Automorphic forms, representations, and L-functions, Part 2

c}(z, cp) = z(c)q n C¢)

I:

ue(J)* mod naCx)

( )

z(u) !!__ C

which follows from (3.2.6. 1) and (3.2.6.2) where the notation is explained. Thus in the nonarchimedean case we can define, for V and ¢ over any field E of charac­ teristic 0 (an open subgroup of I acting trivially on V, and ¢ trivial on some n;n @), an e1( V, ¢) E E*, in a unique way such that e1( Va, g;a) = e1( V, ¢)a for any homo­ morphism a: E --> E' and such that e1 is the old eb given by (3.6. 10), when E = C. So defined, e1(VE, ¢ · TrE1 F) is inductive in degree 0 (2.3.2) for every field of scalars E of characteristic 0, and e1(V, ¢) will be given by (3.6. 1 1) if V corresponds to a quasi-character X : F* --> E*. In writing these notes I was tempted to shorten things a bit by using only e1(V, ¢) instead of ev(V, ¢, dx), but decided against it because (I) the ev-system avoids all choices and is the most general and flexible-any other system, like eL or e1 can be immediately described as a special case of ev ; (2) the dependence of e on dx shows "why" e is inductive only in degree 0, and (3) in case our local field F is nonarchi­ medean, the ev-system, like the e l > works over any field E of characteristic 0, as soon as one defines the notion of Haar measure on F with values in E (cf. [D3, (6. 1)]). 4. The Weil-Deligne group, .A-adic representations, L-functions of motives. The representations considered in §3 are just the beginning of the story. Those of Galois type are effective motives of degree 0-which Deligne calls Artin motives in his article [D6, §6] in these PROCEEDINGS-with coefficients in C. We cannot discuss the notion of motive here (cf., e.g., [Dl] and [D6] for this) but we do want to discuss the way in which L-functions and e-functions are attached to motives of any degree. Only very special motives of degree # 0 correspond to the representations of WF considered in §3, namely, those of type A0, i.e., those which, after a finite extension E/F, correspond to direct sums of Heeke characters of type A0 over E. (A candidate for a "motivic Galois group" for these is constructed by Langlands in these

19

NUMBER THEORETIC BACKGROUND

PROCEEDINGS [L3].) The simplest motives not of this type are those given b y elliptic curves with no complex multiplication ; their £-functions are the "Hasse-Weil zeta-functions" which are not expressible in terms of Heeke's £-functions. The procedure for attaching £-functions to motives in the form given it by Deli gne [03], [D6] can be outlined schematically as follows : F is a global field . v i s a place of F. E is a field of finite degree over Q . ..\ runs through the finite places of E whose residue characteristic is prime to

char(F). (J is an embedding of E in C. Motive M over F with ex. multn. by E

System (H;.(M)) of ..\-adic representations of GF /-adic coho.

gd. fld . extension

I I v archimedean {!.

Hodge structure Hdg(M.) over F. with ex. multn. by E (cf. (4.4))

cf. (3 .3), (3.4)

II {!.

v nonarchimedean ; cf. (4.2)

Equiv . class V(M• . a) of repns. of the Weil-Deligne group w�. over C, invariant under Aut(C/E)

ll

Equiv. class V(M•. a) of repns. over of the Weil group wF.

c

restriction

System (H;.(M.)) of ..\-adic representations of GF.

Motive M. over F. with ex. multn. by E Hodge theory cf. (4.4)

ll

ll

ll

cf. (4. 1 )

L- (i.e. F-) and e-factors L- and e-factors at the at the archimedean place v nonarchimedean place v In the next sections we discuss some of the steps and concepts indicated in the above chart. We begin with the Weil-Deligne group. This is a group scheme over Q, but what counts, its points in and representations over fields of characteristic 0, can be described naively with no reference to schemes. (4'.1) The Weil-Deligne group and its representations. Let F be a nonarchimedean local field and let P, GF = Gal(F/F), WF (Weil group), and I (inertia group) have their usual meaning. For w E WF, let ll w \1 denote the power of q to which w raises elements of the residue field, as in ( 1 .4.6). Thus we have II w II = 1 for w E I, and \l
nEZ

20

J. TATE

where A n is the ring of locally constant Q-valued functions on ifJnJ. (4.1.1) DEFINITION [D3, (8.3.6)]. The Wei/-Deligne group Wj;. is the group scheme over Q which is the semidirect product of WF by Ga, on which WF acts by the rule wxw- 1 = llw l lx .

Let E be a field of characteristic 0. The group W};.(E) of points of Wj;. with coordinates in E is just E x WF with the law of composition (ah w 1 )(a2, w2) = (a1 + l w1 l l a2, w 1 w2) for ah az E E and wh w2 E WF. Let V be a finite-dimensional vector space over E. A homomorphism of group schemes over E p ' : w;.

xQ

E -- GL( V)

determines, and is determined by, a pair (p, N) as in (4.1.2) below, such that, on points, p' ((a, w)) = exp(aN) · p(w). That is the explanation for the following defini­ tion : (4.1.2) DEFINITION [D3, (8.4.1)]. Let E be a field of characteristic 0. A representa­ tion of Wf. over E is a pair p' = (p, N) consisting of: (a) A finite-dimensional vector space V over E and a homomorphism p : WF � GL(V) whose kernel contains an open subgroup of I, i.e., which is continuous for the discrete topology in GL( V) . (b) A nilpotent endomorphism N of V, such that p(w)Np(w)- 1 = l w iiN, for W E W. (4.1.3) $-semisimplicity. Let p' = (p, N) be a representation of Wf. over E. Define v : WF � Z by l lw ll = q-v <w> . There is a unique unipotent automorphism u of V such that u commutes with N and with p( WF) and such that exp(aN)p(w)u-v <w> is a semisimple automorphism of V for all a E E and all w E WF-1 [D3, (8.5)]. Then p;. = (pu-v, N) is called the $-semisimplification of p', and p' is called $-semisimple if and only if p' = p;., i.e., u = 1, i.e., the Frobeniuses act semisimply. For this it is necessary and sufficient that the representation p of WF be semisimple in the ordi­ nary sense, because p($) generates a subgroup of finite index in p( WF) , and in characteristic 0 a representation of a group is semisimple if and only if its restriction to a subgroup of finite index is semisimple. In his article in these PROCEEDINGS, Borel discusses admissible morphisms W; � LG ; when G = GLm these are just our $-semisimple (p, N)'s. (4.1.4) ExAMPLE. Sp(n) is the following representation (p, N) of Wf. over Q. V = Qn = Qeo + Qe 1 + · · · + Qen-h ( = l l w l l ;e;) , p(w)e; = w,-(w)e; Ne; = e;+ l (0 � i < n - 1), Nen- 1 =

0.

(4.1.5) Given any (p, N), Ker N is stable under WF.· Hence (p, N) is irreducible ¢> 0 and p is irreducible. It is not hard to show that the $-semisimple indecom­ posable representations of Wf. are those of the form p ' ® Sp(n) with p' irreducible. (The ® is defined by (p, N) ® (ph N1) = (p ® Ph N ® 1 + 1 ® N1).) (4.1.6) Let (p, N, V) be a representation of Wf. over E. We put Vk = (Ker N)I and define a local L-factor, a conductor, and a local constant by N=

21

NUMBER THEORETIC BACKGROUND

Z( V, t) a(V)

=

=

e( V) =

and

det (1 - f/Jt I Vft)- 1 , and L(V, s) a(p) + dim V1 - dim Vft, e(p) det ( - f/J I V1/ Vft),

=

Z(V, q-•), when E c C;

e( V, t) = e( V)t a .

Here, for e, the usual cf; and dx are understood, but omitted from the notation. These quantities do not change if we replace V by its f/J-semisimplification ; but note that they are not additive as functions of V, because VN is not. If N = 0, they are the same as before. One of the main reasons for introducing the Weil-Deligne group is the fantastic generalization of local class field theory embodied in : (4. 1 .7) Conjecture. Let F be a nonarchimedean local field and n an integer � 1 . There is a (in fact more than one) natural bijection between isomorphism classes of f/J-semisimple representations of w;.. of degree n, and of irreducible admissible re­ presentations of GL(n, F). For n = I this is local class field theory. For n = 2, it is discussed at length in [D2, (3.2)]. In this conjecture, for any n, the irreducible representations of w;.. (which are just irreducible representations of WF) should correspond to the cuspidal representations of GL(n, F). I understand that Bernstein and Zelevinsky have shown that the way in which arbitrary admissible representations of GL(n, F) are built out of cuspidal ones follows the same pattern as the way in which arbitrary f/J-semisimple representations of w;.. are built up out of irreducible ones. Thus the main problem is now the correspondence between irreducibles and cuspidals. A more general conjecture, involving an arbitrary reductive group G rather than just GL(n), relates admissible representations of G(F) to homomorphisms of w;.. into the "Langlands dual" of G (see Borel's article in these PROCEEDINGS) . This more general conjecture is the nonarchimedean local case of "Langlands' philo­ sophy". (4.2) ).-adic representations. Now suppose I is a prime different from the residue characteristic p of F and let t 1 : IF --+ Q1 be a nonzero homomorphism. (Such a t 1 exists and is unique up to a constant multiple, because the wild ramification group P is a pro-p-group, and the quotient l/P is isomorphic to the product flrtpZ1.) We have t1 (wow- 1 ) = l wll t1(o') , for q e I, w e W, because conjugation by w induces raising to the ll w ll power in I/P. Let f/J be an in­ verse Frobenius element (4. 1 .8). Suppose EA is a finite extension of Q1• A ).-adic representation of WF is a finite-dimensional vector space VA over EA and a homo­ morphism of topological groups pA : WF --+ GLElVA) where GLdVA) has the 'A-adic topology (i.e., the topology given by the valuation). (4.2. 1) THEOREM (DELIGNE [03, §8]). The relationship VA pA((/Jn q) = p(f/Jn q)exp ( tt (q) N),

=

q e /, n e

V and Z,

22

J. TATE

sets up a bijection between the set of ).-adic representations (p;., V;.) of WF and the set of representations (p, N, V) of w;.. over E;.. The corresponding bijection between iso­ morphism classes of each is independent of the choice oft1 and f/J.

To show that every p;. is of this form one uses

(4.2.2) CoROLLARY (GROTHENDIECK). Let (p;., V;.) be a ).-adic representation of WF · There exists a nilpotent endomorphism N of V;. such that p;.(u) = exp( t1 (u) N) for u in an open subgroup of I. A proof of the corollary can be found in the appendix of [ST]. Here is a sketch. Since I is compact, p;.(l) stabilizes a "lattice" L in V;.. Replacing F by a finite exten­ sion we can assume that p;.(l) fixes L (mod 1 2). Then p;.(l) is a pro-1-group, so is a homomorphic image of t 1(/), since Ker t1 is prime to 1. Choose c e Q 1 such that ct1(1) = Z1 • Then there is an a e G L( V;.) fixing L (mod 1 2) such that p;.(u) = act t < u> = exp(tt {u) N)

for all u e I, where N = c log a. Conjugating by p;.(f/J) we find p;.(f/J)Np;.(f/J)-1 = q-1N. Thus the set of eigenvalues of N is stable under multiplication by q-1. Since q is not a root of unity in characteristic 0, it follows that the only eigenvalue of N is zero, i.e., N is nilpotent. (4.2.3) CoROLLARY. If V;. is a semisimp1e 'A-adic representation of WF then some open subgroup of I acts trivially on V;., so V;. can be viewed as an "ordinary " represen­ tation of WF ·

For any V;. the kernel of N is stable under WF because p;.(w)NPA(w)-1 = I w li N. So if V;. is irreducible, then N = 0 , and the statement follows from (4.2.2). A semi­ simple V;. is a direct sum of irreducible subrepresentations. (4.2.4) In view of (4.2.3), e( V;.) and a ( V;.) have meaning if V;. is semisimple. For arbitrary V;., if (p;. , V;.) and (p, N, V) correspond as in (4.2.1), we define the L- and e-factors associated to V;. to be those associated to V. These can be expressed di­ rectly in terms of V;. as follows : Z( V, t)

=

e( V)

=

a( V) =

det(l - f/J I vn = Z( V;., t), a( V�•) + dim ( V�•) I - dim V;. = a( V;.) , det( - f/J I ( V�·) I) e( V�·) det( - f/J 1 vn = e( V;.),

and where v�· is the semisimplification of V;. in the ordinary sense. One can define a "f/J-semisimplification" of V.t. analogous to that of V (4. 1 .3). The quantities on the right do not change if we replace V;. by its f/J-semisimplification, but they are not additive in V;., because Vi is not. (4.2.4) Motives. Suppose now E is a finite extension of Q. Let M be a motive with complex multiplication by E, defined over our nonarchimedean local field F ([Dl], [D6]). Let n be the rank of M. Attached to M will be 1-adic representations H1 (M),

NUMBER THEORETIC BACKGROUND

23

vector spaces of dimension n over Q1 on which GF acts continuously, one for each

I # char(F). The field E will act on these, and for each I we get a decomposition

H1(M) = EBm H;.(M), where for each place A of E above /, we put H;.(M) = E;. ® E(:9Q, H1(M), a vector space of dimension m over E, where m is the rank of M over E, given by n = m [E: Q]. For each I i: p and each A above / let H;(M) be the representation of W � over E;. corresponding to H;.(M) by (4.2. 1). If our motive M lives up to expectations, the system of A-adic representations H;.(M) will be compatible over E in the sense that the system H;(M) is compatible over E in the following naive sense : for any two finite places )., f.l of E not over p and every commutative diagram E;.

E

/ � � /c Ef.!.

the m-dimensional representations of w;,. over C, H;(M) ® E• C and H;(M) ® Ep C, are isomorphic (or at the very least, have isomorphic
24

J. TATE

of (/). Let ¢ : X0 --+ X0 be the Frobenius morphism, and q : k --+ k the Frobenius au­ tomorphism. The composition q> x q acts on X0 = X0 x k by fixing points and by mapping/�-+ fq in the structure sheaf. This map induces (a morphism canonically isomorphic to) the identity on the site (X0).1 , so the action of the Frobenius mor­ phism q> on Hr(X0, Q1) is the same as that of q- 1 , which is the one corresponding to our (/) under the isomorphism ( * ) . That is why Deligne calls (/) the geometric Fro­ benius.

Deligne [D4] has proved Weil's conjecture, that the characteristic polynomial of

q> acting on Hr(X0 , Q 1) has coefficients in Z, is independent of 1, and that its com­

plex roots have absolute value qr 1 2 . From the independence of 1 it follows in this case of good reduction that the f/J-semisimplifications of the H1(M)'s form a com­ patible system ; and the H1(M)'s are known to be f/J-semisimple for r = 1. It is natural to say that a motive M over F has good reduction, or is unramified if and only if H1(M) = H1(M)l, i.e., if V(M) = V(M)N. In case M = H1 (A), A an abelian variety, this is equivalent to A having good reduction (criterion of Neron­ Ogg-Shafaryevitch in [ST]) . Similarly we say M has potential good reduction ¢> N = 0, and M has semi­ stable reduction if V(M) = V(M)l. Clearly this latter can always be achieved by a finite extension of the ground field. (4.4) F archimedean. Let now M, E, n, m be as in (4.2.4), but take F to be archi­ medean, instead of nonarchimedean. Let z : F --+ C be the embedding of F in C if F is real, or one of the two isomorphisms of F on C if F is complex. Such a z gives us a motive M. over C and M. has a "Betti realization" H8(M.) which is an n­ dimensional vector space over Q whose complexification HB(M.) ® C = EB HPq(M,) is doubly graded in such a way that the map 1 ® c (c = complex conjugation) takes HPq to HqP. (For example, if M = Hr(X) as in (4.3), then H8(M.) = Hr(x;n, Q), where x;n is the complex analytic variety underlying the scheme X x F.• C, and the complexification of this space, H r(X�n, C), is doubly graded by Hodge theory.) Let z = c o z: F --+ C be the map conjugate to z. By transport of structure, there is an isomorphism .. : HB(M.) --+ H8(M,) such that '" ® c preserves the bigrading on the complexifications ; hence '" ® 1 carries HPq(M,) onto HqP(M,). The field E of complex multiplications acts on HB(M.) preserving the bigradation on the com­ plexification, and '" is an E-homomorphism. Let q : E --+ C. Putting V,(M..) = H8(M.) ®E, .. c we obtain a bigraded complex vector space of dimension m and a linear isomorphism 't" ® 1 : V.(M..) --+ V2(M..) taking Vf'l to VjP. There is a natural action of the Weil group WF on these spaces as follows : F complex. z : F � C an isomorphism, WF = F * , and WF acts on Vf'l by scalar multiplication via the character z-P(z)-q. Clearly, 't" ® 1 is Wyequivariant, so the two representations V,(M..) and V.(M..) are isomorphic. We let V(M..) denote their isomorphism class. F real. z = z : F --+ C is the embedding, and WF = C* U jC * . This time M, = M,, so we have only one space, V.(M..) = V.(M.. ), and ,. ® 1 is an automorphism of it. The action of WF on it is as follows : u e C * acts as multiplication by u-P(u)-q on Vf'l. j acts as iP+q('t" ® 1) on Vf'l. Again, let V(M..) denote the equivalence class of this representation.

NUMBER THEORETIC BACKGROUND

25

Notice that the representations obtained from motives via Hodge theory are very special, in that the p and q are integers. Finally, define L(M", s) and e(M", s) to be the L- and e-factors associated to the representation V(M") as in §3. For a table making these explicit see [06, 5.3]. (4 . 5) F global. Let F be a global field and M a motive with complex multiplication by E, defined over F. For each place v of F, let Mv denote the restriction of M to Fv. Let (J : E -+ C. The product L(M", s) = Ti v L(Mv. "' s) converges in a right half­ plane. It is conjectured that it is meromorphic in the whole s-plane and satisfies the functional equation L(M", s) = e(M", s) L(M: , 1 s) with M * = Hom(M, Q) and e(M", s) = Ti v e(MVotl, s). In the function field case this conjecture has been proved by Deligne. Let q be the number of elements in the constant field k of F. Grothendieck proved that for any given A-adic representation V of Gp which is unramified at all but a finite num­ ber of places v, the corresponding £-function L(V, s) = TI v L( Vv, s) is a rational function of q-s (even a polynomial if vc = 0 and Vc; = 0, where G is the geometric Galois group, i.e., the kernel of the map of Gp to Gk), and satisfies a functional equation of the form L(V, s) = e( V, s )L( V*, 1 s) with an e which is a monomial in q-s of degree !: v [k(v) :k] a( V.). Later, Deligne showed that Grothendieck's e( V, s) is equal to the product of the local e( V., s)'s if V = V, 0 is a member of a family ( VA)A E.>!' of A-adic representations of Gp for some infinite set of places !£ of a number field E, and the family is compatible in the following weak sense : for each A, f-t E !£ there is a finite set S of places of F such that for v If. S, the represen­ tations VA and VJt are unramified at v and the characteristic polynornals of (/Jv acting on VA and VJt have coefficients in E and are equal. Deligne's method is to prove that Grothendieck's e is congruent to the product of the local e's modulo A for all A e !£ and is therefore equal to that product. By (4.3) any il-adic representa­ tion coming from l-adic cohomology, i.e., from a motive, is a member of a system which is weakly compatible in the above sense. When dim( V) = 2, then by Jacquet-Langlands (resp. Weil), Springer Lecture Notes 1 14 (resp. 1 89), these results show that L( V, s) comes from an automorphic representation of (resp. modular form on) GL2(Ap) . On the other hand, Drinfeld has recently shown that automorphic representations of GL2 give rise to systems of /-adic representations occurring as constituents in tensor products of those corn­ ing from 1-dirnensional l-adic cohomology, hence from motives. Thus for GL2 over function fields, the equivalence between motives, compatible systems of /­ adic representations, and automorphic representations is pretty well established. In this connection it should be mentioned that Zarhin [Z] has proved the isogeny theorem over function fields : if two abelian varieties A and B over a global function field F give isomorphic /-adic representations, then they are isogenous ; more pre­ cisely, Q1 ® Homp(A, B) = HomcA VtCA), Vt(B)). Over number fields our knowledge is not nearly so advanced. For Artin motives of rank 2, Langlands has made a beginning with the theory of base change (see the remarks (3.5.5)). For elliptic curves M over Q, it is not even known whether -

-

26

J. TATE

L(M, s) has a meromorphic continuation throughout the s-plane, or whether the isogeny theorem is true. For a more detailed account of our ignorance, as well as of a few things which are known, see [84]. REFERENCES [A] E. Artin, Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren, Abh. Math . Sem . Univ. Hamburg 8 (1 930), 292-306 (collected papers, pp. 1 65-1 79). [AT] E. Artin and J. Tate, Class field theory, Benjamin, New York, 1 967. [B] J. Buhler, Icosahedral Galois representations, Ph.D. Thesis, Harvard Univ., (January 1 977). [CF] J. W. S. Cassels and A. Frohlich, Algebraic number theory, Academic Press, London, 1 967. [Dl] P. Deligne, Les constantes des equations fonctionnelles, Seminaire Delange-Pisot-Poitou 1 969(70, 19 bis. [D2] --, Formes modulaires et representations de GL(2), Antwerp II, Lecture Notes in Math., vol. 349, Springer-Verlag, 1 973, pp. 55-106. [D3] , Les constantes des equations fonctionnelles des fonctions L, Antwerp II, Lecture Notes in Math., vol. 349, Springi:lr-Verlag, 1 973, pp. 501-595. [D4] , La conjecture de Wei/. I, Inst. Hautes Etudes Sci. Pub!. Math. 43 (1 974), 273-307. [DS] , Les constantes locales de !"equation fonctionnelle de Ia fonction L d'Artin d'une representation orthogonale, Invent. Math. 35 (1 976), 299-3 1 6. [D6] Valeurs de fonctions L et periodes d'integrales, these PROCEEDINGS, part 2, pp. 3 1 3-346. [DS] P. Deligne and J.-P. Serre, Formes modulaires de poids 1, Ann. Sci. Ecole Norm. Sup. (4) 7 (1 974), 507-530. [K] H. Koch, Classification of the primitive representations of the Galois groups of local fields, Invent. Math. 40 (1 977), 1 95-216. [Ll] R. P. Langlands, On Artin's L-functions, Complex Analysis, Rice Univ. Studies No. 56, 1 970, 23-28 ; also the mimeographed notes : On the functional equation of Artin 's L-functions. [L2] --, Modular forms and 1-adic representations, Lecture Notes in Math., vol. 349, Springer-Verlag, 1 973, pp. 362-498. [L3] --, Automorphic representations, Shimura varieties, and motives. Ein Miirchen, these PROCEEDINGS, part 2, pp. 205-246. [Sl] J.-P. Serre, Corps locaux, Hermann, Paris, 1 962. --

--

--

--,

[S2]

��-

, Facteurs locaux des fonctions zeta des varietes algebriques (Definitions et conjectures),

Seminaire Delange-Pisot-Poitou, 1 969/70, no. 19. --

[S3] , Modular forms of weight one and Galois representations, Proc. Durham Sympos. on Algebraic Number Fields, Academic Press, London, 1 977, pp. 193-267. [S4] , Representations 1-adiques, Algebraic Number Theory, Papers for the Internat. Sympos. , Kyoto 1 976 (S. Iyanaga, Ed.), Japan Soc. for the Promotion of Science, Tokyo, 1977, pp. 177-193. [STJ J.-P. Serre and J. Tate, Good reductio n of abelian varieties, Ann . of Math. (2) 8 8 (1 968), 492-5 1 7 . [Tl] J. Tate, Fourier analysis in number fields and Heeke's zeta functions, Thesis, Princeton Univ., (1 950) (published in [CF], 305-347). [T2] , Local constants, Proc. Durham Sympos. on Algebraic Number Fields, Academic Press, London, 1 977. [Wl] A. Wei!, Sur Ia theorie du corps de classes, J. Math . Soc. Japan 3 (1951), 1 -35. [W2] , Basic number theory, Springer, New York, 1967. [W3] , Sur les formules explicites de Ia theorie des nombres, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 3-1 8. [Z] Ju. G. Zarhin, Endomorphisms of abelian varieties over fields of finite characteristic, Izv. --

--

--

--

Akad. Nauk SSSR Ser. Mat. 39 (2) (1 975) HARVARD UNIVERSITY

=

Math USSR-Izv. 9 (1 975), 255-260.

Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 2, pp. 27-61

AUTOMORPHIC L-FUNCTIONS

A. BOREL This paper is mainly devoted to the L-functions attached by Langlands [35] to an irreducible admissible automorphic representation rc of a reductive group G over a global field k and to local and global problems pertaining to them. In the context of this Institute, it is meant to be complementary to various seminars, in particular to the GL2-seminars, and to stress the general case. We shall therefore start directly with the latter, and refer for background and motivation to other seminars, or to some expository articles on this topic in general [3] or on some aspects of it [7], [14], [15], [23].

The representation rc is a tensor product rc = @.re. over the places of k, where is an irreducible admissible representation of G(k.) [11]. Accordingly the L-func­ tions associated to rc will be Euler products of local factors associated to the tr:.'s. The definition of those uses the notion of the L-group LG of, or associated to, G. This is the subject matter of Chapter I, whose presentation has been much influ­ enced by a letter of Deligne to the author. The L-function will then be an Euler product L(s, rc , r ) assigned to rc and to a finite dimensional representation r of LG. (If G = GL" ' then the L-group is essentially GLn(C), and we may tacitly take for r the standard representation rn of GLn(C), so that the discussion of GLn can be carried out without any explicit mention of the L-group, as is done in the first six sections of [3].) The local L- and s-factors are defined at all places where G and tr: are "unramified" in a suitable sense, a condition which excludes at most finitely many places. Chapter II is devoted to this case. The main point is to express the Satake isomorphism in terms of certain semi simple conjugacy classes in LG (7 . 1 ). At this time, the definition of the local factors at the ramified places is not known in . general. For GLn and rn o however, there is a direct definition [19], [25]. In the general case, the most ambitious scheme is to associate canonically to an irreducible admissible representation of a reductive group H over a local field E a representa­ tion of the Weil-Deligne group We of E into LH, and then use L- and s-factors associated to representations of We [60] . This problem is the main topic of Chapter III. The L-function L(s, rc, r ) associated to rc and r as above is introduced in § 1 3. In fact, it is defined in general as a product oflocal factors indexed by almost all places of k. It converges absolutely in some right half-plane (13.3 ; 14. 2) . Some of the main rc .

AMS (MOS) subject classifications (1 970). Primary IOD20 ; Secondary 1 2A70, 1 2B30, 14G IO,

22E50. © 1979, American Mathematical Society

27

A.

28

BOREL

conjectural analytic properties (meromorphic continuation, functional equation), and the evidence known so far, are discussed in §14. From the point of view of [35], a great many problems on automorphic represen­ tations and their £-functions are special cases of one, the so-called lifting problem or problem of functoriality with respect to L-groups. It is discussed in Chapter V. It is closely connected with Artin's conjecture (see § 1 7 and the base-change sem­ inar [17]). In § 1 8 brief mention is made of some known or conjectured relations between automorphic £-functions and the Hasse-Weil zeta-function of certain varieties, to be discussed in more detail in the seminars on Shimura varieties [8], [40]. Thanks are due to H. Jacquet and R. P. Langlands for various very helpful com­ ments on an earlier draft of this paper. CONTENTS

I. L-groups . . . . . . . . .. . .. .. . .. . . .. . . 28 1 . Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2. Definition of the L-group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3. Parabolic subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1 4 . Remarks o n induced groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5. Restriction of scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 II. Quasi-split groups; the unramified case . . . . . . . . . . . . . . . . 35 6. Semisimple conjugacy classes in LG . . . . . . . . . . . . . . . . . 35 7. The Satake isomorphism and the L-group. Local factors in the unramified case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 III. Wei/ groups and representations. Local factors . .. . . . .. 39 8. Definition of (/)(G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 9. The correspondence for tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41 10. Desiderata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1 1 . Outline of the construction over R, C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46 12. Local factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 IV. The L-function of an automorphic representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 13. The £-function of an irreducible admissible representation of GA .49 14. The £-function of an automorphic representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 V. Lifting problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 1 5. £-homomorphisms of L-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 16. Local lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 1 7. Global lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 1 8 . Relations with other types of £-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 References 59 . .

.

. .

. . . . .

. .

. . . .

. . .

.

.

. .

.

.

. .

. . . .

. .

. . . .

. .

. . . .

. . .

. . . .

. .

. . . .

. .

.

. . . . . . .

. . . . . . . . . . .

. .

.

. .

.

. .

. . . .

. . . . . . . . . . .

.

. . . . . . . . . .

. .

. .

.

. . . . . . . .

. . . . . . . . . .

.

.

. . .

.

. . . . . .

. . . . . . .

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

CHAPTER I.

L-GROUPS.

k is a field, k an algebraic closure of k, k, the separable closure of k in k, and rk the Galois group of k, over k. G is a connected reductive group, over k in 1 . 1 , 1 .2, 2. 1 , 2.2, over k otherwise.

§§ 1, 2 will be used throughout, §3 from Chapter III on. The reader willing to take on faith various statements about restriction of scalars need not read §§4, 5.

AUTOMORPHIC £-FUNCTIONS

29

1. Classification. We recall first

some facts discussed in [58]. 1 . 1 . There is a canonical bijection between isomorphism classes of connected reductive k-groups and isomorphism classes of root systems. It is defined by as­ sociating to G the root datum c);( G) = (X*(T), rp, X* (T), rp v) where T is a maximal torus of G, X * (T) (X* (T)) the group of characters (!-parameter subgroups) of T and (/) ((/) v) the set of roots (coroots) of G with respect to T. 1 .2. The choice of a Borel subgroup B ::::> T is equivalent to that of a basis L1 of W(G, T). The previous bijection yields one between isomorphism classes of triples (G, B, T) and isomorphism classes of based root data cj;0(G) = (X*(T), Ll, X* (T), LJv) . There is a split exact sequence (1) -------> Int G -------> Aut G -------> Aut cj;0(G) -------> (1). (1) To get a splitting, we may choose Xa E Ga (a E Ll) and then have a canonical bijec­ tion (2)

Two such splittings differ by an inner automorphism Int t (t E T). 1 .3. Given r E rk there is g E G(ks) such that g - r T-g- 1 = T, g - TB · g- 1 = B, whence an automorphism of cj;0(G), which depends only on r · We let flc : rk ---> Aut cj;0(G) be the homomorphism so defined. If G' is a k-group which is isomorphic to G over k (hence over k5), then flc = flc' G, G' are inner forms of each other. 1 .4. Letf: G ---> G' be a k-morphism, whose image is a normal subgroup. Then f induces a map cj;(f) : c);( G) ---> cj;(G') (contravariant (resp. covariant) in the first (last) two arguments). Given B, T c G as above, there exists a Borel subgroup B' (resp. a maximal torus T') of G' such thatf(B) c B', f(T) c T', whence also a map cj;0(f): cJ;o( G) ---> cJ;o( G'). 2. Definition of the £-group. 2. 1 . The inverse system lff "6 to

the based root datum Iff0 = (M, Ll, M*, LJV) is c); "6 = (M*, LJV, M, Ll). To the k-group G we first associate the group LG o over C such that cj;0(LG0) = cj;0(G)V. We let LT o , LB o be the maximal torus and Borel subgroup de­ fined by cj;(j, and say they define the canonical splitting of LG0• Let f be as in 1 .4. Then / also induces a map cJ;'fJ(f) : cj;0(G')V ---> cj;0(G)v. An alge­ braic group morphism of LG'0 into LG o associated to it will be denoted Lf0• Given one, any other is of the form Int f o Lf o o int t' (t E LT o , t' E LT'0), and maps LT'0 (resp. LB'0) into LT" (resp. LB0). 2.2. EXAMPLES. (1) Let G = GLn . Then £ GO = GLn . In fact, let M = zn with { x; } its canonical basis. Let { e; } be the dual basis of M* = zn . Then lf!0(GLn) = (M, Ll, M*, LJV) with Ll = { (x; - X;+ 1 ) , I � i < n } , LJV = { (e; - e;+ l ) , I � i < n}, hence cj;0 = c); "6. (2) Let G be semisimple and lf!0(G) = (M, (/), M* , cpv). As usual, let P(W) c M ® Q be the lattice of weights of (/) and Q(W) the group generated by (/) in M. Define P((/)V) and Q(wv) similarly. As is known G is simply connected (resp. of adjoint type) if and only if P(¢) = M (resp. Q(¢) = M). Moreover P(W)

=

{A E M (8) Qj (A, cpv)

c

Z},

P((/)V)

=

{A E Mx (8) Qj(A, (/)) E Z}.

30

A. BOREL

Therefore : G simply connected ¢> LGo of adjoint type ; G of adjoint type ¢> LGo simply connected. (3) Let G be simple. Up to central isogeny, it is characterized by one of the types A m . . . , G2 of the Killing-Cartan classification. It is well known that the map r/J0{G) -+ rjJ0{G)• permutes Bn and Cn and leaves all other types stable. Thus if G = Sp2n (resp. G = PSp2n), then LGo = S02n+I (resp. LGo = Spinzn+ I ). In all other cases, G ,_ LGo preserves the type (but goes from simply connected group to adjoint group, and vice versa). (4) Let again G be reductive and let/ : G -+ G' be a central isogeny. Let N = coker f* : X* (T) ----+ X* (T') (T' = f(T)), N' = coker f * : X * (T') ----+ X* (T). Then N and N' are isomorphic and ker Lr = Hom(N, C * ) _:. N. In particular, yo is an isomorphism if and only iff is one. (5) Let f : G -+ G' be a central surjective morphism, Q = ker f, and Q0 the identity component of Q. Then ker Lr � Q/Q0• If Q is connected, then T" = T n Q is a maximal torus of Q, and the injectivity of Lfo follows from the fact that the exact sequence I -+ T" -+ T -+ T' -+ I neces­ sarily splits. If Q is not connected, then r : H = G/Q0 -+ G' is a nontrivial separable isogeny, with kernel Q/Q0• yo factors through Lro and, by the first part and (4), ker Lfo = ker Lr o � Q/Q0• In particular, if we apply this to the case where G' = Gad is the adjoint group of G, and use (2), we see that the derived group of LGo is simply connected if and only if the center of G is connected. As an example, let G = GSp2n be the group of symplectic similitudes on a 2n-qimensional space. Then the derived group of LGo is isomorphic to Spinzn+I · In fact, we have LGo = (GL 1 x Spinzn+ I )/A where A = { I , a} and a = (ab a2), with a 1 of order two in GL1 and a2 the nontrivial central element of Spinzn+ l · If n = 2, then Spinzn+ l = Spzn · It follows that if G = GSp4, then LGo = GSp4(C). 2.3. We have canonically Aut Iff0 = Aut lff'-6 . Therefore we may view f.J.c as a homo­ morphism of rk into Aut r/J'-6. Choose a monomorphism (1) as in I . 2(2) . We have then a homomorphism p 'r; : rk ____.. Aut (LG0, LB O , Lro).

The associated group to, or L-group of, G is then by definition the semidirect product (2) with respect to f.J.c · We note that f.J.c is well defined up to an inner automorphism by an element of Lro. The group LG is viewed as a topological group in the obvious way. The canonical splitting of LGo (2. I) is stable under rk. We have a canonical projection LG -+ rk with kernel LG0• The splittings of the exact sequence 1 ____.. LGO ____.. LG � rk ____.. I (3)

AUTOMORPHIC £-FUNCTIONS

31

defined a s in 1.2 via an isomorphism Aut W� -=. Aut(LG0, LB0, LT0, {xa}) are called admissible. They differ by inner automorphisms Int t (t e LT0). Note that if G splits over k, then rk acts trivially on LGo and LG is simply the direct product of LGo and rk.

2.4. REMARKS. (I) So far, we can in this definition take LGo over any field. We have chosen C since this is the most important case at present, but it is occasionally useful to use other local fields. (2) There are various variants of this notion, which may be more convenient in certain contexts. For instance we can divide rk by a closed normal subgroup which acts trivially on wv, hence on LG0, e.g., by rk' if k' is a Galois extension of k over which G splits. Then rk is replaced by Gal(k'/k), and LG is a complex reductive Lie group. We can also define a semidirect product LGo )(I z, for any group Z endowed with a homomorphism into rk, e.g., the Weil group of k, if k is a local or global field. In that case, we get the "Weil form" of LG. (3) Let G' be a k-group which is isomorphic to G over k. Then G and G' are inner forms of each other if and only if LG is isomorphic to LG' over r k· In fact, the first condition is equivalent to f.lG = f.lG' • and the latter is easily seen to be equivalent to the second condition. In particular, since two quasi-split groups over k which are inner forms of each other are isomorphic over k, it follows that if G, G' are quasi­ split and LG ..:. LG' over rk then G and G' are k-isomorphic. 2.5. Functoriality. Let f: G --+ G' be a k-morphism whose image is a normal subgroup. Then /q,0 : ¢0(G) --+ cp0(G') clearly commutes with Fi., hence so does fi¥ : ¢0( G')• --+ ¢0( G) v and Lr : LG'0 --+ LGo. We get therefore a continuous homo­ morphism Lf : LG' --+ LG such that

is commutative, which extends Lr . 2.6. Representations. For brevity, by representation of LG we shall mean a conti­ nuous homomorphism r : LG --+ GLm(C) whose restriction to LGo is a morphism of complex Lie groups. Clearly, ker r always contains an open subgroup of Fk, hence r factors through LGO )(I rk 'lk • where k' is a finite Galois extension of k over which G splits. The group LGo )(I r k lk is canonically a complex algebraic group and r is a morphism of com­ plex algebraic groups. '

3. Parabolic subgroups. 3.1. Notation. We let f/J(Gfk) denote the set of parabolic k-subgroups of G, and

write �(G) for �(Gfk). Let p(G/k) be the set of conjugacy classes (with respect to G(k) or G(k), it is the same) of parabolic k-subgroups, and p(G) = p(G/k). Let p(G) k be the set of conjugacy classes of parabolic subgroups which are defined over k (i.e., if p E (J e p(G)k, then rp E (J for all r E rk). In particular p(Gfk) 4 p(G)k. There is equality if G is quasi-splitfk. 3.2. We recall there is a canonical bijection between p(G) and the subsets of L1.

32

A. BOREL

Then p(Gh corresponds to the Fk-stable subsets of L1 and p(G/k) to those Fk­ stable subsets which contain the set L10 of simple roots of a Levi subgroup of a minimal parabolic k-group. In particular we have p(Gfk) = p(G)k if G is quasi­ split over k. Given P E &P(G), we let J(P) be the subset of L1 assigned to the class of P Since two conjugate parabolic subgroups whose intersection is a parabolic sub­ group are identical, we see in particular that if P is defined over k, P' :::> P, and the class of P' is defined over k, then P' is defined over k. 3.3. Parabolic subgroups of LG. A closed subgroup P of LG is parabolic if rc(P) = rk and p o = LGO n p is a parabolic subgroup of LG0• Then p = NLc(P0). In other words, a parabolic subgroup is the normalizer of a parabolic subgroup p o of LG0, provided the normalizer meets every class modulo LG0• We say P is standard if it contains LB. The standard parabolic subgroups are the subgroups (1) where Lp o runs through the standard parabolic subgroup of LG o such that J(LP 0) c LJV is stable under Fk · Every parabolic subgroup of LG is conjugate (under LG or, equivalently, LG o) to one and only one standard parabolic subgroup. We let &P(LG) be the set of parabolic subgroups of LG and p(LG) the set of their conjugacy classes. The given bijection L1 ...... LJV yields then, in view of 3.2, a bijection (2) We shall say that a parabolic subgroup of LG is relevant if its class corresponds to one of p(G/k) under this map. We let Lgp(LG) be the set of relevant parabolic sub­ groups and Lp(LG) the set of their conjugacy classes, the relevant conjugacy classes of parabolic subgroups. Thus, by definition p(Gjk) .._. Lp(LG). (3) Thus, if G and G' are inner forms of each other, p(LG) and p(LG') are the same, but Lp(LG) and Lp(LG') are not. If G' is quasi-split, then Lp(LG') == p(LG') ; hence we have an injection Lp(LG) c Lp(LG') = p(LG'). (4) If �G is anisotropic over k, then Lp(LG) consists of G alone. 3.4. Levi subgroups. Let P be a parabolic subgroup of LG. The unipotent radical N of p o is normal in P and will also be called the unipotent radical of P. Then P/N _.:, rjN � Fk · In fact, it follows from (1) that P is a split extension of N, and is the semidirect product of N by the normalizer in P of any Levi subgroup M0 of P o . Those normalizers will be called the Levi subgroups of P. Let P E f/J(G/k), M a Levi k-subgroup of P. Let Lp be the standard parabolic subgroup in the class associated to that of P (see (3)). Then LM may be identified to a Levi subgroup of LP. In fact if M corresponds to (X * (T), J, X* (T), Jv), then LM o corresponds to (X* (T), Jv, X * (T), J) and LM o � rk is equal to LM by definition and is a Levi subgroup of LP, as defined above. A Levi subgroup of a parabolic subgroup P of LG is relevant if P is.

AUTOMORPHIC £-FUNCTIONS

33

For the sake of brevity, we shall sometimes say "Levi subgroup in G" for "Levi subgroup of a parabolic subgroup of G." Similarly for LG. 3.5.

LEMMA.

The proper Levi subgroups in LG are the centralizers in LG of tori in

g&(LG0), which project onto rk.

Let M be a proper Levi subgroup in LG. It is conjugate to a subgroup > fl'(D). 4. Remarks on induced groups. (To be used mainly to discuss restriction of scalars in §5 and 6.4.) 4. 1 . Let A be a group, A' a subgroup of finite index of A and E a group on which A' operates by automorphisms. Then we let ( 1) Ind1- (E) = J�,(E) = {! : A --+ E i f(a'a) = a' ·f(a) (a E A ; a' eA ')} . It is a group (composition being defined by taking products of values). It is viewed as an A-group by right translations : (x, a E A). r af(x) = f(xa) (2) fl'(St

For s E A'\A, let E. = {fe /�,(E) if(a) = 0 if a i s} . (3) Then E. is a subgroup, J�,(E) is the direct product of the E.'s (s E A'\A), and these subgroups are permuted by A. The subgroup E8 is stable under A ' and is isomorphic to E as an A' module under the map f f-+f(e). The product of the E. ' s (s e A'\A, s #- e) is also stable under A'. We have therefore canonical homomor­ phisms

34

A. BOREL

(4) E XI A' --+ I(E) XI A' --+E XI A' whose composition is the identity. 4.2. Let B be a group, p :B -+ A a homomorphism. Let B' = f.l- l (A') and assume that f.l induces a bijection : B'\B � A'\A. Let E be a group on which A' operates by automorphisms, also viewed as a B'-group via f.l · Then/ �-+ f.l o f induces an isomorphism (1) whose inverse is p-equivariant. This follows immediately from the definitions. 4.3. Let A, E be as before, C a group and v : C -+ A a homomorphism. Let rp : C -+ E XI A be a homomorphism over A. The map cjJ : C -+ E such that rp(c) = (cjJ(c), v( c)) (c e C) is a 1-cocycle of C in E and rp ...... cjJ induces a bijection Hl (C ; E) ___::__. ffJA(C, E),

where, by definition, ffJA(C, E) denotes the set of homomorphisms rp : C -+ E XI A over A, modulo inner automorphisms by elements of E. 4.4. Let A, A', B, B' and E be as in 4.2. We have a commutative diagram with exact rows 1 --+ I1.(E) --+ 11-(E) XI A --+ A --+ 1 (1) 1 --+ E --+ E IX A' --+ A' ---+ 1 where the vertical maps are natural inclusions (4. 1). Let rp: B -+ J1, (E) XI A be a homomorphism over A. Using 4. 1 (4), we get by restriction a homomorphsim q; : B' -+ E XI A' over A'. 4.5. LEMMA. The map rp ...... q; of 4.4 induces a bijection rpA (B, I1.(E)) �

i

i

i

rp A'(B', E).

We have, using 4.2, 4.3 : ffJA (B, I1, (E )) (1) (/)A,(B', E) (2)

= Hl(B ; 11 - (E)) = HI(B; IB- (E) ) , = H l(B' ; E) .

By a variant of Shapiro's lemma, contained, e.g., in [4, 1 .29] : Hl(B ; IB- (E))

___::__.

HI (B' ; E),

and it is clear that the isomorphisms (1), (2) carry this isomorphism over to rp ...... q;. 5. Restriction of scalars. In this section, k' is a finite extension of k in k., G' is a connected k'-group, and G

= Rk'tk G'.

5. 1 . The Galois group rk' of k. over k' is an open subgroup (of finite index) of rk and zk', k = rk',rk may be identified with the set of k-monomorphisms of k' into k• . We have, in the notation of 4. 1 (with A = Fk, A' = rk,) G(k) = If{. ( G '(k)) = IT 11G' (k). (1) IIEFk• \Fk

AUTOMORPHIC £-FUNCTIONS

35

Assume G' to be reductive. Then we see easily that cf;(G) = (M, tp, M*, tp'V) is related to cf;(G') = (M', tp', M' * , tp'") by tp = U tp' · a. M = IF!. (M'), (2) A A aE '\

Similarly, if L1' is a basis of tp', then (3)

is one for tp. From this i t follows that we have a natural isomorphism (4) We have then a commutative diagram

5.2. The map P ' .... Rk'tk P ' induces a bijection between fJJ(G' /k') and fJJ(G/k). Moreover P ' is a Borel subgroup of G' if and only if Rk'tk P' is one of G. Hence G' is quasi-split/k' if and only if G is quasi-split over k (see [5, §6]). Since G(k) ...:::. G'(k'), we also get a bijection p(G'/k') ...:::. p(G/k). If J' c L1' is stable under rk' then J = U a E A '\Ara(J') is stable under rk. This map is easily seen to yield a bijection between rk .-stable subsets of L1' and rk-stable subsets of L1, whence also canonical bijections CHAPTER II.

QUASI-SPLIT GROUPS. THE UNRAMIFIED CASE.

In this chapter, G is a connected reductive quasi-split k-group. From 6.2 on, G is assumed to split over a cyclic extension k' of k, and a denotes a generator of Gal(k'/k). 6. Semisimple conjugacy classes in LG. 6. 1 . Assume B and T to be defined over k. Then the action of rk on X*(T) or

X* (T) given by f.J.G coincides with the ordinary action. The greatest k-split subtorus Td of T is maximal k-split in G, and its centralizer is T; in particular, Td contains regular elements of G. Hence any element w e W which leaves Td stable is com­ pletely determined by its restrictiOn to Td . It follows that k W may be identified with the subgroup of the elements of W which leave Td stable or, equivalently, with the fixed-point set of rk in W. If we go over to the £-group and identify canonically W with W(LG0, LT0), then kW is also the fixed-point set of rk in W, and it operates on the greatest subtorus s of LT" which is pointwise fixed under rk • The group s always contains regular elements ; hence any element of k W is determined by its restriction to S. We let kN be the inverse image of k W in the normalizer N of LT o in LG o . 6.2. LEMMA. Every element w e k W has a representative in kN which is fixed under

(J .

36

A.

BOREL

Write L1• = D 1 U · · · U Dm , where the D;'s are the distinct orbits of F(k' fk) in Ll•. Let o; be the common restriction to S of the elements of D; and S; the identity component of the kernel of o;. Then k W, viewed as a group of automorphisms of S, is generated by the reflections s; to the S; (1 � i � m) , and it suffices to prove the lemma for w = S; (1 � i � m) . [The "reflection" s,. is the unique element # 1 of W which leaves S stable, fixes S; pointwise and is of order two.] We let Lie(M) denote the Lie algebra of the complex Lie group M. For a E Ll•, let, as usual (1) It is one-dimensional. Fix i between 1 and m . By construction of LG, we can find nonzero elements ea E 9« (resp. e_a E 9-« (a E D,.)) which are permuted by (J. We have then (2) where c is # 0, and independent of a E D,. since F(k' /k) is transitive on D;. Here X iLT0) ® c is identified with Lie(Lr o), and a with a ® 1 . The element (3) is fixed under (J. Moreover, since the difference of two simple roots is not a root, we have (4) Using (3) and (4), we get (5) By the transitivity of Gal(k'/k) on D,., the number d = � ( a , �) (6) de D;

is also independent of � E D; ; therefore (7) We claim that d # 0, in fact that d = 1 , 2. The irreducible components of D; are permuted transitively by Gal(k' fk) and have a transitive group of automorphisms. Therefore they are of type A 1 or A2 • Then, accordingly, d = 2 or d = 1 . It follows that h;, /; and /-; span a three-dimensional simple algebra pointwise fixed under (J. Then so is the corresponding analytic subgroup G; of LGo. The group G,. centralizes S; and S n G; is a maximal torus of G;, with Lie algebra spanned by h;. Then the nontrivial element of W (G;, S n G;) is the required element. REMARK. An equivalent statement is proved, in a different manner, in [35, pp. 19-22]. 6.3. We let Y = L(Tdt · The group X * (Td) may be identified to the fixed-point set of Fk in X* (T). The inclusion of X iTd) = X * ( Y) into X* (T) = X * (LT0) induces a surjective morphism LTo � Y, to be denoted v. The map A : t �--+ rl . "t is an endomorphism of LT0, whose differential dA at 1 is (d(J - Id). Let

37

AUTOMORPHIC £-FUNCTIONS

U

(l)

= (ker At,

V = im A.

Then U is pointwise fixed under (J , the Lie algebra of U (resp. V) is the kernel (resp. image) of dA. Since dA is semisimple, they are transversal to each other; hence (2) LTO = U· v, and u n v is finite. Moreover, V = ker v, v(U) = Y. (3) In the rest of this chapter, we let LG stand for the "finite Galois form" LG o >4 Gal(k' fk) of the L-group. We now want to discuss the semisimple conjugacy classes in LG o >4 o with respect to LG0• We have (4) therefore LG0-conjugacy in LG o >4 o is equivalent to o-conjugacy in LG0• 6.4. LEMMA. Let v' : Lro >4 o --+ Y be defined by v' (t x o) = v(t) (t e Lr o). Then v' induces a bijection

( 1) Let n e kN. By 6.2, we m!lY write n = w·s with w = " w and s E LT0 • Then the LT0-component of n- 1 (t >4 o) n is s- l . w-l . t · w"s = s- 1 . 11s · (w-l . t · w) E V· w-l . t · w ;

hence

>4

o) · w . Thus v' is equivariant with respect to the projection kN --+ k W and therefore induces a map of the left-hand side of (l) into the right-hand side of (l), which is obviously surjective. Let t, t' E LT o and assume that v'(t >4 o) = w- 1 • v'(t' >4 o) · w for some w e k W. Then we have v(t) = v(w- 1 · t' · w), where w is a representative of w fixed under o, whence t = v · (w-l . t' · w), with v e V. We can write v = r l . "s for some s e LT0, and get t >4 o = n- 1 (t' >4 o)n, with n = ws. 6.5. LEMMA. Let (LG o >4 o) •• be the set of semisimple elements in LG o >4 o. Then v'(n-l . (t

o) · n

) = v(w-l . t · w) = w-l . v(t) · w = w-l . v'(t

>4

the map

p. : (LT o

>4

o)/IntkN -----+ (LG o x o) ••/Int LG o ,

induced by inclusion is a bijection.

