f )g(πn ) + t m−1 f (πn )g(πn ) + t m Ek+1,n (f )Dg(πn ),

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

243

and hence

n

p (k + 1 − m) res t

m−1

dπn Ek+1,n (f ) · g(πn ) 1 + πn dπn m = res t Ek+1,n (f ) · Dg(πn ) . 1 + πn

Successively applying this formula to m = k, k − 1, . . . , 1 we obtain the lemma. 3.2.3. In this subsection we prove Proposition 3.2.1 for k 0. By Remarks 3.1.5 we can write ∞ χ (γ )−k γ − 1 En,−k (g) = c−k,m D m (g)t m , m=0

where

m+1 ∞ (k + 1)(k + 2) × · · · × (k + i) i χ (γ ) − 1 1 − χ(γ )−1 . c−k,m = (m+1)n k+1 (m + i + 1)! p χ (γ ) i=0

Then by Lemma 3.2.2 one has

res Ek+1,n (f ) · χ (γ )

−k

dπn γ − 1 E−k,n (g) 1 + πn −k−1 D f (πn )D k g(πn ) dπn k = (−1) Ak res π 1 + πn

with Ak =

k m=0

k! p mn c−k,m . (k − m)!

p −n log χ(γ ).

It remains to show that Ak = We prove it by induction on k. If k = 0, then A0 = c0,0 = p −n log χ (γ ) by §3.1.4.4. Suppose now that Ak = p −n log χ(γ ) for some k 0. Taking into account the identity k! p mn c−k,m (k − m)!

m+1 (k + 1)! (m+1)n −n (k − m + 1) × · · · × k χ(γ ) − 1 p = c−k−1,m+1 + p , (k − m)! (m + 1)! χ(γ )k+1

we can write Ak+1 = Ak + c−k−1,0 − p −n χ(γ )−k−1

k m+1 (k − m + 1) × · · · × k . χ(γ ) − 1 (m + 1)!

m=0

244

DENIS BENOIS

In §3.1.4.4 it was shown that c−k−1,0 =

1 − χ(γ )−k−1 . p n (k + 1)

On the other hand, k m+1 (k − m + 1) × · · · × k χ(γ ) − 1 (m + 1)!

m=0

=

k m+1 1 (k + 1) × · · · × (k + 1 − m) χ(γ ) − 1 k +1 (m + 1)! m=0

k+1 1 1 + χ(γ ) − 1 = −1 k +1 =

χ (γ )k+1 − 1 . k +1

Hence Ak+1 = Ak = p −n log χ(γ ), and the proposition is proved for k 0. Lemma 3.2.4. For any k ∈ Z one has

res E−k,n (g) · χ (γ )

k+1

dπn γ − 1 Ek+1,n (f ) 1 + πn dπn −k = res Ek+1,n (f ) · χ(γ ) γ − 1 E−k,n (g) . 1 + πn

Proof. The proof is straightforward. Using the formulas γ ((1 + πn )−1 dπn ) = χ (γ )(1 + πn )−1 dπn and χ (γ )k+1 γ Ek+1,n (f ) ≡ Ek+1,n (f ) (mod K[[πn ]]), we can write dπn res E−k,n (g) · χ (γ ) γ − 1 Ek+1,n (f ) 1 + πn dπn + res Ek+1,n (f ) · χ(γ )−k γ − 1 E−k,n (g) 1 + πn dπn −k k+1 = res χ (γ ) γ − 1 E−k,n (g) · χ(γ ) γ Ek+1,n (f ) 1 + πn dπn + res χ (γ )k+1 γ Ek+1,n (f ) · E−k,n (g) 1 + πn dπn − res E−k,n (g)Ek+1,n (f ) 1 + πn

k+1

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

245

dπn = res χ (γ )γ E−k,n (g)Ek+1,n (f ) 1 + πn dπn − res E−k,n (g)Ek+1,n (f ) 1 + πn dπn = 0, = res(γ − 1) E−k,n (g)Ek+1,n (f ) 1 + πn and the lemma is proved. 3.2.5. Now we can complete the proof of Proposition 3.2.1. It remains to show that the formula is true for k < 0. In this case −k 1 and one has dπn res Ek+1,n (f ) · χ(γ ) γ − 1 E−k,n (g) 1 + πn dπn = − res E−k,n (g) · χ(γ )k+1 γ − 1 Ek+1,n (f ) 1 + πn k −k−1 D g(πn )D log χ(γ ) f (πn ) dπn = (−1)k res . pn π 1 + πn

−k

Thus, the proposition is proved. Remark 3.2.6. Using Proposition 2.2.1 one can write Proposition 3.2.1 in the form

res Ek+1,n (f ) · χ (γ )

−k

dπn γ − 1 E−k,n (g) 1 + πn log χ(γ ) −k−1 = (−1)k D f (ζ − 1)D k g(ζ − 1). p 2n

ζ ∈µpn

§4. Construction of families of points. For a crystalline representation V , PerrinRiou [PR2] constructed some families of integral elements of H 1 (GKn , V (k)) for k $ 0 using the exponential map of Bloch and Kato. For representations of ﬁnite height we extend this construction to all k using complexes C . (D(T (k))). 4.1. Preliminaries 4.1.1. The exponential map of Bloch and Kato [BK]. Let L be a ﬁnite extension of Qp , and let L0 denote the maximal unramiﬁed subﬁeld of L. For a de Rham 0 (V ). Note (see §1.1.2) that one representation V of GL , put tV (L) = DdR (V )/DdR has the fundamental exact sequence + 0 −→ Qp −→ Bcris −→ Bcris ⊕ BdR /BdR −→ 0.

246

DENIS BENOIS

Tensoring this sequence with V , one obtains the long exact cohomology sequence 0 −→ H 0 GL , V −→ Dcris (V ) −→ Dcris (V ) ⊕ tV (L) −→ H 1 GL , V −→ · · · . The last map of this sequence gives rise to the exponential map expV ,L : tV (L) −→ H 1 GL , V with kernel (Dcris (V ))ϕ=1 /H 0 (GL , V ). In particular, let G be a formal group of ﬁnite height over the ring of integers OF , T the Tate module of G, and V = T ⊗ Qp . Then V is a de Rham representation, tV (L) is identiﬁed with the tangent space of G over L, and expV ,L coincides with the classical exponential map arising from the Kummer exact sequence (see [BK, Example 3.10.1]). 4.1.2. The modules Ᏸ(V ) and Ᏼ(V ) (see [PR1], [PR2]). Let s ∈ N. In the rest of this paper, T is a Zp -representation of ﬁnite height of GKs such that V = T ⊗Zp Qp is crystalline. By the theorem of Wach (see §1.2.3), there exists an S-lattice NS ⊂ DS (T ) such that Γs acts trivially on (NS /πNS )(−h) for some h ∈ Z. In all of this section we suppose that h = 0. This assumption is not restrictive because we can replace V by V (−h). In particular, it implies that Fil0 Dcris (V ) = Dcris (V ). The natural embedding ˆ S Sˆ PD )Γs with a W -lattice of Dcris (V ) stable under the Sˆ PD ∈ Bcris identiﬁes (DS (T )⊗ Frobenius ϕ (see §1.2.3). We denote this lattice by M. Note that the ring ᏾ = W [[X]] is equipped with (i) a W [[Γ ]]-module structure given by γ (X) = (1 + X)χ(γ ) − 1, (ii) a σ -semilinear Frobenius map ϕ such that ϕ(X) = (1 + X)p − 1, (iii) a differential operator D = (1 + X)d/dX. One has D ◦ ϕ = pϕ ◦ D. Following Perrin-Riou, deﬁne ᏾ψ = f (X) ∈ ᏾ f ζ (1 + X) − 1 = 0 , ζ ∈µp

Ᏸ(V (k)) = ᏾ψ ⊗W Dcris (V (k)),

k ∈ Z.

It may be shown that ᏾ψ is a free W [[Γ ]]-submodule of ᏾ generated by (1 + X) and that D is invertible on ᏾ψ . Then Ᏸ(V (k)) can be viewed as a W [[Γ ]] -module equipped with the differential operator D = D ⊗ id. Let ᏴK be the subset of K[[X]] consisting of all power series that converge on the maximal ideal mC of C. As above one can deﬁne the action of ϕ on ᏴK . Set

Ᏼ(V (k)) = α ∈ ᏴK ⊗K Dcris (V (k)) | (1 − ϕ)α ∈ Ᏸ(V (k)) . For k = 0 we also introduce the following integral versions of these modules: Ᏸ(T ) = ᏾ψ ⊗W M,

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

247

α ζ (1 + X) − 1 = pα ϕ (X) .

Ᏼ(T ) = α ∈ W [[X]] ⊗W M

ζ ∈µp

Let Ej : Ᏸ(V (k)) → Dcris (V (k))/(1 − p j ϕ)Dcris (V (k)) be the map deﬁned by mod 1 − p j ϕ Dcris (V (k)) , Ej (f ) = D j (f )(0) and let E[0,k] = ⊕kj =0 Ej . Then Ᏸ(V (k)) and Ᏼ(V (k)) are related to each other via the following exact sequence (see [PR2, §2.2.7]): −j 0 −→ log(1 + X)j ⊗ Dcris (V (k))ϕ=p −→ Ᏼ(V (k)) j 0 1−ϕ

−−−→ Ᏸ(V (k))E[0,k] =0 −→ 0. Because Fil0 Dcris (V ) = Dcris (V ), one has Dcris (V )ϕ=p for k = 0 this sequence takes the form

−j

= 0 for j > 0. In particular,

1−ϕ

0 −→ Dcris (V )ϕ=1 −→ Ᏼ(V ) −−−→ Ᏸ(V )E0 =0 −→ 0. Lemma 4.1.3. (i) One has Ᏸ(V ) = Ᏸ(T ) ⊗ Qp and Ᏼ(V ) = Ᏼ(T ) ⊗ Qp . (ii) For any k 0 the diagram D k ⊗ek

Ᏼ(V (k))

1−ϕ

1−ϕ

Ᏸ(V (k))E[0,k] =0

/ Ᏼ(V )

D k ⊗ek

/ Ᏸ(V )E0 =0

is commutative. Proof. (i) It is clear that Ᏸ(V ) = Ᏸ(T ) ⊗ Qp . Let α ∈ Ᏼ(T ) and f = (1 − ϕ)α. Then f ζ (1 + X) − 1 = α ζ (1 + X) − 1 − pϕ(α) = 0, ζ ∈µp

ζ ∈µp

and we obtain that f ∈ Ᏸ(T )E0 =0 . Hence Ᏼ(T ) ⊂ Ᏼ(V ). Suppose now that f ∈ Ᏸ(T )E0 =0 . Then we can write f = (1 − ϕ)f0 + f1 , where f0 ∈ M and f1 (0) = 0. ∞ m Since ϕ(M) ⊂ M, the series f + 0 m=0 ϕ (f1 ) converges to some α ∈ ᏾ ⊗ M and (1−ϕ)α = f . From ζ ∈µp f (ζ (1+X)−1) = 0 it follows that ζ ∈µp α(ζ (1+X)− 1) = pϕ(α). Hence α ∈ Ᏼ(T ), and one has an exact sequence 1−ϕ

0 −→ M ϕ=1 −→ Ᏼ(T ) −−−→ Ᏸ(T )E0 =0 −→ 0. Tensoring this sequence with Qp and taking into account that Ᏸ(T ) ⊗ Qp = Ᏸ(V ), we obtain that Ᏼ(V ) = Ᏼ(T ) ⊗ Qp .

248

DENIS BENOIS

(ii) Let α ∈ Ᏼ(V (k)). As ϕ(ek ) = pk ek and ϕ ◦ D k = p−k D k ◦ ϕ, one has the identity ϕ(D k ⊗ ek )α = (D k ⊗ ek )ϕ(α). Hence (1 − ϕ) D k ⊗ ek (α) = D k ⊗ ek (1 − ϕ)α, and the lemma is proved. 4.1.4. In this subsection we review the construction of elements of H 1 (GKn , V ) given by Perrin-Riou. Fix k ∈ N such that Fil−k Dcris (V ) = Dcris (V ). Denote by ᏴK the set of all power series f (X) ∈ K[[X]] that converge on mC . For f ∈ ᏴK put Rk,n (f ) =

k−1 i=0

Then one has

(−1)i

D i f (πn ) i t ∈ Fil0 Bcris . p in i!

f ζpn − 1 ≡ Rk,n (f ) mod Filk BdR .

Tensoring with Dcris (V ) one obtains a map Rk,n : ᏴK ⊗ Dcris (V ) −→ Dcris (V ) ⊗ Fil0 Bcris . Lemma 4.1.5 [PR2, Prop. 2.3.6]. Let α ∈ Ᏼ(V ). (i) There exists A ∈ Fil0 (Bcris ⊗ Dcris (V )) such that (1 − ϕ)A = Rk,n ((1 − ϕ)α). (ii) The element expV ,Kn (α(ζpn − 1)) coincides with the class of the cocycle g g −→ − Ag − A + Rk,n (α) − Rk,n (α). Sketch of proof. The ﬁrst statement follows from the fundamental exact sequence. To prove (ii) note that mod Dcris (V ) ⊗ Filk BdR . Rk,n (α(πn )) ≡ α ζpn − 1 Since Fil−k Dcris (V ) = Dcris (V ), it implies Rk,n (α(πn )) ≡ α ζpn − 1

+ mod V ⊗ BdR .

Hence

+ mod V ⊗ BdR .

Rk,n (α) − A ≡ α ζpn − 1

On the other hand, one has (1 − ϕ) Rk,n (α) − A = (1 − ϕ)Rk,n (α) − Rk,n ((1 − ϕ)α) = 0. Consider the fundamental exact sequence tensoring with V : f + −→ 0. 0 −→ V −→ V ⊗ Bcris −→ V ⊗ Bcris ⊕ V ⊗ BdR /BdR The last two formulas imply that f (Rk,n (α) − A) = (0, α(ζpn − 1)), and (ii) follows from the deﬁnition of the exponential map.

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

249

4.2. The homomorphisms T ,k,n . Using the functions Ek,n studied in the previous section, we construct here a system of maps T ,k,n : Ᏼ(T ) −→ H 1 GKn , T (k) ,

k ∈ Zp .

Let n s. Fix a generator γn of Γn . Recall that for any Γ -module N we denote by T wkε the map T wkε (n) = n ⊗ ε k . Recall that ˆ S Sˆ PD Γs . M = DS (T )⊗ Since Sˆ PD ⊂ An , then for any f ∈ ᏾ψ and g ∈ Sˆ PD the product gEk,n (f ) is contained in p −1 An . By linearity we obtain a homomorphism Ᏸ(T ) → p−1 DS (T ) ⊗S An . For k ∈ Zp denote by ET ,k,n the composite map T wkε

ET ,k,n : Ᏸ(T ) −→ p−1 DS (T ) ⊗S An −−−→ p −1 DS (T (k)) ⊗S An . By Proposition 3.1.3(iv) one has (γn − 1)ET ,k,n (f ) ≡

χ(γn )k − 1 f (πn ) ⊗ ε k kpn

mod πn DS (T (k)) .

Let α ∈ Ᏼ(T ). Then there exists (not unique) ᏱT ,k,n (α) ∈ DᏻKn (T (k)) such that ᏱT ,k,n (α) ≡ ET ,k,n ((1 − ϕ)α)

mod πn DSn (T (k)) .

(Since p−1 An ⊂ ᏻKn + πn K[[πn ]], it is sufﬁcient, e.g., to take ET ,k,n ((1 − ϕ)α) truncated modulo πn .) The last congruence implies (γn − 1)ᏱT ,k,n (α) ≡

1 − χ(γn )k (ϕ − 1)α(πn ) ⊗ ε k kp n

mod πn DS (T (k)) .

Since operator 1 − ϕ is invertible on πn DS (T (k)), there exists a unique ᏲT ,k,n (α) ∈ DS (T (k)) ⊗S Sn such that (i) ᏲT ,k,n (α) ≡

1 − χ(γn )k α(πn ) ⊗ ε k kp n

mod πn DSn (T (k)) ,

(ii) (ϕ − 1)ᏲT ,k,n (α) = (γn − 1)ᏱT ,k,n (α). Let ᏹn = DᏻKn (T ). Consider the complex C . (ᏹn (k)): 0 −→ ᏹn (k) −→ ᏹn (k) ⊕ ᏹn (k) −→ ᏹn (k) −→ 0. Then the pair (ᏱT ,k,n (α), ᏲT ,k,n (α)) deﬁnes an element of H 1 (C . (ᏹn (k))) which

250

DENIS BENOIS

does not depend on the choice of ᏱT ,k,n (α). Thus we constructed a map Ᏼ(T ) → H 1 (C . (ᏹn (k))). Composing this map with h1 : H 1 (C . (ᏹn (k))) H 1 (GKn , T (k)), one obtains a homomorphism T ,k,n : Ᏼ(T ) −→ H 1 GKn , T (k) , which sends α to h1 ◦ cl(ᏱT ,k,n (α), ᏲT ,k,n (α)). We also consider the homomorphism V ,k,n : Ᏼ(V ) −→ H 1 GKn , V (k) obtained from T ,k,n by ⊗Qp . Theorem 4.3. The homomorphisms T ,k,n satisfy the following properties. (m) (i) For any m ∈ N denote by T ,k,n : Ᏼ(T ) → H 1 (GKn , (T /p m T )(k)) the homom morphism T ,k,n modulo p . Then for any k, l ∈ Zp one has (n−1)

(n−1)

ε T ,k,n = T wk−l ◦ T ,l,n .

(ii) Let j 2 and m = j (n − 1) + 1. Then for any r ∈ Zp one has j (m) k j ε (−1) ◦ resKm /Kn T ,r+k,n = 0. T w−k k k=0

(iii) Let α ∈ Ᏼ(T ). Then

T (−1),k+1,n (D ⊗ e1 )α = −kp n T ,k,n (α).

(iv) For any α ∈ Ᏼ(T ) and n 1 one has

cor Kn+1 /Kn T ,k,n+1 (α) = T ,k,n (σ ⊗ ϕ)α .

(v) Let k 1 and α(X) ∈ Ᏼ(V (k)). Then V ,k,n D k ⊗ ek α = (−1)k (k − 1)!p (k−1)n expV (k),Kn α ζpn − 1 for any n s. 4.4. Proof of Theorem 4.3 4.4.1. Proof of (i). At ﬁrst we show that, for any f ∈ ᏾ψ and g ∈ Sˆ PD , mod πn , p n−1 . gEk,n (f ) ≡ gEl,n Since Sˆ PD is topologically generated by the elements t m /m!, we may assume that g = t m /m!. If m = 0, this congruence follows from Proposition 3.1.3. Let m 1. By the deﬁnition of Ek,n it is sufﬁcient to check that, for i 2, gpn(i−1) t −i D −i (f )

i−1 j =1

(j − k) ≡ 0

mod πn , p n−1 ,

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

251

but this is obvious since gp n(i−1) t −i = p (n−1)(i−2)+n

p m−2 p i−m . m! t i−m

Now by deﬁnition of ET ,k,n we obtain that ε ◦ ET ,l,n ((1 − ϕ)α) ET ,k,n ((1 − ϕ)α) ≡ T wk−l

mod p n−1 , πn .

Hence we can choose ᏱT ,k,n (α) and ᏱT ,l,n (α) such that ᏱT ,k,n (α) ≡ ᏱT ,l,n (α) (mod p n−1 ) and ᏲT ,k,n (α) ≡ ᏲT ,l,n (α) (mod p n−1 ). Thus (i) is proved. 4.4.2. Proof of (ii). We ﬁrst prove that (5)

j k=0

j ε (−1) ◦ ᏱT ,r+k,n (α) = 0 T w−k k k

mod p m ,

m = j (n − 1) + 1,

for a suitable choice of ᏱT ,r+k,n (α). For this it is sufﬁcient to show that g

j j (−1)k ET ,r+k,n (f ) = 0 k

mod p m , πn

k=0

for any f ∈ ᏾ψ and g ∈ Sˆ PD . As before we may assume that g = t i /i!. If i = 0, this congruence follows from Proposition 3.1.3(ii). Let i 1. In §3.1.4.2 it was shown that   j j ∞ l −l−1 (f ) D j j (−1)k · p ln  (−1)k (a − r − k) . Er+k,n (f ) = k k t l+1 l=j

k=0

k=0

a=1

On the other hand, from the identity g

i−1 p l−i t i pln p ln l(n−1)+1 p = = p i! t l+1 i! t l−i+1 t l+1

it follows that gp ln /t l+1 ≡ 0 (mod (p j (n−1)+1 , π)) for l j . Thus   j ∞ l j D −l−1 (f ) ln  ·p (−1)k (a − r − k) ≡ 0 mod p j (n−1)+1 , π , l+1 k t l=j

k=0

a=1

and (5) is proved. The element resKm /Kn T ,r+k,n (α) can be represented by the pair

ᏱT ,r+k,n (α), resKm /Kn ᏲT ,r+k,n (α) ,

252

DENIS BENOIS

where resKm /Kn ᏲT ,r+k,n (α) satisﬁes the following conditions: (i) resKm /Kn ᏲT ,r+k,n (α) ≡

χ r+k (γm ) − 1 α ⊗ ε r+k p n (r + k)

mod DS (T (k)) ⊗ πn K[[πn ]] ,

(ii) (ϕ − 1) resKm /Kn ᏲT ,r+k,n (α) = (γm − 1)ᏱT ,r+k,n (α). From the congruence r+k j j χ (γm ) − 1 ≡0 (−1)k k p n (r + k)

mod p m ,

k=0

we immediately obtain that j k=0

j ε T w−k (−1) ◦ resKm /Kn ᏲT ,r+k,n (α) = 0 k k

mod p m ,

and (ii) is proved. 4.4.3. Proof of (iii). One has M ⊗ e1 = tM. Using the formula tEk+1,n (Df ) = f (πn ) − kp n Ek,n (f ) (see Proposition 3.1.3(iii)), we can write ET (−1),k+1,n (D ⊗ e1 )f = −kp n ET ,k,n (f ) + f (πn ) for any f ∈ Ᏸ(T ). Hence for α ∈ Ᏼ(T ) we can choose ᏱT (−1),k+1,n such that ᏱT (−1),k+1,n (α) ≡ −kp n ᏱT ,k,n (α) + (1 − ϕ)α(πn ) ⊗ ε k

mod DS (T (k)) ⊗ πn K[[πn ]] .

On the other hand, ᏲT (−1),k+1,n (α) ≡ 0 (mod πn DS (T (k))). Take A ∈ πn DS (T (k)) such that A ≡ α(πn ) ⊗ ε k (mod πn ), and put Ᏹ%T (−1),k,n (α) = ᏱT (−1),k,n (α) + (ϕ − 1)A, ᏲT% (−1),k,n (α) = ᏲT (−1),k,n (α) + (γn − 1)A.

Then the pairs (ᏱT (−1),k,n (α), ᏲT (−1),k,n (α)) and (Ᏹ%T (−1),k,n (α), ᏲT% (−1),k,n (α)) are homologous. Moreover, Ᏹ%T (−1),k,n (α) ≡ −kp n ET ,k,n (α) (mod πn ) and (γn − 1)ᏲT% (−1),k,n (α) ≡ χ(γn )k − 1 α ≡ −kp n ᏲT ,k,n (α)

(mod πn ).

Hence cl(Ᏹ%T (−1),k,n (α), ᏲT% (−1),k,n (α)) = −kp n cl(ᏱT ,k,n (α), ᏲT ,k,n (α)), and (iii) is proved.

253

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

4.4.4. Proof of (iv). From an explicit description of the corestriction map it follows that cor Kn+1 /Kn T ,k,n+1 (α) coincides with the class of the cocycle    p−1 j  γn  ᏱT ,k,n+1 (α), ᏲT ,k,n+1 (α) j =0

and, hence, with the class of     p−1 j ϕ ◦  γn  ᏱT ,k,n+1 (α), ϕ ◦ ᏲT ,k,n+1 (α) j =0

which we denote by (Ᏹ%T ,k,n (α), ᏲT% ,k,n (α)). Take A ∈ DSn (T (k)) such that 1 χ(γn )k − 1 α σ (πn ) ⊗ ε k (mod πn ). A ≡ n 1− kp p χ(γn )k − 1 Then by the congruence (v) of Proposition 3.1.3 we have Ᏹ%T ,k,n (α) ≡ ET ,k,n (σ ⊗ ϕ)α + (ϕ − 1)A (mod πn ). Moreover, from the congruence

ᏲT ,k,n (σ ⊗ ϕ)α ≡

it follows that

1 − χ(γn )k σ α (πn ) ⊗ ε k kp n

mod DS (T (k)) ⊗ p −1 A1n

ᏲT% ,k,n (α) ≡ ᏲT ,k,n (σ ⊗ ϕ)α + (γn − 1)A

mod DS (T (k)) ⊗ πn .

As in the proof of (iii) above, these congruences show that the classes of (Ᏹ%T ,k,n (α), ᏲT% ,k,n (α)) and (ᏱT ,k,n ((σ ⊗ ϕ)α), ᏲT ,k,n ((σ ⊗ ϕ)α)) coincide. 4.4.5. Proof of (v). To prove this property we need some auxiliary results. + GK∞ Lemma 4.4.5.1. (i) Let α ∈ (V ⊗ Bcris )GK∞ and β ∈ (V ⊗ Bcris ) . Assume that 0 (γn − 1)α = (ϕ − 1)β. Then there exists u ∈ Fil (V ⊗ Bcris ) such that (1 − ϕ)u = α, and the map

µα,β : GKn −→ V ,

k(g)−1

µα,β (g) = ug − u + 1 + γn + · · · + γn

β,

k(g)

γn

= g|K∞ ,

is a continuous 1-cocycle. (ii) Let (α1 , β1 ) be an another pair satisfying the same condition and such that + + mod V ⊗ πn Bcris mod V ⊗ πn Bcris , β1 ≡ β . α1 ≡ α Then µα1 ,β1 and µα,β are homologous.

254

DENIS BENOIS

Proof. From the fundamental exact sequence it follows that for any a ∈ Bcris there exists b ∈ Fil0 Bcris such that (1 − ϕ)b = a. Hence there exists u ∈ Fil0 (V ⊗ Bcris ) such that (1 − ϕ)u = α. One has k(g)−1 (1 − ϕ)µα,β (g) = (g − 1)(1 − ϕ)u + 1 + γn + · · · + γn (1 − ϕ)β = (g − 1)α + (1 − g)α = 0. Since µα,β (g) ∈ Fil0 (V ⊗ Bcris ), it implies that µα,β (g) ∈ V . An easy computation shows that µα,β is a cocycle. If u% ∈ Fil0 (V ⊗ Bcris ) is another element such that (1−ϕ)u% = α, then u% = u+v for some v ∈ V , and the cocycle µ% α,β (g) = u% g −u% + k(g)−1

(1 + γn + · · · + γn )β is homologous to µα,β (g). + + since for any a ∈ πn Bcris the We note further that 1 − ϕ is invertible on πn Bcris + i % series i 0 ϕ (a) converges. From the congruence α ≡ α (mod (V ⊗ πn Bcris )) it + )) such that (1 − ϕ)u1 = α1 follows now that there exists u1 ≡ u (mod (V ⊗ πn Bcris + and hence µα1 ,β1 (g) ≡ µα,β (g) (mod (V ⊗ πn Bcris )). Since µα,β (g), µα1 ,β1 (g) ∈ V , it implies that µα,β (g) = µα1 ,β1 (g), and the lemma is proved. Lemma 4.4.5.2. Let α ∈ ᏴK ⊗ Dcris (V (k)). Then for any γ ∈ Γn one has (γ − 1)Rk,n (α) ≡ (−1)k−1

χ(γ )k − 1 k+1 α(πn ) mod Dcris (V (k)) ⊗ Bcris . kn k!p

Proof. To simplify notation put κn (γ ) = (χ(γ ) − 1)/p n . By continuity we may assume that α = g(X) ⊗ d with g(X) = K[X] and d ∈ Dcris (V (k)). Since g(X) can be written in the form a ba (1 + X)a , we may suppose in advance that α(X) = (1 + X)a ⊗ d. Then Rk,n (α) = α(πn )

k−1

(−1)i

i=0

ai t i . i!p ni

Hence Rk,n (α)γ = α(πn )(1 + π)aκn (γ )

k−1

(−1)i

i=0

Writing (1 + π )aκn (γ ) in the form i α(πn ) ∞ i=0 cki t with cki = p

−in i

a

∞

j =0 a

j +m=i 0mk−1

jκ

n (γ )

(−1)

j t j /j !,

m χ(γ )

m

χ(γ )i a i t i . i!pni

one obtains that Rk,n (α)γ =

j χ(γ ) − 1 . m!j !

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

The last formula implies that  i   i p −in a  (−1)  i! cki =  a k χ(γ )k − 1   k−1 −kn (−1) p k!

255

if 0 i k − 1, if i = k.

Hence (γ − 1)Rk,n (α) ≡ (−1)k−1

χ(γ )k − 1 k+1 α(π ) mod D (V (k)) ⊗ B , n cris cris k!p kn

and the lemma is proved. nr ∩ W (R). Then the following sequence is exact: Lemma 4.4.5.3. Let S˜n = Oˆ K n 1−ϕ

0 −→ Zp −→ π1−k S˜n −−−→ π −k S˜n −→ 0. Proof. By §1.1.1 one has ker(1 − ϕ) = Zp . To prove the surjectivity of 1 − ϕ we use the following method of Fontaine (see [F3]). Let y ∈ π −k y % with y % ∈ S˜n , and p−1 let v = i=0 [ε]i/p . One has π = vπ1 . By successive approximation modulo p m we show that there exists x % ∈ S˜n such that (6)

v k x % − ϕ(x % ) = y % .

% ∈ S˜ such that Suppose that one has xm n % % v k xm − ϕ xm ≡ y% mod p m S˜n ,

m 0.

% % + p m z and a = p −m (y % − v k x % + ϕ(x % )). Then a ∈ S˜ , and the = xm Put xm+1 m m n m m congruence % % v k xm+1 − ϕ xm+1 ≡ y% mod p m+1 S˜n , m 0,

is equivalent to

v k z − ϕ(z) ≡ am

mod p S˜n .

The left side of this congruence is equal to v¯ k X −X p modulo p. Since R is integrally closed in Fr R, there exists z¯ ∈ R such that v¯ k z¯ − z¯ p = a¯ m . Since v¯ k X − X p = y¯ % is a ˆ nr ˆ nr ˆ nr ˜ ˜ separable polynomial, we ﬁnd that z¯ ∈ R ∩ ᏻˆ nr Kn /p ᏻKn . But R ∩ ᏻKn /p ᏻKn = Sn /p Sn % % by [F2, Prop. 1.8.3], so taking a lifting z ∈ S˜n we ﬁnd xm+1 . Then x = limm→∞ xm is a solution of (6). Put x = π1−k x % . Then (1 − ϕ)x = π −k (v k − ϕ(x % )) = y, and the lemma is proved.

256

DENIS BENOIS

4.4.5.4. We deduce (v) from the previous lemmas. By linearity we may assume that (D k ⊗ ek )α ∈ Ᏼ(T ). It follows from Lemma 4.1.4 that expV (k),Kn (α(ζpn − 1)) coincides with the class of the cocycle k(g)−1 g → − Ag − A − 1 + γn + · · · + γn B, where (1 − ϕ)A = Rk,n ((1 − ϕ)α), A ∈ Fil0 (V (k) ⊗ Bcris ), and B = (1 − γn )Rk,n (α). On the other hand, V ,k,n ((D k ⊗ ek )α) can be represented by the cocycle k(g)−1 ᏲT ,k,n D k ⊗ ek α g → ug − u + 1 + γn + · · · + γn with (1 − ϕ)u = ᏱT ,k,n ((D k ⊗ ek )α). Tensoring the exact sequence from Lemma 4.4.5.3 with T (k) one obtains that u ∈ π1−k S˜n ⊗T (k) and hence u ∈ Fil0 (V (k)⊗Bcris ). It follows immediately from the deﬁnition of ᏱT ,k,n and Rk,n that

ᏱT ,k,n D k ⊗ ek α ≡ (−1)k−1 (k − 1)!p n(k−1) Rk,n ((1 − ϕ)α)

+ ⊗ V (k) . mod πn Bcris

By Lemma 4.4.5.2 we also have the congruence ᏲT ,k,n (α) ≡ (−1)k−1 (k − 1)!p n(k−1) B

+ mod πn Bcris ⊗ V (k) .

Property (v) now follows from Lemma 4.4.5.1, and the theorem is proved. Remark 4.5. The maps T ,k,n depend on the choice of ε. To stress it we often write Tε ,k,n . Using the fact that Ek,n is a Γ -homomorphism, it is easy to verify that τ Tε ,k,n (α) = Tτ ε,k,n (α) = χ(τ )k Tε ,k,n (τ (α)) for any τ ∈ Γ . In particular, let c be the element of Γ deﬁned by c(ε) = ε −1 . Then −1

Tε ,k,n (α) = (−1)k Tε ,k,n (c(α)). These formulas are used in Theorem 5.3.2 below. §5. Explicit reciprocity law 5.1. Cohomological pairing 5.1.1. In this section the main result of the paper is proved. We conserve the notation and conventions of §4. Let T ∗ = HomZp (T , Zp ), and let ᏹ∗ = HomᏻKn (ᏹ, ᏻKn ). Since the category of representations of ﬁnite height is stable under the duals, there exist an integer h 0 and an S-lattice NS∗ ∈ ᏹ∗ such that Γs acts trivially on (NS∗ /πNS∗ )(−h). For any

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

257

k ∈ Zp − [1 − h, 0] deﬁne V ∗ ,k,n : Ᏼ(V ∗ ) −→ H 1 GKn , V (k) , V ∗ ,k,n (β) =

(−1)h p −hn V ∗ (−h),h+k,n D h ⊗ eh β . k · (k + 1) × · · · × (k + h − 1)

Since H 2 (GKn , Qp (1)) Qp , the cup product gives a pairing ∪ ( , )k,n : H 1 GKn , V (k) × H 1 GKn , V ∗ (1 − k) −→ Qp . On the other hand, the natural duality Dcris (V ) × Dcris (V ∗ ) → Qp can be extended by linearity to the map [ , ] : Ᏸ(V ) × Ᏸ(V ∗ ) −→ Qp ⊗ ᏾. Theorem 5.1.2. Let α ∈ Ᏼ(V ), β ∈ Ᏼ(V ∗ ), and let f = (1 − ϕ)α, g = (1 − ϕ)β. Then for any integer k ∈ / [1, h] one has dπn 1 −k V ,k,n (α), V ∗ ,1−k,n (β) k,n = (−1)k Tr K/Qp res D f, D k−1 g . π 1 + πn 5.1.3. Proof of Theorem 5.1.2. It is more convenient to change some notation. Let α ∈ Ᏼ(T ), β ∈ Ᏼ(T (−h)), and let f = (1 − ϕ)α, g = (1 − ϕ)β. Put l = h − k + 1. Then the theorem is equivalent to the following formula T ,k,n (α), T ∗ (−h),l,n (β) k,n h dπn 1 −k −l k nh D f, D ⊗ e−h g = (−1) p (k − m) Tr K/Qp res . π 1 + πn m=1

⊗ε−1

The natural pairing DS (T (k)) × DS (T ∗ (1 − k)) −−−→ S induces a pairing [ , ]D : An ⊗S DS (T (k)) × An ⊗S DS T ∗ (1 − k) −→ An . If m ∈ M and m∗ ∈ M ∗ , then [m ⊗ ε k , m∗ ⊗ eh ⊗ ε l ]D = [m, m∗ ]. By Proposition 1.3.3 and Theorem 2.2.6 one has T ,k,n (α), T ∗ (−h),l,n (β) V ,k,n = − where

pn Tr K/Qp R(α, β), log χ(γn )

R(α, β) = res ᏲT ,k,n (α), γn ᏱT ∗ (−h),l,n (β) D

dπn − ᏱT ,k,n (α), ϕ ᏲT ∗ (−h),l,n (β) D . 1 + πn

258

DENIS BENOIS

Since ᏱT ,k,n (α) ≡ ET ,k,n (f ) (mod A1n ⊗ DS (T (k))), there exists x such that ᏱT ,k,n (α) = ET ,k,n (f ) + (ϕ − 1)x.

Let FT ,k,n (α) = ᏲT ,k,n (α) + (γn − 1)x. Then (1 − ϕ)FT ,k,n (α) = (γn − 1)ET ,k,n (f ). Similarly we can ﬁnd y and FT (−h),l,n (β) = ᏲT ,k,n (β) + (γn − 1)y such that ᏱT (−h),l,n (β) = ET (−h),l,n (g) + (ϕ − 1)y,

(1 − ϕ)FT (−h),l,n (β) = (γn − 1)ET (−h),l,n (g). Substituting these formulas we obtain R(α, β) = res FT ,k,n (α), γn ET ∗ (−h),l,n (g) D dπn − ET ,k,n (f ), ϕFT ∗ (−h),l,n (β) D 1 + πn dπn − res(γn − 1) A, ET ∗ (−h),l,n (g) D 1 + πn dπn − res(γn − 1) ᏱT ,k,n (α), ϕB D 1 + πn dπn − res(ϕ − 1) A, FT ∗ (−h),l,n (β) D 1 + πn dπn − res(ϕ − 1) ᏲT ,k,n (α), γn B D . 1 + πn One has γn ET ∗ (−h),l,n (g) ≡ ET ∗ (−h),l,n (g) (mod A0n ⊗ DS (T (1 − k))). Taking into account the identity dπn dπn Tr K/Qp res(γn − 1) g(πn ) = Tr K/Qp res(ϕ − 1) g(πn ) =0 1 + πn 1 + πn (see, e.g., [H2, Lemma 3.3]), we obtain Tr K/Qp R(α, β) = Tr K/Qp res FT ,k,n (α), ET ∗ (−h),l,n (g) D − ET ,k,n (f ), ϕFT ∗ (−h),l,n (β) D

dπn . 1 + πn

We further note that [ET ,k,n (f ), ϕFT ∗ (−h),l,n (β)]D has a form m u(πn )v(πn )ϕ /π m , where u(X), v(X) ∈ K ⊗ ᏾ψ . Since u(X)v(X)ϕ ∈ K ⊗ ᏾ψ by Lemma 2.2.5.1, one has dπn = 0. Tr K/Qp res ET ,k,n (f ), ϕFT ∗ (−h),l,n (β) D 1 + πn

259

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

Analogously we can replace [FT ,k,n (α), ET ∗ (−h),l,n (g)]D by the power series [(1 − ϕ)FT ,k,n (α), ET ∗ (−h),l,n (g)]D . Since (1 − ϕ)FT ,k,n (α) = (γn − 1)ET ,k,n (f ), we obtain dπn Tr K/Qp R(α, β) = Tr K/Qp res (γn − 1)ET ,k,n (f ), ET ∗ (−h),l,n (g) D . 1 + πn The theorem follows now from Proposition 3.2.1. Namely, let f = fi ⊗ mi and " g= gj ⊗ m∗j ⊗ eh , where fi , gi ∈ ᏾ψ . Since t h El,n (gj ) ≡ phn hm=1 (k − m) · E1−k,n (D −h gj ) (mod πn ), one has Tr K/Qp R(α, β) dπn = Tr K/Qp mi , m∗j res χ(γn )k γn − 1 Ek,n (fi ) · t h El,n (gj ) 1 + πn i,j

= pnh

h

(k − m) Tr K/Qp

mi , m∗j res χ(γn )k γn − 1 Ek,n (fi )

m=1

i,j

· E1−k,n D

−h

dπn gj 1 + πn

= (−1)k−1 p n(h−1) log χ(γn ) h dπn 1 −k −l × (k − m) Tr K/Qp res D f, D ⊗ e−h g . π 1 + πn m=1

Hence T ,k,n (α), T ∗ (−h),l,n (β) k,n h

= (−1)k p nh

m=1

dπn 1 −k −l D f, D ⊗ e−h g (k − m) Tr K/Qp res . π 1 + πn

The theorem is proved. Corollary 5.1.4. Our result can also be written in the form of Coleman:

V ,k,n (α), V ∗ ,1−k,n (β)

= k,n

(−1)k Tr D −k f, D k−1 g (ζ − 1). K/Q p n p ζ ∈µpn

Corollary 5.1.5. Let k 0. Let α ∈ Ᏼ(V ) and β ∈ Ᏼ(V ∗ (k + 1)). Then V ,−k,n (α), expV ∗ (k+1),Kn β ζpn − 1 −k,n 1 =− Tr Kn /Qp D k α, β ⊗ ek+1 ζpn − 1 . n(k+1) k!p

260

DENIS BENOIS

Proof. Note at ﬁrst that from Theorem 4.3 it follows that (−1)k+1 V ∗ ,k+1,n D k+1 ⊗ ek+1 β . expV ∗ (k+1),Kn β ζpn − 1 = k!p kn Hence

V ,−k,n (α), expV ∗ (k+1),Kn β ζpn − 1 −k,n =−

D k f, g ⊗ ek+1 (ζ − 1),

1

k!p

Tr K/Qp n(k+1)

ζ ∈µpn

where f = (1 − ϕ)α, g = (1 − ϕ)β. Thus, we must check that D k f, g ⊗ ek+1 (ζ − 1) = Tr Kn /Qp D k α, β ⊗ ek+1 ζpn − 1 . Tr K/Qp ζ ∈µpn

Since ζ ∈µp α(ζ (1 + X) − 1) = pα ϕ (X) and ζ ∈µp β(ζ (1 + X) − 1) = pβ ϕ (X), one has D k α, ϕ(β) ⊗ ek+1 (ζ − 1) = D k ϕ(α), β ⊗ ek+1 (ζ − 1)

ζ ∈µpn

ζ ∈µpn

= p k+1

D k α σ , β σ ⊗ ek+1 (ζ − 1)

ζ ∈µpn−1

ϕ D k α, β ⊗ ek+1 (ζ − 1). = p ζ ∈µpn

Hence

k

D f, g ⊗ ek+1 (ζ − 1) =

ζ ∈µpn

ϕ h (ζ − 1), 1− p

ζ ∈µpn

[D k α, β

where h(X) = ⊗ ek+1 ]. Then the arguments of Lemma 2.2.5.1 show that ζ ∈µpn (1−ϕ/p)h(ζ −1) is equal to Tr Kn /Qp h(ζpn −1), and the corollary is proved. 5.1.6. In the rest of this paper we always take into account the dependence of T ,k,n on ε. Let α ∈ M ϕ=1 . Then ᏱT ,k,n (α) = 0 and hence (1 − ϕ)ᏲT ,k,n (α) = 0. Since DᏻKn (T (k))ϕ=1 = T (k)GK∞ , it implies that ᏲT ,k,n (α) ∈ T (k)GK∞ . Then Tε ,k,n belongs to the image of the inﬂation map H 1 (Γn , T (k)GK∞ ) → H 1 (GKn , T (k)). Let k∈ / [0, h]. Then zero is not a weight of Hodge of V (k) and therefore V (k)GKn = 0. Hence H 1 (Γn , V (k)GK∞ ) = 0. Then for k ∈ / [0, h] the map Tε ,k,n induces the homomorphism PεV ,k,n : Ᏸ(V )E=0 −→ H 1 GKn , V (k) ,

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

261

deﬁned as follows: PεV ,k,n (f ) = Vε ,k,n (σ ⊗ ϕ)−n (α) ,

where (1 − ϕ)α = f.

Similarly, for k ∈ / [−h, 0] we can deﬁne PεV ∗ ,k,n : Ᏸ(V ∗ )E=0 −→ H 1 GKn , V ∗ (k) by PεV ∗ ,k,n (g) = Vε ∗ ,k,n ((σ ⊗ ϕ)−n (β)), where (1 − ϕ)β = g. Then for an integer k∈ / [0, h + 1], Theorem 5.1.2 can be written in the following form. Corollary 5.1.7. Let f ∈ Ᏸ(V )E=0 and g ∈ Ᏸ(V ∗ )E=0 . Then ε (−1)k D −k f, D k−1 g (ζ − 1). PV ,k,n (f ), PεV ∗ ,1−k,n (g) k,n = Tr K/Qp n p ζ ∈µpn

5.2. Interpolation of exponential maps 5.2.1. In this section we review the theory of Perrin-Riou [PR2]. Recall that Λ = Zp [[Γ ]]. Let ι be the Zp -linear map Λ → Λ deﬁned by ι(τ ) = τ −1 . For any s 1 ∞ m denote by Ᏼs the Qp -subspace of Qp [[X]] consisting of all power series m=0 am X ∞ −s such that limm→∞ |am |m = 0. Put Ᏼ∞ = s=1 Ᏼs . Let Ᏼ∞ (Γ1 ) = {f (γ1 − 1) | f (X) ∈ Ᏼ∞ }, where γ1 is a generator of Γ1 , and let Ᏼ∞ (Γ ) = Ᏼ∞ (Γ1 ) ⊗Zp [[Γ1 ]] Λ. The ﬁeld of fractions of Ᏼ∞ (Γ ) is denoted by ᏷∞ (Γ ). For any Λ-module N the natural homomorphism N → NΓn can be extended to a map Ᏼ∞ (Γ ) ⊗Λ N → Qp ⊗Zp NΓn . We now recall some results on the Iwasawa module associated to T (see [PR1, §2] and [PR2, §3.2.1]). Let Z1∞ (T (k)) = limH 1 GKn , T (k) . ← − cor Then Z1∞ (T (k)) is a Λ-module of rank [K : Qp ] dimQp V with torsion submodule isomorphic to T (k)GK∞ . For any n the natural map Z1∞ (T (k))Γn → H 1 (GKn , T (k)) G is injective, and (T (k))ΓnK∞ is isomorphic to H 1 (Γn , T (k)GK∞ ). So one has a monomorphism 1 H 1 GKn , T (k) Z∞ (T (k)) . πT (k),n : −→ 1 T (k)GK∞ Γn H Γn , T (k)GK∞ Tensoring it with Qp one obtains a map πV (k),n :

Z1∞ (T (k)) T (k)GK∞

H 1 GKn , V (k) , ⊗ Qp −→ 1 H Γn , V (k)GK∞ Γn

262

DENIS BENOIS

which can be extended to a homomorphism

H 1 GKn , V (k) Z1∞ (T (k)) . −→ 1 pr T (k),n : Ᏼ∞ (Γ ) ⊗Λ T (k)GK∞ H Γn , V (k)GK∞

Note that if zero is not a Hodge weight of V (k), then V (k)GKn = 0 and one has H 1 (Γn , V (k)GK∞ ) = 0. The restriction-inﬂation sequence 0 −→ H 1 Γn , V (k)GK∞ −→ H 1 GKn , V (k) −→ H 1 GK∞ , V (k) gives rise to a monomorphism resK∞ /Kn

H 1 GKn , V (k) −→ H 1 GK∞ , V (k) . : 1 G K ∞ H Γn , V (k)

Proposition 5.2.2. Let s, r ∈ N. Let xn,r+k ∈ im πV (r+k),n (n ∈ N, 0 k s) be a family that satisﬁes the following properties. (i) One has cor Kn+1 /Kn (xn+1,i ) = xn,i . (ii) For any 0 j s the sequence j (s−j )n k j ε (−1) ◦ resK∞ /Kn (xn,r+k ) T w−k p k k=0

converges to zero in the group H 1 (GK∞ , V (r)). Then there exists a unique element x ∈ Ᏼs (Γ ) ⊗Λ Z1∞ (T (k))/T (k)GK∞ such that pr T (k),n (x) = xn,k for all n and k. Proof. This is [PR3, Prop. 1.8]; see also [PR2, §1.2]. 5.2.3.

Consider the system of maps

PεV ,k,n : Ᏸ(V )E=0 −→ H 1 GKn , V (k) ,

k∈ / [0, h],

deﬁned in §5.1.6 by the formula

PεV ,k,n (f ) = Vε ,k,n (σ ⊗ ϕ)−n (α) ,

(1 − ϕ)α = f.

Theorem 4.3 implies that for k 1 these maps are related to exponential maps by the formula (7) PεV ,k,n (f ) = (−1)k (k − 1)! expV (k),Kn Ξk,n (f ) , where Ξk,n (f ) = p −n (σ ⊗ ϕ)−n Gk (ζpn − 1) and Gk is a solution of the equation (1 − ϕ)Gk = (D −k ⊗ e−k )(f ). In addition one has cor Kn+1 /Kn PεV ,k,n+1 (f ) = PεV ,k,n (f ). Hence we obtain a compatible system of homomorphisms 1 Z∞ (T (k)) E=0 Fn,k : Ᏸ(V )(k)Γn −→ ⊗ Qp , T (k)GK∞ Γn

263

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

ε Fn,k (f ) = PεV ,k,n T w−k (f ) . In [PR2], Perrin-Riou deﬁned the maps PV ,k,n for k > h directly by (7) and showed that they satisfy the interpolation property in Theorem 4.3(ii). Applying to this system Proposition 5.2.2,1 she proved the following fundamental result. Theorem 5.2.4 (Perrin-Riou). There exists a unique Λ-homomorphism (0),ε

PV

: Ᏸ(V )E=0 −→ Ᏼ∞ (Γ ) ⊗ Z1∞ (T )/T GK∞

such that for any k > h the diagram (0),ε

T wkε ◦PV

Ᏸ(V )E=0 Ξk,n

/ Ᏼ∞ (Γ ) ⊗ Z1 (T (k))/T (k)GK∞ ∞

(−1)k (k−1)! expV (k),n

Kn ⊗K Dcris (V (k))

pr T (k),n

/ H GK , V (k) n 1

is commutative. (0),ε

Theorem 4.3 allows us to give an explicit description of pr V (k),n ◦T wkε ◦ PV (f ) for negative k too. Namely, from the uniqueness of p-adic interpolation we immediately obtain the following result. Corollary 5.2.5. For all k ∈ / [0, h] one has (0),ε

PεV ,k,n = pr V (k),n ◦T wkε ◦ PV

(f ).

5.2.6. For γ ∈ Γ put Dm = m − log γ / log χ(γ ). Note that Dm does not depend on the choice of γ , T wkε Dm = Dm−k , and Dm ≡ m (mod (γn − 1)) for any n. The map (0),ε PV gives rise to a system of map (k),ε

PV (k+1),ε

such that PV

: Ᏸ(V )E=0 −→ ᏷(Γ ) ⊗Λ Z1∞ (T )

(k),ε

= D k PV

. From Theorem 4.3(iii) it follows that

(1),ε

PV

(0),ε

= −T w1ε ◦ PV (−1) (D ⊗ e1 ).

Iterating this equality one obtains (k+1),ε

PV (1) 1 The

(k),ε

= −T w1ε ◦ PV

(D ⊗ e1 ).

referee pointed out an error in [PR2, Prop. 3.1.4] used in the original proof of this theorem (see [Cz1, remark after Prop. II.1.8] and [PR3]).

264

DENIS BENOIS

5.3. The explicit reciprocity law of Perrin-Riou 5.3.1.

Let V ∗ = HomQp (V , Qp ). The pairing ∪ ( , )T ,n : H 1 GKn , T × H 1 GKn , T ∗ (1) −→ Zp

induces a bilinear form , T : Z1∞ (T ) × Z1∞ T ∗ (1) −→ Λ, which can be described explicitly in the following way. If x = (xn ) ∈ Z1∞ (T ) and y = (yn ) ∈ Z1∞ (T ∗ (1)), then the elements τ −1 xn , yn T ,n τ ∈ Zp Γ /Γn τ ∈Γ /Γn

form a compatible system with respect to the maps Zp [Γ /Γn ] → Zp [Γ /Γn−1 ] and deﬁne an element x, yT ∈ Zp [[Γ ]]. In particular, x, yT ≡ τ −1 xn , yn T ,n τ mod (γn − 1) . τ ∈Γ /Γn

It is easy to see that for any λ ∈ Λ one has

# $ λx, yT = λx, yT = x, ι(λ)y T

and

#

$ ε T wkε (x), T w−k (y) T = T w−k x, yT ,

where T w−k : Λ → Λ is the Zp -linear map given by T w−k (τ ) = χ(τ )−k τ for τ ∈ Γ . Hence, , T is a Λ-bilinear form ι , T : Z1∞ (T ) × Z1∞ T ∗ (1) −→ Λ. By linearity this pairing can be extended to ι , V : ᏷(Γ ) ⊗Λ Z1∞ (T ) × ᏷(Γ ) ⊗Λ Z1∞ T ∗ (1) −→ ᏷(Γ ). The formula (1 + X) ∗ (1 + X) = 1 + X deﬁnes a unique product structure on ᏾ψ compatible with the action of Λ. By linearity it can be extended to a bilinear map Ᏸ(V ) × Ᏸ V ∗ (1) −→ Qp ⊗ ᏾, which is denoted by ∗Ᏸ(V ) . Since ᏾ψ is a free Λ-module of rank 1, the involution ι acts naturally on ᏾ψ by the formula −1

ι(1 + X)a = (1 + X)a . We can now state the explicit reciprocity law of Perrin-Riou.

265

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS

Theorem 5.3.2. For any f ∈ Ᏸ(V )E=0 and g ∈ Ᏸ(V ∗ (1))E=0 one has & % (0),ε (1),ε−1 PV (f ), PV ∗ (1) (g ι ) (1 + X) = − Tr K/Qp f ∗Ᏸ(V ) g . V

Proof. Since %

(0),ε

PV

Dιm

= −D−m , one can write this formula in the form

& (0),ε−1 ε−1 (f ), T wh+1 ◦ PV ∗ (−h) D h+1 ⊗ eh+1 g ι (1 + X) V

=

h

D−m ◦ Tr K/Qp f ∗Ᏸ(V ) g .

m=1 pn

Put ωn (X) = (1 + X) − 1. From p-adic interpolation theory (see [PR2, no. 1.2]), it follows that it is sufﬁcient to prove the congruences % & (0),ε (0),ε−1 ε−1 ◦ PV ∗ (−h) D h+1 ⊗ eh+1 g ι (1 + X) D k PV (f ), T wh+1 V h (8) ≡ Dk D−m ◦ Tr K/Qp f ∗Ᏸ(V ) g mod ωn (X) m=1 −1

for all n and k $ 0. Taking into account that T wkε = (−1)k T wkε , D k (h(γ − 1)u(X)) = T wk (h(γ − 1))D k u(X), and T wkε D−m ≡ −(m + k) (mod (γn − 1)), we can write the last congruence in the form % & (0),ε (0),ε−1 ε ε−1 ◦ PV (f ), T wk+h+1 ◦ PV ∗ (−h) D h+1 ⊗ eh+1 g ι (1 + X) T w−k V

≡ (−1)k+h

h

(m + k) Tr K/Qp D k f ∗Ᏸ(V ) D k g

mod ωn (X) .

m=1

By deﬁnition one has % & (0),ε (0),ε−1 ε ε−1 T w−k ◦ PV (f ), T wk+h+1 ◦ PV ∗ (−h) D h+1 ⊗ eh+1 g ι V h+1 −1 ε ε−1 τ PV ,−k,n (f ), PV ∗ (−h),k+h+1 D ≡ ⊗ eh+1 g ι τ ∈Γ /Γn

mod (γn − 1) .

By the formulas in Remark 4.5 one has −1 τ −1 PεV ,−k,n (f ) = χ(τ )k PεV ,−k,n f τ and

−1 PεV ∗ ,k+1,n (D ⊗ e1 )g ι = (−1)k+1 PεV ∗ ,k+1,n c(D ⊗ e1 )g ι .

−k,n

τ

266

DENIS BENOIS

It is easy to see that cD = −Dc and D −1 ι = ιD. Then, by Corollary 5.1.7 one has −1 τ −1 PεV ,−k,n (f ), PεV ∗ (−h),k+h+1 D h+1 ⊗ eh+1 g ι −k,n

=

h (−1)k+h

pn

(k + m) Tr K/Qp

m=1

( ι −1 D k f ζ χ(τ ) − 1 , D k g ζ −1 − 1 ⊗ e1 .

' ζ ∈µpn

But in [PR2, Prop. 4.3.2], it is shown that ( 1 ' k χ(τ −1 ) k ι −1 D f ζ ζ − 1 , D g − 1 ⊗ e 1 pn ζ ∈µpn

is equal to the coefﬁcient of (1+X)χ(τ ) in the interpolation polynomial of D k f ∗Ᏸ(V ) D k g modulo ωn (X). So the congruence (8) is checked, and the theorem is proved.

References [A] [AV]

[AH] [BK]

[Br]

[CE] [C] [CCz1] [CCz2] [Cn1] [Cn2] [Cn3] [Cz1] [Cz2]

V. A. Abrashkin, The ﬁeld of norms functor and the Brückner-Vostokov formula, Math. Ann. 308 (1997), 5–19. Y. Amice and J. Vélu, “Distributions p-adiques associées aux séries de Hecke” in Journées Arithmétiques de Bordeaux (Bordeaux, 1974), Astérisque 24–25, Soc. Math. France, Montrouge, 1975, 119–131. E. Artin and H. Hasse, Die beiden Ergänzungssätze zum Reziprozitätsgesetz der l n -ten Einheitzwurzeln, Abh. Math. Sem. Univ. Hamburg 6 (1928), 146–162. S. Bloch and K. Kato, “L-functions and Tamagawa numbers of motives” in The Grothendieck Festschrift, Vol. 1, Progr. Math. 86, Birkhäuser, Boston, 1990, 333– 400. H. Brückner, “Eine explizite Formel zum Reziprozitätsgesetz für Primzahlexponenten p” in Algebraische Zahlentheorie (Oberwolfach, 1964), Bibliographisches Institut, Mannheim, 1967, 31–39. H. Cartan and S. Eilenberg, Homological Algebra, Princeton Univ. Press, Princeton, 1956. F. Cherbonnier, Représentations p-adiques surconvergentes, thesis, Orsay, 1996. F. Cherbonnier and P. Colmez, Représentations p-adiques surconvergentes, Invent. Math. 133 (1998), 581–661. , Théorie d’Iwasawa des représentations p-adiques d’un corps local, J. Amer. Math. Soc. 12 (1999), 241–268. R. Coleman, Division values in local ﬁelds, Invent. Math. 53 (1979), 91–116. , The dilogarithm and the norm residue symbol, Bull. Soc. Math. France 109 (1981), 373–402. , Local units modulo circular units, Proc. Amer. Math. Soc. 89 (1983), 1–7. P. Colmez, Théorie d’Iwasawa des représentations de de Rham d’un corps local, Ann. of Math. (2) 148 (1998), 485–571. , Représentations cristallines et représentations de hauteur ﬁnie, J. Reine Angew. Math. 514 (1999), 119–143.

ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS [F1]

[F2] [F3] [F4] [F5] [Hen] [H1] [H2] [Ka1] [Ka2]

[KKT] [Ku]

[PR1] [PR2] [PR3] [Se] [Sh] [dSh]

[T] [V] [W] [Win]

267

J.-M. Fontaine, Sur certains types de représentations p-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate, Ann. of Math. (2) 115 (1982), 529–577. , “Représentations p-adiques des corps locaux, I” in The Grothendieck Festschrift, Vol. 2, Progr. Math. 87, Birkhäuser, Boston, 1990, 249–309. , Le corps des périodes p-adiques, Astérisque 223 (1994), 59–111. , Représentations p-adiques semi-stables, Astérisque 223 (1994), 113–184. , Sur un théorème de Bloch et Kato (lettre à B. Perrin-Riou), Invent. Math. 115 (1994), 151–161. G. Henniart, Sur les lois de réciprocité explicites, I, J. Reine Angew. Math. 329 (1981), 177–203. L. Herr, Sur la cohomologie galoisienne des corps p-adiques, Bull. Soc. Math. France 126 (1998), 563–600. , Une nouvelle approche de la dualité locale de Tate, preprint, Univ. Bordeaux 1, 1999. K. Kato, The explicit reciprocity law and the cohomology of Fontaine-Messing, Bull. Soc. Math. France 119 (1991), 397–441. , “Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via BdR , I” in Arithmetic Algebraic Geometry (Trento, 1991), Notes in Math. 1553, Springer, Berlin, 1993, 50–163. K. Kato, M. Kurihara, and T. Tsuji, Local Iwasawa theory of Perrin-Riou and syntomic complexes, preprint, 1996. M. Kurihara, Computation of the syntomic regulator in the cyclotomic case, appendix to: M. Gros, Régulateurs syntomiques et valeurs de fonctions L p-adiques, I, Invent. Math. 99 (1990), 293–320. B. Perrin-Riou, Théorie d’Iwasawa et hauteurs p-adiques, Invent. Math. 109 (1992), 137– 185. , Théorie d’Iwasawa des représentations p-adiques sur un corps local, Invent. Math. 115 (1994), 81–149. , Théorie d’Iwasawa et loi explicite de réciprocité, Doc. Math. 4 (1999), 219–273, available from http://www.mathematik.uni-bielefeld.de/documenta/. J.-P. Serre, Corps locaux, 2d ed., Publ. Inst. Math. Univ. Nancago 8, Hermann, Paris, 1968. I. R. Shafarevic, A general reciprocity law, Amer. Math. Soc. Transl. 4 (1956), 73–105. E. de Shalit, “The explicit reciprocity law of Bloch-Kato” in Columbia University Number Theory Seminar (New York, 1992), Astérisque 228, Soc. Math. France, Montrouge, 1995, 197–221. J. Tate, Relations between K2 and Galois cohomology, Invent. Math. 36 (1976), 257–274. S. V. Vostokov, Explicit form of the law of reciprocity, Math. USSR-Izv. 13 (1979), 557– 588. N. Wach, Représentations p-adiques potentiellement cristallines, Bull. Soc. Math. France 124 (1996), 375–400. J.-P. Wintenberger, Le corps des normes de certaines extensions inﬁnies de corps locaux; applications, Ann. Sci. École Norm. Sup. (4) 16 (1983), 59–89.

Mathématiques et Informatique, Université Bordeaux I, 351 cours de la Libération, 33405, Talence Cedex, France; [email protected]

Vol. 104, No. 2

DUKE MATHEMATICAL JOURNAL

© 2000

REFLECTIONLESS SCHRÖDINGER OPERATORS, THE DYNAMICS OF ZEROS, AND THE SOLITONIC SATO FORMULA J. F. VAN DIEJEN and H. PUSCHMANN

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 2. Reﬂectionless Schrödinger operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 2.1. The equations of motion for the zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 2.2. Hamiltonian structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 2.3. Lax representation and integrals of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 279 3. The action-angle transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 3.1. Spectral properties of the Lax matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 3.2. Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 3.3. The integral curve and action-angle diffeomorphism . . . . . . . . . . . . . . . . . . 288 3.4. Symplectic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 4. The wave function and the tau function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 4.1. A Wilson-type determinantal formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 4.2. The solitonic Sato formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 4.3. The Bethe curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 5. Dynamics and scattering of zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 6. The Korteweg–de Vries hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Appendix A. Proof of the no-crossing Lemma 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 310 Appendix B. Some Cauchy matrix identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 The zeros of the (Jost) eigenfunction of a 1-dimensional Schrödinger operator with a reﬂectionless rapidly decreasing potential are related to the spectral data through a nonlinear algebraic system of Bethe-type equations. We show that the behavior of these zeros (with respect to translations) is governed by a rational RuijsenaarsSchneider particle system with harmonic term. The integration of the particle system— via an explicit construction of the action-angle transform—then provides us with detailed information on the solution curve of the Bethe equations. As a result, we ﬁnd Received 25 May 1999. Revision received 18 November 1999. 2000 Mathematics Subject Classiﬁcation. Primary 37J35; Secondary 35Q40, 35Q51, 37K10, 70H06, 81Q05. van Diejen supported in part by the Fondo Nacional de Desarrollo Cientíﬁco y Tecnológico (FONDECYT) through grant number 1980832 and at the Mathematical Sciences Research Institute (MSRI) by National Science Foundation grant number DMS-9701755. 269

270

VAN DIEJEN AND PUSCHMANN

a Wilson-type determinantal formula for the eigenfunction involving RuijsenaarsSchneider (Lax) matrices, and we furthermore recover the solitonic Sato formula (which parametrizes the eigenfunction explicitly in terms of the spectral data). The ﬂows corresponding to the higher integrals of the rational Ruijsenaars-Schneider system with harmonic term give rise to the soliton solutions of the Korteweg–de Vries (KdV) hierarchy. 1. Introduction. In this paper we consider 1-dimensional Schrödinger operators with eigenfunctions that are elementary in the sense that they factorize as a product of a plane wave and a (monic) polynomial in the spectral parameter. It is shown that the dependence of the zeros of such wave functions on the spatial variable is governed by a rational Ruijsenaars-Schneider system with harmonic term (see [RSc], [R2], [Sc], and [BR]). (Here the zeros are thought of as particles, and the spatial variable of the Schrödinger equation corresponds to the time variable of the Ruijsenaars-Schneider system.) The integration of this dynamical system then provides us with very explicit information regarding the structure of the eigenfunctions of our Schrödinger operator. The same class of (reﬂectionless) Schrödinger operators studied here has been investigated before within the framework of ﬁnite-gap theory (where they arise as a (soliton) limiting case) (see [DMN], [DMKM], and [BBEIM]). It should be stressed, however, that—even though in ﬁnite-gap theory the dynamics of zeros of Schrödinger wave functions plays a key role—no analysis of the corresponding zero motion in terms of Ruijsenaars-Schneider-type Hamiltonian particle mechanics has been considered so far. (See, however, [K] and [Wi] for a study of the dynamics of poles of the rational Baker-Akhiezer function for the Kadomtsev-Petviashvili (KP) hierarchy in terms of Calogero-Moser particle systems, and see [KWZ] for the pole motion of the elliptic Baker-Akhiezer function for the discretized Kadomtsev-Petviashvili equation in terms of discrete-time Ruijsenaars-Schneider models.) Before presenting a more precise description of our results, let us illustrate the main ideas by means of the simplest nontrivial example: that of an eigenfunction with a polynomial part of degree 1 in the spectral parameter. In this situation the RuijsenaarsSchneider system at issue has only one degree of freedom and its integration is completely elementary. Speciﬁcally, we are interested in those Schrödinger equations 2 ∂x + u(x) − z2 ψ(x, z) = 0, −∞ < x < ∞, (1.1) that admit a solution of the form ψ(x, z) = (z − ζ (x)) exp(zx).

(1.2)

(Here z denotes the spectral parameter.) Substitution of ψ(x, z) from equation (1.2) in equation (1.1), and collecting the powers of z, readily entails that the wave function solves the Schrödinger equation if and only if the function ζ (x), characterizing the location of the zero, satisﬁes the second-order differential equation ζ (x) + 2ζ (x)ζ (x) = 0,

(1.3a)

DYNAMICS OF ZEROS AND THE SATO FORMULA

271

and the potential u(x) is related to ζ (x) via the compatibility condition u(x) = 2ζ (x).

(1.3b)

The differential equation (1.3a) can be brought to Hamiltonian form (here we reduce to the submanifold ζ (x) > 0). To this end we employ the Hamiltonian H = exp(ρ) + ζ 2 ,

(1.4)

with a phase space given by = {(ζ, ρ) ∈ R × R} endowed with the standard symplectic form ω = dρ ∧ dζ . (For relevant background material on classical mechanics/symplectic geometry the reader is referred, e.g., to Arnold’s standard monograph [A].) The Hamilton equations ζ = ∂H /∂ρ, ρ = −∂H /∂ζ associated to H of equation (1.4) read ζ = exp(ρ),

ρ = −2ζ.

(1.5)

The system in equation (1.5) reduces to equation (1.3a) upon differentiation of the ﬁrst equation and elimination of the ρ-variable. The Hamilton equations are solved by transforming to the action-angle coordinates ˆζ = exp(ρ) + ζ 2 , (1.6a) ζˆ − ζ . (1.6b) ρˆ = log ζˆ + ζ It is not difﬁcult to check that the action-angle mapping (ζ, ρ) − → (ζˆ , ρ) ˆ between ˆ = {(ζˆ , ρ) our phase space and the action-angle phase space ˆ ∈ R × R | ζˆ > 0} (equipped with the symplectic form ωˆ = dρˆ ∧dζˆ ) is in fact a symplectomorphism (i.e., a symplectic diffeomorphism). In the Darboux (i.e., canonical) coordinates ζˆ , ρˆ the Hamiltonian becomes free Hˆ = ζˆ 2 and the corresponding linear Hamilton equations ζˆ = 0, ρˆ = −2ζˆ have as solution

ζˆ (x) = ζˆ0 ,

ρ(x) ˆ = ρˆ0 − 2x ζˆ0 ,

(1.7)

where ζˆ0 = exp(ρ0 ) + ζ02 and ρˆ0 = log(ζˆ0 − ζ0 ) − log(ζˆ0 + ζ0 ) with ζ0 ≡ ζ (0) and ρ0 ≡ ρ(0). We read off from equations (1.6b) and (1.7) that the zero function ζ (x) is recovered as the unique solution of the algebraic equation ζˆ0 − ζ (x) = exp ρˆ0 − 2x ζˆ0 . ζˆ0 + ζ (x) This immediately produces ζ (x) = ζˆ0

1 − exp ρˆ0 − 2x ζˆ0 . 1 + exp ρˆ0 − 2x ζˆ0

(1.8)

(1.9)

272

VAN DIEJEN AND PUSCHMANN

The upshot is that all Schrödinger operators on the line, with a (regular) positive potential and an eigenfunction of the factorized form given by equation (1.2), are characterized by the 2-parameter family of potentials (cf. equations (1.3b) and (1.9)) u(x) = 2 ∂x2 log τ (x) with τ (x) = 1 + exp ρˆ0 − 2x ζˆ0 (1.10a) and ρˆ0 ∈ R, ζˆ0 > 0. The corresponding wave function reads (cf. equations (1.2) and (1.9)) 1 + z + ζˆ0 / z − ζˆ0 exp ρˆ0 − 2x ζˆ0 (x, z) = exp(zx), (1.10b) 1 + exp ρˆ0 − 2x ζˆ0 where we have renormalized such that (x, z) → exp(zx) for x → +∞. It is immediate from the explicit formula in equation (1.9) that the zero of our wave function increases monotonically from −ζˆ0 to +ζˆ0 as x runs along the real line in the positive direction. Consequently, the wave function becomes proportional to exp(zx) for x → ±∞, and we are thus dealing with a so-called reﬂectionless or Bargmann potential (see, e.g., [DT], [AS], and [M4]). (For a general rapidly decreasing potential the Schrödinger wave function has at both sides (i.e., −∞ and +∞) an asymptotics that is composed of linear combinations of the plane waves exp(zx) and exp(−zx). By forming a suitable linear combination of two fundamental solutions to the Schrödinger equation, it is always possible (for generic z) to achieve that the asymptotics of the wave function is of the form ∼ exp(zx) for x → +∞. A potential is said to be reﬂectionless if one has that at the same time the asymptotics for x → −∞ is also of the form ∼ exp(zx).) By considering the motion of the zero ζ with respect to multiparameter (i.e., multitime) commuting ﬂows associated to the Hamiltonians H m /m, m = 1, 2, . . . (with H taken from equation (1.4)), we recover the 1-soliton solution of the Korteweg–de Vries hierarchy. (See, e.g., [SCM], [AS], [N], and [NMPZ] for relevant preliminaries regarding soliton theory and the KdV equation.) In particular, when the zero function ζ (x, t) is taken to be a solution to the 2-parameter compatible Hamiltonian system ∂H , ∂ρ 1 ∂H 2 ∂t ζ (x, t) = , 2 ∂ρ

∂x ζ (x, t) =

∂H , ∂ζ 1 ∂H 2 ∂t ρ(x, t) = − 2 ∂ζ

∂x ρ(x, t) = −

(1.11a) (1.11b)

(with times x and t), then the above solution method entails for the potential u(x, t) = 2 ∂x ζ (x, t) that u(x, t) = 2 ∂x2 log τ (x, t) with τ (x, t) = 1 + exp ρˆ0 − 2ζˆ0 x − 2ζˆ03 t (1.12) (cf. equation (1.10a)); that is, u(x, t) =

2 ζˆ02 . cosh ζˆ0 x + ζˆ03 t − ρˆ0 /2 2

(1.13)

DYNAMICS OF ZEROS AND THE SATO FORMULA

273

This is the celebrated 1-soliton solution of the KdV equation (see [SCM], [AS], [N], and [NMPZ]) 3 1 ut = uux + uxxx . 2 4

(1.14)

In other words, for a 1-soliton potential of the form in equation (1.13) the position of the zero ζ (x, t) of the corresponding Schrödinger wave function as a function of x and t is governed by the 2-time Hamiltonian system in equations (1.11a) and (1.11b) with H taken from equation (1.4). The aim of the present paper is to lift the above scheme to the case of a Schrödinger operator with an elementary wave function ψ(x, z) that is the product of a plane wave exp(zx) and a monic polynomial in the spectral parameter zN + sN−1 (x)zN−1 + · · · + s0 (x) of arbitrary degree N ∈ N. It is shown in Section 2 that the zeros of a Schrödinger eigenfunction of this type satisfy the equations of motion of a rational Ruijsenaars-Schneider system with harmonic term (see [RSc], [Sc], and [BR]). (Here the spatial variable of the Schrödinger operator plays the role of the time variable of the Ruijsenaars-Schneider system; cf. above.) The relevant equations of motion are then integrated in Section 3 via an explicit construction of the action-angle transform (cf. also [R2] for a construction of the action-angle transform for the RuijsenaarsSchneider systems without harmonic term). In Section 4 we utilize the explicit construction of the zeros as functions of the spatial variable to arrive at closed expressions for both the potential and the wave function of our Schrödinger operator (cf. equations (1.10a) and (1.10b)). This way we ﬁnd a determinantal formula for the reﬂectionless potentials and wave functions in terms of hyperbolic Ruijsenaars-Schneider Lax matrices. This ties in with Shiota’s and Wilson’s recently found determinantal representation in terms of Calogero-Moser Lax matrices (see [M2], [M3], and [P]) for, respectively, the potential and wave function of the linear problem associated to the Kadomtsev-Petviashvili equation in the rational regime (see [Sh] and [Wi]). That is to say, our determinantal formulas may be viewed as KdV soliton counterparts of Shiota’s formula for the rational KP tau function and Wilson’s formula for the rational KP Baker function. From the determinantal representations we furthermore recover Hirota’s formula for the tau function of the reﬂectionless Schrödinger potential (see [H], [SCM], and [N]) as well as the solitonic Sato formula for the corresponding wave function (see [S], [SS], [DKJM], [JM], [SW], [OSTT], [Mo], [BBS], and [DK]). The explicit relation between the zeros of the wave function on the one hand and the action-angle variables for the rational Ruijsenaars-Schneider system with harmonic term on the other hand determines a Bethe-type system of algebraic equations for these zeros that generalizes equation (1.8). (The fact that for N = 1 the Bethe equation (1.8) turns out to be linear in ζ is very deceptive: For N > 2 the Bethe system becomes highly nonlinear.) The action-angle variables of the Ruijsenaars-Schneider system correspond actually to the spectral data of the reﬂectionless Schrödinger operator. The

274

VAN DIEJEN AND PUSCHMANN

Bethe system may, in other words, be interpreted as an algebraic equation for the zeros of the wave function in terms of the spectral data. Such a Bethe system has already appeared before in the physics literature in connection with a quantization problem for the sine-Gordon solitons [BBS]. In our approach we ﬁnd that the projection onto the conﬁguration space of the curve swept out by the Hamiltonian ﬂow of our RuijsenaarsSchneider system describes the unique (with ﬁxed spectral data) solution curve of the Bethe system. (Here the initial conditions for the Ruijsenaars-Schneider ﬂow are in one-to-one correspondence with the spectral data via the action-angle transform.) From the action-angle transform it is possible to deduce very precise information pertaining to the qualitative behavior of the zeros of the wave function as one varies the position variable x along the real axis. A notable feature is, for example, that the zeros do not cross each other (i.e., they always remain simple). Section 5 is devoted to the study of such dynamical issues with an emphasis on the scattering properties of the zeros. In particular, it is seen that the zeros have a constant asymptotics for x → ±∞, which is a manifestation of the fact that, also for N > 1, we are dealing with reﬂectionless Schrödinger operators. The scattering of the zeros turns out to decompose into 1-particle and 2-particle processes (i.e., the relevant phase shifts are built of 1- and 2-particle phase shifts.) We wrap up in Section 6 with a brief explanation of how to encode the N-soliton solutions for the KdV hierarchy in terms of the ﬂows corresponding to the higher integrals of the rational Ruijsenaars-Schneider system with harmonic term. Speciﬁcally, we ﬁnd that—for a Schrödinger potential constituting an N-soliton solution of the KdV hierarchy—the zeros of the wave function move in accordance with the rational Ruijsenaars-Schneider hierarchy with harmonic term. The results in this section ﬁt within the framework of a long line of research on the characterization of special solutions to integrable nonlinear partial differential equations in terms of (integrable) particle systems, starting with the pioneering works of Kruskal, Thickstun, and, especially, Airault, McKean, and Moser (see [Kr], [Th], [AMM], [CC], [Ca], [K], [M3], [M4], [RSc], [P], [Sh], [Wi], and [KWZ]). Our approach should be viewed as an alternative route to tie in the KdV soliton solutions with integrable particle systems, which is different from the previous approaches due to Moser [M4] (using Neumann systems on spheres) and Ruijsenaars-Schneider [RSc] (using hyperbolic RuijsenaarsSchneider systems). 2. Reﬂectionless Schrödinger operators 2.1. The equations of motion for the zeros. We consider the eigenvalue problem associated to a Schrödinger operator on the line: 2 ∂x + u(x) ψ(x, z) = z2 ψ(x, z),

−∞ < x < ∞.

(2.1)

Here the variable z denotes the (complex) spectral parameter, and the function u(x) represents the (real-valued) potential of the problem. It seems quite a natural question

DYNAMICS OF ZEROS AND THE SATO FORMULA

275

to ask oneself for which potentials u(x) the Schrödinger equation (2.1) admits elementary solutions that factorize as a product of an exponential plane wave exp(zx) and a monic polynomial in the spectral parameter with coefﬁcients depending on x: ψ(x, z) = exp(zx)

N

sm (x) zN−m ,

with s0 (x) ≡ 1.

(2.2)

m=0

For u ≡ 0, solutions of this type exist, of course, trivially. Indeed, in this simple situation any (monic) polynomial in z with constant coefﬁcients sm gives rise to an eigenfunction. The simplest choice corresponds to the unit polynomial (i.e., the monic polynomial of degree N = 0), which produces the standard solution ψ(x, z) = exp(zx). This trivial example with u ≡ 0 illustrates very well the more general phenomenon that—starting from a solution to the Schrödinger equation (2.1) having the elementary structure in equation (2.2)—one may pass to other solutions of the same type via multiplication by an arbitrary (monic) polynomial in z with constant coefﬁcients. However, since the solutions thus obtained just differ by an x-independent normalization factor, they are in essence equivalent. Without restriction we may therefore assume from now on that the polynomial part of our wave function is minimal in the sense that overall factors with constant coefﬁcients are divided out. More speciﬁcally, we assume that the wave function is of the form ψ(x, z) = exp(zx)

N

z − ζj (x) ,

(2.3)

j =1

with the zeros ζ1 (x), . . . , ζN (x) being nonconstant functions of x. For technical reasons we furthermore restrict to the situation in which the zeros ζj (x) are real and analytic in the (real) variable x (i.e., for every point x0 on the real line the Taylor series of ζj (x) around x0 exists and converges in a sufﬁciently small neighborhood). The following proposition provides precise criteria to be satisﬁed by the zeros ζ1 (x), . . . , ζN (x) and the potential u(x) so as to guarantee that the wave function ψ(x, z) of equation (2.3) solves the Schrödinger equation (2.1). Proposition 2.1 (The motion of zeros). Let the zero functions ζj : R → R, j = 1, . . . , N , be analytic and nonconstant. Then the wave function ψ(x, z) of equation (2.3) solves the Schrödinger equation (2.1) if and only if the zeros ζ1 (x), . . . , ζN (x) are simple and satisfy the coupled nonlinear system of ordinary differential equations ζj + 2ζj ζj =

1≤k≤N, k=j

2ζj ζk ζj − ζ k

,

j = 1, . . . , N,

(2.4a)

and, furthermore, the potential u(x) is related to the zeros via the compatibility condition u(x) = 2 ζ1 (x) + · · · + ζN (x) . (2.4b)

276

VAN DIEJEN AND PUSCHMANN

(Here the primes refer to the differentiation with respect to the spatial variable x.) Proof. We have from equation (2.3) that

ζk , ∂x ψ(x, z) = ψ(x, z) z − z − ζk 1≤k≤N

ζ + 2zζ 2ζk ζl k k 2 2 + . ∂x ψ(x, z) = ψ(x, z) z − z − ζk (z − ζk )(z − ζl ) 1≤k≤N 1≤k
N ζ + 2zζ k

k=1

z − ζk

k

+

1≤k
2ζk ζl = 0. (z − ζk )(z − ζl )

(2.5)

We observe that the left-hand side of equation (2.5) determines a rational expression in the spectral parameter z. It is clear that, in order for this rational expression to vanish identically, it is necessary that the zero functions ζ1 (x), . . . , ζN (x) be distinct as analytic functions of x. Indeed, if two (or more) of the zero functions ζj (x) coincide, then the last term on the left-hand side of equation (2.5) gives rise to second-order pole(s) in z. From the corresponding numerator(s) we see that the cancellation of such pole(s) would imply that the zero function with multiplicity greater than 1 has a derivative identical to zero, which contradicts our nonconstancy assumption. We conclude that we are in the situation in which our zeros ζj (x) are generically simple, except possibly for special values of x in a certain discrete (by analyticity) subset of R. (Below we see that this discrete set is in fact empty; cf. Lemma 2.2.) A necessary and sufﬁcient condition for the left-hand side of equation (2.5) to be identically zero is now that (i) the residues at the poles z = ζj , j = 1, . . . , N , cancel and that (ii) the expression tends to zero for z → ∞. (Here one uses Liouville’s theorem.) Calculation of the residues at the poles z = ζj , j = 1, . . . , N , gives rise to the nonlinear system (2.4a) and the asymptotics for z → ∞ translates into the compatibility condition (2.4b). In the proof of Proposition 2.1 it was in fact only demonstrated that the zeros of wave function ψ(x, z) from equation (2.3) (solving the Schrödinger equation (2.1)) are simple for generic values of x on the real line. More precisely, at this point we have shown that the zero functions ζ1 (x), . . . , ζN (x) constitute a solution to the differential system (2.4a) with possible (multiple) collisions between the ζj occurring for certain special values of x in a discrete set ⊂ R. The following lemma shows that such collisions actually do not occur (i.e., is empty), whence the zeros are simple for all x ∈ R.

DYNAMICS OF ZEROS AND THE SATO FORMULA

277

Lemma 2.2 (No crossing). Let ζ1 (x), . . . , ζN (x) be real-analytic and nonconstant functions of x ∈ R satisfying the nonlinear system of differential equations given by equation (2.4a). Then we have that ζj (x) = ζk (x) for all x ∈ R when j = k. Proof. The proof of this lemma is a bit technical. It consists of a detailed local analysis of the differential equation (2.4a) around a point in the supposedly nonempty collision set . This produces a contradiction: The asymptotic behavior of the lefthand side of the differential equation turns out to be incompatible with that of the right-hand side around a point at which two (or more) of the zeros ζj (x) would cross each other. Thus it is concluded that the collision set must in fact be empty; that is, crossings do not occur. For the precise details we refer to Appendix A. For future reference we also need the following lemma. It says that our zero functions ζ1 (x), . . . , ζN (x) are strictly monotoneous in x. Lemma 2.3 (Monotonicity). Let ζ1 (x), . . . , ζN (x) be nonconstant, real-analytic functions of x ∈ R satisfying the nonlinear system of differential equations given by equation (2.4a). Then ζj (x) = 0 for all x ∈ R and j = 1, . . . , N . Proof. Suppose that there exists a solution function ζi (x), i ∈ {1, . . . , N }, for which ζi (x) vanishes for a certain value of x, say, x = 0. Then the ith differential equation of the system (2.4a) implies that also ζi (0) = 0. (Here it is used that for k = i one has that ζk (0) = ζi (0) in view of Lemma 2.2.) Further differentiation of the differential equation now entails that actually all derivatives of ζi (x) vanish at x = 0. But then—by analyticity—the function ζi (x) has to be constant, which is in contradiction with our nonconstancy assumption. 2.2. Hamiltonian structure. It is helpful to think of the differential equations of equation (2.4a) as the equations of motion for an N-particle system on the line (by viewing the independent x variable as a time parameter). The particle system at issue describes the motion of the zeros of the wave function as one moves (linearly) along the x-axis. Alternatively, the dynamical system may also be interpreted as characterizing the motion of the zeros ζ1 , . . . , ζN at a ﬁxed point on the x-axis, while performing a uniform translation of the potential. It so happens that the particle system in question has already popped up (albeit in a different context) in a paper by Calogero [Ca] published in 1978 (cf. equation (3.1.23)). (Notice, however, that the additional constraints imposed there (cf. equation (3.1.24)) correspond from our viewpoint to the trivial case of a vanishing Schrödinger potential.) Calogero’s system may be viewed as a special case of a dynamical system commonly referred to as the (rational) Ruijsenaars-Schneider particle system (with harmonic term) (see [RSc], [Sc], [BC], and [BR]). To compare with the works of Ruijsenaars and Schneider, it is convenient to cast the equations of motion into a Hamiltonian form. To this end it is convenient to ﬁrst rewrite the differential equations (2.4a) as a system of ﬁrst-order equations

278

VAN DIEJEN AND PUSCHMANN

ζj = ηj ,

ηj = −2ηj ζj + 2

1≤k≤N k=j

(2.6a) ηj ηk , ζj − ζ k

(2.6b)

j = 1, . . . , N . We study the solutions of equations (2.6a) and (2.6b) restricted to the manifold

ᏹ = (ζ , η) ∈ RN × RN | ζ1 < · · · < ζN , η1 , . . . , ηN > 0 . (2.7) (It is seen later that, for initial conditions in ᏹ, our solutions are indeed analytic in x (cf. Proposition 3.8). The integral curve thus stays within ᏹ in view of Lemmas 2.2 and 2.3.) Let us now turn to the Hamiltonian form of the differential equations. For the Hamiltonian we take N H= exp(ρl ) |ζl − ζk |−1 + ζl2 , (2.8) l=1

1≤k≤N k=l

with a phase space of the form

= (ζ , ρ) ∈ RN × RN | ζ1 < · · · < ζN .

(2.9)

The Hamilton equations of motion ζj =

∂H , ∂ρj

ρj = −

∂H , ∂ζj

j = 1, . . . , N,

(2.10)

become, in this speciﬁc situation, ζj = exp(ρj )

|ζj − ζk |−1 ,

(2.11a)

1≤k≤N k=j

ρj

= −2ζj +

ζj + ζk

1≤k≤N k=j

ζj − ζ k

(2.11b)

(j = 1, . . . , N ). The Hamiltonian H of equation (2.8) is identiﬁed as an inﬁnite coupling limit of the rational Ruijsenaars-Schneider Hamiltonian (see [RSc] and [R2]) placed in a harmonic potential; the Hamiltonian arises as the rational degeneration of a hyperbolic Ruijsenaars-Schneider N -particle Hamiltonian with an external potential introduced by Schneider (see [Sc] and [BR]). The equations (2.6a) and (2.6b) are related to the Hamilton equations (2.11a) and (2.11b) via the change of variables ηj = exp(ρj ) |ζj − ζk |−1 . (2.12) 1≤k≤N, k=j

DYNAMICS OF ZEROS AND THE SATO FORMULA

279

(Alternatively, differentiation of equation (2.11a) with respect to the independent variable leads us back to equation (2.4a), after elimination of the ρ variables and their derivatives by means of equations (2.11a) and (2.11b).) This shows that for the phase space at hand the Hamilton equations (2.11a) and (2.11b) are equivalent to our original system of differential equations given by equation (2.4a). Remark 2.4. The Hamiltonian form of the differential equations for the zeros allows us to read off their qualitative asymptotic behavior as x → ±∞, assuming the Hamilton ﬂow is complete on our phase space. It is shown in Section 3 that the integral curve for the equations of motion is analytic in the curve parameter x (cf. Proposition 3.8 and Corollary 3.9). This means that the ﬂow generated by the Hamiltonian H of equation (2.8) is indeed complete on the phase space of equation (2.9) in view of Lemmas 2.2 and 2.3 (cf. the paragraph just after equation (2.7)). It is therefore immediate from the fact that the Hamiltonian is a conserved quantity that the zeros ζj must be bounded as functions of x (because the Hamiltonian dominates the harmonic term ζ12 +· · ·+ζN2 from above). The limits limx→±∞ ζj (x) then exist by monotonicity (ζj > 0). We conclude that the potential u(x) = ζ1 (x) + · · · + ζN (x) → 0 and that the wave function ψ(x, z) = exp(zx)(z − ζ1 (x)) · · · (z − ζN (x)) ∼ exp(zx) as x → ±∞. That is to say, we are indeed dealing with Schrödinger operators with reﬂectionless potentials as indicated by the title of this section. 2.3. Lax representation and integrals of motion. The ﬂow associated to the Ruijsenaars-Schneider Hamiltonian (2.8) turns out to be integrable. A Lax pair can be gleaned from Bruschi and Ragnisco [BR]: Lj,k = (ηj ηk )1/2 + ζj2 δj,k , Mj,k =

1 ≤ j, k ≤ N,

(ηj ηk )1/2 1 − δj,k , ζj − ζ k

1 ≤ j, k ≤ N.