By results of F. Gantmacher [12, Theorem 14], p. is surjective. Let now s, t E LTo and g E LG o be such that g- l . (s >4 o) · g = t >4 o, i.e., such that g-l . s · "g = t. Using the Bruhat decomposition of LG o with respect to LB0, we can write uniquely g = u · n · v, with u, v in the unipotent radical of LB o and n in the normalizer N of LT0• These groups are stable under o, and normalized by LT0• We have then hence "n s - 1 = n · t. Therefore the connected component of n in N is stable under ·

38

A BOREL

i.e., n represents an element of ,. W; hence n e ,.N, and (t XI a) and (s XI a) are conjugate under ,.N. REMARK. This proof was suggested to me by T. Springer. 6.6. If M is a complex affine variety, we let C[M] denote its coordinate algebra. The algebra C[Y] may be identified with the group algebra of X * ( Y) = X* (Td). The quotient Y/,. W is also an affine variety (in fact isomorphic to an affine space) and C[Yf,. W] = C[Y]�w. Let Rep(LG) c C[LG] be the subalgebra generated by the characters of finite dimensional holomorphic representations. Its elements are constant on conjugacy classes. In particular, they define by restriction functions on (LG o XI a)••/Int LG0 • a,

6.7. PROPOSITION. The map

a = p. o v- 1 : Y/,. W -----+ (LG o XI a)../Int LG o is a bijection, which induces an isomorphism of C[ Y/,. W] onto the algebra A of restric­ tions of elements of Rep(LG). REMARK. We shall use 6.7 only when k is a nonarchimedean local field. In that case 6.7 is proved in [35, pp. 1 8-24]. PROOF. That a is bijective follows from 6.4, 6.5. We prove the second assertion as in [35]. Let p be a finite dimensional holomorphic representation of LG and /p the function on Lrc defined by /p(t) = tr p(t XI a). It can be written as a finite linear combination / = I; clA of characters A e X * (LT0). Since tr p is a class function on LG, we have.[p(s- 1 • t · "s) =fp( t) for all s, t e LT0 • By the linear independence of charac­ ters, it follows that if cl :/= 0, then A is trivial on V {cf. 6.3(1)), hence is fixed under a, i.e., may be identified to an element of X * ( Y). Thus we may view /p as an element of C[ Y]. But invariance by conjugation and 6.4 imply that fe C[Y/,.W], whence a map {3 : A -+ C[Y/,. W], which is obviously induced by a. There remains to see that {3 is surjective. Note that C[Y/,. W] is spanned, as a vector space, by the functions (1)

where A runs through a fundamental domain C of k W on XiTd). But it is standard that we may take for C the intersection of X* (Td) with the Weyl chamber of W in X* (T) defined by B. Therefore every A e C is a dominant weight for LG o with respect to LT0• It is then the highest weight of an irreducible representation 'Kl of LG o . Since it is fixed under a , the representation "'Kl : g ...... 'Kl("g) is equivalent to 'Kl · From this it is elementary that 'Kl extends to an irreducible representation ir:l of LG of the same degree as ""l · The highest weight space is one-dimensional, stable under a. Let c be the eigenvalue of a on it. Then the trace gives rise to a function equal to c · lfJl modulo a linear combination of functions lfJ w with p < A, in the usual ordering. That im {3 contains lfJl (A e C) is then proved by induction on the ordering. 7. The Satake isomorphism and the L-group. Local factors. 7. 1 . We keep the previous notation and conventions. We assume moreover k to

be a nonarchimedean local field, k' to be unramified over k, and a to be the image of a Frobenius element Fr in F,.. Let Q be a special maximal compact subgroup of G(k) [61]. We assume Q n T is the greatest compact subgroup of T(k) and Q contains representatives of ,. W. Let

39

AUTOMORPHIC £-FUNCTIONS

H(G(k), Q) be the Heeke algebra of locally constant, Q-bi-invariant, and compactly supported complex valued functions on G(k). The Satake isomorphism provides a canonical identification H --:::. C[ Y/k W], hence also one of Ylk W with the characters of H [6]. By 6. 7, we have now a canonical isomorphism of H with the algebra A of restric­ tions of characters of finite dimensional representations of LG to semisimple £ GO­ conjugacy classes in (LGo ><1 a) , hence also a canonical bijection between characters of H(G(k) , Q) and semisimple classes in LGo ><1 a. Furthermore, each such class can be represented by an element of the form (t, a), with t e LTo fixed under a (and is determined modulo the finite group u n v, in the notation of 6.3). 7 .2. Local factors. Assume now that U is hyperspecial [61]. Let cp be an additive character of k. Let (n, U,,:) be an irreducible admissible representation of G(k) of class I for Q and r a representation of LG. Then the space of fixed vectors of Q in U,. is one-dimensional, acted upon by H via a character X n · To the latter is assigned by 7. 1 a semisimple class Sx in LGo ><1 a. We then put (I)

L(s, n, r)

=

det(l - r((g

><1 a)) ,

q-• )- 1 ,

e(s, n, r, cp)

= I,

where q is the order of the residue field, and (g, a) any element of Sx. CHAPTER

III. WElL GROUPS AND

REPRESENTATIONS. LOCAL FACTORS.

In this section, k is a localfield, Wk (resp. w;) the absolute Wei/ group (resp. Weil­ Deligne group) ofk. /f H is a reductive k-group, then H(H(k)) is the set of infinitesimal equivalence classes of irreducible admissible representations of H(k). G denotes a connected reductive k-group. The main local problem is to define a partition of H(G(k)) into finite sets Hrp,G or Hrp indexed by the set (/J(G) of admissible homomorphisms of w; into LG, modulo inner automorphisms (see §8 for (/J(G)), and satisfying a certain number of condi­ tions. So far, this has been carried out for any G if k = R, C [37], for tori over any k [34] and (essentially) for G = GL2 (cf. 12.2). §9 recalls the results for tori ; § 1 0 describes some o f the conditions t o b e imposed on this parametrization ; § I I

summarizes the construction over R or C. Such a parametrization would allow one to assign canonically local L- and e-factors to any 1r e H(G(k)) and any complex representation of LG. Two elements n , n ' in the same set Hrp would always have the same local factors, and are hence called £-indistinguishable. In the case of GL,. however, local factors have been defined in an a priori quite different way, so that the parametrization problem becomes subordinated to one concerning L- and e-factors. This is discussed in § 1 2. 8. Definition of (/J(G). 8. 1 . Jordan decomposition in w;. If k = R, C, then W� = Wk and, by definition, every element of Wk is semisimple. Let k be nonarchimedean. Then x e W� is said to be unipotent if and only if it belongs to Ga ; the element x is semisimple if either e(x) # 0 or x is in the inertia group. Here e : w; --> Z is the canonical homomorphism w; --> Wk --> k* --> Z. Every element x e w; admits a unique Jordan decomposition x = x, · xu with x, semisimple, Xu unipotent and X5Xu = xuxs [60] .

40

A. BOREL

8.2. The set l/J(G). We consider homomorphisms a : w� --+ LG over rk, i.e., such

that the diagram

is commutative, and which satisfy moreover the following conditions : (i) a is continuous, a(Ga) is unipotent, in LG0, and a maps semisimple elements into semisimple elements (in LG : x = (u, r) is said to be semisimple if its image under any representation (2.6) is so). (ii) If a( W;) is contained in a Levi subgroup of a parabolic subgroup P of LG, then P is relevant (3.3). Such a's are called admissible. We let l/J(G) be the set of their equivalence classes modulo inner automorphisms by elements of LGo. If we write a(w) = (a(w), v(w)) with a(w) e LGo then w ._. a(w) is a 1-cocycle of w� (acting on LGO via w; --+ rk) in LG0• It follows that (1)

Let H be a subgroup of w;. Then a: w; --+ LG is said to be trivial on H if v(H) acts trivially on LGo and a(H) = { 1 }. Note that if v(H) ! acts trivially on LG0, then al8 is a homomorphism. 8.3. Assume G' is an inner quasi-split form of G. Then (1 )

In fact LG

l/J(G) �

c

l/J(G').

LG' and Lp(LG') =:J Lp(LG) ; therefore a

e

l/J(G) = a e l/J(G').

8.4. PROPOSITION. Let k' be a .finite separable extension of k; let G' be a connected reductive k-group and G = Rk'lk G' . Then there is a canonical bijection l/J(G) ....::, l/J(G'). E

We consider the situation of 5.2, 5.4 with A = LG'0• We have the injections (8.2) :

Moreover LGo

=

=

rk,

A'

=

rk'• B

=

w;, B'

=

W� , .

Ifz.(LG'0) (see 5. 1 ) ; whence, by Shapiro's lemma and 5.2 : fll( w� ; LG0) ____::::____, fll( w�, ; LG'0) ,

But it is clear that this isomorphism maps l/J(G) onto l/J(G'). 8.5. Let ZL = C(LG0). If a : wk --+ ZL and b : w; --+ LG O are 1-cocycles, then ab : w ._. a(w)b(w) is again a 1 -cocycle of W� in LG0• If a is continuous and b cor­ responds to (/) e l/J(G), then ab corresponds to an element of l/J(G). We get therefore maps (1 )

(2)

fll( Wk ; ZL)

X

Hl( W� ; LG0) - Hl( W; ; LG0),

H1( Wk ; ZL)

X

l/J(G) - (/) (G),

which define actions of the group Hl( Wk ; ZL) on the sets fll( W; ; LG0) and l/J(G).

41

AUTOMORPHIC £-FUNCTIONS

8.6. PROPOSITION. Let cp : w� --> LG be an admissible homomorphism. Then the Levi subgroups in LG which contain rp( W�) minimally form one conjugacy class with respect to the centralizer of cp( W�) in LG0• Since cp( W;) projects onto a dense subgroup of rk by definition, this follows from

3.6.

REMARK. Formally, this also applies to the archimedean case, but the proof in that case is simpler [37, pp. 78 -79]. In fact, the argument there applies in all cases to admissible homomorphisms of the Weil (rather than Weil-Deligne) group because cp( Wk) is always fully reducible. In this case, the Levi subgroups which contain cp( Wk) minimally are those of the parabolic subgroups which contain cp( Wk) mini­ mally. Those parabolic subgroups form therefore one class of associated groups. 9. The correspondence for tori.

9. 1 . Let T be a complex torus. A continuous homomorphism cp : T --> C* is described by a pair of elements A, p e X*(T) ® C such that A p e X*(D, by the rule cp( t) = tJ. tl'. Similarly, a continuous homomorphism cp :C* --> T is given by p, v e X* (D ® C such that p v e X* (T) ; we have cp(z) = z�'z", meaning that, for any A e X*(T), A o cp : C* --> C* is given by -

-

A (cp(z))

= z :z .

This can also be interpreted in the following way : identify X* (T) ® C with the Lie algebra Lie ( T(C)) . Then the exponential map yields an isomorphism (X* (T) ® C)/21CiX* (T) = T(C). Then p, v e Lie(T(C)) are such that cp(eh) = eh·p+fi·p (h e C) . 9.2 . Let G = T be a k-torus, and I

= dim T.

Any cp e f/J(G) is trivial on Ga ; hence

(1) where Hlt refers to continuous cocycles. On the other hand (2) We have canonically [34, Theorem I] H(G) = f/J(G). (3) In fact, LT and Wk are replaced in [34] by a finite Galois form LTo :><1 rk'tk and a relative Weil group Wk'tk• where k' is a finite Galois extension of k whose Galois group acts trivially on Lro ; this is easily seen not to change f/J(G). The proof then consists in showing first that the transfer from Wk' tk to k' * yields an isomorphism and second that the pairing (5) associated to the evaluation map (t, A) ..... A(t) (t

e

LTo ; A

e

X* (T)) yields an

42

A. BOREL

isomorphism of the first group onto the group of characters of the second group, which is then (3) by definition. For illustrations, we discuss some simple cases. 9.3. k = C. Then Wk = C * and f/J(G) = Hom(C * , LT0). The correspondence follows from 9. 1 since both Hom(C * , LT0) and Hom(T, C*) are canonically identi­ fied with { (A, ,u) I A, ,u E X * (T) (8) C, A - .u E X * (T)}. 9.4. k = R. We have (1) Wn = C * � {z-} with z- 2 = - 1 , z- · z · z--1 = z (z e C * ) . Put C * = S x R+, with S = {z E C * , z · z = 1 } . Then Int z- is the identity on R+, the inversion on S. Write
(2)

= zp. · z"

(z E

C *), 1-' - v E X * (T),

(see 9. 1). We have tp(z) = O' (cp(z)) (z E C * ) ; hence v = O'(f-t). Fix h E X*(T) (8) C such that a = exp 2nih. Then the character 11: associated to

n(ex)

=

exp(( h , x - 0' · x)) · exp(( f-t , x +

0'

·

x) )/ 2 (x E X* (T) (8) C)

p. 27]. Here - denotes the complex conjugation of Lie(T(C)) = X* (T) (8) C with respect to X* (T) (8) R; hence x >-> 0' · .X is the complex conjugation with respect to Lie(T(R)). It follows that e" E T(R) if and only if x - 0' x E 2ni · X* (T). EXAMPLES. (a) Let T be anisotropic over R. Then 0' = - 1 and we may assume a = 1 , h = 0. We have e" E T(R) if x is purely imaginary and then (3) yields 11: = !-' · The fact that cp( - 1) = 1 shows that 1-' E X * (T), confirming that ll(T(R)) = X*(T). (b) Let T be split over R. Then 0' = 1, 1-' = v, cp(z) = (z · z)P., a2 = 1 and h E X*(T)/2. We have e" E T(R) if and only if x - x E 2ni · X * (T). It is then easily checked that 11: is given by 1-' on the connected identity component of T(R), while its restriction to the torsion subgroup of T(R) is the character naturally. defined by h. 9.5. The unramified case. Let k be nonarchimedean, and assume T to split over an unramified extension k' of k. A character X of T(k) is said to be unramified if it is trivial on the greatest compact subgroup O T(k) of T(k). On the other hand,

-> v(t) yields a bijection [37,

•

(1) where Td is the greatest k-split torus of T (this can also be expressed by saying that

AUTOMORPHIC £-FUNCTIONS

the inclusion Td We have then

c

T induces

an isomorphism

Tik)f0 Tik) ..::::. T(k)fOT(k),

43 cf. [6]).

(2) The group rk operates on LTD via the cyclic group rk'/k which is generated by the image O" of a Frobenius element Fr. An unramified qJ is completely determined by qJ(Fr), which can be written qJ(Fr) = (t, Fr), where t E LTD is determined up to con­ jugacy by LTD. Thus qJ unr( T) = (Lye >
(1)

1

----.

D ----. G

----.

G

----.

1.

Since H1(Fk ; D) = 0, the map p. : G (k) � G(k) i s surjective. Let Gsc be the universal covering of the derived group f!§G of G. We have a commutative diagram 1

1

1�

1 ---> Gsc---> G ---> G / Gsc---> 1

�1 11

G

44

A. BOREL

Going 6ver to £-groups, we get

Since LG�c is of adjoint type (2.2), we see that ZL seen that

=

ker Lao. Moreover, it is easily

(2)

This yields a map This allows us to associate to a E Hl( W� ; ZL) a character aa of (G/G.c)(k) which is trivial on D(k), hence a character na of G(k) = G(k)/D(k). It can be shown to be independent of the choice of D. The map a >--+ na is compatible with restriction of scalars [37, 2. 12] and satisfies : (4) 10.3. Conditions on the sets n'P. (l) If n E n'P, then n(z) = x'P(z) . Id (z E C(G)) . If q/ = a . g:> (g:> , g:> ' E rt>(G) , a E H 1 ( W� ; ZL)) (see 6. 5), then n'P' = {na® n l n E U'P } . (3) The following conditions on a set U'P are equivalent : (i) One element of U'P is square-integrable modulo C(G). (ii) All elements of U'P are square-integrable modulo C(G). (iii) g:>( W�) is not contained in any proper Levi subgroup in LG. (4) Assume g:>(Ga) = {1 }. The following conditions on a set U'P are equivalent : (i) One element of U'P is tempered. (ii) All elements of U'P are tempered. (iii) g:>( Wk) is bounded. (5) Let H be a connected reductive k-group and r; : H -+ G a k-morphism with commutative kernel and cokernel. Let g:> E rt>(G) and g:>' = Lr; o g:>. Then any n E U'P, viewed as an H(k)-module, is the direct sum of finitely many irreducible admissible representations belonging to n'P' " 10.4. The unramified case. We say that g:> E rt>(G) is unramified if it is trivial, in the sense of 6.2, on Ga and on the inertia group I. If so, Im g:> may be assumed to be in LT. Therefore, if rt>(G) contains an unramified element, then G is quasi-split (see 8.2 (ii)). Assume now G to be quasi-split, to split over an unramified Galois extension (2)

AUTOMORPHIC L-FUNCTIONS

45

k' of k, and let rp E W(G) be unramified. There exists t E (LTotk such that rp(Fr) = (t, Fr), (1) (9. 5) and we have (w E W�), (2) rp (w) = (t, Fr) d wl where e : W� --. Z is the canonical homomorphism. The element t defines an un­ ramified character X of a maximal k-torus T of a Borel k-subgroup B of G (9. 5). It is then required that ll

LG0• We may assume that Im rp ' is contained in the normalizer of Lr. Let n = rp ('r) and let w E W be the element of W represented by n. Then w2 = I . Let p., l.i E X * (T) ® C be such that (I) (z E C*), p. - l.i E X * (T) rp(z) = zl' · z" (see 9.1). We have zw ·.u . zw·v ; (2) rp(z) = n · rp (z) · n-1 hence p. = w · l.i, l.i = w · f.J.· Assume now that Im rp is not contained in any proper Levi subgroup in LG or, equivalently, that Im rp ' is not contained in any proper Levi subgroup in LG0• Then w = - Id and p. is regular : in fact, the proper Levi subgroups in LGo are the centralizers of nontrivial tori. This implies first that w does =

46

A. BOREL

not fix pointwise any nontrivial torus in LT0, hence w = - Id ; if now f-l were sin­ gular, then the centralizer of p(C * ) would contain a semisimple subgroup H :1- { 1 } stable under Int n , the latter would leave pointwise fixed a torus S :1- { 1 } o f H, and Im rp' would be contained in the centralizer Z(S) of S, a contradiction. Since 1.1 = w · p = - f-l• we have A(rp( - 1)) = ( - 1)<2p, A>, for all A e X* (T). (3) Let o be half the sum of the roots a of G with respect to T such that (p, a) > 0. Then, Lemma 3.2 of [37] implies in particular (4) A(rp'( - 1)) = ( - 1)<26· A>, for all A E x.(T). It follows that (5)

f-l e o +

X*(T).

Therefore f-l is among the elements of X * (T) ® Q which parametrize the discrete series in Harish-Chandra's theorem. We then let Drp be the set of discrete series representations of G(R) with infinitesimal character X" · If G(R) is compact, then nrp consists of the irreducible finite dimensional representation with dominant weight f-l - o. In that case, no proper parabolic subgroup of LG is relevant ; hence (/)(G) consists of the rp considered here. 1 0.6. Let G = GLm k nonarchimedean. Let cjJ be an admissible representation of W�. If it is irreducible, then cp(Ga) = 1 . If it is indecomposable, then it is a tensor product p ® sp(m) , where m divides n, p is irreducible of degree n/m, and sp(m) is m-dimensional, trivial on 1 , maps a generator of the Lie algebra of Ga onto the nilpotent matrix with ones above the diagonal, zero elsewhere, and w e W� onto the diagonal matrix with entries a(w)i (0 � i < n) [9, 3. 1 .3]. If X is a character of Wk (hence of k *), and rp = x ® sp(n) , then Drp consists of the special representation with central character determined by X· In fact, the Weil-Deligne group came up for the first time precisely to fit the special representations of GL2 into the general scheme (see [9]). 11. Outline of the construction over R, C. We sketch here the various steps which yield the sets 0"' when k = R. For the proofs see [37]. We note first that we may always assume rp( Wk) c N(LT0), and we can write (9. 1)

rp(z) = zp zv

(z e C * ; fl.•

1.1 e

X*(T) ® C,

fl. -

1.1

e X * (T)).

1 1 . 1 . LEMMA . Let rp e (/)(G). Assume rp( W8) is not contained in any proper Levi subgroup in LG. Then (i) G has a Cartan k-subgroup C such that (!?)G n S)(R) is compact [28, 3. 1]. (ii) f-l is regular ; rp(C * ) contains regular elements [37, 3.3]. The group LC o may be viewed as a maximal torus of LGo ; hence there is an isomorphism LC -+- L T defined modulo an element .of W. Therefore rp defines an orbit of W in (/)(C), hence, by 9.2, an orbit X"' of W in X(C(R)). [Note that W, which is defined in G(C), operates on C(R), since C(R) n f')G is compact, hence on X(C(R)).] 1 1 .2. Let G0 = C(R)((!?)G)(R)t. Let A0 be the set of representations of G0 which

AUTOMORPHIC L-FUNCTIONS

47

are square-integrable modulo the center, and have infinitesimal character irreducible representa­

X.< (A. E X!D). The induced representations 1r = Ifio?/Mn0) (no E A 0) are [37, p. 50]. By definition, DID is the set of equivalence classes of these tions [37, p. 54].

1 1 .3. Let p E tP(G). Let LM be a minimal relevant Levi subgroup containing Im p. It is essentially unique (8.6). We assume LM # LG ; we may view p as an ele­ ment of 1/>(M). By 1 1 .2, there is associated to it a finite set of ll!f>.M of discrete series representations of M. We may assume LM to be a Levi subgroup of a relevant parabolic subgroup Lp corresponding to P E fli(G/k). Then U = X * (T) ® R = X* (Lro) ® R. Let V be the subspace of elements of U which are orthogonal to roots of LM, and fixed under rk · It may be identified with the dual a; of the Lie algebra of a split compo­ nent A of P. Let � be the character of C(M) defined by the elements of ll!D,M· We may assume that 1 .; 1 E Cl(a;+). Let P1 be the smallest parabolic k-subgroup containing P such that 1 .; 1 , when restricted to ap1 , is an element of the Weyl chamber a g. Let M1 = z(ap) and P' = P n M1. Then P' is a parabolic subgroup of M1 . Moreover the restriction of le i to the split component M1 n Ap of P' is one ; therefore, for each p E n!f>.M • the induced representation Ind'}{l(p) is tempered. Let ll'!D be the set of all constituents of such representations. Then by definition, DID is the set of Langlands quotients J(Pl> lT) with lT E ll� (cf. [37, p. 82]). 1 1 .4. Complex groups. Assume now k = C. Then Wk = C * , and tP(G) may be identified to the set of homomorphisms of C * into LT0, modulo the Weyl group W, i.e., to ( 1) {(A, f.!. ) , where A., fl. E X * ( T) ® C, A. - fl. E X * (T)} modulo the (diagonal) action of W. In this case Im p is in the Levi subgroup LT of LB, which is the Lp of 1 1 .3. The set ll!f>.M consists of one character of T (cf. 9. 1). Choose P 1> M, as in 1 1 .3. Since the unitary principal series of a complex group are irreducible (N. Wallach), the set ll� consists of one element. Hence so does DID . Thus each DID is a singleton. The classification thus obtained is equivalent to that of Zelovenko. 1 1 . 5. Let G = GLn, k = R. In this case, it is also true that the tempered re­ presentations induced from discrete series are irreducible [22] ; therefore each set n� (cf. 9.3) consists of only one element, hence so does n!D and we get a bijection between IP(G) and UG(R). Let n = 2. If p is reducible, then Im p is commutative ; hence p factors through ( WR)•b = R * and is described by two characters f.!. • v of R * . Then DID consists of a principal series representation 7r(f.J., v) (including finite dimensional representations, as usual). In particular there are three p's with kernel C * , to which correspond respectively n(l , 1), n(sgn, sgn) and n( l , sgn), where sgn is the sign character. If p is irreducible, then p(z") may be assumed to be equal to (s0, ..-), where s0 is a fixed element of the normalizer of LTo inducing the inversion on it. p(R+) belongs to the center of LGo , and p(S) is sum of two characters, described by two integers. Then DID consists of a discrete series representation, twisted by a one-dimensional re­ presentation. 1 1 .6. As is clear from these two examples, the main point to get explicit knowl-

48

A. BOREL

edge of the sets nrp is the decomposition of representations induced from tempered representations of parabolic subgroups. This last problem has been solved by A. Knapp and G. Zuckerman [29], [30]. 1 1 .7. Remark on the nonarchimedean case. Langlands' classification [37] is also valid over p-adic fields [57]. In view of 8.6, it is then clear that the last step (1 1 . 3) of the previous construction can also be carried out in the nonarchimedean case. Thus, besides the decomposition of tempered representations, the main unsolved problem in the p-adic case is the construction and parametrization of the discrete series. 12. Local factors. 1 2. 1 . Let 7C E D( G(k)) and r be a representation of LG (2.6). Assume that 7C E Drp for some rp E f/J(G). For a nontrivial additive character cjJ of k, we let

e(s, 'JC, r) = e(s, 'JC , r, cp) = e(s, r o rp , cp), where on the right-hand sides we have the L- and e-factors assigned to the represen­ tation r o rp of Wk [60]. In the unramified situation of 1 0.4, this coincides with the definition given in 7.2. In view of what has been recalled so far, these local factors are defined if k is archimedean, or if k is nonarchimedean in the unramified case, or if G is a torus. 1 2.2. Let now G = GL,.. In this case there are associated to 7C E D ( G(k)) local factors L(s, '!C) and e(s, 'JC , cp) defined by a generalization of Tate's method, in [25] for n = 2, in [19] for any n, which play a considerable role in the parametrization problem and in the local lifting. A natural question is then whether these factors can be viewed as special cases of 1 2. 1 , where r = r,. is the standard representation of GL,., i.e., whether we have equalities (I) L(s, 7C) = L (s, 'JC , r,.), e(s, 'JC , ¢) = e(s, 'JC, r,., cjJ), with the right-hand side defined by the rule of 1 2. 1 . (a) Let n = 2 . It has been shown in [25] that the equivalence class of 7C i s charac­ terized by the functions L(s, 7C ® x), e(s, 7C ® X• cp), where X varies through the characters of k*. In this case, the parametrization problem and the proof of (I) are part of the following problem : (*) Given q E f/J(G), find 7C = 7C(fl) such that (I)

L(s,

'JC ,

(2)

L(s,

q

r) = L(s, r o rp),

® x) = L(s, 7C ® x),

for all x's, and prove that D(G(k)).

q

.....

e(s,

q

® X• ¢) = e(s, 7C ® X• cp)

7C(fl) establishes a bijection between f/J(G) and

This problem was stated and partially solved in [25]. The most recent and most complete results in preprint form are in [62] ; they still leave out some cases of even residual characteristic, although some arguments sketched by Deligne might take care of them (see [63] for a survey). As stated, the problem is local, but, except at infinity, progress was achieved first mostly by global methods : one uses a global field E whose completion at some place v is k, a reductive E-group H isomorphic to G over k, an element p E f/J(H/k) whose restriction to L(Hfkv) = LG is q, chosen so that there exists an automorphic representation 7C(p) with the L-series L(s, p) (see § 1 4 for the latter). This construe-

AUTOMORPHIC £-FUNCTIONS

49

tion relies, among other things, on Artin's conjecture in some cases, and [38] . In fact, it was already shown in [25] that (*) for odd residual characteristics follows from Artin's conjecture, leading to a proof in the equal characteristic case. At present, there are in principle purely local proofs in the odd residue characteristic case [6J] . Note also that the injectivity assertion is a statement on two-dimensional admissible representations of w;, namely, whether such a representation O" is determined, up to equivalence, by the factors L(s, O" o x) and e(s, O" o x , fjl). But, so far, the known proofs all use admissible representations of reductive groups [63] . (b) For arbitrary n, (I) has been proved in the unramified case, for special re­ presentations, and by H. Jacquet for k = R, C [24]. (c) Local L- and e-factors are also introduced for G = GL2 x GL2 in [21 ] , at any rate for products "" x 1e ' of infinite dimensional irreducible representations. Partial extensions of this to GLm x GLn for other values of m, n are known to experts. (d) For n = 3, 1e e D(G(k)) is again characterized uniquely by the factors L(s, "" ® x) and e(s, "" ® x. fjl) [27] , [46] . For n G 4 on, this is false [46] . How­ ever, it may be there are still such characterizations if X is allowed to run through suitable elements of ll(GLn_ 1 (k)) or maybe just ll(GLn_2(k)). 1 2. 3 . Local factors have also been defined directly for some other classical groups, in particular for GSp4 by F. Rodier [48], extending earlier work of M. E. Novodvorsky and I. Piatetskii-Shapiro, for split orthogonal groups, in an odd number 2n + 1 of variables by M. E. Novodvorsky [41 ] . In the latter case LGo = Sp2n, and in the unramified case, the local factors coincide (up to a translation in s) with those associated by 7.2 to the standard 2n-dimensional representation of the £-group. See also [42] . CHAPTER

IV. THE £-FUNCTION OF AN AUTOMORPHIC REPRESENTATION.

From now on, k is a globalfield, o = ok the ring of integers of k, A k or A the ring of adeles of k, V (resp. V.,, resp. V1) the set of places (resp. infinite places, resp. finite places) of V. For v e V, k., o. and Nv have the usual meaning. Unless otherwise stated, G is a connected reductive k-group. 13. The £-function of an irreducible admissible representation of GA. 1 3 . 1 . Let "" be an irreducible admissible representation of GA and r a representa­

tion of LG. There exists a finite Galois extension k' of k over which G splits and such that r factors through LGo ><1 rk' tk · We want to associate to "" and r infinite Euler products L(s, ""• r) and e(s, ""• r), whose factors are defined (at least) for al­ most all places of k. Let v E v. By restriction, r defines a representation r. of L(Gfk.) = LGO )
L(s,

(2)

e(s,

1e,

'JC ,

r) r)

=

=

D.L(s , "" • • r.),

n.e(s, ""·· r., fjl.),

where f/J. is an additive character of k. associated to a given nontrivial additive character of k, and the factors on the right are given by I 2. I(l).

50

A.

BOREL

The local problem is solved for archimedean v's, and for almost all finite v's (see below) so that the factors on the right are defined except for at most finitely many v e V1. For questions of convergence or meromorphic analytic continuation this does not matter, and we shall also denote such partial products by L(s, 7C, r). By I 0.4, q;. is well defined if the following conditions are fulfilled : G is quasi-split over k., G(o.) is a very special maximal compact subgroup of G(k.), k' is unramified over k, and 7Cv is of class one with respect to G(o.). All but finitely many v e V1 satisfy those conditions [61]. 1 3.2. THEOREM [35]. Let 7C be an irreducible admissible unitarizable representation of GA and r be a representation of LG (2.6). Then L(s, 7C, r) converges absolutely for

Re s sufficiently large.

We may and do view r as a complex analytic representation of LG o � r,,,k> where k' is a finite Galois extension of k over which G splits (2.7). We let V1 be the set of v e V1 satisfying the conditions listed at the end of 1 3. 1 . We have to show that (1)

L'

=

v EITVt

L(s, 7Cv, r.),

converges in some right half-plane. Let Fr. be the Froberiius element of rk'v• lk.• where v' E v,, lies over v E vl . We have q;.(Fr.) = (t., Fr.), with t. e LTo (2) and (3) L(s, 7C0 , r.) = (det(I - r((t., Fr.))N;•)) - 1 . To prove the theorem, it suffices therefore to show the existence of a constant a > 0 such that (4) l.u l � (Nv) a for every v e V1 and eigenvalue ,u of r((t., Fr.)). Let n = [k' : k]. Since we may assume t. fixed under r,. (6 . 3), we have t� = (t., Fr.)n ; hence it is equivalent to show (4) for all eigenvalues ,u of r(t.). These are of the fonD. t;, where it runs through the set Pr of weights of r, restricted to LG0• Thus we have to show the existence of a > 0 such that j t. j Re A � (Nv) a for all v E vl and it E P,. (5) Let G' be a quasi-split inner k-form of G. Then LG = LG', and G is isomorphic to G' over k. for all v e V1 . We may therefore replace G by G' ; changing the nota­ tion slightly, we may (and do) assume G to be quasi-split over k. We then fix a Borel k-subgroup B of G and view LT as the £-group of a maximal k-torus T of G. For a cyclic subgroup D of r,,,,, let Vv be the set of v E vl for which r,. is equal to the inverse image of D in F,. The group U = X* ( T) 0 is then the group of one­ parameter subgroups of a subtorus S of T such that S/k. is a maximal k.-split torus of Gfk. for all v e V0• The group (6) Y = Hom(U, C * ) = Hom(X* (T)D, C * ) (v e V0), is independent of v, and is the Y of §6 for Gfk •. The root datum cp(Gfk.), which is determined by the action of D, is also independent of v e V0.

AUTOMORPHIC £-FUNCTIONS

51

Given y e Y, let y0 be a "logarithm" o f y, i.e., an element o f Hom(X* (T) D, C) such that (7) y(u) = NvYo (u) = Nv , for u e XiT) D . This element is determined modulo a lattice, but its real part Re y0 e Hom(U, R), defined by y(u)

(8)

= Nv

is well defined. If y has values in R+, then we choose y0 to be equal to its real part. The space a * is the dual of a = U ® R (the so-called real Lie algebra of Sfkv), and is acted upon canonically by ,. w as a reflection group. We let a * + be the positive Weyl chamber defined by B. Let Pv be the unramified character of T(kv), given by t ...... lo(t)lv, where I l v is the normalized valuation at v and o half the sum of the positive roots. Then its real logarithm p0 is independent of v e VD · In fact, it is a positive integral power of Nv whose exponent is determined by the kv-roots, their multiplicities, and the indices qa of the Bruhat-Tits theory [61]. But those are determined by the previous data and the action of rk. on the completed Dynkin diagram [61], which is also in­ dependent of v e VD. We write p0 instead of Pv. O · We have p0 e a * +. The representation 'lT:v is a constituent of an unramified principal series PS(xv), where Xv is an unramified character of T(kv), or, equivalently, of S(kv), determined up to a transformation by an element of k W. Thus we may assume Xv . o to be con­ tained in the closure �l'{a * +) of a * +. Since 'lT:v is unitary, the associated spherical function is bounded, and hence Re Xv . o is contained in the convex hull of k W(p0), i.e., we have (po - Xv . O • .A. ) � 0, for all .A. e a * +. (9) (See remark following the proof.) For .A. e X * (LT0), let .A.' be the restriction of .A. to X* (TD). In view of 10.4 and our conventions, we have then (10)

Let ;. (I I)

=

,. W(.A.') n �l'{a * +). Since Re Xv . O E �l'(a * +), we have Nv � Nv.

Combined with (9), this implies ( 1 2)

If now .A. runs through P, there are only finitely many possibilities for A, whence (4), with a = sup(p0, J.) (.A. e Pr), for v e VD. Since V1 is a finite union of such sets, this proves (4). REMARK. The relation (9) is proved in [35, pp. 27-29] for the split case. For a general semisimple simply connected group, see I. Macdonald, Spherical functions on a group ofp-adic type, Publ. Ramanujan Institute 2, Madras, Theorem 4. 7 . I , or H. Matsumoto, Lecture Notes in Math., vol. 590, Springer-Verlag, Berlin and New York, Proposition 4.4. 1 1 . In fact, we have used it for a general connected reductive

52

A. BOREL

group but the reduction to the case of simply connected semisimple groups is easily carried out by going over to the universal covering of the derived group. 1 3.3. CoROLLARY. Let P be a parabolic k-subgroup of G, P = M · N a Levi decom­ position over k of P. Assume that n is a constituent of a representation Ind ��(11) inducedfrom a unitarizable irreducible admissible representation 11 of MA , viewed as a representation of PA trivial on NA- Then L(s, n, r) is absolutely convergent in some right half-plane.

We view LM as a subgroup of LG (3.3). Let r' be the restriction of r to LM. Let v E V1 be such that the conditions listed at the end of 1 3. 1 are satisfied by M, G, 11v and n• . Then, by the transitivity of induction, it follows that there exists Xv as in the above proof such that 11v (resp. n v) is the constituent of class 1 with respect to M(o.) (resp. G (o ,) ) of the principal series PS(x.) for M(kv) (resp. G (kv)) . Then L(s, nv , r ) = L(s, rJv, r') (7.2, 10.4). This being true for almost all v's, w e are reduced to 1 3.2. 14. The L-function of an automorphic representation.

14. 1 . A smooth representation of GA is automorphic if it is a subquotient of the regular representation of GA in Gk\GA- It is cuspidal if it consists of cusp forms. If so, it is unitary modulo the center. We let W.(G/k) denote the set of equivalence classes of irreducible admissible automorphic representations of G A- By Proposi­ tion 2 of [39], every n E W.(G/k) is a constituent of a representation induced from some cuspidal rJ E W.(Mjk), where M is a Levi k-subgroup of a parabolic k-subgroup of G. Combined with 1 3.3 this yields the n E W.(G/k) and r be a representation of LG. r) is absolutely convergent in some right half-plane.

14.2. THEOREM (LANGLANDS). Let

Then L(s,

n,

The L-function of an irreducible admissible automorphic representation will also be called an automorphic L-function. 14.3. There are several conjectures on the analytic character of L(s, n, r) for auto­ morphic n, all checked in some special cases, going back to the work of Heeke on L-series attached to Grossencharaktere and to modular forms. (a) If n E W.(G/k), then L(s, n , r) admits a meromorphic continuation to the whole complex plane. (b) Assume that n and G are such that the local solution to the local problem yields factors L and e at all places. It is then conjectured that there is a functional equation L(s, n , r) = e(s, n, r) · L(l - s, it, r), where it is the contragredient re­ presentation to n . (c) In a number of cases, it has been shown that : (* ) If n is cuspidal, r irreducible nontrivial, then L(s, n, r) is entire. Here and there, conjectures to the effect that this should be a general pheno­ menon have been stated. However, there are counterexamples. Heuristically, one sees this is likely to happen if n is lifted from a cuspidal representation of a reduc­ tive group H (in the sense of V below) and the restriction of r to LH contains the trivial representation. 14.4. (a) Let G = GLn and r = rn be the standard representation of GLn(C). Then 14.3(b), (c) are proved in [25] for n = 2, in [19] for n � 2, if L and e are de-

AUTOMORPHIC £-FUNCTIONS

53

fined to be the products of the L- and .s-factors mentioned in 12.4. As recalled in 12.4, these are the same as those considered here at almost all places, and for n = 2, at all places. (b) If G = GL2 x GL2 and r = r2 ® r2, similar results are established by Jac­ quet in [21]. (c) Let G = GL2 • If r: GL2(C) --+ GL3(C) is the adjoint representation, then 14.3(b), (c) are announced in [16] . This extends results of Shimura [54]. If r = Sym3 (r2), Sym4(r2), then 14.3(b) is stated in [15], in the context of the global lifting (see V) ; for Sym3(r2), it is also proved in [51], in the framework of 14.5 below. (d) Let k be a function field, G = GLm x GLn and r = rm ® rn· Let 1r: (resp. 1r:') be a cuspidal automorphic representation of the first (resp. second) factor. By the methods of [19], [26], [27], one can define L and e, and (Jacquet dixit) show 14.3(b), and also the holomorphy, except when m = n and 1r: is contragredient to ' 1r: . These methods also yield further examples for other groups and for other re­ presentations. It is expected that similar results hold over number fields. (e) 14.3(a) has also been checked when G = PSp(4) in some cases in [1], and, in general, in [42]. A functional equation is also established. 14.3(a), (b) are announced in [41] for orthogonal groups in an odd number of variables over functional fields, for the local factors mentioned in 12.3. For a survey and earlier references, see [43]. See also [44]. 14.5. We describe some cases in which 14.3(a) has been verified in [33] (see also [18] for a survey). Let C be a split k-group, of adjoint type, endowed with its canoni­ cal o-structure. Fix a Borel subgroup B of C and a maximal torus T of B defined over o. Let P be a maximal proper standard parabolic subgroup and P = M · N its standard Levi decomposition. Since C is adjoint, it is easily seen that C(M) is a torus. The group MfC(M) is semisimple, split over k, of adjoint type, of rank equal to rk(C) - 1 . We let G = M/C(M). The group LG o is simply connected (2.2(2)). We have a natural inclusion LG --+ LM, and L M is the Levi subgroup of a standard parabolic subgroup Lp = LM · U with unipotent radical U (3.3). Let A be the split component of P in T, and LA o the split component of Lpo in L T0• The group LA o acts on the Lie algebra u of U and its eigenspaces are irreducible LG0-modules. We let Fp denote the set of contragredient representations to these LG0-modules. The £-functions considered in [33] are of the form L(s, 1r:, r) with r e Fp and 1r: an ir­ reducible cuspidal automorphic representation of G. A number of examples are given in which L(s, 1r: , r) admits a meromorphic continuation. This is deduced from the results of [32] : let m be the length of a composition series of u with respect to M. Then, for suitable numbering of the elements of Fp and strictly positive integers a;, there is a relation M(s) = TI L(a;s, 1r: , r;) · L(sa; + 1 , 1r: , r;)- 1 , (I) l�i�m

where M(s) is the intertwining operator occurring in the theory of Eisenstein series with respect to P, and is known to have a merom orphic continuation to the complex plane [32]. If r = 1 , this and 1 3.2 yield the meromorphic continuation. In general, if we have the analytic continuation for all r;'s except one, (1) gives it for the remain­ ing one. 14.6. The converse problem is to what extent automorphic representations can be characterized by analytic properties of their £-functions, or to give analytic

54

A. BOREL

conditions on a given L-function which will insure that it is automorphic. The first main result was Heeke's characterization of the Mellin transform of a parabolic modular form. Then came Weil's extension of this theorem to congruence sub­ groups [64], [65], its generalization in the context of representations in [25], and the extension to GL3 [46], [27]. In those results, conditions are imposed on the L-func­ tions of 11: and of the twists 11: ® X of 11: by characters. However, the analogous statement is false from n = 4 on [46] . It may remain true if one imposes conditions on the twist 11: ® p of 11: by representations of GL,._ 1 or only of GL,._2 • For results in that direction, over function fields, see [45]. Note however that in the general problem outlined here, one wishes rather to turn things around and deduce the analytical properties of some given L-series by show­ ing directly that it is automorphic (see the seminars on base change and on zeta­ functions of Shimura varieties [17], [8], [40]). 1 4.7. Other problems. (I) One "representation the oretic" form of "Ramanujan's conjecture" is the following : if 11: = ®nv is an irreducible nontrivial admissible cuspidal automorphic representation (and G is simple), then each 'lr:v is tempered. It is now well known to be false for certain orthogonal or unitary groups, and even for one split group [20]. (2) Let 11: be a unitary irreducible representation of GA. If G = GL2, then its multiplicity in the space of cusp forms oL2 (G(k)\G(A)) is at most one, "multiplicity one theorem" [25]. In fact there is even a "strong multiplicity one theorem" [38] : given nv for almost all v's, there is at most one constituent 11: of the space of cusp­ forms with those local factors. The multiplicity one theorem has been proved for GL,. [52] and the strong form for GL3 [28]. It is unknown whether it is true for SL2 • On the other hand, there are counterexamples for some inner forms of SL2 [31]. CHAPTER V. LIFTING PROBLEMS.

Although the problems on automorphic L-functions discussed in § 1 4 are only partially solved, the solutions provide practically all cases in which an L-series (automorphic or not) has been proved to have meromorphic or holomorphic analy­ tic continuation with functional equation. This suggests trying, given an L-series and a reductive group G, to see whether G has an automorphic representation with the given L-series. Many instances of such questions can be viewed more precisely as special cases of the "lifting problem" or of the "problem of functoriality with respect to morphisms of L-groups." There is also a local version. For the sake of exposition, we shall start with the latter, but it should be borne in mind that the motivation and requirements stem from the global one, and that local and global are at present inextricably linked in many proofs. These questions were raised by Langlands in [35]. 15. L-homomorphisms of L-groups.

1 5. 1 . Let E be a field and H, G connected reductive £-groups. A homomorphism u : LH -+ LG over rk is said to be an L-homomorphism if it is continuous and if its restriction to L H0 is a complex analytic homomorphism of LH0 into LGO. Let E be local and G quasi-split. If rp e (/)(H), then u rp e (/)(G). In fact, condition 8.2(i) is clearly satisfied, by u rp, and so is 8.2(ii) because every parabolic subgroup of LG a

a

55

AUTOMORPHIC £-FUNCTIONS

is relevant, G being assumed to be quasi-split. Therefore qJ ...... u o qJ defines a map f/J(H) --+ f/J(G), to be denoted f/J(u). 1 5.2. Let E = k be a global field. For v E v, the Galois group rk. is a subgroup of rk ; hence the L-group of G viewed as a k0-group, to be denoted L(G/k0), is a subgroup of LG = L(G/k). Thus, in particular, the £-homomorphism u of 1 5. 1 defines by restriction an £-homomorphism u0 : L(Hfkv) --+ L(Gjk0), hence also a map f/J(uv) : f/J(Hfkv) --+ f/J(Gfkv) (v E V). The "lifting problem" is, roughly speaking, whether such maps are mirrored by maps of representations in the local case, or of automorphic representations in the global case. 1 5.3. EXAMPLE : BASE CHANGE. Let H be a split over E, F a finite Galois extension of E, and G = RFIEH. Then LGo is a product of copies of LH0, indexed and per­ muted by rFIE (5. 1). There is then a natural L-homomorphism u which is the iden­ tity on rE and the diagonal map on LH0• If E is a local field, then Wj.. is an open normal subgroup of W£, and the map f/J(u) may be viewed as given by the restric­ tion to Wj... 16. Local lifting.

1 6. 1 . Let k = E be a local field, G quasi-split over E, H a connected reductive E-group and u: LH --+ LG an £-homomorphism. The problem of local lifting is, roughly, to establish a correspondence ll(u) : ll(H(k)) --+ ll(G(k)) which preserves L- and e-factors. If the local parametrization problem of III is solved, then ll(u) is the map between indistinguishable classes which assigns llu•rp, G to llrp . H (({) E f/J(H)). The element ll E ll(G(k)) is said to be a lift of 1r E ll(H(k)) if ll E llu•rp, G• where (/) E f/J(H) is such that 7r E nrp.H · We have then (1)

L(s, ll, r)

=

L(s,

1r ,

r o qJ),

e(s, n, r, cp)

=

e(s, 'lr, r 0 u, cp).

for every representation r of LG. 1 6.2. The local lifting is thus viewed as a map between classes of £-indistinguish­ able representations rather than one between representations. However it is pos­ sible to single out one lifting under assumptions which, in the global case, are satisfied almost everywhere : assume H, G to be quasi-split, split over an unrami­ fied extension F of E, endowed with an oE-structure such that H(oE) and G(oE) are very special maximal compact subgroups, and 1r of class one with respect to H(oE). Then qJ such that 1r e Drp.H• and the set llu orp, G are well defined. Moreover, llu•rp, G contains exactly one element of class one (with respect to G(oE)), to be called the natural lift of 1r . 1 6. 3 . A full solution of the local parametrization problem does not seem to be in sight, and it is conceivable that it may require proving at the same time global results such as Artin's conjecture. Meanwhile, one wants to settle some approxima­ tions to it, notably to be able to prove some cases of Artin's conjecture. Note that if G = GLno then the sets Drp.G are either known or conjectured to consist of one element (1 2.2, 1 2.3). Such a lifting problem can then be stated as one of construct­ ing a map u * : ll(H(k)) = ll(G(k)) satisfying certain conditions. So far, there are two examples : (a) Base change (cf. 1 5.3) when H = GL2 and F is cyclic of prime degree over E [17], [38], [49], [56]. Besides some naturality conditions and 1 6.3, the main require-

56

A. BOREL

ments relate the characters of n and of the hypothetical u * (n). The results also describe the fibres and the image of u * . [Note that the results of [38] on this problem are used in [62], so that we cannot invoke the solution of the local parametrization problem (1 2.4) for GLdust to use the map U(u) of 16. 1 . If we could, then the local questions [38] would be mainly to relate the characters of n and ll(u)(n).] (b) H = GL2, G = GL3 , and (1) is given by the adjoint representation of LHo (see [16]). In this case, n = u * (n) must be trivial on the center of LG o and be such that the L- and e-factors of u * (n) ® X (X character of E *) are certain given functions. There is at most one such U ( 12.4(d)) . In [16], n is stated to exist, except possibly if E has even residual characteristic and n is "extraordinary. " 16.4. I n 16.3(a), the lifting problem was connected with the existence o f relations between characters. This is a direct connection between U(H) and ll(G), which is of great importance for the use of the trace formula in proving or using the local or global lifting. We now mention two other examples of such relations. Assume that G is a quasi-split inner form of H. There is then an isomorphism u : L H ...::. LG and an embedding C/J(u) : C/J(H) c: C/J(G). If f: H --+ G is a k,-isomorphism such that J- 1 · rJis an inner automorphism of G for every r e rk . then /establishes a bijection between conjugacy classes which are stable under rk . Using results of Steinberg [59], one then sees easily that maximal k-tori in H are isomorphic over k to maximal k-tori in G. This allows one in some cases to assign regular semisimple classes in G(k) to such classes in H(k), so that it makes sense to compare values of characters of H(k) and of G(k) on such classes. (a) Let k be either R or nonarchimedean with odd residual characteristic. Let G = GL2 and H be the group of invertible elements in the quaternion algebra over k. The sets Drp are singletons, C/J(u) assigns to a (finite dimensional) irreducible representation n of H(k) a discrete series representation n' of G(k). In this case, the semisimple classes of H(k) correspond to the elliptic classes in G(k). It is proved in [25] that the characters of n and n' differ only by a sign on those classes. (b) Let k = R. For rp e C/J(H), C/J(G), let X rp be the sum of the characters of the elements in nrp. Choose 'P E C/J(H) such that Drp consists of tempered representations. Then X rp and Xu•rp are equal on the regular semisimple classes of H(k), up to a sign depending only on H and G [53, 6.3]. 16.5. We could also take the Weil forms of the L-groups. In that case an L­ homomorphism, restricted to WE, is assumed to satisfy the obvious analogue of 8.2(i). Take in particular the case where H = { 1 }. Then u is just an element of C/J(G). The lifting problem in this case is part of the local problem of III. 17. Global lifting.

17 . 1 . Assume G to be quasi-split. Let H be a reductive k-group and u : LH --+ LG an L-homomorphism. Let uv : L(Hfkv) --+ L(Gfkv) and C/J(uv) : C/J(Hfk.) --+ C/J(Gfk.) be the associated maps ( v e V) (see 1 5. 1). Let n = ®. n. (resp. U = ®.D.) be an irreducible admissible representation of HA (resp. GA). Then Dis said to be a lift of n if n. is one of n. for every v e V (16.1). If that is the case, then, for every representation r of LG, we have (1)

L(s, ll, r)

=

L(s, n, r o u),

e(s, n, r)

=

e(s,

1C ,

r 0 u).

AUTOMORPHIC £-FUNCTIONS

57

It is also usually requested that n. be the natural lift (16.2) of 'R: v for almost all v's. The question is then whether every automorphic 1r: has a lift, which is automorphic, or, somewhat more ambitiously, whether there is a map u * : IJ!(H/k) --+ IJ!(G/k) with reasonable properties, which sends 1r: e W.(Hfk) onto a lift of 1r: . One also wants to describe the fibres and the image of u*. In that degree of generality, the problem appears to be inaccessible at present. However, there are many results, old and recent, which are very striking illustra­ tions of this principle, some of which will be extensively discussed in various semi­ nars. Here, for orientation, and to give an idea of the scope of the problem, I shall list briefly some special cases, referring to the literature or to other seminars for more details. REMARK. Let r be a representation of L H of degree n. Then it defines an £-homo­ morphism u : LH --+ LGLn = GLn{C) x rk in the obvious way. A positive answer to the lifting problem would imply in particular that if 1r: is an automorphic repre­ sentation of H, then L(s, 1r , r) = L(s, D, rn) where D is an automorphic representa­ tion of GLn and r n the standard representation. This would therefore to a large extent reduce the study of automorphic L-functions to those of GLn, with respect to the standard representation. 17.2. Let H = { 1 }, G = GLn. Then an L-homomorphism u is just a continuous complex n-dimensional representation of Fk . The question is then whether the Artin L-series L(s, u) is an automorphic L-series of GLn (with respect to the stand­ ard representation of GLn{C)), which should be cuspidal if u is irreducible. In view of known results on GLn (cf. 14.4) this would imply Artin's conjecture. For n = I, a positive answer is given by class-field theory. For n = 2, 3, a posi­ tive answer is equivalent to Artin's conjecture, since there are converses to Heeke theory [25], [65], [27], [46]. For n = 2, it has been proved for dihedral or tetrahedral representations of rk , and for some others over Q (see [38], [17], [15]). 17.3. Let k' be a Galois extension of k, n the degree of k' over k. Take H = Rk '/kGLh G = GLn. There is a natural homomorphism f: LHO XI rk 'lk into the normalizer of a maximal torus LTo of LG0• Since the former group is a quotient of LH, and LG = LGO X rk , we can define an L-homomorphism u : LH --+ LG by u(h , r) = (f(h), r) (h E LH0, r E rk). An automorphic representation of H is a Grossencharakter X of k'. The problem is then whether the Artin L-series L(s, x) is the L-series of an automorphic representation of G. If n = 2, k = Q, and k' is imaginary, this was proved by Heeke ; 1r: is associated to a cuspidal holomorphic automorphic form. If n = 2, k = Q, and k' is real quadratic, this was established by H. Maass. 1r: is then as.s ociated to a nonholomor­ phic automorphic form. For n = 3, this is proved in [26], [27] . 17.4. Base change. This is the global counterpart to 1 6.3(a). Let k' be a finite Galois extension of k. Assume H to be k-split and G = Rk'lkH. There is again an L-homomorphism u : LH --+ LG whose restriction to L H0 is a diagonal map. In this case G(A) and G(k) are canonically isomorphic to H(Ak,) and H(k') ; therefore the problem is to associate an automorphic representation of H(Ak' ) to an automorphic representation of H(Ak). Again, it should be a counterpart to the restriction to Wk' of homomorphisms wk --+ L H0 • If H = GL2 and k' is cyclic of prime degree, the lifting map u* for representa­ tions is constructed in [38], which also gives a description of its image and fibres.

58

A. BOREL

This extends work of Doi-Naganuma, Jacquet [21] (on the quadratic case) and of Saito [49], Shintani [55], [56] (cf. [17]). 1 7.5. Let G be quasi-split, and H an inner form of G. Then L H = LG and f/J(H/kv) c f/J(Gfkv) for all v's (8.3). Moreover, for almost all v's, H and G are isomorphic over k•. ; hence f/J(Hfkv) = f/J(G!kv) and D(H(kv)) = D(G(kv)). The question is then, given 1r = ®v1rv, is there an automorphic representation D = ®vllv of G such that Dv = 7rv for almost all v's ? If G = GL2 and H is the group o f invertible elements o f a quaternion algebra D over k, a positive answer is given by Jacquet-Langlands [25]. Note that, in that case, because of the "strong multiplicity one theorem," at most one D may be associated to a given 1r in this way. The possible D's are in fact the cuspidal automorphic representations for which Dv belongs to the discrete series for all v's over which D does not split (loc. cit.). 1 7.6. If G = GL2, G = GL3 and u is given by the adjoint representation, as in 13.4, the global lifting problem has been solved by Gelbart-Jacquet [16], the "local lifting" being the one of 1 6.2(b). 1 7. 7. Let M be a Levi k-subgroup of a parabolic k-subgroup P of G. Then L M imbeds naturally into LG (3.3), whence an L-homomorphism u : LM --. LG. If 1r is cuspidal, then the analytic continuation and residues of Eisenstein series [32] are known to yield a unitary u * (1r) in many cases, and, conjecturally, in general. 18. Relations with other types of L-functions.

1 8. 1 . In 1 7.2, the lifting problem amounts to identifying an Artin L-function with an automorphic L-function on GLn- One can also include in this problem more general representations of Weil groups if one passes to the Weil form of the L-groups. For simplicity, let us limit ourselves to relative Weil groups Wk'lk• where k' is a finite Galois extension of k over which H and G split. An L-homomorphism u : LHO � wk'/k --> LGO � wk'/k is then a continuous homomorphism compatible with the projections on Wk'lk• whose restriction to LHo is a complex analytic homo­ morphism into LGo , and such that, for w E Wk'lk• u(w) = (u'(w), w) with u(w) semi­ simple (cf. 8.2(i)). If H = { 1 }, an L-homomorphism is said to be an admissible homomorphism of Wk,lk into LG. In analogy with the definition of f/J(G) in the local case, we can con­ sider the set f/Jk,1iG) of equivalence classes of such homomorphisms, modulo inner automorphisms of LGo, and then pass to a suitable limit f/J(G) over k'. The lifting problem asks in this case to associate to any rp E f/J(G) an automorphic representation 1r, such that, for any representation r of LG, L(s, 1r, r) is equal to the Artin-Hecke L-series of r o u. In particular, is every Artin-Hecke L-series that of an automorphic representation of GLno with respect to the standard representation? If G is a torus, then [34] provides a positive answer. In fact, in this case the irredu­ cible admissible automorphic representations of G are the characters of G(k)\G(A), and [34] gives a homomorphism with finite kernel of f/Jk,1iG) onto the set of such characters. 1 8.2. In the same vein, it is natural to ask whether Hasse-Weil zeta-functions (or even L-functions of compatible systems of /-adic representations of Galois groups) can be expressed in terms of automorphic L-functions. For elliptic curves over function fields, it is a theorem. That it should be the case for ell iptic curves over

AUTOMORPHIC £-FUNCTIONS

59

Q is the Taniyama-Weil conjecture ; it has been checked in a number of special cases (see [2], [14] for surveys from the classical and representation theoretic points of view respectively). Apart from that, this problem has been pursued mostly for Shimura curves and certain Shimura varieties ; we refer to the corresponding semi­ nars for a description of the present state of affairs. Finally, one may ask whether it is possible to characterize a priori those auto­ morphic representations whose L-series have an arithmetic or algebraico-geome­ tric significance. A necessary condition if k is a number field is that for an infinite place v, ""• should be associated to a representation a. of Wk, whose restriction to C* is rational, C* being viewed as real algebraic group, i.e., be of type A0 in [3, 6.5]. If the L-series of 1r is to be an Artin L-series, then "" should even be of type A00 (loc. cit.), i.e., a. should be trivial on C* . Let k = Q. Then there are three pos­ sibilities for """' (1 1 .5). If """' = 1r(l , sgn), then "" corresponds to 2-dimensional representations of F0 with odd determinant by the theorem of Deligne-Serre [10], [50]. Modulo the Artin conjecture for such representations, the correspondence is bijective. However, I am not aware of any result for the other two possible values of """' · A positive answer would involve nonholomorphic automorphic forms. In [36], it is shown in many cases for GL2 over Q that the L-series of a representation of type A0 is that of a compatible system of /-adic representations of F0• Over a function field, there is no condition such as A0• In fact, for GL2, Drinfeld has shown that all irreducible admissible automorphic representations are associated to /-adic representations (see the lectures on his work by G. Harder and D. Kazhdan). REFERENCES I. A. N. Andrianov, Dirichlet series with Euler product in the theory of Siegel modular forms ofgenus two, Trudy Mat. Inst. Steklov 112 (1971), 73-94. 2. B. J. Birch and H. P. F. Swinnerton-Dyer, Elliptic curves and modular functions of one variable. IV, Lecture Notes in Math., vol. 476, Springer, New York, 1 975, pp. 2-32. 3. A. Borel, Formes automorphes et series de Dirichlet (d'apres R. P. Langlands), Sem. Bourbaki,

Expose 466, (1 974/75) ; Lecture Notes in Math., vol. 5 1 4, Springer, New York, pp. 1 89-222. 4. A. Borel and J.- P. Serre, Theoremes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1 964), 1 1 1 -1 64. 5. A. Borel et J. Tits, Groupes reductifs, Inst. Hautes Etudes Sci. Publ. Math. 27 (1 965), 55-1 5 1 . 6. P. Cartier, Representations of reductive P-adic groups, these PROCEEDINGS, part 1 , pp. 1 1 1-155. 7 . W. Casselman, GL., Proc. Durham Sympos. Algebraic Number Fields, London, Academic Press, New York, 1 977. 8. -- , The Hasse- Wei/ '-function of some moduli varieties of dimension greater than one, these PROCEEDINGS, part 2, pp. 141-163. 9. P. Deligne, Formes modulaires et representations de GL,, Modular Functions of One Variable, Lecture Notes in Math., vol. 349, Springer, New York, 1973, pp. 55-106. 10. P. Deligne and J.-P. Serre, Formes modulaires de poids 1, Ann. Sci. Ecole Norm. Sup. (4) 7(1 974), 507-530. 11. D. Flath, Decomposition of representations into tensor products, these PROCEEDINGS, part 1 , pp. 1 79-1 83. 12. F. Gantmacher, Canonical representations of automorphisms of a complex semi-simple group, Mat. Sb. 5 (1 939), 101-142. (in English) 13. S. Gelbart, Automorphic functions on adele groups, Ann. of Math. Studies, no. 83, Princeton Univ. Press, Princeton, N. J., 1 975. 14. --, Elliptic curves and automorphic representations, Advances in Math. 21 (1 976), 235292.

60

A. BOREL

15. S. Gelbar t , Automorphic forms and, Artin's conjecture, Lecture Notes in Math. , vol. 627, Springer-Verlag, Berlin and New York, 1977, pp. 241-276. 16. S. Gelbart and H. Jacquet, A relation between automorphic forms on GL, and GL, Proc. Nat. Acad. Sci . U.S.A. 73 (1 976), 3348-3350. 17. P. Gerardin and J.-P. Labesse, The solution of a base change problem for GL, (following Langlands, Saito, Shintani), these PROCEEDINGS, part 2, pp. 1 1 5-133. 18. R. Godement, Fonctions automorphes et produits eu/eriens, Sem. Bourbaki (1 968/69), Expose 349 ; Lecture Notes in Math . , vol. 1 79, New York, 1 970, pp. 37-53. 19. R. Godement and H. Jacquet, Zeta-functions of simple algebras, Lecture Notes in Math. , vol. 260, Springer, New York, 1972. 20. R. Howe and I. I. Piatetski-Shapiro, A counterexample to the 'generalized Ramanujan con­ jecture' for (quasi-) split groups, these PROCEEDINGS, part 1 , pp. 31 5-322. 21. H. Jacquet, Automorphic Forms on GL(2). II, Lecture Notes in Math . , vol . 278, Springer, New York, 1 972. 22. , Generic representations, Non-Commutative Harmonic Analysis, Lecture Notes in Math. , vol. 587, Springer, New York, 1 977, pp. 91-101 . 23. , From GL, to GL., U.S.-Japan Seminar on Number Theory (Ann Arbor, Mich., --

1975).

--

24. , Principal £-functions of the linear group, these PROCEEDINGS, part 2, pp. 63-86. 25. H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Math. , vol. 1 14, Springer, New York, 1 970. 26. H. Jacquet, I. Piatetskii-Shapiro and J. Shalika, Construction of cusp forms on GL3, Univ. of Maryland Lecture Notes, no. 1 6, 1 975. 27. , Heeke theory for GL3, Ann. of Math. (to appear). 28. H. Jacquet and J. Shalika, Comparaison des representations automorphes du groupe lineaire, C. R. Acad. Sci. Paris Ser. A 284 (1 977), 741-744. 29. A. Knapp and G. Zuckerman, Classification of irreducible tempered representations ofsemi­ simple Lie groups, Proc. Nat. Acad. Sci. U.S.A. 73 (1 976), 21 78-21 80. 30. , Normalizing factors, tempered representations and L-groups, these PROCEEDINGS, part 1 , pp. 93-105. 31. J. P. Labesse and R. P. Langlands, £-indistinguishability for SL, Institute for Advanced Study, Princeton, N. J. (preprint). 32. R. P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Math., vol. 544, Springer, New York. 33. --, Euler products, Yale Univ. Press, 1 967. 34. , Representations of abelian algebraic groups, Yale Univ., 1968 (preprint). 35. , Problems in the theory of automorphic forms, Lectures in Modern Analysis and Applications, Lecture Notes in Math. , vol. 170, Springer, New York, 1970, pp. 1 8-86. 36. , Modular forms and 1-adic representations, Modular Functions of One Variable. II, Lecture Notes in Math., vol. 349, Springer, New York, 1973, pp. 361-500. 37. , On the classification of irreducible representations of real algebraic groups (preprint). 38. --, Base change for GL, : The theory of Saito-Shintani with applications, Notes, lnst. Advanced Study, Princeton, N. J., 1 975. 39. , On the notion of an automorphic representation, these PROCEEDINGS, part 1, pp. 203--

--

--

--

--

--

--

207.

40.

--

--

, Automorphic representations, Shimura varieties, and motives. Ein Miirchen, these part 2, pp. 205-246. 41. M. E. Novodvorsky, Theorie de Heeke pour les groupes orthogonaux, C. R. Acad. Sci. Paris Ser. A, 285 (1 975), 93-94. 42. , Automorphic £-functions for the symplectic group GSp(4), these PROCEEDINGS, part 2, pp. 87-95. 43. M. E. Novodvorsky and I. I. Piatetskii-Shapiro, Rankin-Selberg method in the theory of automorphic forms, Proc. Sympos. Pure Math., vol . 30, no. 2, Amer. Math. Soc. , Providence, R. I., 1 977, pp. 297-301 . 44. I. Piatetski-Shapiro, Euler subgroups, Lie Groups and Their Representations, Proc. Summer School, Budapest, 1 97 1 . PROCEEDINGS,

--

AUTOMORPHIC L-FUNCTIONS

61

45. --, Zeta-functions of GL., Notes, Univ. o f Maryland, College Park, Md. 46. --, Converse theorem for GL3, Notes, Univ. of Maryland, College Park, Md. 47. , Multiplicity one theorems, these PROCEEDINGS, part 1, pp. 209-212. 48. F. Rodier, Les representations de GSp(4, k), oil k est un corps local, C. R. Acad. Sci. Paris 283 (1976), 429-431 . 49. H . Saito, A utomorphic forms and algebraic extensions of number fields, Lecture Notes in --

Math, vol. 8, Kinokuniya Book Store Co., Ltd., Tokyo, Japan, 1975. 50. J.-P. Serre, Modular forms of weight one and Galois representations (prepared in collabora­ tion with C. J. Bushnell), Proc. Durham Sympos. Algebraic Number Fields (London), Academic Press, New York, 1977, pp. 193-268. 51. F. Shahidi, Functional equations satisfied by certain L-functions, Compositio Math. (to appear). 52. J. A. Shalika, The multiplicity one theorem for GLm Ann. of Math. (2) 100 (1974), 171-193. 53. D. Shelstad, Characters and inner forms of a quasi-split group over F (to appear). 54. G. Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. (3) 31 (1975), 79-98. 55. T. Shintani, On li/tings of holomorphic automorphic forms, U.S.-Japan Seminar on Number Theory, (Ann Arbor, Mich., 1975). 56. --, On liftings of holomorphic cusp forms, these PROCEEDINGS, part 2, pp. 97-1 10. 57. A Silberger, The Langlands classification /or reductive p-adic groups (to appear). 58. T. A. Springer, Reductive groups, these PROCEEDINGS, part 1 , pp. 3-27. 59. R. Steinberg, Regular elements of semi-simple algebraic groups, Inst. Hautes Etudes Sci. Pub!. Math. 25 (1965), 49-80. 60. J. Tate, Number theoretic background, these PROCEEDINGS, part 2, pp. 3-26. 61. J. Tits, Reductive groups over local fields, these PROCEEDINGS, part 1, pp. 29-69. 62. J. B. Tunnell, On the local Langlands conjecture for GL(2), Thesis, Harvard Univ. 63. , Report on the local Langlands conjecture for GL,, these PROCEEDINGS, part 2, pp. 1 35-1 38. 64. A. Wei!, Ueber die Bestimmung Dirichletscher Reihen durch Funktionalg/eichungen, Math. Ann. 168 (1967), 149-167. 65. , Dirichlet series and automorphic forms, Lecture Notes in Math., vol. 1 89, Springer, New York, 197 1 . --

--

THE INSTITUTE FOR

AovANCED STUDY

Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 63-86

PRINCIPAL L-FUNCTIONS OF THE LINEAR GROUP

HERVE JACQUET Regard the group Gn =; GL(n) as an algebraic group over some local or global field F. Then LG� = GL(n, C). Let r n denote the natural representation of this last group on Cn ; the £-functions attached to rn play a central role in "Langlands philosophy". We review certain aspects of their theory, taking into account recent results on classification of representations. 1. Local nonarcbimedean theory. We let F be a local nonarchimedean field. When this does not create confusion, we write Gn for Gn(F) and use a similar notation for any F-group. (1 . 1) Let 1C be an admissible representation of Gn on a complex vector space V; we denote by 1t the representation contragredient to 1C, V the space on which it operates, and < · , · ) the canonical invariant bilinear form on V x V. The represen­ tation 1t is admissible, and irreducible if 1C is. Moreover ('it)- � 7C. The functions (v E V, ii E V) (1 . 1 . 1) g f-------> (1C(g)v, ii) and their linear combinations are the coefficients of 1C. Clearly if/is a coefficient of

1C, then the functionfV defined by ( 1 . 1 .2)

is a coefficient of 'it. Let M(p x q, F) be the space of matrices with p rows, q columns and entries in F; denote by .?(p x q, F) the space of Schwartz-Bruhat functions on M (p x q, F). If/ is a coefficient of 1C, f/J is in .?(p x q, f), and s in C set ( 1 . 1 . 3)

Z (f/J, s,J) =

J f/J(g) j det gj • f(g) dxg.

The integral is extended to the group Gn and dx g is a Haar measure on this group. Below (Propositions ( 1 .2), ( 1 .4)) we state again the results of [R. G.-H.J.]. (1 .2) PROPOSITION. Suppose 1C is irreducible. (1) There is s0 such that the integrals (1 . 1 . 3) converge absolutely in the half-plane Re(s) > s0• (2) If the residual field of F has q elements, then the integrals represent rational AMS (MOS) subject classifications

(1 970). Primary 12A85, 22E50, 22E55, 1 2A80, 1 2B35. © 1 979, American Mathematical Society

63

64

HERVE JACQUET

functions of q-s ; as such, they admit a common denominator which does not depend on f or fP. (3) Let ¢ =F 1 be an additive character of F. There is a rational function r(s, 11:, ¢) such thatfor all coefficientsf of 11: and all fP Z(fPA , 1 - s + t (n - 1),JV) = r (s, 11: , ¢) Z(fP, s,j),

where fP/\ denotes the Fourier transform offP with respect to ¢ .

In (3) rpA is defined by (1 .2.4)

fPA (x) =

J fP(y) ¢ [tr(yx)] dy,

where dy is the self-dual Haar measure on M(n x n, F). On the other hand the left-hand side in (3) has a meaning by (1) and {2) applied to it. Besides we could formulate Proposition (1 .2) for the pair (n, it) rather than for 11: , the situation being symmetric in 11:, it ; this symmetry will be apparent at each step of the proof anyway. ExAMPLE (1 .2.5). Suppose 11: is cuspidal. This condition is empty and therefore always satisfied if n = I ; in this case 11: is just a quasi-character of Fx and we know Proposition (1 .2) from [J.T. 1] or [A.W. 3] . If n > 1 , the coefficients of 11: are compactly supported modulo the center of Gn ; we can then exploit this fact to obtain Proposition {1.2), the proof being essentially the same as in the case n = 1

([R.G.-H.J.], [H.J. 1])

(1 .3) The proof of Proposition (1 .2) will be given in §2. For the time being, we derive some simple consequences of Proposition (1 .2). If/and fP are as above and h is in G"' then the functions/1 and fP 1 defined by fi(g) = f(gh), fP 1 (x) = fP(xh) (1.3.1) are functions of the same type. Moreover if we assume (1 .2. 1), ( 1 .2.2), then :

( 1 . 3.2) It follows that the subvector space J(n) of C(q-•) spanned by integrals (1 .3.3)

Z(fP, s + f(n - 1), f)

is in fact a fractional ideal of the ring C [q-s, q•]. Furthermore if we take/such that

f(e) =F 0 and rp with support in a small enough neighborhood of e, we find that (1 .3.3) is actually independent of s and =F 0. Thus l(n) contains the constants ; in other words it admits a generator of the form P(q-• )- 1 with P E C [X]. We will

normalize P by demanding that P(O)

=

1 and will set

(1 .3.4) ExAMPLE ( 1 . 3.5). If 11: is supercuspidal then L(s, n) lxl1• Then (loc. cit.) L(s, 11:) = (1 - q-s-t)- 1 .

=

Assume now Proposition (1.2) for (n, it). Because J(n) (which justifies the notation). Set (1.3.6) e(s, 11:, ¢) Then (1 .2.3) reads

= r(s, 11:,

1 unless n =F

= 1

and n(x)

=

0, the factor r is unique

f/J)L (s, n)/L(l - s, it).

PRINCIPAL L-FUNCTIONS

(1 .3.7)

Z (l/J/\, 1 - s +

f(n -

65

1),JV)fL(1 - s, ft) = e(s, 11: , cjJ) Z {l/J , s +

(

!n

- 1),/)/L(s, 11:).

In view of the definition of the £-factors, this implies that both e(s, 11: , ¢) and its inverse are in C[q-•, q•]. Thus e(s, 11: , ¢) is a monomial in q-•. Let also w be the central quasi-character of 11: , that is to say the homomorphism (I) : px --+ ex such that 11:(a) = 11:(a · 1 ,.) = w(a)1 v for a e P .

(1 .3.8)

If the assertions of(l .2.3) are true for one choice of ¢, they are true for any other choice ¢'. In particular (1 .3.9)

e(s, 11: , ¢') = w(b) l b l <•-l l2l " e(s, 11: , ¢) if cjJ '(x) = cjJ(xb).

By exchanging 11: and 1r we find also {if Proposition (1 .2) is true for (11:, 1r)) : e( I - s, ft, cjJ)e(s, 11: , ¢) = w( - 1).

(1 .3. 10)

Moreover let us denote, for any quasi-character X• by 11: ® X the representation g ...... 71:(g)x(det g) of G,. on V. Let also aF or a be the module of F. Then : (1 .3. l l) (1 .3. 1 2)

L(s, 11: ® a t) = L(s + t, 11:),

e(s, 11: ® at , ¢) = e(s + t, 11: , ¢).

The r-factor satisfies similar identities. Again F is a local nonarchimedean field. (2. 1) Let P be an F-parabolic subgroup of G,.. To be specific we take P to be standard of type (nh n2, · · · , n,) , so that the matrices p in P have the form 2. Induced representations.

ml

(2. 1 . 1 )

mz

U;j

m; e

p= 0

G,;

.

m,

We also set U = Up (unipotent radical of P), M = Mp = PfUp. Most of the time we identify M to the subgroup of p e P such that u;i = 0. Then, in a natural way, (1 � i � r ) . (2. 1 .3) If a,. , 1 � i � r, is an admissible representation of G,;, we can form the representa­ tion a = x a,. of M = P/U, regard it as a representation of P trivial on U, and induce it to G,. The representation (2. 1 .2)

(2. 1 .4)

e

= l(G, P; a) = l(G, P; a h az, · · · , a,)

66

HERVE JACQUET

that we obtain is admissible. Its contragredient � is equivalent to f

(2. 1 . 5)

= /(G, P; 0') = J(G, P; 0' 1 > O'z,

···,

O' r) ·

More precisely let W be the space of q . Then the space V of e consists of all func­ tions F from G, to Wwhich are smooth and satisfy F(pg) = op(p)l i ZF(g),

(2. 1 .6)

where op is the module of P. The group G,. operates by right-shifts on V. The space V' of f is defined similarly in terms of the space W of 0' . For F e V, F e V' the function tp(g) = ( F(g), F(g)) satisfies tp(pg) = op(p)tp(g).

(2. 1 .7)

Denoting by tp 1------>

(2. 1 .8)

J

P\G

tp(g) dg

a positive right-invariant form on the space of continuous functions satisfying (2. 1 .7), we may set (F, F) =

(2. 1 .9)

J

P\G

( F(g), F'(g)) dg

and this is a pairing which allows us to identify V' to Let RF be the ring of integers of F and let

V and e '

to �.

(2. 1 . 10)

Since G,.

=

P Km we may take (2. 1 .8) to be ·

(2. 1 . 1 1)

1------>

tp

JKn tp (k) dk.

(2.2) It will be convenient to have a description of the coefficients of e in terms of those of (J. So let/be the coefficient of e determined by Fe V, F E v = V' ((1 . 1 . 1)) ; then the function H: G,. x G, -+ C defined by H(gi> g2) = (F(g 1 ), F(g2)) satisfies the following conditions : (2.2. 1) (2.2.2)

H(ulmgh Uzmgz) = op(m)H(gh gz) for g; E G,., U; E Up, m E Mp;

for any gh g2 the function m is a coefficient of (J ® o}(Z ; H is K,.

(2.2.3)

Moreover/is given by (2.2.4)

f(g) =

J

P \G

x

1------>

H(mgh g2)

K, finite on the right.

H(hg, h) dh =

JK H(kg, k) dk.

Conversely, if H is any function satisfying (2.2. 1)-(2.2.3), then the function f defined by (2.2 . 4) is a coefficient of '/C. The coefficientfV of � is then given by

PRINCIPAL £-FUNCTIONS

67

(2.2.5) H satisfies (2.2. 1 )-(2.2.3) with a instead of q. (2.3) Before formulating the main theorem of this section we remark that if 11: is

and

an admissible representation which is perhaps not irreducible but admits a central quasi-character ((1 .3.8)), then the assertions of(1 .2) make sense for (11: , :it'), although they may fail to be true. Below we assume that each (J; admits a central quasi­ character ; thus � admits also a central quasi-character. (2.3) PROPOSITION. With the notations o/(2. 1 ) suppose the assertions of (l.2) are true for each pair (q; , a,.). Then they are true for (�. � ) . Moreover :

I (�) = IJ J(q;) , r(s, �. cp) = IT r(s, (J;, cp). i

i

The proof will occupy (2.4)-(2.6). (2.4) Letfbe given by (2.2.4). Then, exchanging the order of integrations, we get

JK dk JGn f/J(g) I det g j•+Cn-1) 12H(kg, k) dx g = J dk J f/J(k- 1 g) j det g j•+en-1 ) 12 H(g, k) dxg K Gn = J dk' J f/J(k- 1 pk')j det pj •+Cn-1) 12 H(pk', k) d1p.

Z(f/J, s + f(n - 1 ) , f) = (2.4. 1)

dk

P

If moreover we define (p being as in (2. 1 . 1)) (2.4.2) (2.4.3)

1/f(m�o m2, . . . , m, ; k, k') =

J f/J(k-lpk') ® du,.i,

h(m�o m2, . . . , m, ; k, k') = H(pk ' , k)o pl12(p),

then, after integrating in the variables u,.i, integral (2.4. 1) can be written as (2.4.4)

J dk dk' J 1/f(m�o m2, . . . , m, ; k, k')h(m� o . . . , m, ; k, k') . n j det m,. j•+ Cn;-1)/2 ® dxm,. . '

Because of the K-finiteness of the functions involved, for f/J and H given, (2.4.4) can be written as a sum over a finite set of K x K of the inner integrals. But for given k and k', 1/f(m�o mz, . . . , m, ; k, k') is a finite sum of products n ..w,.(m;) with If!,. E !/(n,. x n,., F) ; similarly h(m�o m2, • · · , m, ; k, k') is a finite sum of products n ,. .fi (m,.) where .fi is a coefficient of (J; ((2.2.2)) . Thus (2.4. 1) is a finite sum of products (2.4.5)

IJZ(f/J,., s + !(n,. j

-

1),/;),

where f/J,. is in !/(n,. x n,., F) and .fi is a coefficient of q ,. . So we have proved (2. 1 . 1 ), (2. 1 .2) for �. and even the inclusion /(�) c n .. /(q;). (2.5) Now we prove the reverse inclusion. Starting with an expression (2.4. 5) we can certainly find f/J E !/(n x n, F) so that with the notations of (2. 1 . 1) :

68

HERVE JACQUET

J f/J(p) ® duii = I,J f/J,{m;).

(2.5.1)

Moreover there are two K-finite functions r; and r; ' on K such that

JJ f/J(k-lxk')r;(k)r;'(k') dk dk' = f/J(x).

(2. 5.2)

Then (2.4.5) is equal, for large Re s, to (2.5.3)

J f/J(k-lpk')i det p i •+Cn-1) 12 I,J .fi(m;)o}t2(p)r;(k)r;'(k') d1p dk

dk' .

Let dh be the normalized Haar measure on the compact group K n P. Changing k to hk, k' to h' k' with h and h' in K n p and then integrating over (K n P) X (K n P), we find that (2.5.3) is equal to (2. 5.4)

J dh dh' J f/J(k-lh-lph' k')i det P i s+Cn-1) 12 ·

IT .fi(m;)o lf,2(p)r;(hk)r;'(h'k') d1p dk dk'. j

Now change p to hph'- 1 and write h, h' in the form (2. 1 . 1) with h; E Kn; > h; E Kn; instead of m; . We see that (2. 5.4) is equal to

J f/J[k-1pk']Hl (p, k, k')i det p i •+Cn-1) 12 d1p dk dk'

(2.5.5)

where H1(p, k, k')

=

JJ r;(hk)r;'(h'k') l} f.{h;

·

m; · h;-1 ) o),12(p) dh dh' .

It is easily verified that there is a function H satisfying (2.2. 1)-(2.2.3) such that (p E P, k E K, k' E K'). H(pk, k') =: H1(p, k, k') If I is the corresponding coefficient of � {(2.2.4)), then, comparing (2.5.4) with (2.4. 1 ), we find that IT Z{f/J;, s + !(n; 1), f.-) = Z{f/J, s + t(n 1), !) . j

-

-

So we have proved that /(�) = 0 ;I(u;) . (2.6) Now we pass to the functional equation. In (2.4. 1), let us replace 1 by 1v and f/J by f/J". Then H is replaced by ii {(2.2. 5)) and the identity (2.4. 3) gives : (2 . 6 . 1 ) h(m!l, mz-1, · · · m;-1 ; k', k) = ii(pk, k')o-p112(p). ,

Similarly (2.4.2) gives (2.6.2)

lf!"(mh m2,

···,

m, ; k', k)

=

J f/J1''(k-1pk') ® du;i,

where lf!" denotes the Fourier transform of 1/! with respect to each one of the vari­ ables m; . Instead of (2.4. 1) and (2.4.4) we have now for Re s large enough

69

PRINCIPAL £-FUNCTIONS

Z(r[JA, s + -!-(n - l),JV) =

(2.6. 3) =

J dk dk' S r[JA(k-1pk') J detpJ s+(n-1l 12H(pk', k) d1p J dk dk' J lf!A(mt. m2, · · · m, ; k', k) ,

" h(mll , mz1, · m;1 ; k', k) n j det m; J s+(n;-1)/Z ® d X m; . " "

i

,

The functional equations for the representations O"; and the remarks we made on h and If! imply now Z(rtJA, I - s + -t (n - l), JV) = fl r(s, a; , sf!)Z(f/J, s + -t (n - 1), /) . i

This concludes the proof of (2.3). (2.7) Again let the notations be as in (2.3). From (2.3) applied to � and � we get (2.7. 1) L(s, �) = fiL(s, a,.) , L(s, �) £

=

fiL(s, 0',). £

Then the last assertion of (2.3) reads e(s, �. ¢)

(2.7.2)

=

fl e(s, 0"; , ¢). i

Let 1C be an irreducible component of � ; then any coefficient of 1C is a coefficient of � and it is an irreducible component of �- It follows that Proposition ( 1 .2) is true for (1C, it). Moreover (2.7.3) I(1e) c /(�), !( it) c /(�), r (s, 1e, ¢) = r(s, �. ¢). Thus there are two polynomials P, P in C [X] such that (2.7.4) L(s, 1e) = P(q-s)L(s, n L(s, it) = P(q-s)L(s, �), P(O) = P(O) = 1 . Note that P(q-s) divides L(s, �) -1 in C[q-s]. Moreover 1C) L(s, �)P(q-s) (s, �. ¢) e(s, 1C , ¢) - r (s, 1C , ¢) L(1L(s, - s, it) - r L(l - s' �) P(q- l+s) (2.7.5) - e(s, �. ¢) P(q-s)s . P(q-l+ ) _

_

Since the e-factors are units of the ring C[q-s , qs], we find that

P(X) = Il (1 a;1qX) . (2.8) In general, if 1C is an arbitrary irreducible admissible representation of G" ' it is a component of an induced representation of the form (2. 1 .4) where the 0"; are cuspidal. Since (1 .2) is true for each O"; ((1 .2.5)), it is true for (�, �) ((2.3)) and thus for (1e, it). So (1 .2) is completely proved. 3. Computation of the £-factor. We have established ( 1 .2) for any irreducible admissible representation 1C of Gn ; we also know r(s, 1e , ¢) ((2.7.2)) . It remains to compute the £-factor. (2.7.6)

-

70

HERVE JACQUET

(3. 1) Square-integrable representations. We have seen that if 71: is cuspidal then L(s, 71:) = 1 unless n = 1 and 71: = at, in which case (3. 1 . 1)

We now compute L(s, 71:) when 71: is essentially square-integrable. Recall that 71: is square-integrable if it admits a central character w and its coefficients (which transform under w) are square-integrable modulo the center. A representation 71: is essentially square-integrable if it has the form 71: = 7l:o ® at where 7l:o is square­ integrable and t real. We now review the work of Bernstein and Zevelenski on the construction of such representations. Let r be a divisor of n so that n = jr. Let P be the standard parabolic subgroup of Gn of type (j, j, . . · , j). Let also -r be a cuspidal representation of G, . Set for 1 ;;;; i ;;;; r, a; = -r ® ai- 1 , Then the induced representation (3. 1 .2)

admits a unique essentially square-integrable component 'Jl:, All essentially square­ integrable representations 71: are bbtained in this way and r, -r are uniquely deter­ mined by 71:. PROPOSITION (3. 1 .3). With the above notations L(s, 71:) = L(s, -r) . Indeed suppose L(s, -r) = 1 . Then L(s, a;) = 1 and by (2.7.1) L(s, �) = 1 . By (2.7.4) we have then L(s, 71: ) = 1 . Suppose now L(s, -r) # 1 . Then j = 1 , r = n, 1: = a.t and our assertion is nothing but Proposition (7. 1 1) of [R.G.-H.J.]. REMARK (3. 1 .4) . If 71: is square-integrable then the poles of L(s, 71:) are in the half-plane Re(s) ;;;; 0. This follows from (3. 1 .3) or from Proposition ( 1 . 3) of [R.G.-H.J.].

(3.2) Tempered representations. Let again P be a parabolic subgroup of type (nb n2, . . . , n,). Let now a,, be a square-integrable representation of Gn,· Then it is

well known that the induced representation (3.2. 1)

is irreducible (cf. for instance [H.J. 2]). The irreducible representations of this type are precisely the tempered ones. Note that if 71: is equivalent to another representa­ tion ' (3.2.2) 71: = I(G, P ' ; a � , a � , . . · , a � , ) where the ai are square-integrable, then P' and P are associate. More precisely

r = r ', P = MUp, P = M' UP' , and there is an inner automorphism of Gn taking M to M' and a = x a; to a' = x a;. ; in other words one passes from (M, (a;)) to ( M', (ai)) by a permutation of the diagonal blocks of M and M'. Conversely if

P' and the a;. are related to P and the a; in this way, then 71:' is equivalent to 'Jl:. By (2.3), L(s, 71:) = IT L(s, a;), L(s, 7t) = IT L(s, ii;), (3.2.3)

i

t"

.s(s, 71:, ¢) = Il .s (s, a, , ¢). i

71

PRINCIPAL £-FUNCTIONS

REMARK (3.2.4). It follows from (3.2.3) and (3. 1 .4) that the poles o f L(s, n ) for n tempered are contained in the half-plane Re(s) � 0. A representation n is said to be essentially tempered if it has the form n = no ® at where no is tempered and t is real. Then (3.2.5) L(s, n) = L(s + t, n0). (3.3) Langlands construction. We now review the work of Silberger and Wallach which extends the results of Langlands to the p-adic case. Let Q = MQ UQ be a parabolic subgroup of type (PI> p2 , . . . , p,) and for each i, I � i � r, -.,. an irreducible essentially tempered representation of Gm;· Set -. = x -.,. . Then -.,. = -.,., 0 ® at; where -.,., 0 is tempered and t,. real. We assume that (3.3.1) t 1 > t2 > . . . > t,.

Then the induced representation (3.3.2) 'lJ = l(G, Q ; -. b -.z, . . . , -. ,) has a largest proper subrepresentation 'l}' (possibly {0}) ; the irreducible representa­ tion n = 'l}/'1)' is noted (3.3.3) Every irreducible representation n has the form (3.3.3) where P (standard) and the -.,. are uniquely determined. The space V' of '1)1 may be described explicitly. Let W be the space of -. and W the space off. Then V' is the space of F in the space V of t; such that (3.3.4)

J (F(ug),

v)

du

=

o,

The integral is absolutely convergent and extended to ()

=

()Q

=

UQ, where

Q = t Q is the parabolic subgroup opposed to Q. It easily follows that the coef­ ficients of n can be obtained by integrals similar to (2.2.4). Namely let H: Gn x --->

C be a function satisfying the following properties : H(u 1 mgb u2mg2) = H(gb g2), u 1 E UQ, u2 E DQ, m E MQ ; (3.3.5) Gn

(3.3.6)

for any gb g2 the function m ,_. H(mgb g2) is a coefficient of ,. ® o}j2 ;

H is Kn x Kn finite on the right. (3.3.7) Then the function f defined by the convergent integral

(3.3.8)

f(g) =

JMIG H(hg, h) dh = JU- xK H (ukg, k) dk du

is a coefficient of n and all coefficients of n can be obtained in this way for suitable

H ( see (3.6.6) and (3.6.7) for convergence questions) . Instead of Q we may consider Q. Then if condition (3.3. 1) is replaced by the

similar condition with the inequalities reversed, the representation analogous to (3.3.3) is defined. For instance the quotient

72

HERVE JACQUET

(3.3.9) of the induced representation (3.3.10) is defined. Its coefficients are given by integrals (3.3.8) where H satisfies (3. 1 .5)­ (3. 1 .7) with (Q, i') instead of (Q, -r) . In particular suppose f is the coefficient of 7C defined by (3.3.8) where H satisfies (3.3.5)-(3.3.7) for (Q, -r). Then (3.3. 1 1) with H(gb g2)

JV(g) = = H(g2, g 1 ) .

jV is a coefficient of 7C' and

JMIG H(hg, h) dh,

But H indeed satisfies (3.3.5)-(3.3.7) for (Q,

i').

Thus

7C' = if:. (3.3. 1 2) Of course Q is also conjugate to the standard parabolic subgroup Q' of type (n" nr-b . . · , nl ) so that (3.3. 1 3) (3.4) THEOREM. Let the notations be as in (3.3). Then L(s, 7C)

=

IT L(s, ..-,.) , L(s, if:)

c(s, 7C, ¢)

=

=

IT L(s, i';),

IT c(s, -r; , ¢).

It is enough to prove the assertion relative to L(s, 7C) ; indeed the one for L(s, if:) can be obtained by exchanging 7C and if: and the last assertion follows then from (2.7) and (cf. (2.3)) r(s, r;, ¢) = IT ; r(s, ..-,., ¢). The proof will occupy the rest of this section and (3.5). By (2.3) and (2.7) there are polynomials P and P such that L(s, 7C) = P(q-s) IT L(s, ..-,.) , L(s, if:) = P(q-s) IT L(s, i';) .

It follows from (3.2.4) and (3.3.1) that L(s, ..- 1 )- 1 and IT £(1 - s, i';) prime. Thus P(q-s) is prime to L(s, ..-1 )- 1 ((2.7.5)). So if P # 1 then (3.4. 1)

-

1

are relatively

IT L(s, -r;)

2�i�r

is not in 1(7C). Therefore it will suffice to show that (3.4. 1) is indeed in 1(7C). (3.5) The identity L(s, 7C) = IT L(s, -r;) (3.5. 1) l�i�r

is trivial for r = 1 . Assuming it is true for r - 1, let us prove it for r. As we have just seen, it suffices to show (3.4. 1 ) is in 1(7C). Set n 1 = P h n2 = p2 + · · · + p,. Let Q' be the parabolic subgroup of type (p2, · · · , p ,) in G"2. Then the representation (3.5 . 2) is defined. Set also 0" 1 = -rb O" = 0" 1 x O"z and let P be the parabolic subgroup of type (nb n2) in Gn. Then 7C is actually a quotient of

PRINCIPAL £-FUNCTIONS

73

(3.5.3) More precisely it is easy to see that the coefficients of 77: are given by the integrals (3.3.8) where H is any function satisfying (3.3.5)-(3.3.7) with (P, a) instead of (Q, -.). If/is a coefficient of 77: defined in this way, then Z(�, s + }(n - 1),/) (3.5.4) = dk du l det g i•Hn-l)/2 �(k-l u-lg)H(g, k) dxg, G. where u is integrated over (] = Up. This is also

JJ

Js

J

JJ l det , 1 det �; �J-1 12 [(; �J k', J . JJ [ (: � ) ]

dk dk'

. a

(3. 5.5)

m l i •Hn - ) /2 1

l

n

� k-

1

1 m2

l

m2 1 •+
xy

k' dx dy .

Here m; ranges over Gn;- We are going to see that given �2 e 9'(n2 x n2, F) and a coefficient /2 of o-2 there are H and � so that Z( �2, s + t(n2 - I), /2) is equal to (3. 5.5). Since, by the induction hypothesis, L(s, o-2) is equal to (3.4. 1), it will follow that (3.4. 1) belongs to 1(77:) which will conclude the proof. There is a coefficient/1 of 77: 1 and a �1 in 9'(n1 x n1, F) with support in a compact neighborhood of e such that (3.5.6) There are also �12 e 9'(n 1 x n2, F), �2 1 e 9'(n2 x nh F), with supports in a neigh­ borhood of 0 so that the function � e 9'(n x n, F) defined by m (3.5.7) � 1 y = � � (m i )�I 2(Y)�21(x) �2(m2)

(

m2

x

)

satisfies (3.5.8) There are also two K-finite functions � and e on K such that

JJ

(3.5.9) Then : Z(�2,

s+ =

(3.5.10)

-!(n

�(k-lzk') � (k)f(k') dk dk' = �(z).

- 1 ) , /2 )

JJ l det

l

m l •+ (n ,-l ) /2

· dx m1dx m2 ·

JJ

JJ { (; k

-l

l det m2 1•+
t!zl m2

� xy) k' ] dx dy.

HERVE JACQUET

74

The proof is then finished as in (2.5). Namely let dh and dh' be the normalized Haar measures on K n P and K n P respectively. For h E K n P, h' E K n P set h

=

hi =

(�1 �J

* \. (0h;_ h;) '

then (3.5.10) is also equal to

JJ dk J J l det m1 l•+
(3.5. 1 1) ·

JJ ,.,[''lk-l(mxm1 1 m2 Y+ xy) k'] dx dy

where

[(

) ] (� �J 1 2JJ �(hk)�'(h'k')f1(h1m1h;_-1)fz(hzmzh�-l) dh dh' .

l 0 H1 m 0 mz , k ' k' (3.5. 1 2) = op 1 -

There is H satisfying (3.3.5)-(3.3.7) for (P, u) so that 0 m 0 (3.5. 1 3) H Om 1 m2 k' ' k = H1 0 1 mz ' k' k' · Comparing (3.5. 1 1) to (3.5.5) we obtain our assertion. (3.6) Unramified representations. Suppose 7C is unramified, that is contains the unit representation of K. Then Bn denoting the parabolic subgroup of type (1 , 1 , 1 , . . . , 1), there are u; e C so that 7C is the unique unramified component of

[(

(3.6. 1) We may even assume u 1 � n-tuple of the f.J.; in the form

) ]

u2 �

• • •

)

[(

� un .

]

Changing notations we may write the

where n = n 1 + n2 + . . . + n , 1J{ = a1i).{, A.{ is a character, ti is real and t1 > t2 > · · · > t,. Then 7: i = l(Gni• Bn i ; lJ { , lJ� , , lJ�) is an essentially tempered unramified representation of Gni and in fact 7C = J(G, Q ; 't:I o 7: z, . . . , 7: ,), (3.6.2) if Q is the parabolic subgroup of type (n 1o n2, . . , n ,) . This follows from an explicit computation : if H satisfies the conditions (3.3.5)-(3.3.7) for (Q, X 7:i) and H(k, k' ) = 1 then • • ·

•

(3.6.3)

J H(u, e) du

=f.

o.

Thus the "J-component" ofrJ = l(G, Q; 1: 1o 7:z, . . . , 1:,) is unramified ; since � � 1J we arrive at (3.6.2).

PRINCIPAL £-FUNCTIONS

Thus (3.6.4)

75

L(s, 11:) = ITL(s, p;), L(s, it) = ITL(s, pi1), j

e(s, 11: , ¢) = Ile(s, p;, ¢) ( = I ifthe exponent of¢ is zero). j

For more precise results see [R.G.-H.J., §6]. REMARK (3.6.5). Let the notations be as in (2.3). Suppose U ; is irreducible un­ ramified. Then � admits a unique unramified component 11: . It easily follows from (3.6.4) that L(s, 11:) = TI L(s, u;) with similar relations for L(s, it) and e. REMARK (3.6.6). Let 11: be a tempered representation of Gn. Let Pn = I(Gm Bn ; 1 , 1 , · · · , 1) and Bn be the spherical function attached to p. Then any co­ efficient of 11: is majorized by a multiple of Bn. Let the notations be now as in (3.3). Then the coefficients of 7:; are majorized by a multiple of Bn; ® at;. Moreover if H satisfies (3.3.5)-(3.3.7) for (Q, 1:) then IHI � cH0 where H0 satisfies (3.3.5) and H0 (mk, k') = o}{Z(m) ij Bn;(m;) . I

Thus as far as convergence is concerned we may replace 7:; by Pn; ® at; and H by H0 > 0. Together with the next lemma, this takes care of all problems of conver­ gence. LEMMA (3.6. 7). If {) is a bounded set of Y(n x n, F), then there is (/)0 � 0 in Y(n x n, F) so that (/)0(k 1 xk2) = (/)0(x) for k; E K and 1 (/) 1 � (/)0 • We may assume that {) contains also all functions x �---+ (/)(k1 xk2) with (/) E {). Then there is (/) 1 � 0 so that 1(/) 1 � (/) 1 for all (/) in {) [A.W. 1 , Lemma 5]. It suffices to take (/)0(x) = JJrp1(k1 xk2) dk1 dk2 where dk; is the normalized Haar measure on K. (3.7) According to "Langlands' philosophy" there should be a "natural bijec­ tion" u �--+ 11: (u) between the n-dimensional semisimple representations of the Weil­ Deligne group w;.. and the irreducible admissible representations of Gn(F). More­ over, one should have for 11: = 11:(u) L(s, 11:) = L(s, u) , 1t" (u) = 11:(0'),

e(s, 11:, ¢) = e(s, u , ¢), 11: (u ® x) = 11: (u) ® x ·

It is clear that if the map u �---+ 11: (u) could be defined for the irreducible representa­ tions of the Weil group, then the conjecture would be proved (cf. §5 below). In §§4, 5 the ground field F is local archimedean. (4. 1) We let Kn be O(n, R) if F = R and U(n) if F = C. We donote by @n the Lie

4. Local archimedean theory.

algebra of the real Lie group Gn(F). We consider only admissible representations of the pair (@" ' Kn) {as in [N.W.]) although we will often allow ourselves to speak of a representation of the group Gn(F) ; in any case, we assume the reader to be thor­ oughly familiar with the relation between representations of the group Gn(F) and of the pair (@m Kn). Let 11: be an admissible representation of (@"' Kn) ; then one can define the con­ tragredient representation it, the coefficients of 11: , and its central quasi-character, even though 11: is not a representation of the group (cf. [H.J.-R.L., §§5, 6]). Again if /is a coefficient of 11: thenjV ((1 . 1 .2)) is a coefficient of it.