(2.13a) (2.13b)

(Here δj,k represents the Kronecker delta.) This Lax pair amounts—up to a gauge transformation—to the rational degeneration of a Lax pair for the hyperbolic generalization of the Hamiltonian H from equation (2.8) due to Schneider [Sc]. The matrices L of equation (2.13a) and M of equation (2.13b) should be thought of as real-analytic matrix-valued functions on the manifold ᏹ of equation (2.7) or—upon employing the transformation (2.12)—on the phase space of equation (2.9). It is not difﬁcult to check that the Hamilton equations (2.11a) and (2.11b) are equivalent to the Lax equation d L = [M, L]. dx Indeed, we have that Lj,k =

1 ηk 1/2 1 ηj 1/2 ηj + ηk + 2ζj ηj δj,k , 2 ηj 2 ηk

(2.14)

280

VAN DIEJEN AND PUSCHMANN

[M, L]j,k =

N

Mj,l Ll,k − Lj,l Ml,k

l=1

=

ηk ηj

1/2 1≤l≤N l=j

1/2 ηj ηl ηj ηk ηl + ζj − ζ l ηk ζk − ζ l 1≤l≤N l=k

+ (ζj + ζk )(ηj ηk )1/2 (δj,k − 1). Comparison of the respective matrix elements reduces the Lax equation (2.14) to the equations of motion (2.6a) and (2.6b) (and thus to equations (2.11a) and (2.11b)). The Lax equation is in fact the ﬁrst of a hierarchy of equations: m−1

m−1

d m n m−1−n n L = L LL = L [M, L]Lm−1−n dx n=0

m

= [M, L ],

n=0

(2.15)

m = 1, . . . , N.

It is immediate from this Lax hierarchy that the evolution of the Lax matrix is isospectral: d Tr(Lm ) = 0, dx

m = 1, . . . , N.

(2.16)

This means, in particular, that the coefﬁcients of the characteristic polynomial of the Lax matrix are integrals for the Ruijsenaars-Schneider Hamiltonian H from equation (2.8). We can compute these integrals explicitly: N det(L − λ 1) = ζ12 − λ · · · ζN2 − λ + ζk2 − λ ηj j =1

=

N

(2.17a)

1≤k≤N k=j

(−λ)N−m Hm ,

(2.17b)

m=0

with N ηj σm−1 ζ12 , . . . , ζj2 , . . . , ζN2 , Hm = σm ζ12 , . . . , ζN2 +

(2.18)

j =1

m = 1, . . . , N (and H0 ≡ 0). Here σm denotes the mth elementary symmetric function yj1 · · · yjm = yj (2.19) σm (y1 , . . . , yM ) = 1≤j1 <···<jm ≤M

J ⊂{1,...,M} j ∈J #J =m

DYNAMICS OF ZEROS AND THE SATO FORMULA

281

(with σ0 ≡ 1) and the “hat” indicates that the j th argument should be omitted. To infer the explicit expression for the characteristic polynomial given by equation (2.17a), one observes that the Lax matrix is of the form L = Z2 + t η1/2 ⊗ η1/2 , where Z2 denotes 1/2 1/2 the square of the diagonal matrix Z = diag(ζ1 , . . . , ζN ), η1/2 = (η1 , . . . , ηN ), and t η 1/2 is the 1-column matrix transposed to the 1-row matrix η 1/2 . Hence, we have that det(L − λ1) = det Z2 + t η1/2 ⊗ η1/2 − λ 1 = det(Z2 − λ 1) det 1 + (Z2 − λ 1)−1t η1/2 ⊗ η1/2 (2.20) N 2 2 ηj . = ζ 1 − λ · · · ζN − λ 1 + 2 j =1 ζj − λ (In passing from the second to the third line one observes that the second factor of the right-hand side determines a rational function in λ that has simple poles at λ = ζj2 , j = 1, . . . , N , and converges to 1 for λ → ∞; the corresponding numerators of the partial fraction expansion then follow from a calculation of the residues at the poles.) We pass from equation (2.20) immediately to equation (2.17a) and then by expanding in λ to equations (2.17b) and (2.18). After endowing the phase space of equation (2.9) with the standard Poisson bracket N ∂F ∂G ∂G ∂F {F, G} = (2.21) − ∂ζj ∂ρj ∂ζj ∂ρj j =1

associated to the symplectic form ω=

N

dρj ∧ dζj

(2.22)

j =1

(where F and G are smooth functions on ), we are in the position to formulate the following proposition, which says that the functions H1 , . . . , HN of equation (2.18) are integrals for our Hamiltonian H of equation (2.8) (equal to H1 ). Proposition 2.5 (Integrals of motion). The functions H1 (ζ , ρ), . . . , HN (ζ , ρ) on from equation (2.9), given by equation (2.18) (with η1 , . . . , ηN taken from equation (2.12)), Poisson-commute with the Hamiltonian H of equation (2.8) {H, Hn } = 0,

1 ≤ n ≤ N.

Proof. It is immediate from the isospectrality of the Lax matrix with respect to . . . , HN are conserved and hence Poisson-commute the H -ﬂow that the functions H1 , 2 with the Hamiltonian H = H1 = N j =1 (ηj + ζj ). It is shown below that the integrals are, in fact, in involution (cf. Proposition 3.12); that is, our particle system is integrable.

282

VAN DIEJEN AND PUSCHMANN

3. The action-angle transformation. In this section we construct the linearizing action-angle transform for the N-particle Hamiltonian H of equation (2.8). The construction makes essential use of some (customized versions) of results from the analytic perturbation theory for ﬁnite-dimensional linear operators (see [Ka]). Result 1. Let T(x) be a real symmetric (N × N)-matrix that is analytic in x around x = x0 ∈ R. Then there exist real eigenvalues λ1 (x), . . . , λN (x) and a corresponding real orthogonal matrix U(x) of orthonormal eigenvectors such that T(x) = U(x) diag(λ1 (x), . . . , λN (x))U−1 (x) with U(x) and λj (x), j = 1, . . . , N , being analytic in x at x = x0 . In the case of simple spectrum, the statement admits a multivariate generalization. Result 2. Let T(x) be a real symmetric (N × N)-matrix that is analytic in x around x = x0 ∈ RM and has simple spectrum at x = x0 . Then there exist simple (at least in a neighborhood of x0 ) real eigenvalues λ1 (x), . . . , λN (x) and a corresponding real orthogonal matrix U(x) of orthonormal eigenvectors such that T(x) = U(x) diag(λ1 (x), . . . , λN (x))U−1 (x) with U(x) and λj (x), j = 1, . . . , N , being analytic in x at x = x0 . Proofs for these two statements can be found in [Ka, Ch. II] (cf. Theorems 5.16 and 6.1). 3.1. Spectral properties of the Lax matrix. The symmetric Lax matrix L of equation (2.13a) is manifestly positive deﬁnite on the manifold ᏹ of equation (2.7). Hence, for each (ζ , η) ∈ ᏹ there exists an orthogonal matrix U(ζ , η) such that ˆ 2 U−1 L = UZ

(3.1)

with Zˆ = diag(ζˆ1 , . . . , ζˆN ) and 0 < ζˆ1 ≤ ζˆ2 ≤ · · · ≤ ζˆN . We wish to show that the eigenvalues ζˆ12 , . . . , ζˆN2 are simple (i.e., that in fact 0 < ζˆ1 < ζˆ2 < · · · < ζˆN ). For this purpose we introduce the open dense submanifold ᏹ0 of ᏹ given by

ᏹ0 = (ζ , η) ∈ ᏹ | ζj + ζk = 0 for 1 ≤ j < k ≤ N , (3.2) and we consider the following rational identity in λ (cf. equation (2.20)): 2 N ζˆ1 − λ · · · ζˆN2 − λ ηj 2 = 1+ 2 . 2 ζ −λ ζ1 − λ · · · ζN − λ j =1 j

(3.3)

It is clear that for (ζ , η) ∈ ᏹ0 the poles of equation (3.3) are simple. Hence, in this generic situation the zeros ζˆ12 , . . . , ζˆN2 must interlace the simple poles ζ12 , . . . , ζN2 , because the right-hand side of equation (3.3) varies between −∞ and +∞ (and thus passes through zero) as λ varies between two subsequent simple poles. (This is because all numerators ηj in the partial fraction decomposition on the right-hand side

DYNAMICS OF ZEROS AND THE SATO FORMULA

283

are positive.) In other words, the zeros are separated by poles, and the spectrum of the Lax matrix is simple on the open dense submanifold ᏹ0 . We furthermore observe that on the complement set ᏹ \ ᏹ0 at most double poles in λ can occur. (In principle such double poles may arise when ζj = −ζk for some j = k.) However, from a continuity argument it is seen that the conﬂuence of the simple poles—as we tend from (ζ , η) ∈ ᏹ0 to (ζ , η) ∈ ᏹ \ ᏹ0 —can in fact never produce such double poles, since a corresponding zero necessarily emerges in the numerator (as on ᏹ0 there always sits a zero between two subsequent poles). Thus, after dividing out coalescing common zeros in numerator and denominator of the left-hand side arising from the transition to a point in ᏹ \ ᏹ0 , we can apply the same reasoning as for (ζ , η) ∈ ᏹ0 to conclude that the spectrum of L remains simple on ᏹ \ ᏹ0 . This proves the following lemma. Lemma 3.1 (Simple spectrum). The eigenvalues of the positive deﬁnite Lax matrix L of equation (2.13a) are simple on the manifold ᏹ of equation (2.7) and depend analytically on the variables (ζ , η). (The statement that the eigenvalues are given by analytic functions on ᏹ is a consequence of the second result from the analytic perturbation theory stated at the beginning of this section, combined with the observation that the matrix elements of L of equation (2.13a) are clearly analytic on ᏹ of equation (2.7).) Let us next analyze in some further detail the mapping ( (3.4) (ζ , η) −→ ζ , ζˆ from ᏹ of equation (2.7) into RN ×RN . Here the components of ζˆ = (ζˆ1 , . . . , ζˆN ) are again deﬁned as the square roots of the eigenvalues of the Lax matrix L of equation (2.13a) subject to the ordering 0 < ζˆ1 < ζˆ2 < · · · < ζˆN . It is helpful to introduce the following collection of open interval unions Ij = ζˆj −1 , ζˆj ∪ − ζˆj , −ζˆj −1 , j = 1, . . . , N

(3.5)

(3.6)

(where ζˆ0 ≡ −ζˆ1 ) and their closures I j with the endpoints included. We are now in the position to describe the manifolds ᏺ and ᏺ0 , which serve as respective target manifolds of ᏹ from equation (2.7) and ᏹ0 from equation (3.2) with respect to the mapping ( from equation (3.4). Deﬁnition 3.2. Let ᏺ be the manifold consisting of (ζ , ζˆ ) ∈ RN × RN such that (i) ζ1 < · · · < ζN and 0 < ζˆ1 < · · · < ζˆN , (ii) #({ζ1 , . . . , ζN } ∩ I j ) > 0 for j = 1, . . . , N , 1 for j = 1, . . . , N − 1, (iii) #({ζ1 , . . . , ζN } ∩ {ζˆj }) = #({ζ1 , . . . , ζN } ∩ {−ζˆj }) ≤ 0 for j = N;

284

VAN DIEJEN AND PUSCHMANN

and let ᏺ0 be the open dense submanifold of ᏺ determined by the conditions (i ) ζ1 < · · · < ζN and 0 < ζˆ1 < · · · < ζˆN , (ii ) #({ζ1 , . . . , ζN } ∩ Ij ) = 1 for j = 1, . . . , N . ( Proposition 3.3 (Action variables). (i) The action (ζ , η) − → (ζ , ζˆ ) deﬁnes a realanalytic map from ᏹ of equation (2.7) into ᏺ. (ii) The map ( restricts to a real-analytic diffeomorphism between the open dense submanifolds ᏹ0 of equation (3.2) and ᏺ0 . The (real-analytic) inverse mapping −1

( (ζ , ζˆ ) −−→ (ζ , η) from ᏺ0 onto ᏹ0 is determined explicitly by 2 2 ˆ 1≤l≤N ζl − ζj 2 , j = 1, . . . , N. ηj = 2 1≤l≤N, l=j ζl − ζj

(3.7)

(

Proof. The mapping (ζ , η) − → (ζ , ζˆ ) is real analytic on ᏹ by Lemma 3.1. To see that ( maps ᏹ into ᏺ it is noted that ζ12 , . . . , ζN2 < ζˆN2 . (This is because the right-hand side of equation (3.3) varies between −∞ and +1 when λ runs from max1≤j ≤N {ζj2 } to +∞.) But then, since for (ζ , η) ∈ ᏹ0 the poles and the zeros of equation (3.3) are simple and interlacing each other (cf. above), it follows that on ᏹ0 each interval union Ij contains exactly one of the elements of {ζ1 , . . . , ζN }. This proves that the open dense submanifold ᏹ0 gets mapped into ᏺ0 ⊂ ᏺ. As we move from a point in ᏹ0 to a point in ᏹ \ ᏹ0 , the position ζj may reach the boundary of its interval union Im (say). In fact, this should happen for at least two positions, say, ζj and ζk (j < k). The movement to the boundary corresponds to the coalescence of the poles ζj2 and ζk2 with a zero ζˆl2 in equation (3.3) (l ∈ {1, . . . , N − 1}). It means that in this situation ζk = −ζj = ζˆl . One concludes that the positions ζ1 , . . . , ζN can only reach the endpoints of the interval unions in even pairs, whence ( maps ᏹ into ᏺ. To determine the inverse of (|ᏹ0 it is noted that for (ζ , ζˆ ) ∈ ᏺ0 the poles of equation (3.3) are simple. A calculation of the residue at λ = ζj2 then produces the inversion relation (3.7). It is furthermore not difﬁcult to infer that the right-hand side of equation (3.7) is positive for (ζ , ζˆ ) ∈ ᏺ0 . Indeed, the interlacing property (ii ) of Deﬁnition 3.2 guarantees that for each j ∈ {1, . . . , N } the number of ζl2 and ζˆl2 smaller than ζj2 is equal (and we stay away from zeros and poles). Hence, the map ( restricts to a real-analytic diffeomorphism between ᏹ0 and ᏺ0 , with the inverse map being governed by equation (3.7). It is convenient to think of the pair (ζ , ζˆ ) ∈ ᏺ0 as real-analytic coordinates for the open dense submanifold ᏹ0 ⊂ ᏹ. These coordinates are related to the standard coordinates (ζ , η) ∈ ᏹ0 via the real-analytic diffeomorphism ( : ᏹ0 → ᏺ0 (see equation (3.4)). Remarkably, in the local (ζ , ζˆ )-coordinates we can perform the diagonalization of the Lax matrix completely explicitly. Lemma 3.4 (Diagonalization). Let L(ζ , ζˆ ) be given by equation (2.13a) with η1 , . . . , ηN taken from equation (3.7). Then one has for (ζ , ζˆ ) ∈ ᏺ0 that

285

DYNAMICS OF ZEROS AND THE SATO FORMULA

ˆ 2 U−1 , L = UZ ˆ = diag(ζˆ1 , . . . , ζˆN ) and with Z 1/2 2 1/2 2 ˆ ζˆl2 − ζj2 1 1≤l≤N ζk − ζl 2 , 2 2 ˆ2 ˆ ζj2 − ζˆk2 1≤l≤N, l=j ζl − ζj 1≤l≤N, l=k ζk − ζl

Uj,k =

1≤l≤N

1 ≤ j , k ≤ N. Furthermore, the matrix U is real analytic in (ζ , ζˆ ) ∈ ᏺ0 and orthogonal: U−1 = t U. Proof. It follows from the interlacing property (ii ) of Deﬁnition 3.2 that all the matrix elements Uj,k (ζ , ζˆ ) are real analytic for (ζ , ζˆ ) ∈ ᏺ0 . Speciﬁcally, we stay away from zeros in numerator and denominator, and the arguments of the square roots are positive (as #{1 ≤ l ≤ N | ζˆl2 < ζj2 } = #{1 ≤ l ≤ N | ζl2 < ζj2 } and #{1 ≤ l ≤ N | ζl2 > ζˆk2 } = #{1 ≤ l ≤ N | ζˆl2 > ζˆk2 }). In Appendix B (cf. Corollary B.4 and Proposition B.6), it is shown that the stated matrix U is orthogonal and that it diagonalizes the Lax matrix L(ζ , ζˆ ). It is clear from Proposition 3.3 and Lemma 3.4 that the matrix elements Uj,k are real analytic when viewed as functions on ᏹ0 of equation (3.2). From the explicit expressions for the matrix elements, one might be inclined to think that U becomes singular on ᏹ \ ᏹ0 . This, however, turns out to be not true: The matrix elements Uj,k extend to real-analytic functions on ᏹ of equation (2.13b) by Lemma 3.1 and by the second result from the analytic perturbation theory cited at the beginning of this section. Similarly, the right-hand side of equation (3.7) extends from a real-analytic function on ᏹ0 to a real-analytic function on ᏹ (because the left-hand side does so). Notice, on the other hand, that for N > 1 the inverse mapping (−1 : ᏺ0 → ᏹ0 does not extend analytically to ᏺ. (The expressions for ηj in equation (3.7) become singular on ᏺ \ ᏺ0 unless N = 1.) 3.2. Linearization. The coordinates ζˆ1 , . . . , ζˆN serve as the action variables for our Ruijsenaars-Schneider Hamiltonian H of equation (2.8). To construct the corresponding angle variables, it is convenient to consider the matrix-valued function ˆ −1/2 U−1 ZUZˆ −1/2 1 + Z ˆ −1/2 U−1 ZUZˆ −1/2 −1 N = 1−Z

(3.8)

ˆ and the orthogonal matrix U are deﬁned as on ᏹ, where the diagonal matrices Z, Z in Subsections 2.3 and 3.1. The following lemma provides an explicit expression for the matrix elements of N in the local (ζ , ζˆ )-coordinates. Lemma 3.5 (Angle variables). (i) The symmetric matrix N from equation (3.8) is real analytic and positive deﬁnite on ᏹ of equation (2.7).

286

VAN DIEJEN AND PUSCHMANN

(ii) In the coordinate patch (ᏺ0 , (−1 ) of the open dense subset ᏹ0 from equation (3.2) of ᏹ, the matrix elements of N are given explicitly by Nj,k =

1/2 1/2 2 ζˆj ζˆk

ζˆj + ζˆk

1/2 1/2

ηˆ j ηˆ k ,

1 ≤ j, k ≤ N,

(3.9a)

with ηˆ j =

ζˆj − ζl ζˆj + ζˆl , ζˆj + ζl 1≤l≤N ζˆj − ζˆl 1≤l≤N

j = 1, . . . , N.

(3.9b)

l=j

Proof. By the analyticity of Z, Zˆ 1/2 , and U, it is clear that the matrix N from equation (3.8) is analytic in ᏹ outside the locus

(ζ , η) ∈ ᏹ | det 1 + Zˆ −1/2 U−1 ZUZˆ −1/2 = 0 . We now show that this locus is in fact empty. Let us ﬁrst consider the case (ζ , η) ∈ ᏹ0 . With the aid of the explicit expressions for U in the local (ζ , ζˆ )-coordinates from Proposition 3.3, we can compute the matrix elements of N from equation (3.8) explicitly. This entails the formulas in (3.9a) and (3.9b). For the details of the calculation the reader is referred to Appendix B (see Proposition B.8). It is not difﬁcult to infer that the interlacing property (ii ) of Deﬁnition 3.2 guarantees that on ᏹ0 the quantities ηˆ j of equation (3.9b) are analytic and positive. Indeed, we have that ηˆ j =

1≤l≤N

ζˆj2 − ζl2 2 ζˆj + ζl 1≤l≤N l=j

2 ζˆj + ζˆl ζˆ 2 − ζˆ 2 j

l

with #{l = 1, . . . , N | ζl2 > ζˆj2 } = #{l = 1, . . . , N | ζˆl2 > ζˆj2 }, and we stay away from zeros in the numerator and denominator. It thus follows that N is real analytic and positive deﬁnite on ᏹ0 of equation (3.2). (To decide on the positivity of N we have used at this point that 0 < ζˆ1 < · · · < ζˆN and that the principal minors are easily computed using the Cauchy determinant formula (see [We, p. 202]); cf. also Corollary B.3). To generalize the result to the whole manifold ᏹ of equation (2.13b), it sufﬁces to show that the ηˆ j of equation (3.9b) extend to real-analytic positive functions on ᏹ, or, equivalently, that the functions ζˆj − ζl , ζˆj + ζl

j = 1, . . . , N,

(3.10)

1≤l≤N

extend to real-analytic functions on ᏹ without zeros. This is seen by recalling that the matrix U, with elements Uj,k given by Lemma 3.4, extends analytically to an

DYNAMICS OF ZEROS AND THE SATO FORMULA

287

orthogonal matrix on ᏹ (by Lemma 3.1 and Result 2). Indeed, after squaring Uj,k and dividing by ηj of equation (3.7) (> 0), we conclude that for all j, k ∈ {1, . . . , N } the expression 2 −1 ζˆj − ζk2 ζˆj2 − ζl2 (3.11) 1≤l≤N, l=k

extends analytically to ᏹ. Since on ᏹ with a ζi → ζˆj there always corresponds a ζi → −ζˆj and vice versa (see Subsection 3.1), it follows from the analyticity of the expressions in equation (3.11) that in such a situation the fractions (ζˆj +ζi )/(ζˆj −ζi ) and (ζˆj − ζi )/(ζˆj + ζi ) extend analytically to the point of conﬂuence. Hence, the expressions in equation (3.10) extend to real-analytic functions on ᏹ without zeros, which completes the proof. Let us deﬁne the action-angle manifold

ˆ = ζˆ , ηˆ ∈ RN × RN | 0 < ζˆ1 < · · · < ζˆN , ηˆ 1 , . . . , ηˆ N > 0 ᏹ

(3.12)

ˆ deﬁned by the mapping and consider the action-angle transformation : ᏹ → ᏹ (ζ , η) −→ ζˆ , ηˆ

(3.13)

(where ζˆ = (ζˆ1 , . . . , ζˆN ) contains the ordered square roots of the Lax matrix from equation (2.13a) and where ηˆ = (ηˆ 1 , . . . , ηˆ N ) is determined by equation (3.9b)). Proposition 3.6 (Action-angle transformation). The action-angle transformation ˆ of equaˆ is a real-analytic map from ᏹ of equation (2.7) into ᏹ → (ζˆ , η) (ζ , η) − tion (3.12). Proof. This is immediate from Lemmas 3.1 and 3.5. The mapping of equation (3.13) turns out to linearize the equations of motion (2.6a) and (2.6b). Proposition 3.7 (Linearization). Let us assume that a continuously differentiable curve (ζ , η)(x), x ∈ R, lies in the manifold ᏹ of equation (2.7) and provides a solution to the equations of motion in equations (2.6a) and (2.6b). Then the image ˆ of the form (ζˆ , η)(x), ˆ of this curve with respect to the action-angle map : ᏹ → ᏹ x ∈ R, satisﬁes the linear system of differential equations ζˆj = 0, j = 1, . . . , N. (3.14) ηˆ = −2ζˆj ηˆ j , j

Proof. The equation ζˆj (x) = 0 is immediate from the isospectrality (2.16) of the Lax matrix. It is furthermore clear that for generic x (i.e., except for values in a

288

VAN DIEJEN AND PUSCHMANN

discrete subset of R) we have (ζ , η)(x) ∈ ᏹ0 of equation (3.2). (This is because ζj = ηj > 0.) It means that for such generic values of x the vector (ζ , ζˆ )(x) lies in ᏺ0 given Proposition 3.3. The differentiation of ηˆ j (x) from equation (3.9b) and the invoking of the equations of motion then produces ηˆ j

= −ηˆ j

N l=1

= −2ζˆj ηˆ j

(2.6a)

= −2ζˆj ηˆ j

(3.7)

ζl

ζˆj − ζl

+

ζl

ζˆj + ζl

N

ηl 2 ζˆ − ζl2 l=1 j

N l=1

ζˆk2 − ζl2 2 2 1≤k≤N, k=l ζk − ζl

1≤k≤N, k=j

= −2ζˆj ηˆ j . In the last step we used the identity of Lemma B.5 to infer that the summation on the right-hand side of the third line produces 1. This proves the stated differential equation for ηˆ j (x) for the generic values of x such that (ζ , η)(x) ∈ ᏹ0 . The extension to all x ∈ R then follows by analyticity. 3.3. The integral curve and action-angle diffeomorphism. The linearized equations (3.14) are solved trivially: ηˆ j (x) = ηˆ j (0) exp − 2x ζˆj (0) , j = 1, . . . , N. (3.15) ζˆj (x) = ζˆj (0), Proposition 3.7 thus tells us that if (ζ , η)(x) satisﬁes the equations of motion given by equations (2.6a) and (2.6b), then the matrix N of equation (3.8) evolves as ˆ ˆ Inversion of the relation in equation (3.8) allows us to exp(−x Z)N(0) exp(−x Z). reconstruct Z from N, ˘ −1 Z = UZU

ˆ 1/2 (1 − N)(1 + N)−1 Z ˆ 1/2 , with Z˘ = Z

(3.16)

and thus solve the equations of motion. Proposition 3.8 (Integral curve). Let (ζ0 , η0 ) ∈ ᏹ of equation (2.13b), and let ˆ of equation (3.12) be the image of (ζ0 , η0 ) with respect to the action-angle ˆ ∈ᏹ (ζˆ , η) ˆ = diag(ζˆ1 , . . . , ζˆN ), and let N be the map of equation (3.13). Furthermore, let Z 1/2 ˆ ˆ ˆ matrix with entries Nj,k = 2(ηˆ j ηˆ k ζj ζk ) /(ζj + ζˆk ), 1 ≤ j , k ≤ N . Then we have that the ordered eigenvalues ζ1 (x) ≤ · · · ≤ ζN (x) of the matrix ˆ 1/2 (3.17) ˆ N exp − x Zˆ 1 + exp − x Z ˆ N exp − x Zˆ −1 Z Zˆ 1/2 1 − exp − x Z deﬁne a real-analytic curve (ζ , ζ )(x), x ∈ R, that lies in ᏹ of equation (2.13b) and solves the system of differential equations (2.6a) and (2.6b) subject to the initial condition (ζ , ζ )(0) = (ζ0 , η0 ).

DYNAMICS OF ZEROS AND THE SATO FORMULA

289

Proof. By Lemma 3.5 one has that N is positive deﬁnite. Hence, the symmetric matrix in equation (3.17) is analytic in x. But then so are its eigenvalues ζ1 (x), . . . , ζN (x), by the ﬁrst result from the analytic perturbation theory cited at the beginning of this section. This shows that our curve (ζ , ζ )(x), x ∈ R, is real analytic in x. The general theory of ordinary differential equations guarantees that locally (i.e., for x sufﬁciently small) the (Hamiltonian) system (2.6a) and (2.6b) with initial condition (ζ0 , η0 ) has a unique continuously differentiable integral curve in ᏹ. It is clear from Proposition 3.7 and the inversion formula (3.16) that this local solution coincides with the local restriction of the curve (ζ , ζ )(x). This proves that the curve (ζ , ζ )(x) solves the differential equations (2.6a) and (2.6b) with initial condition (ζ0 , η0 ) for x sufﬁciently small. But then, by analyticity, the differential equations must actually hold for all x ∈ R. The fact that the integral curve (ζ , ζ )(x) indeed stays within ᏹ for all x ∈ R is now a consequence of Lemmas 2.2 and 2.3. Notice that Proposition 3.8 says, in particular, that the eigenvalues of the matrix in equation (3.17) are simple: ζ1 (x) < · · · < ζN (x) for x ∈ R. The following corollary is an immediate consequence of Proposition 3.8. Corollary 3.9 (Completeness). The ﬂow generated by the Ruijsenaars-Schneider Hamiltonian with harmonic term H of equation (2.8) is complete on the phase space of equation (2.9) and analytic in the ﬂow parameter x ∈ R. ˆ of equation ˆ ∈ᏹ The map of equation (3.13) associates an image point (ζˆ , η) (3.12) to a point (ζ , η) ∈ ᏹ of equation (2.13b). We recall that to compute the image point one ﬁrst constructs the Lax matrix L of equation (2.13a) and determines its (simple) eigenvalues ζˆ1 < · · · < ζˆN . The components of ηˆ are next derived from equation (3.9b) (ﬁrst for (ζ , η) in the submanifold ᏹ0 of equation (3.2) by directly evaluating the stated expression for ηˆ j , and then for (ζ , η) ∈ ᏹ by analytic continuation). It is not difﬁcult to see that is an injection. Indeed, the inversion formula (3.16) ˆ For (ζˆ , η) ˆ in (ᏹ0 ) (cf. equation provides the components of ζ as functions of (ζˆ , η). (3.2)) the components of η now follow from equation (3.7). More generally one ˆ as the vector tangent to the real-analytic curve ζ (x), obtains η in terms of (ζˆ , η) x ∈ R—swept out by the ordered eigenvalues of the matrix (3.17)—at the special point x = 0. The inverse mapping −1 from (ᏹ) onto ᏹ turns out to be real analytic. To see this, one employs Result 2 to infer that the diagonal ζ of the matrix ˆ For (ζˆ , η) ˆ in the open dense Z from equation (3.16) is analytic in the variables (ζˆ , η). subset (ᏹ0 ) ⊂ (ᏹ) the analyticity of η is now immediate from equation (3.7). The ˆ ∈ (ᏹ) follows because the poles in ηj of equation (3.7) extension to general (ζˆ , η) originating from the denominator are compensated by zeros in the numerator. (This is clear from the above-mentioned alternative characterization of η as the tangent vector to the curve ζ (x) at x = 0.) The following proposition says that the map constitutes a real-analytic diffeoˆ (i.e., (ᏹ) = ᏹ ˆ ). morphism between the manifolds ᏹ and ᏹ

290

VAN DIEJEN AND PUSCHMANN

Proposition 3.10 (Action-angle diffeomorphism). The action-angle transformation of equation (3.13) deﬁnes a real-analytic diffeomorphism between ᏹ of equaˆ of equation (3.12). tion (2.13b) and ᏹ Proof. In view of the Proposition 3.6 and the paragraph preceding the current ˆ is surjective. To this end we consider proposition, it sufﬁces to show that : ᏹ → ᏹ ˆ of equation (3.12). The positivity of the corresponding ˆ in ᏹ an arbitrary point (ζˆ , η) matrix N (cf. Proposition 3.8) leads us to conclude (by a perturbation argument) that, for x sufﬁciently large, the j th eigenvalue ζj (x) of the matrix in equation (3.17) tends monotonically from below to ζˆj as x → +∞. Hence, we have that (ζ , ζ )(x) ∈ ᏹ of equation (2.13b) for x sufﬁciently large. But then, by Proposition 3.8, it follows that ˆ = (ζ , ζ )(x) stays in ᏹ for all x ∈ R. In particular, (ζ , ζ )(0) ∈ ᏹ and thus (ζˆ , η) ((ζ , ζ )(0)), which proves the surjectivity. 3.4. Symplectic structure. The variables (ζ , ρ) in of equation (2.9) determine a system of real-analytic canonical coordinates on ᏹ of equation (2.13b) via the identiﬁcation in equation (2.12). We can easily compute the standard symplectic form ω of equation (2.22) in the (ζ , η)-coordinates (using equation (2.12)) and in the local (ζ , ζˆ )-coordinates (using equation (3.7)): ω=

dρj ∧ dζj

(3.18a)

1≤j ≤N

=

1≤j ≤N

ηj−1 dηj ∧ dζj +

1≤j =k≤N

dζj ∧ dζk ζj − ζk

dζˆj ∧ dζk dζˆj ∧ dζk = . + ζˆj − ζk ζˆj + ζk 1≤j,k≤N

(3.18b)

(3.18c)

ˆ of equation (3.12) with canonical Similarly, we can equip the action-angle manifold ᏹ coordinates by setting

ηˆ j = exp ρˆj

1≤l≤N, l=j

ζˆj + ζˆl , ˆ ζj − ζˆl

j = 1, . . . , N,

(3.19)

ˆ in with (ζˆ , ρ) ˆ =

ζˆ , ρˆ ∈ RN × RN | 0 < ζˆ1 < · · · < ζˆN .

(3.20)

ˆ of equation (3.20) reads in the (ζˆ , ρ)ˆ The standard symplectic form coming from ˆ ˆ coordinates, the (ζ , η)-coordinates, and the local (ζ , ζˆ )-coordinates:

DYNAMICS OF ZEROS AND THE SATO FORMULA

ωˆ =

dρˆj ∧ dζˆj

291 (3.21a)

1≤j ≤N

=

1≤j ≤N

ηˆ j−1 dηˆ j ∧ dζˆj +

1≤j =k≤N

dζˆj ∧ dζˆk ζˆj − ζˆk

ˆ dζj ∧ dζk dζˆj ∧ dζk = , + ζˆj − ζk ζˆj + ζk 1≤j, k≤N

(3.21b)

(3.21c)

respectively. We see that in the local (ζ , ζˆ )-coordinates both ω and ωˆ read the same. This proves that the action-angle mapping of equation (3.13) is a symplectomorphism. Proposition 3.11 (Canonicity). The action-angle mapping of equation (3.13) is a symplectomorphism (i.e., a symplectic diffeomorphism) between (ᏹ, ω) and ˆ , ω). ˆ (ᏹ It is instructive to compute the Poisson brackets between the coordinate functions of the various coordinate systems. (1) The (ζ , ρ)-coordinates (with values in of equation (2.9)) give {ζj , ζk } = 0,

{ζj , ρk } = δj,k ,

{ρj , ρk } = 0.

(3.22)

(2) The (ζ , η)-coordinates (with values in ᏹ of equation (2.7)) give {ζj , ζk } = 0,

{ζj , ηk } = ηj δj,k ,

{ηj , ηk } =

2ηj ηk 1 − δj,k . ζj − ζk

(3.23)

(3) The local (ζ , ζˆ )-coordinates (with values in ᏺ0 , cf. Deﬁnition 3.2) give 2 2 2 ˆ2 1 ˆ

1 1≤l≤N ζj − ζl 1≤l≤N ζk − ζl ˆ 2 ζj , ζk = − 2 , 2 ˆ ˆ2 ζj + ζˆk ζj − ζˆk 1≤l≤N, l=j ζj − ζl 1≤l≤N, l=k ζk − ζl (3.24a)

{ζj , ζk } = 0, ζˆj , ζˆk = 0. (3.24b) ˆ of equation (3.12)) give ˆ (4) The (ζˆ , η)-coordinates (with values in ᏹ

ζˆj , ζˆk = 0,

ζˆj , ηˆ k = ηj δj,k ,

2ηˆ j ηˆ k ηˆ j , ηˆ k = 1 − δj,k . ζˆj − ζˆk

ˆ of equation (3.20)) give ˆ (5) The (ζˆ , ρ)-coordinates (with values in

ζˆj , ζˆk = 0, ρˆj , ρˆk = 0. ζˆj , ρˆk = δj,k ,

(3.25)

(3.26)

As a consequence of Proposition 3.11 we arrive at the integrability of the RuijsenaarsSchneider Hamiltonian H of equation (2.8).

292

VAN DIEJEN AND PUSCHMANN

Proposition 3.12 (Integrability). The functions H1 , . . . , HN given by equation (2.18) are in involution with respect to the symplectic form ω: {Hm , Hn } = 0,

1 ≤ m, n ≤ N.

Proof. The Poisson commutativity follows from the canonicity of the action-angle ˆ map in Proposition 3.11 and from the fact that in the action-angle coordinates (ζˆ , ρ) the Hamiltonians become Hm = σm ζˆ12 , . . . , ζˆN2 , m = 1, . . . , N (where σm is the mth elementary symmetric function). Indeed, in action-angle coordinates the vanishing of the Poisson brackets between the integrals Hm is immediate from equation (3.26). 4. The wave function and the tau function. In this section we return to our original problem of characterizing the 1-dimensional Schrödinger operators with eigenfunctions that are the product of a plane wave and a polynomial in the spectral parameter. We use the solution to the equations of motion for the rational RuijsenaarsSchneider system with harmonic term, from the previous section, to derive closed expressions for the potential and the eigenfunction of the Schrödinger operators under consideration. ˆ be a point in the manifold 4.1. A Wilson-type determinantal formula. Let (ζˆ , η) ˆ of equation (3.12). We consider the matrix ᏹ ˜ ˆ 1 − exp − 2x Z ˆ N 1 + exp − 2x Zˆ N −1 , Z(x) =Z (4.1) with Zˆ = diag(ζˆ1 , . . . , ζˆN ) and Nj,k = 2(ηˆ j ηˆ k ζˆj ζˆk )1/2 /(ζˆj + ζˆk ), 1 ≤ j, k ≤ N. From ˜ Proposition 3.8 it follows that the eigenvalues of Z(x) from equation (4.1) constitute a real-analytic solution to the differential equations in equation (2.4a). Hence, by Proposition 2.1, it is clear that the wave function ˜ ψ(x, z) = exp(zx) det z1 − Z(x) (4.2a) solves the Schrödinger equation (2.1) with potential ˜ u(x) = 2∂x Tr Z(x) .

(4.2b)

In equation (4.2a) the wave function is normalized such that the polynomial factor is monic. Often, though, it is somewhat more comfortable to work with the (Jost) wave function ˜ det z1 − Z(x) (x, z) = exp(zx) (4.3) , ˆ det z1 − Z which is normalized such that (x, z) → exp(zx) for x → +∞.

DYNAMICS OF ZEROS AND THE SATO FORMULA

293

ˆ of equation ˆ ∈ᏹ Theorem 4.1 (Wilson-type formula). Let us assume that (ζˆ , η) (3.12). We have that the wave function ˆ N ˆ z1 − Zˆ −1 exp − 2x Z det 1 + z1 + Z (x, z) = exp(zx) ˆ N det 1 + exp − 2x Z solves the Schrödinger equation 2 ∂x + u(x) ψ(x, z) = z2 ψ(x, z),

−∞ < x < ∞,

with a potential of the form ˆ N . u(x) = 2∂x2 log det 1 + exp − 2x Z ˆ and N are as deﬁned above and z ∈ C \ {ζˆ1 , . . . , ζˆN }. Here the matrices Z Proof. The main point is the above-mentioned observation that (x, z) of equation (4.3) solves the Schrödinger equation with potential u(x) of equation (4.2b). Elementary manipulations then reveal that ˜ det z1 − Z(x) (x, z) = exp(zx) ˆ det z1 − Z ˆ 1 − exp − 2x Z ˆ N 1 + exp − 2x Zˆ N −1 det z1 − Z = exp(zx) ˆ det z1 − Z ˆ 1 − exp − 2x Z ˆ N det z 1 + exp − 2x Zˆ N − Z = exp(zx) ˆ det 1 + exp − 2x Z ˆ N det z1 − Z ˆ − z1 + Z ˆ exp − 2x Z ˆ N det z1 − Z = exp(zx) ˆ N det z1 − Zˆ det 1 + exp − 2x Z ˆ z1 − Z ˆ −1 exp − 2x Z ˆ N det 1 + z1 + Z = exp(zx) ˆ N det 1 + exp − 2x Z and that ˜ u(x) = 2∂x Tr Z(x) ˆ 1 − exp − 2x Zˆ N 1 + exp − 2x Z ˆ N −1 = 2∂x Tr Z ˆ N = 2∂x2 Tr log 1 + exp − 2x Z ˆ N , = 2∂x2 log det 1 + exp − 2x Z which completes the proof.

294

VAN DIEJEN AND PUSCHMANN

From the determinantal formula for the potential given in the theorem, it is read off that we are dealing with the well-known class of reﬂectionless Schrödinger potentials (or Bargmann potentials) (see [DT], [AS], and [M4]). The parameters 0 < ζˆ1 < · · · < ζˆN and ηˆ 1 , . . . , ηˆ N > 0 correspond to the so-called spectral data of the potential. Speciﬁcally, for the spectral values z = −ζˆj , j = 1, . . . , N , the wave function (x, z) is square integrable. These spectral values constitute the discrete spectrum of our Schrödinger operator. The parameters ηˆ j , j = 1, . . . , N , govern the associated normalization constants: ∞ 1 2 x, −ζˆj dx = ηˆ j , j = 1, . . . , N. (4.4) ˆ 2ζj −∞ As it turns out, the determinantal formula for the reﬂectionless wave function from Theorem 4.1 can also be derived in a completely different way using ﬁnite-gap machinery (see [DMKM]) or inverse scattering theory (see [DT], [ELZ], and [DK]). When rewritten in appropriate coordinates our matrix N, with elements Nj,k = 2(ηˆ j ηˆ k ζˆj ζˆk )1/2 /(ζˆj + ζˆk ), 1 ≤ j , k ≤ N , becomes equal to the Lax matrix of a hyperbolic Ruijsenaars-Schneider system [RSc]. Its symmetric functions, which are easily computed with the aid of the Cauchy determinantal formula (see [We, p. 202]; cf. also Corollary B.3), form a complete set of commuting integrals for this system. More explicitly, we have that det(N − λ1) =

N

(−λ)N−m Sm ζˆ , ηˆ ,

(4.5)

m=0

with Sm ζˆ , ηˆ =

J ⊂{1,...,N} j ∈J #J =m

ηˆ j

ζˆj − ζˆk 2 , ζˆj + ζˆk

m = 1, . . . , N

(4.6)

j,k∈J j
(and S0 ≡ 1). By passing to the canonical coordinatization cosh(xj − xk ) ˆζj = exp(2xj ), ηˆ j = exp(pj ) sinh(x − x ) , j k

(4.7)

1≤l≤N, l=j

j = 1, . . . , N (with (x, p) ∈ RN × RN such that x1 < · · · < xN ), the matrix N is seen to go over into the hyperbolic Lax matrix of [RSc] (where the coupling parameter is chosen to be a half-period), and the symmetric functions S1 , . . . , SN become of the form cosh(xj − xk ) pj j ∈J e (4.8) Sm (x, p) = sinh(x − x ) , m = 1, . . . , N. J ⊂{1,...,N} #J =m

j ∈J k∈J

j

k

295

DYNAMICS OF ZEROS AND THE SATO FORMULA

It was shown in [RSc] and [R1] that the Hamiltonians S1 , . . . , SN are in involution with respect to the symplectic form ω˜ = dpj ∧ dxj (4.9a) 1≤j ≤N

=

dηˆ j dζˆj dζˆj ∧ dζˆk ∧ + . ηˆ j ζˆj ζˆ 2 − ζˆk2 1≤j ≤N 1≤j =k≤N j

(4.9b)

It is interesting to note that the latter symplectic structure is not the same as the one that turned our action-angle transformation into a symplectomorphism (cf. ωˆ of equation (3.21b)). The upshot is that Theorem 4.1 may be interpreted as providing an explicit parametrization of the reﬂectionless Schrödinger wave functions in terms of the Lax matrix of a hyperbolic Ruijsenaars-Schneider system. This amounts to a KdV soliton analog of a recent formula due to Wilson, who expressed the (stationary) Baker function of the linear problem associated to the KP equation for the rational regime in terms of Calogero-Moser Lax matrices (see [Wi]). (See also Section 6 for further connections with soliton equations.) 4.2. The solitonic Sato formula. By invoking the Cauchy determinant formula (see Corollary B.3), it is not difﬁcult to explicitly evaluate the determinantal expression ˆ of ˆ ∈ for (x, z) of Theorem 4.1. Upon passing to the canonical coordinates (ζˆ , ρ) equation (3.20) via equation (3.19), this entails the following theorem. ˆ of equation ˆ be a point in Theorem 4.2 (Solitonic Sato formula). Let (ζˆ , ρ) (3.20) and let z ∈ C \ {ζˆ1 , . . . , ζˆN }. We have that the wave function (x, z) = exp(zx) ζˆj + ζˆk z + ζˆj z − ζˆj ζˆj − ζˆk exp ρˆj − 2x ζˆj ×

J ⊂{1,...,N} j ∈J

J ⊂{1,...,N} j ∈J, k∈J

j ∈J,k∈J

j ∈J

ζˆj + ζˆk ζˆj − ζˆk exp ρˆj − 2x ζˆj

solves the Schrödinger equation 2 ∂x + u(x) ψ(x, z) = z2 ψ(x, z),

j ∈J

−∞ < x < ∞,

with potential u(x) = 2∂x2 log τ (x),

ζˆj + ζˆk ˆ ρˆj − 2x ζj . τ (x) = ˆ exp ζ − ζˆk j ∈J J ⊂{1,...,N} j ∈J, k∈J j

296

VAN DIEJEN AND PUSCHMANN

Theorem 4.2 provides an explicit formula for the (Jost) wave function of the reﬂectionless Schrödinger operator on the line. This formula turns out to be closely related to a formula for the wave function originating from Sato theory (see [S], [SS], [DKJM], [JM], [SW], [OSTT], [Mo], [BBS], and [DK]). We come back to this point in Section 6. Multiplication of (x, z) by a factor (z − ζˆ1 ) · · · (z − ζˆN ) eliminates the denominators in z and brings us back to a wave function of the form ψ(x, z) = exp(zx)pN (z; x) with pN (z; x) a monic polynomial of degree N in z: ψ(x, z) = exp(zx) z + ζˆj z − ζˆk ×

J ⊂{1,...,N} j ∈J

k∈J

J ⊂{1,...,N} j ∈J, k∈J

j ∈J, k∈J

ζˆj + ζˆk ζˆj − ζˆk exp ρˆj − 2x ζˆj j ∈J

ζˆj + ζˆk ζˆj − ζˆk exp ρˆj − 2x ζˆj

.

j ∈J

(4.10) Collecting powers of z then results in a closed expression for coefﬁcients of the polynomial factor pN (z; x) (cf. [BBS]): ψ(x, z) = exp(zx)

N

sm (x) zN−m ,

(4.11a)

m=0

with sm (x) = (−1)m

τ ρˆ1 , . . . , ρˆl1 + iπ, . . . , ρˆlm + iπ, . . . , ρˆN ; ζˆ1 , . . . , ζˆN | x ζˆl1 · · · ζˆlm × τ ρˆ1 , . . . , ρˆN ; ζˆ1 , . . . , ζˆN | x 1≤l1 <···
(s0 (x) ≡ 1) and τ ρˆ1 , . . . , ρˆN ; ζˆ1 , . . . , ζˆN | x =

ζˆj + ζˆk exp ρˆj − 2x ζˆj . ˆ ˆk − ζ ζ j j ∈J J ⊂{1,...,N} j ∈J

k∈J

(4.11c) 4.3. The Bethe curve. In this section we have so far utilized the solution of the equations of motion for the rational Ruijsenaars-Schneider Hamiltonian with harmonic term H of equation (2.8) to derive closed expressions for the (Jost) wave functions of Schrödinger operators on the line with Bargmann potentials. This led to the Wilson-type determinantal formula of Theorem 4.1 and the Sato-type formula of Theorem 4.2. We now apply the explicit information on the Ruijsenaars-Schneider dynamics to characterize the locations of the zeros of the wave functions in question.

DYNAMICS OF ZEROS AND THE SATO FORMULA

297

ˆ of equation ˆ be a point in the manifold Theorem 4.3 (Bethe curve). Let (ζˆ , ρ) ˆ ˆ (3.20), and let the tau function τ (ρˆ1 , . . . , ρˆN ; ζ1 , . . . , ζN | x) be deﬁned in accordance with equation (4.11c). Then the wave function ψ(x, z) = exp(zx)

N

z − ζj (x)

j =1

solves the Schrödinger equation 2 ∂x + u(x) ψ(x, z) = z2 ψ(x, z),

−∞ < x < ∞,

with a reﬂectionless potential of the form u(x) = 2∂x2 log τ ρˆ1 , . . . , ρˆN ; ζˆ1 , . . . , ζˆN | x , if and only if the zeros ζ1 (x), . . . , ζN (x) are simple, real analytic in x, and they constitute a solution to the following nonlinear system of Bethe equations ζˆj − ζl (x) = (−1)N−j exp ρˆj − 2x ζˆj , j = 1, . . . , N. (4.12) ˆζj + ζl (x) 1≤l≤N Furthermore, the unique (up to permutations of the ζj (x)) real-analytic (in x) solution to the Bethe equations (4.12) is given by the projection onto the conﬁguration space of the integral curve for the rational Ruijsenaars-Schneider system with harmonic term (ζ , ρ)(x) = −1 (ζˆ , ρˆ − 2x ζˆ ), x ∈ R. The components of the projected curve ζ (x), x ∈ R, are given explicitly by ordered eigenvalues ζ1 (x) < · · · < ζN (x) of the matrix ˜ ˆ 1 − exp − 2x Z ˆ N 1 + exp − 2x Z ˆ N −1 , Z(x) =Z (4.13) ˆ = diag(ζˆ1 , . . . , ζˆN ) and where N is the (N ×N)-matrix with elements Nj,k = where Z 2(ηˆ j ηˆ k ζˆj ζˆk )1/2 /(ζˆj + ζˆk ) and ηˆ j = exp(ρˆj ) 1≤k≤N, k=j |ζˆj + ζˆk |/|ζˆj − ζˆk |. Proof. We know from Subsections 4.1 and 4.2 that the wave function with zeros ˜ sweeping out the real-analytic curve given by the eigenvalues of the matrix Z(x) of equation (4.13) satisﬁes the reﬂectionless Schrödinger equation stated in the theorem. The relevant curve is precisely the position part of the integral curve (ζ , ζ )(x), x ∈ R, for the rational Ruijsenaars-Schneider system with harmonic term (cf. Proposition 3.8), which is obtained as the inverse image of the line (ζˆ , ρˆ −2x ζˆ ) with respect to the ˆ action-angle map of equation (3.13). (Here we have passed from (ζˆ , η)-coordinates ˆ to (ζˆ , ρ)-coordinates; cf. equation (3.19).) From the explicit relation between the components of ζ (x) and the components of (ζˆ , ρˆ − 2x ζˆ ) given by the inverse of the action-angle map (cf. equations (3.9b) and (3.19)), it now follows that the zeros of the wave function satisfy the Bethe equations (4.12).

298

VAN DIEJEN AND PUSCHMANN

To prove the reverse statement we consider the algebraic system ζˆj − ζl = .j , j = 1, . . . , N, ζˆj + ζl 1≤l≤N for the unknowns ζ1 , . . . , ζN . It is clear that for .1 , . . . , .N = 0 the unique (up to permutation) solution to the equations is given by ζj = ζˆj , j = 1, . . . , N . Furthermore, Jacobi matrix ∂/∂ζ , with components ∂.j /∂ζk = −2.j ζˆj /(ζˆj2 − ζk2 ), specializes on the solution (ζ , ) = (ζˆ , 0) to the invertible diagonal matrix ζˆj − ζˆl ∂.j 1 δj,k . =− ∂ζk ζ =ζˆ 2ζˆj 1≤l≤N, l=j ζˆj + ζˆl By the inverse-function theorem, we thus have that locally the algebraic system deﬁnes ζ = (ζ1 , . . . , ζN ) uniquely as a smooth function of = (.1 , . . . , .N ) such that ζ ()|=0 = ζˆ . This shows that, for x sufﬁciently large, the Bethe equations (4.12) deﬁne ζ1 (x) < · · · < ζN (x) uniquely as a smooth function of x such that ζj (x) → ζˆj for x → +∞. The upshot is that the solution curve to the Bethe equations swept-out ˜ ordered eigenvalues of the matrix Z(x) of equation (4.13) constitutes the unique realanalytic global solution to the Bethe equations. That with these zeros the wave function satisﬁes our Schrödinger equation is again clear from Subsections 4.1 and 4.2. Theorem 4.3 tells us that—as the value of the position variable x varies along the line—the zeros of the (Jost) wave function of the Schrödinger operator with Bargmann potential sweep out a curve parametrized by the spectral data. This curve is characterized as the unique solution to the system of Bethe equations of equation (4.12). A parameter represention of the solution curve is given in terms of the ordered ˜ eigenvalues of the matrix Z(x) of equation (4.13) with x ∈ R. From equations (4.11a)– (4.11c) we read off explicit formulas for the elementary symmetric functions (cf. equation (2.19)) of the zeros ζ1 (x), . . . , ζN (x): σm ζ1 (x), . . . , ζN (x) τ ρˆ1 , . . . , ρˆl1 + iπ, . . . , ρˆlm + iπ, . . . , ρˆN ; ζˆ1 , . . . , ζˆN | x ζˆl1 · · · ζˆlm , = ˆ1 , . . . , ζˆN | x τ ρ ˆ , . . . , ρ ˆ ; ζ 1 N 1≤l1 <···
DYNAMICS OF ZEROS AND THE SATO FORMULA

299

equation (4.11c). With the aid of equation (4.14) it is possible to perform the differentiations more explicitly. Indeed, it follows from our construction that the potential at issue may be expressed in terms of the zeros as u(x) = 2 N j =1 ζj (x) (cf. Proposition 2.1). From the Hamilton equations in equation (2.11a) and the conservation of the 2 ˆ2 Hamiltonian H from equation (2.8), we then deduce that u(x) = 2 N j =1 (ζj −ζj (x)), which entails upon rewriting in terms of elementary symmetric functions u(x) = 4σ2 ζ1 (x), . . . , ζN (x) − 2σ12 ζ1 (x), . . . , ζN (x) + 2σ1 ζˆ12 , . . . , ζˆN2 . (4.15) Combining with equation (4.14) renders this formula into an explicit expression for the Bargmann potential u(x) = 2∂x2 log τ (x) = 2τ (x)/τ (x) − 2(τ (x)/τ (x))2 evaluated directly in terms of tau functions (i.e., not involving derivatives of tau functions): τ ρˆ1 , . . . , ρˆj + iπ, . . . , ρˆk + iπ, . . . , ρˆN ; ζˆ1 , . . . , ζˆN | x u(x) = 4 ζˆj ζˆk τ ρˆ1 , . . . , ρˆN ; ζˆ1 , . . . , ζˆN | x 1≤j
(4.16) Remark 4.4. By equations (2.6a) and (3.7) it follows that, given a solution to the Bethe equations (4.12) for a ﬁxed value of x (say, x = 0), the full solution curve may in principle be retrieved as the solution to the system of ﬁrst-order differential equations 2 2 ˆ 1≤l≤N ζl − ζj (x) 2 , j = 1, . . . , N. ζj (x) = (4.17) 2 1≤l≤N, l=j ζl (x) − ζj (x) These equations are the solitonic limiting case of differential equations due to Dubrovin for the zeros of the ﬁnite-gap Schrödinger wave functions (see [DMN], [DMKM], [BBEIM], and [BBS]). In a nutshell, the framework of ﬁnite-gap theory (see [DMN], [DMKM], [BBEIM], and [GK]) associates to a hyperelliptic (spectral) curve y 2 = R(E) =

2N+1

E − Ej ,

(4.18)

j =1

a Schrödinger equation with spectral parameter E that admits a Bloch solution with zeros γ1 (x), . . . , γN (x) determined by the Dubrovin equations 2 R(γj (x)) , j = 1, . . . , N. γj (x) = (4.19) 1≤l≤N, l=j γl (x) − γj (x)

300

VAN DIEJEN AND PUSCHMANN

The Bargmann potentials correspond to a degenerate case of the ﬁnite-gap potentials for which the hyperelliptic curve becomes singular with 2N of the roots of R(E) (cf. 2 equation (4.18)) becoming double: R(E) = E N j =1 (E − Ej ) (see [DMN], [Mc], and [DMKM]). The Dubrovin equations (4.19) then pass over into our ﬁrst-order system (4.17) by means of the identiﬁcations γj (x) = ζj2 (x) and Ej = ζˆj2 . (Notice in this connection also that here our spectral parameter is of the form E = z2 .) ˜ We thus have that the ordered eigenvalues of the matrix Z(x) in equation (4.13) provide an explicit solution to the degenerate Dubrovin equations (4.17). In general the solution of the Dubrovin equations (4.19) is obtained through a linearization procedure involving the Abel map associated to the hyperelliptic curve (4.18) (see [DMN], [BBEIM], and [GK]). It is instructive to note that one can also derive the degenerate Dubrovin equations directly from the Bethe equations (4.12). To this end, one differentiates the Bethe system with respect to x, which leads to the relations

ζl (x)

ζˆ 2 − ζl2 (x) 1≤l≤N j

= 1,

j = 1, . . . , N.

(4.20)

These relations may be interpreted as a linear algebraic system for the quantities ζ1 (x), . . . , ζN (x) involving the Cauchy matrix [(ζˆj2 − ζl2 (x))−1 ]1≤j,l≤N . Solving for ζj (x), j = 1, . . . , N (cf. Lemma B.5), readily reproduces the degenerate Dubrovin equations (4.17). 5. Dynamics and scattering of zeros. In summary, it was demonstrated in the ˆ of equation (3.20) deterˆ in the phase space previous section that a point (ζˆ , ρ) mines the spectral data of a Schrödinger operator with Bargmann potential. As the position variable varies along the real line, the locations of the zeros of the associated Jost eigenfunction move along a trajectory governed by the Hamilton ﬂow of the rational Ruijsenaars-Schneider system with harmonic term. The initial condition, determining the positions ζj (x) and velocities ζj (x) of the zeros at x = 0, is in one-to-one correspondence with the spectral data via the action-angle map from equation (3.13) (written in the canonical coordinatization of equations (2.12) ˆ The full trajectory of the zeros is then recovered and (3.19)): (ζ , ρ)(0) = −1 (ζˆ , ρ). as the position part of the inverse image with respect to the action-angle map of the ˆ from equation (3.20). More pragmatically, it means that line (ζˆ , ρˆ −2x ζˆ ), x ∈ R, in ˜ the trajectory of the zeros is given by the eigenvalues of the matrix Z(x) in equation (4.13), x ∈ R. We now exhibit some key properties of these trajectories. The next theorem describes their asymptotical behavior for x → ±∞. For a study of dynamical and asymptotical properties of the zeros of reﬂectionless Schrödinger wave functions in the context of ﬁnite-gap theory (cf. Remark 4.4), the reader is referred to the paper [DMKM] and references therein. Our results below provide a characterization of the zero dynamics in terms of Hamiltonian particle mechanics.

DYNAMICS OF ZEROS AND THE SATO FORMULA

301

ˆ of ˆ be a point in the manifold Theorem 5.1 (Asymptotics of zeros). Let (ζˆ , ρ) equation (3.20), and let (ζ , ρ)(x), x ∈ R, be the integral curve in of equation (2.9) for the Hamilton equations (2.11a) and (2.11b) with initial condition (ζ , ρ)(0) = ˆ that is, (ζ , ρ)(x) = −1 (ζˆ , ρˆ −2x ζˆ ), x ∈ R. Then our integral curve has −1 (ζˆ , ρ); the following straight-line asymptotics for x → ±∞: ζj (x) = ζj± +O e−.|x| , . > 0, ρj (x) = −2xζj± + ρj± +O e−.|x| , where

ζj+ = ζˆj ,

ρj+ = ρˆj + j ζˆ , with

and

ζj− = −ζˆN+1−j ,

ρj− = −ρˆN+1−j + N+1−j ζˆ ,

N (1 + δj,k ) log ζˆj + ζˆk , j ζˆ = k=1

j = 1, . . . , N . Proof. It is immediate from the determinantal formula of Proposition 3.8 that limx→+∞ ζj (x) = ζˆj and that limx→−∞ ζj (x) = −ζˆN+1−j (j = 1, . . . , N ), where the convergence is exponentially fast. To determine the asymptotics of ρj (x), we employ the identities (cf. equations (2.12) and (3.7)) 2 2 ˆ 1≤k≤N ζk − ζj (x) 2 ηj (x) = (5.1a) 2 1≤k≤N, k=j ζk (x) − ζj (x) ζj (x) − ζk (x)−1 = exp(ρj (x)) (5.1b) 1≤k≤N k=j

and (cf. equations (3.9b), (3.15), and (3.19)) ζˆj − ζk (x) ζˆj + ζˆk ηˆ j (x) = ζˆj + ζk (x) 1≤k≤N ζˆj − ζˆk 1≤k≤N

(5.2a)

k=j

ζˆj + ζˆk , = exp ρˆj (x) ˆ ζj − ζˆk

(5.2b)

1≤k≤N k=j

with ρˆj (x) ≡ ρˆj − 2x ζˆj , j = 1, . . . , N . Speciﬁcally, the equality of the right-hand sides of equations (5.2a) and (5.2b) provides us with the leading asymptotics of ζj (x) − ζˆj and ζj (x) + ζˆN+1−j for x → +∞ and x → −∞, respectively. Plugging

302

VAN DIEJEN AND PUSCHMANN

the resulting asymptotics into the equality between the right-hand sides of equations (5.1a) and (5.1b) now readily entails the asymptotics of ρj (x) for x → ±∞ stated by the theorem. The expressions in Theorem 5.1 for the asymptotic quantities (ζ + ,ρ + ) and (ζ − ,ρ − ) ˆ may be interpreted as deﬁning two in terms of the action-angle coordinates (ζˆ , ρ) ˆ → ± , between the action-angle phase space ˆ in symplectomorphisms W± : + + + N equation (3.20) and the asymptotic phase spaces = {(ζ , ρ ) ∈ R × RN | + + 0 < ζ1+ < · · · < ζN+ } equipped with the symplectic form ω+ ≡ N j =1 dρj ∧ dζj and − = {(ζ − , ρ − ) ∈ RN × RN | ζ1− < · · · < ζN− < 0} with the symplectic form − − ω− = N j =1 dρj ∧ dζj , respectively. From the maps W± one obtains the so-called

scattering map S = W+ W−−1 : − → + , linking the straight-line asymptotics of the Ruijsenaars-Schneider integral curve (ζ , ρ)(x) = −1 (ζˆ , ρˆ − 2x ζˆ ), x ∈ R for x → −∞ and x → +∞. (See, e.g., [RS] and [Tr] for further background material on scattering theory.) We then have the following corollary of Theorem 5.1. Corollary 5.2 (Scattering of zeros). The scattering map S for the rational Ruijsenaars-Schneider dynamics with harmonic term characterized by the Hamiltonian H in equation (2.8) is given by the symplectomorphism S : − −→ + with the action (ζ + , ρ + ) = S(ζ − , ρ − ) being deﬁned as  ζ + = −ζ − j N+1−j , j = 1, . . . , N. − ), ρ + = −ρ − + 2 (−ζ j j N+1−j

(5.3a)

(5.3b)

(Here the notation is in correspondence with Theorem 5.1.) Proof. The relation between the asymptotic quantities (ζ + , ρ + ) and (ζ − , ρ − ), deﬁning the action of the scattering map S (= W+ W−−1 ), is obtained from the formulas ˆ from the expression in Theorem 5.1 by eliminating the action-angle coordinates (ζˆ , ρ) for (ζ + , ρ + ) with the aid of the expression for (ζ − , ρ − ). The fact that the resulting map S deﬁnes a symplectomorphism between (− , ω− ) and (+ , ω+ ) is obvious. One sees from the scattering map that the asymptotic actions ζ ± get reﬂected in the origin as one passes from x = −∞ to x = ∞, and one also sees that the phase shifts j (−ζ − ) relating the asymptotic angles ρ + , ρ − break up into a 1-particle contribution of the form 2 log(−2ζj− ) and 2-particle contributions of the form log(−ζj− −ζk− ), k = j . A similar analysis of scattering behavior for the rational Ruijsenaars-Schneider system without harmonic term can be found in [R2] (see also [M1], [M2], and [M3] for corresponding results pertaining to the scattering process of the nonperiodic Toda chain and the rational Calogero-Moser particle system, respectively).

DYNAMICS OF ZEROS AND THE SATO FORMULA

−ζˆ3 | | | | | | |

ζ1 ζ1 ζ1

−ζˆ2 | | | ζ1 | | |

ζ2

ζ1

−ζˆ1 | ζ2 | | | ζ1 |

ζ3 ζ2 ζ2 ζ2 ζ1

ζˆ1 | ζ3 | | | ζ2 |

ζ3

ζ2

ζˆ2 | | | ζ3 | | |

ζ3 ζ3 ζ3

ζˆ3 | | | | | | |

303

ζ32 < ζ22 < ζ12 ζ32 = ζ22 < ζ12 ζ22 < ζ32 < ζ12 ζ22 < ζ32 = ζ12 ζ22 < ζ12 < ζ32 ζ22 = ζ12 < ζ32 ζ12 < ζ22 < ζ32

Figure 1. Schematics of the trajectories of the zeros ζj (x) for N = 3 as the position variable x runs along the real line in the positive direction. The conﬁguration on the ﬁrst line corresponds to x 0 and that on the last line to x ! 0.

Let us now analyze the qualitative behavior of the trajectories of the zeros as functions of the position variable x. We select a Bethe curve ζ (x), x ∈ R, characterized ˆ We know from Subsection 3.1 that by the initial condition (ζ (0), ζ (0)) = −1 (ζˆ , η). ˆ ˆ −ζN < ζ1 (x) < · · · < ζN (x) < ζN . It means that the motion of the zeros ζj (x) is conﬁned to the open interval ] − ζˆN , ζˆN [. Let us divide the half-open interval ] − ζˆN , ζˆN ] in subinterval unions I˜1 , . . . , I˜N of the form (cf. equation (3.6)): I˜j = I˜j+ ∪ I˜j− , with I˜j+ = ζˆj −1 , ζˆj , I˜j− = − ζˆj , −ζˆj −1 , (5.4) − for x 0 and that and ζˆ0 ≡ 0. It is clear from Theorem 5.1 that ζj (x) ∈ I˜N+1−j + ˜ ζj (x) ∈ Ij for x ! 0. The following dynamical picture emerges: As the value of x increases, the zeros ζ1 (x), . . . , ζN (x) increase monotonically (cf. Proposition 2.3) in such a way that at every instant each interval union I˜j , j = 1, . . . , N , contains precisely one zero (cf. Subsection 3.1). This means that for each 1 ≤ j ≤ N − 1 a zero can shift from the interval I˜j−+1 into the (right-) neighboring interval I˜j− if and only if simultaneously a zero shifts from the interval I˜+ into the (right-) neighboring

interval I˜j++1 :

j

I˜j−+1 −→ I˜j− ⇐⇒ I˜j+ −→ I˜j++1 .

(5.5)

In other words, the zeros shift from subinterval to right-neighboring subinterval in even pairs. Figure 1 depicts the dynamics of the zeros for the case N = 3. A typical plot of the N = 3 trajectories is contained in Figure 2. It is very instructive to translate the above characteristics regarding the trajectories of the zeros ζj (x) into dynamical properties of the corresponding Dirichlet eigenvalues ζj2 (x), j = 1, . . . , N . (For ζj (x0 ) = 0 the value ζj2 (x0 ) is an eigenvalue of our Schrödinger operator with vanishing Dirichlet boundary condition at

304

VAN DIEJEN AND PUSCHMANN

2 1 −4

−2

2

x

4

−1 −2

Figure 2. The trajectories of the zeros ζj (x) for N = 3 (with ζˆ1 = 1, ζˆ2 = 2, ζˆ3 = 2(1/2) and ηˆ 1 = 1, ηˆ 2 = 102 , ηˆ 3 = 104 ). The zeros cross the intermediate asymptotic values in even pairs.

x = x0 in the Hilbert space L2 (] − ∞, x0 ]) if ζj (x0 ) > 0 and in the Hilbert space L2 ([x0 , ∞[) if ζj (x0 ) < 0.) On the one hand, we have for large positive x that 0 < ζ1 (x) < ζˆ1 < ζ2 (x) < ζˆ2 < · · · < ζN (x) < ζˆN and thus that ζ12 (x) < · · · < ζN2 (x). On the other hand, we have for large negative x that −ζˆN < ζ1 (x) < −ζˆN−1 < ζ2 (x) < · · · < −ζˆ1 < ζN (x) < 0 and thus that ζN2 (x) < · · · < ζ12 (x). It means that, as x runs from −∞ to +∞ along the real axis, the Dirichlet eigenvalues cross each other. See Figure 3 for a plot of the Dirichlet eigenvalues associated to the zeros of Figure 2. To study these level crossings let us pick a value of x in general position on the line. We may then have both positive and negative zeros: ζ1 (x) < · · · < ζM (x) < 0 ≤ ζM+1 (x) < · · · < ζN (x) with M ∈ {0, . . . , N }. Hence, 2 (x) < · · · < ζ 2 (x) and that ζ 2 2 we have in this situation that ζM 1 M+1 (x) < · · · < ζN (x). The corresponding intertwining of the discrete spectrum ζˆ12 , . . . , ζˆN2 and the Dirichlet eigenvalues ζ12 (x), . . . , ζN2 (x) becomes therefore of the form (suppressing the argument x) ζσ21 ≤ ζˆ12 ≤ ζσ22 ≤ · · · ≤ ζˆj2−1 ≤ ζσ2j ≤ ζˆj2 ≤ ζσ2j +1 ≤ · · · ≤ ζˆσ2N−1 ≤ ζσ2N ≤ ζˆN2 , where σ denotes a permutation of the indices {1, . . . , N } for which σj > σk if j < k with σj , σk ∈ {1, . . . , M} and σj < σk if j < k with σj , σk ∈ {M + 1, . . . , N }. It means, in particular, that the ordering of the Dirichlet eigenvalues reads: ζσ21 (x) ≤ ζσ22 (x) ≤ · · · ≤ ζσ2N (x). An equality ζσ2j (x) = ζσ2k (x) (σj > σk ) occurs if and only if ζσj (x)+ζσk (x) = 0; this can happen only when σj > M ≥ σk and |j −k| = 1. (Notice that M = σ1 if ζσ1 (x) < 0 and M = σ1 − 1 if ζσ1 (x) ≥ 0.)

DYNAMICS OF ZEROS AND THE SATO FORMULA

305

6 5 4 3 2 1 −4

−2

0

2 x

4

Figure 3. The trajectories of the Dirichlet eigenvalues ζj2 (x) for N = 3 (with ζˆ1 = 1, ζˆ2 = 2, ζˆ3 = 2(1/2) and ηˆ 1 = 1, ηˆ 2 = 102 , ηˆ 3 = 104 ). The level crossings occur at the asymptotic values ζˆj2 , j = 1, . . . , N − 1 (which, together with the extremal (ground-state) eigenvalue ζˆ 2 , constitute the discrete spectrum of Schrödinger N

operator in the Hilbert space L2 (] − ∞, +∞[)).

Let us encode the conﬁguration ζσ21 (x) < ζσ22 (x) < · · · < ζσ2N (x) by the permutation (σ1 σ2 · · · σN ). A level crossing ζσ2j (x −.) < ζσ2j +1 (x −.) → ζσ2j +1 (x +.) < ζσ2j (x +.) (with . > 0), which is possible if and only if σj ≥ σ1 > σj +1 , now manifests itself as a transposition of the form σ1 · · · σj σj +1 · · · σN −→ σ1 · · · σj +1 σj · · · σN . (5.6) By varying x along the real axis, the consecutively ordered conﬁgurations of the Dirichlet eigenvalues give rise to a path of permutations from (N N − 1 · · · 2 1) (for x 0) to (1 2 · · · N − 1 N) for (x ! 0). Two subsequent elements in a path are related by a transposition of the form in equation (5.6) with σj ≥ σ1 > σj +1 (or by a composition of such transpositions in the case of simultaneous crossings). In Figure 4 the possible permutation paths for N = 3 and N = 4 are given, assuming simultaneous crossings do not occur. The cases of simultaneous crossings are obtained via contraction; for example, “(3 2 1) → (2 3 1)” + “(2 3 1) → (2 1 3)” = “(3 2 1) → (2 1 3)”. For general N a rich branching structure of the possible paths emerges, corresponding to the different orders in which the sequence of level crossings can occur. For given N, the total number of distinct conﬁgurations generated by these paths is given by N N −1 = 2N−1 . (5.7) M −1 M=1

306

VAN DIEJEN AND PUSCHMANN

(3 2 1)

(4 3 2 1)

/ (3 4 2 1)

/ (2 3 1)

/ (2 1 3)

/ (1 2 3)

(2; 3 4 1) HH HH vv v HH v v HH v v H# v v / (3 2 4 1) (2 3 1 4) HH v; HH v v HH vv HH vv H# v v (3 2 1 4)

/ (2 1 3 4)

/ (1 2 3 4)

Figure 4. Permutation paths for N = 3 and N = 4, encoding the possible consecutively ordered conﬁgurations of the Dirichlet eigenvalues ζj2 (x) as they appear when x varies along the real line in the positive direction. Here the permutation (σ1 σ2 · · · σN ) corresponds to the conﬁguration ζσ21 (x) < ζσ22 (x) < · · · < ζσ2N (x), and the arrows represent the respective level crossings σj ↔ σj +1 .

The above dynamical considerations condense into the following properties of the Dirichlet spectrum of reﬂectionless Schrödinger operators. Proposition 5.3 (Location of zeros). Let (x, z) be the Jost eigenfunction of a Schrödinger operator with a Bargmann potential on the line, whose discrete spectrum is characterized by the data 0 < ζˆ1 < · · · < ζˆN (cf. Theorems 4.1 and 4.2 for explicit parametrizations of the potentials and Jost functions under consideration). Then the roots z = ζj (x), j = 1, . . . , N , of the equation (x, z) = 0 are distributed over the open interval ] − ζˆN , ζˆN [ in such a way that each of the interval unions I˜1 , . . . , I˜N in equation (5.4) contains precisely one root. Furthermore, the roots ζj (x) increase monotonically as a function of x. As a consequence of Proposition 5.3, we recover a well-known result pertaining to the interlacing of the Dirichlet eigenvalues and the discrete spectrum (see, e.g., [CL], [DT], and [ELZ]). Corollary 5.4 (Dirichlet eigenvalues). For the Schrödinger operators of Proposition 5.3, the Dirichlet eigenvalues ζ12 (x0 ) ≤ · · · ≤ ζN2 (x0 ) interlace the discrete spectrum ζˆ12 < · · · < ζˆN2 : ζ12 (x0 ) ≤ ζˆ12 ≤ ζ22 (x0 ) ≤ ζˆ22 ≤ · · · ≤ ζN2 (x0 ) ≤ ζˆN2 , with equality holding if and only if ζj2 (x0 ) = ζˆj2 = ζj2+1 (x0 ). Notice that in Corollary 5.4 we have renumbered the Dirichlet eigenvalues such that they are ordered from small to large.

307

DYNAMICS OF ZEROS AND THE SATO FORMULA

6. The Korteweg–de Vries hierarchy. It is well known that there is a close relation between Schrödinger equations with reﬂectionless potentials and the soliton solutions of the Korteweg–de Vries equation (see [DT], [AS], [N], [NMPZ], and [DMKM]). In this section we exploit this connection to characterize the motion of the zeros of the Jost function for a Schrödinger operator with a Korteweg–de Vries soliton potential in terms of Ruijsenaars-Schneider dynamics. It follows from Proposition 3.12 that the Hamiltonians Ᏼm =

1 Tr(Lm ), m

m = 1, . . . , M,

(6.1)

Poisson-commute amongst themselves. As a consequence, the Hamilton ﬂows generated by Ᏼ1 , . . . , ᏴM commute. In other words, the multitime Hamiltonian system ∂ζj ∂ Ᏼm = , ∂tm ∂ρj

∂ρj ∂ Ᏼm =− , ∂tm ∂ζj

(6.2)

with 1 ≤ j ≤ N and 1 ≤ m ≤ M, is compatible and has a unique solution ζj (t1 , . . . , tM ) and ρj (t1 , . . . , tM ) given the initial condition ζj (0, . . . , 0) = ζj (0) and ρj (0, . . . , 0) = ρj (0). In action-angle coordinates the Hamiltonians from equation (6.1) become N

Ᏼm =

1 2m ζˆj , m

m = 1, . . . , M.

(6.3)

j =1

The transformed equations ∂tm ζˆj = ∂ρˆj Ᏼm and ∂tm ρˆj = −∂ζˆj Ᏼm are thus linear: ∂ ζˆj = 0, ∂tm

∂ ρˆj = −2ζˆj2m−1 , ∂tm

m = 1, . . . , M,

(6.4)

and have as solution ζˆj (t1 , . . . , tM ) = ζˆj (0), ρˆj (t1 , . . . , tM ) = ρˆj (0) − 2

(6.5a) tm ζˆj2m−1 (0),

(6.5b)

1≤m≤M

j = 1, . . . , N . Transforming back with the inverse of the action-angle transformation in equation (3.13) readily produces the solution to the system in equation (6.2) (cf. Subsection 3.3). For the associated Schrödinger potential u(t1 , . . . , tM ) = 2∂t1 (ζ1 (t1 , . . . , tM ) + · · · + ζN (t1 , . . . , tM )) we then obtain in the same way as before (cf. Theorems 4.1 and 4.2) u(t1 , . . . , tM ) = 2∂t21 log τ (t1 , . . . , tM )

(6.6a)

308

VAN DIEJEN AND PUSCHMANN

with

τ (t1 , . . . , tM ) = det 1 + exp −2

M

ˆ 2m−1 N tm Z

(6.6b)

m=1

ζˆj + ζˆk exp ρˆj t1 , . . . , tM , = ˆ ζ − ζˆk j ∈J J ⊂{1,...,N} j ∈J, k∈J j

(6.6c)

where ρˆj (t1 , . . . , tM ) = ρˆj − 2

M m=1

tm ζˆj2m−1 .

(6.6d)

(Here the notation is the same as in Theorems 4.1 and 4.2 and we have suppressed the argument of ζˆj (0) and ρˆj (0).) Furthermore, the corresponding wave function, solving the Schrödinger equation (∂t21 + u(t1 , . . . , tM ) − z2 ) = 0 subject to the boundary 2m−1 ) for t → +∞, is given by (cf. condition (t1 , . . . , tN ; z) → exp( M 1 m=1 tm z equation (4.3)) M M z − ζj (t1 , . . . , tM ) 2m−1 tm z . (6.7a) (t1 , . . . , tM ; z) = exp z − ζˆj m=1

j =1

Invoking of our solution to the Hamiltonian system (6.2) entails (cf. Theorems 4.1 and 4.2) (t1 , . . . , tM ; z) = exp

M

M 2m−1 det 1 + z1 + Z ˆ z1 − Z ˆ −1 exp − 2 ˆ N tm Z

tm z2m−1

m=1

= exp

M

det 1 + exp − 2

×

m=1

M m=1

ˆ 2m−1 N tm Z (6.7b)

tm z2m−1

m=1

J ⊂{1,...,N} j ∈J

z+ζˆj

z−ζˆj

J ⊂{1,...,N} j ∈J, k∈J

ζˆj +ζˆk ρˆj (t1 , . . . , tM ) ζˆj −ζˆk exp j ∈J, k∈J

j ∈J

ζˆj + ζˆk ρˆj (t1 , . . . , tM ) ζˆj − ζˆk exp

.

j ∈J

(6.7c) The above formula for u(t1 , . . . , tM ) is precisely the Hirota formula for the celebrated N -soliton solution of the KdV hierarchy written in terms of the tau function

DYNAMICS OF ZEROS AND THE SATO FORMULA

309

(see [H], [SCM], and [N]). In particular, for M = 2 with t1 = x and t2 = t, we have that u(x, t) of equations (6.6a)–(6.6d) amounts to the N-soliton solution of KdV equation 3 1 ut = uux + uxxx . 2 4

(6.8)

The corresponding wave function (t1 , . . . , tN ; z) is often referred to as the so-called Baker function of the associated linear problem. It is well known from the work of the Kyoto school that this Baker function can also be expressed in terms of the tau function by means of a formula originally due to Sato (see [S], [SS], [DKJM], [JM], [SW], [OSTT], and [Mo]). Our formulas in equations (6.7b) and (6.7c) are equivalent to the Sato formula (upon specialization to the solitonic regime of the KdV hierarchy) (see [BBS] and [DK]). In summary, we have derived the following theorem. Theorem 6.1 (Zero motion for the KdV hierarchy). Let u(t1 , . . . , tM ) be an N soliton solution of the KdV hierarchy. Then the Schrödinger equation 2 ∂t1 + u(t1 , . . . , tM ) − z2 ψ = 0 (6.9a) has a solution of the form M tm z2m−1 z − ζ1 (t1 , . . . , tM ) · · · z − ζN (t1 , . . . , tM ) , ψ(t1 , . . . , tN ; z) = exp m=1

(6.9b)

with the zeros ζ1 (t1 , . . . , tM ), . . . , ζN (t1 , . . . , tM ) moving in accordance with the equations of motion (6.2) for the rational Ruijsenaars-Schneider hierarchy with harmonic term. Reversely, if a Schrödinger equation (6.9a) has a solution of the form (6.9b) with the zeros ζ1 (t1 , . . . , tM ), . . . , ζN (t1 , . . . , tM ) moving in accordance with the rational Ruijsenaars-Schneider hierarchy with harmonic term (6.2), then the potential u(t1 , . . . , tN ) (= 2∂t1 (ζ1 (t1 , . . . , tN ) + · · · + ζN (t1 , . . . , tN ))) constitutes an N-soliton solution of the Korteweg–de Vries hierarchy. Theorem 6.1 describes a relation between the KdV hierarchy and the rational Ruijsenaars-Schneider system with harmonic term. Different relations between the motion of KdV solitons and ﬁnite-dimensional integrable systems have been considered previously by Moser [M4] (who connects the KdV solitons to the motion of a Neumann system on a sphere) and by Ruijsenaars and Schneider [RSc] (who encode the KdV solitons in terms of the motion of a hyperbolic Ruijsenaars-Schneider system). Our considerations produce a formula for the solitonic tau function and Baker function of the KdV hierarchy in terms of the Lax matrix of the hyperbolic Ruijsenaars-Schneider system (cf. Subsection 4.1). In the case of the tau function,

310

VAN DIEJEN AND PUSCHMANN

such a formula was obtained previously by Ruijsenaars and Schneider [RSc]. Similar formulas for the tau function and the associated Baker function for the rational solutions of the full KP hierarchy in terms of rational Calogero-Moser Lax matrices were presented recently by Shiota [Sh] (for the tau function) and Wilson [Wi] (for the Baker function). Appendices Appendix A. Proof of the no-crossing Lemma 2.2. In this appendix we prove Lemma 2.2, which says that a real-analytic, nonconstant solution to the differential system ζj + 2ζj ζj

2ζj ζk

=

1≤k≤N, k=j

ζj − ζ k

,

j = 1, . . . , N,

(A.1)

is such that ζj (x) = ζk (x) for all x ∈ R and j = k. In other words, the solution functions ζ1 (x), . . . , ζN (x) do not cross each other. To prove the lemma we demonstrate that the assumption that crossings do occur entails a contradiction. Without restriction one may assume that the crossing occurs for x = 0 and that the solution functions ζj (x) are distinct for x ∈] − ., .[\{0} with . > 0 sufﬁciently small. (Two (or more) solution functions cannot be identical as analytic functions of x because then the differential system (A.1) would require them to be constant.) More precisely, we may assume that ζj (x) = ζk (x),

1 ≤ j < k ≤ N, for x ∈] − ., .[\{0}

and, furthermore, that there is an index set J ⊂ {1, . . . , N } containing at least two elements such that ζj (0) = c,

j ∈ J,

and

ζj (0) = c,

j ∈ J.

By analyticity there exists an i ∈ J such that for all j ∈ J , |ζi (x) − c| ≤ |ζj (x) − c|

for x ∈] − δ, δ[

(A.2)

with 0 < δ < . sufﬁciently small. Basically, ζi (x) is the solution function that converges fastest to the crossing point c as x → 0. (Notice, however, that such a solution function may not be unique; indeed, in principle it might happen that we have two solution functions ζi (x), ζi (x) satisfying the estimate (A.2), which symmetrically approach the crossing point c from above and below such that ζi (x) + ζi (x) = 2c.) By analyticity and nonconstancy we have for j = 1, . . . , N that ζj (x) = ζj (0) +

x mj (mj ) ζ (0) mj ! j

+ o x mj ,

mj ≥ 1,

DYNAMICS OF ZEROS AND THE SATO FORMULA (mj )

with ζj

(0) = 0, and we have for j ∈ J that x nj (nj ) (n ) ζj (x) − ζi (x) = ζj (0) − ζi j (0) nj !

(n )

311

+ o x nj ,

(n )

with ζj j (0) = ζi j (0) and nj ≥ mi ≥ mj (cf. equation (A.2)). Armed with these Taylor expansions we determine the asymptotic behavior of the ith equation of the system (A.1). We get for the left-hand side:  + o(1) if mi = 1, ζi (0) + 2ζi (0)ζi (0) ζi (x) + 2ζi (x)ζi (x) = (A.3)  mi −2 (mi ) x ζi (0)/(mi − 2)! + o x mi −2 if mi ≥ 2, and for the right-hand side: 2 ζ (x)ζ (x) i

1≤k≤N, k=i

k

ζi (x) − ζk (x)

(mi ) (m ) m +m −2 ζi (0) ζk k (0) 2 x mi +mk −2 i k = +o x (mi − 1)!(mk − 1)! ζi (0) − ζk (0) k∈J 2 mk x mi −2 (m ) m −2 i i ζ (0) + o x − (mi − 1)! i

(A.4)

k∈J mk <mi

2 nk ! x 2mi −2−nk ζ (mi ) (0) ζ (mk ) (0) 2m −2−n i k i k +o x + . (n ) (n ) ((mi − 1)!)2 ζ k (0) − ζ k (0) k∈J, k=i mk =mi

i

k

We observe that it follows from our deﬁnition of ζi that (mk )

ζk

(0)

(n ) (n ) ζi k (0) − ζk k (0)

<0

for k ∈ J \ {i} with mk = mi .

(A.5)

Indeed, the estimate (A.2) implies that for 0 < x < . we are in one of the following (m ) (n ) situations: (i) c < ζi (x) < ζk (x) or ζi (x) < c < ζk (x) ⇒ ζk k (0) > 0 and ζi k (0) < (n ) (m ) (n ) ζk k (0), (ii) ζk (x) < ζi (x) < c or ζk (x) < c < ζi (x) ⇒ ζk k (0) < 0 and ζi k (0) > (n ) ζk k (0). By comparing the lowest-order behavior of the left-hand side and right-hand side for x → 0—using the inequality (A.5) and nk ≥ mi —one concludes that the righthand side contains nonzero terms of order O(x mi −2 ) and O(x 2mi −2−nk ) originating from the third and fourth lines of equation (A.4). These terms do not match with the O(1) or O(x mi −2 ) terms on the left-hand side given by equation (A.3). (The signs of the O(x mi −2 ) terms on both sides are not compatible.) We conclude that the set J must be empty, that is, that the solution functions cannot cross each other without violating our analyticity or nonconstancy assumption.