76

HERVE JACQUET

As in the nonarchimedean case let Y(p x q, F) be the space of Schwartz func­ tions on M(p x q, F). We also introduce the subspace Yo = Y0(p x q, F) of functions (/) of the form (P being a polynomial), if F = R, 1J(x) = P(x,.i)exp( - n- L: xri)

1J(z) = P(z,.i, zii)exp( - 2n- L: z;Ai), if F = C. We can consider, for a given n-, the integrals ( l . 1 .3) where (/) is in Y(p x q, F) and f is a coefficient of n- . (4.2) PROPOSITION. Suppose n- is an irreducible admissible representation of (@m Kn). (1) There is s0 so that the integrals ( 1 . 1 .3) converge absolutely in the half-plane Re(s) > s0• (2) For (/J in Y0(n x n, F), the integrals are meromorphicfunctions ofs in the whole complex plane . More precisely they can be written as polynomials in s times a fixed meromorphic function of s . (3) Let ¢ # 1 be an additive character of F; there is a meromorphic function r(s, n-, ¢) so that,for all coefficients! of n- and all (/) E Y0(n x n, F), Z(1JA, 1 - s + l (n - l), JV) = r(s, n-, ¢)Z(1J, s, f). Again (/)!\ is defined by (1 .2.4). If (/) is in Y0(n

x

n, F) and ¢ is defined by

¢(x) = exp( ± 2in-x), if F = R, ¢(z) = exp[ ± 2in-(z + z)], if F = C, then (/)!\ is still in Y0 ; the left-hand side of (4.2.3) has then a meaning by (4.2.2) applied to i't. If ¢ is not given by (4.2.4) then some adjustment has to be made, that we leave to the reader (cf. (1 .3. 1 1), (1 .3. 12)). From now on, we assume ¢ is given by (4.2.4). EXAMPLE (4.2.5). If n = 1, then n- is just a quasi-character of px and (4.2) is proved in [J.T.] or [A.W. 2]. (4.3) The proof of Proposition (4.2) will be given below in (4.4). For the time being, we derive some simple consequences of (4.2). By Lemma (3.6.7), the convergence of (1 . 1 .3) for Re s > s0 is actually uniform for (/) in a bounded set ; thus, for Re s > s0, (4.2.4)

(4.3.1)

1J �--+ Z((!J, s, f)

is a distribution, depending holomorphically on s . Since Y0(n x Y(n x n, F), it follows that iff # 0 then there is at least a (/) E Yo � 0.

n , F) is dense in so that Z((/J, s, f)

Suppose (4.2.2) has been proved. Let L(s) be a meromorphic function of s so that, for all (/) in Y0, Z(1J, s + t(n - l ), f)/L(s ) is a polynomial ; let also /be the subvector space ofC[s] spanned by those ratios. By what we have just seen I # {0}. Moreover if (/) is in Y0, so is the function (/J' defined by (4.3.2)

77

PRINCIPAL £-FUNCTIONS

Let w be the central quasi-character of 70 ; if we differentiate with respect to t the identity (4.3.3)

Z((/), s + -! (n - 1),/) =

and then set t

J (/)(xe-1)f(x) idet xi •+
x

w

0, we arrive at a relation of the form a i= 0. (4.3.4) (as + b) Z((/), s + !(n - 1), f) + Z((/)', s + -! (n - 1), f) = 0, This shows that I is an ideal. Let P0 be a generator of I and L(s, 70) = P0(s) L(s). We see that the ratios Z((/J, s + -!(n - 1),/)/L{s, 70) are again polynomials, but this time they span C[s] as a vector space. Up to multiplication by a constant, these properties characterize L{s, 70). Assume now (4.2) proved for the pairs (70, :ir). Then r is uniquely determined and one can define a factor e as in (1 .3.5) ; then the functional equation (4.2.3) reads as in (1 .3.6). In view of the definition of the £-factors, this implies that both e(s, 70, ¢) and its inverse are in C[s]. Thus e(s, 70, ¢) is just a constant if cjJ is given by(4.2.4) ; otherwise it is an exponential factor ((1 .3.9)). REMARK (4.3.5). For the time being the L- and e-factors are defined up to multi­ plication by constants ; of course these constants are related since r is intrinsically defined. For n = 1 , one may take the factors to be those given in [J.T.l , 2]. REMARK (4.3.6). Relations (1 .3.9)- (1 .3. 12) apply to the archimedean case. (4.4). Let the notations be as in (2. 1), the ground field F being now R or C; it goes without saying that a; is now an admissible representation of (®n; • Kn,.). The induced representation .;((2. 1 .4)) can still be defined [N.W.] and its coefficients are given by the rule of (2.2), the only difference being that in (2.2.2) g; must be in Kn and in (2.2.3) H must be C"" on Gn x Gn . The remarks made before (2.3) apply and, as in (2.3), we have : PROPOSITION (4.4. 1). With the notations of (2. 1) suppose that each a; admits a central quasi-character and the assertions of (4.2) are true for each pair (a;, fi;). Then they are true for (�, �). Moreover, r(s, �. ¢) = TI ;r(s, a;, ¢) and one can take =

L(s, �) = TI L(s, f7;),

L(s, �) = TIL {s, a;), i

i

e(s, �. ¢) = Tie(s, a;, ¢). i

The proof is similar to the proof of (2.3). It suffices to observe that if (/) is in !/0(n x n ; F) the functions x �--+ (/)(k1 xk2), k; E Kno span a finite dimensional vector space of !/0 and for each k, k' in Kn the functions (2.4.2) belong to the space ® !/0(n; x n;, F). Notations being as in (4.4. 1 ), let 70 be an irreducible component of �. Then any coefficient of 70 is a coefficient of � and :ir is a component of �. Thus (4.2) is true for (70, :ir). More precisely there are polynomials R and R such that L(s, 70) = R(s) TI L(s, a;) , L(s, :ir) = R(s) TI L{s, fi;) (4.4.2) i

i

78

HERVE JACQUET

while (4.4.3) Since e(s, (4.4.4)

r(s, 11:, cp) = Tir(s, a,. , cp). i

11:,

cp) and e(s, a, cp) are constants R and R have the form R(s) = TI (s - s,), R(s) = cTI{l - s - s,). i

i

Finally any given irreducible admissible representation 11: of Gn is a component of some induced representation � as in (4.4. 1) with n,. = I . Since (4.2) is then true for each (a;, a,.) , it is true for (11:, :it') and (4.2) is proved. (4.5) PROPOSITION. For all s, L(s, 11:) #- 0. PROOF. Let 11: be an irreducible admissible representation of Gn so that 11: is an irreducible representation of some induced representation � as in (4.4. 1 ) with n,. = I . We first show that for any (/) in Y(n x n, F) the ratio (4.5.1) Z(f/>, s + f(n - 1), f)/L(s, 11:) continues to an entire function of s. Indeed let s0 be such that for Re(s) > s0 (resp. Re(s) < I - s0) the integrals Z(f/>, s + f(n - 1), f) (resp. Z(f/>, I - s + t(n - 1 ), /V)) converge absolutely. Select s 1 o s2 such that s0 < s 1 < s2 • Let also Q be a polynomial such that Q(s )L(s, �) has no pole in the strip 1 - s2 � Re(s) � s2• Let f/>; be a sequence of Yo converg­ ing to some element (/) of Y. Then, for a given f and a given polynomial R, the following limits are uniform in the domain s 1 � Re(s) � s2 : lim R(s)Z( f/>,., s + -!(n - 1 ), f) = R(s )Z( f/>, s + t(n - 1 ), f ), ( 4. 5.2) i-+oo

(4.5.3)

i, j-+oo

lim R(s)Z(f/>; -

f/)b

s+

t(n

- 1), f) = 0.

Similarly r/Jt;' - (/)If � 0 so that� uniformly in the domain I - s2 � Re(s) � 1 lim R(s)Z((/)If' - (/)If, 1 - s + t(n - I) , JV) = 0. (4.5.4)

St .

i, j-oo

Indeed by using repeatedly (4.3.4) one can reduce these assertions to the case R 1 ; they are then obvious. On the other hand, Q(s)Z(f/>,. - f/>i> s + !(n - 1), f)

;� �� �) Q(s)Z((/)If' - f/>J, 1 - s +

= e(s, �. cp) - 1 L( The classical formula

F(x + iy) r(x' - iy)

�

I Y I "'-x'


�

!(n

=

- 1), fV).

+ oo )

shows that Q(s)L(s, �)/L(l - s, �) is bounded by a polynomial in the previous strip. Thus we also find, uniformly for 1 - s2 � Re(s) � I - sb lim Q(s)Z(f/>,. - (/)b s + t(n - 1) , / ) = 0. (4.5.5) i, j-+oo

79

PRINCIPAL L-FUNCTIONS

Now fixj and let c for s 1 � Re(s)

�

>

0 be given ; choose N so that for i, j

�

J Q(s)Z(qJ; - qli, s + -!-(n - 1), / J �

c,

N

s2 or 1 - s2 � Re(s) � 1 - s 1 . But now

Q(s)Z(qJ; - qli, s + -!-(n - 1), / ) = Q(s)R(s)L(s, .;) where R is another polynomial. Since L(s, .;) decreases rapidly, on any vertical strips, we find that given (i, j) there is M so that, for j l m(s) l � M, (4.5.6)

J Q(s)Z(qJ; - qli, s + -!-(n - 1) , / )J �

s.

Thus by the maximum principle (4.5.6) is satisfied for i, j � N and l - s2 � Re(s) � s2• It follows that Q(s)Z( qJ;, s + -!-(n - 1 ),f) is a Cauchy sequence for the topo­ logy of uniform convergence on the strip 1 - s2 � Re(s) � s2• Its limit Z(s) is holomorphic on 1 - s2 < Re(s) < s2 and coincides with Q(s)Z(qJ, s + -!-(n - l ),f) on s 1 � Re(s) < s2• Since s2 is arbitrarily large the holomorphy of (4 . 5.1) follows. Suppose now L(s0, n") = 0 and / :f. 0. Then for each qJ, the meromorphic func­ tion Z( qJ, s + -!-(n - I), f) vanishes at s0 ; but if qJ has compact support contained in Gn then Z(qJ, s + -!-(n - 1), f) is defined for all s by the convergent integral (1 . 1 .3). So we find that (1 . 1 .3) vanishes for s = s0 and qJ E C�(Gn) ; therefore/ = 0, a contradiction. Thus L(s0, n) :f. 0. Q.E.D. COROLLARY (4.5.8). Suppose .; is as in (4.4. 1) and n is an irreducible component of .; . Let R and R be as in (4.4.2). Then if s; is a zero of order m of R (resp. R) it is a pole of order � m of L(s, .;) (resp. L(s, �) ). 5. Computation of the £-factor ; arcbimedean case. Recall there is a "natural bijection" A �--+ n{A) between the set of classes of semisimple representations of the Weil group Wp of degree n and the admissible irreducible representations of Gn(F) ([R.L.], [A.K.-G. Z] , [N.W.]). In particular n(A)- = n-0). (5. 1) THEOREM. For any A of degree n, let n be n(A). Then L(s, n)

L(s, A), s(s, n, ¢)

=

=

L(s, it) = L(s, h s (s, A, ¢).

The proof of this theorem will occupy all of §5. Since the £-factor has not been normalized, it would be more correct to say that one can take L(s, n) to be L(s, A) and that r (s, n, ¢) = s(s, A, cj;)L(l - s, � )/L(s, A). (5.2) Square-integrable representations. Suppose A is irreducible. Then n{A) is essentially square-integrable and conversely. More precisely, if n = 1 then A is a quasi-character of px , A = n, and our assertion follows from the definition of the factors L and c for A. Suppose n :f. 1 ; then n = 2 and our assertion has been proved in [H.J.-R.L., Theorem (1 3. 1), Lemmas ( 1 3.24) and (5. 1 7)] ; actually in this case, the proof is a refinement of (4.4. 1). (5.3) Tempered representations. Suppose A is unitary ; then n is tempered and conversely. More precisely A = EB A; where A; is irreducible of degree n; . Set rr; • '=

·

80

HERVE JACQUET

n(A;). Then n = l(G, P ; a1o a2, (nb nz , · · · , n,). By (4.4. 1)

(5.3 . 1 )

· · · , a ,)

where P is the parabolic subgroup of type

L(s, n) = IJ L(s, a;) = IJ L(s, A;) = L(s, A) .

The factors L(s, i) and e(s, n, ¢) are computed similarly so our assertion follows again. The case of the essentially tempered representations follows from (1 .3. 1 1), ( 1 .3 . 1 2), and the relation n(A ® x) = n(A) ® X · (5.3.2) REMARK (5.3.3). It follows from (5.3.1) that, when n is tempered, the poles of L(s, n) are in the half-plane Re(s) � 0. (5.4) General case. In general, we may write A = E9 p.;, p.; = p.;, o ® at; where p.;, o is unitary of degree p;, t; real. We set then -r;, o = n(p.;,0), -r; = n(p.;) so that by (5.3.2) -r; = -r;, o ® a1 i . We may assume (3.3 . 1 ) is satisfied. Then if Q is the para­ bolic subgroup of type (Pb p2, • • • , p,) the induced representation r; of (3.3.2) admits a unique irreducible quotient noted as in (3.3.3). That quotient is n = n(A) . By (4.4; 1) and (5.3) we already know that r(s, n, ¢) = IJ r(s, -r;, ¢) = t(s, A , ¢)L(1 - s, �)/L(s, A) . i

So it will suffice to compute L(s, n). For exchanging then A and � we will get L (s, i) (since 1r = n(�) ) and the t-factor from the r-factor. So it will suffice to show that L(s, n) = TI 1:;;; :;; , L(s, -r;). This is trivial if r = 1 . So we may assume r > 1 and our assertion true for r - 1 . A priori, we have L (s, n) = P(s) IJ L(s, -r;), L(s, i) = P(s) IJ L(s, f;)

where P and P are polynomials related by (4.4.4). Let s0 be a zero of order u of P. By (4.5.8) it is a pole of order � u of TI ;L(s, -r;) and by (4.4.4) a pole of order � u of TI ;L(l - s, f;). But L(s, -r1) and TI L( l - s, f;) cannot have a common pole ((5.3.2), (5.3.3), (3.3. 1)). Thus it will suffice to show that any pole of order u of TI ;;;:;2 L(s, -r;) = TI ;;;:; 2 L(s, p.,-) which is not a pole of L(s, -.1) is a pole of order � u of L(s, n). This will be proved in (5.5) and (5.6). (5.5) Let now the notations be as in (3. 5). So set n1 = PI > n2 = n - n1o a1 = n(p.1) = -r1o O"z = n(p.z EB • . . $ p.,), a = a1 x a2. By the induction hypothesis (5.5. 1 )

L(s, O"z)

=

2 �i� r

IT L(s, p.;).

Again n is a quotient of (3.5.3) where P = MU is the parabolic subgroup of type (nh n2). The coefficients of n are given by the absolutely convergent integrals (5.5.2)

f(g) =

JAnG H(hg, h) dh = JKx U H(ukg, k) dk

where D = Up, P = tp and H: Gn ing properties : (5.5.3)

_

x

Gn --+ C is any function satisfying the follow­ u1 e U, u2 e D, m e M;

PRINCIPAL £-FUNCTIONS

o}(2,

81

(5.5.4) for k; E Km the function m f-> H(mkb k2) is a coefficient of 0' (8) H is Coo and Kn x Kn finite on the right. (5.5.5) This being so iff is given by (5.5.2) then Z(
=

J l det md s+ Cn1-l l 1 det m21 s+ Cnz-ll/2Ji (m1) y 1 f2Cm2)
2

This multiple integral converges absolutely if Re(s) is large enough. We are going to show that (5.5.7) For any
0,

(5.5.8) Then there is a function H satisfying (5.5.3) -(5.5.5) such that (5.5.9)

H [(�l �2) k', kJ o¥2(�1 �J J Jfl (hlmlh�-l)f2 (h2m2h�-l)�'(h'k')�(hk) dh dh' . =

If/is the coefficient of n corresponding to H by (5.5.2) we get, as in (3. 5), (5. 5. 10)

A(
=

Z(
This establishes (5.5.7) for

(; l �J :r
0

and is in Y'0 if

0

(5. 5. 1 2)

B(
=

r(s, n , cf;)A(
where B is obtained by replacing in the definition of A the pair (P, 0') by the pair (P, 6'). It is then clear that (5.5.7) can be proved as the holomorphy of (4.5. 1) (cf. proof of(4.5)). To begin with it is clear that it can be obtained from (P, 6') as n is obtained from

82

HERVE JACQUET

(P, a). (Cf. (3. 3). ) More precisely iff is a coefficient of 77: given by (5.5.2) then JV is given by (3.3. 1 1) and H (loc. cit.) satisfies (5. 5.3)-(5.5.5) for (P, a). This being so set B(f/J , s, /1 , /2) =

(5. 5. 1 3)

J j det m1 j •+Cn1-1l 12 j det m2 j •+C"2-1l 12 f1 (m 1) /2(m2) (/J[(m 1 ; yx �: 2)J dxm 1 dxm2 dx dy,

each time f.· is a coefficient of a,.. Then for (/) E Yo (5.5.7) applies to B with L(s, ft) instead of L(s, 77:). Now let (/) E Yo and a coefficient /; of 0"; be given ; choose � . e and H as above so that (5.5.9) is satisfied as well as (5.5. 10) where f is given by (5.5.2). Then

JJ (JJA[k-lzk']e(k)�(k') dk dk' = (/JA(z). Moreover n - being given as in (3.3. 1 1) by H(gh g2) = H(g2, g1), we find

(� �2) JJf�[hlmlh�-1]f'/[h2m2h2-1] �(h'k')�'(hk) dh dh'.

= o}(2 1

SincefV is given by (3.3. 1 1) we get B (f/Ji\, 1 - s,f �, f '{)

=

Z(f/JA, 1 - s + t(n - 1), JV).

Comparing with (5.5.10) this concludes the proof of (5.5. 1 2) and (5.5.7). (5.6) Let d be the space of functions on X = Gn 1 x {M(n2 x n2 , F)} which, in an obvious sense, are of compact support with respect to the first variable and of Schwartz type with respect to the second variable. We give to d the obvious topolo­ gy. In particular C;'(X) is a dense subset of d. On the other _hand we may regard d as a subspace of the space of Schwartz functions on {M (n 1 x nh F)} x {M (n2 x n2, F)}. For f/J1 in C;'(Gn 1), f/J12 in C';'(M (n 1 x n2, F)), f/J21 m C;'(M (n2 x nh F)), and f/J2 in Y(n2 x n2, F) we may set (5.6. 1) We let do be the subspace of d spanned by these functions ; it is easily checked that do contains a dense subspace of C;'(X) so is dense in d. For 7/f in d and coefficientsfh /2 of 0"1> a 2 we set (5.6.2)

U (7/f, s, fl , /2) =

JJ 7/f(mh m2) j det md s+ (nJ-1)/2 j det m2 j s+ Cnz-1)/2

· Ji (m 1)/2(m2) dxm 1 d x m2.

The integral converges absolutely for Re(s) large enough. By (5.5.7) we know that (5.6.3) For 7/f in d0, the ratio U(7/f, s, /1 , /2) /L (s, 77:) continues to an entire function ofs. On the other hand, we are going to prove that

PRINCIPAL £-FUNCTIONS

E-3

(5.6.4) For 1/f in d, the ratio U(l/f, s, fi> f2)/L(s, O'z) continues to a holomorphic function of s. As such, it is a continuous function of(l/f, s) on d x C. Let us observe that the first assertion of (5.6.4) is trivial for 1/f in the (algebraic) tensor product (5.6.5) For if 1/f = 1/f 1 ® 1/f2 the ratio is the product of an absolutely convergent integral by Z(l/f2, s + -}(n2 - 1 ), /2 ) /L(s, O'z) which is a polynomial. Moreover assume s0 is a pole of order u of L(s, 0'2) ; then there aref1 , f2 and 1/f E d1 so that the ratio of (5.6.4) has a pole of order u at s0. Taking at the moment (5.6.4) for granted, assume s0 is not a pole of L(s, 0' 1 ) and is a pole of order < u of L(s, n). Since do is dense in d, it follows from (5.6.3) that the distribution U( , s,fb f2), which depends meromorphically on s, has a pole of order < u at s0 • The same is true of the (scalar) meromorphic functions U(l/f, s, fi> f2) . By taking 1/f in d1 we get a contradiction. This proves (5. 1). It remains therefore to prove (5.6.4). As noted, the first assertion is trivial for 1/f in d1, the ratio being then the product of a polynomial by a function bounded in vertical strips. Now there is a relation (5.6. 6) a #- 0, (as + b) U(f/J, s, /1, /2) + U(f/J', s, /1 , /2) = 0, where (/J' is in d1 if (/J is and depends continuously on f/J. Next introduce for Re s sufficiently small the integral V(f/J, s, /1 , /2) ·

=

J 1/f(mb m2) i det mds+ (n!-1) /2 i det m211-s+ (nz-l) /2f1(m1)f2V(m2) dxm1 dxm2.

Then, for 1/f E d 1o V (1/f�'� , s, fl , fz) = r(s, 0'2, ¢) U(f/J, s, /1 , /2). Again V satisfies a relation like (5.6.6). Finally, d 1 is dense in d. These remarks being made the proof of (5.6.4) is similar to the proof of the holomorphy of (4.5. 1). This concludes the proof of (5. 1). 6. Global theory. The field F is now an A-field. (6. 1) Let n = ®v nv be an irreducible admissible representation of Gn(A) ; again n may be a representation of some other object than the group [I.P.S.]. Then ir = ®irv · Set (6. 1 . 1) L(s, ir) = IJL(s, ir0). L (s, n) = IT v L(s, n0), v Suppose that the central quasi-character w of n is trivial on px . Then set (6. 1 .2) nv, ¢ v). c(s, n) = Ilc(s, v Here ¢ = TI ¢v is a nontrivial character of A/F; almost all factors in (6. 1 .2) are one and the product does not depend on ¢. (6.2) THEOREM. Suppose n is automorphic. Then the infinite products (6. 1 . 1) con­ verge absolutely in some right half-space. They continue to meromorphic functions of

84

HERVE JACQUET

s in the whole complex plane . As such they satisfy L(s, n) = s(s, n)L(l - s, it). If F is a number field, they have finitely many poles and are bounded at infinity in vertical strips . If F is a function field whose field of constants has 0 elements, they are rationalfunctions of o-s. PROOF. If n is cuspidal this is Theorems 3, 4, 5, VII, §6 of [A.W. 2] for n = 1 , and Theorem (1 3.8) of [R.G.-H.J.] for n > 1 . If n > 1 and n i s not cuspidal, there is a standard parabolic subgroup of type (nb n2, · · · , n,) , and, for every i, an irredu­ cible automorphic cuspidal representation a; of Gn; such that the following condi­ tions are satisfied : for any place v, n. is an irreducible component of the induced representation e. = /(G., P. ; a1., a2., · · · , a,.) . (Cf. [R.L.].) Thus, for any place v, L (s, n.)

=

P.(s) IT L(s, a .- . •) ,

L (s, it.)

=

P.(s) IT L(s, a.- . •) .

r(s, n., ¢.)

=

IT r(s, a.- . •• ¢.),

j

j

j

where P. is a polynomial in s if F. = R or C, and a polynomial in q;;• if F. is non­ archimedean with residual field of cardinality q • . Moreover, for almost all v, n . and the a.- . • are unramified so that ((3.6.5)) P. = P. = 1 , s(s, n., ¢.) = 1 , almost all v. Thus L (s, n)

=

L(s, it)

=

P.(s) IT L(s, a,-) , IT v j

ITv P.(s) ITj L(s, a,.),

Pv(1 - s) s(s, n) - IT IT. s(s, a,-) . v pv(s) ' _

Our assertions are then obvious, except the one on the boundedness of L(s, n) in the number field case, which follows from the Phragmen-Lindelof principle. BIBLIOGRAPHY [A.A.] A. Andrianov, Zeta functions of simple algebras with non abelian characters, Uspehi Mat. Nauk 23 (1968), 1-66 = Russian Math. Surveys 23 (4) (1968), 1-65. [1.8.-A.Z. 1] I. N. Bernstein and A. V. Zelevinsky, Representations of the group GL(n, F) where F is a non archimedean local field, Uspehi Mat. Nauk 31 (3) (1976), 5-70 = Russian Math. Surveys 31 (3) (1976), 1-68. [1.8.-A.Z. 2] -- , Induced representations of the group GL(n) over a p-adic field, Funkcional Anal. i Prilozen 10 (3) (1 976), 74-75 = Functional Anal. Appl. 10 (3) (1976), 225-227. [1.8.-A.Z. 3] -- , Induced representations of reductive p-adic groups. II, preprint. [M.E.] M. Eichler, Allgemeine Kongruenzklasseneinteilungen der /deale einfacher A lgebren uber algebraischen Zahlkiirpern und ihrer L-Reihen, J. Reine Angew. Math. 179 (1 938), 227-25 1 . [G.F. 1 ] G . Fujisaki, On the zeta-functions of a simple algebra over the field of rational numbers. Fac. Sci. Univ. Tokyo, Sect. 1 7 (1 958), 567-604. [G.F. 2] --, On the £-functions of a simple algebra over the field of rational numbers, Fac. Sci. Univ. Tokyo, Sect. 1 9 (1 962), 293-3 1 1 .

PRINCIPAL £-FUNCTIONS

85

[S.G.) S . Gelbart, Fourier analysis o n matrix spaces, Mem. Amer. Math. Soc., vol. 108, Provi­ dence, R.I., 1 971 . [R.G.) R. Godement, Les fonction C des algebres simples. I, II, Sem. Bourbaki, 1 957/1 958, Exp. 1 7 1 , 1 76. [R.G.-H.J.) R . Godement and H. Jacquet, Zeta functions of simple algebras, Lecture Notes in Math., vol. 260, Springer-Verlag, 1972. [E.H.) E. Heeke, Uber Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produkten­ twicklung. I, II, Math. Ann. 114 (1937), 1-28 ; 3 1 6-35 1 . [K.H.) K. Hey, Analytische Zahlentheorie in Systemen hyperkomplexer Zahlen, Dissertation, Hamburg, 1929. [K.I.) K. Iwasawa, A note on £-functions, Proc. Intemat. Congr. Math., Cambridge, Mass., 1 950, p. 322. [H.J. 1) H. Jacquet, Zeta functions of simple algebras (local theory), Harmonic Analysis on Homogeneous Spaces, Mem. Amer. Math. Soc., Providence, R.I., 1973, pp. 381-386. [H.J. 2] , Generic representations, Non-Commutative Harmonic Analysis, Lecture Notes in Math., vol. 587, Springer-Verlag, Berlin, 1 977, pp. 91-101 . [H.J.-R.L.) H. Jacquet and R. Langlands, Automorphic forms on GL{2), Lecture Notes in Math., vol. 1 14, Springer-Verlag, Berlin, 1 970. [M.K.) M. Kinoshita, On the �-functions of a total matrix algebra over the field of rational num­ bers, J. Math. Soc. Japan 17 (1 965), 374-408. [A.K.-G.Z.] A. W. Knapp and G. Zuckerman, Classification theorems for representations of semi-simple Lie groups, Non-Commutative Harmonic Analysis, Lecture Notes in Math., vol. 587, Springer-Verlag, Berlin, 1977, pp. 1 38-1 59. [R.L.) R. P. Langlands, On the classification of irreducible representations of real algebraic groups, mimeographed notes, Institute for Advanced Study, Princeton (1 973). [H.M. 1) H. Maass, Uber eine neue Art von nicht-analytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Functionalgleichungen, Math. Ann. 121 (1 949), 1411 83. [H.M. 2) , Zetafunktionen mit Grossencharakteren und Kugelfunktionen, Math. Ann. 134 (1957), 1-32. [G.M. 1) G. N. Maloletkin, Zeta functions of a semi-simple algebra over the field of rational numbers, Dokl. Akad. Nauk SSSR 201 (3) (1971), 535-537 Soviet Math. Dokl. 12 (6) (1971), 1 678-1681 . [G.M. 2) , Zeta /unctions of parabolic forms, Mat. Sb. 86 (1 28) (1971), 622-643 = Math. USSR-Sb. IS (4) (1971), 61 9-641. [A.S.) A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. 20 (1956), 47-87. [H.S.) H. Shimizu, On zeta functions of quaternion algebras, Ann. of Math. (2) 81 (1965), 1 66193. [G.S.] G. Shimura, On Dirichlet series and abelian varieties attached to automorphic forms, Ann. of Math. (2) 76 (1962). [E.S.] E. Stein, Analysis in matrix spaces and some new representations of SL(N, C), Ann. of Math. (2) 86 (1 967), 461-490. [T.T.] T. Tamagawa, On the zeta functions of a division algebra, Ann. of Math. (2) 77 (1963), 387-405. [J.T. l) J. Tate, Fourier analysis in number fields and Heeke's zeta-functions, Algebraic Number Theory, edited by J.W.S. Cassels and A. Frohlich, Thompson, 1967, pp. 305-347. [J.T. 2) --, Number theoretic background, these PROCEEDINGS, part 2, pp. 3-26. [N.W.) N. Wallach, Representations of reductive Lie groups, these PROCEEDINGS, part 1, pp. 7186. [G.W.) G. Warner, Zeta functions o n the real genera/ linear group, Pacific J. Math. 4S (1973), 681-691 . [A.W. 1) A. Weil, Fonctions zeta et distributions, Sem. Bourbaki, no. 3 1 2, (1965-66), 9 pp. [A.W. 2) --, Sur certains groupes d'operateurs unitaires, Acta Math. 111 (1 964), 144-21 1 . [A.W. 3) , Basic number theory, Springer-Verlag, Berlin, 1967. --

--

=

--

--

86

HERVE JACQUET

[A.W. 4] A. Weil, Dirichlet series and automorphic forms, Lecture Notes in Math., vol. 1 89, Springer-Verlag, Berlin, 1 97 1 . [E.W.] E . Witt, Riemann-Rochscher Satz und �-Funktion im Hyperkomplexen, Math. Ann . 110 (1 934), 1 2-28. [A.Z. 1] A. V. Zelevinskii, Classification of irreducible non-cuspidal representations of the group GL(n) over p-adic fields, Funkcional Anal. i Prilozen 11 (1) (1 977), 67 Functional Anal. Appl. 1 1 (1) (1 977). [A.Z. 2] Classification of representations of the group GL(n) over a p-adic field, preprint. (Russian) =

-- ,

COLUMBIA UNIVERSITY

Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 2, pp. 87-95

AUTOMORPHIC L-FUNCTIONS FOR SYMPLECTIC GROUP

G Sp(4)

MARK E. NOVODVORSKY* 0. Introduction. The paper represents a talk delivered at the Summer Institute of the American Mathematical Society at Corvallis in 1 977. Let k be a global field, P the set of its normalized valuations, A = 11 pE P kp the adele ring of k, G a reductive algebraic group over k, GA = Il pEP Gp the group of ideles of G, 1t: = ®pEP 7t:p a class of isomorphisms of irreducible admissible repre­ sentations of GA• p a finite dimensional representation of Langlands' dual group LG. R. P. Langlands [1] defined £-function L(1r:, p, s) as a certain Euler product, L(1r:, p, s) = pITP L(7t:p, p, s), (0. 1) E convergent for all complex numbers s with sufficiently large real part if 1r: is uni­ tarizable. He conjectured that if 1t: is the class of an automorphic representation, then L(1r:, p) can be continued to a meromorphic function on the whole complex plane which satisfies the functional equation (0.2) L(1r:, p, s) = c(7t:, p, s) L(1r: * , p, 1 - s), 1r:* being the contragradient representation to 1t: and c(7t:, p, s) = pEIT c(7t: P• p, s) P another Euler product whose existence is a part of the conjecture. However, the definition of the local factors L(7t:p , p, s) depends on a parametrization problem (cf. Borel's paper, Automorphic L-function, these PROCEEDINGS, part 2, pp. 27- 61) and, therefore, does not work at present until all the irreducible representations 1t: P of the groups G P corresponding to nonarchimedean valuations p are of class 1 . Therefore, in order to prove Langlands' conjecture one must first construct the Euler factors L(1r:p) and c(7t:p). A general approach to this problem was suggested by I. I. Piatetski-Shapiro [2]. Here we develop the ideas of [2] for the symplectic group G = G Sp(4) and prove Langlands' conjecture for the standard 4-dimensional representation p of its Lang­ lands dual group LG = G Sp(4, C). We consider also the group G = G Sp(4) x GL(2) and prove Langlands' conjecture for generic cuspidal automorphic repre­ sentations 1r: (cf. Piatetski-Shapiro's paper Multiplicity one theorems, these PRoAMS (MOS) subject classifications (1 970). Primary lOD lO, 22E55 .

* Supported by NSF Grant MCS76-021 60Al . © 1979, American Mathematical Society

87

88

MARK E. NOVODVORSKY

CEEDINGS, part 1 , pp. 209-21 2) and the standard 8-dimensional representation p of its Langlands dual group G Sp(4, C) x GL(2, C) (i.e. p is the tensor product of the standard 4- and 2-dimensional representations of the factors). Our results are complete only for functional global fields k . For number fields k we obtain meromorphic continuations of these automorphic L-functions but, because of certain difficulties with archimedean valuations, not the functional equations (0.2). Since the methods of [2] are based on generalized Whittaker models, we consider generic and hypercuspidal automorphic representations separately (cf. the quoted paper of Piatetski-Shapiro). The results for generic representations of G Sp(4) (§ 1) can be extended to all split orthogonal groups of Dynkin type Bn (Sp(4) covers S0(5)) ; in this form they were announced by the author (for char k > 0) in [10] and [11]. The results for hyper­ cuspidal representations of G Sp(4) (§2) were obtained by I. I. Piatetski-Shapiro and the author. The results for G Sp(4) x GL(2) (§3) were announced by the author (for char k > 0) in [12]. In this paper we assume that char k "# 2. 1. G Sp(4), !generic representations. Let G � G Sp(4) be the group of similitudes of the bilinear form

of four variables over k, Z the subgroup of all upper triangular unipotent matrices from G, cjJ a nondegenerate l character of the group ZA of ideles of Z which is trivial on principal ideles Zk, p a nonarchimedean valuation of the field k. The space of all locally constant complex functions W on the group Gp which satisfy the equation (1 . 1 ) W(zg) = cp(z) W(g) 'V z E Zp, g E Gp, is denoted "'f/'q,(Gp) ; Gp acts in this space by right translations. A class 11:p of isomor­ phisms of irreducible admissible representations of the group Gp is called non­ degenerate if it contains the subrepresentation of GP in an irreducible Gp-invariant subspace of "'f/'q,( Gp) ; this subspace then is called a Whittaker model of 11:p and de­ noted by "'f/'q,(71:p) · THEOREM (1. M. GELFAND, D. A. KAZDAN (3], F. RODIER (4]). A Whittaker model of a nondegenerate class 71:p is unique . Let 11: p be a nondegenerate class of isomorphisms of irreducible admissible re­ presentations of the group Gp. Its restriction to the center Cp of Gp is proportional to a character, denote it wp . In view of the canonical isomorphism of the center C of the group G Sp(4) and the multiplicative group, (1 .2)

C3

c J a

a

- a e k',

'That is, rp is not trivial on the ideles of any horocycle subgroup of Z.

AUTOMORPHIC £-FUNCTIONS FOR G SP (4)

89

OJp can be considered as a character of k0 . The contragradient class �; coincides with �P ® OJp(up) where u is the unique homomorphism of G Sp(4) into k* whose square is determinant (u is the factor of similitude). Therefore, 'if'"¢C�;) coincides with the set of functions W*(g) = W(g) · OJj;1 (up(g) ), (1 .3) We introduce the subgroups

and consider the integral ( 1 .4)

,fp( W, s) =

J (U·Hl p W(uh) ii a il •-1 1 2d(uh),

1. The integral (1 .4) is absolutely convergent in a vertical half-plane and defines ,Ip( W) as a rational function in qp•, qp being the number of ele­ ments in the residue field of the valuation p. All the functions ,/p( W), W e 'if'"¢(�p), admit a common denominator. There exists a rational in qp• function r(�p) such that ,fp( W, s) = r(�p. s),fp{(.8( W))*, 1 - s), THEOREM

Re s

( 1 . 5)

�

0

{3 =

1 0 0 1

1 0 0 1 0

We denote by Qp(� p) the polynomial in q"js with constant term 1 which is the com­ mon denominator of smallest degree for functions ,1p( W), W e 'if'"¢(�p), and define (1 .6)

L(�p. s) = [Q(�p. q-•)]- 1 , e(�p. s) = r (�p. s) · L(�p. s)/L(�;. 1 - s).

Let � = (?!)p = P �p be the class of isomorphisms of an automorphic generic cuspidal representation of the group G A- Then all its nonarchimedean components �p are nondegenerate and the Euler product (1 .7) L(�. s) = IT L(�p. s), pEP\S

S being the subset of all archimedean valuations from P, is absolutely convergent in a vertical half-plane Re s � ; it coincides with the Euler product of Langlands

90

MARK E. NOVODVORSKY

corresponding to the standard 4-dimensional representation of Langlands' dual group LG � G Sp(4, C) if k is a functional global field and differs from it by a meromorphic (in the whole complex plane in s) factor if k is a number field. The factors e(7rp) are of the form aqj/" and for almost all p E P\S are identically equal to I ; therefore, the Euler product (1 .8)

e(7r, s) = n e(7rp, s) p EP\S

is, in fact, finite and defines e(n') as an entire function without zeros. THEOREM 2. The function L(n) admits meromorphic continuation to the whole complex plane. If char k = p > 0, then this continuation is a polynomial in p• and

p-• satisfying the functional equation

(1.8)

L(n, s)

= e(7r, s)L(n *, I

-

s).

2. G Sp(4), hypercuspidal representations. This case has been investigated in co­ operation with I. I. Piatetski-Shapiro. For hypercuspidal representations, Whittaker models associated to nondegener­ ate characters of maximal unipotent subgroups do not work, and one has to con­ sider other subgroups ai:J.d generalized Whittaker models. It is convenient for our purposes to consider a different realization of the group G = G Sp(4). We choose a 2-dimensional semisimple k-algebra K and a 2-dimensional bilinear skew-sym­ metric form B in 2 variables over K; G can be realized now as the group of simili­ tudes of the form trK 1 kB. Now G contains the subgroup G1 � {g e GL(2, K) : det g e k } (2. I) of all K-linear transformations from G ; the subgroup of all upper triangular uni­ potent matrices from G1 is denoted by U1, the unipotent radical of the normalizer of U1 in G is denoted by U; U is a 3-dimensional commutative horocycle subgroup in G. The algebra K has a unique nontrivial k-automorphism

(2.2) it induces a k-automorphism of the form trKi k B which is denoted T' =

{(� �), 1C E K *}

c

-r, -r E

G. We put

G1 ;

obviously, T' is isomorphic to the multiplicative group K* of the algebra K. The groups G1, U1, and T1 are normalized by -r. Let p be a nonarchimedean valuation of the field k. We take a nontrivial charac­ ter X of the factor group up/ u; and a quasi-character �I of the group r; whose re­ striction on the subgroup

is unitary. If f is -.-invariant, we continue it to a character � of the group Tp , T = Tl (2.3) If �� is not -.-invariant, we put

U

-.T�

AUTOMORPHIC £-FUNCTIONS FOR G SP(4)

T = T', � = e ' (2. 4) In both cases we define Z = T· U, cj;(tu) = �(t) · x(u) (2.5)

V

91

t E Tp, u E Up ;

Z is an algebraic k-subgroup of G, cf; is a quasi-character of the group Zp. As in § 1 , we denote by "If"q,(Gp) the set of all complex locally constant functions W on the group Gp satisfying the equation ( 1 . 1), and an irreducible admissible subrepresent­ ation of Gp in "If" q, by right translations is called the Whittaker model of its class of isomorphisms 1rP and denoted if/'q,('Kp). THEOREM 3 . Every class 1rp of isomorphisms of irreducible admissible representa­ tions of the group Gp is either nondegenerate or has Whittaker model "If"q,(1rp)for some algebra K, subgroup T, and character cf; of the described type; for every fixed cf; this model is unique.

REMARK 1 . One class 1rp can have Whittaker models corresponding to several of the characters cf; described in §§ l and 2. REMARK 2. The group Tp = r; U -. r; is the stabilizer of X in the normalizer of the group Up in Gp; so, X can be lifted to either a character or a 2-dimensional ir­ reducible representation of the group fP Up · Theorem 3 can be reformulated as the theorem of the existence and uniqueness of Whittaker models associated to these 1 - and 2-dimensional representations ; such reformulation might be preferable logically but leads to more bulky constructions for Euler factors in case of 2-dimen­ sional representations. In the case when Z and cf; are defined by the formula (2.3), the uniqueness part of Theorem 3 was proven by I. I. Piatetski-Shapiro and the author [8] and by the au­ thor [9]. For supercuspidal representations 'Kp the uniqueness part of Theorem 3 follows from F. Rodier's Theorem 1 [5]. Now we put •

(2.6)

H=

{(" 1 ),

"E

k*

}

c

G'

c

GL(2, k)

and consider the integral (2.7)

fp( W, s)

=

s Hp W(h) jj h jj • dh,

THEOREM 4. The integral (2. 7) is absolutely convergent in a vert1cal half-plane Re s � and defines fp( W) as a rational function in q-p•. All the functions fp(W}, W E "If"q,('Kp) , admit common denominator.

As in § 1 , we denote by Qp('Kp) the normalized common denominator of the smallest degree and put

(2.8) THEOREM 5. Euler factor Lp('Kp) does not depend on the choice of the subgroup Z and the character cf; of Zp (particularly, if the class 1rp is nondegenerate, the factors L(1Cp) defined in §§1 and2 coincide) .

92

MARK E. NOVODVORSKY

Let 11: = @pEP 11:p be the class of isomorphisms of an automorphic cuspidal representation of the group GA. Choosing for every nonarchimedean p a Whittaker model "'f"q,(11:p). we obtain Euler factors L(11:p) and define L(11: , s)

(2.9)

pEP\S L(11:p, s),

= IT

S being the set of archimedean valuations of k ; this product is absolutely conver­ gent in a vertical half-plane Re s � , it coincides with Langlands' Euler product corresponding to the standard 4-dimensional representation of Langlands' dual group L G � G Sp(4, C) if k is a functional field, and differs from it by a meromor­ phic (in the whole complex plane in s) factor if k is a number field.

THEOREM 6. The function L(11:) admits meromorphic continuation to the whole complex plane . .lf char k = p > 0, then this continuation is a rational in p-• function satisfying the functional equation

L(11: , s) = .>(11: , s)L(11:* , 1

(2. 10)

where .>(11:) is an entire function in s without zeros.

-

s)

In fact, .>(11:) is an Euler product of some factors .>(11:p) appearing in local func­ tional equations ; these equations are rather bulky and will appear in a joint work with I. I. Piatetski-Shapiro. ·

3. G Sp(4) x GL(2), generic representations. In this section we use a modification of H. Jacquet's treatment of induced representations applied in [7] to automorphic £-functions on GL(2) x GL(2). The subgroups C and Z of G Sp(4), the character ¢ of ZA, the space "'f" q,( G Sp( 4, kp)) , p nonarchimedean, Whittaker model "'f" q,(11:p) for a nondegenerate class 11:p of irreducible admissible representations of G Sp(4, kp), the homomorphism a, and the character wp in this section are the same as in § 1 . We define the subgroup

(3. 1 )

H=

{(: g ;)

e G Sp(4) , a , b, c, d e k, g e GL(2)

}

and also its homomorphisms onto the group GL(2, k) : ¢1 :

(3 2) .

¢z :

(: g ;) - g, (: ;) - (: ;} g

We denote by Z the subgroup of all upper triangular unipotent matrices in GL(2,kp). It is the image of H n Z under the homomorphism ¢ �o and the kernel

AUTOMORPHIC £-FUNCTIONS FOR G SP(4)

93

of this homomorphism considered on the group (H n Z) A belongs to its com­ mutator subgroup ; therefore, every character of the group ZA defines under ¢ 1 a character of ZA which we denote with the same symbol. As in § 1 , we denote by "'f/¢(GL(2, kp)) the space of all locally constant complex functions W satisfying the equation W(zg)

(3 .3)

rp(z) W(g) ,

=

z E Zp, g E GL(2, kp)

and by "'f/ �(fc p), fcP a class of an infinite-dimensional irreducible admissible repre­ sentations of GL(2, kp), a subrepresentation of this group in "'f/ sb(GL(2, kp)) which belongs to fcp ; such a subrepresentation exists and is unique (cf. H. Jacquet, R. P. Langlands [6]). We denote by tiJp the restriction of fcp on the center of the group GL(2, kp) ; we consider tiJp as a character of the multiplicative group kt in view of the canonical isomorphism The space "ffl¢{fc;), 1c; � fcp ® w-p1 (det) being the contragradient representation to fcP • consists of the functions W*(g)

(3 .4)

W(g) · w-p1(det(g) ) ,

=

g E GL(2, kp).

For every locally constant complex function � on the plane kp E9 kp we define its Fourier transform iJ(x, y)

(3.5)

=

Jkpffikp �(u, v) a(yu - xv) d(u, v)

where a is a fixed nontrivial character of the group Afk of classes of adeles ; for every complex quasi-character fl. of the multiplicative group kt we put h E Hp.

(3 .6)

If the integral (3 .6) is absolutely convergent, then the function/satisfies, obviously, the equation (3.7)

1((: g ;) h, � ) .

fl.

=

p(d- 1 )/(h , � . r) ,

Therefore, we can consider the integral (3 . S)

.,! ( W, W, � . fi. l (s) , s) = fi.t (K, s)

=

J W(h) W(iflt(h)) f(h, �. fi. l) II ifJz(h) jj s dh , II " 11 25 w(K) tiJ(K),

w E "'f/q,(77:p), w E "'f/vifcp).

and a similar one for .,t ( W* , W *, iJ, p2(s) , s ), pz(K , s) = II " jj Zs w- l (K) w- l (!C). THEOREM 7.

The integrals (3.6) and (3 . 8) are absolutely convergent in a vertical

94

MARK E. NOVODVORSKY

half-plane Re s � and define .f( W, W, �) and .f( W *, W*, fi>) as rational/unctions in q-p•. There exists a rational in q;• function 7(11:P • itp) such that .f ( W, W, �. ,u 1 (s), s) = r('ll:p , itp, s) .f ( W, W*, fP, ,u 2(1 - s), 1 - s) (3 . 9) "f/ W E "'f/¢(11: p), JV E "'f/ ;p(it p), �. All the rational functions .f( W, W, �) (for different W e "'f/¢(11:p). W e "'f/;p(itp), �) admit a common denominator. As in § 1 , we denote by Q(11:P • itp) the normalized common denominator of the smallest degree in q-s and define L('ll:p ® itp, s) = [Q('ll:p , itp, q-•)]- 1 , ( 3.10) e('ll:p ® itp, s) = r('ll:p , itp, s) · L('ll:p ® itp. s)jL(11:; ® it;, 1 - s). If 11: ® it is the class of isomorphisms of a cuspidal generic irreducible representa­ tion of the group G Sp(4, kp) x GL(2, kp), 11: = @pEP 'll:p , it = @pEP itp, then we put (3. 1 1)

L(11: ® it, s)

=

e('ll: ® it, s)

=

IT L('ll:p ® itp, s),

p EP\S

IT e('ll:p ® itp, s),

p EP\S

S being the set of archimedean valuations of k. As before, the first product is ab­

solutely convergent in a vertical half-plane Re s � . For a functional field k it coincides with Langlands' £-function L(11:, p, s) if p is the tensor product of the standard 4- and 2-dimensional representations of the complex groups G Sp(4, C) and GL(2, C) ; for a number field k these £-functions differ by a factor merom or­ phic in the whole complex plane. The second product (3. 1 1) is, actually, finite and defines e('ll: ® it) as an entire function without zeros. THEOREM 8 . The function L(11: ® it) admits meromorphic continuation on the whole complex plane. If char k = p > 0, L(11: ® it) is a rational function in p-• satisfying the equation L(11: ® it, s) = e('ll: ® it, s) L(11:* ® it*, 1 - s). (3. 12) REFERENCES 1. R. P. Langlands, Problems in the theory of automorphic forms, Lecture Notes in Math., vol. 1 70, Springer-Verlag, Berlin and New York, 1 970. 2. I. I. Piatetski-Shapiro, Euler subgroups, Proc. Summer School Bolya-Janos Math. Soc. (Buda­ pest, 1970), Halsted, New York, 1975. 3. I. M. Gelfand and D. A. Kazdan, Representations of the group GL(n, K) where K is a local field, Inst. Appl. Math., no. 242, Moscow, 1 971 . 4. F. Rodier, Modele de Whittaker des representations admissibles des groupes reductifs p-adiques deployes, C. R. Acad. Sci. Paris Ser. A 275 (1 972), pp. 1045-1048. 5. -- , Le representations de G Sp(4, k) on k est un corps local, C. R. Acad. Sci. Paris Ser. A 283 (1 976), pp. 429-43 1 . 6. H . Jacquet and R . P . Langlands, Automorphic forms on GL(2) , Lecture Notes i n Math., vol . 1 1 4, Springer-Verlag, Berlin and New York, 1970.