312

VAN DIEJEN AND PUSCHMANN

Appendix B. Some Cauchy matrix identities. In this appendix we have collected some matrix identities associated to the well-known Cauchy matrix. These identities were essential in the construction of the action-angle transformation for the rational Ruijsenaars-Schneider system with harmonic term presented in Section 3. Most of the identities hinge on the following partial fraction decomposition for rational functions with simple poles that are regular at inﬁnity. Lemma B.1 (Partial fractions decomposition). Let a1 , . . . ,aM and b1 , . . . , bN , with M ≤ N, be M + N points in the complex plane such that bn = bl if n = l. Then one has that N 1≤m≤M (x − am ) 1≤m≤M (bl − am ) = δN,M + (x − bl ) 1≤n≤N, n=l (bl − bn ) 1≤n≤N (x − bn ) l=1

as a rational identity in x. Proof. This is immediate from Liouville’s theorem upon inferring that the residues of the simple poles at x = bn , n = 1, . . . , N , and the asymptotics for x → ∞ are the same on both sides of the equation. Let α = (α1 , . . . , αN ) and β = (β1 , . . . , βN ) denote (complex) vectors with components subject to the genericity conditions: αj = αk , βj = βk for 1 ≤ j < k ≤ N, and αj = −βk for 1 ≤ j , k ≤ N. We form the matrices C(α, β) and D(α, β) with elements given by 1 1≤l≤N (αj + βl ) Cj,k (α, β) = , Dj,k (α, β) = δj,k , (B.1) αj + β k 1≤l≤N, l=j (αj − αl ) 1 ≤ j , k ≤ N . The matrix C(α, β) is commonly referred to as the Cauchy matrix. The following proposition provides a formula for the inverse of the Cauchy matrix. Proposition B.2 (Inverse of Cauchy matrix). We have that C−1 (α, β) = D(β, α)C(β, α)D(α, β) as a rational identity in the components of α and β. Proof. In components, the relation C(α, β)D(β, α)C(β, α)D(α, β) = 1 reads N l=1

(αk + βn ) 1 1 1≤m≤N (βl + αm ) 1≤n≤N = δj,k , αj + βl 1≤n≤N, n=l (βl − βn ) βl + αk 1≤m≤N, m=k (αk − αm )

1 ≤ j , k ≤ N. This identity follows from Lemma B.1 with M = N −1 via the substitutions x = αk , am = αm for m = 1, . . . , j − 1 and am = αm+1 for m = j, . . . , N − 1, and bn = −βn for n = 1, . . . , N .

DYNAMICS OF ZEROS AND THE SATO FORMULA

313

From the formula for the inverse of C(α, β) one obtains the famous Cauchy determinantal formula (and vice versa) (see [We, p. 202]). Corollary B.3 (Cauchy determinant). We have that 1≤j
Uj,k (ζ , ζˆ ) =

1≤l≤N

1 ≤ j , k ≤ N, is orthogonal: U−1 (ζ , ζˆ ) = t U(ζ , ζˆ ) = U(ζˆ , ζ ). Proof. This is immediate from Proposition B.2 and the observation that 2 2 2 U ζ , ζˆ = −D1/2 − ζ 2 , ζˆ C − ζ 2 , ζˆ D1/2 ζˆ , −ζ 2 . (The condition (ζ , ζˆ ) ∈ ᏺ0 ensures that the arguments of the square roots are positive.) Let e = (1, . . . , 1) and η = (η1 , . . . , ηN ) be N-dimensional vectors with 2 2 ˆ 1≤l≤N ζl − ζj 2 ˆ2 2 . ηj = Dj,j − ζ , ζ = 2 1≤l≤N, l=j ζl − ζj

(B.2)

314

VAN DIEJEN AND PUSCHMANN

Here it is again assumed that (ζ , ζˆ ) is a point of the manifold ᏺ0 characterized by Deﬁnition 3.2. The following lemma states in essence that the columns of the matrix D(α, β)C(α, β) sum up to 1. Lemma B.5. One has that 2 η · C − ζ 2 , ζˆ = e, or equivalently,

ζˆl2 − ζj2 2 = 1, 2 1≤l≤N, l=j ζl − ζj j =1

N

1≤l≤N, l=k

k = 1, . . . , N . Proof. The proof is immediate from Lemma B.1 with M = N, x = ζˆk2 , and an = ζˆn2 , bn = ζn2 , n = 1, . . . , N . The next proposition shows that our orthogonalized Cauchy matrix U(ζ , ζˆ ) diagonalizes the Lax matrix for the rational Ruijsenaars-Schneider system with harmonic term. Proposition B.6. Let (ζ , ζˆ ) be a point of the manifold ᏺ0 characterized by Definition 3.2. Furthermore, let L ζ , ζˆ = Z2 + t η1/2 ⊗ η1/2 , where Z = diag(ζ1 , . . . , ζN ) and where η = (η1 , . . . , ηN ) has components given by equation (B.2). Then the orthogonal matrix U(ζ , ζˆ ) of Corollary B.4 diagonalizes the symmetric matrix L(ζ , ζˆ ): ˆ = diag ζˆ1 , . . . , ζˆN . U−1 LU = Zˆ 2 , with Z Proof. We have 2 2 2 LU = Z2 + t η1/2 ⊗ η1/2 D1/2 − ζ 2 , ζˆ C ζ 2 , −ζˆ D1/2 ζˆ , −ζ 2 2 2 2 Lemma B.5 1/2 = D − ζ 2 , ζˆ Z2 C ζ 2 , −ζˆ − t e ⊗ e D1/2 ζˆ , −ζ 2 2 2 2 = D1/2 − ζ 2 , ζˆ C ζ 2 , −ζˆ Zˆ 2 D1/2 ζˆ , −ζ 2 ˆ 2. = UZ The construction of the action-angle coordinates for the rational RuijsenaarsSchneider system with harmonic term is based on the calculation of the matrix elements of B(ζ , ζˆ ) = C−1 (−ζ , ζˆ )C(ζ , ζˆ ). Lemma B.7. Let ζ , ζˆ be N-dimensional vectors with components that satisfy the genericity conditions ζj = ζk , ζˆj = ζˆk for 1 ≤ j < k ≤ N, and ζj = ±ζˆk for 1 ≤

315

DYNAMICS OF ZEROS AND THE SATO FORMULA

j = k ≤ N. The elements of the (N × N)-matrix B(ζ , ζˆ ) = C−1 (−ζ , ζˆ )C(ζ , ζˆ ) are given by ζˆj − ζl ζˆk + ζˆl ˆ Bj,k ζ , ζ = , ζˆk + ζl 1≤l≤N, l=j ζˆj − ζˆl 1≤l≤N 1 ≤ j , k ≤ N. Proof. We get from Proposition B.2 that B ζ , ζˆ = D ζˆ , −ζ C ζˆ , −ζ D − ζ , ζˆ C ζ , ζˆ . In components, this gives Bj,k

N ˆ ζˆj − ζl 1≤m≤N, m=j ζm − ζl ζ , ζˆ = . ˆ ˆ ζˆk + ζl 1≤l≤N, l=j ζj − ζl l=1 1≤n≤N, n=l ζn − ζl

1≤l≤N

To complete the proof, the partial fraction decomposition of Lemma B.1 is employed to rewrite the sum as a product. The relevant substitutions are: M = N − 1, x = ζˆk , am = −ζˆm for m = 1, . . . , j −1 and am = −ζˆm+1 for m = j, . . . , N −1, and bn = −ζn for n = 1, . . . , N . With the aid of Lemma B.7 we can compute the elements of the matrix N in equation (3.8). Proposition B.8. Let (ζ , ζˆ ) be a point of the manifold ᏺ0 characterized by Definition 3.2, and let N be the (N × N)-matrix of the form N = (1 − R)(1 + R)−1 with

ˆ −1/2 U−1 ZUZˆ −1/2 , R=Z

where Z = diag(ζ1 , . . . , ζN ), Zˆ = diag(ζˆ1 , . . . , ζˆN ), and U is taken from Corollary B.4 (here we suppress the arguments). Then the elements of the matrix N are given explicitly by ˆ ˆ 1/2 1/2 1/2 ζj ζk , 1 ≤ j, k ≤ N, Nj,k = 2ηˆ j ηˆ k ζˆj + ζˆk with ηˆ j =

ζˆj − ζl ζˆj + ζˆl , ˆj + ζl ˆj − ζˆl ζ ζ 1≤l≤N 1≤l≤N

j = 1, . . . , N.

l=j

2

2

2

Proof. From the representation U = D1/2 (−ζ 2 , ζˆ )C(ζ 2 , −ζˆ )D1/2 (ζˆ , −ζ 2 ), it ˆ 1/2 R = ZCD1/2 Z ˆ −1/2 , where C = C(ζ 2 , −ζˆ 2 ) and D = D(ζˆ 2 , is clear that CD1/2 Z

316

VAN DIEJEN AND PUSCHMANN

−ζ 2 ). Elementary manipulations then entail that N = (1 + R)−1 (1 − R) ˆ 1/2 −1 CD1/2 Z ˆ 1/2 (1 − R) = (1 + R)−1 CD1/2 Z ˆ 1/2 + ZCD1/2 Z ˆ −1/2 −1 CD1/2 Z ˆ 1/2 − ZCD1/2 Z ˆ −1/2 = CD1/2 Z ˆ 1/2 CZ ˆ −1/2 ˆ + ZC −1 CZˆ − ZC D1/2 Z = D−1/2 Z ˆ 1/2 C−1 − ζ , ζˆ C ζ , ζˆ D1/2 Z ˆ −1/2 . = D−1/2 Z Invoking of Lemma B.7 now completes the proof. Acknowledgments. This paper was written for the most part when the ﬁrst author was visiting the Mathematical Sciences Research Institute (MSRI) at Berkeley, taking part in the program on random matrix models and their applications in spring 1999. It is a pleasure to thank the organizers for their invitation and the institute for its hospitality. We would furthermore like to thank Professors J. Harnad, A. N. Kirillov, M. Jimbo, and K. T.-R. McLaughlin for some useful conversations and communications. Finally, thanks are also due to the referee for constructive remarks and pointing out [DMKM]. References [AS]

[AMM]

[A] [BBS] [BBEIM]

[BC] [BR] [Ca]

[CC] [CL]

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM Stud. Appl. Math. 4, Society for Industrial and Applied Mathematics, Philadelphia, 1981. H. Airault, H. P. McKean, and J. Moser, Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem, Comm. Pure Appl. Math. 30 (1977), 95–148. V. I. Arnold, Mathematical Methods of Classical Mechanics, 2d ed., Grad. Texts in Math. 60, Springer, New York, 1989. O. Babelon, D. Bernard, and F. A. Smirnov, Quantization of solitons and the restricted sine-Gordon model, Comm. Math. Phys. 182 (1996), 319–354. E. D. Belokolos, A. I. Bobenko, V. Z. Enol’skii, A. R. Its, and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1994. M. Bruschi and F. Calogero, The Lax representation for an integrable class of relativistic dynamical systems, Comm. Math. Phys. 109 (1987), 481–492. M. Bruschi and O. Ragnisco, On new solvable many-body dynamical systems with velocity-dependent forces, Inverse Problems 4 (1988), L15–L20. F. Calogero, Motion of poles and zeros of special solutions of nonlinear and linear partial differential equations and related “solvable” many-body problems, Nuovo Cimento B (11) 43 (1978), 177–241. D. V. Chudnovsky and G. V. Chudnovsky, Pole expansions of nonlinear partial differential equations, Nuovo Cimento B (11) 40 (1977), 339–353. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.

DYNAMICS OF ZEROS AND THE SATO FORMULA [DKJM]

[DT] [DK] [DMKM]

[DMN]

[ELZ] [GK]

[H] [JM] [Ka] [K] [KWZ]

[Kr]

[Mc]

[Mo]

[M1]

[M2] [M3]

[M4]

317

E. Date, M. Kashiwara, M. Jimbo, and T. Miwa, “Transformation groups for soliton equations” in Nonlinear Integrable Systems—Classical Theory and Quantum Theory (Kyoto, 1981), ed. M. Jimbo and T. Miwa, World Scientiﬁc, Singapore, 1983, 39–119. P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32 (1979), 121–251. J. F. van Diejen and A. N. Kirillov, A combinatorial formula for the associated Legendre functions of integer degree, Adv. Math. 149 (2000), 61–88. B. A. Dubrovin, T. M. Malanyuk, I. M. Krichever, and V. G. Makhankov, Exact solutions of the time-dependent Schrödinger equation with self-consistent potentials, Soviet J. Particles and Nuclei 19 (1988), 252–269. B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, Non-linear equations of Kortewegde Vries type, ﬁnite-zone linear operators, and Abelian varieties, Russian Math. Surveys 31 (1976), 59–146. N. M. Ercolani, C. D. Levermore, and T. Zhang, The behavior of the Weyl function in the zero-dispersion KdV limit, Comm. Math. Phys. 183 (1997), 119–143. P. G. Grinevich and I. M. Krichever, “Algebraic-geometry methods in soliton theory” in Soliton Theory: A Survey of Results, ed. A. Fordy, Nonlinear Sci. Theory Appl., Manchester Univ. Press, Manchester, 1990, 354–400. R. Hirota, “Direct methods in soliton theory” in Solitons, ed. R. K. Bullough and P. J. Caudrey, Topics Current Phys. 17, Springer, Berlin, 1980, 157–176. M. Jimbo and T. Miwa, Solitons and inﬁnite-dimensional Lie algebras, Publ. Res. Inst. Math. Sci. 19 (1983), 943–1001. T. Kato, Perturbation Theory for Linear Operators, 2d ed., Grundlehren Math. Wiss. 132, Springer, Berlin, 1976. I. M. Krichever, Rational solutions of the Zakharov-Shabat equations and completely integrable systems of N particles on the line, J. Soviet Math. 21 (1983), 335–345. I. M. Krichever, P. Wiegmann, and A. Zabrodin, Elliptic solutions to difference non-linear equations and related many-body problems, Comm. Math. Phys. 193 (1998), 373–396. M. D. Kruskal, “The Korteweg-de Vries equation and related evolution equations” in Nonlinear Wave Motion (Potsdam, N.Y., 1972), ed. A. C. Newell, Lecture Notes in Appl. Math. 15, Amer. Math. Soc., Providence, 1974, 61–83. H. P. McKean, “Theta functions, solitons, and singular curves” in Partial Differential Equations and Geometry (Park City, Utah, 1977), ed. C. I. Byrnes, Lecture Notes in Pure and Appl. Math. 48, Dekker, New York, 1979, 237–254. P. van Moerbeke, “Integrable foundations of string theory” in Lectures on Integrable Systems (Sophia-Antipolis, 1991), ed. O. Babelon, P. Cartier, and Y. KosmannSchwarzbach, World Scientiﬁc, River Edge, N.J., 1994, 163–267. J. Moser, “Finitely many mass points on the line under the inﬂuence of an exponential potential—an integrable system” in Dynamical Systems: Theory and Applications (Seattle, Wash., 1974), ed. J. Moser, Lecture Notes in Phys. 38, Springer, Berlin, 1975, 467–497. , Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math. 16 (1975), 197–220. , “Various aspects of integrable Hamiltonian systems” in Dynamical Systems (Bressanone, 1978), ed. J. Guckenheimer, J. Moser, and S. E. Newhouse, Progr. Math. 8, Birkhäuser, Boston, 1980, 233–289. , Integrable Hamiltonian Systems and Spectral Theory, Lezioni Fermiane, Accademia Nazionale dei Lincei, Rome, 1983.

318 [N]

[NMPZ]

[OSTT] [P] [RS] [R1] [R2] [RSc] [S] [SS]

[Sc] [SCM] [SW] [Sh] [Th] [Tr] [We] [Wi]

VAN DIEJEN AND PUSCHMANN A. C. Newell, Solitons in Mathematics and Physics, CBMS-NSF Regional Conf. Ser. in Appl. Math. 48, Society for Industrial and Applied Mathematics, Philadelphia, 1985. S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method, Contemp. Soviet Math., Consultants Bureau, New York, 1984. Y. Ohta, J. Satsuma, D. Takahashi, and T. Tokihiro, An elementary introduction to Sato theory, Progr. Theoret. Phys. Suppl. 94 (1988), 210–241. A. M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, Vol. I, Birkhäuser, Basel, 1990. M. Reed and B. Simon, Methods of Modern Mathematical Physics, III: Scattering Theory, Academic Press, New York, 1979. S. N. M. Ruijsenaars, Complete integrability of relativistic Calogero-Moser systems and elliptic function identities, Comm. Math. Phys. 110 (1987), 191–213. , Action-angle maps and scattering theory for some ﬁnite-dimensional integrable systems, I: The pure soliton case, Comm. Math. Phys. 115 (1988), 127–165. S. N. M. Ruijsenaars and H. Schneider, A new class of integrable systems and its relation to solitons, Ann. Physics 170 (1986), 370–405. M. Sato, Soliton equations as dynamical systems on an inﬁnite dimensional Grassmann manifolds, S¯urikaisekikenky¯usho K¯oky¯uroku 439 (1981), 30–46. M. Sato and Y. Sato, “Soliton equations as dynamical systems on inﬁnite-dimensional Grassmann manifold” in Nonlinear Partial Differential Equations in Applied Science (Tokyo, 1982), ed. H. Fujita, P. D. Lax, and G. Strang, Lecture Notes Numer. Appl. Anal. 5, North-Holland, Amsterdam, 1983, 259–271. H. Schneider, Integrable relativistic N -particle systems in an external potential, Phys. D 26 (1987), 203–209. A. C. Scott, F. Y. F. Chu, and D. W. McLaughlin, The soliton: A new concept in applied science, Proc. IEE-E 61 (1973), 1443–1483. G. Segal and G. Wilson, Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. 61 (1985), 5– 65. T. Shiota, Calogero-Moser hierarchy and KP hierarchy, J. Math. Phys. 35 (1994), 5844 –5849. W. R. Thickstun, A system of particles equivalent to solitons, J. Math. Anal. Appl. 55 (1976), 335–346. W. Thirring, A Course in Mathematical Physics, Vol. I: Classical Dynamical Systems, Springer, New York, 1978. H. Weyl, The Classical Groups: Their Invariants and Representations, 2d ed., Princeton Math. Ser. 1, Princeton Univ. Press, Princeton, 1946. G. Wilson, Collisions of Calogero-Moser particles and an adelic Grassmannian, with an appendix by I. G. Macdonald, Invent. Math. 133 (1998), 1–41.

van Diejen: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago 1, Chile Puschmann: Facultad de Ingeniería, Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, Chile

Vol. 104, No. 2

DUKE MATHEMATICAL JOURNAL

© 2000

REFLECTION GROUPS OF LORENTZIAN LATTICES RICHARD E. BORCHERDS

Contents 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Notation and terminology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Modular forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 Discriminant forms and the Weil representation . . . . . . . . . . . . . . . . . . . . . . . . . 323 The singular theta correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 Eta quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 Dimensions of spaces of modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 The geometry of 0 (N ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 An application of Serre duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 Eisenstein series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 Reﬂective forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

0. Introduction. The aim of this paper is to provide evidence for the following new principle: Interesting reﬂection groups of Lorentzian lattices are controlled by certain modular forms with poles at cusps. We use this principle to explain many of the known examples of such reﬂection groups and to ﬁnd several new examples of reﬂection groups of Lorentzian lattices, including one whose fundamental domain has 960 faces. We do not give a precise deﬁnition of what it means for a reﬂection group of a Lorentzian lattice to be interesting, mainly because there seem to be occasional counterexamples to almost any precise version of the principle. However, the interesting groups should include the cases when the reﬂection group is coﬁnite or, more generally, when the quotient of the full automorphism group by the reﬂection group contains a free abelian subgroup of ﬁnite index. The main idea of this paper is roughly as follows (and is described in more detail in Section 11). Suppose that L is an even level N lattice in R1,n . Then the idea is that if Received 4 October 1999. 2000 Mathematics Subject Classiﬁcation. Primary 11H56; Secondary 11F11, 20F55, 51F15. Author’s work supported by a Royal Society professorship and National Science Foundation grant number DMS-9970611. 319

320

RICHARD E. BORCHERDS

L has an interesting reﬂection group, there should be a modular form of weight (1 − n)/2 and level N with certain rather mild singularities (called reﬂective singularities) at cusps. The singular theta correspondence of [B5] associates a piecewise linear function on the hyperbolic space of R1,n to this modular form, and the singularities of this piecewise linear function should be reﬂection hyperplanes of the reﬂection group. We do not have any sort of proof that interesting Lorentzian reﬂection groups always correspond to reﬂective modular forms (or even a good deﬁnition of what an interesting group is). However, we can still use this rather vague correspondence to ﬁnd new reﬂection groups, because it is usually easy to list examples of reﬂective forms using the results in the ﬁrst ten sections. In Section 12, we use this to give many examples of reﬂection groups corresponding to reﬂective modular forms. In particular, we show that most of the known examples in dimensions at least 5 can be found by systematically searching for reﬂective forms of small levels. (This does not seem to work so well for Lorentzian lattices of dimension 3 and possibly 4; Nikulin has found many examples that do not obviously correspond to reﬂective forms.) This gives some sort of structure to the collection of all nice Lorentzian reﬂection groups, which previously looked like a miscellaneous collection of unrelated examples found by a half-dozen assorted methods. We also ﬁnd many new examples of Lorentzian lattices with interesting reﬂection groups, including one whose fundamental domain has 960 faces (comfortably beating the previous record of 210 faces). The main problem with these examples is that there are almost too many of them: For small composite levels (4, 6, 8, 9), there are so many cases that we do not try to list them all but just give vague indications of how to list many of them. Somewhere around level 25, the number of examples for each level decreases to a trickle, and above this level we only seem to get a few low-dimensional lattices for each level. However, there are probably still a few occasional examples when the level is a few hundred. So reﬂective modular forms seem to be a useful practical method for ﬁnding interesting reﬂection groups of Lorentzian lattices. On the other hand, there are occasional counterexamples to show that it does not always work. There are one or two highdimensional reﬂection groups that seem well behaved but do not appear to correspond to reﬂective modular forms. On the other hand, there are occasional Lorentzian lattices with nonzero reﬂective modular forms whose reﬂection groups are quite complicated. Moreover, when there is a reﬂective form, the reﬂection group of the Lorentzian lattice can have several different behaviors: For example, it might be coﬁnite, or it might have a free abelian subgroup of ﬁnite index, or it might have a spacelike Weyl vector, or it might have none of these properties. So the relation between reﬂection group and reﬂective forms is useful in practice, but the theoretical side is still rather mysterious. The reader who wishes to tidy up the theory is warned that Section 12 contains counterexamples to several plausible simplifying conjectures. Sections 2 through 10 are mainly a summary of various assorted results that we need, most of which are minor variations of known results. The contents of these sections should mostly be clear from their titles.

REFLECTION GROUPS OF LORENTZIAN LATTICES

321

Correction. Barnard pointed out a mistake in the formula for the Weyl vector in [B5, Theorem 10.4]. In the formula for ρz , the terms for λ = 0 were incorrectly omitted. So the condition (λ, W ) > 0 in the sum should be deleted, and the factor of 1/2 in front of the sum should be replaced by 1/4. Acknowledgments. I would like to thank D. Allcock, E. Freitag, S. Kondo, I. Dolgachev, and D. Zagier for their help. This paper was written in part at the MaxPlanck-Institut für Mathematik in Bonn. 1. Notation and terminology. ˜ A metaplectic double cover of a group. √ √ The principal value of the square root, with −π/2 < arg( ∗) ≤ π/2. The dual of a lattice. A A discriminant form. An The An root lattice or the elements of order n in a discriminant form A. An The nth powers of elements of A. 2 (R). A discrete subgroup of SL Dn The Dn root lattice. The Dedekind delta function η24 . eγ An element of a basis of C[L /L]. e e(x) = exp(2π ix), en (x) = exp(2πix/n). Ek An Eisenstein series (see Section 10) or the Ek root lattice. η The Dedekind eta function. θL A theta function of a lattice L. g The genus of a subgroup of SL2 (R). I Im,n The even unimodular lattice of dimension m + n and signature m − n. L An even lattice. N The level of a modular form or discriminant form. q n e2πinτ or a discriminant form. Q The rational numbers. R The real numbers. Rj The primitive elliptic element ﬁxing the point j . 2 (Z). ρL A representation of SL sign The signature of a lattice or discriminant form. SL A special linear group. Ta/c The primitive parabolic element ﬁxing the cusp a/c. Tr The trace of something. τ A complex number with positive imaginary part or the number of orbits of cusps. χ A character. For χθ and χn , see Section 5. Z The integers.

322

RICHARD E. BORCHERDS

2. Modular forms. In this section we recall the deﬁnition of a vector-valued modular form and set up notation for the rest of the paper. We deﬁne e(x) to be exp(2πix), and we deﬁne en (x) to be exp(2πx/n). Recall 2 (R), whose elements can that the group SL2 (R) has a (metaplectic) double cover SL be written in the form √ a b , ± cτ + d , c d where ac db ∈ SL2 (R). The multiplication is deﬁned so that the usual formulas for the transformation of modular forms work for half-integer weights, which means that

A, f (·) B, g(·) = AB, f B(·) g(·)

2 (Z) is for A, B ∈ SL2 (R) and for f, g as suitable functions on H . The group SL the inverse image of SL2 (Z) ⊂ SL (R) in SL (R). The group SL (Z) is generated 2 2 1 1 2 0 −1 , √τ , with S 2 = (ST )3 = Z, by S and T, where T = , 1 and S = 01 1 0 0 Z = −1 0 1 , i . The center is cyclic of order 4 and is generated by Z. The quotient by Z 2 is the group SL2 (Z). 2 (R) that contains Z and is coﬁnite (this Suppose that is a discrete subgroup of SL means that the quotient space has ﬁnite volume). Suppose that ρ is a representation of on a ﬁnite-dimensional complex vector space Vρ . Choose k ∈ Q. We deﬁne a modular form of weight k and type ρ to be a holomorphic function f on the upper half-plane H with values in the vector space Vρ such that f

aτ + b cτ + d

= (cτ + d)k ρ

a c

b √ , cτ + d f (τ ) d

√ for elements ac db , cτ + d of . The expression (cτ + d)k means, of course, √ 2k √ cτ + d with the principal value of . (We allow singularities at cusps.) A modular form has a Fourier expansion at the cusp at inﬁnity as follows. The Fourier coefﬁcients cn,γ ∈ C of f are deﬁned by f (τ ) =

cn,γ q n eγ ,

n∈Q γ

where q n means e(nτ ) and where the sum runs over a basis eγ of Vρ consisting of eigenvectors of T . Note that n is not necessarily integral; more precisely, cn,γ is nonzero only if n ≡ λγ mod 1, where the eigenvalue of T on eγ is e(λγ ). We say that f is meromorphic at the cusp i∞ if cn,γ = 0 for n 0, and we say f is meromorphic at the cusp a/c if f ((aτ + b)/(cτ + d)) is meromorphic at i∞ for ac db ∈ SL2 (Z). We say that f is holomorphic at cusps if the coefﬁcients of the Fourier expansions at all cusps vanish for n < 0.

REFLECTION GROUPS OF LORENTZIAN LATTICES

323

3. Discriminant forms and the Weil representation. In this section we recall the deﬁnition of the Weil representation of a discriminant form and prove some results about it that are used in Section 11. We let L be a nonsingular even lattice of dimension dim(L) and signature sign(L), with dual L . The quotient L /L is a ﬁnite group whose order is the absolute value of the discriminant of the lattice L. We deﬁne a discriminant form A to be a ﬁnite abelian group with a Q/Z-valued quadratic form γ → γ 2 /2. If L is an even lattice, then L /L is a discriminant form, with the quadratic form of L /L given by the mod 1 reduction of γ 2 /2, and conversely every discriminant form can be constructed in this way. (For the theory of the discriminant form of a lattice, see [Nik1].) This quadratic form on L /L determines the signature mod 8 of L, by Milgram’s formula γ 2 L sign(L) = e . e 2 L 8 γ ∈L /L

We deﬁne the signature sign(A) ∈ Z/8Z of a discriminant form to be the signature mod 8 of any even lattice with that discriminant form. We let the elements eγ for γ ∈ L /L be the standard basis of the group ring C[L /L], so that eγ eδ = eγ +δ . 2 (Z), called A particularly important example ρA of a unitary representation of SL the Weil representation of the discriminant form A, can be constructed as follows. The underlying space of ρA is the group ring C[A] of A, and the action is deﬁned by (γ , γ ) eγ , ρA (T ) eγ = e 2 e − sign(A)/8 ρA (S) eγ = e − (γ , δ) eδ , √ |A| δ∈A 2 (Z). The representation ρA factors where S and T are the standard generators of SL 2 (Z/NZ) of the ﬁnite group SL2 (Z/NZ), where N is a through the double cover SL positive integer such that Nγ 2 /2 is an integer for all γ ∈ L . The smallest such integer N is called the level of A. In particular, the representation ρA factors through a ﬁnite 2 (Z). If L is an even lattice, then we deﬁne ρL to be the representation quotient of SL ρL /L . We summarize some results about discriminant forms from [CS, Chapter 15, Section 7]; for more details see [Nik1] or [CS]. We use a minor variation of the notation of [CS] for discriminant forms. We recall that every discriminant form can be written as a sum of Jordan components (not uniquely if p = 2), and every Jordan component can be written as the sum of indecomposable Jordan components (usually not uniquely). The possible nontrivial Jordan components are as follows. We let q > 1 be a power of a prime p and n a positive integer and t ∈ Z/8Z. We deﬁne antisquare by antisquare(q ±n ) = 0 if q is a square or the exponent is +n, and antisquare(q ±n ) = 1 if q is not a square and the exponent is −n. (See [CS, page 370].)

324

RICHARD E. BORCHERDS

For q odd, the nontrivial Jordan components of exponent q are q ±n for n ≥ 1. The indecomposable components are q ±1 , generated by an element γ with qγ = 0, γ 2 ≡ a/q mod 2, where a is an even integer with pa = ±1. The component q ±n is a sum of copies of q +1 and q −1 , with an even number of copies of q −1 if ±n = +n and an odd number if ±n = −n. These components all have level q. The signature is given by sign(q ±n ) = −n(q − 1) + 4 antisquare(q ±n ). For q even, the odd Jordan components of exponent q are qt±n . If n = 1, then t ≡ ±1 mod 8 if ± = + and t ≡ ±3 mod 8 if ± = −. If n = 2, then t ≡ 0, ±2 mod 8 if ± = + and t ≡ 4, ±2 mod 8 if ± = −. For any n, we have t ≡ n mod 2. The indecomposable components are qt±1 for 2t = ±1, and they are generated by an element γ with qγ = 0, γ 2 ≡ t/q mod 2. (Note that some of these are isomorphic to each other.) These components all have level 2q. The signature is given by sign(qt±n ) = t + 4 antisquare(q ±n ). For q even, the nontrivial even Jordan components of exponent q are q ±2n = qI±2n I . ±2 The indecomposable even Jordan components are q , which are generated by two elements γ and δ with qγ = qδ = 0, (γ , δ) = 1/q, γ 2 ≡ δ 2 ≡ 0 mod 2 if ± = +, γ 2 ≡ δ 2 ≡ 2/q mod 2 if ± = −. These components all have level q. The signature is given by sign(q ±n ) = 4 antisquare(q ±n ). The sum of two Jordan components with the same prime power q can be worked out as follows: We add the ranks, multiply the signs in the exponent, and if any components have a subscript t, we add together all subscripts t. If A is a discriminant form, then we deﬁne An to be the elements of order n. We deﬁne An to be the nth powers of elements of A, so that we have an exact sequence 0 −→ An −→ A −→ An −→ 0, and An is the orthogonal complement of An . We deﬁne An∗ to be the set of elements δ ∈ A such that (γ , δ) ≡ nγ 2 /2 mod 1 for all γ ∈ An , so that An∗ is a coset of An . We easily see that An is the same as An∗ if and only if the Jordan block of type 2k (where 2k || n) is even. In any case, An∗ always contains an element δ with 2δ = 0. Lemma 3.1. Suppose that A is a discriminant form. Then nγ 2 e (γ , δ) − 2

γ ∈A

is zero unless δ ∈ An∗ (in which case, it has absolute value

√

|A||An |).

Proof. The square of the absolute value of this sum is γ1 ,γ2 ∈A

nγ 2 nγ 2 e (γ1 , δ) − 1 − (γ2 , δ) + 2 2 2

=

γ1 ,γ2 ∈A

nγ 2 e (γ1 , δ) − 1 − n(γ1 , γ2 ) 2

REFLECTION GROUPS OF LORENTZIAN LATTICES

= |A|

325

nγ 2 e (γ1 , δ) − 1 . 2

γ1 ∈An

The map taking γ1 to e((γ1 , δ) − nγ12 /2) is a character of An , so this sum is zero unless this is the trivial character, in other words, unless δ ∈ An∗ . This proves Lemma 3.1. 2 (Z) has image a b ∈ SL2 (Z). Then ρA (e0 ) is Lemma 3.2. Suppose that g ∈ SL c d a linear combination of the elements eγ for γ ∈ Ac∗ . Proof. It is sufﬁcient to prove this when g is of the form T m ST n S for some m, n ∈ Z with (N, n) = (N, c), because g is a product of an element of this form by an element of ˜ 0 (N ) (where N is the level of A) and because e0 is an eigenvalue of ˜ 0 (N ). 2 (Z). We We calculate the image of 1 = e0 ∈ C[A] under these elements of SL know that e − sign(A)/8 eγ . S(e0 ) = √ |A| γ ∈L /L

Applying T n shows that e − sign(A)/8 n(γ , γ ) eγ . e T S(1) = √ 2 |A| γ ∈A n

Applying S again shows that e − sign(A)/4 n(γ , γ ) ST S(1) = eδ . e (γ , δ) + |A| 2 n

δ∈A γ ∈A

Using Lemma 3.1, we see that the coefﬁcient of eδ in this expression is zero unless δ ∈ Ac∗ . As all the elements eδ are eigenvectors of T m , the same is true for T m ST n S(e0 ). This proves Lemma 3.2. 4. The singular theta correspondence. We summarize some of the results from [B5]. The main idea is that we can use modular forms with poles at cusps to construct some automorphic forms with singularities. In particular, we can often use this to construct piecewise linear functions on hyperbolic space with singularities along the reﬂection hyperplanes of a reﬂection group, and this gives the connection between modular forms with singularities and nice hyperbolic reﬂection groups. If L is a lattice, then we deﬁne the Grassmannian G(L) to be the set of maximal positive deﬁnite subspaces of L⊗R. It is a symmetric space acted on by the orthogonal group OL (R).

326

RICHARD E. BORCHERDS

The Siegel theta function θL+γ of a coset L + γ of L in L is deﬁned by

θL τ ; v

+

=

e

τ λ2v + 2

λ∈L+γ

+

τ¯ λ2v −

2

for τ ∈ H , v + ∈ G(L). We write 5L for the C[L /L]-valued function eγ θL+γ (τ ; v). 5L (τ ; v) = γ ∈L /L

2 (Z) Siegel’s transformation formula for 5L under SL given by b− /2 aτ + b a b+ /2 5L ρL cτ¯ + d v = (cτ + d) c cτ + d

(see [B5, Theorem 4.1]) is b √ , cτ + d 5L (τ ; v). d

We deﬁne 6(v, F ) by 6(v, F ) =

SL2 \H

¯ L (τ ; v)F (τ )y b+ /2−2 dx dy, 5

as in [B5, Section 6]. By [B5, Theorem 6.2], 6(v, F ) is an automorphic function of v ∈ G(L) whose only singularities are on points of the form γ ⊥ , for γ ∈ L , γ 2 < 0, where there is a nonzero coefﬁcient cγ 2 /2,γ of F . Theorem 4.1. Suppose L is an even lattice of signature (2, b− ), and suppose F is a modular form of weight 1−b− /2 and of representation ρL which is holomorphic on H , meromorphic at cusps, and whose coefﬁcients cλ (m) are integers for m ≤ 0. Then there is a meromorphic function 9L (ZL , F ) for Z ∈ P with the following properties. (1) 9L (ZL , F ) is an automorphic form of weight c0 (0)/2 for the group Aut(L, F ), with respect to some unitary character χ of Aut(L, F ). (2) The only zeros or poles of 9L lie on the rational quadratic divisors λ⊥ for λ ∈ L, λ2 < 0 and are zeros of order 0<x∈R xλ∈L

cxλ

x 2 λ2 2

(or poles, if this number is negative). (3) 9L is a holomorphic function if the orders of all zeros in (2) above are nonnegative. If, in addition, L has dimension at least 5 or if L has dimension 4 and contains no two-dimensional isotropic sublattice, then 9L is a holomorphic automorphic form. If, in addition, c0 (0) = b− −2, then 9L has singular weight so the only nonzero Fourier coefﬁcients of 9L correspond to vectors of K of norm 0.

REFLECTION GROUPS OF LORENTZIAN LATTICES

327

This follows from [B5, Theorem 13.3]. If L is Lorentzian (in other words, if sign(L) = 2 − dim(L)), then the set of all one-dimensional positive deﬁnite subspaces of L is a copy of hyperbolic space of dimension dim(L) − 1. Theorem 4.2. Suppose M is a Lorentzian lattice of dimension 1 + b− . Suppose that F is a modular form of type ρM and of weight (1/2−b− /2, 0) which is holomorphic on H , meromorphic at cusps, and all of whose Fourier coefﬁcients cλ (m) are real for m < 0. Finally, suppose that if cλ (λ2 /2) = 0 and λ2 < 0, then reﬂection in λ⊥ is in Aut(M, F, C). Then Aut(M, F, C) is the semidirect product of a reﬂection subgroup and a subgroup ﬁxing the Weyl vector ρ(M, W, F ) of a Weyl chamber W . This is a special case of [B5, Theorem 12.1]. Both Theorems 4.1 and 4.2 depend on integrating the vector-valued modular form against a vector-valued theta function over a fundamental domain of SL2 (Z). In this paper, we usually start with a complex-valued modular form for 0 (N) rather than a vector-valued form as used in Theorems 4.1 and 4.2. There are two more or less equivalent ways to use these theorems on complex-valued forms of level N. First, instead of integrating a vector-valued form times the vector-valued theta function of a lattice over a fundamental domain of SL2 (Z), we can integrate a scalar-valued modular form times the theta function of a lattice over a fundamental domain of 0 (N ). Alternatively, we can ﬁrst induce the complex-valued modular form for 0 (N) up to a vector-valued modular form for SL2 (Z) and then apply the theorems directly to part of this vector-valued form. For these constructions to work, it is necessary and sufﬁcient for the complex-valued form to be a modular form for some character χ of ˜ 0 (N ), where the scalar-valued theta function of the lattice is a modular form of character χ and level sign(L)/2 for ˜ 0 (N). Several sections of this paper describe how to ﬁnd such modular forms. Note that the singularities of the automorphic form associated to a level N modular form depend on all poles at all cusps of this form, not just the poles at i∞. Theorem 4.2 is very useful in practice for ﬁnding Lorentzian lattices with interesting reﬂection groups, because we just ﬁnd lattices together with modular forms satisfying the conditions of the theorem. However, there is a problem with using it for theoretical purposes: It seems hard to give useful general conditions under which the Weyl vector is nonzero or has positive norm. If the Weyl vector happens to be zero, then of course Theorem 4.2 does not say anything. In practical examples, this does not matter because we can just check in each case to see whether the vector is zero (which does happen occasionally). Note the rather curious fact that in this paper we do not need to use the fact that Theorem 4.2 has been proved (or even that it is true!) because we are only using it to suggest interesting places to look for lattices, and whenever we ﬁnd a lattice using Theorem 4.2, we still have to prove its properties directly because of the possibility that the Weyl vector is zero.

328

RICHARD E. BORCHERDS

5. Theta functions. In this section, we work out the level and character of theta functions of even lattices. Most of the results are known, but there seems to be no convenient reference giving the results in the generality we require. Lemma 5.1. Suppose that N is a positive integer. If 4 N, then two characters of 0 (N ) are the same, provided that they have the same values on the elements such that c > 0, d > 0, and d ≡ 1 mod 4. If 4 | N, then two characters of ˜ 0 (N) are the same, provided that they have the same values on Z and on the elements such that c > 0, d > 0, and d ≡ 1 mod 4. Proof. It is sufﬁcient to show that the images of the elements mentioned above generate 0 (N ). Suppose that ac db ∈ 0 (N). We show how to multiply it by powers of elements of the generating set above so that it becomes an element of the generating set, which proves the lemma. We ﬁrst note that T is in the group generated by the set above, because the generating set is closed under left multiplication by T and is nonempty. If d is even, then c is odd so we can multiply it on the right by T so that d is odd; hence, we can assume that d is odd. Next we arrange that d ≡ 1 mod 4. If 4 | N, we multiply by Z if necessary so that d ≡ 1 mod 4. If 4 N, we multiply on the right by N1 01 , if necessary, to make c not divisible by 4 and then multiply on the right by a suitable power of T so that d ≡ 1 mod 4. We now have to make c and d positive 1 0 (without changing d mod 4). We multiply on the right by a suitable power of 4N 1 to make c positive without changing d. Finally, we multiply on the right by a suitable power of 01 41 to make d positive without changing c. The result is in the generating set, so this proves Lemma 5.1. We deﬁne the symbol dc for all pairs of coprime integers c and d as follows. The symbol is multiplicative in both c and d. If d is an odd prime, it is just the usual Legendre symbol. If d is 2, it is 1 if c ≡ ±1 mod 08 and−1 otherwise. If d = −1, it is 1 if c > 0 and −1 if c < 0. Finally, we deﬁne ±1 = ±1 0 = 1. We now deﬁne some characters χn (for n a positive integer) and χθ of ˜ 0 (N). We suppose that if p is an odd prime occurring an odd number of times in the prime factorization of n, then it divides N. Also suppose that if 2 occurs an odd number of times in the prime factorization of n, then 8 divides N. We deﬁne the character χn of 0 (N ) by d a b = . χn c d n

2 Lemma 5.2. Let θA1 = n∈Z q n be the theta function of the A1 lattice. There is a (unique) character χθ of the metaplectic double cover of 0 (4) such that θA1

aτ + b cτ + d

= χθ

a c

√ b √ , cτ + d cτ + dθA1 (τ ). d

REFLECTION GROUPS OF LORENTZIAN LATTICES

329

(In other words, θA1 is a modular form for ˜ 0 (4) of weight 1/2 and character χθ .) The values are given by  c  if d ≡ 1 mod 4, ±   √ d a b , ± cτ + d = χθ c d  c   if d ≡ 3 mod 4. ±(−i) d In particular, χθ (Z) = −i. Proof. This follows from the theorem on page 148 of [Ko]. Lemma 5.3. Suppose 4 | N . Then the kernel of the character χθ of 0 (N) maps isomorphically onto 1 (4)∩0 (N), and if we identify the kernel with this image, then 0 (N ) is the product (1 (4) ∩ 0 (N)) × Z/4Z (where Z/4Z is its center, generated by Z). The lifting of 1 (4) ∩ 0 (N) to ˜ 0 (N) is given by c √ a b a b , −→ cτ + d . c d c d d Proof. This follows immediately from Lemma 5.2 because χθ is a character whose values are ±1 or ±i, χθ (Z) = −i, and 0 (N) is the product of its center of order 2 (generated by Z) and the subgroup 1 (4) ∩ 0 (N). This proves Lemma 5.3. We deﬁne the group 02 (N ) to be the subgroup of 0 (N) of elements whose diagonal entries are squares in Z/nZ. If 4 N, then 02 (N) is the intersection of the kernels of the characters χp for p an odd prime dividing N . If 4 | N, then 02 (N) can be lifted to a subgroup of ˜ 0 (N ), as in Lemma 5.3, and is the intersection of the kernels of the characters χθ and χp of ˜ 0 (N) for p a prime dividing N/4. Theorem 5.4. Suppose that A is adiscriminant form of level dividing N. If b √ 2 (Z) acts on the Weil and c are divisible by N , then g = ac db , cτ + d ∈ SL representation C[A] by g eγ = χA (g)eaγ , where χA is the character of ˜ 0 (N) given by  −1  sign(A)+(|A| )−1 χ|A|2sign(A) χ θ χA = χ |A|

if 4 | N, if 4 N.

Proof. First assume that A has even signature. Choose an even lattice in a positive deﬁnite space with discriminant form A. Then [Eb, Corollary 3.1] and the discussion on [Eb, page 94] show that (−1)sign(A)/2 |A| a b eγ = eaγ , c d d

330

RICHARD E. BORCHERDS

provided that d is odd and positive. By Lemma 5.1, it is sufﬁcient to check that this is equal to the value of |A|χA when d g = Z and when d ≡ 1 mod 4, d > 0. But in the latter case −1 = 1 and = d d |A| , so this has the same character values as χ|A| . Also if sign(A)

(−1/A)−1

d ≡ 1 mod 4 and sign(A) is even, then χ2sign(A) , χθ , and χθ are all 1 on the element g. Therefore, the two characters coincide on elements with d ≡ 1 mod 4. As Z(eγ ) = (−i)sign(A) e−γ , we see that χA (Z) = (−i)sign(A) . We now check that −1 = the characters are equal on the element Z. If 4 N, this follows from χ|A| (Z) = |A| sign(A)

(−1)sign(A)/2 . If 4 | N, this follows from χ2 (Z) = 1, χθ (Z) = (−i)sign(A) , and −1 −1 −1 1−( ) = (−i)1−(|A|) = χθ |A| (Z). This proves Theorem 5.4 when A has χ|A| (Z) = |A| even signature. We can do the case of odd signature very quickly by reducing it to the case of even signature as follows. If A has odd signature, then 4 | N , and the discriminant form A ⊕ 2 has determinant 2|A| (where 2 is the discriminant form of the A1 lattice and has order 2). Theorem 5.4 for the element γ ∈ A now follows from Lemma 5.2 and Theorem 5.4 applied to the element γ +0 ∈ A⊕2 . This proves Theorem 5.4.

6. Eta quotients. In this section we work out the levels and characters of some eta quotients. We use these results in Section 12 to construct examples of modular forms of given characters. The function η(tτ ) has a zero of order (t, c)2 /24t at the cusp a/c. 1/24 n Lemma 6.1 (Rademacher). Recall that η(τ ) = q n>0 (1−q ) is the Dedekind a b eta function. Suppose that c d ∈ SL2 (Z) with c > 0. Then √ aτ + b a b √ , cτ + d η = χη cτ + d η(τ ), c d cτ + d 2 (Z) with values given as follows: where χη is a character of SL √ a b , ± cτ + d χη c d  d   ± e24 − 3c + bd 1 − c2 + c(a + d) , c odd, c > 0;   c       −d  2  c odd, c < 0;  ± −c e24 3c − 6 + bd 1 − c + c(a + d) , =  c   d odd, c ≥ 0; ± e24 3d − 3 + ac 1 − d 2 + d(b − c) ,    d       −c  ± e24 − 3d − 9 + ac 1 − d 2 + d(b − c) , d odd, c < 0. d

331

REFLECTION GROUPS OF LORENTZIAN LATTICES

Proof. The cases with c ≥ 0 follow from the theorem in [R, page 163]. The cases with c < 0 follow easily from the cases with c > 0 and the fact that χη (Z) = −i. This proves Lemma 6.1. Theorem 6.2. Suppose rδ for that we are given a positive integer N and integers

δ | N and |A| with |A|/ δ|N δ rδ a rational square. Suppose that (1/24) δ|N rδ δ and

(N/24) δ|N rδ /δ are both integers. Then

η(δτ )rδ

δ|N

is a modular form for ˜ 0 (N ) of weight k = 2k+(−1)−1 4 N and to χθ |A| χ22k |A| if 4 | N.

δ rδ /2

and of character equal to χ|A| if

Proof. By Lemma 5.1, it is enough to check that the characters are equal whenever c > 0 and d ≡ 1 mod 4 and also that they are equal on Z if 4 | N. If c > 0 and d ≡ 1 mod 4, then by Lemma 6.1 the character value is given by

e24

δ|N

rδ c/δ c ac 2 1 − d + d bδ − 3d − 3 + δ δ d

 = e24 db

δ|N

 rδ + 3(d − 1) δrδ + a − d − ad 2 c rδ  δ

δ|N

δ|N

δ|N rδ rδ c δ|N δ × d d 2k c |A| = i (d−1)k d d 2k 2k d c d = . 2 d |A| If 4 N, then 2k is even so this is the value of the character χ|A| . If 4 | N , then this is 2k+(−1)−1 the value of χθ2k χ22k |A| , which is the same as the value of χθ |A| χ22k |A| because χθ = ±1 whenever d ≡ 1 mod 4. So both characters have the same value whenever c > 0 and d ≡ 1 mod 4. Finally, we have to check that both characters are equal on Z whenever 4 | N. This follows by the same argument used in Theorem 5.4. This proves Theorem 6.2. Theorem 6.2 generalizes some theorems of Newman (see [N1], [N2]), who did the case of weight 0 and trivial character.

332

RICHARD E. BORCHERDS

7. Dimensions of spaces of modular forms. In this section we recall the formulas for the dimensions of some spaces of modular or cusp forms associated to a 2 (R). For weight at least 2, the representation ρ of a discrete coﬁnite subgroup of SL dimension is given by either the Riemann-Roch theorem or the Selberg trace formula. More generally, if G is a group acting on A, then it also acts on the spaces of cusp forms and we calculate the character of these representations. These results are used in Sections 9 and 12. For weight 1/2 forms, Serre and Stark described an explicit basis as follows. Theorem 7.1. Suppose that χ is an even Dirichlet character mod N. Then a basis for the space of modular forms of weight 1/2 and character χθ χ for 0 (N) is given by the forms 2 ψ(n)q tn , n∈Z

where ψ is a primitive even character of conductor r(ψ), t is a positive integer such that 4r(ψ)2 t divides N, and χ(n) = ψ(n) Dn for all n coprime to N, where D is the √ discriminant of the quadratic ﬁeld Q[ t]. (Note that ψ is determined by t and χ.) Proof. This is [SS, Theorem A, page 34]. The dimensions of spaces of holomorphic modular forms can all be worked out as follows. For weight less than zero, there are no nonzero forms, and weight 0 is trivial as these are just constants. For weight greater than 2, we can work out the dimension using the Selberg trace formula (see below) or the Riemann-Roch theorem; with a bit more care this also works for weight 2 (there are extra correction terms coming from weight 0 forms in this case). For weight 1/2, the Serre-Stark theorem gives an explicit basis, which can be used to do the case of weight 3/2 because the Selberg trace formula gives the difference of dimensions for weights k and 2 − k. This leaves the case of weight 1, which seems to be the hardest case to do. In general, weight 1 forms are closely related to odd two-dimensional complex representations of the Galois group of Q. Fortunately, for the low-level cases we are interested in, the weight 1 forms are usually easy to construct explicitly using Eisenstein series and theta series of two-dimensional lattices (mainly because the exotic Galois representations only occur for higher levels). Now we use the Selberg trace formula to ﬁnd the dimensions of spaces of forms of weight at least 2. If X is a ﬁnite-order automorphism of a ﬁnite-dimensional complex vector space V with eigenvalues e(−βj ) for 1 ≤ j ≤ dim(V ) and 0 ≤ βj < 1, then we deﬁne

δ∞ (X) to be (1/2 − βj ) and we deﬁne δN (X) to be δ∞ (X) − dim(V )/2N. More generally, if g is an endomorphism of V commuting with the action of G, then we deﬁne δρ,∞ (X, g) to be 1 − βj Tr g | V e(βj )X , 2

REFLECTION GROUPS OF LORENTZIAN LATTICES

333

where the sum is over the distinct eigenvalues e(−βj ) of X, and we put δN (X, g) = δ∞ (X, g) − Tr(g)/2N. Lemma 7.2. If ρ is a representation of a group containing X on a ﬁnite-dimensional complex vector space and X N = 1, then

δN (X, g) =

δ∞ (X, g) =

1 N

Tr ρ X j g 0<j
1 − e(j/N)

Tr(g) 1 + 2N N

,

Tr ρ X j g 0<j
1 − e(j/N)

.

Proof. The trace of g on the subspace of ρ on which X −1 has eigenvalue e(k/N) is j jk 1 . Tr ρ gX e N N j modN

Therefore, k j jk 1 1 1 − − × Tr ρ gX e δN (X) = Tr ρ (g) 2 2N N N N 0 ≤ k
0<j
This proves Lemma 7.2. We write ModForm(, k, ρ) for the space of modular forms of weight k and representation ρ for that are holomorphic at cusps. 2 (R) of coﬁnite volume Lemma 7.3. Suppose that is a discrete subgroup of SL and containing Z. Let ρ be a complex representation of of ﬁnite dimension d on which Z acts multiplication by some constant. Choose k ∈ Q with k > 2. Then the dimension of the space of ModForm(, k, ρ) is equal to zero unless Z acts as e(−k/2), in which case it is (k − 1) dim(ρ)ω(F ) k + δρ,νj e δρ,∞ Tj , Rj + 4π 2νj 1≤j ≤ρ

1≤j ≤τ

334

RICHARD E. BORCHERDS

where ω(F ) is the hyperbolic area of the fundamental domain F and is equal to 1 2π 2g − 2 + 1− +τ ; νj 1≤j ≤ρ

g is the genus of the compactiﬁcation of \H ; ρ is the number of elliptic ﬁxed points in a fundamental domain; νj is the order of the j th elliptic ﬁxed point, so the subgroup of ﬁxing the j th ν elliptic ﬁxed point is cyclic, generated by Rj with Rj j = Z; Rj is the primitive elliptic element corresponding to j ; an elliptic element is called primitive if it is conjugate to the “clockwise” element π π cos(π/ν) − sin(π/ν) 2 (R) , sin τ + cos ∈ SL sin(π/ν) cos(π/ν) ν ν ﬁxing i; τ is the number of orbits of cusps of acting on H ; 2 (R) such that the stabilizer Tj is the unique element conjugate to T −1 under SL of the j th cusp is generated by Tj and Z. Proof. If Z acts as multiplication by some constant not equal to e(−k/2), then the transformation of modular forms under the element Z immediately shows that any modular form of weight k is zero. If Z acts as e(−k/2), then we get a “multiplier system” the sense 1.3.4] from the representation ρ by putting χ in √of [F, Deﬁnition χ ac db = ρ ac db , cτ + d , where we choose the value of the square root with √ −π/2 < arg( cτ + d) ≤ π/2. (Note that the variable k used in [F] is half that used here.) By [F, Theorem 2.5.5] the dimension of the space of modular forms is given by d αj (k − 1)dω(F ) d dτ + − − + − βj , 4π 2 2νj νj 2 1≤j ≤ρ

1≤j ≤τ

where d = dim(ρ); αj is the sum of the numbers αjp for 1 ≤ p ≤ d, where the eigenvalues of Rj are e(−(k/2 + αjp )/νj ) and αjp ∈ {0, 1, . . . , νj − 1} (see [F, pages 66–68]); βj is the sum of the numbers βjp (1 ≤ p ≤ d), where the eigenvalues of Tj−1 are e(βjp ) and 0 ≤ βjp < 1. It is easy to check that αj d d k − − = δρ,νj e Rj 2 2νj νj νj

REFLECTION GROUPS OF LORENTZIAN LATTICES

335

and

d − βj = δρ,∞ Tj . 2 Putting everything together proves Lemma 7.3.

Remark. If the weight is greater than 2, then the dimension of the space of all cusp forms is given by subtracting the dimension of the space of Eisenstein series, which is the sum over all cusps j of the dimension of the subspace of ρ ﬁxed by Tj . Corollary 7.4. Suppose that ρ is a ﬁnite-dimensional representation of . Suppose that k ∈ Q and k > 2. Then 1 k ψ Zj , e dim ModForm(, k, ρ) = 4 2 0≤j <4

where ψ(g) is given by ψ(g) =

(k − 1)ω(F ) k Tr ρ (g) + δρ,νj e Rj , g + δρ,∞ Tj , g . 4π 2νj 1≤j ≤ρ

1≤j ≤τ

Proof. Break up ρ into the eigenspaces of Z, and apply Lemma 7.3 to each eigenspace. This proves Corollary 7.4. Corollary 7.5. Suppose that ρ is a ﬁnite-dimensional representation of acted on by a ﬁnite group G. Suppose that k ∈ Q and k > 2. Then the character of ModForm(, k, ρ), considered as a representation of G, is given by 1 jk Tr g | ModForm(, k, ρ) = ψ gZ j , e 4 2 0≤j <4

where ψ is given by the formula of Corollary 7.4 and where g ∈ G. Proof. Note that the dimension of ModForm(, k, σ ) is given by Tr(M | σ ) for some element M in the group ring of G, whenever σ is a representation of satisfying the conditions of Corollary 7.4. Therefore, G χ(g) dim ModForm(, k, ρ) ⊗ χ¯ Tr g | ModForm(, k, ρ) = χ∈Irred(G)

=

χ∈Irred(G)

=

χ∈Irred(G)

G χ(g) dim ModForm , k, ρ ⊗ χ¯ G χ(g) Tr M | ρ ⊗ χ¯

= Tr Mg | ρ .

336

RICHARD E. BORCHERDS

(The sums are over the sets Irred of irreducible representations of G, and χ¯ is the dual of the representation χ .) Therefore, to ﬁnd the trace of g, we just insert an extra factor of g whenever we have a trace in the formula for ψ. This proves Corollary 7.5. 8. The geometry of 0 (N ). We summarize some standard results about the cusps and elliptic points of the group 0 (N). We need this information in order to use the formulas of Section 7. For proofs, see [Sh] or [Mi]. The group 0 (N ) has index N p|N (1+1/p) (where the product is over all primes dividing N). The equivalence class of the cusp a/c of 0 (N) is determined by the invariants (c, N ) (a divisor of N ) and (c/(c, N))−1 a (an element of (Z/(c, N/c))∗ ). A complete set of representatives for the cusps is given by a/c for c | N, c > 0, 0 < a ≤ (c, N/c), (a, c) = 1. The cusp a/c has width N/(c2 , N ). In the rest of this section, we work out the values of the characters χθ and χp on the elements Rj and Ta/c of Section 7, which are associated to elliptic points or cusps of 0 (N ). 0 (N) is given by Lemma 8.1. If (a, c) = 1, then the element Ta/c of 1 + act −a 2 t Ta/c = , c2 tτ + 1 − act , c2 t 1 − act where t = N/(c2 , N). Proof. We can assume c > 0 as the case c < 0 follows from this, and the case c = 0is trivial to check. The element Ta/c is the conjugate of some element of the 1 −t form 0 1 , 1 for t > 0, so it is equal to d −b √ a b √ 1 −t , −cτ + a , , cτ + d ,1 −c a c d 0 1 where t is the smallest positive integer such that this element is in 0 (N), and b and d in Z are chosen so that ad −bc = 1. If we evaluate this element (keeping careful track of the values of the square roots), we ﬁnd it is equal to the expression in the lemma. This is in 0 (N ) if and only if N | c2 t, so t = N/(c2 , N ). This proves Lemma 8.1. Lemma 8.2. If p is an odd prime dividing N, then χp (Ta/c ) = 1. Proof. By Lemma 8.1 we see that χp (Ta/c ) is 1 provided that 1 − act is a square mod p. But this is always true because p must divide either c or t = N/(c2 , N ) if it divides N. This proves Lemma 8.2. Lemma 8.3. Suppose that 8 | N and (a, c) = 1. Then χ2 (Ta/c ) is −1 in the three cases 2 || c and 8 || N, or 4 || c and 8 || N, or 4 || c and 16 || N, and is 1 otherwise.

REFLECTION GROUPS OF LORENTZIAN LATTICES

337

Proof. By Lemma 8.1 we have 1 + act −a 2 t 1 − act 2 tτ + 1 − act = χ2 Ta/c = χ2 c , , 2 c2 t 1 − act so χ2 (Ta/c ) is 1 if 1 − act ≡ 1 mod 8 and is −1 if 1 − act ≡ 5 mod 8 (where t = N/(c2 , N)). Note that act ≡ 1 mod 4 as 8 | N . We do a case-by-case check to show that act is 4 mod 8 in the three cases listed above, and it is 0 mod 8 otherwise. (1) If c is odd, then 8 | t = N/(c2 , N ), so we can assume that c is even (and hence a is odd). (2) If 8 | c, then act ≡ 0 mod 8, so we can assume that 2 || c or 4 || c. (3) If 2 || c and 16 | N, then 4 | t so act ≡ 0 mod 8. (4) If 2 || c and 8 || N , then 2 || t so act ≡ 4 mod 8. (5) If 4 || c and 32 | N, then 2 | t so act ≡ 0 mod 8. (6) If 4 || c and 32 N, then t is odd so act ≡ 4 mod 8. This proves Lemma 8.3. Lemma 8.4. Suppose that 4 | N. If 2 || c and 4 || N, then χθ (Ta/c ) = −i t , with t = N/(c2 , N). If 2 || c and 8 || N , then χθ (Ta/c ) = −1. Otherwise χθ (Ta/c ) = 1. Proof. By Lemmas 8.1 and 5.2, we see that if 4 | act, then 1 + act −a 2 t c2 t 1 − act 2 = . χθ Ta/c = χθ , c tτ + 1 − act = 1 − act t c2 t 1 − act This is equal to 1 unless 2 || t and 2 || c, in which case it is −1 and 8 || N . So Lemma 8.4 is true whenever 4 | act. If c is odd, then 4 | t so 4 | act, and if 4 | c, then 4 | act, so we can assume that 2 || c. If 8 | N and 2 || c, then 2 | t so 4 | act. So we can also assume that 4 || N , which implies that a and t are odd and act ≡ 2 mod 4. Then by Lemmas 8.1 and 5.2, t χθ Ta/c = −i 1 − act 1 − act = −i(−1)(t−1)(1−act−1)/4 t = −i(−1)(t−1)/2 = −i t . This proves Lemma 8.4. Finally we work out the values of characters on elliptic elements. Note that if the characters χθ or χ2 are nontrivial, then 4 | N so there are no elliptic elements. So we only have to do the characters χp on elliptic elements for p an odd prime. Lemma 8.5. Suppose that 0 (N) has an elliptic ﬁxed point of order 2 ﬁxed by the primitive elliptic element R ∈ 0 (N) as in Lemma 7.3, and let p be an odd prime dividing N (so that p ≡ 1 mod 4). Then χp (R) = (−1)(p−1)/4 .

338

RICHARD E. BORCHERDS

b Proof. We know that R 2 = ± 01 01 so R = ac −a for some a, b, c, and χp (R) = a 2 − bc = 1, N | c, and p | N, we see that a 2 ≡ −1 mod p, so that a is an . As −a p element of order exactly 4 in (Z/pZ)∗ , and hence is in the (unique) index 2 subgroup of (Z/pZ)∗ if and only if p ≡ 1 mod 8. But χp (a) = 1 if and only if a is in the index 2 subgroup of (Z/pZ)∗ . This proves Lemma 8.5. Lemma 8.6. Suppose that 0 (N) has an elliptic ﬁxed point of order 3 ﬁxed by the primitive elliptic element R ∈ 0 (N) as in Lemma 7.3, and let p be an odd prime dividing N. Then χp (R) = (−1)(p−1)/2 . Proof. The character χp has order 2, so

−1 χp (R) = χp R 3 = χp (Z) = = (−1)(p−1)/2 . p

This proves Lemma 8.6. 9. An application of Serre duality. In this section we summarize some results from [B7]. We later often need to ﬁnd modular forms with given singularities at cusps. In this section we show how it is sometimes possible to prove the existence of modular forms with given singularities without writing them down explicitly. The main idea is that Serre duality shows that the space of obstructions to ﬁnding a modular form with given singularities at cusps is a space of cusp forms whose dimension can usually be worked out explicitly. If this space of obstructions has smaller dimension than the space of potential solutions, then at least one solution must exist. If κ is a cusp of , let qκ be a uniformizing parameter at κ on \H . For a representation ρ on Vρ , let Vρ∗ denote the dual. Let PowSer κ (, ρ) = C[[qκ ]] ⊗ Vρ be the space of formal power series in qκ with coefﬁcients in Vρ , let Laur κ (, ρ) = C[[qκ ]] qκ−1 ⊗ Vρ be the space of formal Laurent series in qκ with coefﬁcients in Vρ , and let Singκ (, ρ) =

Laur κ (, ρ) PowSer κ (, ρ)

be the space of possible singularities of Vρ -valued Laurent series at κ. The two spaces PowSer κ (, ρ ∗ ) and Singκ (, ρ) are paired into C by taking the residue f, φ = Res f φqκ−1 dqκ , for f ∈ PowSer κ (, ρ ∗ ) and φ ∈ Singκ (, ρ). Here the product of f and φ is deﬁned using the pairing of Vρ and Vρ∗ .

REFLECTION GROUPS OF LORENTZIAN LATTICES

Then the spaces Sing(, ρ) =

κ

and

339

Singκ (, ρ)

PowSer , ρ ∗ = PowSer κ , ρ ∗ , κ

where κ runs over the -inequivalent cusps, are paired by the sum of the local pairings at the cusps. There are maps λ : CuspForm , k, ρ ∗ −→ PowSer , ρ ∗ and

λ : SingModForm , 2 − k, ρ −→ Sing(, ρ)

deﬁned in the obvious way by taking the Fourier expansions of their nonpositive part at the various cusps. We deﬁne the space Obstruct(, k, ρ) of obstructions to ﬁnding a modular form of type ρ and weight k which is holomorphic on H and has given meromorphic singularities at the cusps to be the space Sing(, ρ) . Obstruct(, k, ρ) = λ SingModForm(, k, ρ) Lemma 9.1. Suppose that ρ is a representation of factoring through some ﬁnite quotient of and k ∈ Q with Z = e(−k/2) on ρ. Then Obstruct , 2 − k, ρ is dual to

CuspForm , k, ρ ∗

(and both spaces are ﬁnite-dimensional). Proof. This can be proved in the same way as [B7, Theorem 3.1] (which is really a special case of Serre duality). The only real difference is that in [B7] the space Sing is different because it also includes information about the constant terms of functions. This has the effect of replacing the space of holomorphic modular forms in [B7] by a space CuspForm(, k, ρ ∗ ) of cusp forms. This proves Lemma 9.1. Example 9.2. In one of the examples later on, we need to know that there is a nonzero weight −7 form of character χ3 for 0 (3) whose singularities are a pole of order at most 1 at i∞ and a pole of order at most 3 at 0, and such that the coefﬁcient of q32 at the cusp 0 vanishes. The space of possible singularities is spanned by q −1 at i∞ and by q3−1 and q3−3 at 0, so it is three-dimensional. The space of obstructions

340

RICHARD E. BORCHERDS

is the space of weight 2 − −7 = 9 cusp forms of character χ3 for 0 (3), which has dimension 2. This is less than 3, so the space of forms with the singularities above is at least one-dimensional, so a nonzero form exists. (For an explicit formula, see Section 12.) Warning. This method only gives a lower bound for dimensions of spaces of forms with given singularities. In practice this lower bound is often the exact dimension, but there are occasional examples where the lower bound is zero but nevertheless there is a nonzero form. 10. Eisenstein series. We summarize some standard results about Eisenstein series that we use in Section 12. Lemma 10.1. Assume that k ≥ 2 is even. Put Ek (τ ) = 1 −

2k n k−1 q d . Bk d|n

n≥1

If k > 2, then Ek is a modular form of weight k for 0 (1). If

d|N ad E2 (τ ) is a modular form of weight 2 for 0 (N).

d|N ad /d

= 0, then

Proof. These are just the usual Eisenstein series for SL2 (Z). Lemma 10.2. Assume that k ≥ 2 is integral, and let χ be a nonprincipal Dirichlet character mod N with χ (−1) = (−1)k . Then Ek (τ, χ) =

n≥1

q

n

d

k−1

d|n

n χ d

is a modular form of weight k and character χ for 0 (N). Proof. See [Mi, Theorems 7.1.3 and 7.2.12 and Lemma 7.1.1]. Lemma 10.3. Let χ be a nonprincipal Dirichlet character mod N with χ(−1) = −1. Then n 2 E1 (τ, χ ) = 1 + qn χ , L(0, χ) d n≥1

where L(0, χ) = −B1,χ = −

d|n

χ(n)

0
n N

is a modular form of weight k and character χ for 0 (N). Proof. See [Mi, Theorem 7.2.13 and Lemma 7.1.1].

REFLECTION GROUPS OF LORENTZIAN LATTICES

341

Example 10.4. The Eisenstein series E1 (τ, χ3 ) is a weight 1 modular form for 0 (3) of character χ3 whose power series is E1 (τ, χ3 ) = 1 + 6

n>0

q

n

d|n

n = 1 + 6q + 6q 3 + 6q 4 + O q 7 . χ3 d

This form is also the theta function of the A2 lattice. (It is used in Wiles’s proof of Fermat’s last theorem to show that any weight 1 form is congruent mod 3 to a weight 2 form: Just multiply by E1 (τ, χ3 ).) 11. Reﬂective forms. Suppose that L is a lattice of level dividing N. We deﬁne a singularity at a cusp a/c of 0 (N) to be reﬂective if it is a linear combination of terms q −1/n for n a positive integer, such that every norm −2/n element of (L /L)c∗ has order dividing n. Note that if a norm −2/n element of L /L has order dividing n, then every lift of it to a norm −2/n element of L is a root of L . We say that a modular form is reﬂective for L if it is a meromorphic modular form of weight sign(L)/2, level N, character χL , and its only singularities are reﬂective singularities at cusps. This deﬁnition is the result of a lot of experimentation to ﬁnd a deﬁnition that is easy to use and that also seems to give most of the “interesting” lattices. There are many possible variations of it, some of which we now brieﬂy describe. First of all, we could use vector-valued modular forms rather than scalar-valued modular forms. The main problem with this is practical inconvenience: It is not that easy (though it is possible) to work with modular forms taking values in a space of dimension, say, 29 . Moreover, examples suggest that the most interesting vector-valued modular forms are often invariant under Aut(A). This suggests using invariant vector-valued modular forms instead, but other examples suggest that these are very closely related to scalarvalued modular forms of level N, so we may as well use these. The “allowable” singularities can also be varied. For example, we could also allow singularities of the form q −n such that A has no vectors of norm −2n. One problem with this is that it turns out there are then too many modular forms with these singularities if the p rank of A is 1 for some prime p. Another possibility is to allow singularities of the automorphic form that correspond to characteristic vectors as well as roots. The reason for this is that this would include the reﬂection groups of the lattices I1,20 , I1,21 , I1,22 , I1,23 which were described in [B1]. The main theme of this paper is that lattices of negative signature that have a nonzero reﬂective modular form tend to be “interesting.” The meaning of “interesting” depends on the dimension of the maximal positive deﬁnite sublattice. For example, negative deﬁnite lattices might be similar to the Leech lattice, Lorentzian lattices should have interesting reﬂection groups, and lattices in R2,n should have interesting automorphic forms associated with them. The main reason for the deﬁnition of reﬂective forms is the following lemma.

342

RICHARD E. BORCHERDS

Lemma 11.1. Suppose that f is a reﬂective modular form for the lattice L. Then all singularities of the automorphic form of f on G(L) are orthogonal to negative norm roots of L. Proof. By [B5, Theorem 6.2], the singularities of 6 are orthogonal to vectors γ ∈ L such that γ 2 < 0, and the vector-valued form g(f )ρA (g)(1) 2 (Z)/ 0 (N) g∈SL

of f has a singularity of type q γ /2 eγ . By Lemma 3.2, if the vector-valued form has 2 such a singularity, then f has a singularity q γ /2 at some cusp a/c with the image of γ ∈ Ac∗ . By the deﬁnition of a reﬂective form, this implies that γ is a root of L . This proves Lemma 11.1. 2

Warning. If a level N modular form f is nonzero, it is possible that the corresponding vector-valued modular form is zero. And even if the vector-valued modular form has nontrivial singularities at cusps, it is possible that the corresponding automorphic form is zero. Although we normally expect a singularity of a vector-valued form to give a singularity of the corresponding automorphic form, it is possible that all such singularities correspond to vectors of an empty set, or it is possible that all singularities happen to cancel each other out. (This is related to the well-known problem that the image of a nonzero form under the theta correspondence can be zero.) For lattices of some given level N, it is usually easy to determine the reﬂective singularities, though the answer sometimes involves many different cases. The following lemma gives a simple condition for a singularity to be reﬂective, which in practice covers many cases. Lemma 11.2. Suppose a/c is a cusp of 0 (N), and suppose L is an even lattice of level equal to N. Then a pole of order 1 at a/c is reﬂective if (c, N) is a Hall divisor of N. Proof. The cusp a/c has width h = N/(c2 , N ) = N/(c, N). To show that qh−1 = q −1/ h is a reﬂective singularity at a/c, it is enough to show that all elements α ∈ Ac∗ with (α, α) ≡ −2/ h mod 2 satisfy hα = 0. But this is obvious because Ac∗ = Ac is the set of elements of A of order dividing h because (N, c) is a Hall divisor of N . This proves Lemma 11.2. 12. Examples. In this section we give some examples of lattices with reﬂective modular forms. For a given level, the method for ﬁnding such forms is as follows. (1) Work out the ring of modular forms for 02 (N), paying particular attention to the forms with zeros only at cusps. (2) Work out the possible reﬂective singularities for each possible discriminant form of level N .

REFLECTION GROUPS OF LORENTZIAN LATTICES

343

(3) Try to ﬁnd modular forms with reﬂective singularities for each discriminant form. (4) Try to ﬁnd lattices corresponding to these modular forms. (5) For each lattice with a nonzero reﬂective modular form, see if it is connected with any interesting reﬂection groups, automorphic forms, moduli spaces, or Lie algebras. The examples probably include most interesting cases for small prime level, but they become less and less complete as the level gets larger because the number of cases to consider becomes rather large. What usually seems to happen is that for each level there is some “critical” signature, with the property that almost all lattices up to that signature have nonzero reﬂective modular forms, but beyond that signature there are only a few isolated examples, usually with p-rank at most 2 for some prime p. For example, for level N = 1 the critical signature is −24, corresponding to the Leech lattice, the lattice I I1,25 whose reﬂection group was described by Conway, and so on, while for level N = 2 the critical signature is −16, corresponding to the Barnes-Wall lattice, and so on. There are several other methods for ﬁnding examples of lattices with reﬂective modular forms. First, the inverses of eta quotients of elements of Conway’s group of automorphisms of the Leech lattice are often reﬂective modular forms for various lattices. Second, Hauptmoduls of genus 0 subgroups of SL2 (R) are often reﬂective modular forms for lattices of signature 0; see the case N = 17 below for some examples of this. More generally, several other modular functions of genus 0 subgroups with poles of order 1 at some cusps are reﬂective modular forms for some lattices. (The restriction to genus 0 subgroups is just an observation: Most cases seem to be related to genus 0 subgroups. I do not know of a good theoretical reason for this.) Third, Martin [M] gave a list of many eta quotients with multiplicative coefﬁcients, and again many of these seem to be the inverses of reﬂective modular forms. (Note that there are many multiplicative eta quotients not on Martin’s list that also appear as reﬂective modular forms, because Martin restricted himself to forms of integral weight whose conjugate under the Fricke involution was also multiplicative.) Martin’s list contains many examples of high level, up to level 576, which suggests that there should be many reﬂective modular forms beyond the examples in this section. Many of the calculations with modular forms in this section were done using the PARI calculator (see [BBBCO]). In most of the examples below, the tables have the following meaning. “Group” describes the relevant subgroup of SL2 (Z); “index” gives its index in SL2 (Z); ν2 , ν3 , and ν∞ are the numbers of elliptic points of orders 2 and 3 and the numbers of cusps of a fundamental domain; and “genus” gives the genus of the corresponding compact Riemann surface. The second table in each section is a table of the cusps, with one line per cusp. The “cusps” column gives a representative cusp; “width” is the width of the cusp; “characters” lists the nontrivial values of characters on the normalized generator of the subgroup ﬁxing a cusp; “η” lists an eta function with

344

RICHARD E. BORCHERDS

a zero of order “zero” at this cusp and no other zero and with weight “weight” and character “character.” The Hilbert function is the rational function of x whose coefﬁcients give the dimensions of the spaces of modular forms of various weights. Sometimes we put in extra variables up and uθ , which describe the dimensions of spaces with various characters. We sometimes write qn for e(τ/n). N = 1. All the results we get for this case are well known, but this case is still useful as a warming-up exercise. 2 (Z). The The discriminant form A has order 1, and the group ˜ 0 (1) is just SL character χ is always 1. Table 1 group 0 (1)

index ν2 1

1

ν3

ν∞

genus

1

1

0

Table 2 cusps width i∞

1

η 124

zero weight 1

12

The ring of modular forms is a polynomial ring with generators given by the Eisenstein series E4 and E6 of weights 4 and 6, and the Hilbert function is 1/(1 − x 4 )(1 − x 6 ). The critical weight is k = 12, and the critical form is (τ ) = η(τ )24 . By looking at the forms E4 (τ )n /(τ ), we see that every even lattice L with N = 1 and sign(L) ≥ −24 has a nonzero reﬂective modular form. The signature must be 0 mod 8. Next we ﬁnd some possible reﬂective forms. By Lemma 11.2, poles of order at most 1 at the cusp are reﬂective singularities. The modular forms with poles of order at most 1 at the cusp are exactly those of the form (holomorphic modular form)/. So there are no examples of weight less than −12, and the ones of weights −12, −8, and −4 are multiples of 1/, E4 /, and E42 /. We now look at some of these cases in more detail. For signature −24 we take f to be 1/(τ ) = q −1 +24+· · · . We get an automorphic form for the lattice I I2,26 of singular weight 24/2 = 12. The corresponding Lie algebra is the fake monster Lie algebra. Its Weyl group is Conway’s reﬂection group of the lattice I I1,25 , which has a norm 0 Weyl vector. The critical lattice of this Weyl vector is of course the Leech lattice. The lattice I I4,28 has a reﬂective form, and it is the underlying integral lattice of Allcock’s largest quaternionic reﬂection group (see [A]). The lattices I In,16+n for various values of n > 2 appear in the moduli space of K3 surfaces, possibly with some extra structure such as a B-ﬁeld. The existence of a reﬂective form for these lattices

REFLECTION GROUPS OF LORENTZIAN LATTICES

345

appears to be signiﬁcant in the corresponding moduli spaces as it is usually necessary to discard points of the Grassmannian that are orthogonal to norm −2 vectors. For signatures −8 and −16, we ﬁnd the lattices I I1,9 and I I1,17 whose (arithmetic) reﬂection groups were ﬁrst described by Vinberg [V1]. For signature zero we take f to be j (τ ) − 744 = q −1 + 196884q + · · · . We get an automorphic function for the lattice I I2,2 , which is more or less j (σ ) − j (τ ) in suitable coordinates. The corresponding Lie algebra is the monster Lie algebra. The reﬂection group is not very interesting as it is of order 2 (and is the Weyl group of the monster Lie algebra). N = 2. The discriminant form A has order 22n for some nonnegative integer n. The character χA is always trivial as A always has square order and signature divisible by 4. The group 0 (2) = 02 (2) has two cusps that can be taken as i∞ (of width 1) and 0 (of width 2). It has one elliptic point of order 2, which can be take as the point (1 + i)/2, ﬁxed by 21 −1 −1 . Table 3 group

index ν2

0 (2)

3

1

ν3

ν∞

genus

0

2

0

Table 4 cusps width

η

0

2

116 2−8

i∞

1

1−8 216

zero weight 1

4

1

4

The ring of modular forms for 02 (2) = 0 (2) is a polynomial ring on the generators −E2 (τ ) + 2E2 (2τ ) = θD4 (τ ) = 1 + 24q + 24q 2 + O(q 3 ) of weight 2 with a zero at the elliptic point, and E4 (τ ) of weight 4. The Hilbert function is 1/(1 − x 2 )(1 − x 4 ). All poles of order at most 1 at cusps are reﬂective by Lemma 11.2 as N = 2 is square-free. If A = I I (2−2 ), then A has no nonzero elements of norm 1 mod 2, so a pole of order 2 at the cusp 0 is also reﬂective. (There are also other possible reﬂective singularities for lattices of high 2-rank.) By looking at the form 2+ (τ )−1 , with order 1 poles at all cusps, we see that all level 2 even lattices of signature at least −16 have reﬂective modular forms. The Lorentzian lattices I I1,17 (2+8 ) and I I1,17 (2+10 ) have norm 0 Weyl vectors; their reﬂection groups are not arithmetic but are similar to the case of I I1,25 . The remaining Lorentzian lattices of dimension at most 18 have positive norm Weyl vectors, so their reﬂection groups are arithmetic. The lattice I I1,17 (2+2 ) is the even sublattice of an odd unimodular lattice (see [V1]). The arithmetic reﬂection group of the lattice

346

RICHARD E. BORCHERDS

I I1,17 (2+4 ) appeared recently in Kondo and Keum’s work [KK] as the Picard lattice of the Kummer surface of a generic product of elliptic curves, and it can be obtained as the orthogonal complement of a D42 in I I1,25 . The reﬂection group of I I1,17 (2+6 ) seems to be the highest dimension of a “new” example of an arithmetic reﬂection group in this paper. This lattice has an unusually complicated fundamental domain, with 896+64 sides. It can be described as follows. Let K be the 16-dimensional even lattice in the genus I I0,16 (2+6 ) that has root system A16 1 . It can be constructed by applying construction A of [CS, Chapter 5] to the ﬁrst-order Reed-Muller code of length 16 with 32 elements (see [CS, page 129], which uses construction B applied to this code to construct the Barnes-Wall lattice). Then L = I I1,17 (2+6 ) can be constructed as K ⊕ I I1,1 . The fundamental domain of the reﬂection group R of L has two norm 0 vectors z, z of type K (together with many other norm 0 vectors of other lattices). The generalized Weyl vector is given by ρ = (z + z )/2. The fundamental domain has 64 walls corresponding to norm −2 roots of L and 896 walls corresponding to norm −4 roots of L (or equivalently to norm −1 roots of L ). The norm −2 simple roots split into two groups of 32. The ﬁrst group of 32 have inner product zero with z and −1 with z and correspond to the sixteen coordinate vectors of K and their negatives. The other 32 have inner product −1 with z and zero with z and correspond to the 32 elements of the Reed-Muller code. (There is, of course, an automorphism of the fundamental domain exchanging z and z and the two groups of 32 norm −2 simple roots.) The 896 norm −1 simple roots of L all have inner product 1 with both z and z and correspond to the elements of the dual of the Reed-Muller code whose weight is 2 mod 4. The Reed-Muller code has Hamming weight enumerator x 16 +30x 8 y 8 +y 16 ; so by the MacWilliams identity (see [CS, page 78]), the dual code has weight enumerator (x + y)16 + 30(x + y)8 (x − y)8 + (x − y)16 32 = x 16 + 140y 4 x 12 + 448y 6 x 10 + 870y 8 x 8 + 448y 10 x 6 + 140y 12 x 4 + y 16 , and therefore there are 448 + 448 = 896 elements of length 2 mod 4. Alternatively, L can be constructed as the orthogonal complement of a certain A81 in the Dynkin diagram of I I1,25 . (Note that the Dynkin diagram of I I1,25 contains more than one orbit of subsets isomorphic to A81 . The orbit we use has the special property that it is not contained in an A71 A2 subdiagram.) The 64 norm −2 roots correspond to the 64 ways to extend the A81 to an A91 , and the 896 norm −1 simple roots correspond to the 896 ways of extending it to an A3 A61 diagram. These A3 A61 diagrams are the same as those used by Kondo in [Kon] to describe the automorphism group of a generic Jacobian Kummer surface. The automorphism group of the Dynkin diagram of L has a normal subgroup of order 210 , and the quotient is the alternating group A8 . Unfortunately, I I1,17 (2+6 ) cannot be the Picard lattice of a K3 surface: Kondo pointed out to me that its 2-rank (6) is larger than its codimension (4) in I I3,19 .

REFLECTION GROUPS OF LORENTZIAN LATTICES

347

The lattices I In,n+20 (2−2 ) also have a reﬂective modular form θD4 (τ )/(τ ). In particular, the Lorentzian lattice I I1,21 (2−2 ) has an arithmetic reﬂection group (see [B1, page 149] for a description of its fundamental domain). Esselmann [Es] showed that this is essentially the only example of a coﬁnite reﬂection group of a Lorentzian lattice of dimension at least 21. (Of course, we can ﬁnd trivial variations of this example by taking the Atkin-Lehner conjugate I I1,21 (2−20 ) or by multiplying all norms by a constant.) We can ﬁnd some automorphic forms of singular weight, corresponding to lattices L and reﬂective modular forms f as follows. Case 1: L = I I2,18 (2+10 ), f = η1−8 2−8 . This is the denominator function of a generalized Kac-Moody algebra of rank 18. This example is related to the element 2A of Aut(J), of cycle shape 18 28 . There are 24 lattices in the genus I I0,16 (2+8 ) by [SV], one of which is the Barnes-Wall lattice, and the others all have root systems of rank 16. Case 2: L = I I2,10 (2+2 ), f = 24 η1−16 28 . This example is related to the element −2A of Aut(J). Case 3: L = I I2,10 (2+10 ), f = η18 2−16 . Case 1 is closely related to the reﬂection group of the lattice I I1,17 (2+8 ), whose fundamental domain has a nonzero norm 0 vector ﬁxed by its automorphism group, as in the lattice I I1,25 . For more about this lattice and its reﬂection group, see [B2]. The last two cases are really the same, since the lattices are Atkin-Lehner conjugates of each other, and the automorphic forms we get are more or less the same. This automorphic form is the denominator function of two generalized Kac-Moody superalgebras, and it is also closely related to the moduli space of Enriques surfaces. See [B5, Example 13.7], [Sch], and [B4] for more details. If R is the reﬂection group of the lattice I I1,17 (2+2 ) generated by the reﬂections of norm −2 vectors and if D is its fundamental domain, then Aut(D) has a ﬁnite-index subgroup isomorphic to Z and ﬁxes a nonzero norm 0 vector z. However, there seems to be no reﬂective modular form for 0 (2) corresponding to this reﬂection group. The remaining level 2 cases of signatures −4, −8, and −12 are left to the reader; they all give known arithmetic reﬂection groups, often associated to unimodular lattices as in [V1]. Table 5 group 0 (3)

index ν2 4

0

ν3

ν∞

genus

1

2

0

N = 3. The character χ3 is trivial for forms of even weight and nontrivial for forms of odd weight. The forms of integer weights and arbitrary character are the same as the forms for 1 (3) and trivial character. Note that 0 (3) is the product of 1 (3) and its center of order 2 is generated by Z.