AUTOMORPHIC £-FUNCTIONS FOR G SP(4)

95

7 . H. Jacquet, Automorphic forms on GL(2). Part 2, Lecture Notes in Math., vol. 278, Springer­ Verlag, Berlin and New York, 1 972. 8. M. E. Novodvorsky and I. I. Piatetskii-Shapiro, Generalized Bessel models for the symplectic group ofrank 2, Mat. Sb. 90 (2) (1 973), 246-256. 9. M. E. Novodvorsky, On the theorems of uniqueness of generalized Bessel models, Mat. Sb. 90 (2) (1973), 275-287. 10. , Theorie de Heeke pour les groupes orthogonaux, C. R. Acad. Sci. Paris Ser. A 285 (1977), 93-94. 11. , Fonctions / pour des groupes orthogonaux, C. R. Acad. Sci. Paris Ser. A 280 (1975), 1421-1422. 12. , Fonction / pour G Sp(4), C. R. Acad. Sci, Paris Ser. A 280 (1 975), 1 91-192. --

--

--

PURDUE

UNIVERSITY

Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 2, pp. 97-1 10

ON LIFTINGS OF HOLOMORPIDC CUSP FORMS

TAKURO SHINTANI Introduction. In [10], H. Saito calculated the trace of "twisted" Heeke operators acting on the space of holomorphic cusp forms with respect to the Hilbert modular group over a cyclic totally real abelian field of prime degree. He discovered a strik­ ing identity between his "twisted trace formula" and the ordinary trace formula for the Heeke operators acting on the spaces of elliptic modular forms. Applying his identity he showed that elliptic modular forms are "lifted" to Hilbert modular forms (for the origin of "lifting" type results, see [2]). No less significantly, he char­ acterized the space spanned by lifted forms. In the U.S.-Japan symposium "Ap­ plications of automorphic forms to number theory" which was held at Ann Arbor in June 1 975, the author reported a representation theoretic interpretation and generalization of Saito's work (see [13]). Results presented in the author's talk were immediately generalized by Langlands in [8]. Moreover, Langlands discovered an unexpected application of "lifting theory" to the theory of Artin L-functions. The present paper consists of two sections. In § I , we reproduce (with slight modifications) what the author presented at Ann Arbor. The second section is devoted to a few supplementary remarks. The author wishes to express his hearty thanks to Professor H. Saito, who gave him detailed expositions of [10] before its publication. Two of the author's previous papers [14] and [15] are also motivated by [10]. Notation. For a ring R, R x is the group of units of R. We write G(R) = GR = GL(2, R). For a p-adic field k, o(k) = ok is the ring of integers of k. We denote by qk the cardinality of the residue class field of k. For each t e k set d(tx) = ak(t) dx = i t i k dx, where dx is an invariant measure of the additive group of k. For each locally compact totally disconnected group G, C0(G) is the space of compactly supported, locally constant functions on G. Let k be a field and F be a field extension of k. We denote by N[ the norm mapping from F to k. 1.

1 . 1 . Let k be either a finite algebraic number field or a p-adic field. Let F be a commutative semisimple algebra over k of prime degree /. We assume that the group of automorphisms of F over k contains a cyclic subgroup g of order I gen­ erated by q . Then F is isomorphic either to a cyclic field extension of k or to the direct sum of /-copies of k. Set GF = GL(2, F) and Gk = GL{2, k). We may regard AMS (MOS) subject classifications (1 970). Primary lOOxx ; Secondary 100 1 5 , 10020. © 1979, American Mathematical Society

97

98

TAKURO SHINTANI

g as a group of automorphisms of GF (with the fixed point set Gk) in a natural man­ ner. Two elements x and y of GF are said to be a-twistedly conjugate in GF if there exists a g E GF such that y = g"xg- I . We denote by xGF· " the set consisting of all elements of GF which are a-twistedly conjugate to x in GF. The set of a-twisted conjugacy classes in GF is denoted by '&',(GF). The usual conjugate class in GF containing x is denoted by xG F . For each x E GF, set N(x) = x"1- 1 . x"1-z · · · x" · x.

It is easy to see that N(g" · xg- I) = gN(x)g- I . We denote by '&'(Gk) the set of conjugacy classes in Gk. For the proof of the next lemma see Lemma 3.4 and Lemma 3.6 of [10].

LEMMA 1 (SAITO) . The notation and assumptions being as above, the mapping xGF, " ...... N(x)GF n Gk is an injection from '&' ,( GF) into '&'( Gk). If F is not a field, the mapping is a bijection. In the following, for each c E '&',(GF), we denote by N(c) E '&'(Gk) the image of c under the mapping given in Lemma 1 . An x E Gk is said to be regular if the group of centralizers of x in Gk is a two-dimensional k-torus. We call a conjugate class in Gk regular if it consists of regular elements. A a-twisted conjugacy class c in GF is said to be regular if N(c) is regular. We denote by '&';(GF) (resp. '&''(Gk)) the set of regular a-twisted conjugate classes (resp. regular conjugate classes) of GF (resp. Gk).

1 .2. We keep the notation in 1 . 1 and assume that k is a p-adic field. Let G'F be the semidirect product of GF with g. More precisely, G'F is the group with the underly­ ing set g x GF with the composition rule given by (r, g)(z-', g') = (z-z-', g• 'g') (z-, z- ' E g, g, g' E GF). It is immediate to see (via Lemma I) that (a, g 1 ) and (a, g2) are conjugate in G'F if and only if g 1 and g2 are a-twistedly conjugate in GF· Denote by Gk (resp. GF) the set of equivalence classes of irreducible admissible unitariz­ able representations of Gk (resp. GF). For each R E GF and z- E g, let R• be the re­ presentation of GF given by R•(x) = R(x•). Then R< E GF. Thus, the group g op­ erates on GF. Denote by G} the subset of g-fixed elements of GF. A representation of Gp. is said to be admissible if its restriction to GF is admissible. Let R- be an irre­ ducible admissible unitarizable representation of the group Gp.. As is well known, the restriction R of R- to GF is either irreducible or a direct sum of /-mutually inequivalent irreducible representations of GF. In the former case R- is said to be of the first kind. Assume R- to be of the first kind and set J.,. = R-((a, I)) . Then (V g E GF).

(1 . 1)

Thus R E Gj!.. The relation (1. 1) characterizes the operator J, up to a multiplication by an lth root of unity. Conversely let R E GJ.. Then there exists a unitary operator of order I which satisfies ( 1 . 1). Hence, R is extended to an admissible irreducible unitarizable representation R- of Gp. of the first kind by setting R-((am, g)) = J�R(g). For rp E C0(GF), denote by R-(rp) the operator given by

-

R (rp)

=

LFR-((a, g)) rp(g dg, )

ON LIFTINGS OF HOLOMORPHIC CUSP FORMS

99

where dg is the invariant measure of GF normalized so that the total volume of GocF) is equal to l . It is shown that there exists a locally summable function x(R-) on GF such that trace R-(rp)

=

JGFrp(g)x(R-)(g) dg.

It is proved that x(R-)(g) depends only upon the a-twisted conjugate class of g in GF and that x(R-) is defined only on 'G'�(GF). For each r E Gk, denote by X r the char­ acter of r. Recall that x,(x) depends only upon the conjugate class of x and that Xr is defined only on 'G''(Gk), assume further that the characteristic of the residue class field of k is not equal to 2. THEOREM I . Let the notation and assumptions be as above. (I) For each R E GJ, there exist an extension R- of R to a representation of G"j and an r E Gk such that (1.2) where N is the norm mapping from 'G';(GF) to 'G''(Gk) given in Lemma I . (2) For each r E Gk , there uniquely exists R E GJ whose suitable extension R- to a representation of G"j satisfies ( 1 .2). For each r E Gk, we call R E G�, which is related to r by ( 1 .2), the lifting of r from Gk to G �- (Jacquet introduced in [4] the notion of lifting from a different viewpoint.) REMARK I . An analogue of Theorem 1 for finite general linear groups was given in [14]. Furthermore, an analogue of Theorem I for the case of (F, k) = (C, R) is given in [15] (resp. [8]) by a local (resp. global) method. Let us describe the lifting from Gk to GJ in a concrete manner. If F is isomorphic (as a k-algebra) to the direct sum of I copies of k, GF is isomorphic to the direct product of /-copies of Gk. It is immediate to see that the lifting of r E Gk is given by r@ · · · @ r (the same is true even when the characteristic of the residue class field of k is equal to 2). Next, we consider the case when F is a cyclic field extension of prime degree /. First, let us recall a description of Gk. For details, see §§3 and 4 of Chapter 1 of [5]. For quasi-characters p1 and p2 of kx such that p1p21 i= ai l , let p(f.J.I> p2) be the cor­ responding irreducible representation of Gk in the principal series. If p1p21 = ak , let a(pb p2) be the corresponding special representation of Gk. For a quadratic extension K of k and a quasi-character w of Kx such that w' i= w (' denotes the conjugation of K with respect to k), let n-(w, K) be the corresponding absolutely cuspidal representation of Gk. If the characteristic of the residue class field of k is not equal to 2 (as we are now assuming), it is known that each infinite dimensional irreducible admissible representation of Gk is equivalent to some of p(f.J.I> p2) , o-(f.J. I> p2) and n-(w, K). For each quasi-character f.J. of kx, we denote by p the quasi­ character of Fx given by p = f.J. o N�. For a quadratic extension K i= F of k and a quasi-character w of K x, denote by w the qu isi-character of L x (L = K · F) given by w =

w · Nf.

PROPOSITION I . The notation and assumptions being as above, the lifting R of r E Gk to G� is given as follows :

1 00

TAKURO SHINTANI

(1) Ifr = p(f.i.t> f.1.2) (resp . o{f.i.h f.1.2)) , R = p(!Jh !-'2) (resp. R = o{!Jh !-'2)) . (2) Ifr = n-(w, K) and K =1- F, R = n-(cu, L), where L = F K. (3) Ifr = n-(w, K) and K = F, R = p(w, w'). REMARK 2. The proof of the first part of Proposition 1 is straightforward (even when the characteristic of the residue class field of k is equal to 2). 1 .3. We keep the notation in 1 .2. In particular, k is a JJ-adic field (the residue class field of k may be of characteristic 2). For an x e GF, let Z11(x) be the subgroup of GF consisting of all elements of g of GF satisfying g "xg - 1 = x. Normalize invariant measures on GF and Z11(x) so that total volumes of their maximal compact open subgroups are all equal to 1 . Denote by dg the invariant measure on GF f Z11(x) given as the quotient of the normalized invariant measure of GF by that of Z11(x). Forf e C0(GF), set ·

( 1 . 3) It is shown that the integral is absolutely convergent. Moreover, for c e ((;f;(GF),

A11(f, x) = A11(f, y) for any x, y e c. For each c e ((;!�(G), we put

(1 .4) A11(/, c) = A11(f, x), where x is an arbitrary element of c. For each x e Gk, let Z(x) be the subgroup of centralizers of x in Gk. We normalize invariant measures on Gk and Z(x) so that the total volumes of their maximal compact open subgroups are all equal to 1 . Denote by dg the invariant measure on Gk/Z(x) given as the quotient of the normalized invariant measure of Gk by that of Z(x). For fe C0(Gk), set ( 1 . 5)

A(f, x) =

J Gk!Z (x) f(gxg-1) dg.

It is known that the integral is absolutely convergent. For each c e ((;f'(Gk), we put (1 .6)

A(f, c) = A(f, x),

where x is an arbitrary element of c. It is easy to see that the right side of (1.6) is independent of the choice of x e c. PROPOSITION 2. The notation being as above, for each le C0(GF), there exists an f E C0(Gk) which satisfies the following conditions (I) and (2) : (1) For each regular c E C11(GF), A11(f, c) = A(f, N(c)) . (2) For each c E ((;f ' (Gk) - N (((;f;(GF)), A(f, c) = 0. REMARK 3. In Proposition 2, assume I is the characteristic function of Go (Fl · Assume further that F is either the unramified cyclic extension of degree I of k or is isomorphic to the direct sum of I copies of F. Then one may putfto be the char­ acteristic function of Go (kl . REMARK 4. The correspondence I f--0 f in Proposition 2 is, in a sense, dual to the lifting of irreducible characters of Gk given in Theorem 1 . 1 .4. Let k be a totally real algebraic number field of degree n and let kA (resp. k;l)

ON LIFTINGS OF HOLOMORPHIC CUSP FORMS

101

be the ring of adeles (resp. the group of ideles) of k. Denote by koo (resp. kA. o) the infinite (resp. finite) component of kA- Then kA = koo EB kA, O and k;,j_ = k� x k;4, 0• Moreover, both groups Gkoo and GkAo O are embedded into GkA in a natural manner and GkA. = Gkoo X GkA, O · Let { oo h · · ·, oo n } be the set of infinite places of k. For each g E GkA' set g = googO (goo E Goo, go E GkAo o) and goo = (goo, h . . · , goo,n), where goo, ; E GR is the component of g"" corresponding to the infinite place oo ,. For goo ,; we introduce the following standard parametrization sin 0; goo, ,. = t; + t · 1 X1 ; VY; 1y-:-1 -sC?mS 0; 0'· cos 0t· - ' 'V "

( )( )(

(t;

·

t:

)(

)

> 0, X ; E R, Y ; > 0, 0; E R). Let 0; be the differential operator on GkA given by

For 0 = (O h · · ·, O n) E Rn , we denote by tc(O) the element of Gk"" whose ith component is 0; sin 0, ( -cos sin 0; cos 0;) ·

Let X be a unitary character of the group k;,j_fkX and let r be a function on the set of infinite places of k with values in the set of positive integers. Set r; = r( oo ,) . Denote by S(x, r, kA) the space of C-valued functions f on G kA which satisfy the following conditions ( 1 .7), ( 1 . 8) and ( 1 .9) . ( 1 . 1) f(g) is bounded, smooth with respect to goo and locally constant with respect to g0• Furthermore, OJ = fr,{r; 2)/4 ( 1 � i � n) . -

( 1 . 8) ( 1 .9)

(( ))

! r z z g = x (z)f(g) (V r E Gk,

(

f (gtc(O) ) = exp v' - 1

vz

jj r,o,) J(g)

E k�). (V 0 E Rn).

Denote by p(k) the set of finite primes of k. For each v E p(k), let k. be the comple­ tion of k with respect to v and let o. be the ring of integers of k •. Then the group GkA. o is isomorphic to the restricted direct product n .ak . Normalize the Haar measure of GkAo O so that the volume of IT .Go. is equal to 1 . For each q> E C0(Gk11, 0), we denote by T(q>) the linear operator on S(x, r, kA) given by ( T(q>)f) (g) = fGkA• 0 f(gx)q>(x) dx. It is shown that Trp is of finite rank. Let F be a totally real cyclic extension of k of prime degree /. Set l = x NfJ. Then x is acharacter of the group F},/Fx . Extend r to a function r on the set of infinite places of F by setting r(w) = r( w), where w is any infinite place of F and w is its restriction to k. For each -r E g = Gal{F/k) (the Galois group of F with respect to k) and any function f on GFA' denote by Jrf the function on GFA given by (Jrf) (g) = f(gr) (g E GF) · It is easy to see that Jr leaves the space S{x , r, FA) invariant. For each v E p(k), set F. = k. ®k F. Let o (F.) be the maximal compact subring of F•. The group g operates on F. in a natural manner. If v remains to be a prime in F, F. is a cyclic field extension of k• . If v splits in F, F. is isomorphic to the direct sum of .

a

1 02

TAKURO SHINTANI

/-copies of k •. In both cases, F. is embedded in FA, o in a natural manner. The group GFA, o is the restricted direct product n .aF.. For each g E GFA, O• we denote by Kv E GF. the v-component of g. In the following we choose and fix a generator o of g once and for all. Let fP be an element of Cif'(GFA, ol ) of the form]ffJ(g) = n .lp.(g.), where fPvE Cif'(GF) and fPv is the characteristic function of G•
The notation and assumptions being as above (in particular, fPv and · · · , rn > 2,

'Pv are related by ( 1 ) and (2) of Proposition 1 ) , if rh r2 , ( 1 . 1 0)

REMARK 5. In [10, Theorem 5.6], k is the rational number field and F is a tamely ramified totally real abelian field of degree /. Furthermore, fP is assumed to be unramified. However, the correspondence fPv -+ rp. is explicitly described for each v. In that point, Theorem 5.6 of [10] is more precise than Theorem 2. REMARK 6. If I # 2, S(XX•• r, kA) is isomorphic to S(x. r, kA) as GkA, 0-module. 1 . 5. Let us consider the representation theoretic meaning of both sides of the equality ( 1 . 1 0) of Theorem 2. Via the right regular representation : g �--+ Tg, the group GFA, o acts on S('l,. r, FA)· By a theorem of Jacquet-Langlands (see Proposi­ tion 1 0.9 and Proposition 1 1 . 1 . 1 of [5]), the space S(z, r, FA) decomposes into an algebraic direct sum of irreducible mutually inequivalent GFA, 0-submodules. Denote by GFA, 0(r, 'l,) the set of equivalence classes of irreducible representations of GFA, o realized on irreducible GF A, 0-submodules of S(l, r, FA). For each tr: e GFA, 0(r, 'l,). denote by V(tr:, r, 'l,) the irreducible GFA, 0-submodule of S(l, r, FA) on which tr: is realized. We have S(l, r, FA) = .E " V(tr:, r, 'l,) (an algebraic direct sum), where the summation with respect to tr: is over GFA, 0(r, 'l,). Denote by tr:" the re­ presentation of GFA, o given by tr:"(g) = tr:(g "). The obvious relation J" TK. T�" implies that tr:" e GFA, 0(r, l) for each tr: e GFA, o (r, 'l,) and that V(tr:", r, tr:) = J;;-1 V(tr:, r, 'l,). Take a tr: e GFA, 0(r, 'l,). It is known that, for each v e p(k), there exists tr:. e GF. such that tr: is equivalent to the restricted tensor product ®vEJl (kJ 'Irv (except for a finite number of v, tr:v is an unramified representation of GF.). Denote by GFA, 0(r, l• g) (g = Gal(F/k)) the subset of GFA, 0(r, z) consisting of all tr: such that tr:" � tr:. If tr:" # tr:, J" induces a cyclic permutation among V(tr:", r, 'l,) (-r e g). Thus the trace of the restriction of JaT(fP) to the subspace of S(l, r, FA) spanned by { V(tr:, r, 'l,) ; tr: e GFA. 0(r, 'l,) - GFA, 0(r, t• g)} vanishes. If tr:" � tr:, tr:� � tr:v for each v e p(k). There exists a linear operator Ju{tr: .) of order I on the representation space of tr:v such that =

1rv(g") = Ju(tr: .}- 1tr:v (g)Ju (tr:v) {V g E GF.).

ON LIFfiNGS OF HOLOMORPHIC CUSP FORMS

103

Such an operator J"('rr0) is unique up to a multiplication by an /th root of unity. For unramified 1r0, normalize J" (7r0) so that it fixes 1rv{G.) -invariant vectors in the representation space of 1r0• We may assume that the system of linear operators {Jq(1rv) ; v E p(k)} is normalized so that J(f j v(1r, r, x) � ®vE p (k) Jq(1rv). For q> = Ti v'Pv E C()(GFA, o), set 1rv(f/>v) = fGF. rpv{x)1rv(x) dx. Then 1rv(f/>v) is a linear oper­ ator of finite rank acting on the representation space of 'lrv · Moreover Trace J" T(rp) j v( 1r, r, x) = IJ v trace J" (1rv) 7r:v(f/>v). Hence, (1 . 1 1)

where the summation with respect to 1r is over all GFA. 0(r, l• g). In a similar man­ ner, we have (1 . 1 2) for every (/) = n v(/)v E C()(GkA' o). where the summation with respect to 'If: = ®v'lf:v is over all GkA. 0(r, x) and

J

'lf:v((/)v) = Gk• q; v{X) 7r: 0(X) dx. Recall that in equalities ( 1 . 10), ( 1 . 1 1) and (1 . 12), 'Pv and (/)v are related by the equalities (1) and (2) of Proposition 2 (V v e p(k)) . Thus, it is now natural to infer that a local implication of these equalities is Theorem 1 . Furthermore, a global consequence is the following representation theoretic version of Theorem 3 of [10]. THEOREM 3 . Assume rh . . . , rn > 2. (1) For each 'If: = ®v'lf:v E G�A. 0(r, x) (7r:v E Gk.), there uniquely exists 1r = ®1rv e Gt;A, 0(r, l• g) (1rv e GF) such that for each odd place v of k, 'lrv is the lifting of 'lf:v from Gk. to GJ We call 1r the lifting of 'If: from G�A. o(r, X) to Gt;A, o(r, l• g). (2) If I i= 2, for each 1r e Gt;A, 0(r, l• g), there uniquely exists 'If: e G�A. 0(r, x) such that 1r is the lifting of 'If: . (3) If I= 2, for each 1r e Gt;A· 0(r, l• g), there exists a 'If: e Gt. 0(r, x) or GkA. 0(r, XX I ) ( X I is the character of order 2 of kJ./P which corresponds to F in class field theory) such that 1r is the lifting of 'If: . Moreover, 'lf: I and 'lf:2 E GkA, o(r, x) have the same lifting to GFA. o(r, l· g) if and only if 'lf: I = 1C2 or 1C 1 = 1C2 ® X I (det). 2. In this section we expose the proof of Proposition 2. Then we indicate how Theorem 1 and Theorem 3 are made plausible by Theorem 2 (the proof of Theorem 2 is a (more or less obvious) modification of proofs of Theorem 1 and Theorem 5.6 of [10]). 2. 1 . In the following three subsections we use the notation in 1 . 1 and 1 .3 without further comment. In particular, k is a p-adic field. Assume that F is isomorphic to the direct sum of /-copies of k. Then we may assume that GF is the direct product •.

104

TAKURO SHINTANI

of l copies of Gk and that, for each x = (xh · · · , x1) E GF, :xa is given by xa (x1 , xh · · · , x1_ 1 ). For any f E GF, set, for any x E Gk, (2. 1) f(x) = f((xz , . . . , x1)- 1 x, Xz, . . . , x1 ) dxz . . . dx1 •

=

J (Gk) l - I

I t i s easy to see that jE C0(Gk) and that A a(f, c) = A (J, N(c)) for arbitrary c E c;(GF) · Thus the proof of Proposition 2 is quite straightforward for this case. 2.2. We summarize known results on orbital integrals on Gk. We denote by qk the cardinality of the residue class field of k and by 7T: a generator of the maximal ideal of o(k). Denote by Qk the set of isomorphism classes of two dimensional semisimple algebras over k. For each K E Qk, choose an embedding of K into M(2, k) as a k-algebra. Via the embedding, we identify K with a subalgebra of M(2, k). For each x E K, denote by x' the image of x under the unique nontrivial k-algebra automorphism of K with respect to k. It is well known that � ' (Gk) = U U (t) Gk (disjoint union), K eQk t

where the union with respect to t is over a complete set of representatives of (Kx - kx) with respect to the action of automorphism groups of K with respect to k. Furthermore,

V

�
{ e z ) zC � Yk} u

(disjoint union).

Let o(K) be the maximal o(k)-order of K. For each nonnegative integer m, we denote by o(K) m the unique o(k)-order of K such that [o(K), o(K) m1 = qf. Let o(K); be the group of units of o(K) m . For each t E o(KY - o(kY, there uniquely exists a nonnegative integer i such that t E o(K)f - o(K)ft-1 • Set i = i(t) for t E o(KY - o(kY . For each/ E C 'Q(Gk), set a(z) = f(z) and {J(z) = A (J, z(1 D) (z E kx) (cf. (1. 5)). Then both a and {3 are locally constant, compactly supported functions on kx . Furthermore, for each K E Qk, we denote by IPK a function on (Kx - kx) given by IPK(t) = A(J, t) (note that IP K does not depend upon a choice of an embed­ ding of K into M(2, k)). Then IPK is a locally constant function on Kx - kx . The next lemma is a version of Lemma 6.2 of [7] (see also Theorem 2. 1 . 1 and Theorem 2.2.2 of [12]). We include a proof. LEMMA 2. Let the notation and assumptions be as above. Then a triple { rp K ; (K E Qk), a, {3} satisfies the following conditions : (I) The support of IP K is relatively compact in Kx . (2) (/)K(t) = (/)K(t') ("f/ t E Kx - kx). (3) For each z E kx there exists a neighborhood U(z) of z (s.t. z-1 U(z) c o(KY ) such that (2.2)

(/)K(t)

=

{1 -

}

C(K) a(z) + C(K)q�
for any t E U(z) - U(z) n kx, where we put C(K) = [o(K)X, o(K)f']. PROOF. Choose an w E o(K) so that { 1 , w } is an o(k)-basis of o(K). An o(k)­ basis for o(K)m is given by { 1 , '!T:mw } (m = 0, 1 , 2, . . . ). For each t E K, there uni­ quely exists

105

ON LIFTINGS OF HOLOMORPHIC CUSP FORMS

c (t) =

(��� :��)

such that

Gw) = t(t) (!}

Then the mapping t ,_. c (t) is an embedding of K into M(2, k) as a k-algebra. It is known that every proper o(K)m-ideal in K is principal (see Proposition 1 of [3]). Thus, Gk =

U Go (k) (1

m=O

71:

)

m t(Kx) (disjoint union).

Hence, for anyfE C0(Gk) and any t E Kx - kx , (2. 3) where we set f-(x) = fc co.> f(uxu- 1) du, cm = [o(K) X , o(K),';;] ,

(cf. the proof of Lemma 7.3.2 of [5]). It is sufficient to prove the lemma for z = I . Since f is locally constant and compactly supported, there exists a neigh­ borhood U of I in o(KY such that (m = 0, 1, 2, · · · )

and that (n = 0, 1 ,

for all t E

u - u

· · ·

)

n k x . We note that a{1) = f(lz),

� {1)

and

ncoZ

= I; q-,;n f�

(0I 71:1n) (m = 1 , 2,

Hence we have

{

· · ·

)

.

}

rpK (t ) = I - C(K) a(l) + C(K)q �Ct) -1 f3(l ) qk - 1

for any t E U - U n kx . A triple {a, f3, rpK ; K E Qk} of locally constant compactly supported functions a, f3 on kx and a system {rpK ; K E Qk} of locally constant functions (/JK on Kx - kx is said to be an admissible triple if it satisfies the conditions (1), (2) and (3) of Lem­ ma 2. The next lemma, which is also a version of Lemma 6.2 of [7], follows from Corol­ lary 1 . 1 .4 of [12] and the previous lemma. LEMMA 3. Let { a, f3, rpK ; K E Qk } be an admissible triple. Then there exists an fE C0(Gk) such that rpK(t) = A(f, t) for any t E Kx - kx. Moreover, for such an f, a(z) = f(z 1 2) and �(z) = A (f, z(1 D). ·

106

TAKURO SHINTANI

2.3. Let F be a cyclic field extension of prime degree I of k. For each K E Qk, set L = K ®k F. Then L is a two-dimensional semisimple algebra over F. A pre­ scribed embedding of K into M(2, k) is naturally extended to an embedding of L into M(2, F). Furthermore, the norm mapping N f from F to k is naturally ex­ tended to the norm mapping Nk from L to K. Set L 1 = { t E LX, Nkt = 1 } and F1 = L 1 n F and L' = {t E U ; Nkt E Kx - kx } . The following description of � (GF) is due to H. Saito (see Lemma 3.5 and Remark 3.8 of [10]). LEMMA 4.

..

(1) where the union with respect to t is taken over a complete set of representatives for

L' I L1 with respect to the action of the automorphism groups of L with respect to F. (2) If I "# 2,

��( GF) -

�

..(GF) = zEFUXfF1 (z · 12)GF• " U zE FUXfF1 z ( l � )GF• 11 •

If I = 2,

For each / E C0(GF), we define functions a and (3 on kx as follows : For z E N[P , set z = N[z (z E P) and a (z) = A (f, z · 1 2) and (3(z) = A (f, z(I D) 1 / l k (cf. ( 1 . 3)). For z E P - Nf FX, set (3(z) = 0. If 1 "# 2, set a(z) = 0. If / = 2, set

..

..

It follows from Lemma 1 and Lemma 4 that a and (3 are well-defined functions on kx . Moreover, they are both locally constant and compactly supported. For each K E Qk, we define a function rpK on K x - kx as follows : For t E (Kx - k x ) n Nk L, set t = Nk t (t E U) and rpK(t) = A ..(f, t). For t ¢ (Kx - kx) n NkL x , set rp K(t) = 0. Lemma 1 and Lemma 4 again guarantee that rpK is a well­ defined function on Kx - kx . The proof of Proposition 2 is now reduced to the following Lemma 5. LEMMA 5. The notation being as above, {a, (3; rp K ; K E Qk} is an admissible triple. For a subgroup V of Kx (K E Qk), set (2.4)

The next lemma will be applied for the proof of Lemma 5. LEMMA 6. For a given K E Qk and a given compact subset C1 of GF, there exist an open subgroup V of Kx and a compact subset C2 ofGF such that g " wg- 1 E C 1 for some w E W( V) implies g"g- 1 E C2 • PROOF. There exists an open compact subgroup V1 of Kx which satisfies the fol­ lowing conditions :

I07

ON LIFfiNGS OF HOLOMORPHIC CUSP FORMS

The mapping : x ,__. x1 is an isomorphism from V1 to Vf whose inverse mapping .

���

(2. 5) ( V1 is so small that the series is absolutely and uniformly convergent on V1 ). Denote by c3 the image of cl under the norm mapping : X N(x) (cf. 1 . 1). Since N(g"wg- 1 ) = gN(w)g- 1 ; g "wg- 1 E ch for some w E W(VI) (cf. (2.4)) implies gw1g- 1 E c3. Put c4 = c3 n {gw1g-I ; w E W( VI), g E GF} · On C4 , the binomial series (2. 5) is absolutely and uniformly convergent. Hence the closure C5 of the

......

image of C4 under the mapping :

x ,__. x l i i

is compact. It is easy to see that

gwg- 1 = (gwlg- 1 ) 1 1 1 for w E W( Vl). Hence g"wg- 1 E cl for some w E W( VI) implies g"g- 1 E C1 C51. Thus, we may put C2 = C1 C51 and V = V1 • PROOF OF LEMMA 5. It is easy to see that f/JK satisfies the conditions (I ) and (2). For a t0 E Nf ·FX, take z E px such that t0 = Nfz. Let C0 be the support off and

choose an open compact subgroup V of Kx which has the property described in Lemma 6 for C1 = z- 1 C0• Then there exists a compact subset C2 of GF such that zg"wg- 1 E C0 for w E W(V) implies g"g-I E C2• ( W( V) is given by (2.4).) Let {y,- ; i E I} be a complete set of representatives for the double coset Go C Fl\GFfGk. It follows from Lemma I that the mapping : g ,__. g "g- 1 is a homeomor­ phism from GFfGk onto the closed subset {g E GF, N(g) = I } of GF. Hence, there exists a finite subset /0 of I such that (2.6) zg" wg- 1 E C0 for a w E W( V) implies g E U G(oF)Y;Gk. iEJo

Then, for t E v - v n kx' Zu(t) = K X if v is small enough and I: {-t ;A (f.· , t), f{J K(tot 1) = A 11(/, zt) = iEJ o where tJ.i1 is the invariant volume of Gk n y;:-1 G(oF)Y; in Gk, and f.· E C0(Gk) is

given by

Since /0 is finite, it follows from Lemma 2 that there exists a neighborhood V1 of I in Kx such that, for any t E VI - VI n k and any i E Io,

c

V

{ q��)Jt :(I ) + C(K)qi(t)-l A(.t:. ( I D).

A(.{;, t) = I -

However, (2.6) implies that I: /;( I ) ,u; = Au(!. z) = a(to), iEJo

�o,u.-A( f.· , e D) = Au( !. ze D) = l l l ;l �(to).

There exists a smaller neighborhood Vz c VI such that, for t E Vz - Vz n p ' i(ti) = ord l + i(t). Set U = {t0t 1 ; t E V2}. Then U is a neighborhood of t0 in Kx . We have proved that, for any t E U - U n kx,

108 (2.7)

TAKURO SHINTANI

{ q��\ } a(to) + C(K)q iCtt( / H �(to).

rpx(t) = 1 -

Now take a to E kx - NfP . Set L = F ®k K. If ! -:f. 2, Nf,_P n kx = NfP . Hence there exists a neighborhood U of t0 in K such that U n Nt,_Lx = 0 . Hence, rpx(t) = 0 for any t E u - u n k x . Since a( to) = �(to) = 0, the condition (3) is satisfied. Next assume l = 2. If K is not a field, there exists a neighborhood U of to in KX such that u n NfP = 0. Hence rpx(t) = 0 for any t E u - u n k x . Since � (to) = 0 and C(K) = qk - 1 ' the equality (2. 7) is valid for any t E u - u n

kx.

Now assume that K is a quadratic extension of k . Then there exists s0 E Lx such that t0 = Nf,_s0• Furthermore, Z..(s0) is isomorphic to the multiplicative group of the division quaternion algebra over k. The proof of the following Sublemma is quite similar to that of Lemma 6. SUBLEMMA. The notation and assumptions being as above, for a given compact subset C1 of GF, there exist an open subgroup V of Kx and a compact subset C2 of GF such that g 11s0vg- 1 E C1 for some v E V implies g11s0g- 1 E C2. Normalize invariant measures of Z.,.(s0) and Kx so that volumes of their maximal compact subgroups are all equal to 1 . It follows easily from the Sublemma that there exists a neighborhood U1 of s0 in Lx such that the following integral is absolutely convergent for s E U1 and give� rise to a locally constant function on U 1 : f cF1xx f(g11sg- 1 ) dg. If t = Nf,_s if= k x , the above integral is equal to rpx(t) = A.,.(f, s). If s = s0, the above integral is equal to A.,.(f, s0) x f z.csol !K x die, where die is the quotient measure of the invariant measure of Z.,.(s0) by that of Kx . The volume of Z.,.(s0)/Kx is equal to 1 or 2 according as K is ramified or not. On the other hand, C(K) is equal to qk or qk + I according as K is ramified or not. Since a(t0) = - (qk - l)A.,.(f, s0) and �(t0) = 0, we have shown that there exists a neighborhood U of t0 such that (2. 7) is valid for any t E u - u n k X . The proof of Lemma 5 is now complete. REMARK. The proof of Lemma 5 shows the following further relations between f and f of Proposition 2 when F is a field extension of k. If z = Nfz (z E P), f(z · l z) = A.,.(f, z · 12) and If kx 3 z if= NfP , A(f, z( 1 D) = 0 and f(z · l z) = 0 = - (qk -

( (z 1))

l) A.,. !,

if I

-:f.

2,

if I = 2.

2.4. When F is isomorphic to the direct sum of I copies of k, the correspondence f -+ f of Proposition 2 is made explicit by (2. 1). When F is the unramified cyclic extension of k of degree I, one can make the correspondence explicit iff is bi-G. (F) ­

invariant. In more detail, denote by L0(Gk, G. (k) ) the set of all functions f E C0(Gk) which are right and left G. (k) -invariant. Then L0(Gk, G. (k) ) becomes a commutative

ON LIFTINGS OF HOLOMORPHIC CUSP FORMS

109

algebra with respect to the convolution product. For indeterminates X and Y, let C[X, Y, x-r, y- I ]0 be the subalgebra consisting of all elements of C[X, Y, x-r , y-1 ] which are symmetric with respect to X and Y. For each /E L0(Gk, Go Ckl), put F(f, k)[X, Y] = 1: m, nE Z Cmnxm yn, where

Then it is known that the mapping : f ,.....

F(J, k)

is an algebra isomorphism from

L0(Gk, Go Ckl ) onto C[X, Y, x- I , Y- 1 ]0 (cf. Theorem 3 of [11]).

Let F be the unramified cyclic extension of

L0(GF, Go CFl ), there uniquely exists a 'A(/)

=

F('A(f),k)(X,

Y). The mapping : f ,.....

Lo(GF, GoCFl ) into Lo(Gk, Go Ckl ).

E

k

of degree l. For each f E

L0(Gk, GoCkl ) such that F(f, F) (X1, Yl)

'A(f)

is a C-algebra homomorphism from

The homomorphism 'A is discussed in [6] in a more general context. The following proposition is implicit in Saito [10]. In fact, considerable parts of 3.4-3. 1 3 and 5. 1-5.4 of [10] are devoted to the proof of Proposition 3.

PROPOSITION 3. The notation and assumptions being as above, f E L0(GF, Go CFl) and 'A(f) E L0(Gk, G0) are related by (1) and (2) of Proposition 2.

2.5. Let M be a finite subset of �(k) containing all primes of k which ramify in F. Denote by C0(GFA. 0,M) the subspace of C0(GFA, 0) spanned by q> fl vv (q;v E C0(GF.)) such that, for each v � M, v E L0(GF.• Go (F,l ). For q> = rr vv E CQ'( GFA. 0• M), we choose 1/) = rr viPv E CQ'( GkA, o) as follows : For v E M, we choose IPv E C0(Gk) so that v and IPv are related by (l) and (2) of Proposition 2. For v r/= M, if v remains to be prime in F, set IPv = 'A(q>v), where 'A is the homomor­ phism from L0(GF,• Go cF.l ) to L0(Gk,• Go (k,l ) introduced in 2.4. If v splits in F, denote by IPv the function on Gk, given by the right side of (2. 1) for !__ = v · In both cases, IPv E L0(Gk,• Go Ckl ) (v r/= M). For any unramified nv E Gk,• denote by n;; the lifting of nv from Gk, to GF, (see Proposition 1). It follows from the first part of Proposition 1 that ('<:/ v r/= M).

(2. 8)

Denote by GFA. 0(r, X· M, g) (resp. GkA. 0(r, X· M)) the subset of GFA. 0(r, l• g) (resp. GkA. 0(r, x)) consisting of all n = Q9 nv (resp. n = Q9 nv) such that nv (resp. nv) is an unramified irreducible representation of GF, (resp. Gk,) for any v r/= M. For q> = fl q;v E C0(GFA. o' M), we have, by ( 1 . 10), (1 . 1 1) and (1 . 1 2) that (2.9)

/-1

ll: TIv trace J,lnv) 1rv((/> v) = l:0 l: TI trace nv( l/)v), n

j= 1t V

where the summation with respect to n is over all GFA. 0(r, l• M, g) and the sum­ mation with respect to n is over all GkA, 0(r, XXi • M). For each n = ®nv e GFA. 0 (r, l• M, g) denote by X(n) the subset of U}-;;;1 GkA. 0(r, XXi • M) consisting of all n = ®nv which satisfy the following condition : For each v ¢ M, the lifting of n. from Gk, to G!J.., is n • .

1 10

TAKURO SHINTANI

Then equalities (2.8) and (2.9) together with "the strong multiplicity one theo­ rem" (see Theorem B of [9] and Theorem 2 of [1]) show the following : (2. 10) Here I must confess that I was too optimistic at Ann Arbor. I was erroneously con­ vinced that both Theorem 1 and Theorem 3 are immediate consequences of the equality (2. 10). Actually, highly nontrivial considerations are necessary to derive these theorems from (2. 10). Anyway, far-reaching generalizations of Theorem 1 and Theorem 3 are established in [8] . REFERENCES 1. W. Casselmann, On some results of Atkin and Lehner, Math. Ann. 201 (1 973), 301-3 14. 2. K. Doi and H. Naganuma, On the functional equation of certain Dirichlet series, Invent. Math. 9 (1 969), 1-14. 3. Y. Ihara, Heeke polynomial as congruence �-functions in elliptic modular case, Ann. of Math. (2) 85 (1 967), 267-295. 4. H. Jacquet, Automorphic forms on GL(2). II, Lecture Notes in Math., vol. 278, Springer­

Verlag, 1 972. 5. H. Jacquet and R. Langlands, Automorphic forms on GL(2), Lecture Notes in Math., vol. 1 14, Springer-Verlag, 1 970. 6. R. Langlands, Problems in the theory of automorphic forms, Lecture Notes in Math., vol . 1 70, Springer-Verlag, 1 970, pp. 1 8-61 . 7. , Modular forms and 1-adic representations, Lecture Notes in Math., vol. 349, Springer­ Verlag, 1 973, pp. 361 -500. 8. , Base change for GL(2), the theory of Saito-Shintani with applications, Institute for Advanced Study, Princeton, N.J., 1 975. 9. T. Miyake, On automorphic forms on GL, and Heeke operators, Ann. of Math. (2) 94 (1971), 1 74-1 89. 10. H. Saito, Automorphic forms and algebraic extensions of number fields, Lectures in Mathema­ tics, vol. 8, Kyoto Univ., Kyoto, Japan, 1 975. 11. I. Satake, Theory of spherical functions on reductive algebraic groups over IJ-adic fields, Pub!. Math. No. 18, Inst. Hautes Etudes Sci . , 1 963. 12. J. Shalika, A theorem on semi-simple IJ-adic groups, Ann. of Math. (2) 95 (1 972), 226-242. 13. T. Shintani, On liftings of holomorphic automorphic forms (a representation theoretic in­ terpretation of the recent work of H. Saito), Informal proceedings (available at Univ. Tokyo) for U.S.-Japan Seminar on Number Theory (Ann Arbor, 1 975). 14. , Two remarks on irreducible characters offinite genera/ linear groups, J. Math. Soc. Japan 28 (1 976), 396-414. 15. , On irreducible unitary characters of a certain group extension of GL(2, C), J. Math . Soc. Japan 29 (1 977), 1 65-1 88. --

--

--

--

UNIVERSITY O F

TOKYO

Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 2, pp. 1 1 1-1 1 3

ORBITAL INTEGRALS AND BASE CHANGE

R. KOTTWITZ Let F be a local field of characteristic 0 and let E be either F x x F (I times) or a cyclic extension of F of degree l, where l is a prime. Embed F in F x x F diagonally. Let (r}F be the valuation ring of F, and let (r}E be (r}F x x (r}F if E = Fx x F, and let it be the valuation ring of E if E is a field. If E is a field, let r = Gal(E/F), and let (J be a generator of r. If E = F X X F, let (J be the auto­ morphism (xi> · , Xr) 1-+ (Xz , Xt , xl ) of E, and let r be the group of automor­ phisms of E generated by O" ; again F is cyclic of order l. Let G = GL2 • The action of F on E induces an action of F on G(E). We use the embedding of F in E to identify G(F) with G(E)F. Define a norm map N: G(E) -> G(E) by putting Ng = ga r - ! · · · gag. This map was introduced by Saito [2]. It de­ pends on the choice of O". · · ·

· · ·

· · ·

· · ·

• •

0 0

0 0

·

0

,

LEMMA I. Let g, x E G(E). Then

(i) (ii) (iii) (iv)

N(g-axg) = g-l(Nx)g ; (Nx) " = x(Nx)x- 1 ; det(Nx) = NE;F(det x) ; Nx is conjugate in G(E) to an element of G(F).

The first three statements are easy calculations, and it is not hard to get (iv) from (ii). The equality (i) suggests the following definition : x, y in G(E) are (}"-conjugate if there exists g E G(E) such that y = g-"xg. Statements (i) and (iv) together say that N induces a map from ()"-conjugacy classes in G(E) to conjugacy classes in G(F). This map is always injective, and it is surjective if E = F x · · · x F. It should also be noted that x is ()"-conjugate to y if and only if (O" , x) is conjugate to (0", y) in the semidirect product r IX G(E). Choose Haar measures dg and dgE on G(F) and G(E) respectively. If F is non­ archimedean, normalize dg, dgE so that meas(KF) = meas(KE) = I , where KF = G((r}F), KE = G((r}E). For any element r of G(F), let Gr denote the centralizer of r in G. We now give a definition which is due to Shintani [4]. if

DEFINITION. Let qJ E C;;' (G(E)) andjE C:;'(G(F)). We say thatf is associated to qJ

(A) AMS (MOS) subject classifications (1 970). Primary 22E50. © 19 79, American Mathematical Society

Ill

1 12

R. KOTTWITZ

whenever No

=

T and T is a regular element of G(F) ; f(g-l rg) dg J dt G7(F)\G (F)

(B)

=

o

for every regular element T of G(F) which is not a norm from G(E). In (A) and (B) dt is a Haar measure on GrCF). REMARKS. ( I ) For each regular T e G(F) which is a norm, it is enough to check condition (A) of the definition for only one o. (2) For any element 0 E G(E), let GnE) = {g E G(E) : g-17og = o}. It is not hard to see that if T = No is a regular element of G(F), then Ci'a(E) = G7(F). For T e G(F), define e(r) to be - 1 if T is central and is the norm of an element of G(E) which is not a-conjugate to a central element of G(E), and define e(r) to be 1 otherwise. It can be shown that iff is associated to rp, then

J

(A')

�m�m

f(g- 1 Tg) dg dt

= e(r)

f

�� ��

rp(g-u og) dgE dt'

whenever No = T belongs to G(F), where dt' is a Haar measure on G�(E) which depends only on the Haar measure dt on G7(F) ;

J

(B')

G7(F)\G (F)

f(r1 rg)

'1t

=

o

for every element T of G(F) which is not the norm of some element of G(E). LEMMA 2. (i) Let rp E C�(G(E)). There is at least one fe C�(G(F)) which is

associated to rp. (ii) Let f e C�(G(F)). Then there exists some rp E C�(G(E)) to which f is asso­ ciated (f and only if fc7(F)\G CF) f(g-I rg) dgfdt = 0 for every element T of G(F) which is not a norm from G(E) (or equivalently, for every regular element T of G(F) which is not a norm from G(E)).

Assume that F is nonarchimedean, and let YfF = Yf( G(F), Kp) be the Heeke algebra of complex-valued compactly supported functions on G(F) which are hi­ invariant under Kp. Let YfE = Yf(G(E), KE)· Saito [2] introduced a C-algebra homomorphism b : YfE -+ YfF which we will now describe. A functionfin .Yt'p gives rise to a functionfv on the set Dp of isomorphism classes of unramified irreducible admissible representations of G(F) by putting fV(11:) = Tr 7e(f) for 1C E Dp. The set Dp is an algebraic variety over C (it is isomorphic to C * x C* divided by the action of the symmetric group S2, which acts on the product by permuting the two factors). The map f ...... f V is a C-algebra isomorphism, called the Satake isomorphism, of YfF with the algebra of regular functions on the variety Dp. This discussion applies to E as well ; if E is a field, then DE is again isomorphic to C* X C* divided by s2, and if E is F X X F, then DE is Dp X . • •

X

• . .

Dp.

There is a map of algebraic varieties from DF to DE ; if E = F x · · · x F the map is 1C ...... 1C ® • · • ® 1C, and if E is a field it is 7C(-zo) ...... 7C(Res ��-.) where 7C('Z") is the unramified representation of G(F) corresponding to the unramified representation -. of the Weil group Wp of F, and 7C(Res ��-.) is the unramified representation of

113

ORBITAL INTEGRALS AND BASE CHANGE

G(E) corresponding to the restriction of 1: to the Weil group WE of E. So we get a C-algebra homomorphism from the algebra of regular functions on DE to the algebra of regular functions on DF, and hence also a C-algebra homomorphism b : yt'E -+ yt'F • REMARK. If E = F x · · · x F, then .Yt'E � .Yt'F® · · · ®.Yt'F, and b(fi ® · · · ®fz) = II* . . . *h ·

LEMMA 3. IfF is nonarchimedean and E is unrami.fied over F (E = F x allowed), then b( rp) is associated to rp for all rp E .Yt'E ·

···

x F is

This lemma is easy to prove for E = F x · · · x F. It was first proved by Saito [2] when E is a field ; it was subsequently proved by Langlands [1] using the build­ ings of SL2(F) and SL2{E). Let A be the group of diagonal matrices contained in GL2• To regular elements of A there are associated weighted orbital integrals which appear in the trace formula for GL2 over a global field. We need to introduce a function ).F on G(F) whose logarithm is the weight factor in these integrals. Let g E G(F) and write g = a(� f)k with a E A(F) and k E KF. Then .A.F(g) = I if x E {f)F and .A.F(g) = l x l -2 otherwise. The function .A.E is defined in the same way on G(E) in case E is a field. LEMMA 4. Suppose that F is nonarchimedean and that E is an unrami.fied extension

field of F. Then for any o E A(E) such that No is regular, andfor any rp E :Yt'E, I

J

Amwm

b(rp) (g- 1 (No)g) Iog AF(g)

�!

=

J

'!

rp(g-" og)l og AE(g) E

Amw �

where da is any Haar measure on A(F).

This is proved in §3 of [1]. REFERENCES I. R. P. Langlands, Base changefor GL(2), Notes, Institute for Advanced Study, 1 975. 2. H. Saito, Automorphic forms and algebraic extensions of number fields, Lectures in Math­

--

ematics, Kyoto University, 1975.

3. , Automorphic forms and algebraic extensions of number fields, U.S.-Japan Seminar on Number Theory (Ann Arbor, 1 975). 4. T. Shintani, On /iftings of holomorphic automorphic forms, U.S.-Japan Seminar on Number Theory (Ann Arbor, 1 975) (see these PROCEEDINGS, part 2, pp. 97-1 10). RICHLAND, WASHINGTON

Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 2, pp. 1 1 5-133

THE SOLUTION OF A BASE CHANGE PROBLEM FOR

GL(2)

(FOLLOWING LANGLANDS, SAITO, SIDNTANI)

P. GERARDIN AND J. P. LABESSE These Notes present a survey of the results on the lifting of automorphic repre­ sentations of GL(2) with respect to a cyclic extension of prime degree ofthe ground­ field, and of some of its applications to the Artin conjecture, with some sketches of proofs. §§ 1-5 are devoted to the definitions and results on the lifting, §6 to the proof of the Artin conjecture in the tetrahedral case. The first part ends up with three appendices describing respectively two-dimensional representations of the Weil group (Appendix A), representations of GL(2) over a local field (Appendix B) and a global field (Appendix C). Part II gives some indications on the proof of the results on lifting. The main tools are the orbital and twisted orbital integrals, and a twisted trace formula [Sa]. The main references for the lifting are [Sa], [S-1], [S-2], [L]. As a side remark, we would like to point out that the study of the example of the general linear group over a finite field [S-2] is illuminating. I. DEFINITIONS, THEOREMS, APPLICATIONS 1. Notation. Let F be a local or global field. Then WF is the Weil group of F and, for F a p-field, W� is the Weil-Deligne group of F [T]. We recall that there exists a canonical surjective homomorphism WF -+ CF which identifies W�b with CF.

In all these notes, E is a Galois extension of F, cyclic of prime degree /, and F its Galois group (the "split case", where E = F X F X . . . X F /-times and r is generated by CT : (xh x2, . . . , x1) f-+ (x2, . . . , xr. x1 ) is handled easily and is left to the reader). For F local, choose a nontrivial character ¢ of the additive group of F, and de­ fine ¢E1 F = ¢ o TrE1 F · In the following we shall use systematically the notation given by Borel [B) about £-groups : r/J(G), . . . , and by Tate [T] for Weil groups, the L- and e-factors. 2. Base change for GL( l ) . From abelian class-field theory, there is a commuta­

tive diagram

AMS (MOS) subject classifications (1 970). Primary 100 1 5 , 1 2A70 ; Secondary 20E50, 20E55 . © 1 9 79, American Mathematical Society \

1 15

1 16

P.