348

RICHARD E. BORCHERDS

Table 6 cusps width

η

0

3

19 3−3

i∞

1

1−3 39

zero weight 1

3

1

3

The ring of modular forms for 02 (3) = 1 (3) is a polynomial ring generated by θA2 (τ ) = E1 (τ, χ3 ) = 1 + 6q + 6q 3 + 6q 4 + O(q 7 ) of weight 1 and by E3 (τ, χ3 ) = η(τ )−3 η(3τ )9 = q + 3q 2 + 9q 3 + 13q 4 + 24q 5 + O(q 6 ) of weight 3. The Hilbert function is 1/(1 − u3 x)(1 − u3 x 3 ). Next we ﬁnd some reﬂective singularities. At the cusp i∞, the singularity q −1 is reﬂective. At the cusp 0, the singularity q3−1 is reﬂective. If the discriminant form A is I I (3±1 ) or I I (3−2 ), then A has no nonzero elements of norm 0 mod 2, so the singularities q3−1 , q3−3 are reﬂective at the cusp zero. The forms 5A2 (τ )n /3+ show that all level 3 even lattices of signature at least −12 have nonzero reﬂective modular forms. There are also some other examples of reﬂective modular forms for lattices of small 3-rank. If we take the signature to be −18 and take A to be I I (3+1 ), then the form 5E6 (τ )/(τ ) is reﬂective. This can be used to show that the reﬂection group of the lattice I I1,19 (3+1 ) is arithmetic. This reﬂection group was ﬁrst found by Vinberg [V2] in his investigations of the “most algebraic” K3 surfaces. The lattice can also be constructed as the orthogonal complement of an E6 in I I1,25 , and this gives another proof that the reﬂection group is arithmetic (see [B1]). Next take the signature to be −14, and take A to be I I (3−1 ). We let f be the form E1 (τ, χ3 )5 − 270E1 (τ, χ3 )2 η(τ )−3 η(3τ )9 = q −1 − 216 − 9126q + O q 2 . (τ ) The constant 270 is chosen so that the coefﬁcient of q −2/3 = q3−2 of f (−1/τ ) = −i3−5/2 τ 3 (−9q −1 + 810q −1/3 + 1944 + 53136q 2/3 + O(q)) vanishes. So the automorphic forms with singularities constructed from f have all their singularities orthogonal to roots. However, something unexpected now happens: The C[A]-valued modular form induced from f is identically zero! So the piecewise linear automorphic forms, constructed from f as in Theorem 4.2, have no singularities and are also zero. In spite of this, the reﬂection group of L = I I1,15 (3−1 ) still has a nonzero vector (of norm 0) in the fundamental domain ﬁxed by the automorphism group of the fundamental domain. To see this, we represent L as the orthogonal complement of an A2 in I I1,17 . Then the quotient of Aut + (L) by the reﬂection group can be worked out using [B6, Theorem 2.7], and it turns out to be an inﬁnite dihedral group, which has an index 2 subgroup isomorphic to Z. Next we can classify the primitive norm 0 vectors z of L, and we ﬁnd that there are just two orbits, with the lattice z⊥ /z having

349

REFLECTION GROUPS OF LORENTZIAN LATTICES

root systems E8 E6 and D13 . Fix ρ to be a primitive norm 0 vector corresponding to a lattice with root system D13 . As D13 has a rank of one less than the corresponding lattice, there is a group Z of automorphisms of the fundamental domain ﬁxing z. This group Z has ﬁnite index in the full automorphism group of the fundamental domain, so the full automorphism group of the fundamental domain must ﬁx z. However, z is not quite a Weyl vector, as it has zero inner product with some simple roots (forming an afﬁne D13 Dynkin diagram) and has inner product 1 with the others. There are 10 lattices in the genus I I0,12 (3+6 ) (see [SV]). One is the Coxeter-Todd lattice with no roots, and the others all have root systems of rank 12. There are also some automorphic forms of singular weight corresponding to the following lattices and reﬂective forms: (1) I I2,14 (3−8 ), η1−6 3−6 ; (2) I I2,8 (3+7 ), η13 3−9 ; (3) I I2,8 (3+3 ), 32 η1−9 33 . The lattice I I2,8 (3+5 ) appears in [ACT], where it is the underlying integral lattice of a unimodular Eisenstein lattice. The automorphic forms for this case have been studied in great detail by Freitag in [AF] and [Fr]. In particular, there is one of weight 12 (coming from the function 27η1−9 33 ) whose restriction to the complex hyperbolic space CH 4 vanishes (to order 3) exactly along the reﬂection hyperplanes of a certain complex reﬂection group related to the moduli space of cubic surfaces. So its cube root is an automorphic form of weight 4 with order 1 zeros along all the reﬂection hyperplanes (see [A]). Table 7 group 0 (4)

index ν2 6

0

ν3

ν∞

genus

0

3

0

Table 8 cusps

width characters

η

zero weight character

4

18 2−4

1

2

1/2

1

1−2 25 4−2

1/4

1/2

1/4 = i∞

1

2−4 48

1

2

1=0

χθ = −i

χθ

We have seen above that sometimes the Weyl vector of a reﬂective form unexpectedly vanishes because all the singularities just happen to cancel out. Another way that the Weyl vector can unexpectedly vanish is if the vectors corresponding to the singularities happen not to exist (usually when the p-rank of A is small). For example, for the lattice L = I I2,8 (3−1 ), the automorphic form is constant even though the vector-valued modular form has nontrivial singularities. The singularities of the

350

RICHARD E. BORCHERDS

vector-valued modular form imply that the automorphic form has zeros corresponding to all norm 4/3 vectors of L , but L happens to have no such vectors so the automorphic form is constant. N = 4. The group 0 (4) has three cusps that can be taken as i∞ (of width 1), 0 (of width 4), and 1/2 (of width 1). It has no elliptic points and has genus 0. The double cover of 0 (4) is the product of its center of order 4 (generated by Z) and a subgroup that can be identiﬁed with 1 (4). The ring of modular forms of integral or half-integral weight for 1 (4) is a polynomial ring generated by θA1 (τ ) = 1 + 2q + 2q 4 + O(q 9 ) of weight 1/2 and η(τ )8 η(2τ )−4 = 1 − 8q + · · · of weight 2. The Hilbert function is 1/(1 − uθ x 1/2 )(1 − x 2 ). The ideals of cusp forms vanishing at i∞, 0, or 1/2 are generated by η(2τ )−4 η(4τ )8 , η(τ )8 η(2τ )−4 , η(τ )−2 η(2τ )5 η(4τ )−2 . Note that the last function has a zero of order 1/4 at 1/2. The ideal of cusp forms of even weight is generated by 4+ (τ ) = η(2τ )12 of weight 6. If L is a unimodular positive deﬁnite lattice, then θL (2τ ) is a modular form for 1 (4). Next we ﬁnd some reﬂective singularities. To reduce the number of cases to consider, we assume that A = L /L has exponent 2, so that A is I I (2t+n ) for some t and n. As usual, poles of order 1 are reﬂective singularities at the cusps i∞ and 0. At the cusp 1/2, poles of order 1/4 and 1/2 are reﬂective, because all elements of A2∗ have order 1 or 2. If the parity vector of A does not have norm 0 mod 2, then a pole of order 1 at 1/2 is also reﬂective. Finally, at the cusp 0, poles of order 1 or 2 are reﬂective, and if A has no nonzero vectors of norm 0 mod 2, then poles of order 4 are reﬂective. The form η(τ )−12 η(2τ )−2 η(4τ )4 shows that all level 4 exponent 2 lattices of signature −14 have nonzero reﬂective modular forms. By multiplying this form by θA1 (τ )n for n ≥ 1, we see that the level 4 lattices of signature at least −14 have nonzero reﬂective modular forms. We can ﬁnd many examples of eta quotients that are eigenforms of Hecke operators by ﬁnding eta quotients with poles of order at most 1 at all cusps. This gives ﬁfteen nonconstant examples as follows: η1−2 25 4−2 , η1−4 210 4−4 , η1−8 220 4−8 , η18 2−4 , η16 21 4−2 , η14 26 4−4 , η216 4−8 , η2−4 48 , η1−2 21 46 , η1−4 26 44 , η1−8 216 , η1−8 28 4−8 , η1−6 23 4−6 , η1−4 2−2 4−4 , η2−12 . The inverses of these forms are often reﬂective forms for various lattices. Note that the forms η1−6 215 4−6 , η12 211 4−6 , η1−6 211 42 , η12 27 42 are eigenfunctions of Hecke operators, but as they have a zero of order 3/4 at 1/2, their inverses do not usually give reﬂective automorphic forms (except for rather special discriminant forms). The form η12 27 42 is the highest-weight eta product I know of that is an eigenform and has nonintegral weight. Most of the time lattices of positive signature with reﬂective forms do not seem to be interesting, but there are some exceptions. For example, there is a reﬂective form for the lattice I I2,1 (21+1 ). The corresponding automorphic form is essentially

REFLECTION GROUPS OF LORENTZIAN LATTICES

351

E6 , which is the denominator function of a generalized Kac-Moody algebra. See [B3, Section 15, Example 2] for more details. The lattices I I1,19 (26+2 ), I I1,15 (22+2 ), I I1,11 (26+2 ) have reﬂective forms of type θDn /. They are the even sublattices of odd unimodular lattices and have coﬁnite reﬂection groups, as was ﬁrst found by Vinberg [V1]. +1 The function θE7 / is a reﬂective form for the lattices I In,n+17 (2−1 ). In particular, +1 ) whose reﬂection group we ﬁnd Nikulin’s example of the Lorentzian lattice I I1,18 (2−1 is arithmetic. This example can also be constructed as the orthogonal complement of an E7 in I I1,25 . Yoshikawa told me that he has used the automorphic forms coming from the modular forms η(τ )−8 η(2τ )8 η(4τ )−8 θA1 (τ )k to construct automorphic products (for odd unimodular lattices). These automorphic products are the squares of discriminant forms of various moduli spaces of “generalized Enriques surfaces” and can also be constructed using analytic torsion. Table 9 group

index ν2

0 (5)

6

2

ν3

ν∞

genus

0

2

0

Table 10 cusps width

η

zero weight

0

5

15 5−1

1

2

i∞

1

1−1 55

1

2

N = 5. The ring of modular forms for 02 (5) is not a polynomial ring but is generated by the three-dimensional space of weight 2 forms, which is spanned by η(τ )5 η(5τ )−1 , η(τ )−1 η(5τ )5 (of nontrivial character) and E2 (τ )−5E2 (5τ ) (of trivial character). The Hilbert function is (1 + x 2 )/(1 − u5 x 2 )2 . Remark. The ring of all modular forms of integral weight for 1 (5) is a polynomial ring generated by the weight 1 Eisenstein series 1+(3+i)(q +(1−i)q 2 +(1+i)q 3 − iq 4 + q 5 + O(q 6 )) and its complex conjugate. These correspond to the two complex conjugate order 4 characters of Z/4Z, and each of them has a simple zero at one of the elliptic points and no other zeros. The subring of forms of even weight is the ring of modular forms for 02 (5). Even lattices of level 5 all have signature divisible by 4. The form 5 (τ )−1 shows that all level 5 even lattices of signature −8 and even 5-rank have nonzero reﬂective modular forms. (So does the lattice of 5-rank 1; see below.) If we multiply this form by products of powers of E2 (τ ) − 5E2 (5τ ) and η(τ )5 η(5τ )−1 , we also see that all even level 5 lattices of signature at least −4 have nonzero reﬂective modular forms.

352

RICHARD E. BORCHERDS

For the discriminant forms A = I I (5±1 ) or I I (5+2 ), the only norm 0 element is zero, so q5−1 and q5−5 are all reﬂective singularities at the cusp 0. If we take A to be I I (5−1 ) and take the signature to be −8, then there is a reﬂective form. This gives a Lorentzian lattice I I1,9 (5−1 ) with a reﬂection group of ﬁnite index. If we take L to be I I1,17 (5−1 ), then there is a reﬂective automorphic form. (This is slightly surprising as the space of forms with a pole of order at most 1 at i∞ and a pole of order at most 5 at zero is two-dimensional, so we would normally expect there to be no such forms satisfying the two conditions that the coefﬁcients of q5−2 and q5−3 both vanish. However, it turns out that these two conditions are not independent; in fact the modular form we get has “complex multiplication” (see [Ri]), meaning that the coefﬁcient of q5n is zero whenever n ≡ 2, 3 mod 5.) In spite of the existence of a nonzero reﬂective modular form, the reﬂection group of I I1,17 (5−1 ) is not coﬁnite and does not even have virtually free abelian index. (In particular, this lattice is a counterexample to several otherwise plausible conjectures about Lorentzian lattices with nonzero reﬂective modular forms.) As a substitute for this, the lattice is very closely related to Bugaenko’s largest example of a cocompact hyperbolic reﬂection group. In fact I I1,17 (5−1 ) can be made into a lattice over Z[φ], and Bugaenko [Bu] showed that the corresponding hyperbolic reﬂection group was cocompact. The relationship between Bugaenko’s reﬂection group and the reﬂective form is rather mysterious. The lattice has ﬁve orbits of primitive norm 0 vectors, corresponding to the ﬁve elements of the genus I I0,16 (5−1 ), which have root systems A2 A14 , E7 A9 , E6 D9 , E8 E7 5A1 , D14 A1 5A1 . It is possible to produce some examples of cocompact hyperbolic reﬂection groups from level 5 lattices as follows. Lemma 12.1. Suppose that L is an even Lorentzian lattice of level 5, and suppose that there is a self-adjoint endomorphism φ of L such that φ 2 = φ +1. Let H φ be the hyperbolic space of the Lorentzian eigenspace (L ⊗ R)φ . Then the subgroup of the reﬂection of L acting on H φ is a hyperbolic reﬂection group of H φ . If W is coﬁnite, then W φ is cocompact. Proof. Let H be the hyperbolic space of L, and let H φ be the subspace of it ﬁxed by φ. The main point is that the intersection of any reﬂection hyperplane of W with H φ is a reﬂection hyperplane of the group W φ acting on H φ . To see this, recall that a reﬂection of W is the reﬂection of a norm −2 vector of L or a norm −2/5 vector of L . First suppose that v is a norm −2 vector of L. As v ⊥ intersects H φ , v φ⊥ and v must generate a negative deﬁnite space. This easily implies that v and v ⊥ span a lattice isomorphic to A21 , and the product of two reﬂections of this lattice is the automorphism −1, which commutes with φ. This is a reﬂection of W φ acting on H φ whose reﬂection hyperplane is v ⊥ ∩H φ . The argument when v is a norm −2/5 vector of L is similar. It now follows that W φ is a reﬂection group acting on H φ whose fundamental domain is the intersection of H φ with a fundamental domain of W acting on H .

REFLECTION GROUPS OF LORENTZIAN LATTICES

353

Finally, if W is coﬁnite, then all norm 0 vectors in the fundamental domain of W are rational and therefore cannot be ﬁxed by φ, so the fundamental domain of W φ has no norm 0 vectors in it and is therefore compact. This proves Lemma 12.1. Unfortunately, this lemma does not give the largest examples found by Bugaenko. If we take A to be I I (5+1 ) and take the signature to be −12, then q5−4 is a reﬂective singularity at zero as A has no nonzero elements of norm −4/5 mod 2, and q5−5 is reﬂective as any norm 0 element of A is zero. So A has a reﬂective modular form of weight −6, level 5, and character χ5 whose singularity at i∞ is a multiple of q −1 and whose singularity at zero is a linear combination of q5−1 , q5−4 , and q5−5 . (There is a two-dimensional space of such forms.) This gives a Lorentzian lattice L = I I1,13 (5+1 ) with Aut + (L)/R inﬁnite dihedral. It is the orthogonal complement of an A4 in I I1,17 . This case is similar to I I1,15 (3−1 ). The (level 1) form E6 / has a singularity at zero of the form q −1 = q5−5 so it is a reﬂective modular form for the lattice L = I I1,13 (5−2 ). The corresponding vectorvalued modular form is zero. The lattice L is a module over Z[φ]. It may be the lattice of the orthogonal complement of an I2 (5) in Bugaenko’s lattice, which would imply that it has a cocompact reﬂection group. As in the case N = 3, we also get a few examples of automorphic forms of singular weight coming from the reﬂective forms η1−4 5−4 , η1−1 55 , and η15 5−1 . There are ﬁve lattices in the genus I I0,8 (5+4 ) corresponding to the case I I2,10 by [SV]. One has no roots and by [SH, page 744], the root systems of the other four are A41 5A41 , A22 5A22 , A4 5A4 , D4 5D4 . Problem. Does the lattice I I2,6 (5+3 ) correspond to some nice moduli space in the same way that the corresponding lattices I I2,10 (2+2 ) and I I2,8 (3+5 ), for levels 2 and 3, correspond to the moduli spaces of Enriques surfaces or cubic surfaces? Table 11 group 0 (6)

index ν2 12

0

ν3

ν∞

genus

0

4

0

Table 12 width

η

6

16 2−3 3−2 61

1

1

1/2

3

1−3 26 31 6−2

1

1

1/3

2

1−2 21 36 6−3

1

1

1/6 = i∞

1

11 2−2 3−3 66

1

1

cusps 1=0

zero weight

354

RICHARD E. BORCHERDS

N = 6. The group 0 (6) is the product of 1 (6) and its center of order 2 generated by Z. The forms of trivial character have even weight, and those of nontrivial character have odd weight. The ring of modular forms of integral weight for 0 (6) is a polynomial ring generated by E1 (τ, χ3 ) and E1 (2τ, χ3 ). The Hilbert function is 1/(1 − u3 x)2 . The ideal of cusp forms is generated by 6+ (τ ) = η(τ )2 η(2τ )2 η(3τ )2 η(6τ )2 of weight 4. We can ﬁnd eta quotients with a given integral-order pole at the cusps. In particular, we ﬁnd the following ﬁfteen nonconstant holomorphic eta quotients with zeros of order at most 1 at all cusps: η16 2−3 3−2 61 , η1−3 26 31 6−2 , η1−2 21 36 6−3 , η11 2−2 3−3 66 , η13 23 3−1 6−1 , η1−1 2−1 33 63 , η14 2−2 34 6−2 , η1−2 24 3−2 64 , η17 2−5 3−5 67 , η1−5 27 37 6−5 , η11 24 35 6−4 , η14 21 3−4 65 , η15 2−4 31 64 , η1−4 25 34 61 , η12 22 32 62 . Their inverses give numerous examples of reﬂective forms for various lattices. For example, η1−2 2−2 3−2 6−2 is a reﬂective form for all even level 6 lattices of signature −8, and by multiplying by a power of E1 (τ, χ3 ) we get reﬂective forms whenever the signature is at least −8. The other eta quotients give many examples where roots of certain norms are excluded. We can also ﬁnd many examples of signature less than −8 if we restrict the 2-rank or 3-rank to be at most 2. One example of such a lattice with an arithmetic reﬂection group is the orthogonal complement I I1,15 (2−2 3−1 ) of a D4 E6 root system in I I1,25 . Table 13 group

index ν2

0 (7)

8

0

ν3

ν∞

genus

2

2

0

Table 14 cusps width

η

0

7

17 7−1

i∞

1

1−1 77

zero weight 2

3

2

3

N = 7. The ring of modular forms of integral weight for 02 (7) is generated by E1 (τ, χ7 ) (of weight 1, which vanishes at both elliptic points), 7+ (τ ) (of weight 3, which vanishes at both cusps), and the two weight 3 Eisenstein series. The Hilbert function is (1 + u7 x 3 )/(1 − u7 x)(1 − u7 x 3 ). The ideal of cusp forms is generated by 7+ (τ ) = η(τ )3 η(7τ )3 of weight 3. Note that the ideal of forms vanishing at i∞ is not principal. The function η(τ )4 η(7τ )−4 is a Hauptmodul for 0 (7). Remark. The ring of modular forms for 1 (7) of integral weight has a simpler structure: it is generated by the three weight 1 forms E1 (τ, χ7 ), E1 (τ, χ), E1 (τ, χ), ¯ where χ is a character of Z/7Z of order 6 and χ¯ is its complex conjugate. The ideal of relations between these generators is generated by E1 (τ, χ7 )2 −E1 (τ, χ)E1 (τ, χ). ¯ We can even embed this into a polynomial ring of modular forms: All the zeros of

355

REFLECTION GROUPS OF LORENTZIAN LATTICES

the forms E1 (τ, χ ) and E1 (τ, χ) ¯ have order 2, so their square roots are also modular forms (of half-integral weight for a strange character of 0 (7)), and they generate a polynomial ring whose elements of integral weight are the modular forms for 1 (7). Any even lattice of level 7 and signature at least −6 has a reﬂective form of the form E1 (τ, χ7 )n η1−3 7−3 . The automorphic form associated to η1−3 7−3 and to the lattice I I2,8 (7+5 ) has singular weight and is the denominator function of a generalized Kac-Moody algebra. The reﬂection group of I I1,7 (7−3 ) has a norm 0 Weyl vector. The corresponding genus I I0,6 (7−3 ) has three elements [SH, Proposition 3.4a], and by [SH, Table 1] there is one with no roots (corresponding to the norm 0 Weyl vector), one with root system A3 7A3 , and one with root system A31 7A31 . For the discriminant form I I (7−1 ), the singularities q7−1 and q7−7 are reﬂective. The lattice I I1,11 (7−1 ) has a reﬂective form and is the orthogonal complement of an A6 in I I1,17 . The quotient Aut+ (L)/R is inﬁnite dihedral and ﬁxes a norm 0 vector corresponding to a lattice in the genus I I0,10 (7−1 ) with root system D9 . There is a second lattice in this genus, isomorphic to the sum of E8 , and a two-dimensional deﬁnite lattice of determinant 7, so its root system is E8 A1 7A1 . Table 15 group 0 (8)

index ν2 12

0

ν3

ν∞

genus

0

4

0

Table 16 cusps

width

1=0

8

1/2

2

1/4

1

1/8 = i∞

1

characters

η 14 2−2

zero weight character 1

1

χθ2

χ2 = χθ = −1 1−2 25 4−2

1/2

1/2

χθ

2−2 45 8−2

1/2

1/2

χ θ χ2

4−2 84

1

1

χθ2

χ2 = −1

N = 8. The double cover of 0 (8) is the product of its center of order 4 (generated by Z) and a subgroup that can be identiﬁed with its image 0 (8)∩1 (4). The ring of modular forms of integral or half-integral weight for 02 (8) = 1 (8) is a polynomial ring generated by η(2τ )−2 η(4τ )5 η(8τ )−2 (of weight 1/2 and character χ2 χθ ) and η(τ )−2 η(2τ )5 η(4τ )−2 (of weight 1/2 and character χθ ). The Hilbert function is 1/(1 − uθ u2 x 1/2 )(1 − uθ x 1/2 ). There are many level 8 discriminant forms and many possible reﬂective singularities. Together they give a bewildering number of examples of level 8 lattices with reﬂective forms; they are probably best left to a computer to classify. As examples, we

356

RICHARD E. BORCHERDS

just mention I I1,18 (47+1 ), I I1,16 (41+1 ), I I1,14 (43−1 ), I I1,12 (45−1 ). These are the even sublattices of some of the odd unimodular lattices with coﬁnite reﬂection groups found by Vinberg and Kaplinskaja [VK]. Table 17 group 0 (9)

index ν2 12

0

ν3

ν∞

genus

0

4

0

Table 18 width

η

1=0

9

13 3−1

1

1

1/3, 2/3

1

1−3 310 9−3

1,1

2

1

3−1 93

1

1

cusps

1/9 = i∞

zero weight

N = 9. The group 0 (9) is the product of its center of order 2 and the group so a form has nontrivial character if and only if it has odd weight. The ring of modular forms of integral weight for 02 (9) is a polynomial ring generated by η(τ )3 η(3τ )−1 and η(9τ )3 η(3τ )−1 . The Hilbert function is 1/(1 − x)2 . For the sake of completeness, we also describe generators for the ring of all integral weight modular forms for 1 (9). This is a three-dimensional free module over the ring of modular forms for 02 (9), with a basis consisting of 1 and the two weight 1 Eisenstein series for the two order 6 characters of Z/6Z. Each of these Eisenstein series has zeros of order 1/3 and 2/3 at the cusps 1/3 and 2/3 (not necessarily in that order). Note that the character of a modular form can be read off from the parity of its weight and the fractional part of the order of the zero at 1/3. We can embed this ring in a polynomial ring, generated by the cube roots of the two weight 1 modular forms with poles of order 1 at 1/3 or 2/3. The form η(τ )−3 η(3τ )2 η(9τ )−3 is a reﬂective form for all even level 9 lattices of signature −4, and by multiplying it by a suitable power of, say, θA2 (τ ), we get reﬂective forms whenever the signature is at least −4. A few examples of modular forms that might correspond to automorphic forms of singular weight on some lattices are η3−8 , η1−3 31 , η31 9−3 , and η1−3 32 9−3 .

02 (9),

N = 10. The ring of modular forms of integral weight for 02 (10) is generated by the seven-dimensional space of forms of weight 2, and the ideal of relations between them is generated by ﬁfteen quadratic relations. The Hilbert function is (1 + 5x 2 )/(1 − x)2 . Remark. We can embed the ring of modular forms for 02 (10) in a polynomial ring as follows. The ring of modular forms of integral weight for 1 (10) is generated

REFLECTION GROUPS OF LORENTZIAN LATTICES

357

Table 19 index ν2

group 0 (10)

18

ν3

ν∞

genus

0

4

0

2

Table 20 width

η

10

110 2−5 5−2 101

3

2

1/2

5

1−5 210 51 10−2

3

2

1/5

2

1−2 21 510 10−5

3

2

1

11 2−2 5−5 1010

3

2

cusps 1=0

1/10 = i∞

zero weight

by the four-dimensional space of forms of weight 1. We can ﬁnd two of these forms that have zeros of order 3 at the two elliptic points of order 2. Their cube roots are modular forms of weight 1/3 for characters of order 3 of 1 (10), and they generate a polynomial ring in two variables. The space of modular forms for 02 (10) can be identiﬁed with the polynomials of degree divisible by 6. The cube root of the product of any three of the forms above with zeros only at one cusp is a form with a zero of order 1 at three of the four cusps, and these four forms are a basis of the space of weight 2 forms with nontrivial character. Their inverses are the functions η1−1 2−2 5−3 102 , η1−2 2−1 52 10−3 , η1−3 22 5−1 10−2 , η12 2−3 5−2 10−1 . Any one of these four functions (of nontrivial character and weight −2) shows that any even level 10 lattice of signature −4 and odd 5-rank has a reﬂective form. There is also a weight −2 form of trivial character whose poles and zeros are a pole of order 1 at each cusp and order 1 zeros at the two elliptic points. (Construction: take a linear combination of the weight 2 Eisenstein series with trivial character that vanishes at two cusps (this automatically vanishes at the two elliptic points), then divide it by an eta product with order 2 zeros at these cusps and order 1 zeros at the other two cusps.) This is a reﬂective form for the even level 10 lattice of signature −4 with even 5-rank. So every even level 10 lattice of signature greater than or equal to −4 has a reﬂective form. N = 11. The ring of modular forms of integral weight for 02 (11) is generated by E1 (τ, χ11 ) (of weight 1, which vanishes at both elliptic points), 11+ (τ ) (of weight 2, which vanishes at both cusps), and the two weight 3 Eisenstein series. The Hilbert function is (1 + x 3 )/(1 − x)(1 − x 2 ). The ideal of cusp forms is generated by 11+ (τ ) = η(τ )2 η(11τ )2 of weight 2. The ideal of forms vanishing at i∞ is not principal. The function η(τ )12 η(11τ )−12 has a pole of order 5 at i∞, a zero of order 5 at zero, and is a modular function for 0 (11), showing that zero is a torsion point

358

RICHARD E. BORCHERDS

of order 5 on the modular elliptic curve of 0 (11). (In fact this point generates the subgroup of rational points on this elliptic curve.) Table 21 index ν2

group 0 (11)

12

ν3

ν∞

genus

0

2

1

0

Table 22 η

cusps width

zero weight

0

11

111 11−1

i∞

1

1−1 1111

5

5

5

5

The forms E1 (τ, χ11 )n η1−2 11−2 show that all even lattices of level 11 and signature at least −4 are reﬂective. The lattice I I1,7 (11−1 ) has a reﬂection group of inﬁnite dihedral index in its automorphism group. The corresponding genus I I0,6 (11−1 ) contains just one lattice, which has root system D5 . Table 23 index ν2

group 0 (12)

24

0

ν3

ν∞

genus

0

6

0

Table 24 cusps

width characters

η

zero weight

12

16 2−3 3−2 61

2

1

1/2

3

1−6 215 32 4−6 6−5 122

2

1

1/3

4

1−2 21 36 6−3

2

1

1/4

3

2−3 46 61 12−2

2

1

1/6

1

12 2−5 3−6 42 615 12−6

2

1

1/12 = i∞

1

21 4−2 6−3 126

2

1

1=0

χθ = i

χθ = −i

N = 12. The ring of modular forms is generated by the forms θ(τ ), θ(3τ ), E1 (τ, χ3 ), and E1 (2τ, χ3 ), and the Hilbert function is (1 + x 1/2 + 2x)/(1 − x 1/2 )(1 − x). There are quite a lot of modular forms whose zeros are all zeros of order at most 1 at cusps: We can ﬁnd forms with zeros of order 1/2 at 1/2 and 1/6 and an odd

REFLECTION GROUPS OF LORENTZIAN LATTICES

359

number of zeros at the other cusps, or we can ﬁnd forms whose zeros at 1/2 and 1/6 have orders (0, 0), (1/4, 3/4), (3/4, 1/4), or (1, 1) and that have an even number of zeros at the other cusps. The maximum weight of these forms is 3, attained by the form η(2τ )3 η(6τ )3 with a zero of order 1 at every cusp. The inverses of these forms are reﬂective forms for many lattices of signature up to −6. Table 25 index ν2

group 0 (13)

14

ν3

ν∞

genus

2

2

0

2

Table 26 cusps width 0 i∞

η

zero weight

13

113 13−1

7

6

1

1−1 1313

7

6

N = 13. The Hilbert function is (1+2x 2 +6x 4 +5x 6 )/(1−x 2 )(1−x 6 ). The space of weight 2 forms for 02 (13) is three-dimensional, spanned by E2 (τ ) − 13E3 (13τ ), E2 (τ, χ13 ), and the cusp form E1 (τ, χ)2 − E2 (τ, χ) ¯ 2 , where χ is an order 4 char∗ acter of (Z/13Z) . The ring of modular forms for 1 (13) is generated by the sixdimensional space of forms of weight 1, which has a basis of the six forms E1 (τ, χ) as χ runs through the six odd characters of (Z/13Z)∗ . Each of these weight 1 Eisenstein series has a zero at an elliptic point of order 2, two zeros at elliptic points of order 3, and no other zeros. The function η12 13−2 is a Hauptmodul for 0 (13). There is also a Hauptmodul for 0 (13)+. These give automorphic forms for the lattice I I2,2 (13+2 ) which are the denominator functions for generalized Kac-Moody algebras related to elements of order 13 in the monster group. There are no modular forms of negative weight with poles of order at most 1 at the cusps, as can be seen from the relation index 7 number of zeros = weight × = weight × 12 6 and the fact that the weight is even and there are only two cusps. The space of cusp forms of weight 4 and character χ13 has dimension 2, and as this is the space of obstructions to ﬁnding a form of weight −2 and character χ13 with given singularities, we see that there is a nonzero form of weight −2 and character χ13 whose singularities are a pole of order 1 at i∞ and a singularity at zero with terms involving only q −1 and q −3 . This is a reﬂective form for the lattices I In,4+n (13+1 ). The lattice L = I I1,5 (13+1 ) is one of the lattices with Aut(L)/R(L) inﬁnite dihedral. This is the orthogonal complement of an A12 in I I1,17 . There is a unique

360

RICHARD E. BORCHERDS

lattice in the genus I I0,4 (13+1 ), and it has root system D3 . This can be seen from the fact that all such lattices are the orthogonal complement of an A12 in an even 16-dimensional self-dual negative deﬁnite lattice. N = 14. The group 0 (14) is the product of 02 (14) and its center of order 2 generated by Z. Table 27 group

index ν2

0 (14)

24

0

ν3

ν∞

genus

0

4

1

Table 28 width

η

1=0

14

114 2−7 7−2 141

6

3

1/2

7

1−7 214 71 14−2

6

3

1/7

2

1−2 21 714 14−7

6

3

1/14 = i∞

1

11 2−2 7−7 1414

6

3

cusps

zero weight

The ring of modular forms of integral weight for 02 (14) is generated by E1 (τ, χ7 ), E1 (2τ, χ7 ), and 14+ (τ ). The Hilbert function is (1 + x 2 )/(1 − x)2 . The ideal of cusp forms is generated by 14+ (τ ) = η(τ )η(2τ )η(7τ )η(14τ ) of weight 2. We can construct some automorphic forms on lattices of signature −2 or −4 of singular weight from the modular forms η1−2 21 7−2 141 , η11 2−2 71 14−2 , and η1−1 2−1 7−1 14−1 . Table 29 group

index ν2

0 (15)

24

0

ν3

ν∞

genus

0

4

1

Table 30 width

η

15

115 3−5 5−3 151

8

4

1/3

5

1−5 315 51 15−3

8

4

1/5

3

1−3 31 515 15−5

8

4

1

11 3−3 5−5 1515

8

4

cusps 1=0

1/15 = i∞

zero weight

361

REFLECTION GROUPS OF LORENTZIAN LATTICES

N = 15. This case seems very similar to the case N = 14. The ring of modular forms of integral weight for 02 (15) is generated by E1 (τ, χ3 ), E1 (5τ, χ3 ), and 15+ (τ ). The Hilbert function is (1 + x 2 )/(1 − x)2 . The ideal of cusp forms is generated by 15+ (τ ) = η(τ )η(5τ )η(3τ )η(15τ ) of weight 2. We can construct some automorphic forms on lattices of signature −2 or −4 of singular weight from the modular forms η1−2 31 51 15−2 , η11 3−2 5−2 151 , and η1−1 3−1 5−1 15−1 . Table 31 group 0 (16)

index ν2 24

0

ν3

ν∞

genus

0

6

0

Table 32 η

zero

16

12 2−1

1

1/2

χθ

1/2

4

1−2 25 4−2

1

1/2

χθ

1/4, 3/4

1

2−2 45 8−2

1/2,1/2

1/2

χ 2 χθ

1

1/2

χθ

1

1/2

χθ

cusps 1=0

width characters

χ2 = −1

1/8

1

4−2 85 16−2

1/16 = i∞

1

8−1 162

weight character

N = 16. Note that the isomorphism class of a cusp a/c is no longer always determined by (c, 16). The ring of modular forms of integral or half-integral weight for 02 (16) is generated by the three-dimensional space of forms of weight 1/2, which is spanned by the ﬁve forms listed above. The Hilbert function is (1 + χ2 χθ x 1/2 )/(1 − χθ x 1/2 )2 . As in the case N = 8, there seem to be rather a lot of examples. Table 33 group 0 (17)

index ν2 18

2

ν3

ν∞

genus

0

2

1

N = 17. All modular forms for 02 (17) have even weight. The Hilbert function is (1 + (1 + 2u17 )x 2 + (3 + 4u17 )x 4 + x 6 )/(1 − x 2 )(1 − x 4 ). The group 0 (17)+ has genus 0, and its Hauptmodul q −1 + 7q + 14q 2 + O(q 3 ) has poles of order 1 at all cusps. The Hauptmodul with poles of order 1 at all cusps is a reﬂective modular form for lattices I In,n (17±2m ). For example, we get automorphic forms for the lattice

362

RICHARD E. BORCHERDS

I I2,2 (17+2 ). The automorphic form for I I2,2 (17+2 ) is the denominator function of a generalized Kac-Moody algebra associated with an element of order 17 of the monster group. We get similar statements if we replace 17 by any of the other primes p = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, or 71 such that 0 (p)+ has genus 0. Table 34 group 0 (18)

index ν2 36

0

ν3

ν∞

genus

0

8

0

Table 35 width

η

1=0

18

16 2−3 3−2 61

3

1

1/2

9

1−3 26 31 6−2

3

1

1/3, 2/3

2

1−6 23 320 6−10 9−6 183

3,3

2

1/6, 5/6

1

13 2−6 3−10 620 93 18−6

3,3

2

1/9

2

3−2 61 96 18−3

3

1

1

31 6−2 9−3 186

3

1

cusps

1/18 = i∞

zero weight

N = 18. The Hilbert function is (1 + 2u3 x)/(1 − u3 x)2 , and the ring of modular forms is generated by the four-dimensional space of forms of weight 1, which is spanned by E1 (τ, χ3 ), E1 (2τ, χ3 ), E1 (3τ, χ3 ), and E1 (6τ, χ3 ). There are many eta quotients with poles of order 1 at all cusps: For any set of three or six cusps (with multiplicities) such that cusps with the same denominator have the same multiplicity, there is an eta quotient of weight 1 or 2 with these zeros. This gives twelve such eta quotients of weight 1 and eight of weight 2. There are four of weight 1 with no poles at the cusps 1/3, 2/3, 1/6, or 5/6, and the inverses of these are reﬂective forms for even level 18 lattices of signature −2. Table 36 group 0 (20)

index ν2 36

0

ν3

ν∞

genus

0

6

1

N = 20. The space of weight 1/2 forms is spanned by η1−2 25 4−2 (zero of order 5/4 at 1/2 and order 1/4 at 1/10) and η5−2 105 20−2 (zero of order 1/4 at 1/2 and order 5/4 at 1/10). The space of cusp forms of weight 2 and trivial character is spanned by η22 102 ; it has zeros of order 1 at all cusps.

363

REFLECTION GROUPS OF LORENTZIAN LATTICES

Table 37 cusps

η

width characters

zero weight

110 2−5 5−2 101

6

2

1−10 225 4−10 52 10−5 202

6

2

1=0

20

1/2

5

1/4

5

2−5 410 101 20−2

6

2

1/5

4

1−2 21 510 10−5

6

2

1/10

1

12 2−5 42 5−10 1025 20−10

6

2

21 4−2 10−5 2010

6

2

1/20 = i∞

χθ = −i

χθ = −i

1

The form 11 21 4−1 5−1 101 201 has zeros of order 1/2 at 1/2 and 1/10 and zeros of order 1 at 1 and 1/20. The form 1−1 21 41 51 101 20−1 has zeros of order 1/2 at 1/2 and 1/10 and zeros of order 1 at 1/4 and 1/5. Their inverses are reﬂective forms for even lattices of level 20, signature −2, and even 5-rank. Table 38 group

index ν2

0 (23)

24

ν3

ν∞

genus

0

2

2

0

Table 39 cusps width 0 i∞

η

zero weight

23

123 23−1

22

11

1

1−1 2323

22

11

N = 23. The ring of modular forms of integral weight for 02 (23) is generated by 23+ (τ ) = η(τ )η(23τ ), E1 (τ, χ23 ), and one of the two weight 3 Eisenstein series. The Hilbert function is (1 + x 3 )/(1 − x)2 . The ideal of cusp forms is generated by 23+ (τ ) = η(τ )η(23τ ) of weight 1. This is the lowest level for which there is a cusp form of weight 1. Remark. The modular function η(τ )12 η(23τ )−12 has a pole of order 11 at i∞ and a zero of order 11 at zero, and it shows that the cusp 0 gives a torsion point of order 11 on the modular abelian surface of 0 (23). See [Sh, page 197] for more about this. The forms E1 (τ, χ23 )n /23+ (τ ) show that all level 23 lattices of signature at least −2 have reﬂective forms. The automorphic form of the lattice I I2,4 (23+3 ) and of the function 1/23+ has singular weight and is the denominator function of a generalized Kac-Moody algebra. This generalized Kac-Moody algebra contains the Feingold-Frenkel rank 3

364

RICHARD E. BORCHERDS

Kac-Moody algebra as a subalgebra and can be used to explain why the root multiplicities of the Feingold-Frenkel algebra are often given by values of the partition function. See [Ni] for details. N = 28. The eta product 11 2−1 41 71 14−1 281 has weight 1, character χ7 , and its zeros are order 1 zeros at the cusps 1/1, 1/4, 1/7, and 1/28. (It is nonzero at the cusps 1/2 and 1/14.) So its inverse is a reﬂective form for any level 28 even lattice of signature −2 and odd 7-rank. N = 30. Here are some weight 1 eta quotients with order 1 zeros at six of the eight cusps: 21 31 51 6−1 10−1 301 (nonzero at 1/3, 1/5), 11 3−1 5−1 61 101 151 (nonzero at 1/6, 1/10), 1−1 21 31 51 15−1 301 (nonzero at 1/2, 1/30), 11 2−1 61 101 151 30−1 (nonzero at 1/1, 1/15). These are all modular forms for the character χ15 . So any even lattice of level 30 signature −2, and odd 5-rank has a reﬂective form. (The 3-rank is automatically odd for any such lattice.) The maximal order of an automorphism of the Leech lattice with ﬁxed points is 30, and three of the eta quotients above occur as generalized cycle shapes of such order 30 automorphisms. There is no special reason for stopping at N = 30: There are hints that there might be examples for N up to a few hundred. 13. Open problems. We list a few suggestions for further research. Problem 13.1. Find some analogue of reﬂective forms for other sorts of hyperbolic reﬂection groups. In particular, explain why the complex hyperbolic reﬂection groups found by Allcock [A] have underlying integral lattices with nonzero reﬂective modular forms. Is this true of all complex hyperbolic reﬂection groups (except perhaps in small dimensions)? Is there a relation between the cocompact hyperbolic reﬂection groups found by Bugaenko [Bu] and lattices with reﬂective forms? Problem 13.2. Find some sort of converse theorem that implies that all “interesting” lattices of some sort have nonzero reﬂective forms. Bruinier [Br] has recently proved some related converse theorems, showing that certain sorts of automorphic inﬁnite products always come from modular forms with singularities. Problem 13.3. Are there any other reﬂection groups of high-dimensional lattices with coﬁnite volume other than those listed in Section 12? Problem 13.4. Which of the rank 3 hyperbolic lattices classiﬁed by Nikulin [Nik2] have reﬂective forms? Note that in Part III of Nikulin’s papers, there are many examples where the Weyl vector has negative norm. Problem 13.5. Classify all holomorphic eta quotients whose zeros at cusps are all of order at most 1. This would be suitable for a computer. Problem 13.6. Write a computer program to classify the lattices such that the space of possible reﬂective singularities is greater than the dimension of the space of

REFLECTION GROUPS OF LORENTZIAN LATTICES

365

cusp forms that give obstructions to the existence of such a singularity. This would give a large class of examples of lattices with reﬂective forms, and the examples in Section 12 suggest that this would include most of them. Problem 13.7. The group 0 (242 ) has 48 cusps, all conjugate under its normalizer. The form 1/η(24τ ) has poles of order 1 at all cusps. Are there any lattices for which it is a reﬂective form? References [A] [ACT] [AF] [BBBCO]

[B1] [B2] [B3] [B4] [B5] [B6] [B7] [Br] [Bu]

[CS] [Eb] [Es] [F] [Fr] [KK]

D. Allcock, The Leech lattice and complex hyperbolic reﬂections, preprint, 1997, available from http://www.math.utah.edu/ ˜allcock. D. Allcock, J. A. Carlson, and D. Toledo, A complex hyperbolic structure for moduli of cubic surfaces, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), 49–54. D. Allcock and E. Freitag, Cubic surfaces and Borcherds products, preprint, 1999, available from http://www.rzuser.uni-heidelberg.de/˜t91. C. Batut, K. Belabas, D. Bernardi, H. Cohen, and M. Olivier, User’s Guide to PARI-GP, guide and PARI programs available from ftp://megrez.math.ubordeaux.fr/pub/pari/. R. E. Borcherds, Automorphism groups of Lorentzian lattices, J. Algebra 111 (1987), 133–153. , Lattices like the Leech lattice, J. Algebra 130 (1990), 219–234. , Automorphic forms on Os+2,2 (R) and inﬁnite products, Invent. Math. 120 (1995), 161–213. , The moduli space of Enriques surfaces and the fake monster Lie superalgebra, Topology 35 (1996), 699–710. , Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), 491–562. , Coxeter groups, Lorentzian lattices, and K3 surfaces, Internat. Math. Res. Notices 1998, 1011–1031. , The Gross-Kohnen-Zagier theorem in higher dimensions, Duke Math. J. 97 (1999), 219–233. J. H. Bruinier, Borcherds products and Chern classes of Hirzebruch-Zagier divisors, Invent. Math. 138 (1999), 51–83. V. O. Bugaenko, “Arithmetic crystallographic groups generated by reﬂections, and reﬂective hyperbolic lattices” in Lie Groups, Their Discrete Subgroups, and Invariant Theory, Adv. Soviet Math. 8, Amer. Math. Soc., Providence, 1992, 33–55. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 2d ed, Grundlehren Math. Wiss. 290, Springer, New York, 1993. W. Ebeling, Lattices and Codes: A Course Partially Based on Lectures by F. Hirzebruch, Adv. Lectures Math., Vieweg, Braunschweig, 1994. F. Esselmann, Über die maximale Dimension von Lorentz-Gittern mit coendlicher Spiegelungsgruppe, J. Number Theory 61 (1996), 103–144. J. Fischer, An Approach to the Selberg Trace Formula via the Selberg Zeta-Function, Lecture Notes in Math. 1253, Springer, Berlin, 1987. E. Freitag, Some modular forms related to cubic surfaces, preprint, 1999, available from www.rzuser.uni-heidelberg.de/˜t91. J. H. Keum and S. Kondo, The automorphism groups of Kummer surfaces associated with the product of two elliptic curves, preprint, 1998.

366 [Ko] [Kon] [M] [Mi] [N1] [N2] [Ni] [Nik1]

[Nik2]

[R] [Ri]

[SH] [SV] [Sch] [SS]

[Sh] [V1]

[V2] [VK]

RICHARD E. BORCHERDS N. Koblitz, Introduction to Elliptic Curves and Modular Forms, 2d ed., Grad. Texts in Math. 97, Springer, New York, 1993. ¯ The automorphism group of a generic Jacobian Kummer surface, J. Algebraic S. Kondo, Geom. 7 (1998), 589–609. Y. Martin, Multiplicative η-quotients, Trans. Amer. Math. Soc. 348 (1996), 4825–4856. T. Miyake, Modular Forms, Springer, Berlin, 1989. M. Newman, Construction and application of a class of modular functions, Proc. London. Math. Soc. (3) 7 (1957), 334–350. , Construction and application of a class of modular functions, II, Proc. London Math. Soc. (3) 9 (1959), 373–387. P. Niemann, Some generalized Kac-Moody algebras with known root multiplicities, Ph.D. thesis, Cambridge Univ., Cambridge, 1997. V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 111–177, 238; English transl. in Math. USSR-Izv. 14 (1979), 103–167. , On the classiﬁcation of hyperbolic root systems of the rank three, I, http://www.arXiv.org/abs/alg-geom/9711032; II, http://www.arXiv.org/abs/ alg-geom/9712033; III, http://www.arXiv.org/abs/math.AG/9905150. H. Rademacher, Topics in Analytic Number Theory, Grundlehren Math. Wiss. 169, Springer, New York, 1973. K. A. Ribet, “Galois representations attached to eigenforms with Nebentypus” in Modular Functions of One Variable, V (Bonn, 1976), Lecture Notes in Math. 601, Springer, Berlin, 1977; 17–51. R. Scharlau and B. Hemkemeier, Classiﬁcation of integral lattices with large class number, Math. Comp. 67 (1998), 737–749. R. Scharlau and B. B. Venkov, The genus of the Barnes-Wall lattice, Comment. Math. Helv. 69 (1994), 322–333. N. R. Scheithauer, The fake monster superalgebra, http://www.arXiv.org/abs/math.QA/ 9905113, to appear in Adv. Math. J.-P. Serre and H. M. Stark, “Modular forms of weight 1/2” in Modular Functions of One Variable, VI (Bonn, 1976), Lecture Notes in Math. 627, Springer, Berlin, 1977, 27–67. G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Kanô Memorial Lectures 1, Publ. Math. Soc. Japan 11, Iwanami Shoten, Tokyo, 1971. È. B. Vinberg, “Some arithmetical discrete groups in Lobaˇcevski˘ı spaces” in Discrete Subgroups of Lie Groups and Applications to Moduli (Bombay, 1973), Oxford Univ. Press, Bombay, 1975, 323–348. , The two most algebraic K3 surfaces, Math. Ann. 265 (1983), 1–21. È. B. Vinberg and I. M. Kaplinskaja, The groups O18,1 (Z) and O19,1 (Z) (in Russian), Dokl. Akad. Nauk SSSR 238 (1978), 1273–1275.

Department of Mathematics, University of California at Berkeley, Berkeley, California 94720-3840, USA; [email protected]; http://math.berkeley.edu/˜reb

Vol. 104, No. 3

DUKE MATHEMATICAL JOURNAL

© 2000

DIFFERENTIABILITY PROPERTIES OF ISOTROPIC FUNCTIONS MIROSLAV ŠILHAVÝ

1. Introduction. Let Sym denote the linear space of all symmetric second-order tensors on an n-dimensional real vector space Vect with scalar product. (If Vect is identiﬁed with Rn , then Sym may be identiﬁed with the set of all symmetric n-by-n matrices.) A function f : Sym → R is said to be isotropic if f (A) = f (QAQT ) for all A ∈ Sym and all Q proper orthogonals. An isotropic function has a representation f (A) = f˜(a), where f˜ is a symmetric function on Rn and a = (a1 , . . . , an ) are the eigenvalues of A with appropriate multiplicities. Clearly, f˜(a) = f (diag(a)) in any orthonormal basis, and thus if f is of class C r , r = 0, 1, . . . , ∞, then also f˜ is of class C r . Ball [1] showed that for r = 0, 1, 2, ∞, the converse is also true and conjectured that the converse is true for all r. This was subsequently proved by Sylvester [6] using complex techniques and detailed estimates of the derivatives of eigenvalues. Earlier, Chadwick and Ogden [2], [3] gave formulas for D r f , r = 1, 2, 3, in terms of f˜ and its derivatives assuming the differentiability (see also [1]). In this note, I derive the result of Sylvester by elementary means and give a recursive formula for D r f in terms of f˜ for arbitrary r. I also specialize these formulas to derive the forms of D r f , r = 1, 2, 3, which are equivalent to those by Chadwick and Ogden. 2. Notation. Throughout, the indices i, j, k range the interval {1, . . . , n}, unless stated otherwise. The direct vector notation is used in [4], [5]. In addition to the notation explained in the introduction, we recall that a second-order tensor A is a linear transformation from Vect into Vect, with the product of two tensors being the composition of the linear transformations. Furthermore, Orth+ denotes the proper orthogonal group, and Skew denotes the set of all skew tensors. By a basis in Vect, we always mean an orthonormal basis. Let Sn be the set of all real symmetric n-by-n matrices. Let ei be the canonical basis in Rr . All vector spaces are ﬁnite-dimensional and real. For a vector space X, we denote by Fr (X) the vector space of all symmetric rlinear forms F : X × · · · × X → R on X. The direct notation is used to denote the derivatives (differentials) of functions f deﬁned on a vector space X with values in R. Thus for x ∈ X, the rth derivative D r f (x) is a symmetric r-form on X; that is,

Received 19 May 1999. Revision received 3 June 1999. 2000 Mathematics Subject Classiﬁcation. Primary 74A20; Secondary 74B99. Author’s work supported by grant number 201/00/1516 of the Grant Agency of the Czech Republic. 367

368

MIROSLAV ŠILHAVÝ

D r f (x) ∈ Fr (X) and D r f : X → Fr (X). For each positive integer r and each class C r function f on X and x ∈ Rn , we denote by D [r] f (x) = (Df (x), . . . , D r f (x)). A function f : Rn → R is said to be symmetric if f (P w) = f (w) for every w ∈ Rn and every n-by-n permutation matrix P . We denote by CSr (Rn ) the set of all symmetric functions of class C r on Rn , and we denote by CIr (Sym) the set of all isotropic functions of class C r on Sym. Proposition 2.1. The function f : Sym → R is isotropic if and only if there exists a symmetric function f˜ : Rn → R, such that for each basis {ei } and each A ∈ Sym represented by A = diag(a), a ∈ Rn , f (A) = f˜(a).

(1)

The correspondence ˜: f → f˜ is one-to-one between isotropic functions on Sym and symmetric functions on Rn . This is well known and immediate. The function f˜ is called the representation of f . 3. The main result. For each i = j , we denote by W (ij ) the skew matrix with elements Mkl , where Mij = −Mj i = 1 and Mkl = 0 for all other pairs of indices. For B ∈ Sn , we denote B[ij ] = W (ij ) B −BW (ij ) ∈ Sn . Let (R) ⊂ Rn be an open ball of radius R with the center at the origin. Lemma 3.1. For each positive integer r and each f˜ ∈ CSr (Rn ), there exist functions F s (·, f˜) ∈ C r−s (Rn , Fr (Sn )), s = 1, . . . , r, such that the following hold. (a) If s = 1, . . . , r, [r−s] s D F ·, f˜ C 0 ((R)) ≤ C(r, R)D [r] f˜C 0 ((R)) , (2) where C(r, R) is a constant independent of f˜ and ·C 0 ((R)) is the supremum norm on (R). (b) If s = 1, n 1 ˜ F a, f B = Di f˜(a)Bii1 , 1

a ∈ Rn ,

B 1 ∈ Sn .

(3)

Hijs Bijs ,

(4)

i=1

(c) If 1 < s ≤ r, n 1 F s a, f˜ B 1 , . . . , B s = Gsi Biis + 2 i=1

1≤i=j ≤n

a ∈ Rn , B 1 , . . . , B s ∈ Sn , where for each i, Gsi = Di F s−1 a, f˜ B 1 , . . . , B s−1

(5)

DIFFERENTIABILITY PROPERTIES

369

and for each i, j , 1 ≤ i = j ≤ n, Hijs

=

1 s−1 0

Di F

s−1

k=1

1 k s−1 ˜ a , f B , . . . , B[ij ] , . . . , B dt

t

(6)

with the abbreviation a t := a + t (aj − ai )ei . Proof. For a ﬁxed r by induction on s. For s = 1 we clearly have F 1 (·, f ) ∈ and (2) holds with C(r, R) = 1. Let 1 < s ≤ r be given, let F s be deﬁned by (4) through (6), and let (2) hold with some C(r, R) for all values of s = s less than our s. Since F s−1 (·, f ) is of class C r−s+1 by the induction hypothesis, we see from (5) and (6) that Gsi , Hijs are all of class C r−s . Moreover, a differentiation, the chain rule, and the induction hypothesis (2) provide [r−s] s D Gi C 0 ((R)) ≤ M D [r−s+1] F s−1 ·, f˜ B 1 , . . . , B s−1 C 0 ((R))

≤ MC(r, R)D [r] f˜C 0 ((R)) B 1 · · · B s−1 , C s−1 ((R))

for some M ≥ 1 independent of f˜; that is, [r−s] s

D Gi C 0 ((R)) ≤ C (r, R)D [r] f˜C 0 ((R)) B 1 · · · B s−1 , and similarly [r−s] s

D Hij C 0 ((R)) ≤ C (r, R)D [r] f˜C 0 ((R)) B 1 · · · B s−1 , with possibly a larger value of C (r, R) = MC(r, R). Lemma 3.2. Let f : Sym → R be an isotropic function of class C r . Then (a) for each A, B1 , . . . , Br ∈ Sym and Q ∈ Orth+ we have D r f (A) B1 , . . . , Br = D r f QAQT QB1 QT , . . . , QBr QT ;

(7)

(b) for each A, B1 , . . . , Br−1 ∈ Sym and W ∈ Skew we have r

r−1

D f (A) [W, A], B , . . . , B 1

=−

r−1 k=1

D r−1 f (A) B1 , . . . , W, Bk , . . . , Br−1 . (8)

Here [A, B] = AB − BA. Proof. (a) Differentiate f (A) = f (QAQT ) r times in the directions B1 , . . . , Br . (b) In (7) we replace r by r − 1 and set Q = etW , t ∈ R. A differentiation with respect to t at t = 0 gives the result.

370

MIROSLAV ŠILHAVÝ

Lemma 3.3. For each r ≥ 1, each f ∈ CIr (Sym), each basis {ei }, and each A, ∈ Sym, represented by the matrices A = A = diag(a), a ∈ Rn , B 1 , . . . , B r ∈ Sn , we have D r f (A) B1 , . . . , Br = F r a, f˜ B 1 , . . . , B r . (9) B1 , . . . , Br

Proof. We prove this by induction on r. For r = 1, (9) and (3) represent a wellknown formula for the ﬁrst derivative of an isotropic function (e.g., [5]). Suppose that the assertion of the lemma is true for some particular r −1 ≥ 1. In view of the linearity of D r f (A)[B1 , . . . , Br ] with respect to Br , it sufﬁces to prove (9) only for some special choices of Br and for B1 , . . . , Br−1 ∈ Sym arbitrary. Namely, it sufﬁces to take (i) Br ≡ B r = diag(ei ), i = 1, . . . , n, and (ii) Br ≡ B r = B (ij ) , 1 ≤ i = j ≤ n, where B (ij ) denotes the n-by-n symmetric matrix with elements Mkl , where Mij = Mj i = 1 and all other elements Mkl vanish. First let Br ≡ B r = diag(ei ). By the induction hypothesis, D r−1 f (A + λBr ) B1 , . . . , Br−1 = F r−1 a + λb, f˜ B 1 , . . . , B r−1 , and a differentiation combined with the fact that F r−1 (·, f˜) is of class C 1 (by Lemma 3.1) provides n D r f (A) B1 , . . . , Br = Di F r−1 a, f˜ B 1 , . . . , B r−1 Biir , i=1

which is (9) in this special case. Let Br ≡ B r = B (ij ) , where i, j is a ﬁxed pair, 1 ≤ i = j ≤ n. Assume ﬁrst that ai = aj . Set W = W = (ai − aj )−1 W (ij ) and note that [W, A] = −B (ij ) . The application of (8) and the induction hypothesis give 1 r−1 r−1 k , . . . , B r−1 D f (A) B , . . . , B k=1 [ij ] D r f (A) B1 , . . . , Br = ai − a j (10) r−1 r−1 k , . . . , B r−1 a, f˜ B 1 , . . . , B[ij k=1 F ] = , ai − a j where we have identiﬁed tensors with matrices. Let a t be as in Lemma 3.1 and note that if a = (. . . , ai , . . . , aj , . . . ), then a t for t = 1 equals a 1 = (. . . , aj , . . . , aj , . . . ). Then for A1 = A1 = diag(a 1 ), we have A1[ij ] = 0, and hence from (8) and the induction hypothesis, 0=

r−1 k=1

=

r−1 k=1

k r−1 D r−1 f A1 B 1 , . . . , B[ij ], . . . , B F

k r−1 a 1 , f˜ B 1 , . . . , B[ij . ], . . . , B

r−1

(11)

DIFFERENTIABILITY PROPERTIES

371

Thus the last expression in (10) can be rewritten as the right-hand side of (6), which is (9) in this case. Next assume that ai = aj . The preceding part of the proof shows: r−1 r−1 k , . . . , B r−1 a% , f˜ B 1 , . . . , B[ij 1 k=1 F ] r r D f (A% ) B , . . . , B = (12) % for each % = 0, where A% = A + % diag(ei ), a% = a + %ei , B 1 , . . . , B r−1 ∈ Sn , and B r = B (ij ) . The limit as % → 0 of the left-hand side of (12) is D r f (A)[B 1 , . . . , B r ]; the limit as % → 0 of the right-hand side exists and equals r−1 k=1

k r−1 Di F r−1 a, f˜ B 1 , . . . , B[ij ], . . . , B

by l’Hospital’s rule. Thus (9) also holds if ai = aj . Remark 3.4. We have proved the following alternative expression of H r :  r−1 k , . . . , B r−1  F r−1 a, f˜ B 1 , . . . , B[ij  k=1 ]  if ai = aj ,    ai − a j r Hij = r−1    k r−1  Di F r−1 a, f˜ B 1 , . . . , B[ij if ai = aj .  ], . . . , B 

(13)

k=1

This formula is useful for calculating the derivatives, in contrast to (6), which is useful for examining the differentiability properties of Hijr . Theorem 3.5. Let f : Sym → R be an isotropic function. Then (a) f ∈ C r (Sym), r = 0, 1, . . . , ∞, if and only if f˜ ∈ C r (Rn ); (b) for each basis {ei } and each A, B1 , . . . , Br ∈ Sym, represented by the matrices A = A = diag(a), a ∈ Rn , B 1 , . . . , B r ∈ Sn , we have D r f (A) B1 , . . . , Br = F r a, f˜ B 1 , . . . , B r . Proof. (a) The direct implication is immediate. Let us prove the converse. Note ﬁrst that (2) and (9) imply that for each r ≥ 1 and each R > 0, there exists a constant C(r, R) such that if f ∈ CI∞ (Sym), then (14) D r f C 0 ((R)) ≤ C(r, R)D [r] f˜C 0 ((R)) . Let f be such that f˜ ∈ C r (Rn ) and let R > 0. Let ϕ : Sym → R be a C ∞ molliﬁer of the form ϕ(A) = ψ(|A|), A ∈ Sym; set ϕ% (A) = ϕ(A/%)/% n(n+1)/2 , A ∈ Sym; and for each f ∈ CIr (Sym), r = 0, . . . , let f% (A) = f (B)ϕ% (A − B) dB,

372

MIROSLAV ŠILHAVÝ

where the integral extends over Sym and dB denotes the Lebesgue measure on Sym. Clearly, f% is isotropic. We denote by f˜% the representation of f% and note that f% −→ f,

f˜% −→ f˜,

D [r] f˜% −→ D [r] f˜ as % −→ 0+,

(15)

and the convergence is uniform on compact sets. By (14) and (15) we see that for each s, 1 ≤ s ≤ r, D s f% is a Cauchy sequence in C 0 ((R), Fs (Sym)) and thus there exist M s ∈ C 0 ((R), Fs (Sym)) such that D s f% −→ M s

as % −→ 0+

uniformly on compact sets. Since D s f% are the continuous derivatives of f% , we have f% (A)D s g(A) dA = (−1)s D s f% (A)g(A) dA for each g ∈ C0∞ (Sym). The limit % → 0+ gives f (A)D s g(A) dA = M s (A)g(A) dA. Thus M s are the distributional derivatives of f on Sym. Since M s are continuous functions, elementary considerations show that f ∈ C r (Sym) and D s f = M s , s = 1, . . . , r. (b) This follows from Lemma 3.3. 4. Low-order derivatives. Let {ei } be a basis and let A, B1 , B2 , B3 ∈ Sym be represented by the matrices A = A = diag(a), a ∈ Rn , B 1 , B 2 , B 3 ∈ Sn and let the components of a be distinct. Let f : Sym → R and let the subscripts attached to f˜ denote the partial derivatives of f˜. The application of (9), (3), (4), (5), and (13) provides n 1 f˜i Bii1 Df (A) B = i=1

if f

∈ C 1 (Sym);

further,

n 2 f˜ij Bii1 Bjj + D 2 f (A) B1 , B2 = i,j =1

1≤i=j ≤n

if f ∈ C 2 (Sym) (cf. [3], [1]), and n 2 3 f˜ij k Bii1 Bjj Bkk D 3 f (A) B1 , B2 , B3 = i,j,k=1

f˜i − f˜j 1 2 B B ai − aj ij ij

DIFFERENTIABILITY PROPERTIES n

+

k=1 1≤i=j ≤n

+2

373

f˜ik − f˜j k 1 2 3 2 3 1 2 3 Bij + Bkk Bij Bij Bij Bij Bkk + Bij1 Bkk ai − a j

n i,j,k=1 i=j =k=i

ai f˜j − f˜k + aj f˜k − f˜i + ak f˜i − f˜j 1 2 3 Bki Bkj Bij (ai − aj )(aj − ak )(ak − ai )

if f ∈ C 3 (Sym). The last formula is equivalent to the one given in [3]. References [1] [2] [3] [4] [5] [6]

J. M. Ball, Differentiability properties of symmetric and isotropic functions, Duke Math. J. 51 (1984), 699–728. P. Chadwick and R. W. Ogden, On the deﬁnition of elastic moduli, Arch. Rational Mech. Anal. 44 (1971/72), 41–53. , A theorem of tensor calculus and its application to isotropic elasticity, Arch. Rational Mech. Anal. 44 (1971/72), 54–68. M. E. Gurtin, An Introduction to Continuum Mechanics, Math. Sci. Engrg. 158, Academic Press, New York, 1981. M. Šilhavý, The Mechanics and Thermodynamics of Continuous Media, Texts Monogr. Phys., Springer, Berlin, 1997. J. Sylvester, On the differentiability of O(n) invariant functions of symmetric matrices, Duke Math. J. 52 (1985), 475–483.

Mathematical Institute of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Prague 1, Czech Republic; [email protected]

Vol. 104, No. 3

DUKE MATHEMATICAL JOURNAL

© 2000

A NEW ELLIPSOID ASSOCIATED WITH CONVEX BODIES ERWIN LUTWAK, DEANE YANG, and GAOYONG ZHANG

Corresponding to each origin-symmetric convex (or more general) subset of Euclidean n-space Rn , there is a unique ellipsoid with the following property: The moment of inertia of the ellipsoid and the moment of inertia of the convex set are the same about every 1-dimensional subspace of Rn . This ellipsoid is called the Legendre ellipsoid of the convex set. The Legendre ellipsoid and its polar (the Binet ellipsoid) are well-known concepts from classical mechanics. See Milman and Pajor [MPa1], [MPa2], Lindenstrauss and Milman [LiM], and Leichtweiß [Le] for some historical references. It has slowly come to be recognized that alongside the Brunn-Minkowski theory there is a dual theory. The nature of the duality between the Brunn-Minkowski theory and the dual Brunn-Minkowski theory is subtle and not yet understood. It is easily seen that the Legendre (and Binet) ellipsoid is an object of this dual Brunn-Minkowski theory. This observation leads immediately to the natural question regarding the possible existence of a dual analog of the classical Legendre ellipsoid in the Brunn-Minkowski theory. It is the aim of this paper to demonstrate the existence of precisely this dual object. In retrospect, one may well wonder why the new ellipsoid presented in this note was not discovered long ago. The simple answer is that the deﬁnition of the new ellipsoid becomes obvious only with the notion of L2 -curvature in hand. However, the Brunn-Minkowski theory was only recently extended to incorporate the new notion of Lp -curvature (see [L2], [L3]). A positive-deﬁnite n × n real symmetric matrix A generates an ellipsoid (A), in Rn , deﬁned by (A) = x ∈ Rn : x · Ax ≤ 1 , where x · Ax denotes the standard inner product of x and Ax in Rn . Associated with a star-shaped (about the origin) set K ⊂ Rn is its Legendre ellipsoid 2 K, which is generated by the matrix [mij (K)]−1 , where mij (K) =

n+2 V (K)

K

ei · x ej · x dx,

with e1 , . . . , en denoting the standard basis for Rn and V (K) denoting the n-dimensional volume of K. Received 30 September 1999. Revision received 4 January 2000. 2000 Mathematics Subject Classiﬁcation. Primary 52A40. Authors’ work supported, in part, by National Science Foundation grant number DMS-9803261. 375

376

LUTWAK, YANG, AND ZHANG

We will associate a new ellipsoid −2 K with each convex body K ⊂ Rn . One approach to deﬁning −2 K without introducing new notation is to ﬁrst deﬁne it for polytopes and then use approximation (with respect to the Hausdorff metric) to extend the deﬁnition to all convex bodies. Suppose P ⊂ Rn is a polytope that contains the origin in its interior. Let u1 , . . . , uN denote the outer unit normals to the faces of P , let a1 , . . . , aN denote the areas (i.e., (n − 1)-dimensional volumes) of the corresponding faces, and let h1 , . . . , hN denote the distances from the origin to the corresponding faces. The ellipsoid −2 P is generated by the matrix [m ˜ ij (P )], where N

m ˜ ij (P ) =

1 al ei · ul ej · ul . V (P ) hl l=1

An alternate deﬁnition of the operator −2 is given after additional notation is introduced. The easily established afﬁne nature of the operator 2 is formally stated in the following lemma. Lemma 1. If K ⊂ Rn is star-shaped about the origin, then for each φ ∈ GL(n), 2 (φK) = φ2 K. While more difﬁcult to see, we prove the following lemma. Lemma 1∗ . Suppose K ⊂ Rn is a convex body that contains the origin in its interior. Then for each φ ∈ GL(n), −2 (φK) = φ−2 K. The following theorem is fundamental and goes back, at least, to Blaschke [Bl], John [J], and Petty [P1] (see also Milman and Pajor [MPa1], [MPa2]). We give yet another proof in this paper. Theorem 1. If K ⊂ Rn is star-shaped about the origin, then V 2 K ≥ V (K), with equality if and only if K is an ellipsoid centered at the origin. For our new ellipsoids we establish the following. Theorem 1∗ . Suppose K ⊂ Rn is a convex body that contains the origin in its interior. Then V −2 K ≤ V (K), with equality if and only if K is an ellipsoid centered at the origin.

A NEW ELLIPSOID ASSOCIATED WITH CONVEX BODIES

377

The operator −2 has the following monotonicity property. Theorem 2∗ . Suppose K ⊂ Rn is a convex body that contains the origin in its interior. If E is an ellipsoid centered at the origin such that E ⊂ K, then V −2 E ≤ V −2 K , with equality if and only if E = −2 K. Let S n−1 denote the unit sphere, centered at the origin, in Rn . Let B denote the unit ball, centered at the origin, in Rn , and let ωn = V (B). From Theorem 2∗ we obtain the following. Theorem 3∗ . Suppose K ⊂ Rn is a convex body that is origin-symmetric; then V −2 K ≥ 2−n ωn V (K), with equality if and only if K is a parallelotope. The analog of Theorem 3∗ for the operator 2 is one of the major open problems in the ﬁeld: Finding the maximum of V (2 K)/V (K) as K ranges even over the class of origin-symmetric convex bodies (or even important small subclasses) is difﬁcult (see, e.g., the survey of Lindenstrauss and Milman [LiM]). It is even difﬁcult to show that there exists a√ c (independent of the dimension n) such that [V (2 K)/V (K)]1/n is bounded by c n as K ranges over the class of origin-symmetric convex bodies. This problem was ﬁrst posed by Bourgain [Bo1]. The best-known bounds to date appear to be those of Bourgain [Bo2] (see also Dar [D] and Junge [Ju]). There is an important class of questions in the local theory of Banach spaces which are well known to be equivalent in that an answer to one will immediately provide an answer to the others. Bourgain’s problem is one member of this important class of equivalent problems. See Milman and Pajor [MPa2, Section 5]. We present a version of Theorem 3∗ for arbitrary convex bodies. We then present a classical characterization of the operator 2 and its obvious counterpart for the operator −2 . Finally, we present an analog of Milman’s important notion of “isotropic position” and explore some of its consequences. We have chosen to reprove all the classical results concerning the operator 2 for two reasons. First, we want to show the close connection and interrelationship between the operators 2 and −2 . Second, we believe that new proofs of classical results are almost always enlightening. A serious attempt has been made to present all arguments in a reasonably selfcontained manner. For quick reference, some basic properties of L2 -mixed and dual mixed volumes will be listed. Some recent applications of dual mixed volumes can be found in [G1], [Z1], [Z2], and [Z3]. The L1 -analogs of some of the identities presented may be found in [L1]. For general reference the reader may wish to consult the books of Gardner [G2], Schneider [S], and Thompson [T].

378

LUTWAK, YANG, AND ZHANG

Recall that if K ⊂ Rn is a convex body that contains the origin in its interior, then the polar of K, is deﬁned by K ∗ = x ∈ Rn : x · y ≤ 1, for all y ∈ K .

K ∗,

From the deﬁnition it follows easily that for each convex body K, we have K ∗∗ = K.

(1)

From the deﬁnition of a polar body, it follows trivially that for each convex body K and φ ∈ GL(n) (φK)∗ = φ −t (K ∗ ),

(2)

where φ −t denotes the inverse of the transpose of φ. The radial function ρK = ρ(K, ·) : Rn \ {0} → [0, ∞) of a compact, star-shaped (about the origin) K ⊂ Rn , is deﬁned for x = 0 by ρ(K, x) = max λ ≥ 0 : λx ∈ K . If ρK is positive and continuous, K is called a star body (about the origin). Two star bodies K and L are said to be dilates (of one another) if ρK (u)/ρL (u) is independent of u ∈ S n−1 . From the deﬁnition of radial function, it follows immediately that for a star body K, an x ∈ Rn \ {0}, and a φ ∈ GL(n), we have ρφK (x) = ρK φ −1 x ; (3) φK = {φx : x ∈ K} is the image of K under φ. If K ⊂ Rn is a convex body that contains the origin in its interior, then its support function hK = h(K, ·) : Rn → [0, ∞) is deﬁned for x ∈ Rn by h(K, x) = max x · y : y ∈ K . Since it is assumed throughout that all of our convex bodies contain the origin in their interiors, all support functions are strictly positive on Rn \ {0}. From the deﬁnition of support function, it follows immediately that for a convex body K, an x ∈ Rn , and a φ ∈ GL(n), we have hφK (x) = hK φ t x , (3∗ ) where φ t denotes the transpose of φ. If K is a convex body, then it follows from the deﬁnitions of support and radial functions and the deﬁnition of polar body that hK ∗ =

1 ρK

and

ρK ∗ =

1 . hK

(4)

A NEW ELLIPSOID ASSOCIATED WITH CONVEX BODIES

379

−2 ε ·L is For star bodies K, L, and ε > 0, the L2 -harmonic radial combination K + the star body deﬁned by −2 ε · L, · −2 = ρ(K, ·)−2 + ερ(L, ·)−2 . ρ K+

(5)

For convex bodies K, L, and ε > 0, the Firey L2 -combination K+2 ε ·L is deﬁned as the convex body whose support function is given by 2 h K+2 ε · L, · = h(K, ·)2 + εh(L, ·)2 .

(5∗ )

Note that the “scalar” multiplications “ε · L” in (5) and (5∗ ) are different. The temptation to put a subscript under each “·” was resisted. From (4) we see that the relationship between the two types of combinations is that for convex bodies K, L, and ε > 0, −2 ε · L∗ ∗ . K+2 ε · L = K ∗ + The dual mixed volume V−2 (K, L) of the star bodies K, L can be deﬁned by −2 ε · L − V (K) V K+ n V−2 (K, L) = lim . (6) −2 ε ε→0+ The L2 –mixed volume V2 (K, L) of the convex bodies K, L was deﬁned in [L2] by V K+2 ε · L − V (K) n V2 (K, L) = lim . (6∗ ) 2 ε ε→0+ That this limit exists was demonstrated in [L2]. From the deﬁnitions (5) and (6), it follows immediately that for each star body K, V−2 (K, K) = V (K).

(7)

From the deﬁnitions (5∗ ) and (6∗ ), it follows immediately that for each convex body K, V2 (K, K) = V (K).

(7∗ )

From (3) and the deﬁnition of an L2 -harmonic radial combination (5), it follows immediately that for an L2 -harmonic radial combination of star bodies K and L, and φ ∈ GL(n), −2 ε · L = φK + −2 ε · φL. φ K+ This observation together with the deﬁnition of the dual mixed volume V−2 shows that for φ ∈ SL(n) and star bodies K, L we have V−2 (φK, φL) = V−2 (K, L), or equivalently, V−2 φK, L = V−2 K, φ −1 L . (8)

380

LUTWAK, YANG, AND ZHANG

From (3∗ ) and the deﬁnition of a Firey L2 -combination (5∗ ), it follows immediately that for a Firey combination of convex bodies K and L, and φ ∈ GL(n), φ K+2 ε · L = φK+2 ε · φL. This observation together with the deﬁnition of the L2 –mixed volume V2 shows that for φ ∈ SL(n) and convex bodies K, L we have V2 (φK, φL) = V2 (K, L) or equivalently V2 φK, L = V2 K, φ −1 L . (8∗ ) Deﬁnitions (5) and (6) and the polar coordinate formula for volume give the following integral representation of the dual mixed volume V−2 (K, L) of the star bodies K, L: 1 V−2 (K, L) = ρ n+2 (v)ρL−2 (v) dS(v), (9) n S n−1 K where the integration is with respect to spherical Lebesgue measure S on S n−1 . It was shown in [L2] that corresponding to each convex body K, there is a positive Borel measure S2 (K, ·) on S n−1 such that 1 V2 (K, L) = h2 (u) dS2 (K, u) (9∗ ) n S n−1 L for each convex body L. We require two basic inequalities regarding the mixed volumes V2 and the dual mixed volumes V−2 . The dual mixed volume inequality for V−2 is that for star bodies K, L, V−2 (K, L) ≥ V (K)(n+2)/n V (L)−2/n ,

(10)

with equality if and only if K and L are dilates. This inequality is an immediate consequence of the Hölder inequality and the integral representation (9). The L2 analog of the classical Minkowski inequality states that for convex bodies K, L, V2 (K, L) ≥ V (K)(n−2)/n V (L)2/n ,

(10∗ )

with equality if and only if K and L are dilates. This L2 -analog of the Minkowski inequality was established in [L2] by using the classical Minkowski mixed volume inequality. An immediate consequence of the dual mixed volume inequality (10) and identity (7) that we use is that if for star bodies K, L we have V−2 (Q, K) V−2 (Q, L) = V (Q) V (Q) for all star bodies Q, which belong to some class that contains both K and L, then in fact K = L.

A NEW ELLIPSOID ASSOCIATED WITH CONVEX BODIES

381

It is easy to verify that if A is a positive deﬁnite n × n real symmetric matrix, then the radial and support functions of the ellipsoid (A) = {x ∈ Rn : x · Ax ≤ 1} are given by −2 ρ(A) (u) = u · Au and h2(A) (u) = u · A−1 u for u ∈ S n−1 . Thus, for a star body K, h22 K (u) =

n+2 V (K)

K

(u · x)2 dx,

(11)

for u ∈ S n−1 . The normalization above is chosen so that for the unit ball B, we have 2 B = B. It must be emphasized that our normalization differs from the classical. For the polar of 2 K we write 2∗ K rather than (2 K)∗ . For each convex body K, we can deﬁne the ellipsoid −2 K by 1 −2 ρ−2 K (u) = (u · v)2 dS2 (K, v) (11∗ ) V (K) S n−1 for u ∈ S n−1 . Note that for the unit ball B, we have −2 B = B. For the polar of ∗ K rather than ( K)∗ , and thus −2 K we write −2 −2 1 (u · v)2 dS2 (K, v). h2 ∗ K (u) = −2 V (K) S n−1 It was shown in [L2] that the L2 –surface area measure S2 (K, ·) is absolutely continuous with respect to the classical surface area measure SK and that the RadonNikodym derivative 1 dS2 (K, ·) = . dSK hK Thus, if P is a polytope whose faces have outer unit normals u1 , . . . , uN , and ai denotes the area of the face with outer normal ui , and hi denotes the distance from the origin to this face, then the measure S2 (P , ·) is concentrated at the points u1 , . . . , uN ∈ S n−1 and S2 (P , {ui }) = ai / hi . Thus, for the polytope P , we have ρ−2 (u) = −2 P

N

1 al (u · ul )2 V (P ) hl l=1

for u ∈ S n−1 . If K is a convex body such that ∂K is C 2 and whose Gauss curvature is positive, then it is well known that the measure SK is absolutely continuous with respect to spherical Lebesgue measure (i.e., (n−1)-dimensional Hausdorff measure) S, and the Radon-Nikodym derivative dSK = fK , dS where fK : S n−1 → (0, ∞) is the reciprocal Gauss curvature of ∂K viewed as a

382

LUTWAK, YANG, AND ZHANG

function of the outer normals (i.e., fK (u), for u ∈ S n−1 , is the reciprocal Gauss curvature at the point of ∂K whose outer unit normal is u). Thus for u ∈ S n−1 , 1 (u) = (u · v)2 h−1 ρ−2 K (v)fK (v) dS(v), −2 K V (K) S n−1 or equivalently, (u) = ρ−2 −2 K

1 V (K)

∂K

2 u · ν(x) h−1 K ν(x) dx,

where ν(x) denotes the outer unit normal at x ∈ ∂K and the integration is with respect to the intrinsic measure on ∂K. A connection between the operators 2 and −2 is given in the following identity. Lemma 2. Suppose K, L ⊂ Rn . If L is a convex body that contains the origin in its interior and K is a star body about the origin, then V−2 K, −2 L V2 L, 2 K = . V (L) V (K) Proof. From the integral representation (9∗ ), deﬁnition (11), Fubini’s theorem, deﬁnition (11∗ ), and the integral representation (9), it follows that 1 V2 L, 2 K = h2 (u) dS2 (L, u) n S n−1 2 K 1 n+2 2 = (u · x) dx dS2 (L, u) n S n−1 V (K) K 1 n+2 (u · v)2 ρK (v) dS(v) dS2 (L, u) = nV (K) S n−1 S n−1 V (L) = ρ n+2 (v)ρ−2 (v) dS(v) −2 L nV (K) S n−1 K V (L) = V−2 K, −2 L . V (K) From the integral representation (9), deﬁnition (11), and Fubini’s theorem, we immediately see that if K and L are star bodies, then V−2 K, 2∗ L V−2 L, 2∗ K = . (12) V (K) V (L) From the integral representation (9∗ ), deﬁnition (11∗ ), (4), and Fubini’s theorem, we immediately see that if K and L are convex bodies, then ∗ L ∗ K V2 K, −2 V2 L, −2 = . (12∗ ) V (K) V (L)

A NEW ELLIPSOID ASSOCIATED WITH CONVEX BODIES

383

An immediate consequence of the deﬁnition of the L2 -centroid body (11) and the transformation rule for support function (3∗ ) is that for φ ∈ GL(n), 2 φK = φ2 K. Since, for the unit ball, B, we have 2 B = B, it follows that if E is an ellipsoid centered at the origin, then 2 E = E. The following lemma shows that −2 is also an intertwining operator with the linear group GL(n). Lemma 1∗ . Suppose K ⊂ Rn is a convex body that contains the origin in its interior. If φ ∈ GL(n), then −2 (φK) = φ−2 K. Proof. From Lemma 2, followed by (8∗ ), Lemma 1, Lemma 2 again, and (8), we have for each star body Q, V2 φK, 2 Q V2 K, φ −1 2 Q V−2 Q, −2 φK = = V (Q) V (φK) V (K) V2 K, 2 φ −1 Q V−2 φ −1 Q, −2 K = = V (K) V φ −1 Q V−2 Q, φ−2 K = . V (Q) But V−2 (Q, −2 φK)/V (Q) = V−2 (Q, φ−2 K)/V (Q) for all star bodies Q implies that −2 φK = φ−2 K. Since, for the unit ball B, we have −2 B = B, it follows from Lemma 1∗ that if E is an ellipsoid centered at the origin, then −2 E = E. Thus −2 2 K = 2 K for all K. In Lemma 2 take L = 2 K, use (7∗ ), and get: For each star body K, V−2 K, 2 K = V (K). (13) But (13) and the dual mixed volume inequality (10) immediately yield the following. Theorem 1. If K ⊂ Rn is a star body about the origin, then V 2 K ≥ V (K), with equality if and only if K is an ellipsoid centered at the origin.

384

LUTWAK, YANG, AND ZHANG

In Lemma 2 take K = −2 L, use (7), and get: For each convex body L, V2 L, −2 L = V (L).

(13∗ )

But (13∗ ) and the L2 –mixed volume inequality (10∗ ) immediately yield the following. Theorem 1∗ . Suppose K ⊂ Rn is a convex body that contains the origin in its interior. Then V −2 K ≤ V (K), with equality if and only if K is an ellipsoid centered at the origin. Theorem 2∗ . Suppose K ⊂ Rn is a convex body that contains the origin in its interior. If E is an ellipsoid centered at the origin such that E ⊂ K, then V −2 E ≤ V −2 K , with equality if and only if E = −2 K. Proof. From the integral representation (9∗ ) we see that the mixed volume V2 (K, ·) is monotone with respect to set inclusion. Now from (7∗ ), the monotonicity of the mixed volume V2 (K, ·), Lemma 1∗ and (1), identity (12∗ ), and the L2 –mixed volume inequality (10∗ ), we have 1=

V2 (K, K) V (K)

∗ E∗ ∗ K V2 E ∗ , −2 V2 (K, E) V2 K, −2 ≥ = = V (K) V (K) V (E ∗ ) 2/n

∗ K 2/n V (E ∗ ) −2/n V −2 ωn −2/n ωn ≥ = , ωn ωn V (E) V −2 K where the last equality is a consequence of the fact that, by (2), the product of the volumes of polar reciprocal ellipsoids, which are centered at the origin, is ωn2 . Hence we have V −2 K ≥ V (E) = V −2 E , with equality (from the equality conditions of the L2 –mixed volume inequality (10∗ )) implying that E and −2 K are dilates, which in turn implies that E = −2 K. The inﬁmum of V (−2 K)/V (K) taken over all convex bodies that contain the origin in their interiors is zero. To get a positive lower bound, some restriction must be made on the positions of the bodies (relative to the origin). A fundamental result due to Ball [B] is that if K is an origin-symmetric convex body, then there exists an ellipsoid EK ⊂ K, centered at the origin, such that V (EK ) ≥ 2−n ωn V (K). Barthe [Br] improved Ball’s theorem by showing that for each origin-symmetric convex body K that is not a parallelotope, there exists an

A NEW ELLIPSOID ASSOCIATED WITH CONVEX BODIES

385

ellipsoid EK ⊂ K, centered at the origin, such that V (EK ) > 2−n ωn V (K). Combine this with Theorem 2∗ and the immediate result is the following. Theorem 3∗ . Suppose K ⊂ Rn is a convex body that is origin-symmetric; then ωn V −2 K ≥ n V (K), 2 with equality if and only if K is a parallelotope. Associated with each convex body K is an important afﬁnely associated point called the John point, j (K) ∈ int K. This point is the center of the (unique) ellipsoid of maximal volume that is contained in the body K. (As an aside, the authors note that in their opinions it would be more appropriate to call this point the Löwner point.) The John point is an afﬁnely associated point in that for each φ ∈ SL(n) we have j (φK) = φj (K). A fundamental result due to Ball [B] is that if K is positioned so that its John point is at the origin, then there exists an ellipsoid EK ⊂ K, centered at the origin, such that V (EK ) ≥ n!ωn n−n/2 (n+1)−(n+1)/2 V (K). Barthe [Br] improved Ball’s theorem by showing that if K is positioned so that its John point is at the origin, then unless K is a simplex, there exists an ellipsoid EK ⊂ K, centered at the origin, such that V (EK ) >

n!ωn V (K). n/2 n (n + 1)(n+1)/2

Together with Theorem 2∗ this immediately gives the following. Theorem 4∗ . If K ⊂ Rn is a convex body positioned so that its John point is at the origin, then n!ωn V (K), V −2 K ≥ n/2 n (n + 1)(n+1)/2 with equality if and only if K is a simplex. The volume-normalized version of the operator 2 is the operator that maps each star body K to [ωn /V (2 K)]1/n 2 K. A classical characterization of the volumenormalized version of the operator 2 is as the solution to the following problem: Given a ﬁxed star body K, ﬁnd an ellipsoid centered at the origin E that minimizes V−2 (K, E) subject to the constraint that V (E) = ωn . Existence, uniqueness, and characterization of the solution to the problem are all contained in the following. Theorem 5. Suppose K ⊂ Rn is a star body about the origin and E is an ellipsoid centered at the origin such that V (E) = ωn . Then

V 2 K 2/n V−2 (K, E) ≥ V (K) , ωn with equality if and only if E = λ2 K where λ = [ωn /V (2 K)]1/n .

386

LUTWAK, YANG, AND ZHANG

Proof. From (2) and Lemma 1, followed by (12), and the dual Minkowski inequality (10), we have 2/n V−2 E ∗ , 2∗ K V−2 (K, E) V−2 K, 2∗ E ∗ −2/n = = ≥ ωn V 2 K , V (K) V (K) V (E ∗ ) with equality if and only if E and 2 K are dilates. Suppose K ⊂ Rn is a ﬁxed convex body that contains the origin in its interior. Find an ellipsoid E, centered at the origin, which minimizes V2 (K, E) subject to the constraint that V (E) = ωn . The solution of the problem turns out to characterize the volume-normalized operator −2 . Existence, uniqueness, and characterization of the solution to the problem are all contained in the following. Theorem 5∗ . Suppose K ⊂ Rn is a convex body that contains the origin in its interior and E is an ellipsoid centered at the origin such that V (E) = ωn . Then

V −2 K −2/n V2 (K, E) ≥ V (K) , ωn with equality if and only if E = λ−2 K, where λ = [ωn /V (−2 K)]1/n . Proof. From (2) and Lemma 1∗ , followed by (12∗ ), and the L2 –Minkowski inequality (10∗ ), we have ∗ E∗ ∗ K V2 E ∗ , −2 2/n V2 (K, E) V2 K, −2 −2/n ∗ = = ≥ ωn V −2 K , ∗ V (K) V (K) V (E ) with equality if and only if E and −2 K are dilates. The L1 -analog of the problem solved by Theorem 5∗ was treated by Petty [P2]. Generalizations were considered by Clack [C] and Giannopoulos and Papadimitrakis [GiPap]. A star body K is said to be in isotropic position if 2 K is a ball and V (K) = 1. Note that for each star body there is a GL(n)-transformation that will map the body into one that is in isotropic position. If the star body K is in isotropic √ position, then the isotropic constant LK of K is deﬁned to be the radius of (1/ n + 2)2 K. If K is an arbitrary star body, then deﬁne its isotropic constant by

V 2 K 1/n 1 LK = √ . n + 2 ωn V (K) From Theorem 1, it immediately follows that for each star body K, −1/n

ωn LK ≥ √ , n+2 with equality if and only if K is an ellipsoid centered at the origin. An important

A NEW ELLIPSOID ASSOCIATED WITH CONVEX BODIES

387

question (previously mentioned) asks if sup LK : K is an origin-symmetric convex body in Rn in isotropic position is dominated by a real number independent of the dimension n. A convex body K is said to be in dual isotropic position if −2 K is a ball and V (K) = 1. Note that for each convex body there is a GL(n)-transformation that will map the body into one that is in isotropic position. If K is in dual isotropic position, then deﬁne the dual isotropic constant L∗K to be the radius of −2 K. If K is an arbitrary convex body, we can deﬁne its dual isotropic constant by

V −2 K 1/n ∗ . LK = ωn V (K) Theorems 1∗ and 3∗ immediately give the following. Theorem 6∗ . Suppose K ⊂ Rn is a convex body that contains the origin in its interior. If K is origin-symmetric and in dual isotropic position, then 1 −1/n ≤ L∗K ≤ ωn . 2 Equality on the left-hand side holds if and only if K is a parallelotope and equality on the right-hand side holds if and only if K is an ellipsoid. Let v denote (n − 1)-dimensional volume. For u ∈ S n−1 , let u⊥ denote the 1codimension subspace of Rn that is orthogonal to u. Milman and Pajor [MPa2] showed that if K is origin-symmetric, then √ n n + 2 V (K) V (K) ≤ h2 K (u) ≤ √ √ ⊥ 2(n + 2) v K ∩ u⊥ 2 3 v K ∩u for all u ∈ S n−1 . Equality on the left-hand side holds for K a right cylinder and u orthogonal to its base, and equality on the right-hand side holds for K a double right cone and u along its axis. For u ∈ S n−1 and a convex body K, let K | u⊥ denote the image of the orthogonal projection of K onto u⊥ . Theorem 7∗ . Suppose K ⊂ Rn is a convex body that contains the origin in its interior. If K is origin-symmetric, then for every u ∈ S n−1 v K | u⊥ 2 v K | u⊥ ∗ K (u) ≤ 2 ≤ h−2 . √ V (K) n V (K) Equality on the left-hand side holds for K a double right cone and u along its axis, and equality on the right-hand side holds for K a right cylinder and u orthogonal to its base.

388

LUTWAK, YANG, AND ZHANG

Proof. From (11∗ ), together with the fact that dS2 (K, ·) = h−1 K dSK , and the Hölder inequality we have √

1/2 1 n|u · v| 2 −1 ρ−2 K (u) = hK (v) dSK (v) nV (K) S n−1 hK (v) √ 1 ≥ n|u · v| dSK (v) nV (K) S n−1 2v K | u⊥ = √ , nV (K) which gives the left inequality. To get the right-hand inequality, note that

1/2 1 |u · v| −1 ρ−2 K (u) = |u · v| dSK (v) V (K) S n−1 hK (v)

2v K | u⊥ |u · v| 1/2 ≤ max V (K) v∈S n−1 hK (v)

1/2 2v K | u⊥ ρK (u)−1 = V (K) 2v K | u⊥ , ≤ V (K) where the last inequality follows from the well-known and easily established fact that V (K) ≤ 2v(K | u⊥ )ρK (u). Theorem 8∗ . Suppose K ⊂ Rn is a convex body that contains the origin in its interior. If K is origin-symmetric and in dual isotropic position, then √ v K | u⊥ ≤ n, for all u ∈ S n−1 . Equality holds if and only if K is a cube and u is in the direction of one of its vertices. Proof. Suppose K is not a cube. From the left-hand inequality of Theorem 7∗ , the fact that [V (−2 K)/ωn ]1/n is the radius of the ball −2 K, together with Theorem 3∗ , we have

2v K | u⊥ V (−2 K) −1/n < 2V (K)−1/n . ≤ √ ωn nV (K) Ball conjectured that each origin-symmetric convex body can be GL(n)-transformed into a body for which the inequality of Theorem 8∗ holds. Giannopoulos and Papadimitrakis [GiPap] showed that this can be accomplished by making the body “surface isotropic.” Theorem 8∗ shows that this can also be done by making the body dual isotropic.

A NEW ELLIPSOID ASSOCIATED WITH CONVEX BODIES

389

Acknowledgment. The authors are most grateful to the referee for the extraordinary attention given to our paper. References [B] [Br] [Bl] [Bo1] [Bo2]

[BoM] [C] [D]

[F] [G1] [G2] [GiPap] [J] [Ju] [Le] [LiM]

[L1] [L2] [L3] [MPa1] [MPa2]

[P1]

K. Ball, Volume ratios and a reverse isoperimetric inequality, J. London Math. Soc. (2) 44 (1991), 351–359. F. Barthe, On a reverse form of the Brascamp-Lieb inequality, Invent. Math. 134 (1998), 335–361. W. Blaschke, Afﬁne Geometrie XIV, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.–Phys. Kl. 70 (1918), 72–75. J. Bourgain, On high-dimensional maximal functions associated to convex bodies, Amer. J. Math. 108 (1986), 1467–1476. , “On the distribution of polynomials on high-dimensional convex sets” in Geometric Aspects of Functional Analysis, Lecture Notes in Math. 1469, Springer, Berlin, 1991, 127–137. J. Bourgain and V. Milman, New volume ratio properties for convex symmetric bodies in R n , Invent. Math. 88 (1987), 319–340. R. Clack, Minkowski surface area under afﬁne transformations, Mathematika 37 (1990), 232–238. S. Dar, “Remarks on Bourgain’s problem on slicing of convex bodies” in Geometric Aspects of Functional Analysis (Israel, 1992–1994), Oper. Theory Adv. Appl. 77 (1995), 61–66. W. J. Firey, p-means of convex bodies, Math. Scand. 10 (1962), 17–24. R. J. Gardner, A positive answer to the Busemann-Petty problem in three dimensions, Ann. of Math. (2) 140 (1994), 435–447. , Geometric Tomography, Encyclopedia Math. Appl. 58, Cambridge Univ. Press, Cambridge, 1995. A. Giannopoulos and M. Papadimitrakis, Isotropic surface area measures, Mathematika 46 (1999), 1–13. F. John, Polar correspondence with respect to a convex region, Duke Math. J. 3 (1937), 355–369. M. Junge, Hyperplane conjecture for quotient spaces of Lp , Forum Math. 6 (1994), 617–635. K. Leichtweiß, Afﬁne Geometry of Convex Bodies, J. A. Barth, Heidelberg, 1998. J. Lindenstrauss and V. D. Milman, “The local theory of normed spaces and its applications to convexity” in Handbook of Convex Geometry, ed. P. M. Gruber and J. M. Wills, North-Holland, Amsterdam, 1993, 1149–1220. E. Lutwak, Centroid bodies and dual mixed volumes, Proc. London Math. Soc. (3) 60 (1990), 365–391. , The Brunn-Minkowski-Firey theory, I: Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), 131–150. , The Brunn-Minkowski-Firey theory, II: Afﬁne and geominimal surface areas, Adv. Math. 118 (1996), 244–294. V. D. Milman and A. Pajor, Cas limites dans des inégalités du type de Khinchine et applications géométriques, C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), 91–96. , “Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space” in Geometric Aspects of Functional Analysis, Lecture Notes in Math. 1376, Springer, Berlin, 1989, 64–104. C. M. Petty, Centroid surfaces, Paciﬁc J. Math. 11 (1961), 1535–1547.

390 [P2] [S] [T] [Z1] [Z2] [Z3]

LUTWAK, YANG, AND ZHANG , Surface area of a convex body under afﬁne transformations, Proc. Amer. Math. Soc. 12 (1961), 824–828. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia Math. Appl. 44, Cambridge Univ. Press, Cambridge, 1993. A. C. Thompson, Minkowski Geometry, Encyclopedia Math. Appl. 63, Cambridge Univ. Press, Cambridge, 1996. G. Zhang, Centered bodies and dual mixed volumes, Trans. Amer. Math. Soc. 345 (1994), 777–801. , Intersection bodies and the Busemann-Petty inequalities in R4 , Ann. of Math. (2) 140 (1994), 331–346. , A positive solution to the Busemann-Petty problem in R4 , Ann. of Math. (2) 149 (1999), 535–543.

Department of Mathematics, Polytechnic University, Six MetroTech Center, Brooklyn, New York 11201-3840, USA

Vol. 104, No. 3

DUKE MATHEMATICAL JOURNAL

© 2000

TRACES OF INTERTWINERS FOR QUANTUM GROUPS AND DIFFERENCE EQUATIONS, I PAVEL ETINGOF and ALEXANDER VARCHENKO

0. Introduction. This paper begins a series of papers whose goal is to establish a representation-theoretic interpretation of the quantum Knizhnik-ZamolodchikovBernard (qKZB) equations and to use this interpretation to study solutions of these equations. It was motivated by the recent work on the qKZB equations in [Fe], [FeTV1], [FeTV2], [MuV], [FeV2]–[FeV5] and by the theory of “quantum conformal blocks” that began with the classical paper [FR]. 0.1. The qKZB equations in [Fe] are difference equations with respect to an unknown function f (z1 , . . . , zN , λ, τ, µ, p) with values in V1 ⊗ · · · ⊗ VN ⊗ VN∗ ⊗ · · · ⊗ V1∗ , where Vi are suitable ﬁnite-dimensional representations of the quantum group Uq (g) (g is a simple Lie algebra), zi , p, τ ∈ C, and λ, µ are weights for g. The qKZB equations are a q-deformation of the Knizhnik-Zamolodchikov-Bernard (KZB) differential equations and an elliptic analogue of the quantum KnizhnikZamolodchikov (qKZ) difference equations, which are, in turn, generalizations of the usual (trigonometric) Knizhnik-Zamolodchikov (KZ) equations. It is proved in [FeTV2] (using an integral representation of solutions) that for g = sl2 the monodromy of the qKZB equations is given by the dual qKZB equations, which are obtained from the qKZB equations by interchanging (λ, τ ) with (µ, p). This fact generalizes the monodromy theorems for the KZB and qKZ equations: the monodromy of the KZB differential equations is the trigonometric degeneration of the qKZB equations (which involves dynamical R-matrices without spectral parameter) (see, e.g., [K3]), and the monodromy of the qKZ equations is given by elliptic dynamical R-matrices (but there is no difference equation) (see [TV1], [TV2]; see also [FR]). The self-duality of the qKZB equations leads one to expect that they should have symmetric solutions uV1 ,...,VN (z, λ, τ, µ, p), that is, such that uV1 ,...,VN (z, λ, τ, µ, p) = u∗V ∗ ,...,V ∗ (z, µ, p, λ, τ ), where u∗ is the dual of u (considered as an endomorN 1 phism of V1 ⊗ · · · ⊗ VN ). Such a solution u (for g = sl2 ) was constructed in [FeV3] and [FeTV2] by an explicit integral formula. It is called the universal hypergeometric function. This function has many interesting properties, in particular the SL(3, Z)-symmetry (see [FeV2], [FeV4], and [FeV5]), where the group SL(3, Z) acts on the lattice Z3 generated by the periods 1, τ, p. A consequence of this symmetry Received 31 August 1999. Revision received 29 November 1999. 2000 Mathematics Subject Classiﬁcation. Primary 20G42; Secondary 33D52, 37K10. Etingof’s work partially supported by National Science Foundation grant number DMS-9700477. Varchenko’s work partially supported by National Science Foundation grant number DMS-9801582. 391

392

ETINGOF AND VARCHENKO

is the qKZB heat equation in [FeV3] for the function u, which is a q-deformation of the KZB heat equation in [Ber]. 0.2. A central fact about the KZB and qKZ equations (and one of the main reasons why they are interesting) is that they are satisﬁed by conformal blocks. More precisely, the KZB equations are satisﬁed by conformal blocks of the Wess-Zumino-Witten conformal ﬁeld theory on an elliptic curve (see [Ber]), and the qKZ equations are satisﬁed by quantum conformal blocks on the cylinder (see [FR]). In representation-theoretic terms, conformal blocks on an elliptic curve are traces of products of intertwining operators for afﬁne Lie algebras (weighted by an element from the maximal torus) (see [Ber]), and quantum conformal blocks on the cylinder are highest matrix elements of products of intertwining operators for quantum afﬁne algebras (see [FR]). This representation-theoretic interpretation of the KZB and qKZ equations is not only interesting by itself, but it also allows us to prove nontrivial properties of solutions, for example, monodromy theorems (see, e.g., [K3], [FR]). The goal of this series is to give a similar interpretation of the qKZB equations. In light of the above, the main idea is obvious: one should consider quantum conformal blocks on an elliptic curve or, representation theoretically, one should consider traces of products of intertwining operators for quantum afﬁne algebras, weighted by an element of the maximal torus. It is natural to expect that such traces satisfy a pair of dual qKZB equations. This is actually true, and we plan to give a proof of it in a subsequent part of the series. However, the details of the proof are relatively complicated, and we would like to start with a simpler (“trigonometric”) limiting case, when τ, p → ∞. This limiting case is the main subject of this paper. 0.3. The structure of this paper is as follows. In Section 1, we introduce the main object of the paper—the renormalized universal trace function FV1 ,...,VN (λ, µ) ∈ (V1 ⊗ · · · ⊗ VN )[0] ⊗ (VN∗ ⊗ · · · ⊗ V1∗ )[0], where λ, µ are weights for g. It is obtained from traces of products of intertwining operators for Uq (g) weighted by an element of the maximal torus. At the end of the section, we formulate the main results of the paper, Theorems 1.1–1.5. Theorems 1.1 and 1.2 state that the function FV1 ,...,VN satisﬁes two systems of difference equations, one with shifts of λ and the other with shifts of µ, which go to each other under the transformation λ → µ, µ → λ. In the special case g = sln , N = 1, V1 = S mn Cn , these systems (as was shown in [EK1]) reduce to the trigonometric Macdonald-Ruijsenaars (MR) systems, so we call them the MR system and the dual MR system. Theorem 1.5 states that the function FV1 ,...,VN (λ, µ) is symmetric: FV1 ,...,VN (λ, µ) = FV∗ ∗ ,...,V ∗ (µ, λ), where ∗ is the permutation of components. It follows from TheoN 1 rems 1.1 and 1.2. Theorems 1.3 and 1.4 state that the function FV1 ,...,Vn satisﬁes two additional systems of difference equations—the trigonometric degenerations of the qKZB and the dual qKZB equations, respectively. Theorems 1.1–1.5 are proved in Sections 2–5.

TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS

393

In Section 6, we study the symmetry of trace functions under q → q −1 , and we deﬁne a modiﬁed trace function uV (λ, µ) by renormalizing FV (λ, µ). (This function is introduced to connect our paper with the papers [FeV2]–[FeV5], and here we deﬁne it only for N = 1; we plan to deﬁne it in general in another paper.) Using the q → q −1 transformation properties and Theorem 1.5, we show that the function uV is symmetric. In Section 7, we compute the function FV (λ, µ), uV (λ, µ) explicitly in the case g = sl2 . In Section 8, we compute explicitly the trigonometric degeneration of the function u from [FeV3] in the case N = 1. We show that this function is the same as uV (λ, µ) up to normalization. In Section 9, we explain that Macdonald’s theory for root systems of type An−1 is a special case of the theory developed in this paper for g = sln , N = 1, V1 = S mn Cn . In Section 10, we consider limiting (degenerate) cases of the theory developed in this paper. 0.4. In subsequent papers of the series, we plan the following. (1) We will give a representation-theoretic proof of the qKZB heat equation and the orthogonality relations for the trigonometric degeneration of the function u (see [FeV3]), using the ideas of [EK1], [EK3], and [EK4]. Cherednik’s theory of difference Fourier transform and Macdonald-Mehta identities for root systems of type A is a special case of this theory, corresponding to the situation g = sln , N = 1, V1 = S mn Cn . (2) We will give a representation-theoretic derivation of the resonance relations from [FeV2] in the trigonometric case, using the ideas of [ESt2]. (3) We will generalize all the results to the case of quantum afﬁne algebras. This involves a representation-theoretic deﬁnition of the function u from [FeV3] for generic values of parameters for any simple Lie algebra and representations, and a representation-theoretic proof of its main properties, such as qKZB and MR equations, orthogonality, and modular transformations (e.g., the qKZB heat equation). As a special case, this theory should contain Macdonald’s theory for afﬁne root systems of type Aˆ n−1 , which was originated in [EK2] but has not been developed from an analytic standpoint. In particular, the classical limit (q → 1) of the modular transformation of the function u should yield the result of Kirillov [K1], [K2], which says that the modular transformation S of afﬁne Jack polynomials (which are essentially the 1-point functions of the WZW model in genus 1; see [EK2]) is given by a matrix of special values of Macdonald polynomials at roots of unity. (4) Specializing this theory to the critical level, we will prove that radial parts of the n ) at the critical level corresponding to the representation central elements of Uq (sl S mn Cn are elliptic Ruijsenaars operators (as far as we know, this is known only in the trigonometric degeneration). Acknowledgments. The work of the ﬁrst author was partly done while he was employed by the Clay Mathematics Institute (CMI) as a CMI Prize Fellow. He thanks the Department of Mathematics at the University of North Carolina at Chapel Hill for hospitality. The second author is grateful to the Department of Mathematics at

394

ETINGOF AND VARCHENKO

Harvard University for hospitality. Both authors are grateful to A. A. Kirillov Jr. for useful suggestions on how to improve the paper. 1. Trace functions for Uq (g) 1.1. The trace functions. Let g be a simple Lie algebra over C. Let h be a Cartan subalgebra of g, and let αi be simple roots of g, i = 1, . . . , r. Let (aij ) be the Cartan matrix of g. Let di be relatively prime positive integers such that (di aij ) is a symmetric matrix. Let ei , fi , and hi be the Chevalley generators of g. Let t be a complex number that is not purely imaginary, and let q = et . For any operator A, we denote etA by q A . Let Uq (g) be the Drinfeld-Jimbo quantum group corresponding to g. We use the same deﬁnition of Uq (g) as in [EFK2]; namely, Uq (g) is a Hopf algebra with generators Ei , Fi , i = 1, . . . , r, q h , h ∈ h, (q 0 = 1), with relations q x+y = q x q y ,

q h Ej q −h = q αj (h) Ei ,

x, y ∈ h,

Ei Fj − Fj Ei = δij 1−aij

(−1)k

k=0 1−aij

k=0

(−1)k

1 − aij k

1 − aij k

qi

qi

q h Fj q −h = q −αj (h) Fi ,

q di hi − q −di hi , q di − q −di

1−aij −k

Ej Eik = 0,

i = j,

1−aij −k

Fj Fik = 0,

i = j,

Ei Fi

where qi = q di and we used notation [n]q ! n , [n]q ! = [1]q × [2]q × · · · × [n]q , = k q [k]q ![n − k]q !

[n]q =

q n − q −n . q − q −1

Comultiplication $, antipode S, and counit % in Uq (g) are given by $(Fi ) = Fi ⊗ 1 + q −di hi ⊗ Fi , $ q h = q h ⊗ q h , S(Fi ) = −q di hi Fi , S q h = q −h , S(Ei ) = −Ei q −di hi ,

$(Ei ) = Ei ⊗ q di hi + 1 ⊗ Ei ,

%(Ei ) = %(Fi ) = 0,

%(q h ) = 1.

Let Mµ be the Verma module over Uq (g) with highest weight µ, and let vµ be its highest weight vector. Let V be a ﬁnite-dimensional representation of Uq (g), and let v ∈ V be a vector of weight µv . It is well known that for a generic µ there exists a unique intertwining operator (vµ : Mµ → Mµ−µv ⊗ V such that (vµ vµ = vµ−µv ⊗ v + l. o. t. (here l. o. t. denotes lower order terms). It is useful to consider the

TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS

395

generating function of such operators (Vµ ∈ HomC (Mµ , ⊕Mν ⊗ V ⊗ V ∗ ) deﬁned by (Vµ = v∈B (vµ ⊗ v ∗ , where the summation is taken over a homogeneous basis B of V . Let V1 , . . . , VN be ﬁnite-dimensional representations of Uq (g), and let vi ∈ Vi be vectors of weights µvi , such that µvi = 0. Deﬁne the formal series in V1 ⊗ · · · ⊗ VN [0] ⊗ q 2(λ,µ) C[[q −2(λ,α1 ) , . . . , q −2(λ,αr ) ]] as + v1 ,...,vN (λ, µ) = Tr |Mµ (v1 N ⊗ 1N−1 · · · (vµN q 2λ . (1.1) µ−

i=2 µvi

It follows from [ESt2, Corollary 3.4] that this series converges (in a suitable region of values of the parameters) to a function of the form q 2(λ,µ) f (λ, µ), where f is a rational function in q 2(λ,αi ) and q 2(µ,αi ) , which is a ﬁnite sum of products of functions of λ and functions of µ. The function (1.1) is called a trace function. Deﬁne also the universal trace function, with values in V1 ⊗ · · · ⊗ VN ⊗ VN∗ ⊗ · · · ⊗ V1∗ : ∗ +V1 ,...,VN (λ, µ) = (1.2) + v1 ,...,vN (λ, µ) ⊗ vN ⊗ · · · ⊗ v1∗ , vi ∈Bi

where Bi are homogeneous bases of Vi . It is easy to see that this function takes values in (V1 ⊗ · · · ⊗ VN )[0] ⊗ (VN∗ ⊗ · · · ⊗ V1∗ )[0]. Using the generating functions (Vµ , one can express the universal trace function as (1.3) +V1 ,...,VN (λ, µ) = Tr (V1 N (∗i) ⊗ 1N−1 · · · (VµN q 2λ , µ+

i=2 h

where we label the components Vi by i and Vi∗ by ∗i, and the notation h(k) for a label k was deﬁned in [Fe]: when acting on a homogeneous multivector, h(k) has to be replaced with the weight in the kth component. Example 1 (See Section 7). Let g = sl2 . In this case, let us represent weights by complex numbers, so that the unique fundamental weight corresponds to 1. If N = 1, and V = V1 is the 3-dimensional representation, then

2 q λµ q −2λ −2 . +V (λ, µ) = 1+ q −q 1 − q −2λ 1 − q 2µ 1 − q −2(λ−1) (Since V [0] is 1-dimensional, we view the function +V as a scalar function.) This example is also computed in [ESt1]. 1.2. The main results. It turns out that the trace function +V1 ,...,VN (λ, µ) satisﬁes some remarkable difference equations. These equations are written in terms of socalled dynamical R-matrices. Below we give a brief introduction to the theory of dynamical R-matrices, referring the reader to the expository paper [ESch] for a more detailed discussion of them.

396

ETINGOF AND VARCHENKO

Let V , W be ﬁnite-dimensional representations of Uq (g). Recall from [EV] the deﬁnitions of the fusion matrix and the exchange matrix. The fusion matrix is the operator JW V (µ) : W ⊗ V → W ⊗ V deﬁned by the formula w JW V (µ)(w⊗v) (1.4) . (µ−µv ⊗ 1 (vµ = (µ The exchange matrix RV W (µ) ∈ End(V ⊗ W ) is deﬁned by 21 21 (1.5) RV W (µ) = JV−1 W (µ)᏾ V ⊗W JW V (µ), where ᏾ is the universal R-matrix of Uq (g). We also use the universal fusion matrix J (λ) and the universal exchange matrix R(λ). They take values in a completion of Uq (g) ⊗ Uq (g) and are deﬁned by the condition that they give JV W (λ), RV W (λ) when evaluated in representations V , W (cf. also [ABRR], [JKOS]). Remark. The fusion matrix describes how to “fuse” together two intertwining operators. The exchange matrix describes how to exchange the order of intertwining operators. The fusion matrix satisﬁes the 2-cocycle identity (see below) and is sometimes referred to as a “quasi-Hopf twist.” The exchange matrix satisﬁes the quantum dynamical Yang-Baxter equation and is referred to as a “dynamical R-matrix.” Let J(λ) := J (−λ − ρ), where ρ is the half-sum of positive roots, and let R(λ) = R(−λ − ρ). Let Q(λ) = m21 (1 ⊗ S −1 )(J(λ)), where m21 (a ⊗ b) := ba and where S is the antipode. It is easy to show that Q(λ) is invertible for generic λ. Deﬁne (1.6)

J1,...,N (λ) = J1,2,...,N (λ)J2,3,...,N (λ) · · · JN−1,N (λ),

where, for example, J1,2,...,N stands for (1 ⊗ $n−1 )(J), where $n−1 : Uq (g) → Uq (g)n−1 is the iterated coproduct. We agree that J1 (λ) = 1. Thus, J1,...,N (λ) describes how to “fuse” N intertwining operators. Let

δq (λ) = q 2(λ,ρ) (1.7) 1 − q −2(λ,α) α>0

be the Weyl denominator. Set (1.8)

ϕV1 ,...,VN (λ, µ) = J1,...,N (λ)−1 +V1 ,...,VN (λ, µ)δq (λ).

Finally, introduce the renormalized trace function (1.9)

FV1 ,...,Vn (λ, µ) (∗1) ϕV1 ,...,VN (λ, −µ − ρ). = Q−1 (µ)(∗N) ⊗ · · · ⊗ Q−1 µ − h(∗2,...,∗N)

It is convenient to formulate properties of trace functions using this renormalization.

TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS

397

Example 2 (See Section 7). If g = sl2 , N = 1, and V = V1 is the 3-dimensional representation, then FV (λ, µ) = q −λµ

q 2(λ+µ) − q 2λ−2 − q 2µ−2 + 1 . 1 − q 2λ−2 1 − q 2µ−2

(This formula is obtained after simpliﬁcations from formula (7.20) below when m = 1.) Note that it is seen from this formula that FV is symmetric in λ and µ. The following theorems describe the properties of FV1 ,...,VN (λ, µ). For any ﬁnite-dimensional Uq (g)-module W , deﬁne the difference operator ᏰW acting on functions of λ ∈ h∗ with values in (V1 ⊗ · · · ⊗ VN )[0] given by the formula (2,...,N) 0N (1.10) ᏰW = Tr |W [ν] R01 (λ) Tν , λ + h · · · R W V1 W VN ν

where Tν f (λ) = f (λ + ν) and the component W is labeled by zero. Theorem 1.1 (Macdonald-Ruijsenaars equations). We have that ᏰλW FV1 ,...,VN (λ, µ) = χW q −2µ FV1 ,...,VN (λ, µ), (1.11) dim W [ν]x ν is the character of W , and by ᏰλW we mean the where χW (x) = operator ᏰW acting on F as a function of λ, in components V1 , . . . , VN . Theorem 1.1 is proved in Section 2. Example 3. If g = sl2 , N = 1, V = V1 is the 3-dimensional representation, and W is the 2-dimensional representation, then 1 − q 2λ−4 1 − q 2λ+2 −1 T , ᏰW = T + 1 − q 2λ−2 1 − q 2λ where Tf (λ) = f (λ+1). To prove this, it is enough to check that this operator is the unique operator of the form a(λ)T + b(λ)T −1 such that (1.11) holds for the function FV given above (note that in our case χW (q −2µ ) = q µ + q −µ ). We introduce also the dual Macdonald-Ruijsenaars operators Ᏸ∨ W , acting on functions of µ ∈ h∗ with values in (VN∗ ⊗ · · · ⊗ V1∗ )[0], by the formula 01 (∗1,...,∗N−1) 0N (1.12) Ᏸ∨ = Tr | (µ) Tν , µ + h · · · R R ∗ ∗ W [ν] W WV WV ν

N

1

where Vj∗ is considered as a module over Uq (g) via the antipode. Theorem 1.2 (Dual Macdonald-Ruijsenaars equations). We have that ∨,µ ᏰW FV1 ,...,VN (λ, µ) = χW q −2λ FV1 ,...,VN (λ, µ), (1.13) ∨,µ

∗ ∗ where ᏰW is Ᏸ∨ W acting on F as a function of µ, in components VN , . . . , V1 .

398

ETINGOF AND VARCHENKO

Theorem 1.2 is proved in Section 3. To formulate the next two results, we need to deﬁne some operators acting on functions of λ and µ with values in (V1 ⊗ · · · ⊗ VN )[0] ⊗ (VN∗ ⊗ · · · ⊗ V1∗ )[0]. For j = 1, . . . , N , deﬁne the operators 2 Dj = q −2µ− xi (1.14) q −2 xi ⊗xi ∗j,∗1,...,∗j −1 , ∗j

where xi is an orthonormal basis of h. Also, deﬁne the operators (1.15)

−1 Kj = Rj +1,j λ + h(j +2,...,N) · · · RNj (λ)−1 7j × Rj 1 λ + h(2,...,j −1) + h(j +1,...,N) · · · Rjj −1 λ + h(j +1,...,N) ,

where 7j f (λ) := f (λ + h(j ) ), and hj,...,k acting on a homogeneous multivector has to be replaced with the sum of weights of components j, . . . , k of this multivector. It is easy to check that Dj commute with each other, and it is known that so do Kj (see [Fe]). Remark. As we have mentioned before, these operators are the trigonometric limits of the qKZB operators with spectral parameters, introduced by Felder. Analogously, deﬁne the operators 2 q −2 xi ⊗xi j,j +1,...,N , (1.16) Dj∨ = q −2λ− xi j

and (1.17)

−1 Kj∨ =R∗j −1,∗j µ + h(∗1,...,∗j −2) · · · R∗1,∗j (µ)−1 7∗j × R∗j,∗N µ + h(∗j +1,...,∗N−1) + h(∗1,...,∗j −1) · · · R∗j,∗j +1 µ + h(∗1,...,∗j −1) ,

where 7∗j f (µ) = f (µ + h∗j ). Like Dj , Kj , the operators Dj∨ , Kj∨ commute. Theorem 1.3 (The qKZB equations). The function FV1 ,...,VN satisﬁes the qKZB equations (1.18) FV1 ,...,VN (λ, µ) = Kj ⊗ Dj FV1 ,...,VN (λ, µ). Theorem 1.4 (The dual qKZB equations). We have (1.19) FV1 ,...,VN (λ, µ) = Dj∨ ⊗ Kj∨ FV1 ,...,VN (λ, µ). Example 4. Let g = sl2 , N = 2, V1 = V2 = C2 with standard basis v+ , v− . In this case, V1 ⊗ V2 [0] is 2-dimensional with basis v+ ⊗ v− , v− ⊗ v+ , and the action of the

TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS

399

dynamical R-matrix in this basis is  1   R(λ) =   −1  q −q q 2λ − 1

 q −1/2 q −1 − q  q −2λ − 1  −2λ −2λ  . 2 −2 q −q q −q  2 q −2λ − 1

Therefore, if FV1 ,V2 (λ, µ) is represented by a 2-by-2 matrix with respect to the above basis, then the qKZB equation corresponding to j = 2 has the form  −1  −2λ q −q q − q 2 q −2λ − q −2 2  q −2λ − 1    q −2λ − 1   −1     q −q 1 q 2λ − 1

F11 (λ + 1, µ) F12 (λ − 1, µ) F11 (λ, µ) F12 (λ, µ) q µ 0 . = × 0 q −µ F21 (λ, µ) F22 (λ, µ) F21 (λ + 1, µ) F22 (λ − 1, µ) Here for convenience we took the shift operator 72 from the left side of the equation to the right side. Remark 1. We should warn the reader that the term “qKZB equations” is normally used for equations that contain elliptic dynamical R-matrices with spectral parameters and are difference equations with respect to these parameters z1 , . . . , zN (see, e.g., [FeTV1]). The equations we consider are a limiting case of the “genuine” qKZB equations, when the modular parameter τ goes to inﬁnity and the ratios of the spectral parameters zj /zj +1 go to zero, with e−2π Im τ << |zj /zj +1 | << 1. Namely, the equations considered here are the equations satisﬁed by the limit (if it exists) of a solution of the “genuine” qKZB equations in the described asymptotic zone. Throughout this paper, we abuse terminology and use the term “qKZB equations” to refer to this limiting case. Remark 2. If N = 1, equations (1.18) and (1.19) are trivial: (1.18) says that the V -component of FV has zero weight, and (1.19) says that the V ∗ -component of FV has zero weight. Remark 3. It is not hard to show using the quantum dynamical Yang-Baxter equation for the dynamical R-matrices that the Macdonald-Ruijsenaars operators commute with the qKZB operators. Similarly, the dual Macdonald-Ruijsenaars operators commute with the dual qKZB operators. Remark 4. Theorems 1.1 and 1.3 have the following interpretation.Suppose v1 , . . . , vN are homogeneous vectors in V1 , . . . , VN of weights ν1 , . . . , νN , νi = 0.

400

ETINGOF AND VARCHENKO

Then the function (FV1 ,...,VN (λ, µ), v1 ⊗ · · · ⊗ vN ) is a common eigenfunction of the operators ᏰλW and Kj with eigenvalues equal to χW (q −2µ ) and ;j (µ) = q −2(µ,νj )+(νj ,νj )+2 i<j (νi ,νj ) , respectively. Thus, the trace functions provide a solution of the problem of simultaneous diagonalization of the Macdonald-Ruijsenaars operators ᏰλW and qKZB operators Kj . Remark 5. The problem of deducing equations of type (1.18) for trace functions (in the case of afﬁne Lie algebras) was suggested to the ﬁrst author in 1992 by his adviser, Igor Frenkel, as a topic for his Ph.D. thesis. However, he failed to solve this problem at that time, partly because the adequate framework—the theory of dynamical R-matrices—was not around yet. Theorem 1.5 (The symmetry identities). The function FV1 ,...,VN is symmetric: FV1 ,...,VN (λ, µ) = FV∗ ∗ ,...,V ∗ (µ, λ),

(1.20)

N

F∗

where V1∗ )[0].

1

is the result of interchanging the factors (V1 ⊗· · ·⊗VN )[0] and (VN∗ ⊗· · ·⊗

Theorem 1.4 is proved in Section 4. Theorem 1.5 is proved in Section 5, using Theorems 1.1 and 1.2. Theorems 1.5 and 1.4 obviously imply Theorem 1.3. Remark 1. Theorem 1.3 can also be derived directly, using the method of Frenkel and Reshetikhin of derivation of the Knizhnik-Zamolodchikov equations. However, this derivation is rather long, and we do not give it here. Remark 2. In the special case g = sln , N = 1, V1 = S mn Cn , the function F is closely related to the kernel of Cherednik’s difference Fourier transform for sln (see [C]), and the symmetry theorem above (Theorem 1.5) is closely related to Cherednik’s theorem on the symmetry of the difference Fourier transform (see [C]). Remark 3. In the special case of Remark 2, Theorem 1.5 was proved in [ESt1, Theorem 5.6]. Remark 4. Theorems 1.1–1.5 can be generalized to the case when g is any symmetrizable Kac-Moody algebra, and Vi are highest weight modules over Uq (g). In this case, the functions +, F make sense as formal power series, and (if g is not ﬁnite-dimensional) the operators ᏰW , Ᏸ∨ W are inﬁnite difference operators: they are inﬁnite sums of terms f (λ)Tν . However, one can show that these sums make sense as operators on power series. We plan to discuss this elsewhere. 2. The Macdonald-Ruijsenaars (MR) equations 2.1. Radial parts. Let V be a ﬁnite-dimensional Uq (g)-module. Then we have the following proposition. Proposition 2.1. (i) For any element X of Uq (g) there exists a unique difference

TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS

401

operator DX acting on V [0]-valued functions, such that Tr (Vµ Xq 2λ = DX Tr (Vµ q 2λ . (2.1) (ii) If X is central, then DXY = DY DX for all Y ∈ Uq (g). In particular, if X, Y are central, then DX DY = DY DX . Proof. This is proved in [EK1, Section 6]. (The assumption that g = gln , which is made throughout [EK1], is not important for the proof of this result.) We call the operator DX the radial part of X. Now recall the Drinfeld-Reshetikhin [D], [R] construction of central elements of Uq (g). In this construction, one deﬁnes elements CW corresponding to ﬁnite-dimensional representations W of Uq (g) by the formula CW = Tr |W (1 ⊗ πW ) ᏾21 ᏾ 1 ⊗ q 2ρ . (2.2) The map W → CW deﬁnes a homomorphism of the Grothendieck ring of the category of ﬁnite-dimensional representations of Uq (g) to the center of Uq (g). Deﬁne difference operators ᏹW := DCW . Proposition 2.2. (i) ᏹW ᏹU = ᏹU ᏹW = ᏹW ⊗U . (ii) ᏹW +V (λ, µ) = χW (q 2(µ+ρ) )+V (λ, µ), where χW (x) = ν dimW [ν]x ν is the character of W . Proof. See [EK1]. (The proof for gl(n) generalizes tautologically to other Lie algebras.) This proposition shows that if we can compute ᏹW explicitly, then we will get a system of difference equations for +V . 2.2. The difference equations. Let G(λ) = q −2ρ Q−1 (λ)S(Q)(λ − h). Theorem 2.3. For any V [0]-valued function on h∗ , (2.3) Tr |W [ν] G(λ + h)RW V (λ) f (λ + ν). (ᏹW f )(λ) = ν

The proof of Theorem 2.3 occupies Sections 2.3–2.6. 2.3. The deﬁning property of J . Deﬁne 1 (1) (2) ᏶(λ) := J − λ − ρ + h + h (2.4) . 2 Lemma 2.4 (The Arnaudon-Buffenoir-Ragoucy-Roche (ABRR) equation [ABRR]; see also [JKOS]). Let V , W be ﬁnite-dimensional Uq (g)-modules. Then (2.5) ᏾21 q 2λ 1 ᏶(λ) = ᏶(λ)q xi ⊗xi q 2λ 1 . Moreover, this solution is unique among solutions of the form 1 + N , where N has only summands with positive weight of the second component.

402

ETINGOF AND VARCHENKO

Remark. As pointed out in [EV], this equation is a limiting case of the quantum Knizhnik-Zamolodchikov equation of [FR], obtained when the afﬁne quantum group degenerates into the ﬁnite-dimensional quantum group. Proof. We recall from [ABRR] (see also [EV, Section 9]) that JW V (λ) satisﬁes the equation 2 2 JW V (λ) q 2(λ+ρ)− xi = ᏾V21W q − xi ⊗xi q 2(λ+ρ)− xi JW V (λ). (2.6) 2

2

Using the weight-zero property of J , we get (2.7)

2 JW V (λ) q −2(λ+ρ) 1 q − xi = ᏾V21W q −2(λ+ρ) 1 q − (xi ⊗1+1⊗xi )(1⊗xi ) JW V (λ). 2

Now apply both sides of (2.7) to the subspace W [ν] ⊗ V [µ]. Using the weight-zero condition again, we get (on that subspace) (2.8)

µ+ν ν JW V (λ) q −2(λ+ρ) 1 q1 q 2 = ᏾V21W q −2(λ+ρ) 1 q µ+ν 1 JW V (λ).

Thus, replacing λ with λ + (1/2)(µ + ν), we get (2.9)

1 1 JW V λ + (µ + ν) q −2(λ+ρ) 1 q ν 2 = ᏾V21W q −2(λ+ρ) 1 JW V λ + (µ + ν) , 2 2

which means that

1 (1) h + h(2) 2 1 (1) = JW V λ + h + h(2) q xi ⊗xi q −2(λ+ρ) 1 . 2

᏾V21W q −2(λ+ρ) 1 JW V λ +

(2.10)

This implies the ﬁrst statement. The second statement is straightforward and also follows from [ABRR]. We also need the cocycle identity for ᏶. To deduce it, recall the 2-cocycle identity for J (see [ABRR], [EV]): J 12,3 (λ)J 12 λ − h(3) = J 1,23 (λ)J 23 (λ). (2.11) Thus, the cocycle identity for ᏶ has the form 1 1 ᏶12,3 (λ)᏶12 λ + h(3) = ᏶1,23 (λ)᏶23 λ − h(1) . (2.12) 2 2

403

TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS

2.4. The function ZV . Let V be a ﬁnite-dimensional Uq (g)-module. Consider the following function with values in V ⊗ V ∗ ⊗ Uq (g) (with components labeled as 1, ∗1, 2): (2.13) ZV (λ, µ) = Tr 0 (Vµ ,01 ᏾20 q 2λ . Lemma 2.5. We have ZV (λ, µ) = ᏾21 q 2λ 1 ZV (λ, µ).

(2.14)

Proof. Let us move the R-matrix ᏾20 around the trace. We get (2.15) ZV (λ, µ) = Tr 0 ᏾21 ᏾20 (Vµ ,01 q 2λ = ᏾21 Tr 0 (Vµ ,01 q 2λ ᏾20 , and the lemma follows after interchanging q 2λ with (Vµ and moving it around the trace. Corollary 2.6. We have ZV (λ, µ) = ᏶

12

(2.16)

(1) (λ)+V

1 (2) λ+ h ,µ . 2 (1)

Proof. Both sides of the equation satisfy (2.14) and have the form +V (λ + (1/2)h(2) , µ) + N , where N has positive degree in component 2. It is easy to show that a solution of (2.14) with such property is unique, which implies the lemma. 2.5. The function XV . Deﬁne the function −1 XV (λ, µ) = Tr 0 (Vµ ,01 ᏾20 q 2λ ᏾03 (2.17) , with values in V ⊗ V ∗ ⊗ Uq (g) ⊗ Uq (g) (components labeled 1, ∗1, 2, 3). Lemma 2.7. We have (2.18)

−1 XV (λ, µ) = ᏾12,3 q 2λ 3 XV (λ, µ) q −2λ 3 ᏾23 .

Proof. We have XV (λ, µ) = Tr 0 (2.19)

᏾03

−1

(Vµ ,01 ᏾20 q 2λ

−1 = ᏾13 Tr 0 (Vµ ,01 ᏾03 ᏾20 q 2λ −1 −1 = ᏾13 ᏾23 Tr 0 (Vµ ,01 ᏾20 ᏾03 q 2λ ᏾23 −1 = ᏾13 ᏾23 q 2λ 3 XV (λ, µ) q −2λ 3 ᏾23 .

The lemma is proved.

404

ETINGOF AND VARCHENKO

Lemma 2.8. We have (2.20)

1 (3) 1 (2) (1) (3) λ− h +V λ + h − h , µ ᏶32 (λ)−1 . 2 2

XV (λ, µ) = ᏶

3,12

(λ)᏶

12

Proof. Consider the function X(λ) = ᏶3,12 (λ)−1 XV (λ)᏶32 (λ). (For brevity we suppress the dependence on µ in the notation.) By Lemmas 2.7 and 2.4, this function satisﬁes the equation (2.21)

2λ (x ⊗1⊗x +1⊗x ⊗x ) 1⊗x ⊗x i i i i X(λ) = X(λ) q 2λ q i i. q 3q 3

On the other hand, by the deﬁnition of XV , the function X is of the form ZV (λ − (1/2)h(3) ) + N , where N has negative weights in the third component. As before, a solution of (2.21) with such property is unique, so it must coincide with its highest term. Thus, N = 0, and by Corollary 2.6 we get the lemma. Corollary 2.9. We have (2.22) −1 Tr 0 ᏾20 ᏾03 (Vµ ,01 q 2λ −2λ 3,12 1 (3) 1 (2) (1) 12 (3) = q ᏶ (λ)᏶ λ − h +V λ + h − h , µ ᏶32 (λ)−1 q 2λ 2 . 2 2 2 Proof. This is straightforward from Lemma 2.8. 2.6. The difference operators. To obtain the action of the operators ᏹW on traces, we need to apply the operation m23 (1 ⊗ 1 ⊗ S) to both sides of (2.22), then multiply by q 2ρ and take the trace. Let ᏾ = ai ⊗ bi and ᏾−1 = ai ⊗ bi . Let u = S(bi )ai be the Drinfeld element. Let ᏶(λ) = ck ⊗ dk (λ) and ᏶−1 (λ) = ck ⊗ dk (λ). Then we get

(2.23)

1 ᏹW +V (λ, µ) = Tr 2 λ + h(2) 2 (2) 2λ 2ρ × +V λ + h , µ dk (λ)q S ck S(ci )q 2 . (1) di (λ)cj

Let us simplify the expression for

(2) ⊗ q −2λ di (λ)dj

dk (λ)q 2λ S(ck ), which enters in (2.23).

Lemma 2.10. We have 2 (2.24) dk (λ)q 2λ S(ck ) = q xi P (λ)S(u)q 2λ , where P (λ) :=

di (λ)S −1 (ci ).

TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS

405

Proof. From Lemma 2.4 we get ᏶−1 (λ) q −2λ 1 = q − xi ⊗xi q −2λ 1 ᏶−1 (λ)᏾21 . Applying m21 (S ⊗ 1) on both sides, we get 2 (2.25) di (λ)ai S(bi )S(ci )q 2λ , dk (λ)q 2λ S ck = q xi which implies the lemma, since ai S(bi ) = S(u). Substituting (2.24) into (2.23) and using the cyclic property of the trace, we get ᏹW +V (λ, µ)

(2.26)

(1)

di (λ)cj +V (λ + ν, µ) 1 2 (2) × Tr 2 q xi P (λ)S(u)q 2λ S(ci )q 2ρ q −2λ di (λ)dj λ + ν 2 (1) = di (λ)cj +V (λ + ν, µ) 1 2 (2) × Tr 2 q xi P (λ)q 2λ S −1 (ci )q −2λ di (λ)dj λ + ν S(u)q 2ρ . 2

=

Lemma 2.11. We have (1) (2) di (λ) ⊗ q 2λ S −1 (ci )q −2λ di (λ) (2.27) (1) (1) (2) (2) = aj dk (λ) q − (xi )2 ((xi )1 +(xi )2 ) S −1 (ck )S −1 bj aj dk (λ). 1

Proof. By Lemma 2.4, we have 23,1 −1 1,23 2λ 1,23 −2λ (xi )1 ((xi )2 +(xi )3 ) (2.28) = ᏾ ᏶ (λ)q . q 1 ᏶ (λ) q 1 Applying m13 (S −1 ⊗ 1 ⊗ 1) to both sides, we get the lemma. Now we use the relations (1) (2) (2.29) ak ⊗ ubk , ai ⊗ S −1 bi ai = ᏾23 ᏶1,23 = ᏶1,32 ᏾23 ,

(2.30) and (2.31)

(1) (2) S(ci )di (λ) ⊗ di (λ) = S(Q)

1 λ − h(2) 2

−1

᏶ 1

1 ˆ (1) λ+ h . 2

The ﬁrst relation follows from the coproduct rule for the R-matrix, the second one is the functoriality of ᏶, and the third one is obtained if one applies m12 (S ⊗ 1 ⊗ 1) to the cocycle relation (2.12).

406

ETINGOF AND VARCHENKO

Using (2.29)–(2.31) and the fact that (1 ⊗ S)(᏾−1 ) = ᏾, we get from (2.27), (1) (2) di (λ) ⊗ q 2λ S −1 (ci )q −2λ di (λ) (2) (1) (2.32) = dk (λ)aj 1 u−1 2 q − (xi )2 ((xi )1 +(xi )2 ) S(ck )dk (λ)bj 2 21 −1 1 1 ˆ (2) − (xi )2 ((xi )1 +(xi )2 ) −1 ᏶ ᏾, u 2 S(Q) λ − h λ+ h =q 2 2 2 where hˆ (2) should be replaced by the weight of the second component of the tensor product after the action of ᏶−1 (i.e., 1 ˆ (2) 1 −1 −1 λ+ h (v ⊗ w) = ᏶ ᏶ν λ + wt(w) + ν (v ⊗ w), 2 2 ν −1 which shifts the where wt(w) is the weight of w and where ᏶−1 ν is the part of ᏶ weight of the second component by ν). Substituting (2.32) into (2.26) and using the identity u−1 S(u) = q −4ρ , we obtain ˜ (2.33) ᏹW +V (λ, µ) = Tr |W [ν] G(λ)R W V (λ) +V (λ + ν, µ), ν

where (2.34)

˜ G(λ) = q −2ρ P (λ)S(Q)(λ).

Thus, Theorem 2.3 follows from the following lemma. ˜ Lemma 2.12. We have P (λ) = Q−1 (λ + h). Hence G(λ) = G(λ + h). Proof. The lemma is obtained by applying m321 (S −1 ⊗ 1 ⊗ S −1 ) to the cocycle identity for J. Theorem 2.3 is proved. 2.7. Calculation of G(λ) Proposition 2.13. Let G(λ) = q −2ρ Q−1 (λ)S(Q)(λ − h). Then (2.35)

G(λ) =

δq (λ) . δq (λ − h)

The rest of this subsection is the proof of Proposition 2.13. Lemma 2.14. We have (2.36) $ G(λ) = J(λ) G λ + h(2) ⊗ G(λ) J−1 (λ). Proof. The element S(Q)(λ) coincides with K (−λ − ρ), where the element is deﬁned in [EV, Section 4.2]. By [EV, Lemma 27], one has w, K (λ)w ∗ = Bλ,W (w, w ∗ ), where the bilinear form Bλ,W : W ⊗ ∗ W → C of weight zero is deﬁned w∗ ∗ by the property (1⊗( , ))◦(w λ−λw∗ (λ = Bλ,W (w, w ) IdMλ . Recall the functoriality K (λ)

TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS

407

property of Bλ,W (see [EV]): (2.37)

BU ⊗W ◦ JU W ⊗ J∗ W ∗ U = BU ◦ BW

(for suitable λ-parameters, which are omitted for brevity). This property implies that −1 (2.38) $ Q(λ) = (S ⊗ S) J21 (λ)−1 Q(λ) ⊗ Q λ − h(1) J λ − h(1) − h(2) , which implies the lemma. Set q = eγ . Then J (λ/γ ) is a power series in γ of the form 1+O(γ ), whose terms lie in Uq (g) ⊗ Uq (g) ⊗ F , where F is the ﬁeld of trigonometric functions. m be a power series of weight zero Lemma 2.15. Let X(λ, γ ) = ∞ m=0 Xm (λ)γ satisfying the equation λ λ $ X(λ, γ ) = J X λ + γ h(1) , γ ⊗ X(λ, γ ) J−1 (2.39) , γ γ where Xi ∈ Uq (g) ⊗ F , such that X0 (λ) = 1. Then (i) if X = 1 on highest-weight vectors of all ﬁnite-dimensional representations, then X = 1; (ii) in general, X = f (λ, γ )/f (λ − γ h, γ ), where f = 1 + γf1 + γ 2 f2 + · · · , fi ∈ F . Proof. (i) X1 is a primitive element, so it lies in h ⊗ F . Since it acts trivially on all highest-weight vectors, X1 = 0. Similarly, Xi = 0 for i > 1 by induction, using the same argument. (ii) Let ωi be fundamental weights, and let ηi (λ) be the eigenvalues of X(λ) on the highest-weight vector of the representation with highest weight ωi . It is easy to see that ηi (λ + γ ωj )ηj (λ) = ηj (λ + γ ωi )ηi (λ), which implies that there exists g ˜ = X(λ)g(λ)/g(λ+γ h). Then X˜ satisﬁes such that g(λ+γ ωi ) = ηi (λ)g(λ). Let X(λ) the conditions of (i), so by (i) X˜ = 1. This proves (ii). (We can set f (λ) = g(λ+γ h).) Corollary 2.16. G(λ) = f (λ)/f (λ − h) for a suitable function f . Proof. The proof follows from Lemmas 2.14 and 2.15. To conclude the proof of Proposition 2.13, it remains to show that in Corollary 2.16, one can take f (λ) = δq (λ). To do this, we apply Theorem 2.3 and part (ii) of Proposition 2.2 in the case V = C. In this case +V (λ, µ) = q 2(µ+ρ,λ) /δq (λ), RW V = 1. So, using Corollary 2.16, we obtain (2.40)

f (λ + ν) q 2(µ+ρ,λ) q 2(µ+ρ,λ+ν) dimW [ν] = χW q 2(µ+ρ) . W [ν] f (λ) δq (λ + ν) δq (λ) ν

408

ETINGOF AND VARCHENKO

This equation is obviously satisﬁed if f = δ. Since the validity of this equation for all µ completely determines f (λ + h)/f (λ), we get the lemma. 2.8. A modiﬁcation of Theorem 2.3. Let ϕV (λ, µ) = +V (λ, µ)δq (λ). Theorem 2.17. We have (2.41)

ᏰλW ϕV (λ, µ) = χW q 2(µ+ρ) ϕV (λ, µ).

Proof. The proof follows from Theorem 2.3 and Propositions 2.2 and 2.13. 2.9. Proof of Theorem 1.1. The theorem follows from Theorem 2.17 and the fusion identity

(2.42)

J1,...,N (λ)−1 R0,1,...,N (λ)J1,...,N λ + h(0) = R01 λ + h(2,...,N) · · · R0N (λ),

which is easily checked from the deﬁnition. 3. The dual MR equations 3.1. The map LW V ∗ (µ). Let W be a ﬁnite-dimensional representation of Uq (g). λ the isotypic component of type M in M ⊗ W . For generic µ, we Denote by HµW λ µ λ given by w ⊗ v → (w v. have a natural isomorphism η : W [λ − µ] ⊗ Mλ → HµW λ Consider the intertwining operator (3.1)

PV ⊗V ∗ ,W ᏾V W (Vµ ⊗ IdW : Mµ ⊗ W −→ Mµ ⊗ W ⊗ V ⊗ V ∗ .

Restricting this operator to the isotypic component of Mµ+ν , from the intertwining property, we obtain (3.2)

PV ⊗V ∗ ,W ᏾V W (Vµ ⊗ IdW W [ν]⊗M

µ+ν

= LW V ∗ (µ) IdW ⊗(Vµ+ν ,

where LW V ∗ (µ) : W ⊗ V ∗ → W ⊗ V ∗ is a uniquely determined linear map. Our task now is to ﬁnd this linear map explicitly. 3.2. Computation of LW V ∗ Proposition 3.1. We have (3.3)

LW V ∗ (µ) = RW V (µ + ν)t2 ,

where t2 means transposition in the third component.

409

TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS

Proof. Let us apply both sides of (3.2) to w ⊗ y. Let B be a homogeneous basis of V . Then we get (3.4) v∈B

∗ LW V ∗ (µ)(vµ+ν w ⊗ y ⊗ v ∗ . η−1 PV W ᏾V W (vµ ⊗ 1 (w µ+ν y ⊗ v = v∈B

Moving η−1 and ᏾ to the right, and using the notation LW V ∗ =

pi ⊗ qi , we get

(3.5) v∈B

21 −1 pi w ∗ (vµ ⊗ 1 (w (µ+ν−wt(v) (vµ+ν y ⊗ qi v ∗ , µ+ν y ⊗ v = PV W ᏾V W i

v∈B

where wt(v) denotes the weight of v. Simplifying (3.5), we get (3.6)

v∈B

J (µ+ν)(v⊗w)

(µ+ν

y ⊗ v∗ =

i

v∈B

PV W (᏾21 )−1 J (µ+ν)(pi w⊗qi∗ v)

(µ+ν

y ⊗ v∗.

This implies that on W [ν] ⊗ V , one has t1 J (µ + ν) = ᏾−1 J 21 (µ + ν) L21 W V ∗ (µ) .

(3.7) That is, (3.8)

t LW V ∗ (µ) = J −1 (µ + ν)᏾21 J 21 (µ + ν) 2 = RW V (µ + ν)t2 .

3.3. Proof of Theorem 1.2. Let us multiply both sides of (3.2) on the right by q 2λ (acting in Mµ ⊗ W ) and on the left by q − xi |V xi |W , and compute the trace in the Verma modules, summing over all ν. After multiplication by δq (λ) and using Proposition 3.1 and the fact that ϕV has zero weight in V , we obtain (3.9) Tr |W (ν) RW V (µ + ν)t2 ϕV (λ, µ + ν). χW q 2λ ϕV (λ, µ) = ν

This is equivalent to (3.10) Tr |W ∗ (−ν) RW V (µ + ν)t1 t2 ϕV (λ, µ + ν). χW ∗ q −2λ ϕV (λ, µ) = ν

Rewriting the last equation in terms of FV (λ, µ), we obtain (3.11) χW ∗ q −2λ FV (λ, µ) Tr W ∗ (ν) Q−1 (µ) V ∗ RW V (µ + ν)t1 t2 Q(µ + ν) V ∗ FV (λ, µ + ν). = ν

410

ETINGOF AND VARCHENKO

The expression RW V (µ)t1 t2 can be computed from (2.38): (3.12) RW V (µ)t1 t2 = Q(µ) ⊗ Q µ − h(1) RW ∗ V ∗ µ − h(1) − h(2) Q−1 µ − h(2) ⊗ Q−1 (µ) . This expression and (3.11) imply Theorem 1.2. 4. The dual qKZB equations Proof of Theorem 1.4. Using [EV, formula (3)], we get w vk −1 k (µ−µv ⊗ 1 (vµ = 1 ⊗ ᏾W (µ−µw ⊗ 1 (w µ , V PV W

(4.1)

k

k

where

(4.2)

vk ⊗ wk = RV W (µ)(v ⊗ w).

Therefore, using simpliﬁed notation, we have (4.3)

V −1 ∗ ∗ −1 V W (W R21 (µ)(Vµ+h(W ∗ ) (W ∗ ( =᏾ µ = ᏾21 R (µ) (µ+h(W ∗ ) (µ . µ+h(V ) µ

Let us now take the j th intertwining operator in (1.3) and move it to the right, permuting it with other operators. We have (4.4) +V1 ,...,VN (λ, µ)

= ᏾j +1j · · · ᏾Nj q

2λ

j

 ∗  Rjj +1 µ +

× Tr

N

−1 h(∗i) 

i=j +2

· · · Rj∗N (µ)−1

V V 2λ Vj N (V1 N (∗i) · · · ( j −1N (∗i) ( j +1(∗j ) N · · · (Vµ+h (µ (∗j ) q µ+ i=2 h µ+ i=j h µ+h + i=j +2 h(∗i)

= ᏾j +1j · · · ᏾Nj q

2λ

j

 ∗  Rjj +1 µ +

× Tr (V1

N i=j +2

−1 h(∗i) 

· · · Rj∗N (µ)−1 7∗j

V V Vj · · · ( j −1N ( j +1 · · · (VµN q 2λ (µ+ (∗i) (∗i) (∗i) µ+ N (∗i) µ+ N h µ+ h h i =j h i=2, i =j i=j +1 i=j +2

(In the last equality, we use that

.

h(∗i) = 0.) Using the cyclic property of the trace,

411

TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS

we can now put the j th operator into the beginning and move it to the right to its original place, thus completing the cycle. This yields −1 +V1 ,...,VN (λ, µ) = ᏾j +1j · · · ᏾Nj q 2λ j ᏾j−1 1 · · · ᏾jj −1 

∗  × Rjj +1 µ +

(4.5)



h(∗i) 

i=j +2

N

∗  × R1j µ+

−1

N

i=2, i =j

· · · Rj∗N (µ)−1 7∗j





h(∗i)  · · · Rj∗−1j µ +

N

 h(∗i) 

i=j +1

× +V1 ,...,VN (λ, µ). Multiplying both sides of (4.5) by δq (λ), replacing µ by −µ − ρ, and multiplying by J1,...,N (λ)−1 and the product of Q−1 ’s, we obtain (4.6) −1 1,...,N FV1 ,...,VN (λ, µ) = J1,...,N (λ)−1 ᏾j +1j · · · ᏾Nj q 2λ j ᏾j−1 (λ) 1 · · · ᏾jj −1 J (∗1) × Q−1 (µ)(∗N) ⊗ · · · ⊗ Q−1 µ − h(∗2,...,∗N)  × R∗jj +1 µ − 

i=2, i =j

h(∗i) 

i=j +2

N

× R∗1j µ −

−1

N

−1 · · · R∗j N (µ)−1 7∗j





h(∗i)  · · · R∗j −1j µ −

× Q(µ)(∗N) ⊗ · · · ⊗ Q µ − h

N

 h(∗i) 

i=j +1

(∗2,...,∗N) (∗1)

FV1 ,...,VN (λ, µ).

Inverting (4.6), we get (4.7) −1 1,...,N FV1 ,...,VN (λ, µ) = J1,...,N (λ)−1 ᏾jj −1 · · · ᏾j 1 q −2λ j ᏾Nj · · · ᏾j−1 (λ) +1j J (∗1) × Q−1 (µ)(∗N) ⊗ · · · ⊗ Q−1 µ − h(∗2,...,∗N)  × R∗j −1j µ −

N i=j +1

−1 h(∗i) 

 · · · R∗1j µ −

N i=2, i =j

−1 h(∗i) 

7∗j

412

ETINGOF AND VARCHENKO

 × R∗j N (µ) · · · R∗jj +1 µ −

N

 h(∗i) 

i=j +2

(∗1) × Q(µ)(∗N) ⊗ · · · ⊗ Q µ − h(∗2,...,∗N) FV1 ,...,VN (λ, µ). Using formula (3.12), it is not difﬁcult to check that, on zero-weight vectors,

(∗1) Q−1 (µ)(∗N) ⊗ · · · ⊗ Q−1 µ − h(∗2,...,∗N)  × R∗j −1j µ −

−1

N

h(∗i) 

× R∗j N (µ) · · · R∗jj +1 µ −

N

· · · R∗1j µ −

i=j +1



(4.8)



N



−1 h(∗i) 

7∗j

i=2,i =j

h(∗i) 

i=j +2

(∗1) × Q(µ)(∗N) ⊗ · · · ⊗ Q µ − h(∗2,...,∗N) = Kj∨ . So it remains to check that (4.9)

−1 1,...,N · · · ᏾j−1 (λ) = Dj∨ J1,...,N (λ)−1 ᏾jj −1 · · · ᏾j 1 q −2λ j ᏾Nj +1j J

(on zero-weight vectors). To check (4.9), observe that it can be written in the form (4.10)

1,...,N J1,...,N (λ)−1 ᏾j,1,...,j −1 q −2λ j ᏾j−1 (λ) = Dj∨ , +1,...,N,j J

which implies that it is enough to check (4.9) for N = 2 and N = 3. Let Ej be the left-hand side of (4.9). Since E1 E2 E3 = D1∨ D2∨ D3∨ = 1 for N = 3, it is enough to prove (4.9) for N = 2 and N = 3, j = 1, 3. But the last two cases follow from the case N = 2, so it is enough to check only this case. If N = 2, identity (4.9) easily follows from Lemma 2.4. Theorem 1.4 is proved.

5. The symmetry identity Proof of Theorem 1.5. In this section we prove Theorem 1.5 using Theorems 1.1 and 1.2. Let V be a ﬁnite-dimensional representation of Uq (g), and let v ∈ V be a homogeneous vector. Consider the operator (vµ as a linear operator Uq (n− ) → Uq (n− ) ⊗ V , identifying the Verma modules with Uq (n− ).

TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS

413

Lemma 5.1. The operator (vµ is a rational function of variables q 2(µ,αi ) of the ˜ vµ is an operator-valued polynomial in xi := q 2(µ,αi ) and ˜ vµ /D(µ), where ( form ( where D(µ) is a polynomial in xi with nonzero constant term. Proof. This follows from the arguments in the proof of Proposition 2.2 in [ESt2]. Corollary 5.2. The matrix elements of the universal trace function + can be expanded in formal series belonging to the space q 2(λ,µ) C[[q −2(λ,αi ) ]] ⊗ C[[q 2(µ,αi ) ]]. Remark. Note that the tensor product in Corollary 5.2 is algebraic, that is, not completed. Let K = q −2(λ,µ) C[[q −2(λ,αi ) ]] ⊗ C[[q −2(µ,αi ) ]]. Denote by L the space (V1 ⊗ · · · ⊗ VN )[0] ⊗ K. Lemma 5.3. For any vector v ∈ V1 ⊗ · · · ⊗ VN [0], the function FV1 ,...,Vn , v belongs to L. Proof. It is clear from [ESt2, Proposition 2.2] and the deﬁnition of J that for any V , W the function JV W (λ) is a rational function of yi = q −2(λ,αi ) , which can be represented as a ratio of an operator-valued polynomial of yi whose constant term is invertible, and a scalar polynomial of yi with nonzero constant term. The same statement, for the same reason, applies to Q(λ). These facts and Corollary 5.2 imply the lemma. Let L0 ⊂ L be the space of solutions of equations (1.11) whose highest term (as of a series in xi = q −2(λ,αi ) ) is vq −2(λ,µ) , where v ∈ (V1 ⊗· · ·⊗VN )[0] is independent of µ. By Theorem 1.1 and Lemma 5.3, for any v ∈ (V1 ⊗ · · · ⊗ VN )[0] the inner product FV1 ,...,VN , v is in L0 . Lemma 5.4. L0 is a complex vector space of dimension d = dim(V1 ⊗ · · · ⊗ VN )[0]. Moreover, if B is a basis of (V1 ⊗· · ·⊗VN )[0], then the collection of functions FV1 ,...,VN , v, v ∈ B, is a basis of L0 . Proof. The solutions FV1 ,...,VN , v, v ∈ B, are linearly independent, since, by the arguments in the proof of Lemma 5.3, so are their highest terms. So it remains to show that the dimension of the space L0 is not bigger than d. Let I be the maximal ideal in C[[x1 , . . . , xr ]]. It is clear that for any m ≥ 1 the operator ᏰW − χW (q −2µ ) preserves the subspace I m L, so it is enough to check that the dimension of the space of solutions is at most d on L/I m L for all m. For s ∈ C∗ , deﬁne an automorphism gs of L by gs f (x1 , . . . , xr ) = f (sx1 , . . . , sxr ). Let Ᏸ0W = lims→0 gs ᏰW gs−1 . It is enough to show that the system Ᏸ0W f = χW (q −2µ )f has no more than d linearly independent solutions. The operator Ᏸ0W is easily computed. Namely, since the exchange matrices converge to the usual R-matrices as λ → ∞ (see [EV, Theorem 50]), one gets Ᏸ0W = dim W [ν] Tν . ν

414

ETINGOF AND VARCHENKO

Therefore, the operators Ᏸ0W are diagonal in the monomial basis, and it is obvious that the system Ᏸ0W f = χW (q −2µ )f has no more than d linearly independent solutions. Corollary 5.5. There exists a unique element M(µ) ∈ End((VN∗ ⊗ · · · ⊗ V1∗ )[0]) ⊗ C[[q −2(µ,αi ) ]] such that FV∗ ∗ ,...,V ∗ (µ, λ) = 1 ⊗ M(µ) FV1 ,...,VN (λ, µ). (5.1) N

1

Proof. This follows from Lemma 5.4 and Theorem 1.2. Now we prove Theorem 1.5. By applying Corollary 5.5 twice, we obtain that M(µ)M (λ) = 1 for some function M (λ). This implies that M is in fact independent of µ. Looking at the highest terms of F and F ∗ , one ﬁnds that M = 1. This proves Theorem 1.5. 6. The symmetry of the universal trace function under q → q −1 , and the function uV 6.1. The symmetry identity with q → q −1 . In this section we study the behavior of the functions +V under the transformation q → q −1 . Consider the algebra isomorphism ξ : Uq (g) → Uq −1 (g) given by ξ(Ei ) = Ei q di hi , ξ(Fi ) = q −di hi Fi , ξ(q h ) = q −h . Using this isomorphism, we identify Uq (g) and Uq −1 (g) (as algebras), and regard any module over Uq (g) as a module over Uq −1 (g) and vice versa. We start by proving the following theorem. Theorem 6.1. The function +V1 ,...,VN satisﬁes the equality (6.1)

+V q −1 , −λ, µ = u(q)−1 Q(q, λ)+V (q, λ, µ),

where u(q) denotes the Drinfeld element u of Uq (g). Remark. To avoid confusion, here and below we explicitly specify for every expression whether it is evaluated at q or at q −1 . Proof. It is obvious that ξ is an anti-isomorphism of coalgebras. This implies that for a ﬁnite-dimensional representation V and a vector v ∈ V [0], the operator P (vµ (q) is an intertwining operator Mµ → V ⊗ Mµ for Uq −1 (g); hence the operator (᏾10 )−1 (vµ (q) is an intertwining operator Mµ → Mµ ⊗ V for Uq −1 (g). We have (᏾10 )−1 (q)(vµ (q)vµ = vµ ⊗ v + · · · , so we have (6.2)

᏾10

−1

(q)(Vµ (q) = (Vµ q −1 .

Let us multiply both sides of (6.2) by q 2λ and take the trace.

TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS

415

Using (2.15) and (2.16), we get −1 Tr ᏾10 (q)(Vµ (q)q 2λ = u(q)−1 Q(q, λ)+V (q, λ, µ), (6.3) which implies the theorem. 6.2. The function uˆ V (λ, µ). Now let V be an irreducible Uq (g)-module, and deﬁne the function uˆ V (λ, µ) := δq (λ)+V q −1 , −λ, −µ − ρ . (6.4) Corollary 6.2. The function uˆ V is symmetric: uˆ V (λ, µ) = uˆ ∗V ∗ (µ, λ). Proof. Let ν be the highest weight of V . By Theorem 6.1, we have (6.5)

uˆ V (λ, µ) = q −(ν,ν+2ρ) δq (λ)Q(q, λ)+V (λ, −µ − ρ) = q −(ν,ν+2ρ) Q(q, λ) ⊗ Q(q, µ) FV (λ, µ).

Therefore, the symmetry of uˆ V follows from Theorem 1.5. 6.3. The function uV (λ, µ). The function uˆ V has poles. We would like to get rid of them by multiplying the function uˆ V by its denominator. Following [ESt2], introduce the function V

(6.6)

δV (µ) =

kα

q (α,µ+ρ)−n(α,α)/2 − q −(α,µ+ρ)+n(α,α)/2 ,

α>0 n=1

where for a positive root α we deﬁne kαV := max{n : V [nα] = 0}. Deﬁne the function (6.7)

uV (λ, µ) := δV ∗ (−λ − ρ)δV (−µ − ρ)uˆ V (λ, µ).

Proposition 6.3. The function uV is symmetric: uV (λ, µ) = u∗V ∗ (µ, λ), and it is a product of q −2(λ,µ) and a trigonometric polynomial of λ and µ. In particular, it is holomorphic in λ, µ. Proof. The symmetry of uV follows from Corollary 6.2. The fact that uV is holomorphic in µ is a consequence of Lemma 5.1. The fact that uV is holomorphic in λ follows from the symmetry. Using (6.5), one gets the following expression of uV in terms of FV : (6.8) uV (λ, µ) = q −(ν,ν+2ρ) δV ∗ (−λ − ρ)Q(q, λ) ⊗ δV (−µ − ρ)Q(q, µ) FV (λ, µ). In Section 8 we show that the function uV for Uq (sl2 ) coincides (up to a constant factor) with the trigonometric limit of the universal hypergeometric function u introduced in [FeV3].

416

ETINGOF AND VARCHENKO

7. Calculation of the functions uV and FV for sl2 7.1. Calculation of the trace function. Recall that Uq (sl2 ) is generated by E, F, with relations

q h,

(7.1)

q h Eq −h = q 2 E,

q h F q −h = q −2 F,

EF − F E =

q h − q −h , q − q −1

and the coproduct is deﬁned by (7.2)

$(E) = E ⊗ q h + 1 ⊗ E,

$(F ) = F ⊗ 1 + q −h ⊗ F.

Weights for Uq (sl2 ) can be identiﬁed with complex numbers: we say that a vector v in a Uq (sl2 )-module has weight µ if q h v = q µ v. In this case we have (µ, µ ) = µµ /2. We write q λ for q (λ,α) . Thus, the meaning of q λ in this section is different from the previous sections. Recall also that for any number a the q-number [a]q is deﬁned by [a]q = (q a − q −a )/(q − q −1 ). Consider the function +V (λ, µ), where V is the representation of Uq (sl2 ) with highest weight 2m, m ∈ Z+ . Since the weight space V [0] is 1-dimensional, this function can be considered as a scalar function. Theorem 7.1. The function +V (λ, µ) is given by the formula +V (λ, µ) = q λµ

m

l q l(l−1)/2 q − q −1

l=0

(7.3) ×

[m + l]q ! q −2lλ l , l−1 −2(λ−j ) [l]q ![m − l]q ! j =0 1 − q 2(µ−j ) j =0 1 − q

where [n]q ! = [1]q · · · [n]q . Proof. We ﬁx a generator w0 of V [0]. Let us compute the intertwining operator ( µ : Mµ → M µ ⊗ V . Let wβ , β = m, m − 1, . . . , −m, be the basis of V deﬁned by the condition F wβ = wβ−1 if β = −m. (This basis is unique up to a common scalar.) The image of the highest-weight vector under the operator (µ has the form (7.4)

( µ vµ =

m

cj (µ)F j vµ ⊗ wj ,

j =0

where c0 = 1. Let us compute the coefﬁcients cj (µ). They are computed from the condition that $(E) = E ⊗ q h + 1 ⊗ E annihilates the right-hand side of (7.4). Using the formula (7.5)

ewβ = [m + β + 1]q [m − β]q wβ+1 ,

β = m,

417

TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS

we can rewrite this condition in the form (7.6)

q 2i [i]q [µ − i + 1]q ci (µ) + [m + i]q [m − i + 1]q ci−1 (µ) = 0,

i ≥ 1.

Solving this recurrence relation, we obtain (7.7)

ci (µ) = (−1)i q −i(i+1)

i

[m + i]q ! [µ − i + 1]−1 q . [i]q ![m − i]q ! j =1

F kv

Now we need to compute (µ the q-binomial theorem, we obtain (7.8)

µ.

For this, we need to compute $(F k ). Using

k k k q −lh F k−l ⊗ F l , $ F k = F ⊗ 1 + q −h ⊗ F = l q −2 l=0

where

k i k i=k−l+1 1 − p := l . i l p i=1 1 − p

(7.9)

Now, using the intertwining property of (µ , we obtain ( µ F k vµ = $ F k ( µ v µ

k m k (7.10) −lh k−l l = q F ⊗F cj (µ)F j vµ ⊗ wj , l q −2 l=0

j =0

This double sum reduces to a single sum if we use the following version of the q-binomial theorem: l

k −1 (7.11) x k−l = 1 − pi x . l p k≥l

i=0

Substituting (7.7) into (7.10) and using (7.11), we obtain (7.3). The theorem is proved. Corollary 7.2. We have uV (λ, µ) = q −λµ

m l=0

(7.12) ×

l q −l(l−1)/2 q − q −1

[m + l]q ! q −l(λ+µ) [l]q ![m − l]q !

m

q λ+j − q −λ−j q µ+j − q −µ−j .

j =l+1

Proof. The statement is obtained from Theorem 7.1 and formulas (6.4) and (6.7). The function uV is manifestly symmetric in λ and µ, as predicted by Proposition 6.3.

418

ETINGOF AND VARCHENKO

7.2. A formula for Q(µ). Now we would like to compute the function FV . So it remains to compute the value of Q(µ) on the zero-weight subspace of V . This value is given by the following lemma. Lemma 7.3. The element Q(µ) acts on the zero-weight subspace of V by the formula Q(µ)|V [0] = q −2m

(7.13)

m

q −2µ−2j +2 − q −2m j =1

q −2µ−2j − 1

.

Proof. Denote by Qr,l (µ) the eigenvalue of Q(µ) on the subspace of weight l of the representation of Uq (sl2 ) with highest weight r. It is clear that Q0,0 = 1. The values of Q1,1 and Q1,−1 are easily computed from the deﬁnition. Namely, from the ABRR equation (see Lemma 2.4) we have ᏶(µ) = 1 +

q −1 − q F ⊗E +··· , q −2µ − q −2 q −h ⊗ q h

and, therefore, Q1,−1 = 1,

(7.14)

Q1,1 =

q −2µ − q −2 . q −2µ − 1

Now consider the subspace of weight l in the tensor product C2 ⊗ W , where W has highest weight r. Take the determinant of both sides of (2.38) restricted to this subspace. Using the strict triangularity of J, we can ignore the J terms and obtain (7.15)

Qr+1,l (µ)Qr−1,l (µ) = Qr,l−1 (µ − 1)Qr,l+1 (µ + 1)

q −2µ − q −2 . q −2µ − 1

This implies Qr+1,l (µ) Qr,l−1 (µ − 1) q −2µ − q −2 , = Qr,l+1 (µ + 1) Qr−1,l (µ) q −2µ − 1

(7.16) from which we get (7.17)

Qr+1,l (µ) = Qr,l+1 (µ + 1)

(r+l−1)/2

j =0

q −2µ − q −r−l−1 q −2µ+2j − q −2 = . q −2µ+2j − 1 q −2µ − 1

This yields (7.18)

Qr+1,l (µ) =

(r−l+1)/2

j =0

q −2µ−2j − q −r−l−1 . q −2µ−2j − 1

419

TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS

In particular, Q2m,0 (µ) =

(7.19)

m

q −2µ−2j − q −2m j =0

q −2µ−2j − 1

= q −2m

m

q −2µ−2j +2 − q −2m j =1

q −2µ−2j − 1

,

as desired. 7.3. Calculation of FV (λ, µ) Proposition 7.4. The function FV (λ, µ) is given by (7.20) FV (λ, µ) = q

−λµ

m

m

j =1

l [m + l]q ! q −2µ−2j − 1 2m q q l(l−1)/2 q − q −1 −2µ−2j +2 −2m [l]q ![m − l]q ! q −q

× l

j =1

l=0

q −2lλ 1 − q −2(µ+j )

l

j =1

1 − q −2(λ−j )

.

Proof. Using the deﬁnition of FV , Theorem 7.1, and Lemma 7.3, we get (7.20). Remark. Note that expression (7.20) is not manifestly symmetric. In fact, the separate terms in the sum (7.20) are not symmetric, and it is only the whole sum that has the symmetry λ → µ. 8. Integral representation of the trace function uV for g = sl2 . In [FeV3], G. Felder and the second author, studying the qKZB difference equations, deﬁned the universal hypergeometric function um (λ, τ, µ, p) (depending on a parameter q) with a number of interesting properties. In this section we consider the trigonometric limit of this function and show that it coincides, up to a constant factor, with the function uV (m) deﬁned by (6.8), where V (m) is the irreducible representation of Uq (sl2 ) with highest weight 2m. By deﬁnition, the trigonometric limit of um (λ, τ, µ, p) is the leading coefﬁcient of the asymptotic expansion of um (λ, τ, µ, p) as the modular parameters τ, p tend to i∞. We denote this leading coefﬁcient by um (λ, µ). Sending τ, p to i∞ in the deﬁnition of [FeV3], one obtains the following deﬁnition of the function um (λ, µ). Let q = et with Re(t) > 0, and let 0 < |A| < 1. Deﬁne a function (8.1) Im (λ, µ, A) =q

−λµ

m

Tj A−2 q λ+m − q −λ−m Tj A−2 q µ+m − q −µ−m 1 − Tj A2 1 − Tj A−2 |T1 |=···=|Tm |=1 j =1

2 q −2 1 − Ti Tj−1 dTj × . ∧m √ −1 −2 −1 2 j =1 2π −1Tj 1 − T i Tj q 1 − T i Tj q 1≤i<j ≤m

420

ETINGOF AND VARCHENKO

It is obvious that this function analytically continues to a rational function in A. We denote this analytic continuation also by Im . Deﬁnition. We put (8.2)

um (λ, µ) = Im λ, µ, q m .

Remark. Note that A = q m does not satisfy the condition |A| < 1, which is why we needed to talk about the analytic continuation. The main result of this section is the following theorem. Theorem 8.1. Let V (m) be the irreducible representation of Uq (sl2 ) with highest weight 2m. Then (8.3)

uV (m) (λ, µ) = q (3m−1)m

m [2m]q ! q − q −1 um (λ, µ), m!

where uV (m) is as in Section 6. Proof. The rest of the section is the proof of Theorem 8.1. Lemma 8.2. We have

(8.4)

2 1 − Ti Tj−1 dTj ∧m √ −1 −2 −1 2 j =1 2π −1Tj 1 − T i Tj q |T1 |=···=|Tm |=1 1≤i<j ≤m 1 − Ti Tj q = q −m(m−1)/2

m! . [m]q !

Proof. Let 0 ≤ |p| < 1. Consider the Macdonald denominator of type Am−1 (see [M2]): −1 xi xj , p ∞ $p,t x1 , . . . , xm = (8.5) , −1 i =j txi xj , p ∞

j where (a, p)∞ := ∞ j =0 (1 − ap ). The Macdonald constant term identity (see [M2, pp. 20–21]) says that the constant term of the Laurent series (8.5) (with respect to xi ) is given by j −i

t p, p ∞ t j −i , p ∞ . c. t.($p,t ) = m! (8.6) t j −i+1 , p ∞ t j −i−1 p, p ∞ i<j Setting in this identity p = 0, we get (8.7)

c. t.($0,t ) = m!

(1 − t)m . (1 − t) · · · (1 − t m )

Substituting t = q −2 , we obtain the lemma.

TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS

421

Deﬁne the expression (8.8)

Ik,m = q −k(λ+µ+2m)−k(k−1)/2

k! , [k]q !

0 ≤ k ≤ m,

and the differential form (8.9)

k

Tj A−2 q λ+m − q −λ−m Tj A−2 q µ+m − q −µ−m 1 − Tj A−2 q 2m−2k−2 Ok,m = 1 − Tj A2 1 − Tj A−2 q 2m−2k 1 − Tj A−2 q −2 j =1 2

1 − Ti Tj−1 dTj × . ∧k √ −1 −2 −1 2 j =1 2π −1T T q T q 1 − T 1 − T j i i j j 1≤i<j ≤k Lemma 8.3. We have

(8.10)

|Tj |=A2 q −2(m−k) (1+%)

Ok,m = −kq

−2(m−k)

×

q λ+k − q −λ−k q µ+k − q −µ−k 1 − q −2 1 − A4 q −2(m−k) 1 − q −2(m−k+1)

|Tj |=A2 q −2(m−k+1) (1+%)

Ok−1,m + Ik,m .

Proof. Let us perform the integration with respect to Tk for ﬁxed T1 , . . . , Tk−1 . It is obvious that the differential form Fk,m , as a function of Tk , has two simple poles inside the circle of integration: Tk = A2 q 2k−2m and Tk = 0. Therefore, the integral with respect to Tk is equal to the sum of residues at these two poles. The residue at the ﬁrst pole can be found by a direct computation and equals the ﬁrst term on the right-hand side of (8.10). The residue at zero equals to Ik,m by Lemma 8.2. Lemma 8.3 is proved. Now let us prove Theorem 8.1. Let us move the contour of integration in the deﬁnition of Im from |Tj | = 1 to |Tj | = A2 (1+%), via contours |Tj | = B, A2 (1+%) ≤ B ≤ 1. On the way, we do not run into any poles; therefore, we have −λµ−m(m−1) (8.11) Om,m |A=q m um (λ, µ) = q |Tj |=A2 (1+%)

(in the sense of analytic continuation). So, to prove Theorem 8.1, it is enough to compute |Tj |=A2 (1+%) Om,m |A=q m . We do it by using the recursive relation given in Lemma 8.3. Lemma 8.4. We have (8.12)

|Tj |=A2 q −2(m−k) (1+%)

Ok,m |A=q m =

k j =0

ckj Ij,m ,

422

ETINGOF AND VARCHENKO

where (8.13) ckj = (−1)

k−j

1−q

−2 k−j

k k! j!

i=j +1

q λ+i − q −λ−i q µ+i − q −µ−i 2(i−m) q . 1 − q 2(i+m) 1 − q 2(i−m−1)

Proof. The proof is a straightforward induction in k using Lemma 8.3. Substituting k = m in Lemma 8.4, and using the deﬁnition of Ik,m , we ﬁnd the following expression for um : (8.14) um (λ, µ) = q −m(3m−1)

m

−m −λµ −l(λ+µ)−l(l−1)/2 l m! q − q −1 q q q − q −1 [2m]q ! l=0

×

[m + l]q ! [l]q ![m − l]q !

m

q λ+i − q −λ−i q µ+i − q −µ−i .

i=j +1

Comparing (8.14) with (7.12), we get Theorem 8.1. 9. Trace functions and Macdonald theory. In this section, following [EK1] and [FeV1], we connect the results of this paper with the Macdonald-Ruijsenaars theory. We restrict ourselves to the case of g = sln , N = 1, and we let V be the q-analogue of the representation S mn Cn . The zero-weight subspace of this representation is 1dimensional, so the function +V can be regarded as a scalar function. We denote this scalar function by +m (q, λ, µ). Recall the deﬁnition of Macdonald operators (see [M1] and [EK1]). They are operators on the space of functions f (λ1 , . . . , λn ) that are invariant under simultaneous shifting of the variables λi → λi + c, and they have the form  

tq 2λi − t −1 q 2λj   TI , (9.1) Mr = 2λi − q 2λj q / I ⊂{1,...,n}:|I |=r i∈I,j ∈I / I and TI λj = λj + 1 if j ∈ I . Here q, t are parameters. We where TI λj = λj if j ∈ assume that t = q m+1 , where m is a nonnegative integer. It is known from [M1] that the operators Mr commute. From this it can be deduced that for a generic µ = (µ1 , . . . , µn ), µi = 0, there exists a unique power series fm0 (q, λ, µ) ∈ C[[q λ2 −λ1 , . . . , q λn −λn−1 ]] such that the series fm (q, λ, µ) := q 2(λ,µ−mρ) fm0 (q, λ, µ) satisﬁes difference equations   (9.2) q 2 i∈I (µ+ρ)i  fm (q, λ, µ). Mr fm (q, λ, µ) =  I ⊂{1,...,n}:|I |=r

TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS

423

Remark. The series fm0 is convergent to an analytic (in fact, a trigonometric) function. The following theorem is contained in [EK1]. Theorem 9.1 [EK1, Theorem 5]. We have (9.3)

fm (q, λ, µ) = γm (q, λ)−1 +m q −1 , −λ, µ ,

where (9.4)

γm (q, λ) :=

m

q λl −λj − q 2i q λj −λl .

i=1 l<j

Remark. The exact statement of [EK1, Theorem 5], in our conventions, is that the function fm (q, λ, µ)γm (q, λ) is equal to Tr |Mµ ((Vµ (q −1 )q 2λ ), which is equivalent to Theorem 9.1. Let ᏰW (q −1 , −λ) denote the difference operator, obtained from the operator ᏰW deﬁned in Section 1 by the transformation q → q −1 and by the change of coordinates λ → −λ. Let ;r Cn denote the q-analogue of the rth fundamental representation of sln . Corollary 9.2. We have Ᏸ;r Cn q −1 , −λ = δq (λ)γm (q, λ) ◦ Mr ◦ γm (q, λ)−1 δq (λ)−1 . Proof. This follows from Theorems 9.1 and 1.1. In conclusion of this section, we would like to make several important remarks. Remark 1. Corollary 9.2 is a degenerate (trigonometric) case of [FeV1, Theorem 5.2], which says that the elliptic Ruijsenaars operators are transfer matrices of the elliptic quantum sln acting in V [0]. Thus, Theorems 1.1 and 9.1 immediately imply the trigonometric case of [FeV1, Theorem 5.2] (i.e., the case without spectral parameter). Remark 2. Conversely, the trigonometric case of [FeV1, Theorem 5.2] together with Theorem 1.1 immediately implies Theorem 9.1 (and many other results of [EK1]). This is a “direct” proof of Theorem 9.1, in the sense that it involves (unlike the original proof of [EK1]) a direct computation of the radial parts of the central elements of Uq (g). (Another direct proof of Theorem 9.1 is given in [Mi], where the radial part of the central element corresponding to the vector representation is computed.) Remark 3. The line of argument discussed in Remark 2 can be extended to the elliptic case. Namely, combining an elliptic analogue of Theorem 1.1 (for afﬁne Lie algebras at the critical level) and [FeV1, Theorem 5.2], one can prove an elliptic analogue of Theorem 9.1, which says that the radial parts of the central elements

424

ETINGOF AND VARCHENKO

n ) at the critical level corresponding to evaluation modules ;r Cn (z), acting of Uq (sl on functions with values in V [0], are elliptic Ruisjsenaars operators. This has been a conjecture for a number of years (see, e.g., [Mi, p. 415]). We plan to do this in a subsequent paper of this series. Remark 4. In many arguments of this paper, Verma modules Mµ can be replaced with ﬁnite-dimensional irreducible modules Lµ with sufﬁciently large highest weight, and one can prove analogues of Theorems 1.1–1.5 in this situation (in the same way). ˆ Vµ q 2λ ), where ( ˆ Vµ : Lµ → Lµ ⊗ V ⊗ ˆ m (q, λ, µ) = Tr(( In particular, one may set + ∗ V [0] is the intertwiner with highest coefﬁcient 1. (Such an operator exists if and only if µ − mρ ≥ 0; see [EK1].) Then one can show analogously to Theorem 9.1 ˆ m (q −1 , −λ, µ + mρ) is (see [EK1]) that the function fˆm (q, λ, µ) := γm (q, λ)−1 + 2λ the Macdonald polynomial Pµ (q, t, q ) with highest weight µ (µ is a dominant integral weight). In this case, Theorem 1.1 says that Macdonald’s polynomials are eigenfunctions of Macdonald’s operators, Theorem 1.2 gives recursive relations for Macdonald’s polynomials with respect to the weight (for sl(2), the usual 3-term relation for orthogonal polynomials), and Theorem 1.3 is the Macdonald symmetry identity (see [M1]). (This representation-theoretic derivation of the symmetry identity is somewhat different from the one in [EK3], where a pictorial argument is used.) 10. Limiting cases. In this section we discuss various degenerations of the function FV1 ,...,VN (q, λ, µ) and the corresponding degenerate versions of Theorems 1.1– 1.5. The main limiting cases we are interested in are the classical limit and the rational limit. The classical limit corresponds to passing from Uq (g) to g in the trace construction; in this limit the function F depends rationally on µ but trigonometrically of λ. This limit corresponds to the theory of spherical functions on the Lie group G associated with g, which is discussed in [EFK1]. In the rational limit, which corresponds to the theory of spherical functions on g rather than G, the function F becomes rational in both λ and µ, restoring the symmetry. In this limit, the function F is the Baker-Akhiezer function for a multivariable bispectral problem (see [Be]). 10.1. The classical (KZB) limit. Let (10.1)

λ FVc1 ,...,VN (λ, µ) = lim FV1 ,...,VN q = et , , µ . t→0 2t

We call this limit the classical limit. The existence of this limit follows from Proposition 10.1. Remark. Here and below we write the dependence of functions on q explicitly, since in this section q is allowed to vary. Example 1. If g = sl(2), N = 1, and V = V1 is the 3-dimensional representation, we have 1 1 + eλ µ FVc (λ, µ) = e−λµ/2 1− (10.2) . µ−1 µ 1 − eλ

TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS

425

The classical limit is obtained when, as in the situation of Section 1, we take the ordinary enveloping algebra U (g) instead of the quantized one Uq (g). More precisely, let (Vµ be intertwining operators for U (g) deﬁned as in Section 1, and set +Vc 1 ,...,VN (λ, µ) = Tr (V1 N (∗i) ⊗ 1N−1 · · · (VµN eλ . (10.3) µ+

i=2 h

Also, set (10.4)

δ(λ) = e(λ,ρ)

1 − e−(λ,α) ,

α>0

and let Qc (µ) be the limit of Q(µ) as q → 1 (i.e., it is deﬁned as in Section 1 from representation theory of U (g)). Then we have the following proposition. Proposition 10.1. We have (10.5) FVc1 ,...,VN (λ, µ) (∗1) c = δ(λ) Qc−1 (µ)(∗N) ⊗ · · · ⊗ Qc−1 µ − h(∗2,...,∗N) +V1 ,...,VN (λ, −µ − ρ). Proof. The proof is straightforward. Let us now look at the degenerations of the properties of FV1 ,...,VN in the classical limit. We start with the analogue of Theorem 1.1. First of all, we have the following analogue of Proposition 2.1, which is proved analogously to Proposition 2.1. Proposition 10.2. (i) For any element X of U (g), there exists a unique differential operator DX acting on V [0]-valued functions, such that Tr (Vµ Xeλ = DX Tr (Vµ eλ . (10.6) (ii) If X is central, then DXY = DY DX for all Y ∈ U (g). In particular, if X, Y are central, then DX DY = DY DX . The operator DX can be computed explicitly for any element X, but in general the answer is complicated. However, if X is the quadratic Casimir C, the answer is easy to write down. Namely, deﬁne D˜ X = δ(λ)DX δ(λ)−1 . Then we have (see [E], [ESt1]): (10.7)

D˜ C = $h −

f α eα

α>0

2 sinh (1/2)(λ, α) 2

− (ρ, ρ),

where fα , eα are root generators such that (eα , fα ) = 1, and $h is the Laplacian on the Cartan subalgebra associated with the standard invariant form.