GERARDIN AND J. P. LABESSE

where (}" is a generator of r. The one-dimensional representations of WF are given by the quasi-characters of Wfl-b = CF ; by composing a quasi-character of CF with the norm, a map is defined called the lifting (or more precisely, the base change lift) : x >--> XEI F = X o NEI F which sends the set d(F) of quasi-characters of CF in the set d(E) of quasi­ characters of CE · The group r acts on the set of quasi-characters of cE by : (rO)(z) = O(zr),

the group f of characters of r can be identified with the set of characters of CF which are trivial on NE 1 FCE ; this group f acts on the set of quasi-characters of CF by multiplication : X >--> x(, ( E f . Then the exactness of the second line of the above diagram implies the following result : PROPOSITION I . The lifting X >--> X EI F defines a bijection from the orbits of f in d(F) onto the invariant elements by F in d(E) ; moreover, the following relations hold: L(X EI F) = fl L(x (), E C F

e(X E 1 F) = IT. e(x () for F global, CE F

e(X EI F• ¢ El F) = J...E I F(¢) - 1 IT. e(x(, ¢) for F local. r;, E r

3. Base change for GL(2) on the L-groups. 3 . 1 . Let G be the group GL(2) over F, and GE 1 F be the group over F defined by restriction of scalars of the group GL(2) over E, so that GE 1 F(F) = GL(2, E) ; the group GE 1 F is quasi-split, and there is a natural map : LG = GL(2, C) X Fr----- LGE I F = GL(2, cy � FF ;

here GL(2, C)F is the set of applications from F in GL(2, C), and FF = Gal(F/F) acts by permutations of the coordinates [B, §5]. Given any p E W(G), its restriction to W£ defines an element PE l F E W(GE 1 F) ; the set W(GE 1 F) can be identified with the set W(G/E) [B], and the above map defines the application (/)(G) ---> (/)(GEl F) = (/)(GIE), p

f------->

PE l F

called the base change. For F local, let FrF be a Frobenius element in FF ; when E is unramified, then FrE = Fr� E rE is a Frobenius element for E. Moreover, if p is unramified and defined by FrF >--> s, where s is a semisimple element in GL(2, C), then PEl F is unramified and is given by FrE >--> s 1 • The properties of L- and e-factors with respect to induction [T] show that : L(pEI F) = fl. L(p ® (), C EF

e(PEI F) = IT e(p ® (), r.Er

for F global,

e(pEI F• ¢EI F) = J...E I F(¢) - 2 IT e(p ® ( , ¢ ), for F local. CEF

BASE CHANGE PROBLEM FOR

GL(2)

1 17

3.2. From the classification of the two-dimensional admissible representations of

w;.. (see Appendix A), one has the following result :

PROPOSITION 2. (a) The lifting p E l/J(G) t-> PE I F E (/J(GE I F) has for image the set of F-invariants in l/J(GE 1 F). (b) In the following cases, the lifting is given by

(f-l EB )))E l F = /-lEI F EB ))E l F• {lnd �k O)EI F = Ind ��E OKEI K for E =f. K, (Ind �k O)EI F = 0 EB (JO for E = K, 1 =1- a E r, (x ® sp(2))E I F = XEIF ® sp(2) (for F nonarchimedean). (c) Given a nondecomposable p E l/J(G), the representations which have the same lifting are the p ® ' for all ' E f ; for p = il EB f-l• the representations which have the same lifting are the il' EB 1-l'' for all ,, ,, E f.

4. Base change over a local field. In this section, we denote by a a generator of F = Gal(E/F). 4. 1 . Let ll(G) be the set of classes of admissible irreducible representations of G(F). There is a conjectural bijection l/J(G) � ll(G) [B]. The base change map l/J(G) -+ l/J(GE1F) must reftect a map ll(G) -+ ll(GE1F) · The definition of base change for representations of G(F) will be given in 4.3 ; since the image of l/J(G) is the set of F-invariant elements in l/J(GE1 F), one studies first the admissible irreducible repre­ sentations of G(E) equivalent to their conjugates by r. 4.2. Let 7t be an admissible irreducible representation of G(E) such that t1 7t � 7t ; then there exists an operator C on the space o f 7t such that c- 1 7t(z)C = 7t(zt�) , z E G(E), and CZ = Id. This operator is determined up to an lth root of unity. The mapping 7t' : (am , z) �--> Cm 7t(z) defines an extension 7t' of 7t to the semidirect pro­ duct F � G(E). PROPOSITION 3. This representation has a character given by a locally integrable function Tr 7t' on F � G(E).

On a x G(E), the character Tr 7t' defines a a-invariant distribution on G(E), i.e., invariant under a-conjugation : z �--> y-t�zy, y, z E G(E). Let us state some properties of the a-conjugation. For z E G(E), put or simply N(z) if no confusion can arise. PROPOSITION 4. (a) NE1 F . tJz is conjugate in G(E) to an element of G(F) ; (b) z �--> NE I F , t1z defines an injection of the set of a-conjugacy classes of G(E) into

the set of conjugacy classes of G(F) ; (c) the elliptic classes of G(F) obtained by NE / F , tJ are those with determinant in NE l FEx ; the hyperbolic classes of G(F) obtained are those whose eigenvalues are norms of Ex ; any unipotent class of G(F) is in the image of N. 4.3. Definition of the base change for GL(2) over a local field. Let 7r: E ll(G) and 7t

1 18

P. GERARDIN AND J. P. LABESSE

be an irreducible admissible representation of G(E) which is equivalent to its con­ jugate by a. Then it is called a base change lift of 11:, or a lifting of 11:, if either (a) 11: = 7r:(f.t , v) and it = 7r: (f.tEIF • vE l F), or (b) there exists an extension it' of it to F 1>< G(E) such that Tr it'(a x z) = Tr 11:(x) for z e G(E) whenever NE I F.a z is conjugate in G(E) to a regular semi­ simple element x e G(F). Some of the notation in the following theorem is explained in Appendix B. THEOREM 1 (BASE CHANGE FOR GL(2) OVER A LOCAL FIELD). (a) Any 1C E fl(G)

has a unique lifting 11:E I F e fl(GE1F), and any 11: e D(G)fixed by F is a lifting ; (b) the lifting is independent of the choice of the generator a of F; (c) 1CE I F = 1T:eJF � 11:' = 11: ® C for a C e f, or 11: = 7r:(f.t , v), 11:' = 7r:(f.t ', v') with f.l-lll' and v-I v' in F; (d) w"EIF = (w")E I F • (11: ® X)E I F = 11:E I F ® XEI F for any one-dimensional repre­ sentation X of FX , (11:E I F)v = (11:Y)E I F (contragredient representations) ; (e) for E => F => k with E and F Galois over k, 7(11:EIF) = (r11:)E I F for any r E Gal(E/k), with image r in Gal(F/k) ; (f) at least for p e C/>(G) not exceptional, 11:(p)E I F = 11:(pE I F) · S. Global base change. 5. 1 . Let D(G) be the set of classes of irreducible admissible automorphic repre­ sentations of G(AF) = GL(2, AF), where AF is the ring of adeles of the number field ·

F. From the principle of functoriality [B], the base change on £-groups should re­ flect a map from D(G) to D(GEl F), the set of irreducible admissible automorphic re­ presentations of G(AE) = GL(2, AE) = GE1 F(AF) ; such a map must be compatible with the local data. For any place v of F, put Ev = E ® F Fv ; it is a cyclic Galois extension of F or a product of 1 copies of F; in this latter case, define the lifting 7r:Er!F. of 11: e ll(G) by 1CEr!Fv = 1C ® · · · ® 11: (I times). 5.2. Definition of the global base change for G L(2) Let 11: e D(G), it e D(GE1F) ; then it is called a lifting of 11: (or more precisely a base change lift of 11:) if, for every place v of F, itv is the lifting of 11: 0 • 5.3. The notations used in the following theorem are those of Appendix C. THEOREM 2 {GLOBAL BASE CHANGE FOR GL(2)). (a) Every 1C E ll(G) has a unique lifting 1CE I F E ll( GEIF) ; (b) a cuspidal it e D(GE 1 F) is a lifting if and only if it is fixed by r, and then, it is a .

lifting of cuspida/ representations ; a cuspidal 11: e D(G) has a lifting which is cuspidal except for I = 2 and 11: = 11:(Ind �k 8) and then 11:E I F = 11:(8, 118) ; (c) for a cuspida/ 11: e ll(G), the representations 11:', which have 11:E1 F for lifting are the 11:' = 11: ® C with C e F; (d)

(J)"EIF = (w1r)E I F (11: ® X)EIF = 1CEJF ® X E!F (1CEIF) V = (1i")EI F

(central quasi-characters), (twisting by a quasi-character), (contragredient representations) ;

(e) for E => F => k with E and F Galois over k, 7 (11:E I F) Ga!(F/k) image of f e Gai(E/k) ;

= (711:)EI F for

any r e

BASE CHANGE PROBLEM FOR

GL(2)

1 19

(f) i/ 11: = 7r:(p) for some p e cp(G), then 71:E IF = 71:(PE I F). REMARK. There are examples of noncuspidal n e /l(GE1F), fixed by F which are not liftings (cf. [L, § 10]). 6. Artin conjecture for tetrahedral type. Let p be a two-dimensional admissible representation of the Weil group of the number field F; we assume that its image modulo the center : Wp ---+ GL(2, C) ---+ PGL(2, C) is the tetrahedral group �4. This group is solvable : 1 ---+ D4 ---+ 2f4 ---+ C3 ---+ I . The action of the cyclic group C3 on the dihedral group D4-the so-called mattress group-is given by the cyclic permutations of its nontrivial elements. The inverse image of D4 in WF is a normal subgroup of index 3, hence is the Weil group WE of a cubic Galois extension E of F:

1

--->

WE ---> WF --->

1

___,

n

14

___,

1

2f4

___,

Gal(E/F) ---> 1

11

c3 -----. 1

The restriction PE l F of p to WE has for image the dihedral group D4 ; it is induced from a one-dimensional representation of a subgroup of index 2 in WE• so that there is a corresponding cuspidal automorphic representation 11:(pE 1 F) of GL(2, AE). The inner automorphisms of 2f4 give an action of C3 on D4 , and the action of r = Gal(E/F) fixes the class of PE l F• hence also the class of 11:(pE 1 F). From Theorem 2 (5.3), this representation is the base change lift of exactly three classes of irreducible cuspidal automorphic representations of GL(2, AF), and their central character has for base change lift the central character of 11:(pE 1 F), which is equal to det PE l F = (det p)E1 F ; there is only one of them, say 11:, with, the central character det p. The Artin conjecture for the representation p is the holomorphy of the corres­ ponding L-function : s ..... L(s, p). According to Jacquet-Langlands [J-L, Theorem 1 1 . 1, p. 3 50], the L-function s ..... L(s, 11:) corresponding to cuspidal 11: is holo­ morphic ; hence, the Artin conjecture will be proved for p if we show the equality of these two L-functions. From [J-L, pp. 404--407], this will be done if the following assertion is shown to be true : (c0) 71:v = 11:(p.) for each archimedean place v of F, and for almost all v. As E is cubic over F, each infinite place v of F splits in E, so for wl v, we have the equations : which are more generally true for any place v of F which splits in E. Now if v does not split and is unramified in E, and if moreover Pv is unramified, then so is the restriction PEviFv of Pv to wE. • and the representation (71:E I F)v = 7r: (PE.rF) is unramified ; since Ev/F. is unramified this representation is the base change lift of unramified representations. This shows that 71:v is unramified ; call p"• a two-dimensional representation of WE. • such that 71:v = 11:(p".). We have shown that our assertion (c0) is equivalent to : (c 1 ) if v does not split and is unramified for p and E, Pv and Prr. are equivalent. The adjoint representation of PGL(2) defines an injection of PGL(2) in GL(3),

1 20

P.

GERARDIN AND J. P. LABESSE

hence a morphism A : GL(2) --+ GL(3). We observe now that the condition (c1 ) is equivalent to the apparently weaker condition : (cz) if v is unramified for p and E, the three-dimensional representations Ap. and Ap"• are equivalent. In fact, call a (resp. b) the image of a Frobenius in WF. through p. (resp. p") ; if (c2) is satisfied, a E Cxb ; but det a = det b, hence a = ± b. If a = - b then, since p. and p"• have the same restriction to WE.• as is conjugate to - as, that is Tr{a3) = 0. Hence A(a3) is of order two, and this means that A(a) is of order 6 ; but the image of Ap. is in the tetrahedral group which has no element of order 6 ; so we have a = b, hence p. = p"• ' The introduction of Ap is motivated by the crucial observation, due to Serre, that this three-dimensional representation is induced by a one-dimensional representation of WE ; in fact, the tetrahedral group leaves invariant the set of the three lines joining the middles of the opposite edges of the tetrahedron. This means that Ap is induced by the one-dimensional represen­ tation () of the stabilizer of one of these lines (obtained by restriction of Ap) ; but this stabilizer is the pull-back of the dihedral group D4 c �4 in WF, which is the subgroup WE : Ap = Ind �� (}, From [J-PS-S), to such a three-dimensional irreducible monomial representation

Ap of WF is associated an irreducible cuspidal automorphic representation 11:(Ap) of GL(3, AF). On the other hand, the morphism A reflects a lifting from irreducible

cuspidal automorphic representations of GL(2, AF) to automorphic representations of GL(3, AF) ; and, here, the representation A11: corresponding to 11: is cuspidal [G-J-2]. To prove (c2) it suffices to show the condition : (c3) the lifting A11: is equivalent to 11:(Ap). There is a practical criterion given by [J-S] to prove the equivalence of such repre­ sentations : 11: 1 and 11:2 are equivalent if and only if L(s, 11: 1 x 7r2) has a pole at s = 1 , where L is the L-function attached to the representation of GL3(C) x GL3(C) in GL9(C) given by the tensor product. We shall prove that almost all local factors of L(s, A11: x 11:(Ap) Y) and of L{s, 11:(Ap) x 11:(Ap)v) are equal ; by nonvanishing properties of local factors [J-S], this is enough to prove that L(s, A11: x 11:(Ap)V) has a pole at s = I, and hence (c3) will be proved. If v is split, then 11:. = 11:(p.) and, at least when 11: v and P v are unramified, (A11:). = 11:(Ap.) : the local L-factors are then equal. If v does not split in E and is unramified for E and p, the two local L-functions are those associated to the nine-dimensional representation of WF given by Ap"• ® ((Ap) ';') and Ap. ® ((Ap) ';') ; but we know that Ap. = Ind ��= () . . Now if U (resp. V) are representations of a group G (resp. a subgroup H) one has U ® Ind]} V � Ind ]}( V ® Res ]}U) ; also recall that the two representations p"• and Pv have equivalent restrictions to WE. ; hence Ap"• ® (Ap)'j is equivalent to (Ap.) ® (Ap)'j so that they have the same L-factor. This concludes the proof of the Artin conjecture for the tetrahedral case. The above proof is taken from a letter of Langlands to Serre (December 1 9 75) ; it also contains some indications on a method to handle the octahedral case ; however the latter requires some results on group representations which are not ; yet available. Still, a partial result is obtained [L, § 1 ] :

BASE CHANGE PROBLEM FOR

GL(2)

121

Assume that p is of octahedral type ; we use the fact that @;4 has a normal sub­ group W4 of index 2 ; hence there is a quadratic extension E of F for which the res­ triction PE l F is of type w4 . By the above theorem, 7r (PEI F) exists and, by Theorem 2, it is the base change lift of two cuspidal admissible irreducible automorphic representations 1r' and 1r" of G(AF). Assume now that F = Q, that E is totally real and that the complex conjugations in Gal(Q/Q) are sent by p into the class of (A -�) ; in such a case the components of 1r' and 1r" at the real place verify 1r' = 1r::O = 7r (p�) with P� = 1 EEl sign ; then 1r' and 1r" correspond to holomorphic automorphic forms of weight one. This situation has been studied by Deligne-Serre [D-S], and they show that 1r' = 1r(p') and 1r" = 1r(p") for some representations p', p" of WF in GL(2, C) ; now, by Theorem 2, 1r(p')EIF = 1r(p'EIF), and the same is true for 1r" ; this shows that p' and p" are the two representations which lift to PEl F ; hence either p = p' or p". Thus one concludes that either 1r' = 1r(p) or 1r" = 1r(p), and this gives oo

THEOREM 4. For a two-dimensional representation of W0 which is of octahedral type and which sends the complex conjugation on (A -�), and such that the above quadratic field E is real, the Artin conjecture is satisfied.

Appendix A. List of the two-dimensional admissible representations of W� [D).

Notations. F is a local (resp. global) field, CF is Fx (resp. the group of ideles

classes of F) ; W� is the Weil-Deligne group of F [T]. For F global, v a place of F, there is an injection WF, --> WF which defines an application p ,_. Pv from the two-dimensional admissible representations of WF into those of WF, ; if another representation p' of WF satisfies p� Pv for all but a finite number of places, then p' is equivalent to p, and p� Pv for all v. The two­ dimensional admissible representations of WF are classified by the image of the inertia group in PGL(2, C), called the type of the representation. (1) Cyclic type : f.1. E9 v is the sum of the two one-dimensional representations of WF defined by f.1. and v ; f.1. (£) v � v (£) p ; det(p (£) v) = pv ; (f-t (£) v) ® X (f-!.X) EEl (vx) ; (p E9 v)v = f..l.- 1 E9 v - 1 The L- and e-functions verify : �

�

.

e(p (£) v) = e(p)e(v) L(p EEl v) = L(p)L(v), e(p (£) v , ¢) = e(p , ¢)e(v, ¢) (if F is local), f.1. (£\ v = (8) . ( f-t v EEl v.) (if F is global).

(if Fis global),

(2) Dihedral type : r = Ind�f (), where () is a quasi-character of CK , and K a separable quadratic extension of F; r is irreducible if and only if () =F 11() (I =F u E Gal(K/F)) . For () = 110, let X be a quasi-character of CF such that () = X o NK 1 F and let t1 be the character of CF with Kernel NKl FeK· Then Ind�f 0 = x <£l xt1 ; det 1: = t1 . O lcF ; 1: ® X = Ind�f (O x 0 NK I F) ; f = Ind�:;:o - 1 ; L(r) = L( ()) , e(r) = e(O) (F global), e(r, ¢) = AK I F (¢ ) e( () , ¢ o TrK 1 F) (F local) ; for F global, r = @rv with rv = Ind�k!:' Ov for K./Fv quadratic, and rv = O v EEl Ov for K. = Fv x F• . The equivalences are : Ind�l () 1 Ind�f 8 2 ¢> either K1 = K2 and () I > ()2 conjugate by Gal(K/F), or K1 =F K2, 0{1011 and 02"2021 are of order 2 and 01 o NK 1 K2tK 1 = 82 o NK 1 K2tK2• (3) Exceptional type : the image of the inertia group in PGL(2, C) is W4 (tetra­ hedral type), @;4 (octahedral type), or W5 (icosahedral type) ; they occur only for F �

1 22

P. GERARDIN AND J. P. LABESSE

global or F nonarchimedean local of even residual characteristic; in this latter case, the icosahedral type does not occur. (4) Special type (occurs only for F nonarchimedean local) : X ® sp(2) for a quasi-character X of Fx and sp(2) the representation of W.F defined by z E C >->

(� �)

and w E WF >->

( � �) . ' I

Appendix B. List of admissible irreducible representations of GL(2, F), F local field [J-L].

Notations. G = GL(2, F), I I is the absolute value defined by the dilatation of the Haar measure on F: d(ax) = lal dx, and ¢ is a nontrivial character of the additive group of F. Representations. (1) Principal series p(p , ].1), where p , ].1 are quasi-characters of Fx . (a) Definition. Let p(p , ].1) be the representation of G by right translations in the space of smooth functions for G such that f(ang)

=

p(u) ].l(v) luv-1 1 1 /2 f(g)

for any g E G, a = (8 � E G, n E (ij �). When this representation is irreducible, then 1C(p , ].1) is p(p , ].1). When p(p , ].1) is reducible, there are exactly two irreducible subquotients ; one is finite dimensional and 1C(p , ].1) is this one. The other one is denoted a(p , ].1). (b) Equivalences. 1C(p , ].1) 1C(].I, p). For F = C, any irreducible admissible re­ presentation of G is equivalent to a 7C(p , ].1). (c) Finite dimensional representations. F nonarchimedean : they are one-dimen­ sional, and are the 7C(p , ].1) for w-1 = 1 1 ±1 : 7C(p , ].1) (x) = l det x l ± 112].1 (det x), x E G : the corresponding representations a(p , ].1) are called the special representations ; F = R : They are the 7C(p, ].1) with p ].l- 1 (a) = lal±(n+I) · sign(a) n for integers n � 0 ; F = C : they are the 7C(p , ].1) with /-l].l-1 (a) = [an+l (a) m+1 ] ± 1 for integers n � 0, �

m � 0. (d) Other properties.

Restriction to the center : w" c p . v l = p v ; twisting by a quasi-character of P : 7C(p , v) ® x = 1C(f-tX• vx) ; contragredient representation : 1C(p , ].l)v 7C(p - 1 , v-1) ; local factors : L(1C(p , v)) = L(p) L(v), e(7C(p , v) , ¢) = e(p , if;) e(v, ¢) . (2) Wei/ representations. 1C(t"), 1: = IndU::J;' (}, with (} a quasi-character of K x , and K a separable quadratic extension o f F. (a) Definition. Let GK be the subgroup of index two in G defined by those elements which have a norm of Kx for determinant. Fix a nontrivial character ¢ of the additive group of F. Then 7C(7:) is the class of the representation of G induced by the following representation r((}, ¢) of GK, in the space of smooth functions f on Kx such thatJ( t - 1 x) = (} (t)f(x) for t, x E Kx , NKl F t = I : �

(r((}, if;) (� �) f )(x) (r((} , ¢) (� �)! )(x) 1

=

if; (uNK 1 FX) f(x),

= AK t F(¢)

U

E F,

JK J(y)if;K ! F(xya)d�y,

BASE CHANGE PROBLEM FOR

(r(O , ¢) (� �) t )Cx) = O(b)f(bx)

GL(2)

1 23

for a = NK1 pb , b e Kx ,

where ¢ K 1 F = ¢ o TrK 1 F • y" is the conjugate of y by the nontrivial element u of Gal(K/F), dr/Jy is the self-dual Haar measure on K with respect to the character ¢ K 1 F and A.K 1 p(¢) is the unitary part of the local factor e( &, ¢), where (1 is the nontrivial character of px which is trivial on NK 1 pKx . (b) Equivalences. (I ) n(Ind!tk 0) = n(Ind!tk "0) ; ( 2) n(Ind!tk 0) - n(x. xa) if 0 = "0 and x is a quasi-character of px such that O (a) = x(NKI F a) ;

(3) Other equivalences : n(Ind ;;1 fh) = n(Indltk2 02) for K1 =I= K2 , if and only if 01"10}1 and 0 2"2 021 are of order 2 and satisfy 0 1 o NK1 KztK1 = 0 2 o NK 1K?JK2• (c) Characterization. If a nontrivial character X of px fixes a class 1C of irreducible admissible representations of G : n ® X = n, then X is of order 2, attached to a separable quadratic extension K of F and 1C = 1C(z') where 1: = Ind!tk 0 for a suitable 0, and conversely (cf. [L, §5]). (d) Other properties. For F = R, or nonarchimedean with odd residual characteristic, any irreducible admissible representation is a 7C(A., p) of a 7C(Ind�k 0) ; restriction to the center : m" < • > = (1 · O lpx ; twisting by a quasi-character of px : 1C(7:) ® x = 7C(Indltk 0 . x a NKl p), contragredient representation : 7C(f) = n(Indltk o- 1 ), local factors : L (1C(7:)) = L(O) , e (n(7:), ¢) = AK I F(¢)e(O , sVKIF)· (3) Exceptional representations. They occur only for F nonarchimedean of residual characteristic 2, and, up to twisting by quasi-characters of px, their number is 4( 1 21 - 2 - 1)/3 for F of characteristic 0, infinite for F of characteristic 2. They are supercuspidal (see complements below) (cf. [Tu]). (4) Special representations. They occur only for F nonarchimedean, and are the infinite dimensional subquotient u(p , ))) of the reducible p(p, ))), that is for p))-1 = I 1 ± 1 ; one has u(p, ))) ""' u()), p), u(p, ))) = xu( l l 1 1 2 , 1 1-1 1 2) for x = p i 1 - 1 1 2 . Complements.

(1) Representations u(p , ))). When the induced representation p(p, ))) is not irreducible, u(p, ))) denotes any representation equivalent to the unique infinite dimensional subquotient of p(p, ))) : for F = R, the representations u(p, ))) are the representations 7C(Ind �� 0), 0 =1: "0. (2) For a two-dimensional admissible representation p of Wp, there is at most one irreducible admissible representation 1C of G often denoted 1C(p) when it exists such that m,. = det p , L(1C ® x) = L(p ® x) and e (n ® X• ¢) = e(p ® X• ¢) for any quasi-character X of px ; for p reducible, p = p EB )) , then 1C = 1C(p, ))) ; for p = X ® sp(2) then n = u(p, ))) with f.t = x l p n, )) = X 1 1-1 1 2 ; for p dihedral p = Indltk 0, then 1C = 7C(Indltk 0) ; in the remaining cases, that is when p is ex­ ceptional, the existence of n is still not completly settled, but base change techni­ ques were used to prove it in many instances. Conversely, any 1C should be a n(p). Appendix C. Irreducible admissible automorphic representations of GL(2) [J-L]. Notations. F is a global field, Ap its ring of adeles, Cp = A'j./Fx the group of ideles classes, I I the absolute value on Ap.

1 24

P. GERARDIN AND J. P. LABESSE

(I) Noncuspidal representations. (1 . 1) :n:(A, fl.) for A, fl. Grossencharakters of F: they are the following represen­ tations : :n:(A, fl.) = ®.:n:(A., f.J. v) ; the one-dimensional representations are the :n:(A, fl.) for Af.J.- 1 = 1 1± 1 . (1.2) Any noncuspidal irreducible admissible automorphic representation :n: of GL(2, AF) has the following form : there are two Grossencharakters A and fl. of F,

and a finite set S of places of F, such that the components :n:. of :n: are given by V ¢ S, V E S, :7t' v = O' (A., fl.v ), :7t' 0 = :n: (A v, f.J.v) ,

where u(A. , f.J. v) denotes the infinite dimensional subquotient of the reducible p(A., fl. v) (Appendix B). (2) Cuspidal representations (examples). (2. 1) :n:{'L') with .... = IndJH () for a separable quadratic extension K of F and a

Grossencharakter () of K, not fixed under Gal(K/F), is the representation :n:{'L') = ® :n:(Ind�k'• o.) •

= () . Ei?> () . for K. = F. x F• . Properties. (1) Let :n: e D(G) ; in order that there exist a nontrivial GrosseD­

with Ind�k' () .

charakter X of F such that :n: ® X = :n: , it is necessary and sufficient that there exist a separable quadratic extension E ofF and a Grossencharakter () of E such that (a) X is the character of CF with Kernel NErFCE ; (b) :n: = :n:(Ind�� ()) (in particular :n: = :n:('L', 'L'X) if () = a(), where .... is a GrosseD­ charakter of Fwhich has () for lifting to E) ; (2) :n:(Ind��� 0 1 ) = :n:(Ind�f2 02) <=> either K1 = K2 and ()2 = 0 1 or a()h or K1 =1: K2 then () 1 , a1 0 1 1 and 02 · a()21 are of order 2, and (0 1 ) o NK1K2rK1 = (02) o NK1 Kz/Kz· {3. 1) More generally let p be a two-dimensional admissible representation of WF ; we say that :n: = ®:n: . is :n:(p) if :n:. � :n:(p.) for all v. The existence of such :n: when p is irreducible is related to the Artin conjecture for the p ® X• where X is any quasi-character of CF [J-L, § 1 2]. (3.2) Of course there are many other, more complicated, types of cuspidal representations : think of the classical L1 for example. II. BASE CHANGE FOR G4, A SKETCH OF THE PROOF. 1. The trace formula. In all the following we shall use notations close to those of (G-J-1] in these PROCEEDINGS. Let F be a number field, E a cyclic extension of prime degree, put Z1 = NElF Z(AE), where AE denotes the ring of adeles of E, and Z1 {F) = z1 n Z(F) ; as usual Z is the center of G4 = G and is identified with the multiplicative group. Since E/F is cyclic one has Z1 (F) = NErFZ(E) . Choose a character co of Z1 /Z1 (F) and consider the space £2 (Z1 · G(F)\G(A), co) = £2 of functions on G(F)\G(A), which transform on Z 1 according to co : rp(zrg)

=

co(z)rp(g),

z e zh r e G(F),

and square-integrable on Z1 · G(F)\G(A). In such a situation, which is slightly more general than the one studied in [G-J-1]

BASE CHANGE PROBLEM FOR

GL(2)

1 25

(where E = F), one defines in an obvious way the spaces L5 and L�p · The re­ striction of the natural representation of G(A) in L2 to L5 EB L�P will be denoted by r. Iff E 'tf':,"'(Z1 \G(A), w- 1 ) , the space of smooth functions on G(A) compactly sup­ ported modulo Z1 which transform according to w- 1 on Zh the operator r(f) is of trace class. The Haar measures being chosen as in [G-J-1, §§6-7] we assume more­ over that vol (Z1 · Z(F)\Z(A)) = I. Then tr r(j) is the sum of the expressions (i)-(vii) below (we assume thatfis a tensor product of local functions/.) (cf. [L, §8]). (i)

I;

zEZ! (F)\Z(F)

vol (Z1 • G(F)\G(A)) · /(z),

(ii) where Iff is a set of representatives of the conjugacy classes of elliptic elements (i.e., whose eigenvalues are not in F) taken modulo Z1(F), and e (r) is 1- (resp. 1) if the equation ()- 1 ro = zr has (resp. has not) a solution in z E Z1(F) - { 1 } . (iii) where D o is the set of pairs r; = (f.-1, ).!) of characters of A x jp such that f.-1).! in­ duces w on Zh where M(r;) and JC� are defined in [G-J-1, §4], and with JC�(f) = JzM
(v)

where ).0 and other notations are defined in [G-J-1, §7-B]. The remaining terms will not be written as in [G-J-1] since they are there expressed by noninvariant local distributions. For example the local distribution t A 1 (r, f. )

=

-

.:J(r)

) log

J lxvl>l ( fvK

a (a - b)x. o b

jx. l dx.

where r = (g 2), .:J .(r) = j (a - b)Zjab j �12 and ff(g) = JK. f.(k- 1gk) dk, can be written A 2(r, f.) + A 3(r , f.) where A 2(r, f.) is invariant, and A 3 fits with other terms to provide an invariant expression. More precisely one takes for a nonarchimedean place v : A 2(rJ.)

=

log j (a

- b) f a i .F(r J.) +

J

.:J.(r)f,K(z) lxvl>l log jx. l dx. - l afb l 1 12 · 1 w .l

(z (� �)), =

126

P. GERARDIN AND J. P. LABESSE

where F(r. f.) = �Mr) JA.1c. fv(g- 1rg) for F• . If v is archimedean one takes A2(r , f.)

=

log I I -

� I.

dg

F(r. J.) -

This yields the term (vi)

and

w.

is a uniformizing parameter

�'g : �:�2 J v

z.N.,G. J.

(g - 1 znog) dg .

w*- v

It can be checked that r .... A3(r. J.) extends to a continuous map on A (F.) and one sees that the terms 6.34 in [G-J-1] minus our (v) plus 6.35 minus our (vi) yield the term This can in turn be transformed by a kind of Poisson summation formula to

One has to add the term 6.36 of [G-J-1] minus our (iv) to get the final term : (vii) 2. The twisted trace formula. Here we shall use definitions and results of the Kott­ witz lecture (see [K] in these PROCEEDINGS) . As above we follow closely [L, §8]. Let L2 (Z(AE)\G(AE), w) = l2 where w = (t) 0 NE l F· The Galois group r = Gal(E/F) acts on £ 2 by up(x) = p(xu) for p E £ 2, X E G(AE) and fJ E F. Let Rd denote the restriction of the natural representation of G(AE) in £ 2 to the discrete spectrum l5 EB L;P. The projection commutes with the action of the Galois group ; hence Rd can be extended to a representation R� of r !>( G(AE). Let ¢ E 0'�(Z(AE)\G(AE), w- 1 ) ; then Ri
tr(R�(fl)Ri
=

J

Z (AE) \G (AE)

K(
Assuming, as usual, that ¢ is a tensor product : ¢ = @¢., one can proceed as in [G-J-1] to compute this integral ; if fJ -:f. 1 it is the sum of the following terms

(1)-(7) : (1)

I; vol(Z(A)G�(E)\G�(AE)) •

J

z�q�w�


where the sum runs over the (]-conjugacy classes of elements o such that N(o) is central, and G� is the (]-centralizer of o (cf. [K]). (2)

BASE CHANGE PROBLEM FOR

GL{2)

1 27

where C(j is a set of representatives of the a-conjugacy classes that are not a-con­ jugate to a triangular matrix, taken modulo Z(E) in G(E), and e(o) is t or I accord­ ing as the equation -.-qo-r = zo has or not a solution in Z(E) with z ¢ Z(£)1- u. (3)

where r;v = (v, f-l) if r; = (f-l, v) is a pair of characters of Aj;/E x with f-lV = w. The representation 7r:7J is realized in a space of functions on G(AE), the action of a E r defines an operator n�(a) from the space of 7r:7J to the space of 7r:11" ; then ME( 117J) intertwines 7r:17" and n••• but 177Jv = r;. Hence the product M(117J) n�(a) nkp) is a well­ defined operator in the space of n.7J., and the above expression is meaningful. (4)

where ij = (f-l o NE1F, v o NE 1F) if r; = (f-l, v). There are / 2 elements ij giving rise to the same ij. The reader should be aware that our notation ij has not the same meaning as in [L]. Ao TI v 017(0, ifJv),

(5)

where 017(0, r/Jv) = L(l , l v)-1 J J J ifJv(k-11t-11n-qn0ntk)t-2P dn dt dk, with k the standard maximal compact subgroup of GL2(E ® Fv), t =

(� �),

�-2p =

1 � r-l

and flo

=

E

Kv,

(� D

such that trE; F z = I . The integration is on Kv x Z(Ev)\A(Ev) x N(Fv)\N(Ev), where Ev = E ® Fv· (6)

-

;._ 1 A�(o, r/Jv) TI P(o, r/Jw), 1: 1: / OE!F v w *v

where §' = { o E A1-u(E)Z(E)\A(E) iN(o) ¢ Z(F)},

P (o, ifJw) = Llw(r) Jz cE.l A CF.)\G CE.lifJw(g-u og) dg

and r = N(o). An explicit definition of A�(o, r/Jv) will not be given here ; we shall simply say that A�(o, ifJv) = IA 2 (r, fv) if fv is associated to ifJv under the base change correspondence (see [K]). As above the remaining term can be written (7) For a definition and a detailed study of distributions A� and B" the reader is referred to [L, §7] (where the subscript a is omitted) and to [K, Lemma 4]. 3. The comparison.

We assume now on that the function f

0'� (Z1 \ G(A), w-1 ) and the function ifJ = ®ifJv

E

= ®fv

E

0'�(Z(AE)\G(AE), w) are such

that ifJv and fv are "associated" in the sense defined in [K] ; we consider .@ 1 = I tr (R�(a) RiifJ) ) - tr r(f).

I 28

P. GERARDIN AND J. P. LABESSE

PROPOSITION I .

�1 =

� Svo I: (1B"(7Jv• r/Jv) o 2 2

where

V

�

7J= <.u. • .ul

* .u ; •.u.u=w

- I:_

n,.. 7J

)

fJ tr 7r:7Jw(fw) d7J B{7]. , /.) WoFV

tr(M("7J) 7r:�(u) 7r:7J(rp) )

o1 . 2 = I if I = 2, = 0 if I =f. 2 .

The proof amounts to the comparison term by term of the expressions for tr(r(f)) and I · tr(R�(u)R�( rp) ) . For example to prove that 1 · (1) = (i) note that we work there with a sum over elements o e G(E) such that N(o) is central in G{F) ; hence �(E) is the set of F­ points of a twisted inner form of G. Since we use Tamagawa measures, I vol(Z(A)�( E)\G�(AE)) = I vol(Z(A)G(F)\ G(A) ) = vol(Z1G(F)\G(A)) ;

the expressions to be compared are products of the local analogues, and now, using properties A' and B' in [K] for associated functions and the fact that the number of places where a minus sign occurs is even, we obtain the desired results. To prove I · (2) = (ii) is even simpler since in that case G�(E) = Gr(F), where r = N(o) : the twisting is trivial since Gr is abelian. To compare I · (3) and (iii) is slightly more complicated ; we must distinguish two cases : (a) I =f. 2 ; then "7J = 7Jv implies "7J = 7J = 7Jv and in such a case one has M(7J) = - I . One the other hand one should note that tr 7r:7J.(f.) = Jz.,A. F(a, f.)7] .(a) da and that for associated functions f. and ,P. one has F(a, f.) = F"(b , ,P.) if a = N(b). Moreover tr 7r:�.(U) 7r: 'ij.(rp.) =

J Zv\Av F"(b, r/J.)f;.(b) db,

where z. = Z(E ® F.) and A. = A (E ® F.) . Then tr(1r:�(u) 7r:'ii (rp)) = tr(1r71(f)) i f rp andfare associated and if f; = (f.J. o NEIF• v o NE1F) and 7J = (f.J. , v). This yields I · (3) = (iii). (b) If I = 2, the same arguments apply i f 7J = 7Jv = "7J, but there are other terms corresponding to 7J = (f.J. , v) with f.1. = "v =f. "f.1. and then 1 · (3) - (iii) = -

�

.u *".u ;n=

• .ul ; • l'.u=w tr(M("7]) 7r:�(u) 7r: 7J( rp)).

fi:

To prove that I · (4) = (iv) we use the previous remarks and the fact that mE(fJ)l = fl71,.. 'ij m(7J), where 7J 1-+ ij means that 7J = (f.J. , V) and ij = (f.J. o NEIF• V o NEI F) = (p., P) and by definition : m(7J) = L( I , f.J.- I v) and mE(7]-) = L( I , p.-Ip) . L( l , v lf.J.)

L( I , p.v i)

Now I · (5) = (v) follows from the comparison of orbital and twisted orbital in­ tegrals on unipotent elements.

BASE CHANGE PROBLEM FOR

GL(2)

1 29

To conclude the proof of Proposition 1 it is enough to show that 1 · (6) = (vi), which in turn follows from the equality :

A 2(r, f.)

A �(o, ¢.) if r

N(o). 4. The main theorem. The representation r is a discrete sum of unitary irreducible representations of G(A) with multiplicity one : r =

=

L:

iE /

11:;

=

=

L: (8)n; . •

i E[

and tr r{ f) = ,E ;E1 TI • tr 71: ;• •(/.) for some set I. Let v be a nonarchimedean place. Given f. in the Heeke algebra of G(F.) then tr 11:; , .( !.) is zero unless 11:; • • is unramified and hence corresponds to the conjugacy class of some semisimple element t;. • e GL2{C), the connected component of the L-group. Any function f. in the Heeke algebra defines a rational class function f'j on GL2(C) such that f';{(t; . •) = tr n; .•{f.). Choose a finite set V of places of F containing archimedean ones and assumef. is in the Heeke algebra for v ¢ V. Then one has LEMMA

I . tr r{f)

= ,E ; TI v E V tr 71:; , .(/.) TI v
The representation Rd can also be written

Rd = L; lli = L; ®Dj . v jEJ

jEJ

V

where lli is an automorphic representation of G(AE) and lli. v a representation of G(F. ® E). Since aRd � Rd and is multiplicity free, (J permutes the lli. If •lli � lli one can restrict the operator R�((J) to the space of lli , and denote this restriction by llj((J) . The nonfixed lli do not contribute to the trace of R�((J)Rd(
L: tr llj((J) lli(
•Dt'"Di

Moreover lli = (8)lli· • and •lli. v � lli. v ; we can define ll}, v ((J) up to lth roots of I . If lli. v is unramified, there is a canonical choice. If ¢. and/. are associated in the Heeke algebras, we choose a semisimple element ti · • e GL2(C) such that tr n;, .((J)llj . v(¢.) = tr nj. .<¢.) = f';{( tj. v),

and then for some big enough finite set V of places of F, we have :

LEMMA 2. tr R�((J)Ri
REMARK. If v is nonarchimedean and split in E, then any f. is associated to some ¢. ; in fact in such a case ¢. may be taken to be fw, ® fw2 ® . . . ® fw, where the W; are the places of E above v, and/. = fw, * fW2 * . . . * fw, can be any smooth function on G(F.) ; the conjugacy class of ti · • is well defined by lli . v · If v does not split in E, is unramified, and if ¢. and f. are associated in the Heeke algebras, one can define a function ¢';{ on GL{2, C) as above ; we have

f';{(t) = ifJ';{(t') for t semisimple in

GL2{C).

In such a case the conjugacy class of the ti. v above is not uniquely defined.

1 30

P. GERARDIN AND J. P. LABESSE

If / #:- 2 let R = /Rd. Assume for a while I = 2 ; if p. is a character of A>t;/ Ex such that 11p.p. = w and 11p. '# p., we consider 1:"' = tc71 , where 7J = (p., 11p.). As was said above, the operat­ or M(117J)1C�(o") = 1:�(u) maps the space of tc71 into itself. One defines in such a way a representation 1:� of r � G(AE). One should remark that 1:� � ..;w Let us denote by f7 the set of such representations (modulo equivalence) and let R' = IR� $ I; 1:�. � �E$'"

An analogue of Lemma 2 can be stated. We can now state the main theorem (cf. [L, Theorem 9. 1]). THEOREM 1 .

Assume! and ifJ are associated; then tr R'(u)R(ifJ) = tr r(f).

(Recall that the definition of the correspondence ''f and ifJ are associated" de­ pends on the choice of a 0' E F - {1 }.) The proof ofthe theorem can be carried out as follows : consider the expression tr R'(u)R(ifJ) - tr r(f).

(a)

Thanks to Proposition 1 above, this is equal to

(b) Using properties of some weighted orbital integrals [K, Lemma 4], one can prove that /B(fjv , ifJv) - I; B(7Jv o fv) = 0, 7Jv...,. 7iv

at least when v is nonarchimedean, unramified in E, with ifJv and fv associated and in the Heeke algebras. Hence there is a finite set of places V such that {b) can be written (b') with some nice function {3. Now choose a place v0 � V split in E; then (b') reduces to an absolutely con­ vergent integral : (b")

b

qis 0) o 0 bq-is J-+oooo o(s)fvv(a

ds

where a, depends on the central character m vo· All we need to know is that o(s) is some continuous, bounded and integrable function on the real line. On the other hand (a) can be written using Lemmas 1 and 2 above (a") with ak e C and tk e GL2(C) semisimple elements corresponding to inequivalent unitary representations of GL2(F0J. The series, as the integral, is absolutely con­ vergent. Now fv0 is arbitrary in the Heeke algebra (since v0 is split in E) and the

BASE CHANGE PROBLEM FOR GL(2)

131

Heeke algebra separates inequivalent unramified representations. The Stone­ Weierstrass theorem and easy majorations prove that all a, are zero (which is a stronger statement than Theorem 1). All the desired resu lts can now be xtracted from Theorem I and from some results on the characters of representations of r 11< GLz(E.) (cf. [L, §5]). We shall try to explain some of the steps.

5. Existence of weak liftings. Choose a finite set V of places of F containing all archimedean places and all places ramified in E. Assume that if>v and f. are as­ sociated and in the Heeke algebras for v ¢ V. One can, using Lemmas I and 2, choose element t,.,. e GLz(C) for v e V and n e N such that

tr R;(u)Riif>) = I; a,.(if>) IT f';j(t,.,.), n

v!tV

n

v$V

n

v!tV

tr ��(u)'t: p.(if>) = E {J,.(rp) IT f';j(t,., .),

tr r(f) = I; r ,.(rp) IT f';/(t,. , .) ;

we may assume moreover they are chosen such that the functions T,. : (r/J.)v!t v ...... IT v !t v f';f(t,.,.) on the product for v ¢ V of the Heeke algebras are distinct. (Recall the remark after Lemma 2.) Let ;;,. = Ia,. + {J,. - r.. ; then the above theorem can be restated in the following form Another use of density arguments, the T,. being distinct, yields PROPOSITION 2.

For all n one has o,.(rp) = 0.

This can be read Ia,. + {J,. = r ,.. Assume that there exist a representation U = ®U., unramified outside V, occurring in [0 such that tr U.(rp .) = f';j(t,.,.) for some n with a,. not zero, and any v ¢ V; then using the strong multiplicity one theorem [C] one concludes that such a U is unique and satisfies U � 119. The fact that L(s, U) is entire allows one to conclude that no other U occurring in [�P or in g- has the property that tr n.(¢>.) = f';/(t,.,.), v ¢ v. Then a,.(if>) = IT V E V tr n;(u)U.(if>.) and {J,.(if>) = 0 ; since Ia,. + {J,. = r.. we conclude that r,.(rp) is not identically zero and hence there exists (at least) one 11: in L� ® L� such that 11: = ®n. and tr n.(f.) = f';f(t,.,.) for v ¢ V and then n. is P the lifting of n. for v ¢ V. We shall say that U is a weak lifting of 11: if n. is a lifting of n. for almost all v. We then have proved

If n is a cuspidal automorphic representation of GLz(AE) such t1 n, then n is the weak lifting of some automorphic representation 71: of

THEOREM 2.

that n � GL2(A).

A direct study of L�P allows one to prove :

PROPOSITION 3. Any 11: occurring in L�P lifts to a U in [�P and all u-fixed represen­ tations in [�P are obtained in this way.

In the case I = 2, assume that U = n(p., 11p.) occurs in g- and that p.. is unramified for v ¢ V; one can show using Proposition 2 and results on L-functions on GL2 x

1 32

P. GERARDIN AND J.P. LABESSE

GL2 that ll is the weak lifting of 11: = n(p) where p = lnd�� p. and that there exists n E N such that This can be used to show that at all places of F: tr ll� (O')ll.(f/J.) = tr n.{f.), and hence THEOREM 3. Let the fields E and F be either local or global. Then n(p., "p.) is the lift­ ing ofn(p) where p = Ind�� p..

6. Any cuspidal n has a weak lifting. Assume for a while that some 11: occurring in � L has no weak lifting. Let V be a finite set of places of F including archimedean places and those where E or 11: are ramified. Let / = ®f. be associated to some ifJ and such that f. is in the Heeke algebra for v 1/: V. Consider the 11:k in L� EB L';P such that tr nk .• (f.) = tr n.(f.) if v 1/: V. Since the 11:k have no weak lifting, Proposition 2 shows that the sum of the 11:k gives a zero contribution to tr r(f) :

IT tr 11:k . vUv) IT tr 11:v(fv) = 0 ; I; k v eV v \EV

hence there is a set V1 c V such that � k D v e v, tr n k . v(f.) = 0. One has to prove that this is impossible unless the sum is empty. The idea of the proof (by induction on the cardinality of V1 ) is that characters of inequivalent representations are linearly independent, but the proof is complicated here by the fact that fv cannot assume all values, since fv must be associated to some f/J. (cf. [L, §9, pp . 25-30]). This yields THEOREM 4.

Any cuspidal n has a weak lifting.

7. Local liftings. To finish the proof of the global theorem on base change for cuspidal representations, one must show that the above weak liftings are liftings at all places. Let ll � "ll, occurring in L�, unramified outside a finite set V chosen as before ; there is an n such that, according to Proposition 2, lan(ifJ) = r.lifJ), which can be written, for some V1 c V, I IT tr n;(O')ll.(f/J.) = I: IT tr 11:k , vUv) , k v e v, v ev,

where the nk = ® nk. v have ll as weak lifting. One then proves [L, §9, pp. 33-34] : LEMMA 3. lf.for some v the representation n. is the lifting of some 11:v then it is the lifting of 11:k , v for all k, that is tr n;(O')ll.(f/J.) = tr 11:k, v (f.).

Then one may assume that V1 does not contain such places. On the other hand existence and properties of local liftings are easy to prove, or are deduced from Theorem 3 above, except for some supercuspidal representations (exceptional ones). Lemma 3 and this remark show that all desired local or global results (cf. part I of this paper) can be deduced from THEOREM 5 [L, §9, PROPOSITION 9.6]). (a) Every supercuspidal nv has a lifting.

(b) lf llv

�

" ll. and is supercuspidal, then n. is a lifting.