426

ETINGOF AND VARCHENKO

Thus, we have the following classical analogue of Theorem 1.1. Theorem 10.3. For any X in the center of U (g), let pX be the symmetric polynomial on h∗ such that X|Mµ = pX (µ + ρ). Then we have λ c D˜ X FV1 ,...,VN (λ, µ) = pX (−µ)FVc1 ,...,VN (λ, µ).

(10.8) In particular,

(10.9)

$h −

f α eα

2 sinh2 (1/2)(λ, α) α>0

FVc1 ,...,VN (λ, µ) = (µ, µ)FVc1 ,...,VN (λ, µ).

Formula (10.9) was obtained in [E] and [ESt1], but it can also be derived by taking the classical limit in Theorem 1.1. Now let us consider the classical analogue of Theorem 1.2. Let Ᏸ∨,c W denote the difference operators deﬁned by formula (1.12) for q = 1 (i.e., R(µ) are the exchange matrices for U (g) with µ replaced by −µ − ρ). Then we have the following result, obtained by passing to the limit in Theorem 1.2. Theorem 10.4. We have ∨,c,µ

ᏰW

(10.10)

FVc1 ,...,VN (λ, µ) = χW e−λ FVc1 ,...,VN (λ, µ).

Example 2. In the case of Example 1, Theorems 10.3 and 10.4 have the form 2 1 ∂ µ2 c c − F (λ, µ) (λ, µ) = F V 4 V ∂λ2 2 sinh2 (λ/2) and

(µ − 2)(µ + 1) −1 c T+ T FV (λ, µ) = eλ/2 + e−λ/2 FVc (λ, µ) µ(µ − 1)

(where T is the shift by 1 in µ), which is easily checked from (10.2). Now consider the classical limit of Theorem 1.3. For this purpose introduce the classical dynamical r-matrix r(λ), which is the classical limit of the exchange matrix R(q, λ). This matrix is deﬁned by the formula λ R q = et , = 1 − 2r(λ)t + O t 2 (10.11) 2t and is equal to

1 1 1 cotanh (λ, α)eα ∧ fα , r(λ) = − O + 2 2 2 α>0

where O is the Casimir tensor (see [EV]). Taking the quasi-classical limit in Theorem 1.3 and using that r(−λ) = r 21 (λ), we obtain the following result.

TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS

427

Theorem 10.5. For any j = 1, . . . , N , we have     ∂ − rlj (λ) − rj l (λ) FVc1 ,...,VN (λ, µ) ∂h(j ) l<j l>j (10.12) (∗j ) 1 2 (∗l) = µ+ FVc1 ,...,VN (λ, µ), xi + xi ⊗ xi 2 ∗j l<j

where (∂/∂h(j ) )X(λ) = (∂/∂ν)X(λ) if X is a tensor-valued function whose j th component has weight ν. The last equation is the trigonometric limit of the KZB equation, which is why the classical limit is called “the KZB limit.” Let us now consider the classical limit of Theorem 1.4. Let Kj∨,c be the difference operators deﬁned by formula (1.17) for q = 1 (i.e., R(µ) are exchange matrices for U (g) with µ replaced by −µ − ρ). Then we have the following result, obtained by passing to the limit in Theorem 1.4. Theorem 10.6. We have (10.13)

Kj∨,c FVc1 ,...,VN (λ, µ) = eλ j FVc1 ,...,VN (λ, µ).

Finally, Theorem 1.5 does not have an analogue in the classical limit. In this limit, the symmetry between λ and µ is destroyed, since F c is a product of e−(λ,µ) with a function that is trigonometric in λ but rational in µ. 10.2. The rational limit. The rational limit is a further degeneration of the classical limit. Namely, let µ (10.14) . FVr1 ,...,VN (λ, µ) = lim FVc1 ,...,VN λγ , γ →0 γ We call this limit the rational limit. The existence of this limit and the fact that det(F r ) = 0 can be deduced from [ESt1, Corollary 3.3]. Example 3. If g = sl(2), N = 1, and V = V1 is the 3-dimensional representation, we have 2 r −λµ/2 1+ (10.15) . FV (λ, µ) = e λµ r The degeneration of Theorem 1.1 in this limit is the following theorem. Let D˜ X r be the rational limit of D˜ X ; that is, D˜ X (λ) is the leading coefﬁcient of D˜ X (γ λ) as γ → 0. For instance, 2fα eα (10.16) . D˜ Cr = $h − (λ, α)2 α>0

428

ETINGOF AND VARCHENKO

Theorem 10.7. For any X in the center of U (g), let pX be the symmetric polyr be the top degree component of nomial on h∗ such that X|Mµ = pX (µ + ρ). Let pX pX . Then we have r,λ r r D˜ X FV1 ,...,VN (λ, µ) = pX (−µ)FVr1 ,...,VN (λ, µ).

(10.17) In particular,

2fα eα $h − FVr1 ,...,VN (λ, µ) = (µ, µ)FVr1 ,...,VN (λ, µ). (λ, α)2

(10.18)

α>0

The degeneration of Theorem 1.2 looks as follows. Theorem 10.8. Equations (10.17) and (10.18) are satisﬁed for the function FVr,∗∗ ,...,V ∗ . N

1

Example 4. In the situation of Example 3, Theorems 10.7 and 10.8 have the form 2 ∂ 2 µ2 r r F (λ, µ), F − (λ, µ) = V 4 V ∂λ2 λ2 2 2 λ2 r ∂ r F (λ, µ), − (λ, µ) = F V 4 V ∂µ2 µ2 which is easily checked from (10.15). Using the asymptotics of FVr1 ,...,VN (λ, µ) at inﬁnity, similarly to arguments of Section 5, one can deduce from Theorems 10.7 and 10.8 the following analogue of Theorem 1.5 (the symmetry theorem). Theorem 10.9. The function FVr1 ,...,VN is symmetric: (10.19)

FVr1 ,...,VN (λ, µ) = FVr,∗∗ ,...,V ∗ (µ, λ). N

1

Thus, the symmetry, lost in the ﬁrst limit, is restored after taking the second limit. Remark 1. Another proof of Theorem 10.8 is based on representation of the above sequence of two limits as a single limiting procedure, which is symmetric in λ and µ. Namely, one can show that λ µ FVr1 ,...,VN (λ, µ) = lim FV1 ,...,VN q = est/2 , , (10.20) , s,t→0 t s after which Theorem 10.9 follows from Theorem 1.5. Remark 2. Theorems 10.7 and 10.8 show that the function FVr1 ,...,VN (λ, µ) is a solution of the matrix bispectral problem in several variables (on the bispectral problem, see, e.g., [DuG], [G]). The Baker-Akhiezer function of the rational Calogero

429

TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS

system of type A, which is a known solution of the multidimensional bispectral problem (see [VeStC]; see also [Be]), is a special case of FVr1 ,...,VN (λ, µ) (N = 1, V = V1 = S mn Cn ). Finally, let us consider the rational limit of Theorems 1.3 and 1.4. To formulate the analogue of Theorem 1.3, introduce the rational limit of the classical dynamical r-matrix, r 0 (λ) = limγ →0 γ r(γ λ). It has the form r 0 (λ) =

(10.21)

eα ∧ f α α>0

(λ, α)

.

Taking the rational limit in Theorem 10.4, we get the following. Theorem 10.10. For any j = 1, . . . , N , we have (10.22) 

  ∂  − rlj0 (λ) − rj0l (λ) FVr1 ,...,VN (λ, µ) = µ∗j FVr1 ,...,VN (λ, µ). ∂h(j ) l<j

l>j

The analogue of Theorem 1.4 is the following. Theorem 10.11. Equation (10.22) is satisﬁed for the function FVr,∗∗ ,...,V ∗ . N

1

10.3. The qKZ and KZ limits. Assume that |q| < 1. The qKZ limit is deﬁned by (10.23)

qKZ

FV1 ,...,VN (λ, µ) =

lim

(λ,αi )→−∞

q 2(λ,µ) FV1 ,...,VN (λ, µ).

It is easy to check using [EV, Theorem 50] that qKZ

(10.24)

FV1 ,...,VN (λ, µ) (∗1) = Q−1 (µ)(∗N) ⊗ · · · ⊗ Q−1 µ − h(∗2,...,∗N) ! " × (V1 N (∗i) ⊗ 1N−1 · · · (VµN µ+

i=2 h

(∗1) 1,...,N = Q−1 (µ)(∗N) ⊗ · · · ⊗ Q−1 µ − h(∗2,...,∗N) J (µ)∗ , where , denotes the highest-matrix element. (The last expression is an endomorphism of (VN∗ ⊗· · ·⊗V1∗ )[0], which is regarded as an element of (V1 ⊗· · ·⊗VN )[0]⊗ (VN∗ ⊗ · · · ⊗ V1∗ )[0].) In particular, this function is independent on λ. Let us now consider the behavior of the equations given by Theorems 1.1–1.5 in the qKZ limit. The MR equations given by Theorem 1.1 become trivial. Namely, when (λ, αi ) → −∞, we have J(λ) → 1, and hence R(λ) → ᏾21 . The matrix ᏾21 is triangular, so

430

ETINGOF AND VARCHENKO

only its diagonal part contributes to the trace. Inspection of this diagonal part shows that lim(λ,αi )→−∞ ᏰW = ν dimW [ν] Tν , and the limiting equation is

dimW [ν] q −2(ν,µ) Tν F qKZ = χW q −2µ F qKZ

ν

(where Tν is the shift of λ), which is a trivial consequence of the fact that F qKZ is independent on λ. The dual MR equations given by Theorem 1.2 have a slightly more interesting limit. It is easy to see that the only term on each side of (1.11) which survives in the limit is the term corresponding to the lowest weight νW of W . Therefore, the limiting equation has the form (10.25) qKZ ∗ (∗1,...,∗N−1) vW ⊗ 1, R01 · · · R0N WV ∗ µ+h W V ∗ (µ)(vW ⊗ 1) FV1 ,...,VN (µ + νW ) N

1

qKZ

= FV1 ,...,VN (µ). The qKZB equations become the trigonometric limit of the qKZ equations. Namely, for all j = 1, . . . , N , we have (10.26)

qKZ

qKZ

−1 −1 −2µ ᏾j,j ᏾ · · · ᏾j −1,j ⊗ Dj FV1 ,...,VN (µ) = FV1 ,...,VN (µ). +1 · · · ᏾j N q j 1j

If N = 2, these equations are closely related to the ABRR equation (see Lemma 2.4). In general, they are essentially the N-component version of the ABRR equation. Remark. The dual qKZB equations do not seem to have a reasonable qKZ limit. Also, the symmetry relation (Theorem 1.5) does not hold in the qKZ limit since the function depends on µ and not on λ. The KZ limit is obtained from the qKZ limit as q → 1, in which case the qKZ equations degenerate into the trigonometric KZ equations (the quasi-classical limit of the ABRR equation). We leave it to the reader to derive the limiting equations in this case. We plan to consider these limits in more detail in another paper in the more interesting case of afﬁne Lie algebras and quantum afﬁne algebras. References [ABRR] [Be]

D. Arnaudon, E. Buffenoir, E. Ragoucy, and Ph. Roche, Universal solutions of quantum dynamical Yang-Baxter equations, Lett. Math. Phys. 44 (1998), 201–214. Yu. Berest, “Huygens’ principle and the bispectral problem” in The Bispectral Problem (Montreal, 1997), CRM Proc. Lecture Notes 14, Amer. Math. Soc., Providence, 1998, 11–30.

TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS [Ber] [C] [D] [DuG] [E] [EFK1] [EFK2] [EK1] [EK2] [EK3] [EK4] [ESch] [ESt1] [ESt2] [EV] [Fe]

[FeTV1]

[FeTV2] [FeV1] [FeV2]

[FeV3] [FeV4] [FeV5] [FR]

431

D. Bernard, On the Wess-Zumino-Witten models on the torus, Nuclear Phys. B 303 (1988), 77–93. I. Cherednik, Macdonald’s evaluation conjectures and difference Fourier transform, Invent. Math. 122 (1995), 119–145. V. G. Drinfeld, On almost cocommutative Hopf algebras, Leningrad Math. J. 1 (1990), 321–342. J. J. Duistermaat and F. A. Grünbaum, Differential equations in the spectral parameter, Comm. Math. Phys. 103 (1986), 177–240. P. I. Etingof, Quantum integrable systems and representations of Lie algebras, J. Math. Phys. 36 (1995), 2636–2651. P. Etingof, I. Frenkel, and A. A. Kirillov Jr., Spherical functions on afﬁne Lie groups, Duke Math. J. 80 (1995), 59–90. , Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations, Math. Surveys Monogr. 58, Amer. Math. Soc., Providence, 1998. P. I. Etingof and A. A. Kirillov Jr., Macdonald’s polynomials and representations of quantum groups, Math. Res. Lett. 1 (1994), 279–296. , On the afﬁne analogue of Jack and Macdonald polynomials, Duke Math. J. 78 (1995), 229–256. , Representation-theoretic proof of the inner product and symmetry identities for Macdonald’s polynomials, Compositio Math. 102 (1996), 179–202. , On Cherednik-Macdonald-Mehta identities, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 43–47, available from http://www.ams.org/era/. P. Etingof and O. Schiffmann, Lectures on the dynamical Yang-Baxter equations, preprint, http://xxx.arXiv.org/abs/math.QA/9908064. P. Etingof and K. Styrkas, Algebraic integrability of Schrödinger operators and representations of Lie algebras, Compositio Math. 98 (1995), 91–112. , Algebraic integrability of Macdonald operators and representations of quantum groups, Compositio Math. 114 (1998), 125–152. P. Etingof and A. Varchenko, Exchange dynamical quantum groups, Comm. Math. Phys. 205 (1999), 19–52. G. Felder, “Conformal ﬁeld theory and integrable systems associated to elliptic curves” in Proceedings of the International Congress of Mathematicians (Zürich, 1994), Birkhäuser, Basel, 1995, 1247–1255. G. Felder, V. Tarasov, and A. Varchenko, “Solutions of the elliptic QKZB equations and Bethe ansatz, I” in Topics in Singularity Theory, Adv. Math. Sci. 34, Amer. Math. Soc. Transl. Ser. 2 180 Amer. Math. Soc., Providence, 1997, 45–76. , Monodromy of solutions of the elliptic quantum Knizhnik-Zamolodchikov-Bernard difference equations, preprint, http://xxx.arXiv.org/abs/q-alg/9705017. G. Felder and A. Varchenko, Elliptic quantum groups and Ruijsenaars models, J. Statist. Phys. 89 (1997), 963–980. , Resonance relations for solutions of the elliptic QKZB equations, fusion rules, and eigenvectors of transfer matrices of restricted interaction-round-a-face models, Commun. Contemp. Math. 1 (1999), 335–403. , Quantum KZB heat equation, modular transformations and GL(3, Z), I, preprint, http://xxx.arXiv.org/abs/math.QA/9809139. , The elliptic gamma function and SL(3, Z)×Z3 , preprint, http://xxx.arXiv.org/abs/ math.QA/9907061. , Quantum KZB heat equation, modular transformations, and GL(3, Z), II, preprint, 1999. I. Frenkel and N. Reshetikhin, Quantum afﬁne algebras and holonomic difference

432 [G] [JKOS] [K1] [K2] [K3] [M1] [M2] [Mi] [MuV]

[R] [TV1]

[TV2] [VeStC]

ETINGOF AND VARCHENKO equations, Comm. Math. Phys. 146 (1992), 1–60. F. A. Grünbaum, “Some bispectral musings” in The Bispectral Problem (Montreal, 1997), CRM Proc. Lecture Notes 14, Amer. Math. Soc., Providence, 1998, 31–45. M. Jimbo, S. Odake, H. Konno, and J. Shiraishi, Quasi-Hopf twistors for elliptic quantum groups, Transform. Groups 4 (1999), 303–327. A. A. Kirillov Jr., On an inner product in modular tensor categories, J. Amer. Math. Soc. 9 (1996), 1135–1169. , On an inner product in modular tensor categories, II, Adv. Theor. Math. Phys 2 (1998), 155–180. , Traces of intertwining operators and Macdonald’s polynomials, Ph.D. thesis, Yale Univ., 1995, http://xxx.arXiv.org/abs/q-alg/9503012. I. G. Macdonald, A new class of symmetric functions, Sém. Lothar. Combin. 20 (1988), available from http://cartan.u-strasbg.fr:80/ ˜slc. , Symmetric Functions and Orthogonal Polynomials, Univ. Lecture Ser. 12, Amer. Math. Soc., Providence, 1998. K. Mimachi, Macdonald’s operator from the center of the quantized universal enveloping algebra Uq (gl(N)), Internat. Math. Res. Notices 1994, 415–424. E. Mukhin and A. Varchenko, Solutions of the qKZB equation in tensor products of ﬁnite-dimensional modules over the elliptic quantum group Eτ,η sl2 , preprint, http://xxx.arXiv.org/abs/q-alg/9712056. N. Yu. Reshetikhin, Quasitriangle Hopf algebras and invariants of tangles, Leningrad Math. J. 1 (1990), 491–513. V. Tarasov and A. Varchenko, Geometry of q-hypergeometric functions as a bridge between Yangians and quantum afﬁne algebras, Invent. Math. 128 (1997), 501–588. , Geometry of q-hypergeometric functions, quantum afﬁne algebras and elliptic quantum groups, Astérisque 246 (1997), 1–135. A. P. Veselov, K. L. Styrkas, and O. A. Chalykh, Algebraic integrability for the Schrödinger equation and ﬁnite reﬂection groups, Theoret. and Math. Physics 94 (1993), 182–197.

Etingof: Department of Mathematics, Room 2-165, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA; [email protected] Varchenko: Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599, USA; [email protected]

Vol. 104, No. 3

DUKE MATHEMATICAL JOURNAL

© 2000

ON THE FINITE-GAP ANSATZ IN THE CONTINUUM LIMIT OF THE TODA LATTICE A. B. J. KUIJLAARS

1. Introduction. The ﬁnite, nonperiodic Toda lattice is a dynamical system given by the Lax pair dL/dt = BL − LB for tridiagonal n × n matrices 

a1

b  1  L=    0

b1

0

a2

b2

b2

a3 .. .

.. ..

. .

bn−1



   ,   bn−1 



0

−b  1  B =   

an

b1

0

0

b2

−b2

0 .. .

0

.. ..

. .

−bn−1



   .   bn−1  0

This corresponds to the system of equations dak 2 = 2 bk2 − bk−1 , k = 1, . . . , n, dt dbk = bk (ak+1 − ak ), k = 1, . . . , n − 1, dt

(1.1) (1.2)

with b0 = bn = 0. The Toda lattice is an integrable system that is solved explicitly by the inverse spectral method (see [M]). Deift and McLaughlin [DM] studied the continuum limit of the Toda lattice. Here the size n tends to inﬁnity, and we write ak (t; n) and bk (t; n) to indicate the dependence on n. For given continuous functions a0 (y) and b0 (y) > 0 for y ∈ (0, 1), Deift and McLaughlin [DM] take initial values

k k ak (0; n) = a0 , bk (0; n) = b0 (1.3) n n and study the limiting behavior of a[xn] (tn; n),

b[xn] (tn; n)

as n → ∞ with ﬁxed x ∈ (0, 1) and t > 0. Received 7 June 1999. 2000 Mathematics Subject Classiﬁcation. Primary 37J35, 37K10; Secondary 31A15, 35Q53. Author’s work supported in part by research project number G.0278.97 and by a research grant from the Fonds voor Wetenschappelijk Onderzoek–Vlaanderen (FWO). 433

434

A. B. J. KUIJLAARS

This continuum limit has many similarities to the zero dispersion limit of the Korteweg–de Vries (KdV) equation ut − 6uux + 2 uxxx = 0,

u(x, 0) = u0 (x).

The limit 0 was ﬁrst studied by Lax and Levermore [LL]. Working with singlewell initial data u0 (x) < 0 and using the inverse scattering transform method for the KdV equation, they obtained for each (x, t) a quadratic, constrained minimization problem over density functions in the spectral parameter. Their analysis revealed the signiﬁcance of the set where the constraints are not effective. Under the assumption that this set is a ﬁnite number of intervals, Lax and Levermore described the weak limit of u(x, t; ) as 0 in terms of the endpoints of the intervals. The number N + 1 of intervals depends on (x, t). In a region of space-time where N is constant, it was moreover shown that the endpoints satisfy a hyperbolic system of equations, identical to the multiphase modulation equations derived earlier by Flaschka, Forest, and McLaughlin [FFM]. The nature of rapid, small-scale oscillations that lead to the weak limit if N ≥ 1 was later examined by Venakides [V]. These higher-order asymptotics are constructed out of the theta function associated with the hyperelliptic Riemann surface of genus N based on the 2N + 2 endpoints. See [DVZ] and [ELZ] for recent advances on the zero dispersion limit of the KdV equation. Also in the continuum limit of the Toda lattice, a quadratic minimization problem with constraints plays a prominent role. The nature of the support of the minimizer is again basic in the description of the singular limit. Based on a ﬁnite-gap ansatz (i.e., under the assumption that the set where the constraints are not valid is a ﬁnite union of intervals), weak limits are described in terms of the endpoints of these intervals (see [DM]). A similar approach was developed to describe the semiclassical limit of the defocusing nonlinear Schrödinger equation (see [JLM]). The zero-gap ansatz (i.e., N = 0) was established in all of the above cases at small times t < tb , where tb is the shock time for the formal limit, which is, for example, ut − 6uux = 0,

u(x, 0) = u0 (x)

in the case of the KdV equation, and the system at = 4bbx ,

bt = bax ,

a(x, 0) = a0 (x),

b(x, 0) = b0 (x)

(1.4)

in the case of the Toda lattice. In a number of cases a global zero or one-gap ansatz was established (see [DM], [T]). However, a general result was lacking until the work of Deift, Kriecherbauer, and McLaughlin [DKM] for the continuum limit of the Toda lattice. They proved that the ﬁnite-gap ansatz holds for real analytic spectral data provided that only one constraint is active. In general there are two constraints for the minimizer ψ, namely, ψ ≥ 0 and ψ ≤ φ, where φ is determined by the initial data. The result of [DKM] applies to the case in which the upper constraint φ is not active. It is the purpose of this paper to extend this result to the case in which both the nonnegativity constraint and the upper constraint are active. It turns out that the

ON THE FINITE-GAP ANSATZ

435

methods of [DKM] cannot be easily generalized to this case—at least this author was unable to do it. Instead a number of new ideas are used, coming from potential theory and ﬁnding their origin in approximation theory and the theory of orthogonal polynomials (see, e.g., [DK], [DS], [KM], [ST], and [To]). We expect that a combination of the ideas of [DKM] and this paper may be used to obtain the ﬁnite-gap ansatz for the zero dispersion limit of the KdV equation with real-analytic initial data. The next section contains a more detailed description of the extremal problem as it arises in the continuum limit of the Toda lattice. The main result of the paper is stated in Section 3, and its proof is given in Section 4. 2. The extremal problem. We describe the extremal problem as it appears in the continuum limit of the Toda lattice (see [DM]). For general information on this kind of extremal problem, we refer the reader to [D] and [ST]. The problem associated with the Toda lattice involves two “spectral functions” V and φ that are obtained from the initial data a0 and b0 of (1.3). We write α(y) = a0 (y) − 2b0 (y),

β(y) = a0 (y) + 2b0 (y).

(2.1)

Note that α(y) < β(y) for y ∈ (0, 1), since b0 (y) > 0. The following assumptions are made on the functions α(y) and β(y) (see [DM]): • α(y) has at most one critical point that, if it exists, is a minimum; • β(y) has at most one critical point that, if it exists, is a maximum; • α(0) = β(0) and α(1) = β(1). In [DM] it is assumed that α is strictly decreasing, but the same analysis goes through under the above assumptions (see also [KVA1]). Let A := min α(y),

B := max β(y).

y∈[0,1]

y∈[0,1]

It follows from the assumptions that for each x ∈ [A, B], the set {y ∈ [0, 1] : α(y) ≤ x ≤ β(y)} is a closed interval, which is denoted by [y− (x), y+ (x)]. Then V0 and φ are deﬁned on [A, B] by the formulas

y− (x)

V0 (x) :=

cosh 0

1 φ(x) := π

y+ (x)

y− (x)

x − a0 (y) dy, 2b0 (y)

(2.2)

1 dy β(y) − x x − α(y)

(2.3)

−1

(see [DM]). Note that V0 was denoted by θ in [DM]. We are also using a different normalization for φ. We refer to the function V0 as the external ﬁeld. The function φ is nonnegative and is called the constraint.

436

A. B. J. KUIJLAARS

For an arbitrary, continuous V : [A, B] → R and for a ﬁnite, nonnegative Borel measure µ on [A, B], the energy in the presence of V is 1 IV (µ) := dµ(x) dµ(y) + 2 V (x) dµ(x). (2.4) log |x − y|

For a positive Borel measure σ on [A, B] and a constant c ∈ (0, dσ ), we let σ,c ᏹ := µ : 0 ≤ µ ≤ σ, dµ = c (2.5) be the collection of positive Borel measures µ on [a, b] which are dominated by σ and have total mass c. We refer to c as the normalization constant. Associated with V , σ , and c is the energy minimization problem µ,c EV = inf IV (µ) : µ ∈ ᏹσ,c . (2.6) σ,c such It is known (see [DM], [DS]) that there exists a unique measure µ = µσ,c V in ᏹ σ,c σ,c that EV = IV (µV ). The minimizer is called the equilibrium measure (or extremal measure) associated with V , σ , and c. We introduce an operator L acting on measures by Lµ(x) := log |x − y| dµ(y).

If µ has a density ψ, we also write Lψ instead of Lµ. Then the equilibrium measure is characterized by the variational inequalities Lµ(x) − V (x) ≤ l

if x ∈ supp(σ − µ),

(2.7)

Lµ(x) − V (x) ≥ l

if x ∈ supp(µ),

(2.8)

which are satisﬁed for some constant l, which depends on V , σ , and c (see [DM], [DS]). Here and in what follows, supp denotes the closed support of a measure. In the description of the continuum limit of the Toda lattice presented in [DM], the external ﬁeld V depends on time V (x) = Vt (x) = V0 (x) − tx,

x ∈ [A, B],

with V0 given by (2.2), and dσ (x) = φ(x) dx is the measure supported on [A, B] with

B density φ given by (2.3). The constraint φ is such that A φ(x) dx = 1. Therefore, the equilibrium measures µσ,c Vt exist for 0 < c < 1 and t ≥ 0. Of interest are the sets I 0 (c, t) := x ∈ [A, B] : Lµσ,c Vt − Vt (x) = l , I + (c, t) := x ∈ [A, B] : Lµσ,c Vt − Vt (x) > l , I − (c, t) := x ∈ [A, B] : Lµσ,c Vt − Vt (x) < l , where l is the constant in the variational inequalities (2.7) and (2.8) which also depends

437

ON THE FINITE-GAP ANSATZ

on c and t. Every interval in I 0 is called a band. The bands in I 0 are separated by gaps. It is known that at time t = 0, we have I 0 (c, 0) = [α(c), β(c)] for every c ∈ (0, 1); this means that the zero-gap ansatz holds at time t = 0. The zero-gap ansatz continues to hold until the shock time for the system (1.4) (see [DM]). To analyze the continuum limit beyond the time of shock formation, it is important that the set I 0 (c, t) is a ﬁnite union of intervals. The endpoints of these intervals then give important information on the behavior of the Toda lattice in the continuum limit, as already indicated in the introduction. The ﬁnite-gap ansatz says that there are only ﬁnitely many gaps. Knowing the gaps, the extremal problem can be turned into a Riemann-Hilbert problem, which can be analyzed explicitly. For various cases, it was veriﬁed in [DM] that the ﬁnite-gap ansatz holds. Deift, Kriecherbauer, and McLaughlin [DKM] obtained the ﬁnite-gap ansatz for real-analytic V in the absence of a constraint. More precisely, if we denote by µcV the minimizer for the problem to minimize IV (µ) among all measures in the class c ᏹ := µ ≥ 0 : supp(µ) ⊂ [A, B], dµ = c , (2.9) then it was shown in [DKM] that for C 2 external ﬁelds V , the minimizer µcV has a density ψ given by ψ(x) = where

1 − q (x), π

V (x) − V (y) c dµV (y) x −y 1 c2 − 2 V (y)(x + y) dµcV (y) − (B − x)(x − A)

2 q(x) = V (x) − 2

(2.10)

(2.11) if x ∈ [A, B],

and q − denotes the negative part of q. See [J] for related results. Note that the V used in [DKM] differs from ours by a factor of −2. If V is real-analytic in a neighborhood of [A, B], then q is real-analytic; it follows that ψ is supported on a ﬁnite number of intervals. This result also establishes the ﬁnite-gap ansatz if V is real-analytic and the constraint σ is not effective, that is, if I + (c, t) is empty. Remark 2.1. The extremal problem (2.6) with constraint was also introduced recently in the theory of orthogonal polynomials to describe asymptotics for polynomials that satisfy a discrete orthogonality relation (see [DS], [KVA2], [R], and [KR] for a survey). Some of the techniques used in this paper are based on ideas originating in this area. 3. Statement of main result. We state and prove our main theorem for external ﬁelds and constraints that are deﬁned on the full real line.

438

A. B. J. KUIJLAARS

For a continuous external ﬁeld V : R → R, we impose the growth condition lim

|x|→∞

V (x) = +∞. log |x|

(3.1)

Given such V and a positive Borel measure σ on R with a positive continuous density φ, the extremal problem is deﬁned in a way analogous to (2.6). Thus, µ,c EV = inf IV (µ) : µ ∈ ᏹσ,c , where now the measures µ in ᏹσ,c may have support anywhere on the real line. Because of the growth condition on V , the minimizer µσ,c V exists and is unique. It is called the equilibrium measure associated with V , σ , and c. The equilibrium measure has a compact support (see [DS]). Since it is dominated by σ , it has a density, which we denote by ψVσ,c , or simply ψ, and 0 ≤ ψ ≤ φ. The inequalities (2.7) and (2.8) characterize the equilibrium measure. We refer to the constant l in (2.7) and (2.8) as the equilibrium constant. Following [JLM] and [LL], we partition R into three sets: (3.2) I 0 = x ∈ R : Lψ(x) − V (x) = l , + (3.3) I = x ∈ R : Lψ(x) − V (x) > l , − (3.4) I = x ∈ R : Lψ(x) − V (x) < l . These sets depend on V , σ , and c. Since φ is continuous and 0 ≤ ψ ≤ φ, it is known that Lψ is a continuous function (see, e.g., [DS], [R]). Since V is also assumed to be continuous, it is clear that I + and I − are open sets that are separated by the closed set I 0 . From (2.7) and (2.8), it follows that I + is the set where the constraint φ is active, and I − is the set where the nonnegativity constraint ψ ≥ 0 is active (i.e., ψ = φ on I + and ψ = 0 on I − ). Also, since ψ has compact support, we have that I 0 and I + are bounded, and I − is unbounded. It is the aim of this paper to establish the ﬁnite-gap ansatz in all cases where both V and φ = dσ/dx are real-analytic on R. The following is our main result. Theorem 3.1. Let V be a real-analytic function on R such that V (x) = ∞, x→±∞ log |x| lim

and let σ be a measure on R with positive, real-analytic density φ. Then for each c ∈ (0, dσ ), the following hold. (a) The constrained equilibrium measure µσ,c V has a continuous density ψ. − (b) The sets I = {x : Lψ(x) − V (x) < l} and I + = {x : Lψ(x) − V (x) > l} are both ﬁnite unions of open intervals. (c) The density ψ is real-analytic on the open set {x : 0 < ψ(x) < φ(x)}.

ON THE FINITE-GAP ANSATZ

439

(d) The density ψ has the representation ψ(x) =

1 π

q1− (x),

x ∈ I 0 ∪ I −,

where q1− is the negative part of a function q1 deﬁned on I 0 ∪ I − , which is realanalytic in the interior of I 0 ∪ I − . The function q1 is positive on I − , so that ψ ≡ 0 on I − . (e) The density ψ has the representation ψ(x) = φ(x) −

1 π

q2− (x),

x ∈ I 0 ∪ I +,

where q2− is the negative part of a function q2 deﬁned on I 0 ∪ I + , which is realanalytic in the interior of I 0 ∪ I + . The function q2 is positive on I + , so that ψ ≡ φ on I + . If the conclusions of the theorem hold, we say that the ﬁnite-gap ansatz holds for the triple (V , σ, c). The representations (d) and (e) are the extensions of (2.10) to the constrained case. It is clear from (d) and (e) that for x ∈ I 0 , 1 1 − q1 (x) + q2− (x) = φ(x), π π which shows that there exists a strong connection between the functions q1 and q2 . From the continuity of ψ, as claimed in Theorem 3.1(a), it follows that the sets {x : ψ(x) = 0} and {x : ψ(x) = φ(x)} are closed. Since they are disjoint and the latter is compact, they have a positive distance. Thus the density ψ cannot change abruptly from zero to φ. There is always a continuous (in fact, real-analytic) transition between zero and φ. The main problem in the proof of Theorem 3.1 is to exclude the possibility of such a sudden transition from zero to φ and also to exclude the possibility of an inﬁnite number of transitions. There is no problem if over some interval (a, b) the constraint φ is not effective, since then we may use arguments as in [DKM], based on formulas such as (2.10) and (2.11) to conclude that ψ is real-analytic on those parts of (a, b) where it is positive. Similarly, it can be shown that there is no real problem on an interval where the nonnegativity condition is not active. The proof of Theorem 3.1 is in Section 4. Remark 3.2. The reader may have noticed that Theorem 3.1 does not correspond exactly to the extremal problem associated with the continuum limit of the Toda lattice, as described in Section 2. Indeed, the extremal problem (2.6) is restricted to the ﬁnite interval [A, B], while Theorem 3.1 covers problems deﬁned on the whole real line.

440

A. B. J. KUIJLAARS

The problem on [A, B] is somewhat more difﬁcult because of the endpoint effects. If the support of the equilibrium measure µσ,c V stays away from the endpoints A and B, then the analysis leading to Theorem 3.1 goes through; thus the ﬁnite-gap ansatz holds, provided of course that V and φ are real-analytic. It is possible to obtain the same conclusion if one (or both) of the endpoints is in the support and if both V and φ are real-analytic in a neighborhood of [A, B]. The proof of this result would involve a number of additional technicalities. For the sake of clarity of presentation, it is not included here. Remark 3.3. Based on the previous remark, we conclude that if V0 given by (2.2) and if φ given by (2.3) are real-analytic in a neighborhood of [A, B], then the ﬁnitegap ansatz holds for Vt , φ, and c ∈ (0, 1), for all t ≥ 0. Recall that Vt (x) = V0 (x)−tx. It is possible to construct initial data a0 (y) and b0 (y) giving rise to real-analytic V and φ. However, the fact that a0 and b0 themselves are real-analytic in a neighborhood of [0, 1] is not enough for the real-analyticity of V0 and φ. In fact, real-analyticity of a0 and b0 only implies that the spectral functions are real-analytic on (A, B) \ {α(0), α(1)}, where α is deﬁned by (2.1). From this, it is possible to prove that there are only ﬁnitely many gaps in each compact subinterval of (A, B)\{α(0), α(1)}, but these gaps could possibly accumulate near one of the points A, B, α(0), or α(1). In speciﬁc cases it seems likely that one might be able to use special properties to exclude such accumulation of gaps. Then the true ﬁnite-gap ansatz would hold. Remark 3.4. Let V , σ , and c satisfy the assumptions of Theorem 3.1. We say that the triple (V , σ, c) is regular (otherwise, singular) if the following hold. • The function q1 from part (d) of Theorem 3.1 does not vanish on the interior of I 0 , and it has a simple zero at each of the endpoints of I − . • The function q2 from part (e) of Theorem 3.1 does not vanish on the interior of I 0 , and it has a simple zero at each of the endpoints of I + . • The set I 0 has no isolated points. This notion was introduced for the situation without constraint σ in [DKMVZ], where it was used in the asymptotic analysis of orthogonal polynomials, which is different for the regular and singular cases. Generically the regular case holds. This was shown by McLaughlin and the author (see [KM]), who proved that for a ﬁxed V , the regular case holds for all c, with at most a countable number of exceptions. We expect a similar result for the case with constraint. 4. The proof of Theorem 3.1 4.1. Outline. Since the proof of Theorem 3.1 is rather long, we ﬁrst present an outline. In the proof, the external ﬁeld V and constraint σ are ﬁxed, and the problem is considered depending on the normalization constant c. To simplify notation, we write µc instead of µσ,c V , and we denote its density by ψc . The equilibrium constant from (2.7) and (2.8) is denoted by lc .

ON THE FINITE-GAP ANSATZ

441

As a function of c, the measure µσ,c V increases with c. This is a very useful result. For extremal problems without constraint, it is well known (see, e.g., [ST], [To]). We also need certain continuity properties of µσ,c V in the parameter c. These results are included in Proposition 4.1 and are stated for more general V and σ , as in Theorem 3.1. We use σ |K to denote the restriction of a measure σ to the set K. Proposition 4.1. Let V be a continuous function on R such that limx→±∞ V (x)/ log |x| = +∞, and let σ be a nonnegative Borel measure on R such that L(σ |K ) is continuous for every compact set K ⊂ R. Then the following hold. σ,c for c ∈ (0, dσ ). That is, if 0 < c1 < c2 <

(a) The map c → µV isσ,cincreasing 1 is a positive measure. dσ , then the difference µV 2 − µσ,c V (b) For every c0 ∈ (0, dσ ), we have σ,c0

lim µσ,c V = µV

c→c0

(4.1)

in the sense of weak∗ convergence of measures. Moreover, σ,c0

lim Lµσ,c V = LµV

c→c0

(4.2)

uniformly on compact subsets of R, and lim lc = lc0 ,

c→c0

(4.3)

where lc is the equilibrium constant corresponding to normalization c. For the proof of Theorem 3.1, we consider the collection

Ꮿ := c0 ∈ 0, dσ : the ﬁnite-gap ansatz holds for (V , σ, c) for every 0 < c ≤ c0 . (4.4) Recall that the ﬁnite-gap ansatz holds if the conclusions of Theorem 3.1 are satisﬁed. The proof then consists of three main parts. The ﬁrst part shows that the collection Ꮿ is nonempty. This is done by showing that for c small enough, the constraint σ is not active. Then the ﬁnite-gap ansatz holds by [DKM], as noted earlier. The second part shows that the collection Ꮿ is open. The third part shows that if c0 < dσ is such that (0, c0 ) ⊂ Ꮿ, then c0 ∈ Ꮿ. The three parts together prove that Ꮿ = (0, dσ ), which establishes the theorem. For the proofs of the three parts, we need a number of auxiliary results, which we introduce as lemmas. Lemma 4.8 is crucial and states that the ﬁnite-gap ansatz holds, once we can break up the real axis into a ﬁnite number of intervals on which one of the constraints (either the nonnegativity constraint or the upper constraint φ) is not active. Another key issue is addressed in Lemma 4.10. It says that if for some c0 ∈ Ꮿ the density ψc0 is strictly less than the constraint φ on some closed interval,

442

A. B. J. KUIJLAARS

then this continues to hold for ψc with c slightly bigger than c0 . This lemma would be trivial if we knew that the densities ψc depend continuously on c in the topology of uniform convergence on compacts. We know from Proposition 4.1(b) that Lψc depends continuously on c, but this is not good enough. Our proofs throughout the paper make use of potential theory in the complex plane. Good references for this theory are [Ran] and [ST]. We start the proof of Theorem 3.1 in Section 4.2 with the proof of Proposition 4.1. Then we show in Section 4.3 that Ꮿ is nonempty. A number of lemmas are in Section 4.4, which lead to the important Lemma 4.8. This lemma is used in Section 4.5 to show that the collection Ꮿ is open and ﬁnally in Section 4.6 to show that Ꮿ is

closed in the interval (0, dσ ). 4.2. Proof of Proposition 4.1 Lemma 4.2. Let σ be a measure on R such that L(σ |K ) is continuous for every compact K. Suppose µ1 and µ2 with µ 1 = µ2 are

compactly supported positive measures such that µ1 ≤ σ , µ2 ≤ σ , and dµ2 ≥ dµ1 . If x0 is such that L µ2 − µ1 (x0 ) ≤ min L(µ2 − µ1 )(x) : x ∈ supp(µ2 ) ∩ supp(σ − µ1 ) , (4.5) then x0 ∈ supp(µ2 ) ∩ supp(σ − µ1 ) and equality holds in (4.5). Proof. Deﬁne & := C \ (supp(µ2 ) ∩ supp(σ − µ1 )). On &, we have µ2 − µ1 ≤ 0, and therefore L(µ2 −µ1 ) is superharmonic on & (see [ST]). L(µ2 −µ1 ) is continuous and bounded from below at inﬁnity. The minimum principle for superharmonic functions then implies that L(µ2 −µ1 ) assumes its minimum on supp(µ2 )∩supp(σ −µ1 ). Moreover, if L(µ2 − µ1 ) assumes its minimum at some other point as well, then L(µ2 − µ1 ) would be constant on the component of & containing that point. Since supp(µ2 ) is a compact subset of R, & is connected, and it would then follow that L(µ2 − µ1 ) is constant on & and, by continuity, constant on C. This is impossible, since µ1 = µ2 (see [ST, Theorem II.2.1]).

1 Proof of Proposition 4.1. (a) Let 0 < c1 < c2 < dσ . Put µ1 := µσ,c and µ2 := V σ,c2 µV . Also let l1 and l2 be the equilibrium constants from the inequalities (2.7) and (2.8) corresponding to c1 and c2 , respectively. Then Lµ2 − V ≥ l2 on supp(µ2 ) and Lµ1 − V ≤ l1 on supp(σ − µ1 ). Thus L(µ2 − µ1 ) ≥ l2 − l1

on supp(µ2 ) ∩ supp(σ − µ1 ),

and therefore by the proof of Lemma 4.2 (note that dµ2 > dµ1 ), L(µ2 − µ1 ) ≥ l2 − l1

on C.

We also conclude from (2.7) and (2.8) that L(µ2 − µ1 ) ≤ l2 − l1

on supp(µ1 ) ∩ supp(σ − µ2 ),

(4.6)

ON THE FINITE-GAP ANSATZ

443

so that by (4.6), L(µ2 − µ1 ) = l2 − l1

on supp(µ1 ) ∩ supp(σ − µ2 ).

(4.7)

The relations (4.6) and (4.7) imply that µ2 ≥ µ1 on supp(µ1 ) ∩ supp(σ − µ2 ) (see [ST, Theorem IV.4.5]). On the complement of supp(µ1 ) ∩ supp(σ − µ2 ), we have either µ1 = 0 or µ2 = σ ; in both cases, it is clear that µ2 ≥ µ1 . Thus µ2 ≥ µ1 everywhere, which completes the proof of part (a). (b) The limit (4.1) is an immediate consequence of part (a). ∗ For the proof of (4.2) and (4.3), we write µc instead of µσ,c V . Let x ∈ R, and let xc be real numbers such that lim xc = x ∗ .

(4.8)

c→c0

From (4.1) and the principle of descent for logarithmic potentials (see [ST, Theorem I.6.8]), it follows that Lµc0 (x ∗ ) ≥ lim sup Lµc (xc ). c→c0

(4.9)

Let K be a compact set such that supp(µc ) ⊂ K for some c > c0 . Then µc ≤ σ |K for all c sufﬁciently close to c0 , and we have lim σ |K − µc = σ |K − µc0 c→c0

in the sense of weak∗ convergence for positive measures. The principle of descent combined with the continuity of L(σ |K ) (see the assumptions of the proposition) then yields Lµc0 (x ∗ ) ≤ lim inf Lµc (xc ). c→c0

(4.10)

Hence by (4.9) and (4.10) we have Lµc0 (x ∗ ) = lim Lµc (xc ), c→c0

and this holds for every x ∗ and every xc satisfying (4.8). This implies that the limit (4.2) holds uniformly on compact subsets of R. For (4.3) we choose a point x ∗ from supp(µc0 ) ∩ supp(σ − µc0 ). Note that this is possible, since supp(µc0 ) and supp(σ − µc0 ) are nonempty closed sets whose union is R. Their intersection is then nonempty since R is connected. We have, by (2.7) and (2.8), Lµc0 (x ∗ ) − V (x ∗ ) = lc0 . Since the measures µc increase to µc0 as c increases to c0 by part (a), it follows that x ∗ ∈ supp(σ − µc ) for all c < c0 ; so by (2.7), Lµc (x ∗ ) − V (x ∗ ) ≤ lc ,

if c < c0 .

444

A. B. J. KUIJLAARS

Taking the limit c c0 , we then ﬁnd because of (4.2) that lc0 ≤ lim inf lc .

(4.11)

cc0

Next, we note that it follows from part (a) and (4.1) that supp µc0 = supp(µc ).

(4.12)

c
Since x ∗ ∈ supp(µc0 ), we can ﬁnd for each c < c0 a point xc ∈ supp(µc ) such that xc → x ∗ as c c0 . Then by (2.8), Lµc (xc ) − V (xc ) ≥ lc for every c < c0 . Letting c c0 and using (4.2) and the fact that V is continuous, we ﬁnd that the left-hand side tends to Lµc (x ∗ ) − V (x ∗ ) = lc0 . Thus lc0 ≥ lim sup lc .

(4.13)

cc0

Combining (4.11) and (4.13), we see that limcc0 lc = lc0 . In a similar way, using supp σ − µc0 = supp(σ − µc ) c>c0

instead of (4.12), we prove limcc0 lc = lc0 and the limit (4.3) follows. The following more precise version of Proposition 4.1(a) is used in the proof of Lemma 4.10. 1 2 and µ2 = µσ,c Lemma 4.3. Let V and σ be as in Proposition 4.1. Let µ1 = µσ,c V V with 0 < c1 < c2 < dσ . Let ω be the equilibrium measure of the set supp(µ2 ). Then (4.14) µ1 + (c2 − c1 )ω E ≤ µ2 ,

where E = supp(µ1 ) ∩ supp(σ − µ2 ). Proof. As in the proof of Proposition 4.1(a) (see (4.6) and (4.7)), we have Lµ1 ≤ Lµ2 + l1 − l2

on C

with equality on E. The potential of ω is constant on supp(µ2 ), say, Lω = C on supp(µ2 ). Then (4.15) L µ1 + (c2 − c1 )ω ≤ Lµ2 + l1 − l2 + (c2 − c1 )C on supp(µ2 ) with equality on E. Since µ1 + (c2 − c1 )ω and µ2 are measures with the same total mass and both are supported on supp(µ2 ), the principle of domination for logarithmic potentials (see [ST, Theorem II.3.2]) yields that (4.15) holds on C. Then, by a theorem of de la Vallée Poussin (see [ST, Theorem IV.4.5]), the inequality (4.14) holds.

ON THE FINITE-GAP ANSATZ

445

4.3. Proof that Ꮿ is nonempty. For an external ﬁeld V and normalization c > 0, we use µcV to denote the equilibrium measure (without constraint). That is, µcV minimizes IV (µ) among all positive measures on R with total mass c. To prove that Ꮿ, deﬁned in (4.4), is nonempty, we show that for c small enough, the measure µcV is dominated by σ . We show this for the more general class of C 1+ external ﬁelds, where an external ﬁeld is said to be of class C 1+ if it is differentiable with a derivative satisfying a Hölder inequality with exponent > 0. In this more general version, the lemma is used in the proof of Lemma 4.10. The lemma also contains some additional facts that we need later. Lemma 4.4. Let W be a C 1+ function for some > 0 with lim|x|→∞ W (x)/ log |x| = ∞. Let µcW be the equilibrium measure with external ﬁeld W and normalization constant c > 0. Then the following hold. c with respect to Lebesgue (a) For every c > 0, the measure µcW has a density ψW c measure, and ψW is a bounded function. (b) For every m > 0, there is c0 > 0 such that for every c ∈ (0, c0 ], c ψW (x) ≤ m

for all x ∈ R.

(c) Let * = {x ∈ R : W (x) = minW }. Then supp µcW = *. c>0

(d) Assume that W is convex in a neighborhood of *. Then for every c > 0, the support supp(µcW ) contains * in its interior. Proof. Take a bounded interval [a, b] containing the support of µcW in its interior. The integral equation Lv(x) − W (x) = l with supp(v) ⊂ [a, b],

b a

if x ∈ [a, b]

v(t) dt = c, is equivalent to the singular integral equation PV

b a

v(t) dt = W (x), x −t

where PV denotes the Cauchy principal value. It is well known (see, e.g., [G, Section 42.3] that for Hölder-continuous W , there is a unique solution b 1 W (s) 1 v(x) = √ (b − s)(s − a) ds if x ∈ [a, b], c + PV π π (b − x)(x − a) a s −x and v is Hölder-continuous in the open interval (a, b). From [KD, Lemma 3] we c and ψ c ≤ v + . Here know that dµcW (x) ≤ v + (x) dx, so that µcW has a density ψW W c + v denotes the positive part of v. The support of ψW is strictly contained in [a, b] by

446

A. B. J. KUIJLAARS

c . It follows the choice of [a, b], so that v is ﬁnite and continuous on the support of ψW c that ψW is bounded from above. This proves part (a). c is bounded, we have that ψ c is in Lp for every p > 1, and this is enough Since ψW W for the analysis of [DKM, Theorem 1.34]. It follows that 1 c (x) = qc− (x) (4.16) ψW π

with qc− the negative part of the function 2 W (x) − W (y) c ψW (y) dy qc (x) = W (x) − 2 x −y

(4.17)

(see also [DKMVZ]). We have by the Hölder continuity of W , W (x) − W (y) ≤ M|x − y| if x, y ∈ [a, b], for some M > 0. Thus from (4.17), − qc (x) ≤ 2M

c (y) ψW dy |x − y|1−

if x ∈ R.

c (y) → 0 for almost every y. Then it is easy to see that for c As c 0, we have ψW c (x) ≤ m by (4.16), sufﬁciently small, we have qc− (x) ≤ π 2 m2 for all x ∈ R. Then ψW and this proves part (b). Part (c) is essentially contained in [BR]. Indeed, from the proof of [BR, Lemma 4] we have *⊂ supp µcW . c>0

Conversely, if

x0 ∈ supp(µcW )

for every c > 0, then

LµcW (x0 ) − W (x0 ) = lc

for every c > 0.

Since limc→0 lc = − min W (see [BR, remark after Theorem 2]) and since the measures µcW decrease to zero as c → 0, it follows that W (x0 ) = minW , that is, x0 ∈ *. This proves part (c). To prove part (d), we assume that W is convex in a neighborhood of *. Let c x0 ∈ *, c0 > 0, and suppose that x0 is not an interior point of the support of µW0 . Since W is convex in a neighborhood of x0 , say, in [x0 −, x0 +], it follows by [ST, Theorem IV.1.6] that /c := supp µcW ∩ x0 − , x0 + is an interval for every c. Since x0 ∈ supp(µcW ) by part (c), we have x0 ∈ /c . Since the supports are increasing with the parameter c, we see that x0 is not an interior

ON THE FINITE-GAP ANSATZ

447

point of any /c with c < c0 . It follows that x0 is either a right endpoint of all /c with c ≤ c0 or a left endpoint. Suppose for deﬁniteness that x0 is a right endpoint of all /c with c ≤ c0 . Then there is δ > 0 such that for every c ∈ [c0 /2, c0 ], x0 − δ, x0 ⊂ /c ⊂ supp µcW

and

x0 , x0 + δ ∩ supp µcW = ∅. (4.18)

A formula of Buyarov and Rakhmanov [BR, Theorem 2] says that c0 c ωsupp(µcW ) dc, µW0 =

(4.19)

0

where ωS denotes the equilibrium measure (without external ﬁeld) of the set S. Let c [a, b] be an interval containing the support of µW0 . Then by (4.18) we have for c ∈ [c0 /2, c0 ], supp µcW ⊂ [a, x0 ] ∪ x0 + δ, b ; this implies that

ωsupp(µcW ) |supp(µcW ) ≥ ω[a,x0 ]∪[x0 +δ,b] |supp(µcW ) .

By (4.18) we then have for c ∈ [c0 /2, c0 ], dωsupp(µcW ) dx

(x) ≥

dω[a,x0 ]∪[x0 +δ,b] (x), dx

x ∈ [x0 − δ, x0 ].

Since the density of the equilibrium measure of the union of the two intervals [a, x0 ]∪ [x0 + δ, b] is inﬁnite at x0 , we see that dωsupp(µcW ) c0 (x0 ) = ∞, for c ∈ , c0 . dx 2 c

Because of (4.19) we then get ψW0 (x0 ) = ∞. This is a contradiction with part (a), c since ψW0 is bounded. This contradiction proves part (d). Proposition 4.5. The collection Ꮿ is nonempty. Proof. Since φ is positive and continuous, and supp(ψ1 ) is compact, there exists a constant m > 0 such that φ > m on supp(ψ1 ). By Lemma 4.4(b), there is c0 such that ψVc ≤ m for all c ∈ (0, c0 ]. Here ψVc is the density of the minimizer without upper constraint. We may assume that c0 ≤ 1. Let c ∈ (0, c0 ]. Then supp(ψVc ) ⊂ supp(ψc ) by [DS, Theorem 2.6], and supp(ψc ) ⊂ supp(ψ1 ) by Proposition 4.1(a). Therefore, ψVc ≤ m < φ on supp(ψVc ). It follows that the constraint φ is not effective if c ≤ c0 . Then (V , σ, c) satisﬁes the ﬁnite-gap ansatz by [DKM, Theorem 1.38], and so c0 ∈ Ꮿ.

448

A. B. J. KUIJLAARS

4.4. Auxiliary lemmas. Before we continue with the proof of Theorem 3.1, we state and prove a number of lemmas that are used later in the paper. The ﬁrst is a straightforward extension of the result of Deift, Kriecherbauer, and McLaughlin [DKM]. Lemma 4.6. Let V be continuous on [a, b] and real-analytic on (a, b). Let µcV be the equilibrium measure with external ﬁeld V and normalization c > 0; that is, µcV minimizes IV (µ) among all positive Borel measures on [a, b] with total mass c. Then µcV is absolutely continuous with respect to Lebesgue measure dµcV (x) = ψ(x) dx,

(4.20)

and there exists a real-analytic function q on (a, b) such that ψ(x) =

1 − q (x) for x ∈ [a, b], π

where q − is the negative part of q. The function q satisﬁes 2 V (x) − V (y) c dµV (y) q(x) = V (x) − 2 x −y

1 2 c + c − 2 V (y)(x + y) dµV (y) . (x − a)(x − b)

(4.21)

(4.22)

Proof. This follows by considering V as an external ﬁeld on [a + , b − ], where we can use [DKM, Theorem 1.36] since V is real-analytic in a neighborhood of [a + , b − ], and by letting → 0. Our second lemma gives estimates on the density ψ of the equilibrium measure. It is important that the estimates do not depend on the normalization c. Lemma 4.7. Let V be a real-analytic function on an interval I . Let σ be a measure with real-analytic density φ. Let ψ be the density of the minimizer with external ﬁeld V , constraint σ , and a certain normalization. Suppose a < b in I are such that [a, b] ⊂ supp(ψ) ∩ supp(φ − ψ) and ψ vanishes at a and b. (a) If ψ ≡ 0 outside (a, b), then ψ(x) ≤

b−a max V (y) if x ∈ (a, b). 2π y∈[a,b]

(b) If ψ ≡ 0 on (b, ∞) and on (A, a) with A < a and A ∈ supp(ψ), then ψ(x) ≤

b−a max V (y) π y∈[A,b]

if x ∈ (a, b).

(4.23)

ON THE FINITE-GAP ANSATZ

449

(c) If ψ ≡ 0 on (−∞, a) and on (b, B) with B > b and B ∈ supp(ψ), then ψ(x) ≤

b−a max V (y) π y∈[a,B]

if x ∈ (a, b).

(d) If ψ ≡ 0 on (A, a) and on (b, B) with A < a, B > b, and A, B ∈ supp(ψ), then ψ(x) ≤

b−a max V (y) π y∈[A,B]

if x ∈ (a, b).

Proof. (a) If ψ vanishes outside (a, b), then by (4.23) ψ is the density of the minimizer of the extremal problem without constraint. Since the support of ψ is equal to the interval [a, b] and ψ vanishes at a and b, we have for x ∈ (a, b) (cf. [G, Section 42.3]), b 1 ds V (x) − V (s) ψ(x) = (b − x)(x − a) , (4.24) √ π x − s π (b − s)(s − a) a and (a) immediately follows. (b) Suppose ψ vanishes to the right of b, and let A < a be as in part (b). Because of (4.23) and the choice of A, we have for some constant l that Lψ −V ≤ l on [A, b] with equality on [a, b] and also at A. Let ψ0 denote the restriction of ψ to [a, b], and deﬁne A V0 (x) := V (x) − log |x − t|ψ(t) dt for x ∈ [A, b]. (4.25) −∞

Then Lψ0 − V0 ≤ l

on [A, b],

(4.26)

with equality on the support of ψ0 and at A. It follows that ψ0 is the density of the minimizer with external ﬁeld V0 and certain normalization. Thus for x ∈ (a, b), b V0 (x) − V0 (s) 1 ds ψ(x) = ψ0 (x) = (b − x)(x − a) √ π x − s π (b − s)(s − a) a (4.27) b−a max V (y). ≤ 2π y∈[a,b] 0 Let M = maxy∈[A,b] |V (y)|, and suppose, to get a contradiction, that ψ(x) > (b − a)/π ·M for some x ∈ (a, b). Then it follows from (4.27) that there is y ∈ [a, b] such that 2M < V0 (y). Since V0 (x) = V (x) +

A

ψ(t) dt 2 −∞ (x − t)

if x ∈ [A, b]

(4.28)

450

A. B. J. KUIJLAARS

and |V (y)| ≤ M, it follows that

A

∞

Since y ≥ a and

ψ(t) dt > M. (y − t)2

(4.29)

A

ψ(t) dt 2 ∞ (x − t) is decreasing for x > A, it follows from (4.29) that A ψ(t) dt > M if x ∈ [A, a]. (x − t)2 ∞ Using (4.28) and the deﬁnition of M, we then have that V0 (x) > 0 for x ∈ [A, a]. Thus, V0 is convex on the interval [A, a]. Since ψ0 has no support on (A, a), we get that Lψ0 is strictly concave on [A, a]. Now we have a contradiction, since according to (4.26) the inequality Lψ0 − V0 ≤ l holds on [A, a] with equality at a and A. This contradiction completes the proof of part (b). (c) Part (c) is proved similarly. (d) Let A < a and B > b be as in part (d). Let ψ0 be the restriction of ψ to [a, b], and let V0 = V −L(ψ −ψ0 ). As in the proof of part (b), we have that ψ0 is the density of the minimizer with external ﬁeld V0 on [A, B] and with a suitable normalization; as in (4.27) we obtain the estimate b−a max V (y) if x ∈ (a, b). 2π y∈[a,b] 0

ψ(x) = ψ0 (x) ≤

Let M = maxy∈[A,B] |V (y)|, and suppose that ψ > (b − a)/π · M somewhere in (a, b). Then we have 2M < V0 (y) for some y ∈ [a, b], and in the same way we obtained (4.29) in the proof of part (b), we now ﬁnd ψ(t) dt > M. (4.30) 2 R\(A,B) (y − t) The function

ψ(t) dt 2 R\(A,B) (x − t)

is convex for x ∈ (A, B). Because of (4.30) with y ∈ [a, b], this leads to the fact that ψ(t) dt > M (4.31) (x − t)2 R\(A,B) holds, either for every x in the interval [A, a] or for every x in [b, B]. Suppose (4.31) holds for x ∈ [A, a]. Then we have as in the proof of part (b), V0 (x) > 0

if x ∈ [A, a],

451

ON THE FINITE-GAP ANSATZ

and we see that V0 is convex on [A, a]. This leads to a contradiction in the same way as before. Similarly, if the inequality (4.31) holds on [b, B], then V0 is convex on [b, B] and we get a contradiction as well. This proves part (d). The third lemma shows that it is enough for Theorem 3.1 to be able to cover the real line with a ﬁnite number of closed intervals, such that on each of these intervals, either the constraint σ or the lower bound zero is not active. Lemma 4.8. Let V , σ , and φ be as in Theorem 3.1. Let µ = µσ,c V be the equilibrium measure corresponding to some c ∈ (0, dσ ). Suppose there exists a ﬁnite sequence of points (ak )2n k=1 with a1 < · · · < a2n , such that (a) (−∞, a1 ] ⊂ supp(σ − µ), (b) [a2j −1 , a2j ] ⊂ supp(µ) for j = 1, . . . , n, (c) [a2j , a2j +1 ] ⊂ supp(σ − µ) for j = 1, . . . , n − 1, and (d) [a2n , ∞) ⊂ supp(σ − µ). Then the ﬁnite-gap ansatz holds for (V , σ, c). Proof. Let ψ be the density of µ, and let l be the equilibrium constant in the relations (2.7) and (2.8). Write a0 := min supp(µ) and a2n+1 := max supp(µ). For j = 0, 1, . . . , n, deﬁne V2j (x) := V (x) −

R\[a2j ,a2j +1 ]

log |x − t|ψ(t) dt

if x ∈ a2j , a2j +1 .

From the assumptions of the lemma, (2.7), and (2.8), it follows that Lψ − V ≤ l on [a2j , a2j +1 ], with equality on supp(ψ) ∩ [a2j , a2j +1 ]. Thus, if ψ2j denotes the restriction of ψ to [a2j , a2j +1 ], we have Lψ2j − V2j ≤ l

on a2j , a2j +1

with inequality on the support of ψ2j . This implies that ψ2j is the density of the minimizer of the unconstrained problem with external ﬁeld V2j and normalization [a2j ,a2j +1 ]

ψ(t) dt.

Since V2j is continuous on [a2j , a2j +1 ] and real-analytic on the interior, Lemma 4.6 yields ψ(x) = ψ2j (x) =

1 − q2j (x) if x ∈ a2j , a2j +1 π

for a real-analytic function q2j on (a2j , a2j +1 ).

(4.32)

452

A. B. J. KUIJLAARS

Next, we deﬁne for j = 1, . . . , n, a2j V2j −1 (x) := − V (x) + log |x − t|φ(t) dt −

a2j −1

R\[a2j −1 ,a2j ]

log |x − t|ψ(t) dt

if x ∈ a2j −1 , a2j .

On the interval [a2j −1 , a2j ], by (2.7), (2.8), and the assumption (b) of the lemma, we have the inequality Lψ − V ≤ l with equality on supp(φ − ψ) ∩ [a2j −1 , a2j ]. If we let φ2j −1 and ψ2j −1 be the restrictions to [a2j −1 , a2j ] of φ and ψ, respectively, then it follows that L φ2j −1 − ψ2j −1 − V2j −1 ≤ l on a2j −1 , a2j with equality on the support of φ2j −1 − ψ2j −1 . Thus φ2j −1 − ψ2j −1 (which is nonnegative) is the density of the minimizer with external ﬁeld V2j −1 and normalization a2j φ(t) − ψ(t) dt. a2j −1

Since φ is real-analytic, it is easy to show that log |x −t|φ2j −1 (t) dt is real-analytic on (a2j −1 , a2j ), so that V2j −1 is real-analytic there. It is also continuous on the closure. Hence, Lemma 4.6 gives 1 − q2j −1 (x) (4.33) φ(x) − ψ(x) = π with q2j −1 a real-analytic function on (a2j −1 , a2j ). From (4.32) it follows that supp(ψ)∩[a2j , a2j +1 ] consists of an at most countable number of intervals, which, if inﬁnite, can accumulate only at a2j and a2j +1 . Similarly, from (4.33) it follows that supp(φ −ψ)∩[a2j −1 , a2j ] consists of an at most countable number of intervals, which, if inﬁnite, can accumulate only at a2j −1 and a2j . Let us now assume that for some j = 1, . . . , n, a2j − , a2j +1 ⊂ supp(σ − µ) for some > 0. Then it is clear that the representation (4.32) extends to the bigger interval [a2j − , a2j +1 + ] and that q2j is real-analytic on (a2j − , a2j +1 ). Since [a2j −, a2j ] ⊂ supp(ψ), it then follows that there is no accumulation of components of supp(ψ) near the point a2j . Similarly, there is no accumulation of components of supp(φ − ψ) near a2j . The same conclusion would hold if we would assume that a2j −1 , a2j + ⊂ supp(µ). Similar considerations apply to the points a2j +1 .

ON THE FINITE-GAP ANSATZ

453

Thus, to obtain the ﬁnite-gap ansatz, it is enough to show that for each j = 1, . . . , 2n, there exists an > 0 such that we have either aj , aj + ⊂ supp(µ) ∩ supp(σ − µ) or

aj − , aj ⊂ supp(µ) ∩ supp(σ − µ).

To establish this, we consider without loss of generality an odd-numbered point a2j +1 , which, for convenience, we assume is zero, and we assume that a2j < −1 and a2j +2 > 1, which can be achieved by a rescaling. We then have for some η > 0, − 1 − η, 0 ⊂ supp(σ − µ), 0, 1 + η ⊂ supp(µ), (4.34) and we want to show that for some > 0, either [0, ] ⊂ supp(σ − µ)

(4.35)

[−, 0] ⊂ supp(µ).

(4.36)

or

We assume that (4.35) does not hold. We claim that in such a case we have 1 ψ(t) dt = ∞. (4.37) t 0 The relation (4.37) certainly holds if for some > 0, (0, ] ∩ supp(σ − µ) = ∅,

(4.38)

since then ψ(x) = φ(x) for x ∈ (0, ]. If both (4.35) and (4.38) do not hold, then supp(σ − µ) ∩ (0, 1] is the disjoint union of a sequence of closed intervals [αj , βj ] accumulating at zero. Choose a compact interval I containing the support of µ in its interior. On I , we consider the external ﬁeld W = −V + L(σ |I ), where σ |I denotes the restriction of σ to I . Since σ has a real-analytic density, W is real-analytic on the interior of I . From the variational inequalities (2.7) and (2.8), it follows that L σ |I − µ (x) − W (x) ≤ −l if x ∈ supp(µ), L σ |I − µ (x) − W (x) ≥ −l if x ∈ supp(σ − µ).

454

A. B. J. KUIJLAARS

This implies that σ |I − µ is the minimizer with external ﬁeld W on I , constraint σ , and normalization d(σ − µ). I

For this extremal problem, the constraint σ is not effective on the support of µ, and so by (4.34) it is not effective on the interval (0, 1). The minimizer σ |I − µ has a density that vanishes at αj and βj for every j . Then we can apply Lemma 4.7 to each of the intervals [αj , βj ], and it follows that β j − αj M if x ∈ αj , βj , π where M is the maximum of the absolute value of the second derivative of W on [0, 1]. Since the length of the intervals [αj , βj ] tends to zero as j → ∞, it then easily follows that ψ(x) is bounded away from zero in some neighborhood of zero. Hence the claim (4.37) follows also in case (4.38) does not hold. Thus, assuming that (4.35) does not hold, we have found that (4.37) holds. We now want to establish (4.36). Consider V (x) − log |x − t|ψ(t) dt (4.39) φ(x) − ψ(x) ≤

R\[−1,0]

as an external ﬁeld on [−1, 0]. With a suitable normalization, the equilibrium measure for (4.39) is the restriction of µ to [−1, 0]. The second derivative is ψ(t) V (x) + dt, (x − t)2 R\[−1,0] which tends to +∞ as x → 0− because of (4.37). Thus the external ﬁeld is convex in an interval [−δ, 0] with δ > 0, and therefore the intersection of supp(µ) with [−δ, 0] is an interval (see [ST, Theorem IV.1.10]). Since there is equality at zero in the variational inequalities associated with the extremal problem on [−1, 0], it follows that supp(µ) ∩ [−δ, 0] = [−, 0] for some ≥ 0. If = 0, then ψ = 0 on [−δ, 0). Then ψ(t) lim (Lψ) (x) = lim dt = −∞ x→0− x→0− x −t by (4.37). Thus Lψ −V is strictly decreasing in some left neighborhood of zero. This is incompatible with the relation Lψ − V ≤ l since

on [−δ, 0]

Lψ − V (0) = l.

Thus = 0 cannot hold. Therefore > 0, and it follows that [−, 0] ⊂ supp(µ) with > 0. This proves that (4.36) holds. This completes the proof of Lemma 4.8.

ON THE FINITE-GAP ANSATZ

455

4.5. Proof that Ꮿ is open Proposition 4.9. The collection Ꮿ is open. Proof. Because of the deﬁnition (4.4) of Ꮿ, it is enough to prove that for every c0 ∈ Ꮿ, there is c1 > c0 also belonging to Ꮿ. Thus, let c0 ∈ Ꮿ. Then the support of ψc0 consists of a ﬁnite number of closed nondegenerate intervals, say, n supp ψc0 = a2j −1 , a2j j =1

with a1 < a2 < · · · < a2n . It follows from Proposition 4.1(a) that a2j −1 , a2j ⊂ supp(µc ) for j = 1, . . . , n,

(4.40)

for every c > c0 .

Now we want to show that there exists c1 ∈ (c0 , dσ ) such that for every c ∈ (c0 , c1 ), (−∞, a1 ] ⊂ supp(σ − µc ),

(4.41)

a2j , a2j +1 ⊂ supp(σ − µc ) for j = 1, . . . , n − 1,

(4.42)

[a2n , ∞) ⊂ supp(σ − µc ).

(4.43)

and

If we could show (4.41)–(4.43), then the conditions of Lemma 4.8 would be satisﬁed, and it would follow that the ﬁnite-gap ansatz holds for (V , σ, c). Then c1 ∈ Ꮿ, and Ꮿ would be open. Hence what remains to be done is to prove (4.41)–(4.43). Since ψc0 = 0 < φ on each of the intervals (−∞, a1 ], [a2j , a2j +1 ], and [a2n , ∞), the inclusions (4.41)–(4.43) follow from Lemma 4.10, which we state as a separate lemma since it is also used in the proof of Proposition 4.11. Lemma 4.10. Let c0 ∈ Ꮿ and a < b. Suppose that ψc0 < φ on [a, b]. Then there exists c1 > c0 such that ψc < φ on [a, b] for every c ∈ (c0 , c1 ). Proof. On [a, b] we have Lψc0 − V ≤ lc0 with equality on [a, b] ∩ supp(ψc0 ). If Lψc0 − V < lc0 on the full interval [a, b], then by compactness and continuity in the parameter c (see Proposition 4.1(b)), there is c1 > c0 such that Lψc −V < lc on [a, b] for every c ∈ (c0 , c1 ). Then for such c, ψc = 0 on [a, b] and the lemma follows. Therefore, in the rest of the proof, we assume * := x ∈ [a, b] : Lψc0 (x) − V (x) = lc0 = ∅. (4.44) Observe that V − Lψc0 is differentiable and its derivative is Hölder-continuous with

456

A. B. J. KUIJLAARS

exponent 1/2. It is possible to ﬁnd an auxiliary external ﬁeld W such that W (x) = V (x) − Lψc0 (x) if x ∈ [a, b], W (x) > max − lc0 , V (x) − Lψc0 (x) if x ∈ R \ [a, b],

(4.45) (4.46)

W (x) = +∞, |x|→∞ log |x| lim

and such that W is differentiable with a Hölder-continuous derivative. In addition, we may assume that W is convex in a neighborhood of *. We let µδW be the minimizer with external ﬁeld W and normalization δ, and we let δ be its density (see Lemma 4.4(a)). We let l ψW W,δ be the equilibrium constant in the δ variational conditions for µW . Let > 0 be such that ψc0 < φ on [a − , b + ]. It is possible to ﬁnd such an since ψc0 and φ are continuous. Deﬁne φ(x) − ψc0 (x) , (4.47) m := min x∈[a−,b+]

which is a positive number. Since W assumes its minimum −lc0 on * ⊂ [a, b] only, we have by Lemma 4.4(c), supp µδW = * ⊂ [a, b]. δ>0

It follows that there is δ0 > 0 such that supp µδW ⊂ a − , b +

(4.48)

δ of the for all δ < δ0 . By Lemma 4.4(b), there is δ1 ≤ δ0 such that the density ψW δ measure µW satisﬁes δ ψW <m

if δ < δ1 .

(4.49)

Let c = c0 + δ with δ ∈ (0, δ1 ). We are going to apply Lemma 4.2 with σ,c

0 µ1 := µσ,c V − µV ,

µ2 := µδW ,

σ,c

0

and upper constraint σ˜ = σ − µV . Note that µ1 and µ2 are positive measures with dµ1 = dµ2 , µ1 ≤ σ˜ , and also µ2 ≤ σ˜ because of (4.47)–(4.49). For x ∈ supp(µ2 ), we have Lµ2 (x) = W (x) + lW,δ ≥ V (x) − Lψc0 (x) + lW,δ ,

where we used (4.46), and for x ∈ supp(σ˜ − µ1 ) = supp(σ − µc ), we have Lµ1 (x) = Lψc (x) − Lψc0 (x) ≤ V (x) + lc − Lψc0 (x).

ON THE FINITE-GAP ANSATZ

457

Thus, min L(µ2 − µ1 )(x) : x ∈ supp(µ2 ) ∩ supp σ˜ − µ1 ≥ lW,δ − lc .

(4.50)

Next, if x0 ∈ [a, b] ∩ supp(µ2 ) ∩ supp(ψc ), then Lµ2 (x0 ) = V (x0 ) − Lψc0 (x) + lW,δ and Lµ1 (x0 ) ≥ V (x0 ) + lc − Lψc0 (x0 ), so that L(µ2 − µ1 )(x0 ) ≤ lW,δ − lc .

(4.51)

Then by (4.50), (4.51), and Lemma 4.2, we have [a, b] ∩ supp(µ2 ) ∩ supp(ψc ) ⊂ supp σ˜ − µ1 = supp(σ − µc ). Since clearly R \ supp(ψc ) ⊂ supp(σ − µc ), it follows that [a, b] ∩ supp(µ2 ) ⊂ supp(σ − µc ).

(4.52)

Since W is convex in a neighborhood of *, we see from Lemma 4.4(d) that the support of µ2 contains * in its interior. Then we have by the deﬁnition (4.44) of *, Lψc0 − V > lc0

on [a, b] \ supp(µ2 ).

By continuity in c (see Proposition 4.1(b)), there is c1 ∈ (c0 , c0 + δ1 ] such that Lψc − V > lc

on [a, b] \ supp(µ2 )

for every c ∈ (c0 , c1 ], which implies [a, b] \ supp(µ2 ) ⊂ supp(σ − µc ).

(4.53)

Then if c ∈ (c0 , c1 ], both (4.52) and (4.53) hold and it follows that [a, b] ⊂ supp(σ − µc ).

(4.54)

Note that (4.54) does not imply that ψc < φ on [a, b], since there may be isolated points in [a, b] where equality holds. However, we claim that equality can happen only for c = c1 and not for c < c1 . To see this, we recall from Lemma 4.3 that for c < c1 , we have ψc (x) + (c1 − c)

dω (x) ≤ ψc1 (x) if x ∈ supp(µc ) ∩ supp σ − µc1 , dx

458

A. B. J. KUIJLAARS

where ω is the equilibrium measure of the support of µc1 . This measure has a positive density on supp(µc1 ). Since supp(µc ) ∩ supp σ − µc1 ⊂ supp µc1 , we see that ψc (x) < ψc1 (x) if x ∈ supp(µc ) ∩ supp σ − µc1 .

(4.55)

Now, if x ∈ [a, b], then we have either ψc (x) = 0, in which case it is clear that ψc (x) < φ(x), or ψc (x) > 0. In the latter case, we have x ∈ supp(µc ) and also x ∈ supp(σ − µc1 ) because of (4.54). Hence (4.55) implies that ψc (x) < ψc1 (x) ≤ φ(x). This completes the proof of Lemma 4.10.

4.6. Proof that Ꮿ is closed in (0, dσ )

Proposition 4.11. Let c0 < dσ be such that c ∈ Ꮿ for every c < c0 . Then c0 ∈ Ꮿ. Proof. For c < c0 , the ﬁnite-gap ansatz holds. Thus, the support supp(µc ) consists of a ﬁnite number of disjoint closed intervals. We let Nc be the number of those components of supp(µc ) on which ψc meets the constraint φ in at least one point, and we denote these intervals by [a2j −1 (c), a2j (c)], j = 1, . . . , Nc , with a1 (c) < a2 (c) < · · · < a2Nc (c). We also put *c :=

Nc

a2j −1 (c), a2j (c) for c < c0 .

(4.56)

j =1

By Proposition 4.1(a), the sets *c grow as c increases. First, we show that c → Nc is continuous from the right. To this end, let c < c0 . In each of the intervals a2j (c), a2j +1 (c) , a2Nc (c), ∞ , (4.57) − ∞, a1 (c) , the density ψc satisﬁes ψc < φ. Then Lemma 4.10 implies that ψc+δ < φ on the intervals (4.57) for every δ > 0 sufﬁciently small. In addition, we have that ψc vanishes identically on some subinterval of each of the intervals (4.57). Thus, Lψc − V < lc somewhere in each of the intervals. By continuity, then also Lψc+δ − V < lc+δ somewhere in each of the intervals (4.57), for δ > 0 small enough. This prevents the intervals in *c from growing together in going from c to c + δ. Thus, Nc+δ = Nc for δ > 0 small enough. Next, we want to show that Nc remains bounded as c increases to c0 . If the number Nc would be unbounded, then it follows from the above that there would be inﬁnitely

ON THE FINITE-GAP ANSATZ

459

many c < c0 such that Nc−δ < Nc for all δ > 0 small enough. We show that there can be at most ﬁnitely many such c. Thus, let c < c0 be such that Nc−δ < Nc for δ small enough. Then there is an interval in *c in (4.56) which has no intersection with *c−δ if δ > 0. Thus, there is a j , such that [a2j −1 (c), a2j (c)] is disjoint from *c−δ for all δ > 0. Then ψc vanishes in a left neighborhood of a2j −1 (c), in a right neighborhood of a2j (c), and is equal to φ at a ﬁnite number of points of (a2j −1 (c), a2j (c)) only. Indeed, because of realanalyticity, the density cannot be equal to the constraint on a set with an accumulation point. We have a2j −1 (c), a2j (c) ⊂ supp(ψc ) ∩ supp(φ − ψc ). From Lemma 4.7, it follows that ψ(x) ≤

a2j (c) − a2j −1 (c) max V (y) y∈I π

if x ∈ a2j −1 (c), a2j (c) ,

(4.58)

where I is the convex hull of supp(µc0 ). Let m := min φ, I

M := max V , I

and note that these numbers do not depend on c and j . Since ψ(x) = φ(x) ≥ m for some x ∈ (a2j −1 (c), a2j (c)), we get from (4.58), a2j (c) − a2j −1 (c) ≥

πm =: /, M

(4.59)

and / > 0 is independent of c and j . Since (a2j −1 (c), a2j (c)) is disjoint from *c−δ for every δ > 0, we get from (4.59) that |*c | ≥ |*c−δ | + /, where |*c | denotes the Lebesgue measure of *c . Thus, each time we have Nc > Nc−δ , the Lebesgue measure of *c increases by an amount of at least /. Since all the sets *c are contained in *c0 and are increasing with c and since *c0 has ﬁnite Lebesgue measure, such an increase can happen only a ﬁnite number of times. It follows that Nc > Nc−δ for every δ > 0 small enough can happen for only ﬁnitely many c < c0 . Thus, the numbers Nc remain bounded as c increases to c0 . It also follows that Nc is nonincreasing for c close to c0 , and since these are nonnegative integers, it follows that Nc is in fact constant for c sufﬁciently close to c0 . Say, Nc = N if c ∈ (c0 − δ, c0 ) with δ > 0. Then the aj (c) are monotone functions of c ∈ (c0 − δ, c0 ), decreasing if j is odd, and increasing if j is even. Thus, the limits aj := lim aj (c) for j = 1, . . . , 2N c→c0 −

460

A. B. J. KUIJLAARS

exist. Since the measures µc increase to µc0 if c increases to c0 and since [a2j −1 (c), a2j (c)] ⊂ supp(µc ) for c ∈ (c0 − δ, c0 ), it follows that a2j −1 , a2j ⊂ supp µc0 . (4.60) Next, let j be such that

a2j , a2j +1 ⊂ supp µc0 ,

(4.61)

which implies in particular that a2j < a2j +1 . We have a2j , a2j +1 ⊂ a2j (c), a2j +1 (c) ⊂ supp(σ − µc ). Thus, Lψc − V ≤ lc on [a2j , a2j +1 ] for every c < c0 , and so, by continuity, Lψc0 − V ≤ lc0 on [a2j , a2j +1 ]. It follows that the restriction of ψc0 to [a2j , a2j +1 ] is the density of the extremal measure for the external ﬁeld V (x) − log |x − t|ψc0 (t) dt, x ∈ a2j , a2j +1 . R\[a2j ,a2j +1 ]

This external ﬁeld is real-analytic on the interior. Then we get from Lemma 4.6 that ψc0 has the form ψc0 (x) =

1 − q (x) if x ∈ a2j , a2j +1 π

with q real-analytic on (a2j , a2j +1 ). Thus ψc0 is real-analytic on {x ∈ (a2j , a2j +1 ) : ψc0 (x) > 0}. Also ψc0 ≤ φ. Since φ is positive and real-analytic, it follows that ψc0 hits the constraint φ only in a number of isolated points in (a2j , a2j +1 ) (if at all). Therefore, (4.62) a2j , a2j +1 ⊂ supp σ − µc0 if (4.61) holds. In a similar way, we show that (−∞, a1 ] ⊂ supp σ − µc0 , [a2N , ∞) ⊂ supp σ − µc0 .

(4.63) (4.64)

Now we delete from the sequence a1 < a2 < · · · < a2N the points a2j and a2j +1 if j is such that [a2j , a2j +1 ] ⊂ supp(µc0 ). We renumber the remaining points as a1 < a2 < · · · < a2M with M ≤ N. Then (4.60) holds for j = 1, . . . , M and (4.62) holds for j = 1, . . . , M −1. Also (4.63) holds, and (4.64) holds with N replaced by M. By Lemma 4.8, the ﬁnite-gap ansatz then holds for (V , σ, c0 ). Thus c0 ∈ Ꮿ. 4.7. Conclusion of the proof of Theorem 3.1. Combining Propositions

4.5, 4.9, and 4.11, we see that Ꮿ is a nonempty, open and closed subset of (0, dσ ). Thus Ꮿ = (0, dσ ), and (V , σ, c) satisﬁes the ﬁnite-gap ansatz for every c < dσ . This completes the proof of Theorem 3.1.

ON THE FINITE-GAP ANSATZ

461

Acknowledgments. I wish to thank Ken McLaughlin and Walter Van Assche for their interest in this work and for many stimulating discussions. I am grateful to the anonymous referee for the detailed remarks and suggestions which helped to improve the manuscript. References [BR]

[DK]

[D] [DKM]

[DKMVZ]

[DM] [DVZ]

[DS]

[ELZ] [FFM]

[G] [JLM] [J] [KD] [KM]

[KR]

[KVA1]

V. S. Buyarov and E. A. Rakhmanov, Families of equilibrium measures in an external ﬁeld on the real axis (in Russian), Mat. Sb. 190 (1999), 11–22; English transl. in Russian Acad. Sci. Sb. Math. 190 (1999), 791–802. S. B. Damelin and A. B. J. Kuijlaars, The support of the equilibrium measure in the presence of a monomial external ﬁeld on [−1, 1], Trans. Amer. Math. Soc. 351 (1999), 4561–4584. P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lec. Notes Math. 3, Courant Institute, New York, 1999. P. Deift, T. Kriecherbauer, and K. T.-R. McLaughlin, New results on the equilibrium measure for logarithmic potentials in the presence of an external ﬁeld, J. Approx. Theory 95 (1998), 388–475. P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math. 52 (1999), 1335–1425. P. Deift and K. T.-R. McLaughlin, A continuum limit of the Toda lattice, Mem. Amer. Math. Soc. 131 (1998), no. 624. P. Deift, S. Venakides, and X. Zhou, New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems, Internat. Math. Res. Notices 1997, 286–299. P. D. Dragnev and E. B. Saff, Constrained energy problems with applications to orthogonal polynomials of a discrete variable, J. Anal. Math. 72 (1997), 223– 259. N. M. Ercolani, C. D. Levermore, and T. Zhang, The behavior of the Weyl function in the zero-dispersion KdV limit, Comm. Math. Phys. 183 (1997), 119–143. H. Flaschka, M. G. Forest, and D. W. McLaughlin, Multiphase averaging and the inverse spectral solution of the Korteweg–de Vries equation, Comm. Pure Appl. Math. 33 (1980), 739–784. F. Gakhov, Boundary Value Problems, Pergamon Press, Oxford, 1966. S. Jin, C. D. Levermore, and D. W. McLaughlin, The semiclassical limit of the defocusing NLS hierarchy, Comm. Pure Appl. Math. 52 (1999), 613–654. K. Johansson, On ﬂuctuations of eigenvalues of random Hermitian matrices, Duke Math. J. 91 (1998), 151–204. A. B. J. Kuijlaars and P. D. Dragnev, Equilibrium problems associated with fast decreasing polynomials, Proc. Amer. Math. Soc. 127 (1999), 1065–1074. A. B. J. Kuijlaars and K. T.-R. McLaughlin, Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external ﬁelds, Comm. Pure Appl. Math. 53 (2000), 736–785. A. B. J. Kuijlaars and E. A. Rakhmanov, Zero distributions for discrete orthogonal polynomials, J. Comput. Appl. Math. 99 (1998), 255–274; Corrigendum, J. Comput. Appl. Math. 104 (1999), 213. A. B. J. Kuijlaars and W. Van Assche, The asymptotic zero distribution of orthogonal

462

[KVA2] [LL]

[M]

[R]

[Ran] [ST] [T] [To] [V]

A. B. J. KUIJLAARS polynomials with varying recurrence coefﬁcients, J. Approx. Theory 99 (1999), 167–197. , Extremal polynomials on discrete sets, Proc. London Math. Soc. (3) 79 (1999), 191–221. P. Lax and C. D. Levermore, The small dispersion limit of the Korteweg–de Vries equation, I, Comm. Pure Appl. Math. 36 (1983), 253–290; II, 571–593; III, 809–829. J. Moser, “Finitely many mass points on the line under the inﬂuence of an exponential potential—an integrable system” in Dynamical Systems: Theory and Applications (Seattle, 1974), ed. J. Moser, Lecture Notes in Phys. 38, Springer, Berlin, 1975, 467–497. E. A. Rakhmanov, Equilibrium measure and the distribution of zeros of extremal polynomials of a discrete variable (in Russian), Mat. Sb. 187 (1996), 109–124; English transl. in Russian Acad. Sci. Sb. Math. 187 (1996), 1213–1228. T. Ransford, Potential Theory in the Complex Plane, London Math. Soc. Stud. Texts 28, Cambridge Univ. Press, Cambridge, 1995. E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Grundlehren Math. Wiss. 316, Springer, Berlin, 1997. F. R. Tian, Oscillations of the zero dispersion limit of the Korteweg–de Vries equation, Comm. Pure Appl. Math. 46 (1993), 1093–1129. V. Totik, Weighted Approximation with Varying Weight, Lecture Notes in Math. 1569, Springer, Berlin, 1994. S. Venakides, The Korteweg–de Vries equation with small dispersion: Higher order Lax-Levermore theory, Comm. Pure Appl. Math. 43 (1990), 335–361.

Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, Leuven, Belgium; [email protected]

Vol. 104, No. 3

DUKE MATHEMATICAL JOURNAL

© 2000

HOLOMORPHIC DIFFERENTIAL INVARIANTS FOR AN ELLIPSOIDAL REAL HYPERSURFACE S. M. WEBSTER 0. Introduction. A Levi nondegenerate real hypersurface M 2n−1 in a complex nmanifold has a complete system of local differential invariants, under biholomorphic mappings. These invariants may be described either by the coefﬁcients in a normal form or by the curvature of a connection. For a smooth, bounded, strictly pseudoconvex domain D in complex space Cn , the invariants of M = ∂D are intimately related to the global complex analytic invariants of D. Despite their importance, until now these invariants have been fully computed, to our knowledge, only in the case of the unit ball D = B n , where they all vanish! This is partially explained by the fact that if D is not holomorphically equivalent to B n , then the automorphism group of M is greatly reduced. Our aim is to carry out the computation of invariants in a signiﬁcant case where there is virtually no symmetry. The simplest and most useful of the boundary invariants is the fourth-order pseudoconformal curvature tensor S of Chern and Moser [6]. For n ≥ 3 it vanishes identically if and only if M is locally biholomorphically equivalent to the sphere ∂B n , in the strictly pseudoconvex case. For n = 2 it vanishes identically by default—its role being taken by Cartan’s sixth-order invariant (see [4]). For further information on these invariants and their application, we refer to [1], [2], and [10], for example. In this paper we compute the fourth-order curvature tensor S for an ellipsoidal real hypersurface M (see [12] and [14]), M = z ∈ Cn | r(z, z) = 0 ,

r(z, ζ ) =

n j =1

zj ζj + Aj zj2 + ζj2 − 1,

(0.1)

which we call generic, if 1 0 < A1 < · · · < A n < . 2

(0.2)

A generic ellipsoid (and the domain that it bounds) has only a ﬁnite group of holomorphic symmetries (see [14]), which are of little use in computing differential invariants. Previously in [13], we were able to compute S only at points of intersection of M with the coordinate axes, but the methods of [13] were too ungainly for the more complete computation. Here we proceed by the method of complexiﬁcation, which Received 23 June 1999. Revision received 6 January 2000. 2000 Mathematics Subject Classiﬁcation. Primary 32V40; Secondary 32V05. Author’s work partially supported by the National Science Foundation. 463

464

S. M. WEBSTER

reveals the essential geometric structure, as it does in a number of other problems involving real submanifolds. In particular, we prove the following result. Theorem 0.1. Let M ⊂ Cn , n ≥ 3, be a generic ellipsoidal real hypersurface. Then the fourth-order pseudoconformal tensor S does not vanish at any point of M. Thus, in the terminology of [6], M is nowhere umbilic. This result was rather unexpected. It raises the general question of the behavior of the umbilic set under perturbation. The higher-order invariants of an ellipsoid, especially for n = 2, should also be accessible by our methods, but we do not go into their computation here. These methods, based on the developments of [15], [16], and [17], are at least as interesting as the above result. The Chern-Moser tensor S (= Sαβρσ ) can be interpreted as a relatively invariant (2, 2)-tensor on the holomorphic tangent bundle H (M). Its norm S θ , relative to the Levi form of the contact form θ = −i∂r, is multiplied by u−1 , if θ is multiplied by a positive function u on M. Thus, the theorem gives an everywhere deﬁned, holomorphically invariant contact form , the principal contact form, for which S = 1. The principal characteristic vector ﬁeld V on M is deﬁned by ιV = 1,

ιV d = 0.

(0.3)

It is a holomorphically invariant inﬁnitesimal contact transformation of the structure (M, ). Theorem 0.1 and the method of proof lead to the following result. Theorem 0.2. Let M ⊂ Cn , n ≥ 3, be a generic ellipsoidal real hypersurface. Then the ﬂow of the principal characteristic vector ﬁeld is completely integrable. The term “completely integrable” is necessarily somewhat vague, as it sometimes is even in the more familiar Hamiltonian case. Roughly speaking, it means that, off a proper subvariety, M is foliated by n-dimensional, V -invariant submanifolds on which the ﬂow is susceptible to a simple, explicit description. The details of this are given in Section 4. The proofs of Theorems 0.1 and 0.2 make essential use of the complexiﬁcation ᏹ of M, ᏹ = (z, ζ ) ∈ C2n | r(z, ζ ) = 0 , (0.4) M = FP(ρ), ρ(z, ζ ) = ζ , z , where FP means “ﬁxed point set.” The proofs also make use of the Segre polar varieties Qζ = z ∈ Cn | r(z, ζ ) = 0 . (0.5) By Levi nondegeneracy there is locally a unique such complex hypersurface through a given point and tangent to a given hyperplane. The map (z, ζ ) → (z, Tz Qζ ) locally

465

ELLIPSOIDAL INVARIANTS

identiﬁes ᏹ with the space of holomorphic contact elements. This map also identiﬁes the complexiﬁcation of the Chern-Moser invariants of M with the invariants of Tresse, Cartan, and Hachtroudi (see [3], [8], [5], and [7]) associated with the family of complex hypersurfaces {Qζ }. For the generic ellipsoid (see [15] and [16]), the family of complex quadrics ᏽ = {Qζ } sits in a linear family Pn+1 as a nondegenerate n-dimensional quadric. For each point z ∈ Cn , the linear condition z ∈ Qζ determines a point in the dual space P∗n+1 . The set of all such points determines another nondegenerate n-dimensional quadric in P∗n+1 , which we identify with its projective dual ᏿ ⊂ Pn+1 . Under the birational polar transform (z, ζ ) −→ (z, Qζ ) ≡ (ξ, η) −→ (η, l),

l = [ξ η],

(0.6)

ᏹ corresponds to the set ᏸ0 of pointed lines (η, l) in Pn+1 with η ∈ l ∩ ᏽ and l tangent to ᏿ (at ξ ). As in [15] we study ᏹ ∼ = ᏸ0 by means of moving ᏽ-frames adapted to ᏿. We

use the Maurer-Cartan forms of the complex orthogonal group determined by the complex quadric ᏽ to construct the Cartan-Hachtroudi connection. This allows us to compute the tensor S in terms of the invariants of the quadric ᏿⊥ᏽ ∩ l ⊥ᏽ , relative to the quadric ᏽ ∩ l ⊥ᏽ , where ⊥ ᏽ means dual with respect to the nonsingular quadric ᏽ. (This brings to mind the computation of the intrinsic curvature of a surface in real Euclidean space in terms of the invariants of the second fundamental form relative to the ﬁrst fundamental form.) The vanishing of the tensor S at a point translates into the condition l ⊥ᏽ ⊂ ᏿⊥ᏽ , which contradicts nondegeneracy of ᏿⊥ᏽ , if n ≥ 3. This summarizes the proof of Theorem 0.1. Again following [15], we next use the confocal family of n-quadrics ᏽλ ⊂ Pn+1 , of ᏽ relative to ᏿, to construct special “confocal” coordinates on ᏹ, via ᏸ0 . These coordinates are used in the proof of Theorem 0.2. This is achieved by showing that the n-dimensional variety of pointed lines (η, l), with l tangent to n − 1 additional ﬁxed confocal quadrics ᏽλ , is invariant by the ﬂow of the meromorphic continuation of V to ᏹ ∼ = ᏸ0 . A generalized hyperelliptic Abel-Jacobi map takes this variety to the quotient of Cn by a lattice of rank 2n − 1, with the ﬂow of V going to a linear ﬂow. This work provides another example of a general fact about ellipsoids. Certain problems tend to be explicitly solvable for ellipsoids, regardless of the underlying geometry, by means of the appropriate confocal theory. Thus, the main result of [15] on the dynamics of double-valued reﬂection is the analogue of the complete integrability of billiards in an ellipsoidal domain in real Euclidean space. Theorem 0.2 now appears to be the proper complex analogue of Jacobi’s theorem [9] on the complete integrability of the geodesic ﬂow on an ellipsoid in Rn . The analogy is perhaps clearer if, in Jacobi’s case, we pass to the unit cotangent bundle and take the restriction of the canonical 1-form, since its characteristic vector ﬁeld then generates the geodesic ﬂow.

466

S. M. WEBSTER

1. Local structure and complexiﬁcation. As in [6, Section 4] we describe the local structure of a nondegenerate real hypersurface M by means of local coframes. The 1-form θ = −i∂r is a real contact form on M and is determined up to a nonzero real factor. We choose local complex forms θ α , 1 ≤ α ≤ n − 1, so that {θ, θ α } and α {θ, θ } span the (1,0)- and (0,1)-forms restricted to M, respectively. Then the Levi form (relative to θ ) is representated by the hermitian matrix gαβ deﬁned by β

dθ = i∂∂r = igαβ θ α ∧ θ + θ ∧ ϕ.

(1.1)

Here, and in what follows, repeated Greek indices are summed from 1 to n − 1. In preparation for complexiﬁcation, it is convenient to pass to a “bar-free” notation, β so we set θα = igαβ θ . Then, as in [6], the integrability conditions for the two systems α {θ, θ }, {θ, θα } give dθ = θ γ ∧ θγ + θ ∧ ϕ, dθα = ϕαβ − δαβ ϕ ∧ θβ + ψα ∧ θ,

dθ α = θ β ∧ ϕβα + θ ∧ ψ α , dϕ = ψγ ∧ θ γ + ψ γ ∧ θγ + θ ∧ ψ,

(1.2)

for certain auxiliary 1-forms, ϕβα , ψ α , ψα , ψ. The ﬁrst equation is just (1.1). Taking its exterior derivative gives the form of the last two (see [6]). For ﬁxed forms {θ, θ α , θα , ϕ}, the auxiliary forms are determined, by Cartan’s lemma, up to changes of the form ϕαβ − ϕα0β = Bαβ θ, ψα − ψα0 = Bαβ θβ + Bα θ,

ψ α − ψ 0α = Bβα θ β + B α θ, ψ − ψ 0 = −Bα θ α − B α θα + Bθ.

(1.3)

Here the forms with “0” are any initial choices satisfying (1.2), and the coefﬁcients B are to be chosen so that certain further normalizations hold. If M and the coframe are real analytic, the above continues locally to the complexiﬁcation ᏹ in (0.4). Then we have θ = −i∂z r(z, ζ ), up to a factor, and the two systems are spanned by the dzj and the dζ j , respectively. The (complexiﬁed) Levi-degeneracy locus is 0 rζ Ꮾ = (z, ζ ) ∈ ᏹ | det (z, ζ ) = 0 . (1.4) rz rzζ It is disjoint from M in the present (nondegenerate) case. On the set ᏹ − Ꮾ, the map (z, ζ ) → (z, Tz Qζ ) ∈ Cn × P∗n−1 is locally invertible. Thus, the varieties {Tz Qζ | z ∈ Qζ } give locally a foliation of the space of holomorphic contact elements. Another foliation is given by the ﬁbers z = const. Associated to such a structure is the invariant connection of Cartan [3] and Hachtroudi [8]. We brieﬂy explain this in the next section. 2. The Cartan-Hachtroudi connection. This may be motivated by ﬁrst considering the “ﬂat case,” which consists of the family of pointed hyperplanes in Pn , that is,

467

ELLIPSOIDAL INVARIANTS

the subset of Pn × P∗n satisfying the incidence relation. To such a conﬁguration we attach a projective frame (basis of Cn+1 ), Z = (Z0 , Zα , Zn ), det Z = +1, so that [Z0 ] is the point, and [Z0 ], . . . , [Zn−1 ] span the hyperplane. If Z = Z(t) depends smoothly on some parameters t, then exterior differentiation gives j

dZ = π Z ⇐⇒ dZi = πi Zj , j

tr π = πii = 0, j

dπ = π ∧ π ⇐⇒ dπi = πik ∧ πk .

(2.1) (2.2)

In this section Latin indices run from 0 to n, Greek indices from 1 to n − 1, and the corresponding summation conventions are used. The geometric point [Z0 ] remains ﬁxed if the forms {π0n , π0α } vanish, while the hyperplane [Z0 , . . . , Zn−1 ] is constant if {π0n , παn } vanish. This singles out two integrable systems. An SL(n + 1, Cn ) Cartan connection manifests itself locally by such a matrix π of 1-forms, tr π = 0. In place of (2.2), we have the structure equation dπ = π ∧ π + ,,

tr , = 0.

(2.3)

The connection is torsion free if the matrix of 2-forms , satisﬁes j

,0 = 0,

,ni = 0.

(2.4)

Then tr , = ,αα = 0.

(2.5)

Taking the exterior derivative of (2.3) gives the Bianchi identity 0 = , ∧ π − π ∧ , + d,,

(2.6)

which together with (2.4) gives, in particular, 0 = ,ki ∧ πkn ,

j

0 = π0k ∧ ,k .

(2.7)

Now consider the complexiﬁed real hypersurface ᏹ and local coframe of Section 1. To associate a connection matrix π to this data, we ﬁrst set π0n = θ,

π0α = θ α ,

παn = θα .

(2.8)

Comparison of (1.2) with (2.3) and (2.4) suggests the relations ϕ = πnn − π00 ,

ϕαβ = παβ − δαβ π00 ,

ψ α = πnα ,

ψα = πα0 ,

ψ = πn0 . (2.9)

Applying Cartan’s lemma to (2.7) and using (2.8) and (2.5) gives βσ ρ ,βα ≡ Sαρ θ ∧ θσ mod θ,

(2.10)

468

S. M. WEBSTER

where the coefﬁcients satisfy the symmetry and trace conditions βσ σβ βσ = Sαρ = Sρα , Sαρ

ασ Sαρ = 0.

(2.11)

In fact, a matrix π satisfying all these conditions exists. Further conditions, namely, the full trace condition (2.5) and one further trace condition, are needed to determine π uniquely. However, this is not needed here (see [6] and [8]). We carry out the normalizations only to the point needed to determine S. This amounts to determining Bβα in (1.3). S is a relatively invariant tensor. If we change the contact form θ → uθ , change {θ α , θα } → {θ α , uθα }, and follow this through the structure equations, then we see β that ,α is unchanged. Hence, S → u−1 S. We deﬁne the contraction of S by αρ

βσ Sβσ , S, S = Sαρ

(2.12)

which transforms as S, S → u−2 S, S. Where (2.12) is not zero, we can choose the contact form locally to make it identically 1. On M the sign of θ is determined by requiring a positive deﬁnite Levi form, in the strictly pseudoconvex case. This is the principal contact form. 3. Polar transform and curvature tensor. We describe the map (0.6) and its image. This was ﬁrst carried out explicitly in [15], to which we refer for full details. −1/2 By the change zj → Aj zj , we pass from (0.1) to the deﬁning function (3.1) r(z, ζ ) = A−1 z · ζ + z · z + ζ · ζ − 1,

where the dot product is z · w = nj=1 zj wj and where A is the diagonal matrix with the eigenvalues in (0.2). Then the varieties Qζ are complex spheres. The family of all spheres is an (n + 1)-dimensional linear family denoted Pn+1 . In terms of homogeneous coordinates ξ = (ξ0 , ξ , ξ∗ ) ∈ Cn+2 (∗ = n + 1), equation (0.6) is given by ξ = F (z) = (1, z, z · z), 1 A−1 ζ, ζ · ζ − 1 . η = G(ζ ) = 1, − 2

(3.2) (3.3)

We have F (Cn ) ⊆ ᏿ = {s(ξ, ξ ) = 0} and G(Cn ) ⊆ ᏽ = {q(ξ, ξ ) = 0}, where the two quadrics are deﬁned by the symmetric bilinear forms 1 s(ξ, η) = ξ · η − ζ0 η∗ + ξ∗ η0 , (3.4) 2 1 ζ0 η∗ + ξ∗ η0 . (3.5) q(ξ, η) = 4A2 ξ · η − ξ0 η0 − 2 We write q(ξ, η) = s(Ꮽξ, η), where Ꮽ is a q-symmetric operator.

ELLIPSOIDAL INVARIANTS

469

Since r(z, ζ ) = −2s F (z), G(ζ ) = −2s(ξ, η), ᏹ corresponds to the set ᏸ0 of (ξ, η) ∈ ᏿ × ᏽ such that the line l = [ξ η] is tangent to ᏿ at ξ . The set Ꮾ (deﬁned in (1.4)) corresponds to q(ξ, η) = 0. This just means that l is also tangent to ᏽ at η. For the contact form we may take θ = −2is(ξ, dη). The system θ = θ α = 0 corresponds to dξ ∧ ξ = 0, and θ = θα = 0 corresponds to dη ∧ η = 0. As in [15] we study ᏸ0 via moving q-frames e adapted to ᏿. By deﬁnition, a q-frame e = (e0 , eα , en , e∗ ) is a basis of the homogeneous coordinate space Cn+2 , 1 ≤ α ≤ n−1 as before, and ∗ = n+1, such that e0 , e∗ ∈ l ∩ ᏽ, and eα , en span l ⊥q . It is said to be adapted to ᏿, if l is also tangent to ᏿ at Ꮽen and if l ⊥q is tangent to ᏿⊥q at en . Algebraically, the conditions deﬁning a q-frame may be speciﬁed by setting q(ei , ej ) = gij ,

(3.6)

where g0∗ = gαα = gnn = 1 and where all the other gij are zero. The condition that e be adapted to ᏿ is equivalent to aij = aj i = q Ꮽei , ej , (3.7) aαn = ann = 0, where the second equation deﬁnes the coefﬁcients aij . Adapted frames exist at any point off Ꮾ, where we can choose e0 = e∗ . By changing (3.2) and (3.3) by multiples, which is immaterial in homogeneous coordinates, we may take η = e0 ,

ξ = Ꮽ en .

(3.8)

For moving q-frames, exterior differentiation gives, in (n + 2) × (n + 2) -matrix notation, de = ωe,

dω = ω ∧ ω.

(3.9)

So, for example, de0 = ω00 e0 + ω0α eα + ω0n en + ω0∗ e∗ and dω0α = ω00 ∧ ω0α + ω0β ∧ ωβα + ω0n ∧ ωnα + ω0∗ ∧ ω∗α . Differentiating (3.6) and (3.7) gives 0 = ωg + gωt

(3.10)

da = ωa + aωt .

(3.11)

and

470

S. M. WEBSTER

More explicitly, (3.10) is equivalent to ωα∗ = −ω0α , ω∗∗ = −ω00 ,

ωn∗ = −ω0n ,

ω∗0 = ω0∗ = 0,

ω∗α = −ωα0 ,

ωαβ = −ωβα ,

ω∗n = −ωn0 ,

ωαn = −ωnα ,

ωnn = 0. (3.12)

From the ﬁrst two equations in (3.7) and (3.11), we get 0 = daj n = ωj k akn + aj k ωnk ,

1 ≤ j ≤ n,

where the index k is summed over the n + 2 values from 0 to ∗ = n + 1. This yields, using (3.12) and (3.7), an∗ ω0n , an0 aαβ an∗ aα0 an∗ − an0 aα∗ ω0α − ωnβ − ω0n . ωα0 = an0 an0 (an0 )2 ωn0 =

(3.13) (3.14)

We may further normalize the frame so that s(e0 , e0 ) = 1, off the set ᏿ ∩ ᏽ. Differentiation of this condition gives −ω00 = s(e0 , eα )ω0α + s(e0 , en )ω0n .

(3.15)

From s(ξ, dη) = s(Ꮽen , de0 ) = q(en , de0 ) = ω0n and dω0n = ω0j ∧ ωj n = ω00 ∧ ω0n + ω0α ∧ ωαn ,

(3.16)

we see that we may take θ = −2iω0n ,

θ α = ωnα ,

θα = −2iω0α ,

ϕ = −ω00 .

(3.17)

From (3.16) we get the ﬁrst equation in (1.2). Similarly computing dωnα , dω0α , and dω00 shows that we get the rest of the equations in (1.2), if we choose 1 an∗ ωα0 + ψ 0α = − ψ 0 = 0. ω0α , ψα0 = −ωnα , ϕα0β = ωαβ , 2i an0 (3.18) The next structure equation in (2.3) is, using (2.8) and (2.9), β

,βα = dπαβ − πα0 ∧ π0 − παγ ∧ πγβ − παn ∧ πnβ

= dϕαβ − ϕαγ ∧ ϕγβ − ψα ∧ θ β − θα ∧ ψ β + δαβ θ γ ∧ ψγ + θ ∧ ψ .

(3.19)

Computing this expression modulo the ideal of θ with the forms (3.18) gives 0βσ ρ ,0β α ≡ Sαρ θ ∧ θσ mod θ, −1 0βσ Sαρ = 2ia0n aαρ δ βσ .

(3.20) (3.21)

ELLIPSOIDAL INVARIANTS

471

Observe that a0n = q(ξ, η) is nonzero off the set Ꮾ. Making the substitution (1.3) into (3.19) gives (2.10), after a little algebra, with βσ 0βσ βσ Sαρ = Sαρ + Eαρ ,

(3.22)

βσ = Bαβ δρσ + Bασ δρβ + Bρβ δασ + Bρσ δαβ . Eαρ

(3.23)

β

We want to choose Bα to achieve the trace condition in (2.11); that is, ασ ασ 0 = Sαρ = Sρ0σ + Eαρ = Sρ0σ + Bαα δρσ + (n + 1)Bρσ ,

(3.24)

where we have deﬁned −1 0ασ Sρ0σ = Sαρ = 2ia0n aρσ .

(3.25)

0 = Sγ0γ + 2nBγγ .

(3.26)

Contracting ρ and σ gives

Hence, we choose Bαβ =

−1 1 Sα0β − Sγ0γ δαβ . n+1 2n

(3.27)

To maintain the trace condition in (2.11), we must restrict to changes (1.3) with β Bα = 0. Such changes do not affect S. Hence, we have proved the following. Lemma 3.1. On the subset ᏹ − Ꮾ of the complexiﬁed ellipsoid, the fourth-order curvature tensor S, computed relative to the 1-form θ in (3.17), is given by (3.22) and (3.23) with (3.27), (3.21), and (3.25). Next we compute the contraction of S,

S, S = S 0 , S 0 + 2 S 0 , E + E, E. We readily ﬁnd, using symmetry of the indices,

0 0 0γ 2 S , S = Sγ ,

0 2 0αρ 2 S , E = 8Sβσ Bαβ δρσ = −8(n + 1)−1 Sα0β Sβ0α − (2n)−1 Sγ0γ , 2 αρ E, E = 4Eβσ Bαβ δρσ = 4 (n + 1)Bαβ Bβα + Bγγ 2 = 4(n + 1)−1 Sα0β Sβ0α − (2n)−1 Sγ0γ .

(3.28)

(3.29) (3.30)

(3.31)

Hence, S, S =

−4 0β 0α n(n + 1) + 2 0γ 2 S S + Sγ . n+1 α β n(n + 1)

(3.32)

472

S. M. WEBSTER

4. Proof of the theorems Proof of Theorem 0.1. Suppose that S vanishes at a point (z, ζ ) of ᏹ − Ꮾ. Then, near the image point (ξ, η) ∈ ᏿ × ᏽ, we can choose an adapted frame e and apply the results of Section 3. Taking α = ρ = β = σ in (3.22) (which is possible, since n − 1 ≥ 2) gives ββ 0ββ 0 = Sαα = Sαα ,

(4.1)

β

so that aαα = 0, 1 ≤ α ≤ n − 1 by (3.21). Also, Bα = −(n + 1)−1 Sα = 0, if α = β. Now taking β = ρ = σ gives ββ

0 = Sαβ = Sα0β + 2Bαβ =

0β

n − 1 0β S . n+1 α

(4.2)

It follows that aαβ = 0, for 1 ≤ α, β ≤ n − 1. Since we already have aαn = ann = 0 by (3.7), it follows that the (n − 1)-dimensional space spanned by e1 , . . . , en lies on the nondegenerate, n-dimensional quadric given by q(Ꮽξ, ξ ) = 0. This contradicts a basic property of quadrics for n > 2, and it ﬁnishes the proof. Proof of Theorem 0.2. We must determine precisely the principal contact form . By (3.17) and the remarks after (2.12) it can be written as = ±i S, Ss(ξ, dη), (4.3) since ω0n = q(en , de0 ) = s(ξ, dη). We express this form in terms of the confocal coordinates of [15], which we brieﬂy recall. The family of quadrics in Pn+1 confocal to ᏽ relative to ᏿ is deﬁned by ᏽλ = η | s (λ − B)−1 η, η = 0 , (4.4) where λ ∈ C is not an eigenvalue of the operator B = Ꮽ−1 . The (nonhomogeneous) confocal coordinates λj , 0 ≤ j ≤ n, of a point η of Pn+1 are those values of λ for which η ∈ ᏽλ . The quantities −1 xj = xj (ξ, η) = s λj − B ξ, η ,

λj = λj (η)

(4.5)

are “conjugate coordinate pairs,” in that they give the Pfafﬁan canonical form n

s(ξ, dη) = −

1 xj dλj . 2

(4.6)

j =0

In general, the point η lies on (n + 1) of the quadrics ᏽl , while the line l = [ξ η] is tangent to n of them, and ᏽµ , µ = µ1 , . . . , µn . Since, for ᏹ ∼ = ᏸ0 , η ∈ ᏽ = ᏽ0

473

ELLIPSOIDAL INVARIANTS

and l is tangent to ᏿ = ᏽ∞ , we set λ0 ≡ 0, µn ≡ ∞. Then (λ1 , . . . , λn , µ1 , . . . , µn−1 ) generically provide local “confocal” coordinates on ᏹ. By q-duality (see [15]), the plane l ⊥q is tangent to the duals of the quadrics ᏽ1 , . . . , ᏽn at points that we take to be e1 , . . . , en in our q-frame. Then relative to such a frame, we have, in addition to (3.7), as in [15], aαβ = 0,

α = β;

aαα = µ−1 α .

(4.7)

If we normalize so that s(e0 , e0 ) = 1 (away from e0 ∈ ᏽ ∩ ᏿), then by [15, Prop. 7.2], the conjugate coordinate pairs satisfy p λ, µ " 2 λj , xj ∈ Cµ" ≡ (λ, x) | x + =0 , (4.8) f (λ) n−1 (λ−µα ). where f (λ) is the characteristic polynomial of B and where p(λ, µ) " = α=1 The hyperelliptic curve Cµ" generically has genus n − 1, since f has one double root λ = 1. Thus −p 0, µ " a0n = q(ξ, η) = x0 = ± , (4.9) f (0) so that, relative to confocal coordinates, (3.32) has the form  n−1 2  n−1 −f (0) n(n + 1) + 2 ,  S, S = µ−2 µ−1 α − α 4n (n + 1)p 0, µ " α=1

(4.10)

α=1

which is symmetric in µ1 , . . . , µn−1 . Hence, we have the necessary form in [15] for , =

n

aj λ j , µ " dλj .

(4.11)

j =1

The (meromorphic) characteristic vector ﬁeld V = satisﬁes ιV d = 0, or n

n−1

j =1

α=1

∂µα aj vj = 0,

j vj ∂/∂λj

wα ∂µα aj = 0.

+

α wα ∂/∂µα

(4.12)

We claim that the matrix (∂µα aj ) generically has rank n − 1. This forces wα = 0, so that V is tangent to the n-dimensional varieties ᏸ µ " = (η, l) | l tangent to ᏽµ1 , . . . , ᏽµn−1 , (4.13) on which the µα are constant.

474

S. M. WEBSTER

To prove the rank statement, put " xj , aj = ih µ

−1 ∂µα aj = −2−1 λj − µα + hα µ " aj

(4.14)

for functions h, hα of µ1 , . . . , µn−1 . For generic choices of these values of µ, and of λ1 , . . . , λn−1 , the determinant −1 det λj − µα + hα µ " is not zero, since the functions of one variable λ → (λ−µα )−1 +hα (µ) " are independent. As in [15] the mapping (ξ, η) → ((λj , xj ))nj=1 takes ᏸ(µ) " several-to-one to the (n)

symmetric product Cµ" . The equations for the integral curves of V are Abelian differential equations of the third kind via (4.12). The generalized Jacobi inversion theorem applied to the corresponding Abelian sums gives a basically one-to-one (n) correspondence of Cµ" with a quotient Cn /?µ" , where the lattice ?µ" has rank 2n−1. n The isogeny C /(2?µ" ) → Cn /?µ" factors through ᏸ(µ), " and the integral curves of V on ᏸ(µ) " are the images of straight lines as in [15] and [17]. Restricting back to the real locus M ﬁnishes the proof of Theorem 0.2. It should be possible to give explicit parametrizations of these principal curves by means of (generalized) theta functions. This would be the analogue of the results of Weierstrass [18] and Knörrer [11], for the geodesics of the ellipsoid in real Euclidean space. References [1] [2]

[3] [4] [5] [6] [7] [8] [9] [10]

M. Beals, C. Fefferman, and R. Grossman, Strictly pseudoconvex domains in Cn , Bull. Amer. Math. Soc. (N.S.) 8 (1983), 125–322. D. Burns and S. Shnider, “Real hypersurfaces in complex manifolds” in Several Complex Variables (Williamstown, Mass., 1975), Proc. Sympos. Pure Math. 30, Part 2, Amer. Math. Soc., Providence, 1977, 141–168. É. Cartan, Sur les variétés à connexion projective, Bull. Soc. Math. France 52 (1924), 205– 241. , Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes, Ann. Mat. Pura Appl. (4) 11 (1932), 17–90. S. S. Chern, On the projective structure of a real hypersurface in Cn+1 , Math. Scand. 36 (1975), 74–82. S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219–271; Erratum, Acta Math. 150 (1983), 297. J. J. Faran V, Segre families and real hypersurfaces, Invent. Math. 60 (1980), 135–172. M. Hachtroudi, Les espaces d’éléments à connexion projective normale, Actualités Sci. Indust. 565, Hermann, Paris, 1937. C. G. J. Jacobi, Gesammelte Werke: Vorlesungen über Dynamik, supplemental volume, Chelsea, New York, 1969. H. Jacobowitz, An Introduction to CR Structures, Math. Surveys Monogr. 32, Amer. Math. Soc., Providence, 1990.

ELLIPSOIDAL INVARIANTS [11] [12] [13] [14] [15] [16]

[17] [18]

475

H. Knörrer, Geodesics on the ellipsoid, Invent. Math. 59 (1980), 119–143. S. M. Webster, On the mapping problem for algebraic real hypersurfaces, Invent. Math. 43 (1977), 53–68. , Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom. 13 (1978), 25–41. , Some birational invariants for algebraic real hypersurfaces, Duke Math. J. 45 (1978), 39–46. , Real ellipsoids and double valued reﬂection in complex space, Amer. J. Math. 120 (1998), 757–809. , “Segre polar correspondence and double valued reﬂection for general ellipsoids” in Analysis and Geometry in Several Complex Variables (Katata, 1997), Trends Math., Birkhäuser, Boston, 1999, 273–288. , Stationary curves and complete integrability in the complex domain, to appear in Proceedings of the 1997 Lelong Conference, Birkhäuser. K. Weierstrass, “Über die geodätischen Linien auf dem dreiaxigen Ellipsoid” in Mathematische Werke, Vol. 1: Abhandlungen, 1, Mayer & Müller, Berlin, 1894, 257–266.

Department of Mathematics, University of Chicago, Chicago, Illinois 60637, USA; [email protected]

Vol. 104, No. 3

DUKE MATHEMATICAL JOURNAL

© 2000

RATIONAL POINTS ON QUARTICS JOE HARRIS and YURI TSCHINKEL

Contents 1. 2. 3. 4.

5. 6. 7. 8.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 Fano 3-folds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 A Chow ring calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 The argument via monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 4.1. Generalities about quartic surfaces with a line . . . . . . . . . . . . . . . . . . . . . . . 486 4.2. Analysis of the points of CH ∩ L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 Rational points on quartic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 An example: The Fermat quartic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 Quartic 3-folds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 Other elliptic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496

1. Introduction. Of all the possible extensions to higher dimensions of Faltings’s theorem, probably the most fundamental is the following conjecture. Conjecture 1.1 (Weak Lang conjecture). Let X be a variety deﬁned over a number ﬁeld K. If X is of general type, then the set X(K) of K-rational points of X is not Zariski dense. (While the name “weak Lang conjecture” has become standard usage—in part to distinguish it from the “strong Lang conjecture” below—we should point out that as stated here it was ﬁrst ventured by Bombieri for surfaces (see, e.g., [28]) and by Vojta in [32].) We ask now whether a converse to this statement might hold. As it stands, the converse to the weak Lang conjecture cannot possibly be true. For example, if we take the product X = P1 × C of a rational curve and a curve C of genus g ≥ 2, we get a surface that is not of general type; but by Faltings’s theorem the rational points of X must lie in a ﬁnite union of ﬁbers of X over C. The point is that the Kodaira dimension of a variety is not a sufﬁciently sensitive measure of the positivity or negativity of its canonical bundle. One possible modiﬁcation, if we hope to have a plausible converse to the weak Lang conjecture, is to Received 25 February 1999. Revision received 30 December 1999. 2000 Mathematics Subject Classiﬁcation. Primary 14G05. Harris’s work partially supported by the National Science Foundation. Tschinkel’s work partially supported by the National Security Agency. 477

478

HARRIS AND TSCHINKEL

replace the hypothesis “X is of general type” with “X admits a dominant rational map to a positive-dimensional variety of general type”; or, given the counterexamples to this found by Colliot-Thélène, Skorobogatov, and Swinnerton-Dyer [10], with “there exists a ﬁnite étale cover Y → X and a dominant rational map Y → Z to a positive-dimensional variety of general type.” In other words, we may make the following conjecture, which was suggested to us by Dan Abramovich and Jean-Louis Colliot-Thélène. Conjecture 1.2. Let X be a smooth, connected projective variety deﬁned over a number ﬁeld K. There exists a ﬁnite extension K of K such that the set X(K ) of K -rational points of X is Zariski dense if and only if no ﬁnite étale cover Y → X admits a dominant rational map Y → Z to a positive-dimensional variety of general type. Alternatively, we can give up trying to ﬁnd an if-and-only-if statement and simply ask what sort of condition on the canonical bundle of X ensures that X has a Zariski dense collection of rational points over some ﬁnite extension of K: For example, we may make the following conjecture. Conjecture 1.3. Let X be a smooth, connected projective variety deﬁned over a number ﬁeld K. If the canonical bundle KX of X is negative (i.e., −KX is ample), then for some ﬁnite extension K of K the set X(K ) of K -rational points of X is Zariski dense. This conjecture is easily seen to be true for curves and surfaces, where the hypothesis ensures that X is rational. The ﬁrst real test cases are thus Fano 3-folds. In this paper, we examine the available evidence for this conjecture and add to it by analyzing one further class of Fano 3-folds, the smooth quartic hypersurfaces in P4 . Speciﬁcally, we prove the following theorem. Theorem 1.4. Let S ⊂ Pn be a smooth quartic hypersurface deﬁned over a number ﬁeld K. If n ≥ 4, then for some ﬁnite extension K of K the set S(K ) of K rational points of S is Zariski dense. In Section 7 we show that Theorem 1.4 follows as a straightforward corollary of a result about quartic surfaces. Theorem 1.5. Let S ⊂ P3 be a smooth quartic surface deﬁned over a number ﬁeld K, and let L be a line in P3 contained in S, likewise deﬁned over K. Then (a) for some ﬁnite extension K of K the set S(K ) of K -rational points of S is Zariski dense; and (b) if we assume further that L does not meet six or more other lines contained in S, then in fact the set S(K) of K-rational points of S is Zariski dense. Our proof of Theorem 1.5 is based on an analysis of the ﬁbration of S over P1 given by projection from the line L and on an analysis of the trisection of S → P1

RATIONAL POINTS ON QUARTICS

479

given by the points of L itself. For any plane H ⊂ P3 containing L, let CH ⊂ H be the cubic residual to L in the intersection of H with S. The key point in our argument has to do with the relation (or lack thereof) between the points p of intersection of the curves CH with L and the hyperplane class in Pic(CH ). The basic result, which we establish subject to various hypotheses in the following sections, states that, for all but ﬁnitely many H and any point p ∈ CH ∩ L, the classes of p and the line bundle ᏻCH (1) are linearly independent in Pic(CH ); that is, no multiple of the point p is linearly equivalent to any multiple of the hyperplane class on CH . This implies the desired density of rational points—it sufﬁces to pull back the elliptic ﬁbration to L. The new ﬁbration S → L has a section of inﬁnite order. By a result of Néron, rational points are Zariski dense on S and consequently on S. It is worth mentioning that Conjecture 1.3 is not the strongest possible converse to the weak Lang conjecture. It may well be that we do not need the canonical bundle to be negative—as Theorem 1.5 shows—but only nonpositive in a suitable sense. Thus, for example, we could make the following stronger conjecture. Conjecture 1.6. Let X be a smooth projective variety deﬁned over a number ﬁeld K. If the anticanonical bundle −KX of X is nef (e.g., −KX has nonnegative degree on every curve C ⊂ X), then for some ﬁnite extension K of K the set X(K ) of K -rational points of X is Zariski dense. The ﬁrst interesting test case for this conjecture is K3 surfaces. In fact, we prove it for a large class of K3 surfaces, but the question remains open for general K3 surfaces deﬁned over a number ﬁeld.1 We should also mention here the strong Lang conjecture. Conjecture 1.7 (Strong Lang conjecture). Let X be a variety deﬁned over a number ﬁeld K. If X is of general type, then there exists a proper subvariety X such that, for any ﬁnite extension K of K, #(X \ )(K ) < ∞; that is, the set of K -rational points of X lying outside of is ﬁnite. The converse to this statement seems plausible. For one thing, if φ : A → X is any nonconstant map from a rational or abelian variety to X, it is not hard to see that the image of φ has to be contained in the Langian exceptional subvariety . Moreover, as a consequence of the theorem of Kollár, Miyaoka, and Mori [21] and of Campana [7], we know that varieties that are not of general type admit a rational quotient, namely, a maximal ﬁbration X → Y , where the generic ﬁber of X → Y is rationally connected. The variety Y may admit a further ﬁbration with generic ﬁber 1 Building on ideas of the present paper, Bogomolov and the second author have recently proved Conjecture 1.6 for Enriques, elliptic K3 surfaces, and K3 surfaces with inﬁnite automorphism groups (see [4] and [5]). In particular, part (a) of Theorem 1.5 is now a special case of [5].

480

HARRIS AND TSCHINKEL

of Kodaira dimension zero (see [21], [7], and [11]). Thus the converse to the strong Lang conjecture hinges on whether a variety of Kodaira dimension zero possesses a dense collection of images of rational and/or abelian varieties. Since this is known for curves and surfaces, the converse to the strong Lang conjecture is likewise known for all curves and surfaces, and for all 3-folds except for Calabi-Yau 3-folds, which represent the ﬁrst real test. In general, however, it remains very much open. It is also worth mentioning that there are conjectures describing asymptotics for the number of rational points of bounded height. For example, let us consider a smooth quartic hypersurface in P4 . Then it is expected that the number of rational points, contained in some appropriate Zariski open subset and deﬁned over a sufﬁciently large ﬁnite extension of the ground ﬁeld of bounded height (induced from a standard height on P4 ), grows linearly with the height (cf. [13]). Acknowledgment. We are very grateful to the referee for many helpful comments. 2. Fano 3-folds. In this section we give a brief survey of known classiﬁcation and rationality results for Fano 3-folds over an algebraically closed ﬁeld of characteristic zero (cf. [25], [26], [3], and [23]). More details and references can be found in the recent book [19]. For our purposes it sufﬁces to consider minimal Fano 3-folds (not isomorphic to a blow-up of a Fano variety). The main invariants of Fano 3-folds are: r(X)—the index, deﬁned as the maximal r ∈ Z such that −KX = rL for some L in the Picard group Pic(X); ρ(X)—the rank of Pic(X) and the normalized degree δ(X) = (−KX )3 /r(X)3 . Group I: r ≥ 2, ρ = 1. (1) We have P3 (r = 4). (2) Q3 is the nonsingular quadric hypersurface in P4 (r = 3). The remaining ﬁve families have r = 2. They are indexed by δ. Let H be a line bundle such that |2H | = | − KV |. (3) We have that φH : V1 −→ P2 is a rational map with one indeterminacy point and with irreducible elliptic ﬁbers. V1 can be realized as a double cover of the Veronese cone in P6 whose branch locus is a smooth intersection of this cone and a cubic hypersurface not passing through the vertex of the cone. Another realization is as a hypersurface of degree 6 in the weighted projective space P(1, 1, 1, 2, 3). The general V1 is nonrational. Unirationality is unknown. (4) We have that φH : V2 −→ P3 is a double covering ramiﬁed along a smooth quartic surface. All are unirational, and the general V2 is nonrational.

RATIONAL POINTS ON QUARTICS

481

(5) We have that φH : V3 −→ P4 is a smooth cubic hypersurface. All are unirational, and all are nonrational. (6) We have that φH : V4 −→ P5 is a smooth intersection of two quadrics. All are rational. (7) V5 is birational to a smooth quadric Q3 . Group II: r = 1, ρ = 1. (1) We have that φ−KV : W2 −→ P3 is a double covering ramiﬁed along a smooth sextic surface. They are nonrational, and unirationality is unknown. (2) We have that φ−KV : W4 −→ P4 is a smooth quartic. All are nonrational, and some are unirational. In general, unirationality is unknown. (3) We have that φ−KV : W6 −→ P5 is a smooth complete intersection of a quadric and a cubic. All are unirational and nonrational. (4) We have that φ−KV : W8 −→ P6 is a smooth complete intersection of three quadrics. All are unirational and nonrational. (5) W10 are all unirational, and rationality is unknown: The general one is nonrational. Geometrically, it is a section of Gr(2, 5) in its Plücker embedding by a subspace of codimension 2 and a quadric. (6) W12 , W16 , W18 , and W22 are all rational. (7) W14 are birational to a cubic 3-fold. All are unirational and all are nonrational. Group III: ρ = 2, 3 Theorem 2.1 [26, p. 104]. If ρ(X) = 3, then X is a conic bundle over P1 × P1 and has either a horizontal divisor D P1 × P1 or another conic bundle structure over P1 × P1 . In particular, all varieties in this group are unirational. Nonrational varieties are pointed out in the following list. There are four types of minimal Fano 3-folds with ρ = 3: (1) double cover of P1 × P1 × P1 ramiﬁed in a (2, 2, 2)-divisor, all nonrational;

482

HARRIS AND TSCHINKEL

(2) smooth member of |L⊗2 ⊗ᏻP1 ×P1 ᏻ(2, 3)| on PP1 ×P1 (ᏻ ⊕ ᏻ(−1, −1)⊕2 ) such that X ∩ Y is irreducible (here L is the tautological line bundle and Y ∈ |L|); (3) P1 × P1 × P1 ; (4) PP1 ×P1 (ᏻ ⊕ ᏻ(1, 1)). The remaining varieties in this group have Picard number ρ = 2, and all are conic bundles over P2 : (5) double cover of P3∗ (i.e., P3 blown up in one point), ramiﬁed in a divisor in | − KP3∗ |, all nonrational; (6) double cover of P1 × P2 ramiﬁed in a (2, 2)-divisor; (7) double cover of P1 × P2 ramiﬁed in a (2, 4)-divisor, all nonrational; (8) hypersurface of bidegree (2, 2) in P2 × P2 , all nonrational; (9) hypersurface of bidegree (1, 2) in P2 × P2 ; (10) hypersurface of bidegree (1, 1) in P2 × P2 ; (11) P1 × P2 ; (12) PP2 (ᏻ ⊕ ᏻ(2)); (13) PP2 (ᏻ ⊕ ᏻ(1)). There are many more forms of Fano varieties over nonclosed ﬁelds. The main result of this paper together with the above classiﬁcation implies: If X is a Fano variety deﬁned over some number ﬁeld K such that X is not isomorphic over C to (a blow-up of) V1 or W2 , then there exists a ﬁnite extension K /K such that the set X(K ) is Zariski dense.2 One may ask for conditions that ensure unirationality. Already, for (minimal) Del Pezzo surfaces of degree 1 (i.e., smooth minimal over K surfaces S with ample anticanonical bundle and KS2 = 1), it is unknown whether or not rational points are Zariski dense. 3. A Chow ring calculation. In this section we work over C. We use the notation of the introduction: S ⊂ P3 is a smooth quartic surface containing a line L. We consider hyperplanes H ⊂ P3 passing through L, and we denote by CH the cubic curve residual to L in the intersection of H with S—that is, S · H = L + CH as divisors on H —and let DH = CH ∩L be the intersection of CH with the line L. Note that projection from the line L ⊂ P3 gives a regular map π from the surface S to the line M ∼ = P1 parametrizing planes through L, and note that the curves CH are simply the ﬁbers of this map. Note also that for any point p ∈ L, the point p lies in CH if and only if H is the tangent plane to S at p. Thus the restriction of the map π : S → M to L ⊂ S is simply the restriction to L of the Gauss map on S, mapping L onto the line in (P3 )∗ dual to L. Since the general plane H containing L is tangent to S at the three points of DH , this map has degree 3. The divisors DH are the ﬁbers of the restriction of this map, and so in particular the divisors DH form a linear system on L of degree 3. 2 The

case of V1 was treated in the recent paper [6].

RATIONAL POINTS ON QUARTICS

483

Similarly, for any point p ∈ L, let Tp S ⊂ P3 be the tangent plane to S at p, and let Cp = CTp S ⊂ Tp S be the cubic residual to L in the intersection of Tp S with S. Let Dp = DTp S = Cp ∩ L be the intersection with L; note that p ∈ Dp tautologously. We begin by establishing a weak form of our basic result. We show that if L meets no other line of S (in particular, all CH are irreducible), then for all but countably many H ⊃ L the points of CH ∩ L are not rationally related to ᏻCH (1) in Pic(CH ). The proof is a relatively elementary argument using a calculation in the Néron-Severi group of an associated surface. In the following section, we give a more reﬁned analysis, which allows us to conclude the same statement subject only to the weaker hypothesis that L does not meet six or more lines of S; while the present argument is superseded by that one, the argument here is useful for its (relative) simplicity and its applications to similar situations. (We see some of these in Section 8.) Theorem 3.1. Assume that no other lines lying on S meet L. For every positive integer n, there are only ﬁnitely many points p ∈ L(C), such that the classes of p and the line bundle ᏻCp (n) satisfy 3n · p ∼ ᏻCp (n). Proof. We begin by introducing a basic surface associated to this conﬁguration. The incidence correspondence T = (p, q) : q ∈ Cp ⊂ L × S. To see T more clearly, note ﬁrst that projection from the line L gives a regular map φ : S → M of S to the line M ∼ = P1 parametrizing the pencil of planes containing L; the curves CH are the ﬁbers of this map. Similarly, the divisors cut on L by the curves CH form a base-point-free pencil of degree 3 on L; the restriction φ = φ|L is the map associated to this pencil and correspondingly has degree 3. In these terms, the surface T is simply the ﬁber product T = L ×M S. In particular, T is a 3-sheeted cover of S, branched over the union of the ﬁbers CH of S → M such that CH is tangent to L. Note that the surface T need not always be smooth. It can be singular when some curve CH is simultaneously singular and not transversal to L. At worst, however, it has isolated singularities, since by the hypothesis that L meets no other lines lying on S, no curve CH can have a multiple component. Note that T → L has a tautologous section = (p, p) : p ∈ L ⊂ T ; this is just the intersection of T = L ×M S ⊂ S ×M S with the diagonal ⊂ S ×M S. As a Weil divisor, the pullback ν ∗ (L) of the line L under the 3-sheeted covering ν : T → S is thus a sum ν ∗ (L) = + R

484

HARRIS AND TSCHINKEL

with R ⊂ T ﬂat of degree 2 over L. Note that since the divisors DH form a basepoint-free linear series, all but ﬁnitely many divisors DH are reduced; in particular, R does not contain . The curve R is reducible if and only if the covering L → M is cyclic. Now, suppose that the conclusion of Theorem 3.1 is false; that is, for some n, we have inﬁnitely many p ∈ L(C) with Cp smooth and 3n · p ∼ ᏻCp (n). Fixing a plane # ⊂ P3 , there is thus for inﬁnitely many p a rational function on Cp with a pole of order 3n at p and with zeroes of order n at the points of intersection # ∩ Cp , and this rational function on Cp is nonzero and regular everywhere else. It follows in turn that there is a rational function f on T with divisor (f ) = −3n · + n · ν ∗ # + D, where D is supported on a ﬁnite union of ﬁbers of T → L. Since the hypothesis that L meets no other line of S ensures that all ﬁbers of T → L are irreducible, D must consist of a sum of ﬁbers Cp of T → L. Since all ﬁbers of T → L are linearly equivalent, Theorem 3.1 thus follows from the next lemma. Lemma 3.2. The classes σ , γ , and φ ∈ A1 (T ) of the divisors , ν ∗ #, and C are independent in the group A1 (T ) of Weil divisors modulo linear equivalence on T . Proof. We need to begin with a basic fact (due to Mumford), whose proof is mapped out in Fulton [14, Examples 7.1.16 and 8.3.11]. Lemma 3.3. Let T be a reduced, irreducible, normal, and projective surface. We may deﬁne, for every point p ∈ T and Weil divisors D, E ∈ Z1 (T ) whose supports have no common component in a neighborhood of p, an intersection multiplicity j (p, D · E) ∈ Q, bilinear in D and E, and a bilinear intersection pairing (· , · ) : A1 (T ) × A1 (T ) −→ Q on the group A1 (T ) of Weil divisors on T modulo rational equivalence, with the following properties. (1) If D and E are effective and both contain p, then j (p, D · E) > 0. (2) If D is locally principal at p, that is, D = (f ) for some rational function f in a neighborhood of p, and E is effective and irreducible, then j (p, D · E) = ordp (f |E ). (3) If D and E have no common components, then ([D] · [E]) = j (p, D · E). p∈D∩E

RATIONAL POINTS ON QUARTICS

485

We may now establish Lemma 3.2 by calculating the matrix of intersection numbers of the classes σ , γ , and φ ∈ A1 (T ) and by showing that this matrix is nonsingular. All but one of these numbers are readily calculated. To begin with, φ is the class of a ﬁber of the map T → L, so of course φ 2 = 0; inasmuch as σ is the class of a section of that map, we have (φ · σ ) = 1. Next, γ is the pullback of the hyperplane class under the map ν : T → S *→ P3 ; since the map T → S has degree 3, we have γ 2 = 3 · deg(S) = 12. Since the curves Cp map forward to plane cubics under the map ν, moreover, we have (γ · φ) = 3, and similarly, since the curve maps one-to-one onto the line L ⊂ S, (γ · σ ) = 1. In sum, then, we have Table 1. Table 1 Intersection Numbers γ

φ

σ

γ

12

3

1

φ

3

0

1

σ

1

1

σ2

The only mystery is the self-intersection σ 2 of the curve on T . To ﬁnd this, we use a relation of linear equivalence between and a curve not containing . As we saw above, ν ∗ L = + R, so that if ρ = [R] ∈ A1 (T ) is the class of R, we have σ 2 = (σ · [ν ∗ L] − ρ). Now, by the projection formula (which is easily checked on some desingularization of the surface), ( · ν ∗ L)T = (ν∗ · L)S = (L · L)S = −2, and so σ 2 = −2 − (σ · ρ).

486

HARRIS AND TSCHINKEL

Alternatively, if we choose the plane # ⊂ P3 to contain L, we see that the inverse image in T consists of , R, and the three ﬁbers of the map T → L over the points of intersection of C# with L. Thus γ = σ + ρ + 3φ and

σ 2 = σ · γ − 3φ − ρ = −2 − (σ · ρ).

At this point, we may readily complete the calculation for smooth T —the curves and R intersect transversely over the points p, where Cp is tangent to L at p. The curves CH cut out a pencil of degree 3 on L, which by Riemann and Hurwitz have four branch points; thus ( · R) = 4. For arbitrary S and L, however, T may be singular at the points of intersection of with R—it is so exactly when a curve CH has a singularity at a point of L—and we can no longer say precisely what the intersection multiplicity is. All we do know, in fact, is that must meet R somewhere, so that ( · R) > 0 and, correspondingly, σ 2 < −2. Now, we may calculate the determinant of the matrix of pairwise intersection of the classes σ , γ , and φ. It is −12 + 3 − 9σ 2 + 3 = −6 − 9σ 2 > 0. The matrix is thus nonsingular, the classes σ , γ , and φ are independent, and Lemma 3.2 is proved. Thus, Theorem 3.1 is also proved. 4. The argument via monodromy. We continue to work over C. A ﬁner analysis of the ﬁbration π : S → M ∼ = P1 , speciﬁcally of the monodromy of the family on the torsion points in the Jacobians of ﬁbers, yields a stronger result. 4.1. Generalities about quartic surfaces with a line. To begin, we recall some basic facts about the ﬁbration S → M. First, since S is smooth, the Gauss map Ᏻ : S → (P3 )∗ is regular and, hence, is ﬁnite. It follows that no plane H ⊂ P3 can be tangent to S along a curve; in other words, every hyperplane section of S is reduced. The same is thus true of the ﬁbers CH of the ﬁbration π : S → M. We may thus list the possible singular ﬁbers of π. They are • a cubic with one, two, or three nodes, that is, an irreducible nodal curve, the union of a line and a conic meeting transversely, or the union of three nonconcurrent lines, called ﬁbers of type Ib with b = 1, 2, or 3, respectively; • a cuspidal cubic, called a ﬁber of type II; • the union of a line and a tangent conic, called a ﬁber of type III; or

RATIONAL POINTS ON QUARTICS

487

• the union of three concurrent lines, called a ﬁber of type IV. Note that ﬁbers of type Ib correspond to poles of order b of the j -function on M associated to the elliptic ﬁbration S → M. By contrast, ﬁbers of type II and IV correspond to zeroes of j —after base changes of order 6 and 3, respectively, we may replace the singular curve by an elliptic curve C˜ of j -invariant zero—and ﬁbers of type III correspond to points where j = 1728. After a base change of order 4 we may replace the singular curve by an elliptic curve C˜ of j -invariant 1728. The monodromy action on the homology of the smooth ﬁbers around a ﬁber of type Ib is thus given by the Picard-Lefschetz transformation, while in the case of ﬁbers of type II, III, and IV the monodromy is just the action of the automorphism given by ˜ We list here these actions in Table 2; see the original paper of the base change on C. Kodaira or the discussion in Barth, Peters, and van de Ven for details (cf. [20], resp., [1, pp. 150–160]). Table 2 Type Ib II

III

Monodromy 1 b MIb = 0 1

1 1 MII = −1 0

0 1 MIII = −1 0

IV

0 1 MIV = −1 −1

There is one constraint on the number of singular ﬁbers CH . By a standard Euler characteristic calculation, χ(S) = 24 = χ(CH ). H ∈M

The Euler characteristics of ﬁbers of type Ib , II, III, and IV are b (with b = 1, 2, or 3 in our case), 2, 3, and 4, respectively, giving a linear relation on the numbers of ﬁbers of each type. In particular, we see that there must be at least six singular ﬁbers CH . A related issue is the count of curves CH that are not transverse to L. Given that the divisors DH cut out on L by the curves CH form a pencil of degree 3, it follows by Riemann and Hurwitz that there must be a total of four branch points, counting multiplicity; that is, either two curves CH having a point of intersection multiplicity

488

HARRIS AND TSCHINKEL

3 with L, one such curve and two others having a double point of intersection with L, or four curves CH having a double point of intersection with L. If the points p of CH ∩ L differ from each other by torsion in Pic(CH ) for almost all H , then since sections of an elliptic ﬁbration that differ by torsion in the generic ﬁber can intersect only at singular points of ﬁbers, the multiple point p of intersection of CH0 with L must be a singular point of CH0 . On the other hand, since there can be at most four ﬁbers CH having multiple points of intersection with L, and each can contribute at most 4 to the Euler characteristic, we may draw one conclusion in particular that turns out to be vital to the following analysis: There must be singular ﬁbers CH that intersect L transversely. 4.2. Analysis of the points of CH ∩ L Theorem 4.1. Let S ⊂ P3 be a smooth quartic surface and L ⊂ S a line in P3 contained in S; assume that L does not meet six or more other lines contained in S. Let n be any positive integer. For all but ﬁnitely many p ∈ L(C), 3n · p ∼ ᏻCp (n). The rest of this section is devoted to the proof of this fact. Let H be a plane containing L, and let p1 , p2 , and p3 be the three points on intersection of CH with L. Assume that some multiple of pi is linearly equivalent to a multiple of the hyperplane section of CH , and let n be the smallest positive integer such that 3n · pi ∼ ᏻCH (n). Note that since the monodromy on the three points pi as H varies is at least transitive, this hypothesis holds for one pi if and only if it holds for all three, and the value of n is the same for all three. In this case, the pairwise differences αi,j = pi − pj ∈ Pic0 (CH ),

i = j

are torsion, we let m be their order (the curve CH can be singular). Again, since the monodromy is transitive on the three pairs {±αi,j } = {αi,j , αj,i }, the value of m is the same for all i and j . Note that the classes αi,j are all nonzero, but they need not be distinct. If m = 2, of course, we have αi,j = αj,i ; while if m = 3, we could have α1,2 = α2,3 = α3,1 . If m > 3, however, we can see from the transitivity of the monodromy and the fact that α1,2 + α2,3 + α3,1 = 0 that they must be distinct. Note ﬁnally that if m = 2, the monodromy on the points pi must be cyclic, rather than the symmetric group S3 . A transformation ﬁxing p1 , for example, and exchanging p2 and p3 , would exchange α1,2 and α1,3 and send α2,3 to −α2,3 ; given that α1,2 − α1,3 + α2,3 = 0, this implies 2α2,3 = 0. It follows, in particular, that in case m = 2, there are exactly two ﬁbers CH not transverse to L, and each has a triple point of intersection with L. To carry out the further analysis of the monodromy action on the points pi and the classes αi,j , we consider in turn three potential cases: m > 3, m = 3, and m = 2.

RATIONAL POINTS ON QUARTICS

489

Case a: m > 3 Lemma 4.2. Every ﬁber not transverse to L must be of type IV. Proof. For each such ﬁber, the subgroup G ⊂ Pic0 (CH ) spanned by (any one of) the αi,j is an eigenspace for the action of the monodromy on the points of order m in Pic0 (CH ). The monodromy action MIb associated to a singular ﬁber of type Ib has only one eigenspace, with eigenvalue 1. Since the monodromy on the points pi is necessarily nontrivial and cyclic, and since m = 3, it follows that the action on the classes αi,j is nontrivial. Thus a ﬁber of type Ib singular at a point of L cannot occur. Next, suppose that we have a singular ﬁber of type II. The monodromy MII has the characteristic polynomial p(λ) = λ2 − λ + 1. Suppose λ is a root of this polynomial modulo m. Then we have λ2 ≡ λ − 1 mod m and hence λ3 ≡ λ2 − λ ≡ −1

mod m.

Alternatively, we can just multiply out and see that MII3 = −1. Either way, we see that MII3 cannot ﬁx any element of Pic0 (CH ) of order m = 2, and so no ﬁber of this type can occur. A similar analysis shows that no ﬁber of type III can occur (or we could just observe that the union of a line and a tangent conic cannot have a point of intersection multiplicity 3 with L). This concludes the proof of the lemma. By the discussion above, each ﬁber that is not transverse to L has a triple point of intersection with L and (since the classes αi,j are torsion) this triple point of intersection has to be the singular point of the ﬁber. Therefore, L intersects at least six other lines on S. We do not know of any examples of this case. Case b: m = 3. The same analysis shows that the ﬁbers CH not transverse to L must both be of type IV. None of the other transformations MIb , MII , and MIII has a 3-cycle on the points of order 3. Now we turn to ﬁbers that are transverse to L. Suppose that the classes αi,j do not lie in a cyclic subgroup (for almost all ﬁbers). This could be the case if every singular ﬁber CH transverse to L is of type I3 . But if there were δ such ﬁbers, the formula for the Euler characteristic gives 24 = 2 · 4 + δ · 3, which has no solution. Thus the classes αi,j lie in a cyclic subgroup, which means that we must have α1,2 = α2,3 = α3,1 . This implies that 3pi ∼ p1 + p2 + p3 ∼ ᏻCH (1); that is, n = 1, or, in other words, all three points pi are ﬂexes of CH for almost all planes H . Again, we do not know if this is possible. In any event, we see again that this case can occur only when six or more other lines of S meet L.

490

HARRIS AND TSCHINKEL

Case c: m = 2. This case gives us the least amount of control over the behavior of the singular ﬁbers CH not transverse to L but happily the most over the behavior of those that are. Very simply, in this case the classes αi,j cannot lie in a cyclic subgroup. They must comprise all three classes of order 2 in Pic0 (CH ). It follows that the monodromy associated to each ﬁber CH transverse to L must be trivial on the points of order 2 in Pic0 (CH ), which means that every singular ﬁber CH transverse to L must be of type I2 ; that is, they must consist of the union of a line and a conic. Now, as far as we can tell, the ﬁbers CH not transverse to L can be of type II, III, or IV, but in any event the total contribution of such ﬁbers to the Euler characteristic can be at most eight (if we have either four ﬁbers of type III or two of type IV). The remaining singular ﬁbers must therefore contribute at least 16 to the Euler characteristic, which means we must have at least eight ﬁbers of type I2 ; in particular, L must intersect at least eight other lines of S. 5. Rational points on quartic surfaces. In this section we work over a number ﬁeld K (and we ﬁx an embedding of Q into C). We assume that S and L are deﬁned over K. To deduce part (b) of Theorem 1.5 from the analysis carried out so far, we need one more ingredient. Brieﬂy, Theorems 3.1 and 4.1 assure us (subject to their hypotheses) that for a very general point—that is, for all but countably many points p ∈ L(C)—the point p ∈ Cp (C) is not rationally related to the hyperplane class in Pic(Cp ). If such a point p lies in L(K), the cubic curve Cp has positive rank over K and, hence, a dense set of rational points. It seems reasonable to expect that for “most” of the points p ∈ L(K) this would be true. But there are countably many points p ∈ L(Q) for which p is rationally related to ᏻCp (1)—for each n, there is a ﬁnite subset 2n ⊂ L(Q) such that 3n · p ∼ ᏻCp (n) in Pic0 (Cp )—and it is still a logical possibility, if not a plausible one, that all the points of L(K) lie in the union of these sets. There are two ways of eliminating this possibility. The ﬁrst is to invoke an extremely powerful theorem due to Merel [24]. Theorem 5.1 (Merel). Let K be any number ﬁeld. There is an integer n0 = n0 (K) such that no elliptic curve deﬁned over K has a K-rational point of order n > n0 . This theorem assures us that for n > n0 , the subset 2n is disjoint from L(K); so that for all but ﬁnitely many p ∈ L(K), the point p is not rationally related to ᏻCp (1) in Pic(Cp ). A second way to arrive at this fact uses a theorem that is easier to prove, if less simple to state. Suppose that E → B is any family of elliptic curves and σ a section of E → B, both deﬁned over a number ﬁeld K. For each point t ∈ B, let hB (t) be the height of t (relative to any divisor of degree 1 on B), let h(σt ) be the canonical height of the value σt of the section in the ﬁber Et of the family over t, and let h(σ ) be the canonical height of the section σ in the ﬁber of E over the generic point

RATIONAL POINTS ON QUARTICS

491

of B. We have then the following theorem of Dem’janenko [12], Manin [22], and Silverman [31, Th. B, p. 197]. Theorem 5.2. We have lim

hB (t)→∞

h(σt ) = h(σ ). hB (t)

In particular, to say that σ is not a torsion point in the ﬁber of E over the generic point η of B is to say that h(σ ) > 0, and we can deduce the following corollary. Corollary 5.3. If ση is not a torsion point in the ﬁber Eη of E over the generic point η ∈ B, then there are only ﬁnitely many points t ∈ B(K) such that σt is a torsion point in Et . Applying this to the family T → L of elliptic curves introduced in Section 3 (with the origin given by the tautologous section and with the section σ given by the divisor class ᏻCp (1) in each ﬁber), we deduce again that (subject to the hypotheses of Theorem 4.1) for all but ﬁnitely many p ∈ L(K) the point p is not rationally related to ᏻCp (1) in Pic(Cp ). In fact, a weaker version, due to Néron, sufﬁces—he proved that there are inﬁnitely many points in B(K) with ﬁber containing inﬁnitely many K-rational points (see Serre’s book [30, p. 153]). Now, S ⊂ P3 is a smooth quartic surface, and L ⊂ S is a line in P3 contained in S (both deﬁned over K); assume that L does not meet six or more other lines contained in S (not necessarily deﬁned over K). For each point p ∈ L(K) and for each integer n there are unique points qn and rn ∈ Cp such that qn + (3n − 1) · p ∼ ᏻCp (n) and −rn + (3n + 1) · p ∼ ᏻCp (n), also deﬁned over K. Moreover, for all but ﬁnitely many p ∈ L(K) we have 3n · p ∼ ᏻCp (n) for every n, so that these points are all distinct. We have, accordingly, an inﬁnite collection of K-rational points on Cp , so that Cp is contained in the Zariski closure of S(K); since this is true of inﬁnitely many curves Cp , it follows that the Zariski closure of S(K) is all of S. As for the proof of part (a) of Theorem 1.5, given part (b) this requires only one further trick, and it is a relatively simple one. Lemma 5.4. Let S ⊂ P3 be a smooth quartic surface and L, L , and L ⊂ S three lines in P3 contained in S; assume that L does not meet either L or L , but that L and L do meet. For each plane H containing L, let qH = CH ∩L and rH = CH ∩L .

492

HARRIS AND TSCHINKEL

For all but ﬁnitely many H (deﬁned over C) containing L, the difference qH − rH is not torsion in Pic0 (CH ). Proof. This is easy. Note ﬁrst that since L does not meet L and L , no plane H containing L can be the tangent plane to S at any point of L or L ; in other words, qH and rH are smooth points of the curve CH for all H . Now let T be as in the proof of Theorem 3.1. The assignment to each point p ∈ L of the points qp and rp ∈ Cp gives sections of the map T → L, which we claim do not differ by a translation of ﬁnite order in the ﬁber Cp (for almost all p). But two sections of an elliptic ﬁbration that differ by torsion in the generic ﬁber can intersect only at singular points of ﬁbers. This concludes the proof. Proof of part (a) of Theorem 1.5. Now suppose we have a quartic surface S and a line L ⊂ S. If fewer than six other lines of S meet L, we may apply part (b) of Theorem 1.5 to conclude that the points of S rational over the ﬁeld of the deﬁnition of L are Zariski dense. Suppose conversely that every line of S meets at least six other lines of S. We claim in that case that S must contain a conﬁguration of lines as in Lemma 5.4. To see this, start with any line L0 ⊂ S. Since no more than four lines on S can pass through a single point of S, we have to consider only two possibilities. Case 1: Three pairwise skew lines L1 , L2 , and L3 ⊂ S meet L0 . In this case, let M1 , . . . , M5 ⊂ S be ﬁve other lines meeting L1 (in addition to L0 ). If any one of them fails to meet both L2 and L3 , we are done. If Mi fails to meet Lj , we take for our conﬁguration L = Lj , L = L1 , and L = Mi . Otherwise, assuming that all six lines L0 , M1 , . . . , M5 meet all three lines L1 , L2 , and L3 , we see that all of them have to lie on the unique quadric surface Q ⊂ P3 containing L1 , L2 , and L3 —but then we have nine lines in Q ∩ S, contradicting Bézout (among others). Case 2: Two triples of concurrent lines {L1 , L2 , L3 } and {M1 , M2 , M3 } meet L0 . This is even easier. Just let N be any line meeting L1 and skew to L0 . N cannot meet all three lines M1 , M2 , and M3 . (It is coplanar with them and hence meets L0 .) If it misses Mi , we take L = Mi , L = L1 , and L = N . To complete the proof of Theorem 1.5, suppose now that L, L , and L ⊂ S are a conﬁguration as in Lemma 5.4; let K be any ﬁeld over which S and all three lines are deﬁned. For each plane H containing L and for each integer n, there is a unique point xn ∈ CH such that xn + n · qH ∼ (n + 1) · rH , also deﬁned over K . Moreover, by Merel’s theorem and Lemma 5.4 (or also by Silverman’s theorem), for all but ﬁnitely many H these points are all distinct. We thus have for inﬁnitely many H an inﬁnite collection of K -rational points on CH , and once more it follows that the Zariski closure of S(K ) is all of S. 6. An example: The Fermat quartic. In light of the analysis above, it might seem unlikely that there is any quartic surface S and line L such that, for all planes

RATIONAL POINTS ON QUARTICS

493

H deﬁned over C and containing L, the C-points of CH ∩ L are all rationally related to the hyperplane class in Pic(CH ). In fact, however, it does occur. We describe here the unique example we know of, the Fermat surface. See also the analysis of Piatetski-Shapiro and Shafarevitch [29]. To begin with, we take S ⊂ P3 the quartic given by the equation X4 − Y 4 + Z 4 − W 4 = 0 and L the line given by the equations X=Y

and

Z = W.

Any plane containing L, other than the plane Z = W , can be realized as the span of L and a third point of the form [a, −a, 1, −1] for some scalar a, that is, the plane given parametrically by [U, V , T ] −→ U + aT , U − aT , V + T , V − T . Restricting the equation of S to H gives the equation (U +aT )4 −(U −aT )4 +(V +T )4 −(V −T )4 = 8aU 3 T +8a 3 U T 3 +8V 3 T +8V T 3 , so the equation of CH is simply Fa (U, V , T ) = aU 3 + a 3 U T 2 + V 3 + V T 2 = 0. The points of CH ∩ L are given by the further equation T = 0; that is, they are the points [U, V , T ] = [1, b, 0], where b3 = a. Note that the monodromy on these as a varies is cyclic. What are the singular ﬁbers CH of the ﬁbration S → M? The plane Z = W corresponding to a = ∞ is certainly one, consisting of three concurrent lines meeting a point of L, and the plane X = Y corresponding to a = 0 is another singular ﬁber of type IV. To ﬁnd the remaining ones we have simply to write out the partial derivatives of Fa (U, V , T ) and equate them all to zero. The equations 3aU 2 + a 3 T 2 = 3V 2 + T 2 = 2V T + 2a 3 U T = 0 together imply that a 4 = ±1 and that 1 U=√ T −3

and

a V = √ T. −3

There are thus eight singular ﬁbers apart from a = 0 and a = ∞, each having two singular points (that is, consisting of a line and conic).

494

HARRIS AND TSCHINKEL

We now claim that the three points pi of the intersection of CH with L differ by torsion of order 2 and that each satisﬁes 6 · pi ∼ ᏻCH (2). The second statement follows from the ﬁrst, given that p1 + p2 + p3 ∼ ᏻCH (1), and the ﬁrst is readily checked. We simply observe that two points p and q on a plane cubic curve C differ by torsion of order 2 if and only if the point of intersection Tp C ∩Tq C of the tangent lines to C at p and q lies on C. Now, the equation of the tangent line to CH at a point [µ, ν, τ ] is 3aµ2 + a 3 τ 2 · U + 3ν 2 + τ 2 · V + 2ντ + 2a 3 µτ · T = 0, and at the point [1, b, 0] for some cube root b of a this is −3b3 · U + 3b2 · V = 0. For any two distinct cube roots b of a, the resulting linear forms in U and V are independent, so that the point of intersection of any two of the tangent lines to CH at the points of CH ∩ L is just [U, V , T ] = [0, 0, 1], which is a point of CH . Again, we do not know of any other examples of a quartic surface S and a line L ⊂ S such that, for almost all planes H containing L, the points of CH ∩ L are all rationally related to the hyperplane class in Pic(CH ); nor do we know any examples at all where the points differ from each other by torsion of order greater than 2. 7. Quartic 3-folds. In this section “general” means outside a Zariski closed subset. We now use Theorem 1.5 to deduce Theorem 1.4. This is relatively simple. We just have to check that if X ⊂ Pn is any smooth quartic hypersurface and L ⊂ X any line, then for a general 3-plane P3 ⊂ Pn containing L, the surface S = X ∩ P3 and the line L satisfy the hypotheses of part (b) of Theorem 1.5. It is enough to do this in case n = 4, and it requires only a straightforward geometric argument. We start with a basic fact. Lemma 7.1. If X ⊂ P4 is a smooth quartic hypersurface, the Fano variety F1 (X) ⊂ G(1, 4) of lines on X has pure dimension 1. Proof. To begin with, the homogeneous quartic polynomial F ∈ Sym5 (C5 ) on P4 deﬁning X gives rise to a section τF of the ﬁfth symmetric power Sym5 (S ∗ ) of the dual of the universal subbundle S on G(1, 4), and the zero locus of this section is the Fano scheme F1 (X). This shows that it has dimension at least 1 everywhere and, since the top Chern class c5 (Sym5 S ∗ ) = 320σ3,2 = 0, that it is nonempty; it remains only to see that it cannot be 2- or higher-dimensional. To do this, let 2 = P(S|F1 (X) ) = {(L, p) : p ∈ L} ⊂ F1 (X) × X

RATIONAL POINTS ON QUARTICS

495

be the universal projective line bundle over the Fano variety F1 (X) ⊂ G(1, 4), and let ρ : 2 → X ⊂ P4 be the projection map. The tangent space to F1 (X) at a point L ∈ F1 (X) may be identiﬁed with the space of sections H 0 (L, NL/X ). In these terms, at all sufﬁciently general points (L, p) ∈ 2, the image of the differential dρ(L,p) : TL F1 (X) → Tp X mod Tp L is simply the image of the map H 0 (L, NL/X ) → (NL/X )p = Tp X/Tp L given by evaluation at p. Since NL/X has exactly one summand of nonnegative degree, this image is always 1-dimensional (mod Tp L), and so we may conclude that the image of the map ρ : 2 → X—that is, the union of the lines on X—is always exactly 2-dimensional. But X contains no 2-planes, and no surface in projective space other than a 2-plane may contain ∞2 lines, so we may conclude that X contains only ∞1 lines. On the basis of a naive dimension count, we expect the map ρ from the 2-dimensional variety 2 to the 3-fold X to have a 1-dimensional double point locus, that is, ∞1 pairs of distinct lines L, L ⊂ X that meet. We thus expect that each line of X meets ﬁnitely many others. (We calculate the number in just a moment.) Accordingly, we call a line L ⊂ X exceptional if it meets inﬁnitely many other lines of X. We denote by F1e (X) ⊂ F1 (X) the locus of exceptional lines. By a straightforward dimension count, a general quartic 3-fold X has no exceptional lines, and a general X containing an exceptional line contains only ﬁnitely many; in particular, it contains nonexceptional lines as well. Our situation is that we are able to apply Theorem 1.5 directly to the surface S = H ∩X, where H is a general hyperplane section containing a nonexceptional line L, and so our concern is whether an arbitrary X may contain only exceptional lines. In fact, this is possible; the Fermat quartic is one example (we do not know any other examples). What we want to do, accordingly, is to say as much as we can about quartic 3-folds that have positive-dimensional families of exceptional lines. So, let X be a smooth quartic 3-fold, and let A ⊂ F1e (X) be an irreducible component of the Fano variety of lines on X consisting entirely of exceptional lines. For each line L ∈ A, the locus of lines meeting L contains one or more irreducible components of F1 (X), and so we must have one of the following two situations: (1) there are two irreducible components A, A ⊂ F1 (X) such that every pair of lines L ∈ A and L ∈ A meet; or (2) there is an irreducible component A ⊂ F1 (X) such that every pair of lines L, L ∈ A meet. In the ﬁrst case, the surface S ⊂ X swept out by the lines of A has two rulings by lines; but the only surface in projective space with two rulings by lines is a quadric surface in P3 , and since Pic(X) = Z"ᏻX (1)#, X contains no such surfaces; thus the ﬁrst case cannot occur. In the second case, let L, L ∈ A be two general lines, meeting at a point p. A third general line L ∈ A must meet both L and L ; if it does not pass through p it must lie in the plane spanned by L and L , and so this plane would have to be contained in X. Since X contains no 2-planes, we conclude that all the

496

HARRIS AND TSCHINKEL

lines L ∈ A have a common point p. It follows, in particular, that all the lines L ∈ A are contained in the tangent plane Tp X and, hence, that X ∩ Tp X is simply the cone with vertex p over an irreducible plane quartic curve. Finally, since X is smooth, the Gauss map Ᏻ cannot be constant on any curve of X. Thus X ∩ Tp X can have at most isolated singularities, and hence the curve C must be smooth. We have thus established the following proposition. Proposition 7.2. Let X be a smooth quartic 3-fold, and let A ⊂ F1e (X) be an irreducible component of the Fano variety of lines on X consisting entirely of exceptional lines. Then there is a point p ∈ X such that X ∩ Tp X is the cone with vertex p over a smooth plane quartic curve, and A is simply the ruling of this cone. In particular, A ⊂ G(1, 4) ⊂ P9 has as underlying reduced scheme a smooth plane quartic curve. It is a nice exercise to check directly that in this case the component A of the Fano scheme F1 (X) has multiplicity 2. Since by the proof of Lemma 7.1 the Fano scheme is a curve of degree 320 in G(1, 4) ⊂ P9 , it follows that a quartic 3-fold X contains only exceptional lines if and only if it has exactly 40 hyperplane sections consisting of cones over quartic plane curves. Again, this is the case for the Fermat quartic; we do not know if there are others. In any event, Theorem 1.4 now follows readily from Theorem 1.5. Let X ⊂ P4 be any quartic 3-fold deﬁned over a ﬁeld K. Suppose ﬁrst that X contains a nonexceptional line L; say, L is deﬁned over a ﬁeld K ⊃ K. Then for a general hyperplane H∼ = P3 ⊂ P4 containing L, the surface SH = X ∩H contains no other lines meeting L, and by part (b) of Theorem 1.5 we may deduce that SH (K ) is Zariski dense in SH ; hence X(K ) is Zariski dense in X. If on the other hand every line of X is exceptional, then by Proposition 7.2 the Fano variety of lines on X consists of a union of curves supported on plane quartic curves. If L ⊂ X is a line corresponding to a general point [L] ∈ A ⊂ F1 (X) of a component A of F1 (X)—in particular, if it is not a point of intersection of A with another component of F1 (X)—then L can meet only ﬁnitely many lines L ⊂ X corresponding to points [L ] ∈ / A. It follows that, for a general hyperplane H ∼ = P3 ⊂ P4 containing L, the surface SH = X ∩ H contains exactly three other lines meeting L. So once again the hypothesis of part (b) of Theorem 1.5 is satisﬁed, and we conclude that the points of X rational over the ﬁeld of deﬁnition of L are Zariski dense. 8. Other elliptic surfaces. As suggested in Section 3.1, the approach via the calculation of intersection numbers in the Néron-Severi group of an associated surface works in substantially greater generality. Speciﬁcally, we prove the following theorem. Theorem 8.1. Let S be any smooth irrational surface deﬁned over a number ﬁeld K, and let π : S → P1 be an elliptic ﬁbration (over K) all of whose ﬁbers are irreducible; for λ ∈ P1 (K), let Eλ = π −1 (λ) be the ﬁber of S → P1 over λ. Let

RATIONAL POINTS ON QUARTICS

497

C ⊂ S be any smooth rational or elliptic curve of degree m ≥ 2 over P1 and of n = 0 any integer. Then for all but ﬁnitely many λ ∈ π(C(K)), nm · p ∼ ᏻEλ ⊗ ᏻS (n · C), for all p ∈ (Eλ ∩ C)(K). As before, we may immediately deduce from this the following corollary. Corollary 8.2. Let S be any smooth surface deﬁned over a number ﬁeld K, and let π : S → P1 be an elliptic ﬁbration with irreducible ﬁbers as in Theorem 8.1. (a) Let C ⊂ S be a smooth rational curve. Assume that S, π, and C are deﬁned over a ﬁeld K and that C is rational over K. Then the set S(K) of K-rational points of S is Zariski dense. (b) Now let C ⊂ S be a smooth curve of genus 1, and assume that S, π, and C are deﬁned over a ﬁeld K. Then there is a ﬁnite extension K of K such that the set S(K ) of K -rational points of S is Zariski dense. Part (a) of Corollary 8.2 follows from Theorem 8.1 as before. We see that for all but ﬁnitely many points p ∈ C(K), the ﬁber of S → P1 over π(p) has inﬁnitely many rational points. As for part (b), we have to make an extension of our ground ﬁeld K simply to ensure that the curve C has inﬁnitely many rational points, and then the argument proceeds as before. Of course we can drop the hypothesis in Theorem 8.1 that S is irrational. Proof of Theorem 8.1. The proof is analogous to that of Theorem 3.1. We begin by making a base change. We let T be the incidence correspondence T = (p, q) : q ∈ Eπ(p) ⊂ C × S. As before, T is the ﬁber product T = C ×P1 S. In particular, T is an m-sheeted cover of S, branched over the union of the ﬁbers Eλ of S → P1 such that Eλ is tangent to C. Again, T has, at worst, isolated singularities, since by the hypothesis that S is smooth and all ﬁbers of π are irreducible it follows that all ﬁbers are reduced as well. The surface T is also normal since it is regular in codimension 1 and since it is locally deﬁned by one equation in the smooth irreducible variety C × X. Note that T → C has a tautologous section = (p, p) : p ∈ C ⊂ T . As a divisor, the pullback ν ∗ (C) of the curve C under the m-sheeted covering ν : T → S is thus a sum ν ∗ (C) = + R

498

HARRIS AND TSCHINKEL

with R ⊂ T ﬂat of degree m − 1 over C. As before, since all but ﬁnitely many ﬁbers of C over P1 are reduced, R does not contain . Now let φ ∈ A1 (T ) be the class of a ﬁber of T → C, σ the class of the section , and ρ the class of R. The key ingredient in our proof is the following lemma. Lemma 8.3. The classes σ , ρ, and φ ∈ A1 (T ) are independent in the group A1 (T ) of Weil divisors modulo linear equivalence on T . Proof. We calculate the matrix of intersection numbers of the classes σ , ρ, and φ ∈ A1 (T ). (The intersections are deﬁned even for singular T .) Three of these numbers are readily calculated. To begin with, φ is the class of a ﬁber of the map T → L, so of course φ 2 = 0; and since and R meet each ﬁber in 1 and m−1 points, respectively, we have (φ · σ ) = 1 and (φ · ρ) = m − 1. As before, we do not know anything about c = (σ · ρ) except that it is positive. (If T were smooth, it would have to be 2m − 2.) Finally, let −b be the self-intersection of the curve C on S. By the hypothesis that S is irrational, the canonical class KS is a nonnegative (rational) multiple of the class of a ﬁber of S → P1 and so has nonnegative intersection with C. It follows that b ≥ 2 if C is rational, and g ≥ 0 if C has genus 1. Table 3 Intersection Products φ

σ

ρ

φ

0

1

m−1

σ

1

−b − c

c

ρ

m−1

c

−(m − 1)b − c

Now, to calculate σ 2 and ρ 2 , we use the relation ν ∗ C = + R. It follows that

σ 2 = σ · [ν ∗ C] − ρ ,

and since, by the projection formula (which can be checked on some desingularization), ( · ν ∗ C)T = (ν∗ · C)S = (C · C)S = −b, we have σ 2 = −b − c. Similarly, from the same relation it follows that ρ 2 = ρ · [ν ∗ C] − σ .

RATIONAL POINTS ON QUARTICS

499

By the projection formula, (R · ν ∗ C)T = (ν∗ R · C)S = (m − 1)C · C S = −(m − 1)b and so ρ 2 = −(m − 1)b − c. In sum, then, we have Table 3. The determinant of this matrix is c(m − 1) + (m − 1)b + c + c(m − 1) + (b + c)(m − 1)2 , and since b is nonnegative and c positive, we are done. This concludes the proof of Theorem 8.1 and Lemma 8.3. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

[13] [14] [15] [16] [17] [18]

W. Barth, C. Peters, and A. van de Ven, Compact Complex Surfaces, Ergeb. Math. Grenzgeb. (3) 4, Springer, Berlin, 1984. A. Beauville, Variétés de Prym et jacobiennes intermédiaires, Ann. Sci. École Norm. Sup. (4) 10 (1977), 309–391. , “Variétés rationnelles et unirationnelles” in Algebraic Geometry—Open Problems (Ravello, 1982), Lecture Notes in Math. 997, Springer, Berlin, 1983, 16–33. F. Bogomolov and Yu. Tschinkel, Density of rational points on Enriques surfaces, Math. Res. Lett. 5 (1998), 623–628. , Density of rational points on elliptic K3 surfaces, preprint, http://xxx.lanl.gov/abs/ math.AG/9902092. , On the density of rational points on elliptic ﬁbrations, J. Reine Angew. Math. 511 (1999), 87–93. F. Campana, Connexité rationnelle des variétés de Fano, Ann. Sci. École Norm. Sup. (4) 25 (1992), 539–545. C. H. Clemens, Double solids, Adv. Math. 47 (1983), 107–230. A. Collino, Lines on quartic threefolds, J. London Math. Soc. (2) 19 (1979), 257–267. J.-L. Colliot-Thélène, A. N. Skorobogatov, and P. Swinnerton-Dyer, Double ﬁbres and double covers: Paucity of rational points, Acta Arith. 79 (1997), 113–135. O. Debarre, Variétés de Fano, Astérisque 245 (1997), 4, 197–221, Séminaire Bourbaki 1996/97, exp. no. 827. V. A. Dem’janenko, “Rational points of a class of algebraic curves” in Thirteen Papers on Group Theory, Algebraic Geometry and Algebraic Topology, Amer. Math. Soc. Transl. Ser. 2 66, Amer. Math. Soc., Providence, 1968, 246–272. J. Franke, Yu. I. Manin, and Yu. Tschinkel, Rational points of bounded height on Fano varieties, Invent. Math. 95 (1989), 421–435. W. Fulton, Intersection Theory, 2d ed., Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin, 1998. V. A. Iskovskikh, Fano threefolds, I (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), 516–562, 717. , Fano threefolds, II (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), 506–549. , On the rationality problem for conic bundles, Duke Math. J. 54 (1987), 271–294. V. A. Iskovskikh and Yu. I. Manin, Three-dimensional quartics and counterexamples to the Lüroth problem (in Russian), Mat. Sb. (N.S.) 86 (1971), 140–166.