Part (a) of this theorem is proved by embedding the local situation in an ad hoc

BASE CHANGE PROBLEM FOR

GL(2)

1 33

global one, where the existence of liftings is known at all places except perhaps at one place, and to use the above equation with V�o reduced to one element. Part (b) then follows from the orthogonality relations of [L, §5]. The last paragraph of Langlands paper [L] is devoted to the proof of the exist­ ence of lifting for noncuspidal representations, using their explicit description (cf. Appendix C). BIBLIOGRAPHY [B) A. Borel, Automorphic £-functions, these PROCEEDINGS, part 2, pp. 27-61 . [C] W. Casselman. On some results of Atkin and Lehner, Math. Ann. 201 (1973), 301-334. [D) P. Deligne, Formes modulaires et representations de GL(2), Modular Functions of One

Variable. II, Lecture Notes in Math. , vol. 349, Springer-Verlag, Berlin, 1 973. [D-S] P. Deligne and J.P. Serre, Formes modulaires de poids 1 , Ann. Sci. Ecole Norm. Sup. 4 (1 974), 507-530. [G-J-1) S. Gelbart and H. Jacquet, Forms on GL(2) from the analytic point of view, these P Ro CEEDINGS, part 1 , pp. 21 3-25 1 . [G-J-2] , A relation between automorphic forms o n GL(2) and GL(3), Proc. Nat. Acad. Sci. U. S. A. (1 976) . [J-L] H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Math., vol. 1 14, Springer, Berlin, 1 970. [J-PS-S] H. Jacquet, I. I. Piatetskii-Shapiro and J. Shalika, Construction of cusp forms on GL(3), Lecture Notes, Maryland. [J-S] H. Jacquet and J. Shalika, Comparaison des formes automorphes du groupe lineaire, C. R. Acad. Sci. Paris 284 (1 977), 741 -744. [K) R. Kottwitz, Orbital integrals and base change, these PROCEEDINGS, part 2, pp. 1 1 1-1 1 3 . [L) R. P . Langlands. Base change for GL(2), Institute for Advanced Study, Princeton, N. J., 1975 (preprint). [Sa) H. Saito, Automorphic forms and algebraic extensions of number fields, Lectures in Math., no.8, Kinokuniya Book Store Co. Ltd ., Tokyo, Japan, 1975. [S-1] T. Shintani, On liftings ofholomorphic automorphic forms, U.S.-Japan Seminar on Number Theory (Ann. Arbor, 1 975). [S-2] , Two remarks on irreducible characters offinite genera/ linear groups. J. Math. Soc. Japan 28 (1 976), 396-414. [S-3] , On liftings ofholomorphic cusp forms, these PROCEEDINGS, part 2, pp. 97-1 10. [T] J. Tate, Number theoretic background, these PROCEEDINGS, part 2, pp. 3-26. [Tu] J. Tunnell, On the local Langlands conjecture for GL(2), Invent . Math . 46 (1 978), 1 79-1 98. ­

--

--

--

UNIVERSITE PARIS

VII

Proceedings of Symposia in Pure Mathematics Vol . 33 (1979), part 2, pp. 1 35-1 38

REPORT ON THE LOCAL LANGLANDS CONJECTURE FOR

GL(2)

J. TUNNELL Let ([J(GL(2)/K) be the set of isomorphism classes of two-dimensional F-semi­ simple representations of the Weil-Deligne group W� of a nonarchimedean local field K. The purpose of this report is to discuss the following conjecture of Lang­ lands relating ([J(GL(2)/K) and the set 0(GL(2, K)) of isomorphism classes of irredu­ cible admissible representations of GL(2, K). Conjecture. For each representation e1 in ([J(GL(2)/K) there exists a representation n = n(CT) in O(GL(2, K)) such that the determinant of e1 and the central quasi­ character of n are equal and such that L(n ® X) = L(e1 ® X)

and e(n ® X) = e(CT ® X)

for all quasi-characters X of K * . The map e1 ...... n(e1) is a bijection of ([J(GL(2)/K) with O(GL(2, K)). The existence statement in the conjecture was formulated in [4], where it was shown that for a given representation e1 there is at most one representation n satisfy­ ing the desired conditions. The injectivity statement is equivalent to saying that a two-dimensional F-semisimple representation of W� is determined by its twisted L- and e-factors and determinant. It is straightforward to show directly that redu­ cible two-dimensional representations are in fact determined by the twisted L­ factors alone. The known proofs of the injectivity statement for irreducible repre­ sentations use admissible representation techniques. As discussed in [1, 3.2.3] it follows from the work of Jacquet and Langlands that n(CT) exists when e1 is reducible, and that this establishes a bijection of the set of isomorphism classes of completely reducible (repectively reducible indecompos­ able) two-dimensional F-semisimple representations of W1c with the set of isomor­ phism classes of principal series (respectively special) representations of GL(2, K). Let (/J;,,(GL(2)/K) consist of isomorphism classes of irreducible two-dimensional representations of the Weil group WK, and let Dcusp (GL(2, K)) be the set of iso­ morphism classes of supercuspidal representations of GL(2, K). Members of these sets are characterized by the requirement that their twisted £-factors are all equal to I . To prove the conjecture it is enough to verify that n (CT) exists for all e1 in (/J;,,(GL(2)/K) and to show that this establishes a bijection of (/Ji rr (GL(2)/K) and AMS (MOS) subject classifications (1 970). Primary 1 2B30, 1 2B1 5 ; Secondary 1 0D99. © 1979, American Mathematical Society

1 35

1 36

J.

TUNNELL

Dcusp (GL(2, K) ) . This was first proved for local fields with odd residue characteristic (see §2), which suggested the bijectivity portion of the conjecture in general. The conjecture has been proved for all nonarchimedean local fields except those extensions of Q2 of degree greater than I which do not contain the cube roots of unity. References for the proof are given in the following survey. 1. Existence of n(u). If a is a representation induced from a one-dimensional representation of an index two subgroup of WK there is an explicit construction (due to Weil) of an irreducible admissible representation n(a) of GL(2, K) with the desired properties [4, 4.7]. When K has odd residue characteristic all representations in (/Jirr (GL(2)/K) are induced from proper subgroups [1, 3.4.4], so the construction above applies. For each local field of even residue characteristic there exist irreducible two­ dimensional representations of the Wei! group which are not induced from a proper subgroup [11, paragraph 29]. There is no explicit construction of n(u) known for such representations. The approach of Jacquet and Langlands to the existence of n-(u) in this case is to imbed the local problem in a global one as follows. Let F be a global field and let p be an irreducible continuous two-dimensional complex re­ presentation of the Weil group WF. For each place v of F let p. be the restriction of p to the Wei! group of the local field F• .

THEOREM 1 [4, 1 2.2]. If the global L-functions L(s, p ® X) are holomorphic and bounded in vertical strips asfunctions of the complex variable s for all quasi-characters X of A;/ F* then n-(p.) exists. Cases when the hypotheses of this theorem are met have been described in the discussion of Artin's Conjecture in the base change seminar at this conference. A homomorphism of a group to GL(2, C) will be said to be of type H if the composi­ tion with the quotient map to PGL(2, C) has image isomorphic to H. The results in brief are that Theorem 1 may be applied when p is induced from a proper subgroup (Artin), when F is a field of positive characteristic (Wei!), when p is of A4 type (Langlands-Jacquet-Gelbart), and when F = Q, p is of S4 type and the image of complex conjugation has determinant - 1 (Langlands-Serre-Deligne). THEOREM 2 [10, THEOREM A]. Let K be a nonarchimedean localfield which contains the cube roots of unity if it is a proper extension of Q 2 • Then n-(u) exists for all a in f/J ( GL(2)/K).

This theorem is proved by constructing a global field F, a place v of F such that F. � K, and a representation p of WF satisfying the hypotheses of Theorem 1 such that Pv � a . The nonarchimedean local fields of even residue characteristic which do not contain the cube roots of unity are precisely those for which there exist re­ presentations of the Weil group of S4 type [11, paragraph 25]. The fields excluded in Theorem 2 are those for which there are two-dimensional representations of the Weil group which are not restrictions of global representations known to satisfy the hypotheses of Theorem 1 . Deligne has indicated that if /-adic representations can be associated to certain automorphic representations related to Shimura varie­ ties arising from division algebras over a totally real field F, then the hypotheses of Theorem 1 will hold for representations of WF of S4 type with prescribed behavior

THE LOCAL LANGLANDS CONJECTURE FOR

GL(2)

1 37

at the infinite places. Theorem 2 will then be true for K a completion ofF at a prime dividing two ; since each finite extension of Q2 is the completion of some totally real field, the proposition and Theorem 3 of §4 would then prove the conjecture in all cases. 2. Odd residue characteristic. The Plancherel formula for GL(2, K) shows that the supercuspidal representations of the form n'(o) for o in tf>;rr (GL(2)/K) exhaust the supercuspidal representations if K has odd residue characteristic. References [3], [8] and [9] contain treatments of the representation theory of GL(2, K) and related groups when K has odd residue characteristic. The injectivity statement may be proved by examining the explicit formulas for characters of supercuspidal representations in the case of odd residue characteristic which are given in [7] and the references above (at least for SL(2) and PGL(2)). Sup­ pose that o is induced from a ! -dimensional representation A of the Weil group WE of a quadratic extension E of K. Denote the quasi-character of E * corresponding to A by the same symbol. The restriction of the character function of :n:(o) to a Cartan subgroup isomorphic to E * is given by a formula involving A and its Gal(E/K) conjugate. From the explicit form of the character it can be seen that :n:(o) deter­ mines A up to Gal(E/K) conjugation, and hence determines the induced represent­ ation o. 3. Positive characteristic. The discussion of § I shows that :n:(o) exists for all representations of the Weil group of a local field of positive characteristic. The bijectivity assertion of the conjecture seems to have first been proved by Deligne [2] as a consequence of Drinfeld's results relating automorphic representations of GL(2) over global function fields to 1-adic representations. 4. A method is presented in [10] that gives alternate proofs of Langlands' Con­ jecture for the cases discussed in §§2 and 3 and proves the conjectured bijection for all fields in Theorem 2. PROPOSITION [10, 2.2] . Let o 1 and o2 be two-dimensional representations of the Wei/ group of a localfield, each of which is the restriction of a global representation satisfy­ ing the hypotheses of Theorem 1 . /f :n:(o 1 ) � :n:(oz) then o 1 � Oz .

The proposition above is proved by an inductive application of base change for GL(2). The assumptions of the hypothesis are necessary because the proofs of the base change results for local fields utilize global methods. The following result holds for any nonarchimedean local field K. THEOREM 3 [10, §§4 AND 5]. There are partitions of t!>irr (GL(2)/K) and Dcusp(GL(2, K)) into finite sets tf>;. and U;. respectively (indexed by a common set A) such that ( 1) If o e l/>1 and :n:(o) exists , then :n:(o) e U;.. (2) Card(tf>;.) = Card(U1) for all A e A.

The partition elements are determined by conditions on the Artin conductor and determinant (resp. conductor and central quasi-character) of elements in tf>irr (GL(2)/K) (resp. Dcusp (GL(2, K))). Since the conductor of a representation is de­ termined by the twisted e-factors, the first statement follows from the definitions.

1 38

J. TUNNELL

The computation of the cardinality of the sets f/Jl is done by constructing all irredu­ cible two-dimensional representations of WK as in [11] and counting those with a given Artin conductor and determinant. The sets Dl are studied by utilizing the cor­ respondence between square-integrable representations of GL(2, K) and admissible representations of the group of invertible elements in the quatemion division algebra over K. Theorem 2 together with the injectivity proposition and the counting results of Theorem 3 show that f/Jl and 0;. correspond bijectively by means of o ,_. n(o) for the local fields in the hypotheses to Theorem 2. This gives the known cases of the conjecture stated in the introduction. 5. Remarks. While the statement of Langlands' Conjecture is purely local, the proofs described above in the case of even residue characteristic utilize global methods (base change and cases of Artin's Conjecture). Only in the case of odd residue characteristic are the proofs described above purely local. Cartier and Nobs have indicated a proof of the conjecture for the field Q 2 which is purely local. They calculate the necessary e-factors for irreducible two-dimen­ sional representations of Wll:! and match them with the factors of supercuspidal representations of GL(2, Q2) constructed in [6]. The supercuspidal representations are constructed by inducing finite dimensional representations of subgroups of GL(2, K) which are compact modulo the center. In [5] a similar construction of supercuspidal representations for GL(2, K) is given which should allow, in theory, the calculation of e-factors and matching with the factors of representations of WK to be done in general. REFERENCES

1. P. Deligne, Formes modulaires et representations de GL(2), in Modular functions of one variable, Springer Lecture Notes, No. 349, 1 973, pp. 55-1 06. 2. , Review of" Elliptic A-modules" by V. Drinfeld, Math. Reviews 52�1 976), #5580. 3. I. M. Gelfand, M. I. Graev and I. I . Piatetskii-Shapiro, Representation theory and automorphic functions, Saunders, 1 969. 4. H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Springer Lecture Notes, No. 1 14, 1 970. 5. P. Kutzko, On the supercuspidal representations ofGL,. I, II, Amer. J. Math. 100 (1 978), 43-60. 6. A. Nobs, Supercuspidal representations o/GL(2, Qp), including p 2 (to appear). 7. P. J. Sally, Jr. and J. A. Shalika, Characters of the discrete series of representations of SL(2) over a localfield, Proc. Nat. Acad. Sci. U.S.A. 61 (1 968), 1 231-1 237. 8. , The Plancherel formula for SL(2) over a local field, Proc. Nat. Acad. Sci. U.S.A. 63 (1 969), 661- 667. 9. A. J. Silberger, PGL(2) over the p-adics, Springer Lecture Notes, No. 1 66, 1 970. 10. J. Tunnell, On the local Langlands Conjecture for GL(2), Thesis, Harvard University (1 977). 11. A. Weil, Exercises dyadiques, Invent. Math. 27 (1 974), 1 -22. --

=

--

PRINCETON UNIVERSITY

Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 2, pp. 141-163

THE HASSE-WElL �-FUNCTION OF SOME MODULI VARIETIES OF DIMENSION GREATER THAN ONE

W. CASSELMAN Introduction. Let k be a number field of finite degree, G a connected reductive group over k. By restriction of scalars, one may as well assume k = Q. Let KR � G(R) be the product of the center of G(R) and a maximal compact subgroup. Then, in certain circumstances, one may assign to X = G(R)/KR a G(R)-invariant com­ plex structure ; if r is an arithmetic subgroup of G small enough to contain no torsion then F\X will be a nonsingular algebraic variety. In many cases one can show that it has a model defined over a rather explicit number field, and under certain further assumptions-as Deligne and Langlands may explain-one can choose a canonical model defined over an abelian extension of a special number field E determined, roughly speaking, by G and the complex structure on X. One might expect that the Hasse-Weil �-function of a canonical model is a pro­ duct of £-functions of the sort Langlands associates to automorphic representa­ tions of G(A). This turns out to be false (see the Introduction to [22]), but it is suggestive. The first result of this kind is due to Eichler, who showed that when G = GL2(Q) and F= F0(N), then F\X has a model over Q and its Hasse-Weil �-function is, as far as all but a finite number of factors in its Euler product are concerned, a product of £-functions defined in this case by Heeke. This result was extended by others, notably Shimura, Kuga, Ihara, Deligne, and Langlands, to include : (a) other r in this G, (b) other G, (c) �-functions associated to nontrivial locally constant sheaves, and finally (d) factors of the �-function corresponding to primes where the variety behaves badly. With one exception-some unpublished work of Shimura-all this work is concerned with X of dimension one. (Refer-for a sampling-to [11], [28], [14], [6], and [19].) As a consequence of these results one had a generalization of Ramanujan's con­ jecture, applying the result of Deligne on the roots of the �-function of varieties over finite fields. A further consequence was a functional equation for and analytic continuation of the Hasse-Weil �-function concerned, which turned out-excepting again Shimura's example-to be a �-function associated in a particularly simple way to GL2 or a quaternion algebra. Over the past several years, Langlands has attacked the problem of varieties of dimension > I , and what Milne and I are going to discuss in our lectures is the simplest case he deals with. AMS (MOS) subject classifications (1 970). Primary 1 0D20 ; Secondary 14025, 22E55 . © 1 9 7 9 , American Mathematical Society

141

142

W.

CASSELMAN

To be more precise, let :

F = a totally real field of degree, say, n ; oF = integers in F; B = a quaternion algebra over F;

o8 = a maximal order in B. We recall that this means simply that B is an algebra of dimension four over F such that B ® F = M2(F ), where F is an algebraic closure of F. Since over any locally compact field there exists a unique quaternion division algebra, if v is any valuation of F then B. = B ®F F. is isomorphic either to M2(F.) or to thi s unique division algebra, depending on whether or not B. has 0-divisors. The algebra B is said to be split at v in the first case, ramified in the second. It is in fact split at all but a finite number of valuations ; quadratic reciprocity says this number is even, and another classical result says that for each even set of valuations there is a unique quaternion algebra ramified at exactly those valuations. Thus B itself is a division algebra if and only if it is ramified somewhere. If v is nonarchimedean, the closure of o8 in B. is a maximal compact subring of B.-unique if B is ramified at v, otherwise only unique up to conjugacy by an element of B; . Let G be the algebraic group over Z defined by the multiplicative group of o8 • Thus for any ring R, G(R) = ( o8 ® R)x . In particular, G(Q) � Bx (canonically) and G(R) � GL2(R)1 x (Hx)J (noncanonically), where I is the set of real valua­ tions of F where B is split, J those where it is ramified. For every rational finite prime p over which B does not ramify,

where the product is over all primes p of F dividing p. Of course the simplest case is B = M2(F), but that is unfortunately the case we will not allow-i.e., from now on we assume B to be a division algebra. Furthermore, we will assume B totally indefinite at the real primes ofF-i.e . , that J = 0 . Langlands himself does not make these assumptions, but acknowledges gaps in the argument unless they hold. Let Ll be a finite set of rational primes containing those over which either F or B ramifies, and let K1 be a compact open subgroup of G(Z1) of the form K4 • f h�<� G(Zp), where K"' is a compact open subgroup of Ti pEd G(Zp). Let Z be the center of G, and ZK = (Z(A1) n K1) · Z(R). Consider ex as embedded in G4(R) : a + b .V - 1

�-+

(ab - ab)

and let KR be the image of (Cx)I in G(R). (This is the connected component of the K8 used before.) Then G(R)/K8 is a product of n copies of C - R. Let K be KR · K1. The set KS(C) = G(Q)\G(A)/K is (as will be explained later)

the union of a finite number of compact complex analytic spaces of dimension n ; it will be nonsingular if (as we assume from now on) K1 is small enough. Milne (in his lectures at this Institute) will show that this space is the set of C­ valued points on a certain moduli scheme KS which is defined, smooth, and proper over Spec Z[l/d], where d is the product of primes in Ll. He will also discuss the

143

THE HASSE-WElL �-FUNCTION

structure of its points over finite fields. What I will do is to identify its Hasse-Weil �-function as one of Langlands' ; the basic idea in doing this will be to use the Selberg trace formula to calculate p-factors of Langlands' L-function, and Milne's results and an unorthodox application of the trace formula to calculate p-factors of Hasse-Weil-both for p not dividing .:::1 . {This will be a lot of trouble, and I should explain at the outset that, although when KS has dimension one it is possible to use a congruence relation to obtain the final result, this will not suffice in general.) One may then apply Deligne's results on the roots of the Frobenius to get a gen­ eralization of Ramanujan-Petersson ; however, results about a functional equation for Hasse-Weil are incomplete. In everything both Milne and I do we are essentially reporting on Langlands' work. The result we discuss is only a special case of a more complicated result of his involving subgroups of B x . We avoid, in treating this special case, problems of what he calls "L-indistinguishability," which perhaps he himself will talk about. His result is more general in other ways, too ; I include remarks on this in the last section, where I have collected together a number of substantial parenthetical remarks. My main reference is the summary [21] ; complete statement and proofs are in [22]. Also relevant are Langlands' talk in the Hilbert problems Symposium [20} and his talks at this Institute. 1. Cohomology of KS(C) and representations. 1 . 1 . Let v be the reduced norm : G --> Gm . Since B is a division algebra, the coset space G(Q)\G l (A) is compact, where G1(A) is the kernel of the modulus homomor­ phism I v I : x --> I v (x) 1 . Since the image of Gconn(R), the connected compact of G(R), under l v l is all of Rvos, the set G(Q)\G(A)/Gconn(R) is compact as well. Since K1 is open in G(A1), the set G(Q)\G(A)/Gconn(R)K1 is finite. Therefore there exists a finite set !!£ of elements of G(A) such that G(A) = U G(Q) xGconn(R)K1 (x E !!£). (Strong approximation gives one a better parametrization of !!£, but we won't need that ; see [7].) 1 . 1 . 1 . LEMMA. For any x E G(A), the space G(Q)\G(Q)xGconn(R)K1jK1 as a Gconn(R)-space is isomorphic to Fx\Gconn(R), where Fx is the image in Gconn(R) of G(Q) n Gconn(R) . xK1x-l.

This is because Gconn(R) certainly acts transitively on this space and isotropy subgroup of the coset G(Q)xK1. Let Yl' be the upper half-plane in C. As a consequence of the above :

Fx

is the

1 . 1 .2. PROPOSITION. The Gconn(R)-space G(Q)\G(A)/K1 is isomorphic to a disjoint union of spaces Fx\Gconn(R) (x E !!£). The variety KS(C) is the disjoint union of the Fx\Yf'n.

The same argument as that used to prove Lemma 2 . 1 of [19] may be applied to show that if K1 is only small enough, each rx acts freely and KS(C) is nonsingular.

We assume thisfrom now on. 1 .2. Let do be the space of automorphic forms on G(Q)Z(R)\G(A). It is a direct

sum EBn- of irreducible, admissible, unitary representations of G(A) (an abuse of language since not G(R), but only its Lie algebra g, acts). If g is the Lie algebra of

W.

1 44

CASSELMAN

G (R) = G(R)/Z(R), then in fact the representation on do factors through g. Let t be the Lie algebra of KR KR/KR n ZR. =

1 . 2. 1 . PROPOSITION. The de Rham cohomology H* (KS(C), C) is naturally isomor­ phic to EBH*(g, f . ?roo) ® n-f! .

The cohomology is the relative Lie algebra cohomology. The sum is over all constituents n- of do (with multiplicity if necessary), which one factors as n- = ?roo ® n-1 where ?roo is an irreducible admissible representation of G(R) (more abuse of language as before) and n-1 one of G(A1). PROOF SKETCH. Let d{ft be the subspace of do of functions fixed by elements of K1. Since g commutes with K1, this is a representation of g which is clearly iso­ morphic to EB ?r00 ® n-f t . Thus the proposition amounts to the claim that H* (KS(C), C) is isomorphic to H*(g, f , d{ft). Now from 1 . 1 .2 it follows that d{ft is the direct sum of subspaces d{{{ (x e El"), where each d{{{ is the space d(f'"\(jconn(R)) of KR-finite, Z(g)-finite functions on f'"\(jconn(R) (!'" is the image of r" in G). Thus the proof of 1 . 2. 1 reduces to : 1 .2.2. LEMMA .

There exists a natural isomorphism : H* (f'"\ G/K, C)

�

H* (g, f , d(f'"\ G)).

This sort of thing holds in fact for any semisimple Lie group G , cocompact f' c G and maximal compact K. To see it : the cohomology of X = f'"\ G/ K is that ofthe de Rham complex, whose nth term is the space of coo m-forms on X. Now the projection f'"\ G -+ X is a prin­ cipal bundle, and the bundle ofm-forms is associated to it and the K-space .Am(gf t)t'. Therefore coo m-forms correspond to certain c oo functions from f'"\G to _A m(gj t)t', which by means of an obvious duality may be thought of as K-linear maps from _Am(g/k) to Coo(f'"\G). In short, the de Rham complex on X may be identified with a complex whose mth term is Hom1(.A m(g/ f), Coo(f' "\G)). But this is the mth term of the complex by which H*(g, t , Coo(f'"\G)) is calculated, and it turns out (by an explicit calculation) that the differentials are the same. Therefore 1 .2.2 is true with d(f'"\ G) replaced by Coo(f'\G). To get the final step, roughly, one recalls that ac­ cording to Hodge theory cohomology classes may be represented uniquely by harmonic classes, and observes that these lift in the above process to elements of d. (See Chapter IV of [3] for details on this and other points.) 1 .3. Although the number of representations occurring in the sum in 1 .2. 1 is infinite, all but a finite number of terms vanish. To be more precise, we must say more about the relative Lie algebra cohomology of admissible representations. First of all, since G(R) � P G4(R) 1 (notation as in the introduction), each ?roo factors as ®n-oo," where each ?roo , , is an admissible representation of PG4(R). One has an easy Kiinneth formula : so that the problem for G(R) is reduced to one for PG4(R). (Note that the Lie algebra of PG4 is the same as that of S4.) For this : let C be the Casimir element in U(�l2), and recall that it lies in the center of U(�l2) and is centralized by the maxi-

145

THE HASSE-WEIL �-FUNCTION

mal compact 0(2), hence acts as a scalar on any irreducible admissible representa­ tion of PGLz(R). 1 . 3. 1 . LEMMA. Let (n-, V) be an irreducible, admissible, unitary representation of PGLz(R). Then

if n- (C)

=f. 0,

� Hom,./A* (�l2/�o2), V) if n- (C) = 0.

The idea of the proof here is that if n- is unitary one can put a natural inner pro­ duct on the complex Hom,02(A * (�1/�o2), V) , such that n-(C) = dd* + d*d where d* is the adjoint of d. (This is an observation good for all semisimple groups due to Kuga.) The lemma follows immediately. For PGL2{R), there are only three irreducible admissible representations n- with n-(C) = 0 : (a) the trivial representation C; (b) the character sgn(det(g) ) ; (c) a single discrete series representation n-0, which may be identified with the quotient of the 0{2)-finite functions on Pl (R) by the constant functions. Lemma 1 . 5 and an easy calculation concerning the restriction of these to 0(2) give : 1 . 3.2. COROLLARY. If 7r is an irreducible PGL2{R) other than one of these then H * (n-) = (a) when 7r is C or sgn {det), Hm(n-) � C, � 0, � c, (b) when 7r = n-0,

Hm(7r) � 0,

� C + C, � 0,

admissible unitary representation of H * (�12, �o2, n-) = 0. For these : m

= 0,

m m

=

=

m m m

I, 2; = 0, = =

I, 2.

The assumption of unitarity is not in fact necessary (see [3]). Hence if the irreducible admissible representation n-"' of the original G(R) is cohomologically nontrivial, it must be of the form ®n-"'·" where each n-"'· ' is one of the above three representations, and its cohomology may be calculated accordingly. If n- is an irreducible admissible representation of PGLz(R), set

m(n-)

if n- is cohomologically trivial, if n- is either C or sgn(det), I - 1 if n- is n-0.

= 0 = =

If n-"' = ® n-"'·' is an irreducible admissible representation of G(R) define m(n-"') to be f1 m(n-oo,J. If 7r = n-"' ® 7r 1 is an irreducible admissible representation of G(A) where n-"' is trivial on Z(R), then define m(n-, K) = m(n-"') dim n-ft. Thus the size of m (n-, K) is just dim n-ft and its sign reflects the parity of its contribution to co­ homology. As a final remark let me point out that by applying the strong approximation · theorem one can show that if n- = ®n-. (v over valuations of F) is an irreducible admissible automorphic representation of G(A) and n-. is one-dimensional at a ·

146

W. CASSELMAN

place where B is split, then n. is one-dimensional everywhere. In particular, if one factor of n"" is one-dimensional so are all. As a consequence of this one recovers a result of Matsushima-Shimura [23] which says that the interesting cohomology of KS(C) occurs in the middle dimension. 2. The main theorem and some consequences. 2. I . I recall the L-group attached to G. First of all, if G is considered in the most straightforward way as a group over F -i.e., so that for any extension F' of F, G(F') � (B ® F F')x -then its L-group is just the direct product of LGO = GL2(C) and Gal (F/F). This is because it is an inner twisting of GL2(F). Since G as a group over Q is obtained from this one by restriction of scalars, its L-group LGO is the one in some sense induced from this one : it is the semidirect product of Gal(Q/Q) by LG o = GL2 (C)l, where I is the set of embeddings of F into Q and Gal acts on LGO by permutation of factors. (Since Q may be identified with a subfield of C, this I is essentially the same as before.) Without any serious loss for our purposes we may (and will) replace Gal (Q/Q) by Gal (Fnorm/F), where Fnorm is the smallest extension of F normal over Q. It may be helpful if I point out that it is unramified over Q whenever F is. I recall also that if p is a prime of Q unramified in F and f/J is a Fro­ benius in Gal(Fno rm/F) o ver p, then the local L-group L GQp = LGp may be identified with the subgroup of LGQ whose image in Gal(Fnorm/F) is the cyclic subgroup generated by f/J. Thus one has an exact sequence I ...... GL2(C)l ...... LGp ...... (f/J) ...... I . (Refer to [2] for everything about £-groups.) In order to define £-functions associated to automorphic representations, one must also introduce finite-dimensional representations of LG. There is only one (for each G) that we will be concerned with, and it is defined as follows : the space of this representation p is a tensor product of copies of C 2, one for each element of I; the group LGO = GL2(C)I acts through the standard representation on each factor, and Gal(Q/Q) (or Gal(Fnorm!Q), it makes no difference) acts by permuting the factors. This does indeed define a representation of LG, since it is a semidirect pro­ duct of these two groups. The dimension of this representation is 2n. When F = Q, for example, LG = GLz(C) and p is just the standard representation itself. This particular choice of p may seem arbitrary, but in fact it Wl;!S motivated ori­ ginally (in Langlands' formulation) by general considerations about Shimura varie­ ties (one should refer to the Introduction of [22] for a discussion of this point). 2.2. If X is any smooth, proper scheme over Z[I/d] for some d ;:;; I and p is a prime not dividing d, then the zeta-function of X over Fp is defined to be the func­ tion Zp(s, X), rational in p- s , such that (at least formally) log Zp(s, X)

� oo

=

I

mp m'

(jj:X(Fpm) ) .

As a consequence of the etale theory, this agrees with the definition in terms of /-adic cohomology : 2-d imX det(I - f/J · r'h• cx, Q1) Zp(s, X) = IT = i O

where f/J is the geometric Frobenius. At least as far as factors other than those divid­ ing d are concerned, its Hasse-Weil zeta-function Z(s, X) is the product of these Zp(s, X).

THE HASSE-WElL C-FUNCTION

147

The main theorem of Langlands is :

2.2. 1 . THEOREM. Up to prime factors in Ll, the Hasse-Wei/ zeta-function of KS agrees with n L(s - n/2, 1C, p) m (1<,Kl where the product is over a/I re occurring as constituents of d0. Of course one should consider the rc with multiplicities if necessary. In fact, however, because of the theorem in § 1 6 of [16) relating automorphic forms on G with those for GL2 , together with the result for GL2 in §9 of [16], each rc occurs exactly once. In this product, m(rc, K) = 0 for all but a finite number of rc, since after all the cohomology of KS(C) is finite. What is actually to be proven is a purely local result : for p ¢ Ll, the p-factors of Z(s, KS) and ll L(s - n/2, rc, p) coincide. Now each 1C = @rc. with m(rc, K) i= 0 has the property that rcp for p ¢ L1 is unramified, hence corresponds to an element g(rcp) in the local £-group LGp. The p-factor of L (s - n/2, rc, p) is then det(/ - p(g(rcp) ) jp•-,. 1 2)- 1 . Upon expansion, the theorem reduces to a formal equation (p ¢ L1) : This in turn amounts to an equation of coefficients (for p ¢ Ll, m � 1) : (2. 1) This is the form in which the theorem is actually proven. In these lectures I will give a complete proof only in the simplest case, when F is split over p . 2.3. Before beginning the proof, we give an example and some consequences. The case n = 2 is the first interesting one, in the sense that, as already mentioned, the case n = I is an old result and can be (and has been) done more elementarily by means of a congruence relation. So, suppose for a while that F is quadratic over Q, and consider the possible p-factors occurring in L(s, rc, p). There are two cases : (1) when p = p 1 p2 splits in F and (2) when p = p remains prime. We look at (1) first. In this case G(Qp) � GL2(Fp) x GL2 (Fp2) and an unrami­ fied representation of this must be of the form rc 1 ® rc2 , where each 1C; is an unrami­ fied representation of GL2(Fp,.), hence corresponds to a pair of unramified charac­ ters (a;, {3;) of F� . More precisely, 1C; may be the whole principal series representa­ tion parametrized by the pair when it is irreducible, or the associated one-dimen­ sional character of GLz(Fp,.) when it is not. (By strong approximation, this last happens only when the global representation at hand is also one-dimensional.) In either case, observe that the local L-group LGP is (because p splits) simply the direct product GL2(C) x GLz (C) and that the corresponding element g(rcp) of LGp is a 1 ( p) az(p) f3 1 ( p) • f32(p) . The representation p is simply C 2 ® C 2, so the p-factor of L(s - I , rc, p) is (1 - a 1 (p) az(p)pi -s)- 1 (1 - a 1 ( p) f3z(p)p 1 -s) - 1 ( 1 - f3 1 (p) az(p)pi -s)- 1 (2.2) · (1 - /31 ( P) /32( p)p-•)- 1 .

((

)(

))

148

W.

CASSELMAN

Next case (2). Here np is a single unramified representation of GL2(F), say cor­ responding to the characters (a, f3) of F; Jo; . On the other hand, the local £-group is still a semidirect product of GL2(C) x GL2(C) by the Galois group of order 2, since the Frobenius generates Gal. I leave it as an exercise to verify that the element g(np) of LGP is

f/J

((I0 0I ) (a(p)0 f3(p)0 ))f/J. '

If e 1 , e2 are the standard basis elements of cz then g(np) acts as follows on the space of p (recall that interchanges factors) :

f/J

e1 ® e2 ...... a(p)e2 ® eh e 1 ® e 1 ..... a(p)e 1 ® eh , (p)e ez ® e1 ..... f3 1 ® ez ez ® ez ..... f3(p)ez ® ez . Therefore the p-factor of L(s - 1, n, p) is (2.3) {1 - a(p)p i -s)-1(1 - f3(p)p i -s)- 1 (1 - a(p) f3(p)p2-s)-I. For n > 2, the analogous formulae can be rather complicated. Incidentally, the unpublished result of Shimura we have referred to before is precisely that the p­ factors are of the form (2.2) and (2.3) for certain quadratic fields F. For F = Q (n = 1), one consequence of the expression of the Hasse-Weil zeta­ function as an £-function associated to automorphic forms is that it has a func­ tional equation and analytic continuation. For n > 1 , this consequence is not automatic. For n = 2 it is already a difficult question only very recently settled by Asai [1] under some probably unnecessary restriction on F. Nothing seems to be known for n > 2. What is easier is to obtain a functional equation for the zeta­ function of KS over F itself, at least for certain small degrees. This depends on a generalization of an idea of Rankin due to Jacquet and Shimura (see, for example, [15] and [29]). 3. The analytical trace formula. In this section I will develop a formula for the right side of equation (2. 1 ), in the next one for the left, and in §5 the two will be compared. Beginning in §4 I will assume (although Langlands certainly does not) that p is split in F. This makes things much simpler. The argument will still involve several important ideas, but will avoid what is at once the most complicated and intriguing point of all, the use of paths in a certain Bruhat-Tits building. I hope that what remains is still of some interest. Langlands will presumably say something on this matter in his own talks (see [17]). 3. 1 . The right side of equation (2. 1), (3. 1 ) I; m(n, K) pm n / 2 trace p(g(np)m), 1r

in �o

may be expressed as the trace of a certain operator R1, for some / E C�{G(A)), acting on d0• Recall that do is a discrete direct sum of irreducible, admissible, uni­ tary representations n = n8 ® nP ® np of G(A), and for any f E C�(G(A) f ZK) one has therefore trace R1 = I; ,.. in .w
1r

I; trace n8(f8) trace nP(fP) trace np(fp)

i n .9/o

149

THE HASSE-WElL C -FUNCTION

if/factors as/8 · fP · /p. Recall that m(1r, K) = m(1r8) dim(1rP) KP. A comparison of (3. 1) with (3.2) suggests choosing the three factors off so that (3.3)(a) for any irreducible admissible representation 11: of G(R)/Z(R) ; (3.3)(b) trace 11:(/P) = dim 11:KP for any irreducible admissible representation 11: of G(A�) ; (3. 3)(c) trace 11:(/�ml) = pmn / 2 trace p(g(11:) m) 11: unramified, =0 otherwise. This is just what will be done. The choice of fP is simple : fP = (meas KP)-1 char KP . That of f�ml is not so explicit but just as simple : according to the Satake isomor­ phism [4] there exists a uniquef�ml E .Yl'(G(Qp). G(Zp)) satisfying (3.3)(c). But the matter of/8 is more complicated. 3.2. If the representation 11: of G(R) factors as (8)11:, according to G(R) � GL2(R)I then m(11:) = Ti m(11:,). Iff = Ti le accordingly then trace 11:(/) = II trace 11:(J,). Thus the problem of finding/8 reduces to the problem of finding f E C�(PGL2(R)) such that (3.4)

trace 11:(/) = m(11:)

for any irreducible admissible representation 11: of PGLz(R). One can do this in several ways-for example directly from the Paley-Wiener theorem as in [10], and also from considerations of orbital integrals (see [26]). But Serge Lang has pointed out an argument which is in some sense more elementary than either of these and which I present.below. From this point I follow Lang's book [18, Chapter VI, §7] rather closely. The connected component of PGL2(R) is PSL2(R). Now the maximal compact of SLz(R) corresponding to the one of GL2 (R) at hand is S0(2) ; let e : S0(2) � ex be the fundamental character e

(cos -cossin (}) = cos (} + .J - 1 . sm. sin (} (}

(}

(}.

The single discrete series representation 11:0 of PGL2(R) which is cohomologically nontrivial has the property that its restriction to S0(2) is a direct sum EB en (In I � 2, n even). No other discrete series representation has e2 in its S0(2)-decomposition. I recall that the Heeke algebra of all compactly supported bi-S0(2)-finite dis­ tributions acts on the space of any admissible representation of SL2(R). If D satisfies Lk, Rkz D = em(k 1 )e-n(k2)D for all kh k2 E K, then for any such representation (11:, V) the operator 11:(D) takes all of V into the em-eigenspace and annihilates all but the en-eigenspace. In the special case m = n and D amounts simply to integration . against e-n on K, 11:(D) is the identity on the en-eigenspace. Therefore, by choosing fe C""(SLz(R)) such that

1 50

W. CASSELMAN

and approximating this D closely enough-i.e., choosing f positive, normalized, and with support close enough to K-one may assume that 11:(j) is also the identity on this eigenspace. Apply this to the case m = n = 2, 11: = 11:0 to obtain /1 e C�(SL2(R)) such that (a) /1 (k 1 gkz) = e-2(k1)e- 2(kz)f(g) for all kh k z e S0(2), g e SLz(R) ; (b) 11:o(f1 ) is the identity on the e 2-eigenspace and 0 on any other S0(2)-eigenspace. Because of the remark above about other discrete series representations of PGL2(R) and e2, one even has (3.5)

trace 11:{jj) = I , 0,

11: = 1l:o, 11: is any discrete series representation of PG�(R) other than the 11:0•

At this point recall what Lang calls the Harish transform of an/ e C�(SL2(R) ) :

Here A is the group of all diagonal matrices. This function H1 turns out to lie in C�(A) and even in C�(A)W, where W is the Weyl group. In fact, / � H1 induces an isomorphism of the space of functionsf e C� (S�(R)) hi-invariant under S0(2) with C�(A)W [18, V, §2]. Thus, one may choose /2 e C� (SL2(R)S0(2)) with H1 1 = H�z. Set / = /1 - f2, so H1 = o. Since a discrete series representation possesses no vectors =1- 0 fixed by S0(2), equation (3.5) holds for f instead of fi.. Furthermore, if x is any character of A and 1l:x is the corresponding principal series representation of SL2(R) then (3.7)

trace 11:x(/) =

J H1(a)x(a) da = 0 A

[18, VII, §3]. Since any principal series representation of PGL2(R) restricts to some 1l:

x

on PSL2(R), the same holds for all principal series representations of PGL2(R). Finally, if 11: is any irreducible finite-dimensional representation of PGL2(R) then because its character and that of some unique discrete series representation add up to a principal series representation, trace 11:(/) = - I 11: = C or sgn(det), other finite-dimensional 11:. =0 Define now fa on PGL2(R) to be -f on PSL2(R) and 0 on the other connected component. Equations (3. 5), (3. 7), and (3.8) yield (3.4) for f = fa · 3.3. To recapitulate : the function I = fa . fP . t�m) E coo(G(A)fZK) has the property that trace R1 ldo = I; m(11:, K)p mn / 2 trace p(g(1l:p) m) . (3. 8)

11: in �o

In §5 the trace will also be expressed by means of the Selberg trace formula. I will recall it here in a rather general form for purposes of contrast a little later : Let

THE HASSE-WElL C-FUNCTION

151

W b e any locally compact group, r a discrete subgroup with F\W compact. The conjugacy class of any T E F Will be closed in '§ (a nice exercise) and SO for any F e c:"'(W) the orbital integral J:J"r\!f' F(g 1 rg) dg is defined and finite, where '§7 is the centralizer of r in '§. If F7 = '§7 n F, then clearly F7\W7 is compact also. Under some mild assumption the operator RF is of trace class on L2(F\W) and -

J

meas{F7\W7) r F(g 1 rg) dg !f' \!f'
= 1:

-

4. The algebraic trace. In this and the next section I assume, as mentioned earlier, that p splits completely in F. 4. 1 . There exists a formula for '.Pi KS(Fpm) remarkably similar to that which the Selberg trace formula yields for the right-hand side of equation (2. 1). The set KSCFp) is partitioned naturally by the equivalence relation of isogeny­ i.e., two points lie in the same class when the abelian varieties-plus-structure that they parametrize are isogenous. The isogeny classes are of two types : (1) Those associated to systems (F',{q;}), where F' is a totally imaginary quadratic extension of F which splits at some prime over p and the q; are certain ideals of F'-if !J I > · · · p, are the primes of F which split in F' then q; is one of the two prime factors of !J; . Two systems (F{, {qu}) and (F2, {q2, ;}) determine the same isogeny class if and only if F{ = F2. and {qi.;}, {q2,;} are either equal or conjugate. (2) A single isogeny class associated to the quaternion division algebra D/F which is ramified precisely (a) where B/F is, (b) at all primes of F over p, and (c) at all real valuations of F. This latter case may be thought of as an amalgamation of all the quadratic im­ aginary F'/F which do not split over p. If Y = (F', {q;}) or D, the class of points in KS(Fp) associated to it will be denoted KS( Y). It is stable under the Frobenius ; one may describe rather explicitly its struc­ ture together with this action. (For the parametrization of isogeny classes as well as the structure of each class, refer to Milne's lectures.) To each Y is associated : (I) A certain algebraic group H = Hy defined over Q. If Y = (F', {q;}) then H is (so to speak) the multiplicative group ofF', while if Y = D it is the multiplicative group of D. (2) A certain algebraic group Gy = G defined over Qp . First of all, for each prime p of F over p one has GP = the multiplicative group of F; � (F;) z if Y = (F', { q;}) and p splits in F', = the multiplicative group of D P otherwisei.e., if Y = (F', {q;}) and p does not split in F' or if Y = D. The group (j is n Gpo (3) A coset space X = n Xp , where Xp = Gp(Fp)/Gp(Op). Pn each XP define the transformation (/JP = multiplication by a uniformizing elements of q if Y = (F', { g;}), p splits in F', and q is the q; over p, ,

•

=

multiplication by a uniformizing element of the prime ideal of DP otherwise.

1 52

W. CASSELMAN

Expressing GP as (F;)2 in the first case, with the first factor corresponding to p, lbp is a little more explicitly multiplication by (p, 1). Let X = Xy = TI Xp, lb = l/>y = n lbp · Let H(Q) = H(Q)/H(Q) n ZK, G(Af) (an abuse of language) G(A�) X =

G(Qp)/zK n A'J.

In every case, observe : 4. 1 . 1 . LEMMA . (a) The algebraic group Z is canonically embedded in H ; (b) There exists a canonical class of equivalent embeddings of H(Q) into G(A�)

G(Qp). rendering H(Q) as a discrete subgroup of G(A1).

x

Only the second needs explaining. There clearly exist natural embeddings of

H(Q) into G(A�) and into G(Q p) all equivalent up to inner automorphism. The group H(R)/Z(R) is compact in all cases (no accident) so that since H(Q) is discrete in H(A) = H(A1) x H(R) the group H(Q) is discrete in H(A1)/ZK n A'J, hence in G(A1). Langlands' main result on the structure of KS( Y) is that there is a canonical class ofbijections between its points and the double coset space .

H(Q) \ G(A�)

X

G(Qp)/KP

X

G(Zp)

�

�

H(Q) \ G(AJ)/KP X G(Zp) H(Q) \ G(A�) X X/KP.

The Frobenius on KS( Y) corresponds to (/) acting on this coset space through its action on X. 4.2. The number of points of KS( Y) rational over Fpm is of course the number of fixed points of the mth power of the Frobenius. To express this conveniently requires a digression of some generality. Suppose � to be any locally profinite group, F a discrete subgroup. Then � and hence also C�(�) acts on the space C"'(F\�) through the right regular repre­ sentation. Explicitly, for every F E C�(�). f E C"'(F\�). RFf(x)

= =

sw F(g)f(xg) dg sw F(x- 1y)f(y) dy JF\W KF(x, y)f(y) dy =

where KF(x, y) = .E rF(x 1 ry). The operator RF (or, sometimes, F itself) will be said to possess a formal trace if KF(x, x) has compact support on F\�. and this trace is then defined to be fF\w KF(x, x) dx. One can presumably relate this to other notions of trace, but all that is important here is that the number of fixed points of a power of the Frobenius can be expressed as such a trace and that one has a formula for it. For each T E F, let �r be the centralizer of T in � and F7 = F n �r· If F lies in C�(�), then its orbital integral fw,,w F(g-1rg) dg is defined in many different circumstances, but the most elementary hypothesis that guarantees this is that F(g-1 rg) has compact support on �r\� ; this in turn is assured by the assumption that the conjugacy class of r is closed in �. 4.2. 1 . PROPOSITION. Let F be in C�(�). Suppose that the T E F such that F(g-1rg) #- 0 for some g E � lie in only a finite number of conjugacy classes in F, and that for -

1 53

THE HASSE-WElL �-FUNCTION

each such r the space F7\<§7 is compact and the conjugacy class of r in <§ is closed. Then F has a formal trace which is given by the formula

where the sum is over all conjugacy classes on r. PROOF. Choose a compact open subgroup '§o £;;; <§ small enough so that r n '§0 = { I } and F is hi-invariant under '§0. Thus r acts freely on '§/'§0• Let £( be a set of representatives, SO that <§ is the disjoint union of the Fx <§o (X E £r).

By one assumption on F, for each x E £( the sum :E rF(x-Irx) may be restricted to a finite number of conjugacy classes in r, hence may be written as :E frl L: rN F(x-Io-1rox) where the first sum is over representatives of conjugacy classes and only a finite number need be taken into account. Because F7\<§7 is com­ pact and the conjugacy class of r is closed in '§, the function F(g- Irg) lies in C';'(F7\<§); this implies that the above sum is nontrivial for only a finite subset of £(. Thus F does possess a formal trace, which is given by (meas '§0)

·

I: I: I: F(x- I o- I rox)

xE !!t"

! rl Fr\F

where in fact each sum may be restricted to a finite but arbitrarily large subset. Rearranging this it therefore becomes

4.3. Now take F to be fl(Q), <§ to be G(A1)-both corresponding to some isogeny

class Y. Define

FP

Fp(m)

p�m) p < ml

=

=

=

=

(meas KP)-I char KP, (meas Gp( op) ) - 1 char (
· G( o p) ) ,

TI F�m) , FP . p�m) .

4.3. 1 . LEMMA. The hypotheses of 4.2. 1 are satisfiedfor this choice of r, <§, F with m � I .

=

p<mJ

When Y D , this is clearly so since F\<§ is actually compact. Thus let Y = (F', { q1}). The only serious hypothesis to verify is the finiteness of the number of conjugacy classes with elements r such that F(g- 1 rg) =1= 0 for some g E '§. Now in this case H(Q) (F'Y and G(Qp) = TI Gp(Fp) where GP is (a) (F�Y or (b) n; . In either case p�ml (g- 1rg) F(r), and Fj;"l (r) =1= 0 if and only if (a) at each p,. where F' splits r has order m at q,. and is a unit at q,. ; (b) where F' does not split r is a generator of pj!j. Furthermore, in order that FP(g-1rg) =1= 0 for some g E G(A) it is necessary and sufficient that r be a unit at all primes not dividing p. Thus if p < m l (g- I rg) =1= 0 for some g, the order of r is specified at all primes of F'. Any two such r differ by a unit. But the units of F', modulo ZK, are a finite set. The function p < mJ is important because of: =

=

=

1 54

W. CASSELMAN

4.3.2. PROPOSITION. The formal trace of F(ml is the number of fixed points of the mth power of the Frobenius on xS( Y). I leave this as an easy exercise.

5. The comparison.

5. 1 . From §3 one sees that the right-hand side of equation (2. 1) is

(5. 1 )

I: meas(F7\�7)
J

'�·'"

F(g 1 rg) dg -

where F = G(Q)/G(Q) n Zx, � G(A)/Zx , F f11 fP ff'l . Let rp : G( A) -+ � be the canonical projection. It is easy to see that rp(G7(A)}\�r is always compact, so that one may replace �r by rp(G7(A)} in the above formula. But then one may note that rp(G7(A)}\� � GrCA)\G(A). Using the factorization of G(A) each term may be written as the product of =

=

·

•

meas (G7(Q)Zx\Gr(A)} ,

(5.2) (a)

J Gr (Af'J IG (Ap fP(g-l rPg) dg,

(5.2) (b)

JG,(Qp) \G (Qp) J Cml (g-I rpg) dg, JG7 (11) \G (R) f��cg-1r,g) dg. p

(5.2) (c) (5.2) (d)

From §4 one sees that #xS(Fp ) may be expressed as ..

(5.3) where : Y ranges over all systems (F ', {q;}), identifying a system and its conjugate, and the single D ; r ranges over all conjugacy classes of Hy(Q) ; � (}y(At) G(A�) .x Gy( Qp) ; p (m) FP . p�m) as at the end of §4. Just as above, each term here factors as the product of =

=

=

(5.4)(a) (5.4)(b) (5.4)(c)

meas {H7(Q)(Zx n

A j)\G7(A �) GrCQp)),

J Gr <AP ,G <AP FP(g-lrPg) dg, JG7(Qp) IG (Qp) p <ml(g-I rpg) dg. _

p

The proof of the main theorem reduces to a comparison of measures and orbital integrals. 5.2. Even the comparison of measures is not trivial. For the moment let k be an arbitrary local field, rjJ an additive character of k. If H is any finite-dimensional semisimple algebra over k, Tr its reduced trace, then rjJ Tr is an additive character of H. There exists on H a unique measure dx self-dual under the Fourier transform a

THE HASSE-WElL '-FUNCTION

1 55

determined by this character, and this in turn gives rise to an invariant measure dxx = dxjJxJH On H x . If now k is any global field, cp = TI ¢ v a character of Ak/k, and H a semisimple algebra over k, one obtains thus measures on all the local groups H.x as well as on Hx(Ak), which is called the Tamagawa measure determined by cp (§ 1 5 of [16]). Apply this construction in turn to the field F itself, quadratic extensions of F, and quaternion algebras over F (including M2(F)). I will always assume such measures chosen from now on. One classical result is that the global measure of Hx(A)IA'F · Hx(F) is independent of the quaternion algebra H (§ 1 6 of [16], or the lectures [12] of Gelbart and Jacquet). I will also need a comparison of local measures : 5.2. 1 . LEMMA. Let k be any nonarchimedean local field, H the unique quaternion division algebra over k. Then meas ofi = (q - 1)- 1 meas GL2(ok).

PROOF. Changing cp only multiplies both measures by the same constant; choose

cp to be of conductor ok. Letfbe the characteristic function of M2(ok). Then

/(x) =

J

f(y)cf;(Tr(xy) ) dy

M2 (k)

{1

x E M2(o1), = measMz(ok) o: x ¢ Mz(o,), = (meas M2(o1)) · f(x). Therefore meas M2(o1) = 1 . The group GL2(o1) is open in M2(o1) and since x E M2(o1) lies in GL2(o1) if and only if x mod Pr is nonsingular, meas GL2(o1) = :li GL2(Fq)/:li Mz(Fq) = ( 1 - 1/q) Z ( 1 + 1/q). Now let f be the characteristic function of oH · Then /(x) =

J

OH

cf;(Tr(xy)) dy

= meas oH

{1,

0,

X E P/i,

x ¢ P!i ,

or J = (meas oH)(characteristic function of P.ii) .