500 [19] [20] [21]

[22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

HARRIS AND TSCHINKEL V. A. Iskovskikh and Yu. G. Prokhorov, Algebraic Geometry, Vol. 5: Fano Varieties, Encyclopaedia Math. Sci. 47, Springer, Berlin, 1999. K. Kodaira, On compact analytic surfaces, II, Ann. of Math. (2) 77 (1963), 563–626; III, 78 (1963), 1–40. J. Kollár, Y. Miyaoka, and S. Mori, “Rational curves on Fano varieties” in Classiﬁcation of Irregular Varieties (Trento, 1990), Lecture Notes in Math. 1515, Springer, Berlin, 1992, 100–105. Yu. I. Manin, The p-torsion of elliptic curves is uniformly bounded (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 459–465. , Notes on the arithmetic of Fano threefolds, Compositio Math. 85 (1993), 37–55. L. Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), 437–449. S. Mori and S. Mukai, Classiﬁcation of Fano threefolds with B2 ≥ 2, Manuscripta Math. 36 (1981/82), 147–162. , “On Fano threefolds with B2 ≥ 2” in Algebraic Varieties and Analytic Varieties (Tokyo, 1981), Adv. Stud. Pure Math. 1, North-Holland, Amsterdam, 1983, 101–129. J. P. Murre, “Classiﬁcation of Fano threefolds according to Fano and Iskovskikh” in Algebraic Threefolds (Varenna, 1981), Lecture Notes in Math. 947, Springer, Berlin, 1982, 35–92. J. Noguchi, A higher-dimensional analogue of Mordell’s conjecture over function ﬁelds, Math. Ann. 258 (1981/82), 207–212. I. I. Piatetski-Shapiro and I. Shafarevitch, Torelli’s theorem for algebraic surfaces of type K3 (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530–572. J.-P. Serre, Lectures on the Mordell-Weil Theorem, 2d ed., Aspects Math. E15, Vieweg, Braunschweig, 1990. J. H. Silverman, Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math. 342 (1983), 197–211. P. Vojta, Diophantine Approximations and Value Distribution Theory, Lecture Notes in Math. 1239, Springer, Berlin, 1987.

Harris: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138, USA; [email protected] Tschinkel: Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois 60607, USA; [email protected]

Vol. 104, No. 3

DUKE MATHEMATICAL JOURNAL

© 2000

TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2 CHANG-SHOU LIN 1. Introduction. Let (S 2 , g0 ) be the unit sphere of R3 equipped with the metric g0 induced from the ﬂat metric of R3 . For a positive smooth function f on S 2 , we consider the nonlinear equation

1 f (y)eφ − φ +ρ φ 4π S 2 f (y)e dµ

=0

on S 2 ,

(1.1)ρ

where is the Beltrami-Laplace operator of (S 2 , g0 ), dµ is the volume form with respect to g0 , and ρ > 0 is a constant. Obviously, equation (1.1)ρ is invariant under adding a constant c. Hence, we always seek solutions of (1.1)ρ , which are normalized by φ(y) dµ(y) = 0. (1.2) S2

Equation (1.1)ρ is called the mean ﬁeld equation because it often arises in the context of statistical mechanics of point vortices in the mean ﬁeld limits. Recently, there has been interest in (1.1)ρ because it also arises from the Chern-Simons-Higgs model vortex theory when some parameter tends to zero. (For these recent developments, we refer the readers to [5], [2], [3], [10], [11], [13], [14], [18], [19], [21], [22], and the references therein.) Clearly, equation (1.1)ρ is the Euler-Lagrange equation of the nonlinear functional Jρ (φ) =

1 2

S2

|∇φ|2 dµ − ρ log

S2

f (y)eφ dµ

(1.3)

for φ ∈ H 1 (S 2 ) satisfying (1.2). Here H 1 (S 2 ) denotes the Sobolev space of functions with L2 -integrable ﬁrst derivatives. For ρ < 8π, Jρ (φ) is bounded below, and the inﬁnimum of Jρ can be achieved by the well-known Moser-Trudinger inequality. However, for the case ρ ≥ 8π, the existence of solutions to (1.1)ρ is much more delicate. Recently, under some conditions on f , the existence of an inﬁnimum of J8π has been proved by [10] and [18]. However, the existence of solutions to (1.1)ρ Received 3 August 1999. Revision received 28 January 2000. 2000 Mathematics Subject Classiﬁcation. Primary 35J60. 501

502

CHANG-SHOU LIN

remains open in general for ρ > 8π. In order to study this problem, Y. Y. Li [14] initiated study of the existence of solutions by way of computing the Leray-Schauder topological degree for equation (1.1)ρ . He proved that the concentration phenomenon could occur only when ρ is equal to 8mπ, where m is a positive integer. Therefore, the topological degree is constant in each interval (8mπ, 8(m + 1)π). Furthermore, he showed that the degree is always equal to 1 as long as ρ < 8π. The main purpose of this article is to prove the following theorem. Theorem 1.1. Let f be a positive C 1 function on S 2 , and let d(ρ) denote the Leray-Schauder degree for equations (1.1)ρ and (1.2) for ρ = 8πm, where m is a positive integer. Then (i) d(ρ) = −1 for 8π < ρ < 16π, and (ii) d(ρ) = 0 for 16π < ρ < 24π. An immediate corollary of Theorem 1.1 is that equation (1.1)ρ always possesses a solution when 8π < ρ < 16π . In fact, Theorem 1.1 provides more information about equation (1.1)ρ . For example, Theorem 1.1(ii) implies that if there are solutions to equation (1.1)ρ for 16π < ρ < 24π, then (1.1)ρ always possesses two solutions, at least in the generic situation. Since d(ρ) has a gap at ρ = 8π or 16π, the concentration phenomenon actually occurs there. It is also very interesting to compare Theorem 1.1 with a previous result of Chang-Gursky-Yang (see [4]) for the case ρ = 8π. Note that when ρ = 8π, (1.1)ρ is equivalent to the Gaussian curvature equation on S 2 . Let f be a positive Morse function on S 2 satisfying f (p) = 0 for any critical point p of S 2 . It was proved in [4] that there is an a priori bound for solutions of (1.1)8π , and the Leray-Schauder degree d(8π) can be computed by the formula d(8π) = 1 −

(−1)ind(p) ,

(1.4)

p∈ −

where − = {p ∈ S 2 | p is a critical point of f such that f (p) < 0} and ind(p) is the Morse index of f at p. Note that the expression (1.4) is different from the one in [4], because we have already normalized φ so that (1.2) holds. Thus, for these functions f such that d(8π ) = ±1, equation (1.1)ρ possesses blowing-up solutions for ρi approaching 8π from above and below. The proof of Theorem 1.1 is based on the observation that the concentration phenomenon generally induces symmetry. It has been proved recently that a spherical Harnack inequality holds for blowing-up solutions of either the mean ﬁeld equations on compact Riemann surfaces or the scalar curvature equation on S n . (See [6], [14] for more precise statements.) In the situation when the nonlinear equation is invariant under some group action, it is believed that concentrated solutions should possess a certain symmetry also. In this paper, we study this symmetric property of solutions and apply it to the calculation of the Leray-Schauder degree. First, we consider the case f (y) ≡ 1.

TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2

503

Theorem 1.2. Assume f (y) ≡ 1. Then there is a universal constant C > 0 such that the a priori estimate |φ(y)| ≤ C

for y ∈ S 2

(1.5)

holds for all solutions φ of (1.1)ρ and (1.2) for 8π = ρ ≤ 16π. Furthermore, suppose that φi is a solution of (1.1)ρi and (1.2) such that maxS 2 φi = ∞ and limi→+∞ ρi = 16π . Then ρi > 16π and there exists a direction ni in R3 for each large i such that φi is axially symmetric with respect to the direction ni . Second, we consider the case f (y) ≡ exp(−γ n, y), where n is a unit vector of R3 , n, y is the inner product of n and y ∈ S 2 , and γ is a positive constant. Note that by the Kazdan-Warner identity, equation (1.1)ρ possesses no solutions when f (y) = exp(−γ n, y) and ρ = 8π. Our second symmetry result allows us to compute the topological degree of (1.1)ρ for 8π < ρ < 16π. Theorem 1.3. Let f be described as above. Then for any γ > 0, there exists ρ0 = ρ0 (γ ) > 8π such that for any solution φ of (1.1)ρ with 8π = ρ ≤ ρ0 (γ ), φ is axially symmetric with respect to the direction n. Furthermore, φ is the unique solution for each 8π = ρ ≤ ρ0 (γ ) and γ ≤ 1. Theorems 1.2 and 1.3 allow us to reduce equation (1.1)ρ to an ordinary differential equation when we come to compute the topological degree. For the proofs of both theorems, we will apply the well-known method of moving planes and its variant, the method of moving spheres. The method of moving planes was invented by Alexandrov and was later used to study the radial symmetry for positive solutions of semilinear elliptic equations by Serrin [20], Gidas-Ni-Nirenberg [12], and others. It was applied to study the concentrated behaviors of blowing-up solutions and asymptotic behavior for singular solutions. For these applications, we refer the reader to [6], [14], and the references therein. However, we should remark that for equation (1.1)ρ with those f described above, a solution φ is generally not axially symmetric. The paper is organized as follows. In Section 2, we apply the method of moving planes to prove Theorem 1.2. Here, the isoperimetric inequality due to Bandle in [1] can allow us to start the process of moving planes. For the proof of Theorem 1.3, we ﬁrst apply the method of moving spheres to locate the blow-up point and then apply the method of moving planes to prove axial symmetry. This is done in Section 3. For the case f (y) = exp(−γ n, y), we prove that, if ρi > 8π, then the blow-up occurs at the minimum point of f (y). However, for ρi < 8π, the blow-up can occur at the maximum point of f . It is a very interesting question to study where the blow-up point is actually located. The complete proof of Theorem 1.1 is given in the last section. To calculate the topological degree for Theorem 1.1(ii), we need a theorem due to Wang (see [23]), which states how to count the local degree due to a nondegenerate orbit. The major part of Section 4 is devoted to studying the nondegeneracy of the orbit of solutions. It is interesting to note that in the case of ρ > 16π , the orbit is topologically homeomorphic to RP 2 , the real projective space of

504

CHANG-SHOU LIN

two dimensions. This is the reason why the degree d(ρ) vanishes for 16π < ρ < 24π. This nondegeneracy requires detailed analyses for the linearized equations at blowingup solutions. In particular, for the case with solutions of two blow-up points, a general form of the Pohozaev identity is employed very delicately. 2. The method of moving planes. In this section, we prove the uniform bound of solutions for equation (1.1)ρ with f ≡ 1. First suppose that there is a solution φi of (1.2) and that 1 eφi − = 0 on S 2 , (2.6) φi + ρi φi 4π S 2 e dµ such that max φi −→ +∞, S2

ρi = 8π,

lim ρi = 8π

i→+∞

(2.7) (2.8)

hold. We recall the following result of [15], which is useful in this paper (see [15, Theorems 1.1 and 1.2]). Theorem 2.1. Let f (y) = exp(−γ n, y) for y ∈ S 2 , where γ ≥ 0 and n is a unit vector of R3 . Then (i) If γ = 0 and ρ < 8π , then φ ≡ 0 is the only solution of (1.1)ρ and (1.2). (ii) If γ > 0 and ρ < 8π, then there exists a unique solution of (1.1)ρ and (1.2). Furthermore, this solution is axially symmetric with respect to the direction n. (iii) If 0 ≤ γ ≤ 1 and ρ ≤ 16π, then there exists a unique solution of (1.1)ρ and (1.2) in the class of axially symmetric functions. By Theorem 2.1, we have ρi > 8π.

(2.9)

To yield a contradiction, we apply the well-known method of moving planes. The method can work here mainly due to the concentration phenomenon of φi . To see it, we need a result to describe the concentration of φi , due to Li in [14]. Let φi be a sequence of solutions of (1.1)ρi and (1.2). Assume that {q1 , . . . , qm } is the blow-up set of φi . Set φi (y) f (y)e dµ . (2.10) ξi (y) = φi (y) − log S2

Without loss of generality, we may assume that for any blow-up point q, ξi (Pi ) = max ξi (y) −→ +∞ |y−q|≤δ

for any small δ > 0. In [14], Li proved the following result.

(2.11)

TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2

Theorem 2.2. There exists a constant C > 0 such that eξi (Pi ) ≤C ξi (y) − log 2 ξ (P ) 2 1 + ρi f (Pi )/8 e i i |y − Pi |

505

(2.12)

for |y − Pi | ≤ δ. Furthermore, ξi − ξ¯i −→ 8π

m

G(·, ql ) − 2m

S2

l=1

G(·, y) dµ(y)

(2.13)

2 (S 2 \{q , . . . , q }), where {q , . . . , q } is the blow-up set of φ , G(·, q ) is the in Cloc 1 m 1 m i l Green function with a singularity at ql , and ξ¯i = −S 2 ξi (y) dµ(y) is the integral average of ξi .

An immediate consequence of Theorem 2.2 is the following. Let {q1 , . . . , qm } be the set of blow-up points of φi . Then Theorem 2.2 implies that lim ρi = 8mπ,

i→+∞

and for any δ > 0, there exists a constant c = c(δ) such that ξi (y) + ξi (Pi,j ) ≤ c(δ)

(2.14)

for |y − qj | ≥ δ, where Pi,j is the local maximum point of ξi near qj . In particular, ξi (Pi,j ) − ξi (Pi,l ) ≤ C

(2.15)

for 1 ≤ j , l ≤ m and for some constant C independent of i. We see later that (2.15) is very important for proving symmetry for the case with two blow-up points. Another useful tool in our approach is the isoperimetric inequality due to Bandle (see [1]). Lemma 2.3. Let q(x) be a continuous positive function deﬁned in a simply connected domain ' of R2 . Suppose that q(x) satisﬁes log q(x) + q(x) ≥ 0

for x ∈ '

(2.16)

in the distribution sense and '

q(x) dx ≤ 8π.

(2.17)

Let ω ' be a subdomain such that the ﬁrst eigenvalue + q(x) for the Dirichlet

506

CHANG-SHOU LIN

problem is nonpositive. Then ω

q(x) dx ≥ 4π.

(2.18)

Lemma 2.4. Let v0 (y) = −2 log(1 + (1/8)|y|2 ), and let ϕ(y) satisfy  2, ϕ(y) + ev0 (y) ϕ(y) = 0 in R+ 

ϕ(0, y2 ) = 0

for y2 ∈ R.

(2.19)

Suppose ϕ(y) → 0 as |y| → ∞. Then ϕ(y) = c∂v0 /∂y1 for some constant c ∈ R. Note that v0 is the solution of v0 + ev0 = 0 such that v0 (0) = maxR2 v0 (y) = 0, and equation (2.19) is the linearized equation at v0 . The proof of Lemma 2.3 is elementary and is omitted here. Proof of Theorem 1.2. Let φi be a sequence of solutions of (2.6) and (1.2) such that (2.7) and (2.8) hold. By Theorem 2.2, φi has only one blow-up point. Let Pi be the maximum point of φi . Without loss of generality, we may assume Pi = (1, 0, 0). By using the stereographic projection π of S 2 onto R2 , we set ρi ui (x) = ξi π −1 (x) − log 1 + |x|2 + log(4ρi ), 4π where ξi is given by (2.10). By a straightforward computation, ui (x) satisﬁes l ui (x) + 1 + |x|2 i eui (x) = 0 and

R2

in R2 ,

(2.20)

(2.21)

l 1 + |x|2 i eui (x) dx = 4(2 + li )π,

(2.22)

ρi − 2. 4π

(2.23)

where li =

For simplicity of notation, we denote the maximum point of ui by Pi . Obviously, Pi is contained in a small neighborhood of (1,0). By (2.9) and (2.23), we have li > 0. To apply Lemma 2.3, we set q(x) = (1 + |x|2 )li eui (x) . By elementary calculations, q(x) satisﬁes 4li

log q(x) = ui (x) +

1 + r2

because li > 0.

2 ≥ −q(x)

(2.24)

TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2

507

To start the process of moving planes, we show ui (x) ≥ ui (x − ) for x1 ≥ 0,

(2.25)

where x − = (−x1 , x2 ). To see (2.25), we set w(x) = ui (x) − ui (x − ) for x1 ≥ 0. Then w(x) satisﬁes

where

 w(x) + c(x)w(x) = 0

for x1 ≥ 0,

w(0, x ) ≡ 0 2

for x2 ∈ R,

c(x) = 1 + |x|

2 li

(2.26)

− eui (x) − eui (x ) . ui (x) − ui (x − )

2 = {(x , x ) | x > 0}. Set ' = {x ∈ R 2 | Now suppose w(x) < 0 for some x ∈ R+ 1 2 1 + − − w(x) < 0} and ' = {x | x ∈ '}. Then

l − w(x) + 1 + |x|2 i eui (x ) w(x) ≤ 0

(2.27)

for x ∈ ', which implies that the ﬁrst eigenvalue of +q(x) in '− for the Dirichlet problem is nonpositive where q(x) = (1 + |x|2 )li eui (x) . By (2.19), q(x) satisﬁes (2.16). Thus, we have by Lemma 2.3, l 1 + |x|2 i eui (x) dx ≥ 4π. (2.28) '−

On the other hand, by Theorem 2.2 we have that for any ε > 0, there is i0 = i0 (ε) such that l 2 li ui (x) dx ≥ (2.29) 1 + |x| e 1 + |x|2 i eui (x) dx ≥ 8π − ε. B(Pi ,1/2)

2 R+

Together with (2.28) and (2.29), we have by (2.22), l 4(2 + li )π = 1 + |x|2 i eui (x) dx ≥ 4π + 8π − ε, R2

which obviously yields a contradiction because li → 0. Hence, (2.25) is established. We note that w(x) ≡ 0 because ui has a blow-up point at Pi . Thus, by the strong 2 , and we have maximum principle and the Hopf lemma, we have w(x) > 0 for x ∈ R+ ∂w/∂x1 (0, x2 ) > 0 for x2 ∈ R.

508

CHANG-SHOU LIN

For any λ ∈ R, we let 2λ = {x | x1 > λ}, Tλ = {x | x1 = λ} and let x λ = (2λ − x1 , x2 ) denote the reﬂection point of x with respect to Tλ . As in (2.25), we want to prove ui (x) ≥ ui (x λ ) for x ∈ 2λ and for 0 ≤ λ ≤ 2.

(2.30)

For 0 ≤ λ ≤ 2, we set wλ (x) = ui (x) − ui (x λ ) for x ∈ 2λ . Then wλ (x) satisﬁes l λ wλ (x) + 1 + |x|2 i eui (x) − eui (x ) l l λ = 1 + |x λ |2 i − 1 + |x|2 i eui (x ) ≤ 0

(2.31)

for x ∈ 2λ . Here we note that for x ∈ 2λ , |x|2 − |x λ |2 = 4λ(x1 − λ) ≥ 0 because λ ≥ 0. For 0 ≤ λ ≤ 2, we let λ

l eui (x) − eui (x ) cλ (x) = 1 + |x|2 i . ui (x) − ui (x λ ) By (2.20) and (2.23), we have cλ (x) ≤ c 1 + |x| −4

(2.32)

for some constant c > 0 and |x| ≥ 2. Applying the maximum principle, we can prove by (2.32) that there exists R0 > 0 such that if wλ (x) is negative somewhere in 2λ for 0 ≤ λ ≤ 2, then |xλ | ≤ R0 , where wλ (xλ ) = inf wλ (x)

(2.33)

2λ

(see [6] for the proof of (2.33)). By (2.26), wλ satisﬁes wλ +cλ (x)wλ (x) ≤ 0 in 2λ . Together with (2.33), (2.30) can be proved by the standard argument of the method of moving planes. For details of the proof, we refer the reader to [6]. Clearly, (2.25) yields a contradiction to the fact that ui has a local maximum point at Pi that is near (1, 0). Therefore, the uniform bound is established for ρi → 8π from above. Next, we are going to prove the uniform bound up to 16π. Now suppose φi is a solution of (2.6) and (1.2) with limi→+∞ ρi = 16π, and assume that (2.7) holds also. Then Theorem 2.2 implies that φi has exactly two blow-up points P , Q. By using the same argument as above, we conclude that P is antipodal to Q. Actually, we

TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2

509

want to prove more. Let Pi and Qi be local maximum points of φi near P and Q, respectively; that is, φi (Pi ) =

max φi (y),

|y−P |≤δ0

φi (Qi ) =

max φi (y).

|y−Q|≤δ0

We want to prove that Pi and Qi are antipodal for large i, and φi (y) is axially −−→ symmetric with respect to the direction Pi Qi . Now recall that by Theorem 2.2, there is a constant C > 0 such that φi (Pi ) − φi (Qi ) ≤ C.

(2.34)

−→ Suppose Pi = −Qi . Let n be the direction P Q. By (2.34), we can slightly change the direction n to another direction ni such that the following statements hold. (i) Let Pi∗ and Q∗i be the intersection of the line ni with S 2 where Pi∗ and Q∗i are close to Pi and Qi , respectively. Then eξi (Pi ) |Pi − Pi∗ |2 ∼ eξi (Qi ) |Qi − Q∗i |2 , where for two sequences of positive numbers, ai ∼ bi means that the ratio ai /bi is bounded from below and above by positive constants. (ii) Without loss of generality, we may assume Pi∗ = (0, 0, −1) and Q∗i = (0, 0, 1) and by a rotation if necessary, both Pi and Qi are contained in the half hyperplane {y ∈ S 2 | y1 > 0 and y2 = 0}. Let π be the stereographic projection of S 2 onto R2 . As before, set ρi ui (x) = ξi π −1 (x) − log 1 + |x|2 + log(4ρi ). 4π Then ui (x) satisﬁes (2.21) and (2.22). Clearly, the maximum point of ui is close to the image of Pi . For simplicity of notation, we still use Pi to denote the maximum point of ui ; that is, ui (Pi ) = max ui (x), R2

and we use Qi to denote the maximum point of u∗i , u∗i (Qi ) = max u∗i (x), R2

where u∗i (x) = ui

x − 2(2 + li ) log |x|. |x|2

(Note that 2+li = ρi /4π .) Obviously, both Pi and Qi → 0 as i → +∞. By Theorem 2 . For the two quan2.2, (i), and (ii), we conclude that Pi and Qi are located in R+ ∗ tities eui (Pi ) |Pi |2 and eui (Qi ) |Qi |2 , either they simultaneously have a positive lower bound or they tend to zero simultaneously. For the latter case, if ti and si are the

510

CHANG-SHOU LIN

x1 -coordinates of Pi and Qi , then both |Pi | ∼ ti and |Qi | ∼ si hold. Let vi (y) = ui Pi + e−ξi (Pi )/2 y − ui (Pi ).

(2.35)

2 (R 2 ). By passing to a Then by Theorem 2.2, vi (y) is uniformly bounded in Cloc subsequence, vi converges to v0 (y), where v0 (y) is the solution of  v0 (y) = 0 in R 2 ,  v0 (y) + e (2.36)  v0 (0) = max v0 (y), and ev0 (y) dy = 8π. R2

R2

By a result of Chen and Li in [6], v0 (y) = −2 log(1+(1/8)|y|2 ). Thus, for any small r > 0 and for eξi (Pi ) |x − Pi |2 ≤ r, we have (2.37) ui (x) = ui (Pi ) − eui (Pi ) |x − Pi |2 a + o(1) , where a is a positive constant. To yield a contradiction, we prove ui (x) ≥ ui (x − )

(2.38)

2 . Once (2.38) is established, we follow the same argument of (2.30) to prove for x ∈ R+

ui (x) ≥ ui (x λ )

(2.39)

for x ∈ 2λ and for 0 ≤ λ ≤ 1. Obviously, it yields a contradiction to the fact that ui has a maximum point at Pi . (Note that the standard argument for the method of moving planes still applies for our case because for each i, wλ (x) = ui (x) − ui (x λ ) tends to 0 as |x| → +∞.) To prove (2.38), we divide the argument into two cases. ∗

Case 1. Both eui (Pi ) |Pi |2 and eui (Qi ) |Qi |2 ≥ c > 0. By Theorem 2.2, the scaled function vi (y) = ui Pi + e−ui (Pi )/2 y − ui (Pi ) uniformly converges to v0 (y), the solution of (2.36). Two cases are discussed separately. If eui (Pi ) |Pi |2 → +∞, then by Theorem 2.2, there is a constant C such that ρi u(Pi ) 2 |Pi | ≤ C − ui (Pi ) + 4 log |Pi | (2.40) ui (x) ≤ C + ui (Pi ) − 2 log 1 + e 8 holds for x1 ≤ 0 and |x| ≤ δ0 . By the scaling in (2.35) for any R > 0, ui (x) ≥ ui (Pi ) − 2 log 1 + R 2 − C

(2.41)

TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2

511

for |x −Pi |eu(Pi )/2 ≤ R. Since u(Pi )+2 log |Pi | → +∞, we have by (2.40) and (2.41), ui (x) ≥ ui (x − )

(2.42)

for |x − Pi | ≤ e−ui (Pi )/2 R. The similar inequality holds at Qi also. Now suppose 2 | w (x) < 0} = φ, where w (x) = u (x) − u (x − ). Then ' = {x ∈ R+ 0 0 i i − 2 li ui (x − ) 2 li ui (x) w0 (x) + 1 + |x| e w0 (x) ≤ w0 (x) + 1 + |x| − eui (x ) = 0. e By Lemma 2.3,

l 1 + |x|2 i eui (x) dx ≥ 4π.

2 R−

(2.43)

However, for any ε > 0, there is R0 = R0 (ε) and i0 = i0 (R0 ) such that if i ≥ i0 , then l 2 li ui (x) dx ≥ 1 + |x| e 1 + |x|2 i eui (x) dx ≥ 8π − ε (2.44) Bδ+

B(Pi ,e−ui (Pi )/2 R0 )

0

and

B˜ δ+

1 + |x|

0

2 li ui (x) e

dx =

B(Qi ,e

−u∗ i (Qi )/2

l ∗ 1 + |x|2 i eui (x) dx ≥ 8π − ε (2.45)

R0 )

2 | |x| ≤ δ } and B 2 | |x| ≥ δ −1 }. Clearly, ˜ + = {x ∈ R+ hold, where Bδ+0 = {x ∈ R+ 0 δ0 0 together (2.43), (2.44), and (2.45) yield a contradiction. If eui (Pi ) |Pi |2 ≤ C, then the rescaled function v˜i (y) = ui e−ui (Pi )/2 y − ui (Pi ) 2. tends to v0 (y −y0 ) in any compact set of R2 , where limi→+∞ eui (Pi )/2 Pi = y0 ∈ R+ Since v0 (y) = v0 (|y|) is decreasing in |y| for any R > 0 there is i0 = i0 (R) such that ui (x) ≥ ui (x − ) for |x| ≤ e−ui (Pi )/2 R. Therefore, '∩B(0, e−ui (Pi )/2 R) = φ. The same holds at Qi . By Lemma 2.3, l 1 + |x|2 i eui (x) dx ≥ 4π, '−

where '− = {x | x − ∈ '}. Thus, for any ε > 0, there exists i0 = i(ε) such that for i ≥ i0 , l l 1 + |x|2 i eui (x) dx ≥ 1 + |x|2 i eui (x) dx R2

'−

+ +

|x|≤e−ui (Pi )/2 R

l 1 + |x|2 i eui (x) dx

|x|≤e

−u∗ i (Qi )/2

R

l ∗ 1 + |x|2 i eui (x) dx ≥ 20π − 2ε.

Clearly, it yields a contradiction. Therefore (2.38) is proved for case 1.

512

CHANG-SHOU LIN

∗ Case 2. limi→∞ eui (Pi ) |Pi |2 + eui (Qi ) |Qi |2 = 0. Let Ni = max |w0 (x)| = |w0 (xi )| 2 R+

(2.46)

2 . The maximum can be achieved because w (x) → 0 as |x| → +∞. for some xi ∈ R+ 0 We claim that xi should be in a neighborhood of 0 or ∞. We prove it by contradiction. Suppose there is a small δ0 > 0 and c0 > 0 such that there is a subsequence of ui (still denoted by ui ) such that

|w0 (xˆi )| = sup |w0 (x)| ≥ c0 Ni , Bδc

(2.47)

0

2 | δ ≤ |x| ≤ δ −1 }. Recall that w (x) satisﬁes where Bδc0 = {x ∈ R+ 0 0 0

l − 0 = w0 (x) + c0 (x)w0 (x) = w0 (x) + 1 + |x 2 | i eui (x) − eui (x ) . By Theorem 2.2,

(2.48)

|c0 (x)| = o(1) 1 + |x|−4 ,

2 away from the origin. Set ¯+ where o(1) → 0 uniformly for any compact set of R

wˆ 0 (x) = Ni−1 w0 (x).

(2.49)

2 (R 2 . By the elliptic estimates, w 2 ¯+ Then |wˆ 0 (x)| ≤ 1 for x ∈ R+ ˆ 0 (x) is bounded in Cloc 2 (R 2 \{0}). Since ¯+ \{0}). By passing to a subsequence, wˆ 0 (x) converges to h(x) in Cloc 2 \{0}. c0 (x) → 0, h(x) is a bounded harmonic function that is identical to zero on ∂R+ By the reﬂection x → (−x1 , x2 ), h can be extended to be a bounded harmonic function in R2 \{0}. By the regularity theorem and the Liouville theorem, we conclude h(x) ≡ 0 2 because h ≡ 0 on ∂R 2 . But this yields a contradiction to (2.47). Hence, we in R+ + have proved

sup |w0 (x)| = o(1)Ni . Bδc

(2.50)

0

To yield a contradiction, we ﬁrst suppose that the maximum point xi of w0 is 2 . Set located in Bδ+0 = Bδ0 ∩ R+ w˜ 0 (y) = wˆ 0 e−ui (Pi )/2 y . Then w˜ 0 (y) satisﬁes w˜ 0 (y) + c˜0 (y)w˜ 0 (y) = 0

for |y| ≤ δ0 eui (Pi )/2 ,

(2.51)

TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2

where

c˜0 (y) = c0 e

−ui (Pi )/2

y e

ui (Pi )

= Ki e

−ui (Pi )/2

y

513

evi (y) − evi (y) , vi (y) − vi (y − )

vi (y) = ui (e−ui (Pi )/2 y) − ui (Pi ), and Ki (x) = (1 + |x|2 )li . Since eui (Pi ) |Pi |2 → 0, 2 (R 2 ). Thus, by Theorem 2.2, we have vi (y) converges to v0 (y) of (2.36) in Cloc −4 |c˜0 (y)| ≤ A 1 + |y| (2.52) for |y| ≤ eui (Pi )/2 δ0 and for some constant A independent of i and y. Recall that by (2.50), w˜ i (y) = o(1) for |y| = eui (Pi )/2 δ0 . Thus, by Green’s formulas, we have |w˜ 0 (x)|

− |x − y| 1 c ˜ log (y) w ˜ (y) dy + the boundary term 0 0 2π |y|≤eui (Pi )/2 δ0 |x − y| (2.53) − 1 |x − y| ≤ |c˜0 (y)||w˜ 0 (y)| dy + o(1). log u (P )/2 2π |y|≤e i i δ0 |x − y| =

Let yi = eui (Pi )/2 xi , where xi is the maximum point of wˆ 0 . Then |w˜ 0 (yi )| = 1. By elementary calculations and by (2.52), (2.53) implies −1 (2.54) 1 ≤ c 1 + |yi | for some constant c. Therefore, yi is bounded. Since |w˜ 0 (y)| ≤ 1, after passing to a subsequence if necessary, w˜ 0 (y) converges to 2 (R 2 ), where w ¯+ w˜ in Cloc ˜ satisﬁes  2, w˜ + ev0 (y) w(y) ˜ = 0 in R+ 2  w(y) ˜ =0 on ∂ R+ . 2 . By (2.53) again, we estimate Since yi is bounded, we have w(y) ˜ ≡ 0 in R+ −1 |w˜ 0 (y)| ≤ c 1 + |y| + o(1), (2.55)

which implies |w(y)| ˜ = O(|y|−1 ) at inﬁnity. By Lemma 2.4, we have w(y) ˜ =c

∂v0 (y) ∂y1

(2.56)

for some constant c = 0. Thus, ∂ w˜ ∂ 2 v0 (0) (0) = c = 0, ∂y1 ∂y12

(2.57)

514

CHANG-SHOU LIN

because ∂ 2 v0 (0)/∂y12 = (1/2)v0 (0) < 0. On the other hand, ∂ w˜ 0 ui (Pi )/2 −1 ∂vi ui (Pi )/2 − Pi = N i Pi e e ∂y1 ∂y1 ∂ 2 vi = Ni−1 2 (ηi )eui (Pi )/2 (−2ti ), ∂y1

(2.58)

where ti is the x1 -coordinate of Pi and ηi → 0 as i → +∞. Therefore, by (2.56), we have lim Ni−1 eui (Pi )/2 ti > 0.

i→+∞

Let w0∗ (x) be the Kelvin transformation of w0 ; that is, − x x x ∗ w0 (x) = w0 = ui − ui = u∗i (x) − u∗i (x − ). |x|2 |x|2 |x − |2

(2.59)

(2.60)

Set wˆ 0∗ = Ni−1 w0∗ and w˜ 0∗ (y) = wˆ 0∗ (e−(uˆ i /2)(Qi ) y). By Theorem 2.2 again, (2.58) also holds at Qi ; that is, ∂ w˜ 0∗ u∗ (Qi )/2 ∂ 2 vi∗ (ηi∗ ) u∗ (Qi )/2 Qi = Ni−1 e i (−2si ), e i ∂y1 ∂y12 ∗

where si is the x1 -coordinate of Qi . Since eui (Qi )/2 si ≥ ceui (Pi )/2 ti for some constant 2 . In c > 0, (2.59) implies that w˜ 0∗ converges to w∗ (y), where w ∗ (y) ≡ 0 on R+ ∗ particular, for any R > 0, there exists i0 = i0 (R) > 0 such that w˜ 0 (y) and w˜ 0 (y) > 0 for |y| ≤ R and i ≥ i0 . Let R be large. Then the inequality l 2 li ui (x) 1 + |x| e 1 + |x|2 i eui (x) dx dx ≥ R2

'−

+ +

|x|≤e−ui (Pi )/2 R

l 1 + |x|2 i eui (x) dx

−u∗ (Q )/2 |x|≤e i i R

l ∗ 1 + |x|2 i eui (x) dx

≥ 20π − 2ε yields a contradiction. Therefore, Pi = −Qi is proved. After Pi = −Qi is established, we want to prove that φi is axially symmetric −−→ with respect to the direction ni = Pi Qi . Let π be the stereographic projection of S 2 onto R2 by taking Qi to ∞, and let ui be deﬁned as before. The radial symmetry of ui follows from the same argument as in case 2 above. Following the notation 2 . Then w in case 2, we let w0 (x) = ui (x) − ui (x − ). Suppose w0 (x) ≡ 0 in R+ ˜ 0 (y)

TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2

515

2 (R 2 ). Because of (2.50), (2.54), and (2.55) can be ¯+ also converges to w(y) ˜ in Cloc proved in the same way. Thus, w(y) ˜ ≡ 0 and w(y) ˜ = c(∂v0 /∂y1 )(y) for some c = 0. But ∇ w˜ 0 (0) = Ni−1 e−ui (Pi )/2 2∇ui (0) = 0 because 0 is the maximum point of ui . Obviously, it implies c = 0 and it yields a contradiction again. We conclude that ui is symmetric with respect to x1 , and the radial symmetry follows readily. By Theorem 2.1(iii), the trivial solution φ ≡ 0 is the only solution of (2.6) satisfying (1.2), provided ρ ≤ 16π . Since φi blows up at P and Q, we have ρi > 16π. Therefore, the proof of Theorem 1.2 is ﬁnished.

3. The method of moving spheres. In this section, we begin with the equation −γ n,y φi e 1 e − = 0 on S 2 , (3.1) φi + ρi e−γ n,y φi dµ 4π where n is a unit vector of R3 , γ > 0 is a constant, and limi→+∞ ρi = 8π. We want to prove that φi is axially symmetric with respect to n for large i. If ρi < 8π by Theorem 2.1, then φi is axially symmetric. Therefore, in the following, we always assume ρi > 8π . Since the function e−γ n,y is a monotone function in the variable n, y, by the Kazdan-Warner identity, equation (3.1) has no solution for ρ = 8π. Thus, ξi (Pi ) = max ξi (y) −→ +∞ S2

as i → +∞, where ξi is deﬁned by (2.10). Without loss of generality, we may assume P = limi→+∞ Pi . By Theorem 2.2, P is the only blow-up point of φi . In [8], the authors proved that P must be a critical point of f . Note that ±n are the only critical points of e−γ n,y . In the following, we want to prove a stronger result. Lemma 3.1. P = n. Proof. Without loss of generality, we assume that n = (0, 0, 1). Suppose P = n. Let π(x) be the stereographic projection of R2 onto S 2 , which maps n to +∞. Set ρi ui (x) = ξi π −1 (x) − log 1 + |x|2 + γ + log(4ρi ). (3.2) 4π By a straightforward computation, ui (x) satisﬁes l 2 ui (x) + 1 + |x|2 i e2γ /(1+|x| ) eui (x) = 0 and

in R2 ,

l 2 1 + |x|2 i e2γ /(1+|x| ) eui (x) dx = 4π(2 + li ),

where li =

−2+

ρi 4π

> 0.

(3.3)

(3.4)

516

CHANG-SHOU LIN

For simplicity, we still denote P to be the blow-up point of ui , and Pi is the maximum point of ui near P . Let vi (x) ≡ ui (x) + ui (Pi ). Then vi (x) satisﬁes, by (3.3) and (3.4),

vi (x) + λi Ki (x)evi (x) = 0,

λi

R2

Ki (x)evi (x) dx = ρi ,

where Ki (x) = (1 + |x|2 )li e2γ /(1+|x| ) and where λi = e−ui (Pi ) . By (2.12), vi (x) is 2 (R 2 \{P }) and λ → 0 as i → +∞. Thus, it is easy to see uniformly bounded in Cloc i 2 . By the Liouville that vi (x) converges to a harmonic function G in R2 \{P } in Cloc theorem, G(x) = −4 log |x − P |. To ﬁnd the location of P , we apply the Pohozaev identity. For any unit vector e in R2 , we have 2

λi

Br (P )

e, ∇Ki (x) evi (x) − 1 dx =

|x−P |=r

e, ∇vi

∂vi (e, ν) − |∇vi |2 dσ. ∂ν 2 (3.5)

Recall that λi = e−u(Pi ) → 0 and, by Theorem 2.2, λi Ki (x)evi (x) converges to 8π δ(P ), the delta function at P . Thus, the left-hand side of (3.5) converges to K(P )−1 e, ∇K(P ), and (3.5) implies ∂G ∂G (e, ν) − |∇G|2 dσ ∂ν ∂ν 2 |x−P |=r ∂G 2 1 dσ = 0, = e, ν 2 |x−P |=r ∂ν

8π K(P )−1 e, ∇K(P ) =

e,

where K(x) = lim Ki (x) = e2γ /(1+|x|

2)

i→+∞

because G(x) is symmetric with respect to P . Clearly, zero is the only critical point of K. Hence, P = 0. Now suppose Pi is the maximum point of ui and, by the previous result, Pi → 0 as i → +∞. To yield a contradiction, we want to prove for any a ∈ (0, 1), ui (x) ≥ u∗i (x; a) for |x| ≤ a,

(3.6)

where u∗i (x; a) = ui

a2x |x| − 2(2 + l . ) log i a |x|2

(3.7)

TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2

517

By a straightforward computation, we have 4 2 a a x ∗ ui (x; a) = (ui ) |x| |x|2 4 l a a 4 i 2γ |x|2 /(|x|2 +a 4 ) u∗ |x| 2(2+li ) i =− 1+ 2 e e |x| a |x| li 2 |x| ∗ 2 2 4 2 =− + a e2γ |x| /(|x| +a ) eui . a2 By (3.2), u∗i (x; a) is smooth at x = 0. Thus, u∗i satisﬁes  2 li  ∗ 2 2 4 Du∗ + a 2 + |x| e2γ |x| /(|x| +a ) eui = 0 i a2   ∗ ui (x; a) = ui (x)

for |x| ≤ a,

(3.8)

on |x| = a.

As before, we want to compare u∗i with ui ; that is, we claim wa (x) = ui (x) − u∗i (x; a) ≥ 0

for |x| ≤ a.

(3.9)

Since both li and γ are positive, we have l l |x|2 i 2γ |x|2 /(|x|2 +a 4 ) 2 2 e ≤ 1 + |x|2 i e2γ /(1+|x| ) a + 2 a for |x| ≤ a ≤ 1. Therefore, wa (x) satisﬁes wa (x) + Ca (x)wa (x) l |x|2 i 2γ |x|2 /(a 4 +|x|2 ) 2 2 li 2γ /(1+|x|2 ) u∗i (x;a) = a + 2 e − 1 + |x| e ≤0 e a (3.10) for |x| ≤ a ≤ 1, where

Ca (x) = 1 + |x|

2 li 2γ /(1+|x|2 ) e

∗ eui (x) − eui (x;a) . ui (x) − u∗i (x; a)

To prove (3.9) for a = 1, we suppose w1 (x) < 0 for some x in B1 , the unit ball. Let ' = {x ∈ B1 | w1 (x) < 0}. By (3.8), we have for x ∈ ', l ∗ 2 2 w1 (x) + 1 + |x|2 i e2γ |x| /(1+|x| ) eui (x) w1 (x) l ∗ 2 2 ≤ w1 (x) + 1 + |x|2 i e2γ |x| /(1+|x| ) eui (x) − eui (x) l 2 2 2 = 1 + |x|2 i e2γ |x| /(1+|x| ) − e2γ /(1+|x| ) eui (x) ≤ 0.

518

CHANG-SHOU LIN

Therefore, the ﬁrst eigenvalue of +(1+|x|2 )li e2γ |x| problem on ' is nonpositive. Let

2 /(1+|x|2 )

∗

eui (x) for the Dirichlet

l ∗ 2 2 q(x) = 1 + |x|2 i e2γ |x| /(1+|x| ) eui . Obviously, q(x) satisﬁes 8γ 1 − r 2 4li log q(x) = uˆ i + 2 + 3 ≥ −q(x) 1 + r2 1 + r2 for x ∈ B1 . By Lemma 2.3, we have l 2 1 + |x|2 i e2γ /(1+|x| ) eui dx = |x|≥1

≥

|x|≤1

'

l ∗ 2 2 1 + |x|2 i e2γ |x| /(1+|x| ) eui (x;1) dx

q(x) dx ≥ 4π.

On the other hand, since ui blows up at Pi , we have l 2 1 + |x|2 i e2γ /(1+|x| ) eui dx ≥ 8π(1 − ε) B1

for any ε > 0, provided that i is large. Clearly, it yields a contradiction to l 2 1 + |x|2 i e2γ /(1+|x| ) eui dx = 4π(2 + li ) −→ 8π R2

as i → +∞. Hence, w1 (x) ≥ 0 in B1 . By the strong maximum principle, we have w1 (x) > 0 in B1 . So we start the process of moving spheres at a = 1. Since wa is a supersolution of wa +Ca wa ≤ 0 in Ba , the maximum principle and the Hopf lemma are continuously applied to conclude wa (x) > 0 for 0 ≤ |x| ≤ a and for 0 < a ≤ 1. Since the argument is standard now, we skip the details of the proof. Thus, (3.9) is established. By differentiating wa at the boundary |x| = a, we have ∂ui 2(2 + li ) ∂wa (x) = 2 (x) + ≤0 ∂r ∂r r for |x| = r ≤ 1. But it yields a contradiction because (∂ui /∂r)(Pi ) = 0. Therefore, Lemma 3.1 is proved. Proof of Theorem 1.3. Let φi be a solution of (3.1) with limi→+∞ ρi = 8π. Since ρi > 8π, we know that n is the blow-up point of φi by Lemma 3.1. Without loss of generality, we assume n = (0, 0, −1). Let π be the stereographic projection of S 2

TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2

519

onto R2 such that the north pole is mapped to the inﬁnity. Let ui be deﬁned as in (3.2). Then ui (x) satisﬁes  ui (x) = 0 on R 2 ,  ui (x) + Ki (x)e (3.11)   Ki (x)eui (x) dx = 4π(2 + li ), R2

l 2 2 Ki (x) = 1 + |x|2 i e2γ |x| /(1+|x| ) , li =

ρi − 2. 4π

(3.12) (3.13)

Let Pi be the maximum point of ui and Pi → 0 by Lemma 3.1. Without loss of generality, we assume Pi = (ti , 0) with ti ≥ 0. We ﬁrst claim ti = 0.

(3.14)

w0 (x) = ui (x) − ui (x − ) > 0

(3.15)

Suppose ti > 0. We want to prove

2 . Note that K (x) satisﬁes for x ∈ R+ i

Ki (x) ≥ Ki (x λ )

(3.16)

for x ∈ 2λ and λ ≥ 0, where 2λ and x λ are the same notation as in Section 2. Hence, the difference wλ (x) = ui (x) − ui (x λ ) satisﬁes λ λ wλ (x) + Ki (x) eui (x) − eui (x ) = Ki (x λ ) − Ki (x) eui (x ) ≤ 0

(3.17)

for x ∈ 2λ and 0 ≤ λ ≤ 1. Hence, once (3.15) is established, we start the process of moving planes to show that wλ (x) ≥ 0 for x ∈ 2λ and 0 ≤ λ ≤ 1, which obviously yields a contradiction. 2 | w (x) < 0} = ∅. Recall that w (x) satisﬁes Now suppose that ' = {x ∈ R+ 0 0 w0 (x) + c0 (x)w0 (x) = 0, where

(3.18)

−

eui (x) − eui (x ) c0 (x) = Ki (x) . ui (x) − ui (x − )

To yield a contradiction, we want to show that c0 (x) = o(1)|x|−2 for x ∈ ' and |x| ≤ 1. We prove (3.19) by considering three cases separately.

(3.19)

520

CHANG-SHOU LIN

Case 1. We have that eui (Pi ) |Pi |2 → +∞ as i → +∞. For x ∈ ' we have −

|c0 (x)| ≤ Ki (x)eui (x ) . Since eui (Pi ) |x − − Pi |2 ≥ eui (Pi ) |Pi |2 −→ ∞, by Theorem 2.2, we have ui (x − ) ≤ C − ui (Pi ) − 4 log |Pi − x − | ≤ C − ui (Pi ) + 2 log |Pi | − 2 log |x| because

max |Pi |, |x| ≤ |P − x − |.

Hence, eui (x

−)

−1 ≤ ec eui (Pi ) |Pi |2 |x|−2 = o(1)|x|−2 .

Case 2. We have that eui (Pi ) |Pi |2 ≥ c > 0. In this case, we consider vi (y) = ui e−ui (Pi )/2 y − ui (Pi ). 2 , where v is the solution of Then vi (y) converges to v0 (y − y0 ) for some y0 ∈ R+ 0 (2.36), and

y0 = lim eui (Pi )/2 Pi . i→+∞

2 and ∂v (y − y )/∂y | Since v0 (y − y0 ) ≥ v0 (y − − y0 ) for any y ∈ R+ 0 0 1 y1 =0 > 0, we have for any R > 0, i0 = i0 (R) such that if i ≥ i0 , then

vi (y) ≥ vi (y − ) 2 and |y| ≤ R. By scaling back to u , the above implies holds for y ∈ R+ i

ui (x) ≥ ui (x − ) for |x| ≤ eui (Pi )/2 R. In particular, if x ∈ ', then lim |x − − Pi |2 eui (Pi ) −→ +∞.

i→+∞

By Theorem 2.2, (3.19) follows readily.

TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2

521

Case 3. We have that limi→+∞ eui (Pi ) |Pi |2 = 0. In this case, we want to prove ui (x) ≥ ui (x − )

(3.20)

for |x| ≤ eui (Pi )/2 R and i ≥ i0 , where R is any large number and i0 = i0 (R). By using the same notions of Section 2, we deﬁne wˆ 0 and w˜ 0 as (2.49) and (2.51), respectively. Clearly, (3.20) is equivalent to proving that w˜ 0 (y) converges to w(y) ˜ and w(y) ˜ = c(∂v0 /∂y1 ) for some constant c = 0. The argument just follows from the one in case 2 in the proof of Theorem 1.2. But for our case, we only have a single blow-up point. For the sake of completeness, we sketch the proof. Let Ni = max |w0 (x)|. 2 R+

We want to prove |w0 (xˆi )| = max |w0 (x)| = o(1)Ni , c Bδ

(3.21)

0

2 and |x| ≥ δ }. By using the proof of (2.50), we know that if where Bδc0 = {x | x ∈ R+ 0

|w0 (xˆi )| ≥ c Ni

(3.22)

for some c > 0, then |xˆi | → +∞ as i → +∞. Using the Kelvin transformation as in (2.60), we let y −1 ∗ . wˆ 0 (y) = Ni w0 |y|2 Then wˆ 0∗ (y) satisﬁes wˆ 0∗ (y) + c0∗ (y)wˆ 0∗ (y) = 0

2 in R+ ,

2 . Since |c (x)| = o(1)|x|−4 for |x| ≥ 1, and wˆ 0∗ (y) is smooth up to the boundary of R+ 0 we have y −4 |y| = o(1) |c0∗ (y)| = c0 |y|2

for |y| ≤ 1. Set g(y) = y1α ,

w¯ 0∗ (y) =

wˆ 0∗ (y) g(y)

for 0 < α < 1. By a straightforward computation, w¯ 0∗ (y) satisﬁes w¯ 0∗ (y) + 2∇ log g · ∇ w¯ 0∗ + c0∗ (y) + α(α − 1)y1−2 w¯ 0∗ (y) = 0.

(3.23)

522

CHANG-SHOU LIN

2 , and |y | ≤ 1/|xˆ | → 0 as By (3.22), |w¯ 0∗ (y)| has a maximum point at yi ∈ R+ i i i → +∞. Without loss of generality, we assume w¯ 0∗ (yi ) > 0. By the maximum principle, (3.23) yields

− 2 ∗ 0 ≥ w¯ 0∗ (yi ) = − c0∗ (yi ) + α(α − 1)(yi,1 ) w¯ 0 (yi ) > 0, a contradiction. Thus, (3.21) is established. Therefore, the maximum point xi of |w0 (x)| is near zero. Actually, by setting w˜ 0 (y) = wˆ 0 (e−ui (Pi )/2 y), we prove eui (Pi )/2 xi is bounded. For the details, see (2.54). By (2.55), w˜ 0 (y) converges to c(∂v0 /∂y1 ) for some c = 0. By (2.57) and (2.58), we conclude that c < 0. Thus, w0 (x) > 0 for |x| ≤ e−ui (Pi )/2 R and i ≥ i0 , where R is any large number and i0 = i0 (R). Readily, (3.19) follows. 2 and 0 < α < 1. Since the Kelvin transformaLet w¯ 0 (x) = w0 (x)x1−α for x1 ∈ R+ ∗ 2 tion w0 (y) = w0 (y/|y| ) is smooth at y = 0, |w0 (x)| ≤ c0 x1 holds for 0 ≤ x1 ≤ 1 and x2 ∈ R. Hence, w¯ 0 (x) tends to zero uniformly for x2 ∈ R as x1 → 0. Thus, the inﬁnitum of w¯ 0 is achieved in '; that is, w¯ 0 (xˆi ) = inf w¯ 0 (x) < 0 '

for some xˆi . By a straightforward compution, w¯ 0 satisﬁes w¯ 0 + 2αx1−1

∂ w¯ 0 + c0 (x) + α(α − 1)x1−2 w¯ 0 = 0. ∂x1

(3.24)

By the maximum principle at xˆi , (3.24) yields 0 ≤ w¯ 0 (xˆi ) = − c0 (xˆi ) + α(α − 1)(xˆi,1 )−2 w¯ 0 (xˆi ) < 0 by (3.24), a contradiction. Therefore the claim (3.14) is proved. After (3.14) is established, the radial symmetry of ui can be proved by the same argument at the last stage of the proof of Theorem 1.2. Therefore, the radial symmetry of ui is proved and the uniqueness follows from Theorem 2.1(iii). Thus, Theorem 1.3 is completely proved. 4. Proof of Theorem 1.1. In this section, we apply previous symmetric results to compute the Leray-Schauder degree d(ρ) for 8π < ρ < 16π and 16π < ρ < 24π . Throughout this section, we always consider equation (1.1)ρ with f (y) ≡ exp(−γ n, y) for a ﬁxed unit vector n, and we are only concerned with solutions that are axially symmetric with respect to n. Let π be the stereographic projection of S 2 onto R2 , which maps −n to inﬁnity. For any axially symmetric solution φ of

TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2

523

(1.1)ρ and (1.2) for ρ > 8π , we set u by (3.2). Following conventional notation, we let u(r; s) be the unique solution of   u + 1 u + 1 + r 2 l e2γ r 2 /(1+r 2 ) eu = 0, r (4.1)  u(0; s) = s and u (0; s) = 0, where l > 0 and γ ≥ 0 are constants. To compute d(ρ), it is important to prove the nonsingularity of the linearized equation and to determine the number of negative eigenvalues. The linearized equation of (4.1) at u(r) is called nonsingular if the equation l 2 2 ϕ + 1 + r 2 e2γ r /(1+r ) eu ϕ = 0

in R2

(4.2)

2 2 possesses no bounded nontrivial solutions. For any C function ψ on S satisfying S 2 ψ(y) dµ = 0, we set φ(y) ψ(y) dµ −1 2 f (y)e ϕ(x) = ψ π (x) − S φ(y) dµ S 2 f (y)e

for x ∈ R2 , where f (y) = exp(γ n, y). Then ϕ(x) is a bounded nontrivial solution of (4.2) if and only if ψ(x) is a nontrivial solution of the linearized equation (1.1)ρ : f (y)eφ S 2 f (y)eφ ψ(y) dµ f (y)eφ ψ ψ + ρ − =0 (4.3) φ ( S 2 f (y)eφ dµ)2 S 2 f (y)e dµ on S 2 . Thus, the linearized equation of (4.1) at u is nonsingular if and only if the linearized equation of (1.1)ρ at φ is nonsingular. It is easy to see that u(r; s) exists for all r > 0 and always satisﬁes the asymptotic behavior u(r; s) = −β(s) log r + O(1)

at ∞

(4.4)

for some constant β(s), which continuously depends on the parameter s. Integrating (4.1), β(s) can be computed through the following formulas: l 2 2 (4.5) 1 + |x|2 e2γ |x| /(1+|x| ) eu(|x|;s) dx. 2πβ(s) = R2

Note that by (2.23), we seek solutions u(r; s) such that β(s) = 2(2 + l)

(4.6)

for some s. We recall results about radial solutions from [9] and [16]. First, [9, Theorem 1.1] states the following.

524

CHANG-SHOU LIN

Lemma 4.1. Let u(r; s) be the solution of (4.1) and let β(s) be given by (4.5). Then (i) If 0 < l ≤ 1, then lims→+∞ β(s) = 4 and lims→−∞ β(s) = 2(2 + 2l). (ii) If l > 1, then lims→+∞ β(s) = 4l and lims→−∞ β(s) = 2(2 + 2l). Note that by Lemma 4.1, equation (4.1) always possesses a radial solution satisfying (4.6) for 0 < l < 2. By Theorem 2.1, this solution is unique for 0 ≤ γ ≤ 1 and 0 ≤ l ≤ 2. For γ = 0, by (2.20) and (2.23), the solution u0 (r; l) = −(2 + l) log(1 + r 2 ) + log 4(2 + l) of (4.1) corresponds to the trivial solution φ ≡ 0 on S 2 . Since the Laplacian of S 2 has eigenvalues k(k + 1) for a nonnegative integer k, by the remark above, the linearized equation (4.1) at u0 (r; l) is nonsingular for 0 < l ≤ 2. Thus, u0 (0; l) = log 4(2 + l) is not the minimum of β(s) for 0 < l ≤ 2 because if β(s0 ) achieves the minimum of β(s), then the linearized equation of (4.1) at u(r; s0 ) must be singular (see (4.10) below). Set β0 = min β(s). s∈R

Then we have β0 < 2(l + 2) for l = 2. By the continuity of β(s) on l, we have β0 < 2(l + 2)

(4.7)

for l close to 2. Since by Lemma 4.1, 2(2 + l) < lim β(s) < lim β(s) for l > 2, s→+∞

s→−∞

together with (4.7), we conclude that there are at least two solutions of β(s) = 2(2+l). Hence, (4.1) possesses at least two radial solutions satisfying (4.6) for γ = 0, l > 2, and l close to 2. Hence, we have proved the ﬁrst part of the following result. Corollary 4.2. Fix a unit vector n in R3 . There exists a small δ0 > 0 such that for f (y) = 1 and 16π < ρ ≤ 16π +δ0 , (1.1)ρ possesses at least an axially symmetric nontrivial solution φ satisfying (1.2). Furthermore, any sequence of axially symmetric nontrivial solutions of (1.1)ρi and (1.2) with f (y) = 1 must blow up at ±n. Proof. The existence part has been proved already. Now suppose that φi is an axially symmetric nontrivial solution of (1.1)ρi and (1.2) with ρi > 16π and limi→+∞ ρi = 16π . By Theorem 2.2, either φi blows up at ±n simultaneously or φi is uniformly bounded for all i. Assume that the latter case occurs. By passing to a subsequence, φi converges to φ in C 2 (S 2 ), where φ satisﬁes  eφ 1   =0 − φ + 16π φ 4π S 2 e dµ    φ dµ = 0. S2

on S 2 , (4.8)

TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2

525

By Theorem 2.1, we have φ ≡ 0. Since φi accumulates at φ ≡ 0, the linearized equation of (4.8) at φ = 0 must be singular; that is, there is a nontrivial solution w of  2  w + 4w = 0 on S ,   w dµ = 0. S2

This yields a contradiction to the fact that 4 is not an eigenvalue of the Laplacian on S 2 . Therefore, φi must blow up at ±n. Lemma 4.3. There exists a small 0 < δ1 ≤ δ0 such that axially symmetric nontrivial solutions to (1.1)ρ and (1.2) with f = 1 are unique for 16π < ρ ≤ 16π + δ1 . The proof of Lemma 4.3 is long and requires a delicate application of the Pohozaev identity. Let u(r; s) be the unique solution of (4.1) and let ϕ(r; s) = (∂u/∂s)(r; s). Then ϕ satisﬁes the linearized equation at u,   ϕ + 1 ϕ + 1 + r 2 l e2γ r 2 /(1+r 2 ) eu ϕ = 0, r (4.9)  ϕ(0) = 1, ϕ (0) = 0. By an elementary argument, we prove that for any ﬁxed s, either limr→+∞ (ϕ(r)/log r) = 0 or ϕ(r) is uniformly bounded for r ∈ [0, ∞). Obviously, the latter case is equivalent to limr→+∞ ϕ (r)r = 0. Integrating (4.9), we have by (4.5), ∞ l 2 2 ˙ =− − β(s) 1 + r 2 e2γ r /(1+r ) eu(r;s) ϕ(r; s)r dr = lim rϕ (r; s), (4.10) 0

r→+∞

˙ ˙ where β(s) = dβ/ds. Thus, β(s) = 0 for some s if and only if ϕ(r; s) is uniformly bounded in [0, ∞). The behavior of ϕ(r; s) is described in [16, Lemma 3.1 and Theorem 1.5], which is stated as (i) in the following lemma. Note that (ii) of the lemma is equivalent to (iii) of Theorem 2.1. (See [16, Section 4] for a proof.) Lemma 4.4. (i) For γ = 0 and l > 0, ϕ(r; s) has at least two zeros for all s. Moreover, ϕ(r; s) has exactly two zeros and limr→+∞ ϕ(r; s) = +∞ whenever 4l ≤ β(s) < 4(1 + l). (ii) For 0 < γ ≤ 1 and 0 < l ≤ 2, ϕ(r; s) has exactly two zeros and limr→+∞ ϕ(r; s) = +∞ whenever β(s) = 2(2 + l). From now until the end of the proof of Lemma 4.6, we always assume γ = 0. In this case, we claim There exists a δ1 > 0 such that if 2 < l ≤ 2 + δ1 , s ≥ δ1−1 and ˙ > 0. β(s) ≥ 4 + 2l, then β(s)

(4.11)

526

CHANG-SHOU LIN

Proof of Lemma 4.3. We ﬁrst prove that There exists a small δ1 > 0 such that if 2 < l ≤ 2 + δ1 , s ≥ δ1−1 and ˙ = 0. β(s) ≥ 4 + 2l, then β(s)

(4.12)

Suppose that (4.12) does not hold. Then there is a sequence of solutions ui (r) = u(r; si ) of (4.1) with γ = 0, li → 2 and si → +∞ such that ϕi (r) ≡ ϕ(r; si ) is bounded in [0, ∞) for each i, and β(si ) ≥ 2(2+li ). For simplicity, we let βi = β(si ). By Lemma 4.4(i), ϕi must change sign at least twice. Let ri denote the ﬁrst zero of ϕi (r). By scaling, we let ϕˆi (r) = ϕi (e−ui (0)/2 r). Then ϕˆi satisﬁes   ϕˆ + 1 ϕˆ + 1 + e−ui (0) r 2 li evi (r) ϕi = 0, i i r  ϕˆ (0) = 1 and ϕˆ (0) = 0, i

i

where vi (r) = ui (e−ui (0)/2 r) − ui (0). Since vi (r) converges to v0 (r) ≡ −2 log(1 + 2 , ϕˆ (r) converges to ϕˆ (r) ≡ (8 − r 2 )/(8 + r 2 ) in C 2 , where ϕˆ (r) is r 2 /8) in Cloc i 0 0 loc the solution of   ϕˆ (r) + 1 ϕˆ (r) + ev0 (r) ϕˆ0 (r) = 0, 0 r 0 (4.13)  ϕˆ (0) = 1 and ϕˆ (0) = 0. 0

0

Since ϕˆ0 (r) < 0 for 0 < r < +∞, we have eui (0) ri2 −→ +∞

(4.14)

as i → +∞. To yield a contradiction, we want to prove ri −→ 0

(4.15)

as i → ∞. We prove (4.15) by using the Pohozaev identity. Suppose ri ≥ δ0 > 0 for some δ0 > 0. Set rui rui + 4 + r 2 Ki (r)eui (r) , 2 P˜i (r) = rϕi rui + 2 + r 2 Ki (r)eui (r) ϕi ,

Pi (r) =

where Ki (r) = (1 + r 2 )li . By a straightforward computation, Pi (r) and P˜i (r) satisfy r tKi (t)eui (t) t dt > 0 Pi (r) = 0

(4.16) (4.17)

(4.18)

TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2

527

for r > 0, and P˜i (r) =

r 0

Pi (t)ϕi (t) dt

= Pi (r)ϕi (r) −

r 0

Pi (t)ϕi (t) dt.

(4.19)

Let r = ri . Then by (4.19), ri u (ri ) ri ui + 4 ϕi (ri ) = − − i 2

ri 0

Pi (t)ϕi (t) dt > 0.

(4.20)

Thus, ri ui +4 < 0. By scaling, ϕi (e−ui (0) r) converges to (8 − r 2 )/(8 + r 2 ) uniformly in any compact set of [0, ∞) as i → +∞. From here, it is easy to see ϕi (ri ) → −c < 0 for some c > 0 (actually, it can be proved that c = 1, but this is not important in the following argument) and ri ui (ri ) → −4 as i → ∞. We want to prove ri u (ri ) + 4 ≤ c0 e−ui (0) i

(4.21)

for some constant c0 > 0. To see it, by (4.18), we have r Pi e−ui (0)/2 s = tKi (t)eui (t) t dt 0

= 2li

l −1 t 2 1 + t 2 i eui (t) t dt

0

= 2li e where

r

−ui (0)

s

(4.22)

l −1 τ 2 1 + e−ui (0) τ 2 i evi (τ ) τ dτ,

0

r = e−ui (0)/2 s, vi (τ ) = ui e−ui (0)/2 τ − ui (0).

Note that vi (τ ) → log 1/(1 + (τ 2 /8)2 ). Thus, eui (0) Pi (e−ui (0)/2 s) converges to 2li

s 0

τ3

2 ds.

1 + τ 2 /8

Also, if we set ϕˆi (τ ) = ϕi (e−ui (0) τ ), then ϕi (e−ui (0) r)e−ui (0) = ϕˆi (τ ) → −32τ /(8 + τ 2 )2 . Hence, lim e

i→∞

ui (0)

e−ui (0) R 0

R

= −128 0

Pi (t)ϕi (t) dt

t (8 + t 2 )2

t 0

s3 ds dt. (1 + s 2 /8)2

(4.23)

528

CHANG-SHOU LIN

Let Ri → +∞ such that both vi (s) and ϕˆi (s) uniformly converge for s ≤ Ri . Set ti = e−ui (0)/2 Ri . Thus, for all t ≥ ti , we have eui (0)/2 t → +∞. By Theorem 2.2, eui (t) = O(1)t −4 e−ui (0) for t ≥ ti . Thus, for ri ≥ t ≥ ti , we have ri ri ui (s) tϕ (t) ≤ Ki (s)e s|ϕi (s)| ds ≤ Ki (s)eui (s) s ds i t

≤ c e−ui (0)

t

ri t

(4.24)

s −3 ds = O(1)e−ui (0) t −2 ,

where |ϕi (s)| ≤ 1 are used. Since ri ≥ δ0 > 0 for some δ0 > 0, we have Pi (t) ≤ Pi (ri ) ≤ cri u (ri ) + 4 + O e−ui (0) for 0 ≤ t ≤ ri , where the last inequality is due to (4.16) and ri ≥ δ0 . Hence, by (4.23), (4.24), and (4.20) gives ti ri ri u (ri ) + 4 ≤ c Pi (s) ϕi (s) ds + |Pi (s)| ϕi (s) ds

ti

0

≤ c1 e ≤ c1 e

−ui (0)

−ui (0)

+ ri u (ri ) + 4

ri ti

e

−ui (0) −3

s

ds

−u (0) −2 i + ri u (ri ) + 4 e ti .

Because e−ui (0) ti−2 → 0, the inequality (4.21) follows from the above. By the expression of Pi and ri ≥ δ0 > 0, (4.21) gives Pi (ri ) = O e−ui (0) . On the other hand, since Pi (t) is increasing in t and by (4.22), Pi (ti )eui (0) −→ +∞

as i −→ +∞,

(4.26) obviously yields a contradiction. Thus, we have proved ri → 0. Let u∗i (r) = u(1/r) − βi log r and ϕi∗ (r) = ϕi (1/r). Then u∗i and ϕi∗ satisfy u∗ + and

u∗ ∗ + Ki∗ (r)eui = 0, r

 ϕ∗ ∗  ϕi∗ + i + Ki∗ (r)eui (r) ϕi∗ (r) = 0, r  ϕ ∗ (0) = lim ϕi and ϕ ∗ (0) = 0, i

(4.25)

r→+∞

i

(4.26)

TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2

529

where Ki∗ (r) = (1 + r 2 )li r βi −2(2+li ) . Note limi→+∞ βi − 2(2 + li ) = 0. Since

1 0

∗ Ki∗ (r)eui (r) r dr

∞

= 1

Ki (r)eui (r) r dr −→ 4

as i → +∞, u∗i (0) must tend to +∞. Let ri∗ be the ﬁrst zero of ϕi∗ (r). By an argument similar to that for (4.15), we can prove ri∗ → 0. Hence, ϕi (r) has a zero at (ri∗ )−1 that tends to +∞ as i → +∞. By (4.15), ϕi has a second zero at rˆi . Let ξi denote the zero of ϕi in the interval (ri , rˆi ). Without loss of generality, we may assume ξi is bounded. Otherwise, we may apply the Kelvin transformation and follow the same argument to yield a contradiction. By Theorem 2.2, we have eui (r) ∼ r −4 e−ui (0) for r ∈ [ti , ξi ]. Thus, ri2 Ki (ri )eui (ri ) ≥ c0

e−ui (0) . ri2

(4.27)

We claim eui (0) ri4 ≥ c1

(4.28)

for some positive constant c1 independent of i. To see it, we use the general form of the Pohozaev identity. For any λ > 0, we set Pλ (r; ui ) =

rui (rui + λ) + r 2 Ki (r)eui (r) . 2

(4.29)

By a straightforward computation, Pλ satisﬁes d Pλ (r; ui ) = rKi (r) + (4 − λ)Ki (r) reui (r) . dr

(4.30)

For any δ > 0, we let λ = 4 + δ(e−ui (0) /ri2 ). Thus, d Pλ (r; ui ) ≤ 0 dr

for r ≤ c2 (δ)

e−ui (0)/2 . ri

(4.31)

If eui (0) ri4 → 0 as i → ∞, we have ri ≤ o(1)

e−ui (0)/2 e−ui (0)/2 ≤ c2 (δ) . ri ri

Hence, Pλ (r; ui ) is decreasing in r for r ∈ [0, ri ]. In particular, Pλ (ri ; ui ) < 0.

(4.32)

530

CHANG-SHOU LIN

Recall ri ui (ri ) + 4 < 0 by (4.20). By (4.32), we have ri ui (ri ) ri ui (ri ) + 4 ≤ 0. = Pλ (ri ; ui ) − 2 2

ri u (ri ) e−ui (0) δ ri2 K(ri )eui (ri ) + i 2 ri2 Thus,

ri2 K(ri )eui (ri ) = o(1)

e−ui (0) , ri2

a contradiction to (4.27). Hence, the claim (4.28) is proved. Now, by equation (4.9), r r rϕi ≤ Ki (t)eui (t) |ϕi (t)|t dt ≤ Ki (t)eui (t) t dt ri

≤ c e−ui (0) Hence,

ri

r ri

t −3 dt ≤ c e−ui (0) ri−2 .

ϕi (r) ≤ c e−ui (0) ri−3

for ri ≤ r ≤ ξi .

By (4.28), we have 1 ∼ −ϕi (ri ) = ϕi (ξi ) − ϕi (ri ) =

ξi ri

ϕi (s) ds ≤ c e−ui (0) ri−3 ≤ c1 ri −→ 0,

which is obviously impossible. Therefore, (4.12) is proved. Suppose that (4.11) is false; that is, β (s) < 0 for large s by (4.12). Then β(s) > 4l for 2 < l ≤ 2 + δ1 and for large s. On the other hand, Lemma 4.4 states that ˙ <0 β(s)

whenever 4l < β(s) < 4l + 4.

(4.33)

Since by the Pohozaev identity d Pβ (r; u) = r (rK (r) − (β − 4)K) eu(r) , dr we have 2Pβ (∞; u) =

∞

2rK (r) − (β − 4)K eu(r) r dr,

0

where Pβ (r; u) =

ru (ru + β) + r 2 K(r)eu(r) 2

and

l K(r) = 1 + r 2 .

(4.34)

TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2

531

Set β = 4 + 4l and recall limr→+∞ u (r)r = −β(s). Then (4.34) implies β(s) β(s) − (4 + 4l) =

∞

2rK (r) − 4lK(r) eu(r) r dr

0

∞

=−

4lr 1 + r

2 l−1 u(r) e

(4.35) dr < 0,

0

˙ that is, β(s) < 4 + 4l for all s ∈ R. Together with (4.33), we have β(s) < 0 for all s ∈ R. Thus, 4l < β(s) < 4 + 4l for all s ∈ R. But, u0 (r; l) is the solution with β = 2(2 + l) and 2(2 + l) < 4l for l > 2, which yields a contradiction. Therefore, (4.11) is established. Clearly, uniqueness follows immediately from Corollary 4.2 and (4.11). Lemma 4.5. let ϕ(r; s) be the solution of (4.9) with 2 < l ≤ 2 + δ1 and β(s) = 2(2 + l). Then ϕ(r; s) has exactly three zeros and limr→+∞ ϕ(r; s) = −∞. Proof. By (4.10) and (4.11), we have limr→+∞ rϕ (r) < 0. Hence, limr→+∞ ϕ (r; s) = −∞. Since ϕ(r; s) is known to have two zeros at least, it implies ϕ has three zeros at least. Let τ1 < τ2 < τ3 be the ﬁrst three zeros of ϕ. If δ1 tends to zero, the second zero τ2 must tend to +∞. This can be proved by the same argument as in the proof of Lemma 4.3, where a contradiction results when ξi∗ is assumed to be bounded. Hence, we may assume τ2 ≥ 1. Let u(r) ˆ = u(1/r; s)−2(2 +l) log r, the Kelvin transformation of u. By a straightforward computation, uˆ satisﬁes the same equation as u does. Since u(r) ˆ = −2(2 + l) log r +O(1) at ∞, u(r) ˆ ≡ u(r) follows readily from the uniqueness of Lemma 4.3. It implies that ϕ(r) ˆ = ϕ(1/r; s) satisﬁes (4.9), but not the initial conditions. Hence, ϕ are ϕˆ are linearly independent. By the Liouville theorem, between any two zeros of ϕ(r) ˆ for 0 ≤ r ≤ 1, there must be a zero of ϕ(r; s). Since ϕ(r; s) has only one zero in [0,1], ϕ(r) ˆ has at most two zeros in [0,1]. Therefore, the proof of Lemma 4.5 is complete. Let φ(y; n) denote the unique solution of (1.1)ρ with f ≡ 1 in Lemma 4.3. Since φ(y; −n) is also a solution that is axially symmetric with respect to n, φ(y; −n) ≡ φ(y; n) on S 2 by Lemma 4.3. Clearly, equation (1.1)ρ with f ≡ 1 is invariant under the action of O(3), the group of orthorgonal transformations in R3 . Set O(φ) ≡ {φ(y; n) | |n| = 1} as the orbit of φ. By the remark above, O(φ) is homeomorphic to RP 2 , the real projective space of two dimensions. The orbit O(φ) is called nondegenerate if the null space of the linearized equation (4.3) at φ is equal to the tangent space of O(φ). To analyze (4.3), we may assume n = (0, 0, 1) for simplicity. Let π be the standard stereographic projection of S 2 onto R2 , and let u(r) be the corresponding solution in R2 . To prove nondegeneracy is equivalent to proving that equation (4.2) possesses exactly two independent bounded solutions because O(φ) is a two-dimensional manifold. By separation of variables, it

532

CHANG-SHOU LIN

is equivalent to ﬁnding bounded solutions of ϕk (r)+

l ϕk k 2 ϕk − 2 + 1+r 2 eu(r) ϕk (r) = 0 r r

for r ∈ [0, ∞). By a straightforward computation, we see that 1 + r2 2+l + r w(r) ≡ u (r) 4 2

(4.36)k

(4.37)

always satisﬁes (4.36)k for k = 1, and w(r) is uniformly bounded in [0, ∞) because ru (r) → −2(2 + l) as r → +∞. Hence, w(r) w(r) x1 and x2 are null vectors of r r the linearized equation (4.3) at φ.

(4.38)

To count the number of negative eigenvalues of (4.3), we also count the number of negative eigenvalues of (4.36)k for each integer k. The summation of the number of negative eigenvalues of (4.36)k over k = 1, 2, . . . , is equal to the total number of negative eigenvalues of (4.3). By Lemma 4.5, ϕ(r; s) has exactly three zeros and limr→+∞ ϕ(r; s) = −∞, which gives us that in the class of axially symmetric functions on S 2 , there are exactly three negative eigenvalues. In the following, we should prove that w(r) of (4.37) changes sign only once. Thus, in the class of {ψ(z)x1 , ψ(z)x2 | ψ(z) ∈ C 2 (S 2 )}, there are only two negative eigenvalues, where S 2 = {(x1 , x2 , z) | x12 + x22 + z2 = 1}. Therefore, the following result is expected to hold. Lemma 4.6. let φ(y; n) be the solution of (1.1)ρ and (1.2) in Lemma 4.3. Then the orbit O(φ) is nondegenerate and the linearized equation (4.3) at φ has a twodimensional null space and ﬁve negative eigenvalues, that is, λ5 < 0 = λ6 = λ7 < λ8 . Proof. By Lemma 4.5 and the remark above, we have to prove (i) w(r) of (4.37) changes sign only once; and (ii) for each k ≥ 2 (4.36)k possesses no bounded solutions in [0, ∞), and any solution ϕk satisfying ϕk (0) = ϕk (0) = 0 has no zero in [0, ∞). We note that the nondegeneracy of the orbit follows from Lemma 4.5, (4.38), and the ﬁrst part of (ii). The number of negative eigenvalues comes from the remark above and from the second part of (ii). To prove (i), we note that w(0) = 0 and w (0) = u (0) + (2 + l)/2 = −eu(0) /2 + (2 + l)/2 < 0 if l is close to 2. Thus, w(r) < 0 for r small. By using the identity u(r) ≡ u(1/r)−2(2+l) log r in [0, ∞), we have w(1/r) = −w(r). Hence, w(r) > 0 for large r. Now suppose that w(r) has more than one zero. Then w(r) must have three zeros at least. Let r1 , r3 be the ﬁrst and the last zero of w(r), and let r1 < r2 < r3 be another zero. Since w(r) is bounded near ∞, w(r) and ϕ(r; s) of Lemma 4.5 are linearly independent. Thus, by the Sturm-Liouville comparison theorem, ϕ(r; s)

TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2

533

must have a zero in intervals [0, r1 ), (r1 , r2 ), (r2 , r3 ) and (r3 , ∞). But this yields a contradiction to Lemma 4.5, which states that ϕ has exactly three zeros. Hence, (i) is proved. To prove (ii), we note that for k ≥ 2, the inequality l 2 1 + r 2 eu(r) ≤ 2 1 + o(1) r

(4.39)

holds for 0 ≤ r < +∞, where o(1) → 0 as l → 2. The inequality (4.39) can be checked for 0 ≤ r ≤ e−u(0)/2 R for large R because eu(r) = (1 + o(1))eu(0) (1 + eu(0) r 2 /8)−2 for 0 ≤ r ≤ e−u(0)/2 R, where o(1) → 0 as R → +∞. For e−u(0)/2 R ≤ r ≤ 1, eu(r) = O(1)e−u(0) r −4 = o(1)r −2 by Theorem 2.2. Hence, (4.39) obviously holds. For r ≥ 1, (4.39) follows by the identity u(r) ≡ u(1/r) − 2(2 + l) log r. Note that by (4.39), the coefﬁcient of zero order of (4.36)k is negative for k ≥ 2. Then, the maximum principle, it implies any solution ϕk of (4.36)k never changes sign and |ϕk (x)| → +∞ as |x| → +∞. Thus, Lemma 4.6 is proved completely. Next we consider the case 0 < γ ≤ 1 and 8π < ρ ≤ 8π + ρ0 (γ ). By Theorem 1.3, solutions of (4.1) with β = 2(2 + l) are unique. By Lemma 4.4(ii), the solution ϕ of (4.9) changes sign exactly twice. As in Lemma 4.6, we compute the number of negative eigenvalues of the linearized equation of (4.1) at φ with f (y) = exp(−γ n, y) through counting the number of zeros of ϕk + where K(r) = (1 + r 2 )l e2γ r

ϕk k 2 ϕk − 2 + K(r)eu(r) ϕk = 0, r r

2 /(1+r 2 )

((4.40)k )

.

Lemma 4.7. For any γ > 0, there exists a small ρ0 (γ ) > 0 such that the linearized equation (1.1)ρ with f (y) = exp(γ n.y) and 8π ≤ ρ ≤ 8π + ρ0 (γ ) is nondegenerate. Moreover, λ4 < 0 < λ5 , where λi is the ith eigenvalue of the linearized equation (4.3). Proof. We want to prove that (i) the unique solution ϕ1 of (4.40)1 satisfying ϕ1 (0) = 0 and ϕ1 (0) = 1 has exactly one zero and limr→+∞ ϕ1 (r) = −∞, and (ii) for any integer k ≥ 2, any solution ϕk (r) of (4.40)k with ϕk (0) = ϕk (0) = 0 has no zeros in (0, ∞) and |ϕk (r)| → +∞ as r → +∞. Clearly, the nonsingularity of the linearized equation follows from Lemma 4.4(ii) and the second part of (i) and (ii) above. The number of negative eigenvalues follows from Lemma 4.4(ii) and the ﬁrst part of (i) and (ii) above. Since ϕ1 satisﬁes ϕi (0) = 0 and ϕ1 (0) = 1, ϕ1 (r) > 0 for small r. Now suppose ϕ1 (r) > 0 for all r > 0. Recall that the derivative u (r) satisﬁes u −

u (r) + K(r)eu u (r) + K (r)eu = 0. r2

(4.41)

534

CHANG-SHOU LIN

By (4.40)1 and (4.41), we have ϕ1 (r)u

(r)r − ϕ1 (r)u (r)r

r

ϕ1 u − u ϕ1 t dt

= 0

r

=−

(4.42)

ϕ1 K (t)e

u(t)

t dt < 0,

0

because both K (t) and ϕ1 (t) are positive. If ϕ1 (r) tends to a constant as r → ∞, then rϕ1 (r) → 0 as r → +∞. In this case, the left-hand side of (4.42) tends to zero as r → +∞, which yields a contradiction. If ϕ1 (r) is unbounded as r → +∞, then ϕ1 (r) > 0 for large r. Since u (r) > 0 for large r, the left-hand side of (4.42) is positive for large r, which again yields a contradiction. Hence, ϕ1 (r) must change sign at least once. Now suppose that either ϕ1 (r) has two zeros for r ∈ (0, ∞) or ϕ1 (r) has one zero and ϕ1 (r) remains bounded as r → +∞. In the latter case, we can show that limr→+∞ ϕ1 (r) = 0. Thus, we can assume that ϕ1 (r) has two zeros ri < si for a sequence of li → 0. (Note that si could be +∞.) By Lemma 3.1, ui (x) blows up at the origin. Since eui (0) ri2 → +∞ by a scaling argument, we have by Theorem 2.2,

l o(1) 2 2 1 + r 2 i e2γ r /(1+r ) eui (r) = 2 r

(4.43)

for r ≥ ri , where o(1) → 0 as i → +∞. Since ϕ1 has two zeros at ri and si , ϕ1 must have a local minimum ti ∈ (ri , si ); that is, ϕ1 (ti ) = min ϕ1 (r) < 0. ri ≤r≤si

Applying the maximum principle to (4.40)1 , we have

1 2 li 2γ ti2 /(1+ti2 ) ui (ti ) 0 ≤ ϕ1 (ti ) = 2 − 1 + ti e ϕ(ti ) < 0 e ti by (4.43), which yields a contradiction. Thus, ϕ1 (r) changes sign only once and ϕ1 (r) → −∞ as r → +∞.This proves (i). For the case k ≥ 2, any nontrivial solution ϕk (r) of (4.40)k for k ≥ 2, satisfying ϕk (0) = ϕk (0) = 0, must not change sign, and |ϕk (r)| → +∞ as r → +∞. If ϕk (ri ) = 0 and ri is the ﬁrst zero of ϕi in (0, ∞], then (4.43) holds also, and by appling the maximum principle as above, (4.40)k yields a contradiction. Thus, ϕk (r) does not change sign. This ﬁnishes the proof of Lemma 4.7. To ﬁnish the proof of Theorem 1.1, we need a result of Wang in [23], which tells us how to compute the local degree due to a nondegenerate orbit. Let φ be a nontrivial solution of (1.1)ρ with f ≡ 1. Recall O(φ) is the orbit of φ.

TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2

535

Lemma 4.8. let O(φ) be a nondegenerate orbit. Then the local degree contributed by O(φ) is equal to (−1)m χ (O(φ)), where χ(O(φ)) is the Euler characteristic, and m is the number of negative eigenvalues of the linearized equation. For a proof, we refer the readers to [23]. Now we are in the position to complete the proof of Theorem 1.1. Proof of Theorem 1.1. First we compute d(ρ) for 8π < ρ < 16π for equation (1.1)ρ . Since d(ρ) is independent of the function f , we choose f (y) = e−γ n,y for some direction n and 1 ≥ γ > 0. By Theorem 1.2 and Lemma 4.7, we know that if ρ is close to 8π , then (1.1)ρ possesses a unique solution φρ . Since the linearized equation of φρ is nonsingular, d(ρ) = (−1)m , where m is the number of negative eigenvalues of the linearized operator. Note that φρ is always normalized to satisfy (1.2). Then m = 4 − 1 = 3 by Lemma 4.7. Thus, d(ρ) = −1 for 8π < ρ < 16π. For 16π < ρ < 24π , we consider f (y) ≡ 1 on S 2 . Let ρ ≥ 16π be close to 16π. By Theorem 1.2 and Lemma 4.3, we know that for any two nontrivial solutions φ1 and φ2 of (1.1)ρ and (1.2), there exists a g ∈ O(3) such that φ1 (y) ≡ φ2 (gy) on S 2 . By Lemma 4.3, we have that the orbit O(φ) = {φg | φ is a nontrivial solution of (2.6)} is nondegenerate. Note that O(φ) is homeomorphic to RP 2 . Thus, χ(O(φ)) = 1. By Wang’s theorem and Lemma 4.6, we know the contribution of the degree due to O(φ) is equal to (−1)4 χ (O(φ)) = 1. Thus, d(ρ) = −1 + 1 = 0, where −1 is the local degree due to the trivial solution. Hence, Theorem 1.1 is proved completely. Acknowledgments. I wish to thank Professor Louis Nirenberg for showing me the result of Wang in [23], and the referees for their helpful comments. References [1] [2]

[3] [4] [5] [6]

C. Bandle, Isoperimetric Inequalities and Applications, Monogr. Stud. Math. 7, Pitman, Boston, 1980. E. Caglioti, P.-L. Lions, C. Marchioro, and M. Pulvirenti, A special class of stationary ﬂows for two-dimensional Euler equations: A statistical mechanics description, Comm. Math. Phys. 143 (1992), 501–525. , A special class of stationary ﬂows for two-dimensional Euler equations: A statistical mechanics description, II, Comm. Math. Phys. 174 (1995), 229–260. S.-Y. A. Chang, M. J. Gursky, and P. C. Yang, The scalar curvature equation on 2- and 3-spheres, Calc. Var. Partial Differential Equations 1 (1993), 205–229. S. Chanillo and M. Kiessling, Rotational symmetry of solutions of some nonlinear problems in statistical mechanics and in geometry, Comm. Math. Phys. 160 (1994), 217–238. W. X. Chen and C. Li, Classiﬁcation of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), 615–622.

536 [7] [8] [9] [10] [11] [12]

[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

CHANG-SHOU LIN C.-C. Chen and C.-S. Lin, Estimate of the conformal scalar curvature equation via the method of moving planes, II, J. Differential Geom. 49 (1998), 115–178. , Singular limits of a nonlinear eigenvalue problem in two dimension, preprint. K.-S. Cheng and C.-S. Lin, On the conformal Gaussian curvature equation in R2 , J. Differential Equations 146 (1998), 226–250. W. Ding, J. Jost, J. Li, and G. Wang, The differential equation u = 8π − 8πheu on a compact Riemann surface, Asian J. Math. 1 (1997), 230–248. , Existence results for mean ﬁeld equations, preprint. B. Gidas, W. M. Ni, and L. Nirenberg, “Symmetry of positive solutions of nonlinear elliptic equations in Rn ” in Mathematical Analysis and Applications, Part A, Adv. Math. Supp. Stud. 7a, Academic Press, New York, 1981, 369–402. M. K.-H. Kiessling, Statistical mechanics of classical particles with logarithmic interactions, Comm. Pure Appl. Math. 46 (1993), 27–56. Y. Y. Li, Harnack type inequality: The method of moving planes, Comm. Math. Phys. 200 (1999), 421–444. Y. Y. Li and I. Shafrir, Blow-up analysis for solutions of −u = V eu in dimension two, Indiana Univ. Math. J. 43 (1994), 1255–1270. C. S. Lin, Uniqueness of conformal metrics with prescribed total curvature in R2 , to appear in Calc. Var. Partial Differential Equations. , Uniqueness of solutions of the mean ﬁeld equation on S 2 , to appear in Arch. Rational Mech. Anal. M. Nolasco and G. Tarantello, On a sharp Sobolev-type inequality on two-dimensional compact manifolds, Arch. Rational Mech. Anal. 145 (1998), 161–195. L. M. Polvani and D. G. Dritschel, Wave and vortex dynamics on the surface of a sphere, J. Fluid Mech. 255 (1993), 35–64. J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304 –318. M. Struwe and G. Tarantello, On multivortex solutions in Chern-Simons gauge theory, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1 (1998), 109–121. G. Tarantello, Multiple condensate solutions for the Chern-Simons-Higgs theory, J. Math. Phys. 37 (1996), 3769–3796. Z. Q. Wang, Symmetries and the calculations of degree, Chinese Ann. Math. Ser B. 10 (1989), 520–536.

Department of Mathematics, Chung-Cheng University, Minghsiung, Chia-Yi, Taiwan

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

Recommend Documents