The Fourier transform of this in turn is J = (meas oH)(meas(P/i))f Since meas(P/i) = q 2 · meas oH, meas oH = q- 1 . Reasoning as above, meas ofi = (1 - 1/q )( l + 1 / q )( l /q ) . The lemma follows. One can similarly compare measures on GL2(R) and ax, but that will prove to be unnecessary. 5.2. Again for a while let k be any local field of characteristic 0, '1l either GL2(k), the multiplicative group Hx(k) of the quaternion division algebra over k, PGLz(k),

1 56

W.

CASSELMAN

or n x ;kx . For any X E � define D(x) = I a - ,8 1 2f ia,81 of the eigenvalues of X (over an algebraic closure of k) are a, ,8. If T is any torus of �' then for every regular t e T andfe C;"(�) define OT(f, t) =

JT\<§ f(g-l tg) dg.

The function F(t) = OT(J, t) satisfies F is smooth on the regular elements of T, (5.5) (a) Dl 1 2 F is locally bounded, (5.5) (b) Fhas compact support on T. (5.5) (c) If 1r: is an irreducible admissible representation of � then the character ch" of 1r: also satisfies (5.5) (a)-(b). For any f e C;"(�), f(g) dg = 1 .E D(t)OT( J, t) dt, 2 !Tl T where the sum is over all conjugacy classes of tori in � ' and if 1r: is an irreducible admissible representation then trace 1r: (f) = 21 I; D(t) ch"(t)OT(J, t) dt. (5.6) !Tl T

J

J

J

(For all this see §7 of [16], and for more about orbital integrals see [12] and [26].) 5.2. 1 . LEMMA. Suppose that � = n x ' F a conjugation-invariant function on �reg whose restriction to any torus T satisfies (5.5) (a)-(c). Jf 1 D(t) cha(t)F(t) dt = 0 2 I; !Tl T

J

for all irreducible admissible representations of� then F

=

0.

Fix a torus T. The set of elements of � conjugate to regular elements of Tis open. If f e C;"(T'•g), there exists h e C;"(�reg) with support in this set and such that OT(h) = f Then

J<§ h(g)F(g) dg = JT D(t)OT(h, t)F(t) dt = J D(t)f(t)F(t) dt. T

But the characters of irreducible representations are complete in the space of cen­ tral functions on �' so that/ = .E ia · cha (if k is nonarchimedean one can reduce this question to one about finite groups) and hence by hypothesis the above inte­ gral is 0. In other words, the integral of D · F against any f e C;"(T'•g) is 0, which implies that F itself is 0. One may (and I will) identify conjugacy classes in nx with elliptic and central conjugacy classes in GL2(k). There exists (by § 1 5 of [16] ; see also [12]) a bijective correspondence q +-+ 1r: between (a) irreducible, smooth (hence finite-dimensional) representations q of nx and (b) irreducible, admissible representations 1r: of GL2(k), space-integrable modulo Z, such that

THE HASSE-WElL '-FUNCTION

1 57

ch.. (t) = - ch,.(t)

(5.7)

for all regular elliptic elements t. Representations of nx;kx correspond to repre­ sentations of PGL2(k). When k = R, the trivial representation of Hx corresponds to the single discrete series representation no of PGL2 . For nonarchimedean k, the one-dimensional representations of Hx correspond to the Steinberg (or special) representations and all others correspond to absolutely cuspidal representations of GL2 . 5.2.2. PROPOSITION. Let fn E C�(PGL2(R)) be as in §3. Then for semisimple x E PGL2(R) = '§,

J

x hyperbolic, 0, = (meas ex jRx ) - 1 , x elliptic, = - (meas Hx /Rx)- 1 , x = l. PROOF. The case of hyperbolic x fell out in the very construction offn · Recall thatfn has the property '!lr\'!1

(g- 1 xg) dg =

trace n Cfn) = 1, n = 1 or sgn(det), = - 1, otherwise. = 0, Consider now the constant function hR on nx ;Rx with the value (meas nx;Rx )- 1 . It satisfies trace a(hn)

1, (J = 1 , = 0, otherwise. Thus by (5. 7) and remarks just afterwards trace n(/8) = - trace a(h8) whenever n and a correspond to one another. This and equation (5.6) now yield

J

cx ;R x

D(t)

ch,.(t) dt

J

cx\PGL2 (R)

=

whenever n and a correspond. version on ex IRx),

J

C x\ PGL2 (R)

=

By

fn(g- 1 tg) dg =

J

-

f8(g- 1 tg) dg

J

cx;Rx

D(t)ch.-(t) dt

J

cx\ H x

h8(g- I tg) dt

(5.7) and 5.2. 1 (or in this case just Fourier in­ cx\H x

h8(g- 1 tg) dg =

(meas cx ;Rx)- 1

for all t # ± l E Cx/Rx . Since the orbital integrals offn vanish on hyperbolic elements, /8 and - h8 corres­ pond to one another in the sense of [12] (the section on "matching orbital in­ tegrals"). Hence 5.3. Now let k be nonarchimedean and local, and for the moment let G = GL2(k),

I 58

W. CASSELMAN

K = GL2{ ok)· For each m � 0 IetJ< m > be the unique function in the Heeke algebra Jlt'(G, K) defined according to the Satake isomorphism by trace 7C(J< m > ) = q m 12(am + {J m) whenever 1C = 1C(a, {3) is the unramified principal series represen­ tation associated to the characters x ...... (aord(xl , {Jord (xl) of kx . One can exhibitJ < m > rather explicitly. For every i, j E Z recall the Heeke operator T(pi, W) =

(meas K)- 1 char K

( )K 7J ;

7J

j

where 7J is a generator of p. For m � 0 recall also T(pm) = !: T(pi, w) (i, j � 0, = m). There are the more or less classical Heeke operators and satisfy the relation T(p)T(pm) = T(p m+l ) + qT(p, p) T(pn- 1) which may be written formally as (I - T(p) X + q T(p, p)X2) - 1 = I; T(pm)Xm . (5.7) i +j

5.3. 1 . LEMMA {IHARA). One has

J

=

(I

q 1 t2a X) - 1 + (I

2 . I, J
= T(p), J < m> = T(pm)

_

q T(p, p) T{pm-2)

(m � 2).

PROOF. The case m = 0 is clear. And it is well known that trace 1C(T(p)) = q1 12(a + {3) if 1C = 1C(a, {3), which gives the case m = 1 . Formally, then T(p) +-+ q1 12(a + {3) and, also elementary, T(p, p) +-+ a{J. Now _

_

q 1 t2 [J X) - 1 = I; q m 12(am + [Jm) Xm 00

0

+-+

I;J < m > xm

but this expression also equals (I - q1 12a X) + (I - q 1 12[J X) +-+

I (1 - q 1 t2aX) (I - q1 12[JX) The proposition follows from this and (5.2).

2 - T(p) X - T(p) X + q T(p, p) X 2 "

5.3.2. COROLLARY. If

then

f < m > (x)

= =

let

- (meas K)-1(q - I ) , 0,

m = 2/, otherwise.

Continue to let H be the quaternion division algebra over k. From each m � 0 h <m> =

(meas oH)- 1 char(p7j

- p7}+1 ).

5.3.3. PROPOSITION. Let x be a semisimple element of G, m � 1. Then :

(a) Ifx is hyperbolic,

J G,\G J <m> (g-Ixg) dg (b) If x is elliptic

=

=

7J m

( ) e ) mod

1 . if X = I or 0 otherwise.

7Jm

(K n A),

1 59

THE HASSE-WElL �-FUNCTION

(c) Jfx is central, pm>(x) = - h <m>(x). PROOF. Recall (say from [4]) the relationship between the Satake isomorphism and orbital integrals : for fe :Yt'(G, K) /(a) = D(a)l 1 2

J

JA\G f(g-lag) dg,

trace 11:x
By definition of J<m> , then, j<ml (a)

=

qm l 2 ,

a=

( 1 ) or (I r;m)• rr

otherwise.

=0

Hence (a). Claim (c) is immediate from 5.3.2 and 5.2. 1 . Only (b) is difficult. To begin, consider equation (5.6) as 11: ranges over all essentially square-integrable irreducible representations of GL2 : trace 11:(/<m>)

=

JA D(a)ch (a) da JA\G pm> (g-lag) dg + 1 L; J D(t)ch"(t) dt J J <m> (g-ltg) dg 2 {T) T T\G 1 2

n

where now T ranges over all anisotropic tori. Similarly consider (5.6) as a ranges over all irreducible admissible representations of nx :

J

J

trace a(h <m>) = 21 L; D(t) cha(t) dt h<m>(g-ltg) dg. T\G {T} T Applying 5.2. 1 and equation (5. 7), it thus suffices to prove that

J

J

- trace 71:(j'm> ) + 21 A D(a)ch"(a) da A pm> (g-lag) dg = trace a(h <m>) \G whenever a and 11: correspond. For this : (1) no 11: occurring here possesses K-fixed vectors # 0, so trace 11:(j<m>) = 0 always. (2) If 11: is absolutely cuspidal then chn(a) = 0 unless a E A(o k) (first observed in Lemma 6.4 of [19] ; see also [8] and [5]). According to case (a) already done, OA(f) = 0 on A(o), so the whole left-hand side vanishes. But in this case a is not one-dimensional, hence not trivial on o}j, so that the right-hand side vanishes also. (3) Suppose 11: is special, corresponding to the character (J = X 0 NM of nx . Thus a(h Cml) x(r;m) on the one hand and on the other =

1 2

·

sA D(a)ch"(a) da s A\G J<m>(g-1ag) dg

also, by case (a) again.

=

x(r;m)

5.4. Return to earlier notation. It will now turn out that there is a bijective cor­ respondence between the nonzero terms of the two sums (5. 1) and (5.3), and that corresponding terms agree. This will conclude our p roof.

160

W. CASSELMAN

According to 5.2.2 the orbital integral of fn in (5.2) is 0 unless rn is elliptic or central, so I will assume that to be the case from now on. Suppose first that r E G(Q) is central. The term in (5. 1) corresponding to it is the product of meas(G(Q)ZK\ G( A)), (5.8) (a) (5. 8) (b) J�ml(rp) = ( - l) n (meas Dx(Zp))- 1 , (5.8) (c) fP(rP), (5.8) (d) firn) = ( - l)n (meas Dx(R)/Rx)- 1 . There is exactly one term in (5.7) corresponding to r. among the terms indexed by Y = D. It is the product of meas (Dx(Q)(ZK n Aj)\Dx(AJ)), (5.9) (a) F�ml(rp) = (meas Dx(Zp))- 1 , (5.9) (b) FP(rP). (5.9) (c) These products match since (5.8)(c) and (5.9)(c) are trivially equal and meas (G(Q)ZK\G(A))

= =

meas(Dx(Q)ZK\ Dx(A)) meas(Dx(Q)(ZK n Aj)\Dx(AJ)) meas(Z(R)\Dx(R)).

Now suppose r E G(Q) such that rn is elliptic. Thus F' = F(r) is an imaginary quadratic extension. If this extension splits at no prime of F over p, let Y = Yr = D. Otherwise, Y = Yr is one of the (F', {q;}), and the q; must be specified. Now by the definition ofnmJ as the product ofthefP<mJ and 5.3.3, if the term for r in (5.2) (c) does not vanish then rP is conjugate in GL2(Fp) to

whenever p splits in F'. In other words, rP must have order m at some prime qr of

F' over p and be a unit at the other; set Yr = (F', {qr } ).

In either case the element r clearly gives rise to an element of Hyr(Q), and it turns out to be an easy consequence of 5.2.2 and 5.3.3 that the corresponding terms in (5.2) and (5.4) agree. 6. Supplementary remarks.

6. 1 . When F = Q, the main theorem follows from a congruence relation between Heeke operators and the Frobenius. Although such a relation is likely to hold very generally, it does not yield a formula for the '-function. I want to explain this a bit more carefully. First of all, the integral Heeke operators always define algebraic correspondences on the scheme KS. When F = Q, the congruence relation says that modulo p T(p) = 1/J + 1/J * · T(p, p) where 1/J is the Frobenius and 1/J * its transpose (refer to §7.4 of [28]). An equivalent way of expressing this, as Langlands pointed out to me, is to say that in the ring of algebraic correspondences 1/J is a root of the poly­ nomial xz - T(p)X + pT(p, p). In general one may consider the polynomial det(X p(g)) as a function of the semisimple element g of LG. By the Satake iso­ morphism it corresponds to a polynomial whose coefficients are Heeke operators. -

THE HASSE-WElL �-FUNCTION

161

These will in fact be integral, and it looks not very difficult to show that (/J is a root. In more concrete terms, this will imply immediately that the roots of the Frobenius (acting on /-adic cohomology) lie among the roots of this polynomial, or that factors of Zp(X, KS) lie among the factors of n ,. i n -<*'o det (I - g(g(1t'p))) m (1r, KP) . Now when F = Q, one obtains by a further relation which I have always found a little mysterious that in fact the roots coincide, but it is this second step that does not seem to generalize (I refer to 7. 10(2) of [28] ; see also Piatetskii-Shapiro's argument on pp. 333-...:3 36 of [24] for a representation-theoretic analogue). Langlands has remarked that there is in some sense a good reason for this difficulty, inasmuch as when problems of L-indistiguishability arise one may get in some sense only partial coincidence of the roots of the Frobenius with those of the Heeke poly­ nomial. Incidentally, in his proof of the main theorem in the cases already mentioned, Shimura also used the Selberg trace formula. I might also mention that it is apparently Ihara who first applied the trace for­ mula to matters of this kind-in [14] he showed, modulo some technical problems, that the Hasse-Weil �-function associated to certain sheaves on SLz(Z)\£ is a product of Heeke L-functions. Both he and Shimura used what one might call a global formulation. The first application involving representation theory and local orbital integrals is Langlands' Antwerp talk [19] ; this earlier work seems to have played a role in Langlands' own further development as well as in Drinfeld's (note the remark in [9] on this point). 6.2. It is very little extra work to extend the proof of the main theorem to allow for nontrivial sheaves as well as a refinement which accounts for a factorization of the �-function according to the direct sum decomposition of d0 • Let me first sketch how to deal with sheaves. Suppose � : G � GLn to be a ra­ tional representation of G (i.e., rational over Q) trivial on NJ,.1Q , the kernel of the norm map from px to Qx (considered as algebraic groups) which is canonically embedded as scalars in the centre of G. Then one may associate to � a locally con­ stant Q-sheaf Ee on KS(C) and /-adic sheaves Ee(Q1) on KS (as in a special case in [19]), and consider for each "good" p the p-factor of a �-function

By the Lefschetz trace formula this satisfies

In both these formulas (/J is the Frobenius, and for convenience I have written �;c((/Jm) as the action of(/Jm on the stalk of Ee at the fixed point x of (/Jm . On the other hand one can define numbers m(1t'8, �) analogous to the numbers m(1t'8) in §1 satisfying, for example, m(1t'8, �) = 0 unless Ext � r) (�, 1t'8) "# 0 and set m(l't') = m(1t'8)m(1t'P). The main theorem extends to : Lp(s, K s, �) = . IT L(s - n/2, l't'p , p)m<,.. e) . 1r

1n

do

The proof is almost exactly the same as for � trivial ; one uses an extended version of

1 62

W. CASSELMAN

the trace formula for the trace of R1, / E C�(�), on the representation of � which is £2-induced from � : trace R1

= I: !rl

meas(Fr\�r) Tr Hr)

J

f(g- 1 rg) dg.

(§,\cs

And one also uses a result of Langlands analogous to the one used above which gives not only the structure of xS(Fp) but of E0 on xS(Fp) (the natural guess is correct). Now the second point. As Langlands shows, essentially, in [19], corresponding to the decomposition ..#0 = EB n one has a decomposition H*(xS(C), Q) = EBH:. The only n which occur are those with m(n) #- 0, and for a n = nR ® n1 which does occur the representation n1 may be defined over Q . Corresponding in turn to this decomposition is one of H*(xS, Q1), and hence even of H*(xS x Fp , Q1). Another extension of the main theorem is that for good p 2n Lp(s, n, p) = IT det(l - (tl.l j H�(Q))rs) Hl; . i=O

To prove this : (1) one observes that the subspace H� (Q) is determined in H*(xS,Q) by the effect of operators in Yt'(G(A�), KP) and (2) applies the reason­ ing given above, but allowingfP to be an arbitrary element of this Heeke algebra. It is this result which Piatetskii-Shapiro is concerned with proving (by means of a congruence relation) in [24], with F = Q. 6.3. It has not escaped me that the result in §5 on the measures on GL2 and nx says, essentially, that the measures given by Jacquet-Langlands are multiples, by the same constant, of what Serre calls in [25] the Euler-Poincare measures on each group. (This is not quite true since strictly speaking the E-P measures on these are zero : but there is a clear relationship between E-P measures on SL2 and N1 . ) Does this observation generalize ? For real groups, I would expect a factor card( WcccJ I Wx) , to play a role, as it does in Harder's work [13]. Indeed, I suspect that Harder's paper does essentially relate inner twisting of measures to Serre's measures in the case when G is the inner twist of an anisotropic group. If G is not such an inner twist then Serre's measure is identically 0; how to describe the rela­ tionship of measures then? As for comparison of orbital integrals, Drinfeld has a nice result for GLn and division algebras of degree n (see Kazhdan's talk at this Institute). Diana Shelstad had proven in her thesis [27] many pretty results on real groups. REFERENCES I. T. Asai, On certain Dirichlet series associated with Hilbert modular forms and Rankin's method, Math. Ann. 226 (1 977), 8 1 -94. 2. A. Borel, Automorphic L-functions, these PROCEEDINGS, part 2, pp. 27-61 . 3. A. Borel and N. Wallach, Cohomology of discrete subgroups of semisimple groups, Notes,

lnst. Advanced Study, Princeton, N. J., 1 977. 4. P. Cartier, Lectures on representations of '{J-adic groups : A survey, these PROCEEDINGS, part 1 , pp. 1 1 1-1 55. 5 . W. Casselman, Characters and Jacquet modules, Math. Ann. 230 (1 977), 101-105. 6. P. Deligne, Formes modulaires et representations 1-adiques, Sem. Bourbaki 1968/69, no. 355, Lecture Notes in Math. , vol. 1 79, Springer, New York.

THE HASSE-WElL C-FUNCTION

1 63

7. P . Deligne, Travaux de Shimura, Sem. Bourbaki 1 97017 1 , no. 389, Lecture Notes in Math . , vol. 244, Springer, New York. 8. , Le support du caractere d'une representation supercuspida/e, C. R. Acad. Sci. Paris Ser. A-B 283 (1 976), A1 55-A1 57. 9. V. G. Drinfeld, Proof of the global Langlands conjecture for GL(2) over a function field, Functional Anal. Appl. 61 (3) (1 977), 74-75 . (Russian) 10. M. Dufto and J-P. Labesse, Sur Ia formule des traces de Selberg, Ann . Sci. Ecole Norm. Sup. 4 (1971), 1 93-284. 11. M. Eichler, Introduction to the theory of algebraic numbers and functions, Academic Press, New York, 1 966. 12. S. Gelbart and H. Jacquet, Forms ofGL(2} from the analyticpoint of view, these PROCEEDINGS, part 1 , pp. 21 3-25 1 . 13. G . Harder, A Gauss-Bonnet formula for discrete arithmetically defined groups, Ann. Sci. Ecole Norm. Sup. 4 (1971), 409-455. 14. Y. Ihara, Heeke polynomials as congruence �-functions in elliptic modular case, Ann. of Math. (2) 85 (1 967}, 2 67 2 95. 15. H. Jacquet, Automorphic forms for GL(2} II, Lecture Notes in Math., vol. 278, Springer, New York. 16. H. Jacquet and R. P. Langlands, Automorphic forms for GL(2), Lecture Notes in Math . , vol. 1 14, Springer, New York, 1 973. 17. R. Kottwitz, Combinotorics and Shimura varieties mod p, these PROCEEDINGS, part 2, pp. 1 85-1 92. 18. S. Lang, SL.(R), Addison-Wesley, 1975. 19. R. P. Langlands, Modular forms and 1-adic representations, Lecture Notes in Math., vol. 349, Springer, New York, 1 973 . 20. , Some contemporary problems with origins in the Jugendtraum, Proc. Sympos. Pure Math. , vol. 28, Amer. Math. Soc. , Providence, R.I , 1 976, pp. 401-4 1 8 . 21. , Shimura varieties and the Selberg trace formula, Canad. J. Math. (to appear} 22. , On the zeta-functions of some simple Shimura varieties (to appear). 23. Y. Matsushima and G. Shimura, On the cohomology groups attached to certain vector-valued differential forms on the product of upper half-planes, Ann. of Math. (2) 78 (1 963), 41 7-449. 24. I. I. Piatetskii-Shapiro, Zeta functions of modular curves, Lecture Notes in Math., vol. 349, Springer, New York, 1 973 . 25. J-P. Serre, Cohomologie des groupes discrets, Prospects in Mathematics, Ann. of Math. Studies, no. 70, Princeton Univ. Press, Princeton, N. J., 1 97 1 . 26. D. Shelstad, Orbital integrals for GL,(R), these PROCEEDINGS, part 1 , pp. 107-1 1 0. 27. , Some character relations for real reductive algebraic groups, Thesis, Yale Univ. , 1 974. 28. G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton, 1971 . 29. , On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. 31 (1 975), 79-98. ---

-

---

.

---

---

---

--

UNIVERSITY OF BRITISH CoLUMBIA

Proceedings of Symposia in Pure Mathematics Vol . 33 (1 979), part 2, pp. 1 65-1 84

POINTS ON SIDMURA VARIETIES mod p

J. S. MILNE There is associated to a reductive group G over Q with some additional structure a Shimura variety Sc defined over C. In most cases it is known that Sc has a canonical model SE defined over a specific number field E. For almost all finite primes v of E it is possible to reduce SE modulo the prime and obtain a nonsingular variety Sv over a finite field Fqv" As is explained in [3], in order to identify the Hasse-Weil zeta-function of SE or, more generally, of a locally constant sheaf on SE it is necessary to have a description of (S0(Fq.), Frob) where Sv(Fq.) is the set of points of Sv with coordinates in the algebraic closure Pq. of Fq. and Frob is the Frobenius map Sv(Fq.) -+ Sv(Fq.) which takes a point with coordinates (ah · · · , am) to (aq(• · · · , a'/:{) . To be useful, the description should be directly in terms of the group G. Recently (13] Langlands has conjectured such a description of (Sv(Fq.), Frob) for any Shimura variety S and any sufficiently good prime v. In [12] he has given a fairly detailed outline of a proof of the conjecture for those Shimura varieties which can be realized as coarse moduli schemes for problems involving only abelian varieties, (weak) polarizations, endomorphisms, and points of finite order. (So G(Q) is of the form Aut8(H1 (A, Q), ¢) where B is a semisimple Q-algebra containing an order which acts on the abelian variety A, ¢ is a Riemann form for A whose Rosati involution on End(A) ® Q stabilizes B, and Aut8 refers to B­ linear automorphisms g of H1 (A, Q) such that cj;(gu, gv) = cj;(u, ,u(g)v) with ,u (g) lying in some fixed algebra F contained in the centre of B and fixed by the Rosati involution ; there is also a Hasse principle assumption.) Earlier [8, Conjecture 1] Ihara had made a similar conjecture when S is a Shi­ mura curve and had proved it when G = GL2 [9, Chapter 5]. When G = B x , B a quaternion division algebra over Q, Morita [15] proved Ihara's conjecture for all primes p of E ( = Q) not dividing the discriminant of B. Both he and Shimura have obtained partial results for more general quaternion algebras (unpublished). More recently Ihara has proved his conjecture for all Shimura curves and sufficiently good primes (announcement in [11]). While Ihara bases his proof on the Eichler­ Shimura congruence relations, Morita's method, as described in [10], appears to be quite similar to that of Langlands. In order to give some idea of the techniques Langlands uses in his proof I shall describe it in the case that G is the multiplicative group of a totally indefinite quaAMS (MOS) subject classifications (1 970). Primary 14K1 0 ; Secondary 14010, 14L05. © 1979, American Mathematical Society

1 65

1 66

J. S.

MILNE

ternion algebra over a totally real number field. In § I it is shown that Sc para­ metrizes, in a natural way, a family of abelian varieties with additional structure. The following section describes how Artin's representability criteria may be used to prove the existence of a variety Sg over Q which is a canonical model for Sc and which, when reduced mod p, parametrizes a family of abelian varieties (with additional structure) in characteristic p. Thus the problem of describing (Sp(Fp), Frob) becomes one of describing this family. In §5 the Tate-Honda classification of isogeny classes of abelian varieties over finite fields is used to determine the isogeny classes in the family, and in §6 the individual isogeny classes are described. Since this requires the use of p-divisible groups and their Dieudonne modules, these are reviewed in §3. Notation. F is a totally real number field of degree d over Q, B is a quaternion division algebra over F which is split everywhere at infinity, b ...... b * is a positive F-involution on B, and 08 is a maximal order in B. G is the group scheme over Z such that G(R) = ( OYfP (8) RY for all rings R, where Of!P is the opposite algebra to 08. A is the ring of adeles for Q ; A = R x A1 = R x A� x Qj; ; A1 = Z1 (8) Q, Z1 = proj lim ZjmZ; Z1 = Zf x Zp . K is a (sufficiently small) open subgroup of G(Z1) . .::1 is a product of rational prime numbers such that if p % .::1 then p is unramified in F, B is split at all primes of F dividing p, and K = KPG(Zp) where KP = K n G(Af). Sc = KSc is the Shimura variety over c defined by G, K, and the map h ; ex --> G(R) defined in § 1 ; thus its points in C are Sc(C) = G(Q)\G(A)/K=K where K= is the centralizer of h in G(R). If V = V(Z) is a Z-module then V(R) = V ®z R for any ring R. 1. Sc as a moduli scheme. Recall that an abelian variety over a field k is an alge­ braic group over k whose underlying variety is complete (and connected) ; its group structure is then commutative and the variety is projective. For example, an abelian variety of dimension one is an elliptic curve, and may be described by its equation, which is of the form Y2Z

= X3 +

aXZ2 + bZ3 ,

a, b E k, 4a3 + 27hZ

¥=-

0.

It is impractical to describe abelian varieties of dimension greater than one by equations, but fortunately over C there is a classical description in terms of lattices in complex vector spaces. Let V be a lattice in Cg , i.e., V is the subgroup generated by an R-basis and so V ®z R ""' Cg . Then Cg/ V is a compact complex-analytic manifold which becomes a commutative Lie group under addition. When g = 1 the Weierstrass p-function corresponding to V, and its derivative, define an em­ bedding z >-> (p(z), p'(z), I ) : Cf V 4 P'f: of Cf V as an algebraic subset of the projective plane. Thus Cf V automatically has the structure of an algebraic variety and so is an abelian variety. This is no longer true if g > 1 for there may be too few functions on Cg/ V to define an embedding of it into projective space. Since any meromorphic function on Cg/ V is a quotient

POINTS ON SHIMURA VARIETIES

mod p

167

of theta functions on 0, eKj V will be algebraic if and only if there exist enough theta functions. By definition, a theta function for V is a holomorphic function 0 on 0 such that, for v e V, {}(z + v) = {}(z) exp (2tci(L(z, v) + J(v))) where L(z, v) is a e-linear function of z and J(v) depends only on v. One shows that L(z, v) is additive in v , and so extends to a function L: eg x eg � e which is e-linear in the first variable and R-linear in the second. Set E(z, w) = L(z, w) - L(w, z). Then (a) E is R-valued, R-bilinear, and alternating; (b) E takes integer values on V x V; (c) the form (z, w) ....... E(iz, w) is symmetric and positive. {The symmetry is equivalent to having E(iz, iw) = E(z, w) for all z, w ; the positivity means E(iz, z) � 0 for all z.) A form satisfying these conditions is called a Riemann form for V and it is known that there exist enough functions to define a projective embedding of eKj V if and only if there exists a Riemann form for V which is nondegenerate (and hence such that E(iz, z) is positive definite). If g = 1 we may always define E(z, w) to be the ratio of the oriented area of the parallelogram with sides Ow, Oz to that of a funda­ mental parallelogram for the lattice. Since this form always exists, and is unique up to multiplication by an integer, one rarely bothers to mention it. By contrast, if g > 1 , a nondegenerate Riemann form will not usually exist and when it does, it will not be unique up to multiplication by an integer. However since eKj V is com­ pact the algebraic structure on eKJ V (but not the projective embedding) defined by a Riemann form is independent of the form. Thus, given a lattice in eg for which there exists a nondegenerate Riemann form, we obtain an abelian variety. Conversely, from an abelian variety A of dimension g we can recover a complex vector space W of dimension g and a lattice V in W for which there exists a nondegenerate Riemann form. W can be described (accord­ ing to taste) as the Lie algebra Lie(A) of A, the tangent space tA to A at its zero element, or as the universal covering space of the topological manifold A( e). The lattice V can be described as the kernel of the exponential exp : Lie(A) � A(e), or as the fundamental group of A(e) which, being commutative, is equal to H1 (A, Z). We shall always regard the isomorphism WJ V � A(e) as arising from the exact sequence,

Since H1 (A, Z) is a lattice in tA; we have H1 (A, R) = H1 (A, Z) ® R � tA . Thus A is determined by H1 (A, Z) and the complex vector space structure on H1 (A, R). A complex structure on a real vector space V(R) defines a homomorphism h : ex � AutR( V(R}), h(z) = (v ....... zv), and the complex structure is determined by h. Thus an abelian variety A is uniquely determined by the pair (H1 (A, Z), h) where h : ex � Aut(H1 (A, R)) is defined by the complex structure on H1 (A, R) = tA . Moreover every pair {V(Z), h) for which there exists a Riemann form arises from an abelian variety. Let V(Z) = H1 (A, Z). A point of finite order on A corresponds to an element of V(R) some multiple of which is in V(Z). More precisely, the group of points of

1 68

J. S. MILNE

finite order on A may be identified with V(Q)/ V(Z) c V(R)/ V(Z). For any integer the group A m(C) of points of order m is equal to m-I V(Z)/ V(Z) � V(ZjmZ) � (Z/mZ)2 d im (Al . We define T1A to be proj limm A m (C) = V(Z1) and, for any prime l, T1A to be proj limmA1m(C) = V(Z1) ; thus T1A = fl 1T1A and T1A

m > 0, �

z1 2d im (A) .

A homomorphism A --+ A' of abelian varieties induces a C-linear map tA --+ tA ' such that H1 (A, Z) is mapped into H1 (A', Z). Conversely, if A and A' correspond respectively to ( V, h) and ( V', h') then a map of Z-modules a : V(Z) --+ V'(Z) extend­ ing to a C-linear map a ® 1 : V(R) --+ V'(R) (i.e., such that a ® 1 o h(z) = h'(z) o a ® 1 for all z) arises from a map of complex manifolds A( C) --+ A'(C) and the compactness of A( C) and A'(C) implies that the map is algebraic. We write End(A) for the ring of endomorphisms of A and End0(A) for End(A) ®z Q. Since End0(A) has a faithful representation on H1 (A, Q), it is a finite-dimensional Q­ algebra; it is also semisimple, and its possible dimensions and structures are well understood. To define a homomorphism i : 08 --+ End(A) when A corresponds to (V, h) is the same as to define an action of 08 on V such that h maps ex into Aut080R( V(R)). When such an i is given we say that 08 acts on A provided i(1) = 1 . Such an i induces an injection i : B 4 End0(A). A nondegenerate Riemann form E for A defines an involution a ....... a' of End0(A) by the rule E(az, w) = E(z, a'w) ; this is the Rosati involution, which is known to be positive, i.e., Tr (aa') > 0 for all a # 0 where Tr denotes the reduced trace from End0(A) to Q. Suppose 08 acts on A. We say that two Riemann forms E and E' on A are £-equivalent if there exist nonzero c, c' E OF such that E(u, cv) = E(u, c'v) for all u, v E V(Z), and we define a weak polarization of A to be an £-equivalence class A of nondegenerate Riemann forms. Since F is the centre of B, the Rosati involutions defined by any two elements of su<;h a A induce the same map on i(B). We shall be interested in triples (A, i, A) such that E( i(b)u, v) = E(u, i(b *) v) for u, v E V(R), b E B, E E A, i.e., we require that the Rosati involutions defined by A stabilize i(B) and induce the given involution b ....... b* on B. We next review some notations concerning B. The main involution b ....... b• of B is so defined that under any R-isomorphism B ®F R � M2 (�) , if b corresponds to M = <: ;) then b• corresponds to M' = (�;:) ; thus b + b' = Tr8 1 p(b) and bb' = Nm8 1 p(b). The Skolem-Noether theorem shows that there exists a t E B such that b* = t- l b•t = tb•t-1 for all b E B ; automatically t2 E F and the positivity of b ....... b* implies that t 2 < 0, i.e., t 2 has negative image under all embeddings F 4 R. We fix an isomorphism B ® g R � M2(R) x . . . x M2(R) such that if b +-+ (Ml > ' " • Mn) then b* +-+ (M!r, . . . , M�r) where M:r is the transpose of M1• Since

(

Mtr - 01 _

-

) (

)

1 IM' O - 1 1 0 '

0

-

t maps to an element (c1(� -fi) , . . · , en(� -m with each c1 E R, and t may be chosen so that c1 > 0 . The next lemma implies that if 0 8 acts on a complex manifold cuI V then there is a Riemann form E for V whose corresponding Rosati involution induces b ....... b* on B and any two such forms are £-equivalent, i.e., that there is a unique weak

POINTS ON SHIMURA VARIETIES mod p

1 69

polarization which is compatible with the 08-action and the given involution. LEMMA 1 . 1 . Let V = V(Z) be a free Z-module of rank 4d on which 08 acts. There

is a nondegenerate alternating form cjJ on V(Q) such that : (a) cjJ(u, v) E Z if u, v E V(Z) ; (b) cjJ(ut, u) < O for all u i= 0, u E V(R) ; (c) cjJ(bu , v) = cjJ(u, b*v) for all b E B and u, v E V(Q) ; (d) for any B-automorphism a of V(Q) there exists a f.t(a) E px such that cjJ(au, av) = cf;(u, f.t(a)v) for aU u, v E V(Q). Moreover ' if cjJ' is a second nondegenerate al­ ternating form on V(Q) satisfying (c) then there exists a c E px such that cjJ(u, cv) = cjJ'(u, v) for all u, v E V.

PROOF. V(Q) has dimension one over B and so, after choosing an appropriate basis vector, we may identify V(Q) with B and V(Z) with a left ideal in 08• Define cjJ(u, v) = Tr81iuv ' t) = Tr81Q(utv*). Then cjJ(u, v) = Tr (utv*) = Tr(vt*u*) = Tr ( v( - t)u*) = - cjJ(v, u), and so cjJ is alternating. (a) is obvious, and cjJ(ut, u) = Tr81Q(ut 2u * ) = TrF1Q (t 2Tr8 1 F(uu* ) ) < 0 for u i= 0, which proves (b) and that cjJ is nondegenerate. For (c) we note that cjJ(bu, v) = Tr(utv*b) = Tr(ut(b*v)* ) = cjJ(u, b*v). Finally, any B-automorphism a of V(Q) = B is multi­ plication on the right by an element b E B x . Thus cjJ(a u, a v) = Tr (ubb•v•t) = cjJ(u, f.t(a)v) with fl-(a) = Nm81F(b). For the last part, consider the Q-linear map v ...... cjJ'(l , v) : B -. Q. Since Tr81Q : B x B -> Q is nondegenerate, there is a unique b E B such that cjJ'( l , v) = Tr (btv*) for all v E B. Then cjJ'(u, v) = cjJ' ( 1 , u*v) = Tr (btv*u) = Tr (ubt v *) = Tr(ubv•t). We also have cjJ'(l , v) = - cjJ'(v, 1) = - Tr (vbt) = - Tr (t ' b ' v ') = Tr (b ' v' t) = Tr (b•tv*). Thus b = b•, which implies that it is in F, and we may take c = b . For the remainder of this section V(Z) will be 08 regarded as an 08-module and cjJ will be as in the lemma. For any ring R we may identify G(R) with Aut08C2JR ( V(R )) since any 08 ® R-endomorphism of V(R) = 08 ® R is right multiplication by an element of 08 ® R. Define h to be the homomorphism ex -> G(R) = Aut8C2JR ( V(R)) such that h(i) is right multiplication on V(R) = B ® R --""> M2(R) x . . . x M2 (R) by ((<J_1 fi), · · · , ({)_1 fi)) . Thus Koo = {(Mt. · · · , Md)} with M,. of the form (�b �), a, b E R. The form E = cjJ is a Riemann form for ( V(Z), h), e.g., cjJ(iu, iv) = ¢ (uh(i), vh(i)) = cjJ(u, Nm8 ! F(h(i))v) = cjJ(u, v) and cjJ(iu, u) = cjJ(utj - ( - t 2) 1 1 2 , u) > 0 for u -1= 0. Thus ( V(Z), h) defines an abelian variety A. The action of 08 on V(Z) induces a map i : 08 -> End(A) and the Rosati involution defined by the weak polarization A containing cjJ induces b ...... b* on B. Recall that K is an open subgroup of G(Zr). Two isomorphisms ch, cPz: TrA --""> V(Zr) are K-equivalent if there is a k E K such that (PI = k¢2 . For example, if K = Km = Ker ( G(Zr) -. G(ZjmZ)) then to give a K-equivalence class of isomorphisms TrA -> V(Zr) is the same as to give an isomorphism A m (C) --""> V(ZjmZ), i.e., a level m structure. THEOREM 1 .2. There is a one-one correspondence between the set ofpoints Sc(C) = G(Q) \ G(A)/ KooK and the set of isomorphism classes of triples (A, i, (/>) where A is an abelian variety of dimension 2d, i defines an action of 08 on A, and If> is a K­ equivalence class of 08-isomorphisms TrA --""> V(Zr) ·

1 70

J. S. MILNE

REMARK 1 . 3 . (a) We say that two triples (A, i, �) and (A', i', �') are isomorphic if there exists an isomorphism a : A -+ A' such that a o i(b) = i'(b) o a for all b e 08 and if/ o (T1a) e � for all if/ e �'. (b) Normally when considering families of abelian varieties parametrized by Shimura varieties it is necessary to work with quadruples (A, i, A , �) with A a (weak) polarization. This is not necessary in our case because, as we observed above, A always exists uniquely. PROOF OF .1 .2. We first show how to associate to any g e G(A) a triple (A g , ig , �g). If g = 1 we take (A, i, �) with (A, i) as defined before and � the class of the identity map T1A = V(Z1) .l
POINTS ON SHIMURA VARIETIES

mod p

171

projection V(R) x ( G(A)JK""K) --+ G(A)/K""K. We give G(A)/K""K its usual complex structure and the copy of V(R) over gK""K the complex structure defined by hg. Inside each Vg we have a lattice g V(Z), and these vary continuously with g. Thus we may divide out and obtain a map of complex manifolds .r --+ G(A)/K""K such that the fibre over gK""K is the abelian variety A g . We may now let G(Q) act on both manifolds and divide out again to obtain an analytic family d --+ Sc of abelian varieties. Each fibre A g has the structure defined by (ig, ?>g), and these vary continuously. In fact d --+ Sc is an algebraic family, i.e., d is an algebraic variety and the map is algebraic. If F :F Q the above construction fails because units of F may act on (Ag, ig, cpg) and so the action of G(Q) on .r is not free. However we may "rigidify" the situa­ tion as follows : consider quadruples (A, i, ?> . e) where A, i, and ?> are as before and e is an injection from the unique weak polarization A to px such that e(if/ ) = ce(if;) if if; ' (u, v) = if;(u, cv) . The isomorphism classes of quadruples are classified by px x Sc(C) which is a disjoint union of copies of Sc(C), one for each element of p x , on which px acts by permuting the copies. px x Sc may be regarded as a scheme over C which is an infinite disjoint union of varieties and the previous process gives an algebraic family of d --+ p x x Sc of abelian varieties with structure. References. The most elegant elementary and nonelementary treatments of abelian varieties over C are to be found respectively in [20] and [17, Chapter I]. Families of abelian varieties parametrized by Shimura varieties were extensively studied by Shimura in the 1960 ' s (see his Annals papers of that period). They are also discussed briefly in [4]. 2. S as a scheme over Z [.::1- 1 ), We shall see shortly that Sc has a model Sg over Q, i.e., that there is a scheme Sg over Q whose defining equations, when considered

over C, give Sc. There is no reason to believe that Sg will be unique but Shimura has given conditions which will be satisfied by at most one model ; such a model (when it exists) is said to be canonical. For example, let F' be a quadratic totally imaginary extension of F which splits B and let A0 be the abelian variety of dimen­ sion d defined by the lattice OF ' c: F' ® R. Then OF ' acts on A0 and A0 is said to have complex multiplication by F'. Let A = A0 x A0• If we embed F' in B and choose a basis {eh e2 } for B over F' with e 1 o e2 e 08, then we get a map B 4 M2(F') c: M2 (Endo(A0) ) = End0(A) sending 08 into End(A). Also we get a map T1A = ( O F ' ® O F ,) ® Z1 _L 08 ® Z1 = V(Z1) (in the notation of § 1). The triple (A, i, ?>) defines a point of Sc, and hence a point of Sg with complex coordinates. For Sg to be canonical these coordinates must be algebraic over Q and generate a certain explicitly described class field. For the reasons explained in the introduction we would like to have a scheme S defined by equations in Z[Ll- 1 ) which, when regarded over Q, is the canonical model Sg of Sc, and which is such that it is possible to describe explicitly (S(Fp). Frob ) for any p 1 Ll. Such an S will define a functor R ...... S(R) which associates to any ring R in which Ll is invertible the set of points of S with coordinates in R. (More generally, it associates to any scheme T over spec Z[Ll- 1 ) the set S(T) of maps T --+ S. ) Since the functor determines the scheme uniquely this suggests that in constructing S we should write down a functor !/' such that, in particular, !/'(C) = Sc(C) = G(Q)\G(A)/K""K and try to prove that it is the points functor of a

1 72

J. S. MILNE

scheme. After § 1 it is natural to define sP(R) to consist of isomorphism classes of triples (A, i, rp) where each of the three objects is the analogue over R of the cor­ responding object over C. Thus A is a projective abelian scheme of dimension 2d over R. Intuitively, A can be thought of as an algebraic family of abelian varieties, each of which is defined over a residue field of R. More precisely it is a projective smooth group scheme over spec R with geometrically connected fibres. As before i is to be a homomorphism Os 4 End(A) such that i(I) is the identity map. We assume that A has a polarization whose Rosati involution induces b ...... b* on B. Two problems arise in defining (fi which may be best understood if we write rjJ : T1A -+ V(Z1) as a product IT rp1 : IT T1A -+ IT V(Z1) of maps. Firstly, if p is not invertible in R there will never exist an isomorphism rpp : TpA � V(Zp) ; thus we take rpp to be a map defined only over R[p- 1 ]. Secondly, unless R is an algebraically closed field it is unrealistic to expect there to be an isomorphism rjJ1 : T1A -+ V(Z1) for any I, for this would imply that all coordinates of ali i-power torsion points of A are in R. Instead we assume that K ::::> Km = Ker(G(Z1) -+ G(ZfmZ)) some m, and consider isomor­ phisms rjJ : A m � V(Z/mZ) defined on some etale covering of R, two such isomor­ phisms rjJ 1 and ¢2 being K-equivalent if ¢ 1 = krjJ2 , k e K, locally on spec(R), and we take (fi to be a K-equivalence class in this new sense. It is necessary to put one extra condition on the triple (A, i, (/J) : if the R-algebra R' is such that Os ® R' � M2 (0p ® R') then the two submodules of tA I R corresponding to the idempotents (b 8) and (8 �) should be free OF ® R'-modules of rank 1 locally on spec(R'). (This condition holds automatically if R = C; for examples where it fails in an analogous situation in characteristic p, see [18, 1 .29]. ) Having defined our functor sP we now have to see whether it is the points functor of a scheme. Generally speaking this is a very delicate question but M. Artin has given an often-manageable set of criteria for a functor to be the points functor of an algebraic space. An algebraic space is a slightly more general object than a scheme, but for our purposes it is just as good ; it makes good sense to speak of its points with coordinates in a ring, and the proper and smooth base change theorems in etale cohomology, which are the theorems which allow us to compute Hasse-Weil zeta-functions by reducing modulo a prime, �old for algebraic spaces. (In fact, the algebraic spaces we get are almost certainly schemes, and this surely could be proved by using Mumford's methods [16] instead of Artin's.) Consider first the case that F = Q. Then Artin's criteria may be checked and show that there is an algebraic space S, proper and smooth over Z [J- 1 ], such that S(R) = sP(R) for any ring R in which L1 is invertible. In particular S(C) = sP(C) = Sc(C) and S(Fp) = sP(Fp) for any p not dividing .d. The algebraic family .91 -+ Sc constructed in 1 .4(b) is an element of sP(Sc) = S(Sc), and so gives a map Sc -+ S. This induces a map Sc -+ S x spec C which is an isomorphism. Moreover it is known that S0 is the canonical model. When F '# Q, then a slightly weaker result holds, but one which is just as useful to us. Since there are nontrival automorphisms of (A, i, (fi) there can be no algebraic space S with S(R) = sP(R) for all R. However, there does exist an algebraic space S, proper and smooth over Z [J- 1 ], and a functorial map .:f>(R) -+ S(R) which is an isomorphism whenever R is an algebraically closed field. Thus S(C) = sP(C) = Sc(C) as before, and S0 is the canonical model of Sc. To prove these facts one may "rigidify" the moduli problem as in the second paragraph of 1 .4(b), make the

POINTS ON SHIMURA VARIETIES

mod p

1 73

constructions as in the case F = Q, and then form quotients under the left action by px, or else work directly with stacks. Note that in either case, S (Fp) = f/(Fp) has a description in terms of abelian varieties with additional structure. References. [1] contains a short introduction to Artin's techniques for represent­ ing functors by algebraic spaces and [2] a more complete one. In [5] and [18] these techniques are applied to a situation which is very similar to ours. (In fact, it is almost identical ; see §7 of the Introduction to [5].) The basic definitions concern­ ing abelian schemes can be found in [16]. 3. Finite group schemes, p-divisible groups, and Dieudonne modules. In the remain­ ing sections we shall need to consider the finite subgroup schemes of an abelian variety and so, in this section, we review some of their properties. We fix a perfect field k of characteristic p # 0. Let R be a finite k-algebra (so R is finite-dimensional as a vector space over k) and let N = spec R. For any k-algebra R', a point of N in R' is simply a map of k-algebras R --+ R' ; thus N(R') = Homk_.1g (R, R'). If every N(R') is given the structure of a commutative group in such a way that the maps N(R') --+ N(R") induced by maps R' --+ R" are homomorphisms, then we call N, together with the family of group structures, a finite group scheme over k. As for affine algebraic groups, giving the family of group structures corresponds to giving a comultiplica­ tion map R !!!... R ®k R. ExAMPLE 3. 1 . (a) Any (commutative) finite group M can be regarded in an ob­ vious way as an algebraic group over k and hence as a finite group scheme. Indeed, let R be a product of copies of k, one for each element of M, and let N = spec R. Then N, as a set, is equal to M. The group law on M induces a comultiplication on R which, in turn, induces compatible group structures on N(R') for all R'. If R' has no idempotents other than 0 and 1 , then N(R') = M. (b) f.lpn = spec k[T]f(TP" - 1). Then ppn(R' ) = {I: e R' I�P" = 1 } is a group under multiplication for any R', and these group structures make f.lpn into a finite group scheme. Note that ppn(R) = { 1 } if R has no nilpotents and, in particular, if R is an integral domain. (c) ap = spec k[T]/(TP). Then ap(R') = {a e R' laP = 0}. As (a + b)P = aP + bP in any k-algebra, ap(R') is a group under addition, and these group laws make ap into a finite group scheme. Again ap(R') has only one element if R' has no nilpo­ tents. (d) ZfpZ = spec k[T]f(TP - T). If R' has no idempotents other than 0 and 1 (e.g., R' an integral domain) then (Z/pZ) (R') = Fp , the prime subfield of R', which is a group under addition. This example is a special case of (a), because k[T]/(TP - T) = k[T]/T(T - 1) . . · (T - (p - 1)) � k x . . · x k (p copies). The rank or order of a finite group scheme N = spec R is the dimension of R as a vector space over k. For example the order of the group scheme defined by M in 3. 1(a) is the order of M, while the orders of liP"• ap. and ZfpZ are pn , p and p respectively. A homomorphism from one finite group scheme N1 = spec R 1 to a second N2 = spec R2 is a k-algebra homomorphism R2 --+ R 1 such that the induced maps N1 (R') --+ N2(R') are all homomorphisms of commutative groups.

1 74

J. S. MILNE

From now on we consider only finite group schemes of p-power order. The essential facts are the following. Facts. 3.2.(a) They form an abelian category. Thus we may form kernels, quo­ tients, etc. exactly as if we were working with a category of modules. (b) When k is algebraically closed the only simple objects are f-lP • ap, ZjpZ. This means that any finite group scheme of p-power order has a composition series whose quotients are f.lp• ap, or Z/pZ. There can be no homomorphism from one simple object to another of a different type. (c) The category is self-dual, i.e., there is a contravariant functor N�-> N ( = Car­ tier dual of N) which is an equivalence of the category with itself. More precisely, for each N there is a pairing N x N � Gm ( = GL1 ) such that, for any k-algebra R, the pairing induces isomorphisms N(R) � HomR(N, Gm), N(R) � HomR(N, Gm). For example, (ZjpZ)/\ = f.lp and the pairing (ZjpZ)(R) n X f.lp(R) � Gm(R) is (n, ') I-+ ' ; ap = ap and the pairing ap(R) X ap(R) � Gm ( R) is (a, b) � exp(ab) = 1 + ab + · · · + (ab) P- l f( p - 1) ! . (d) Hom (ZjpZ, ZjpZ) = ZjpZ, Hom (f.lp• f.lp) = ZjpZ, Hom (ap, ap) = k. The statement for Z/pZ is obvious, and that for f.lp follows by Cartier duality. The map ap � ap corresponding to c E k is (T ,_. cT) : k[T]j(TP) � k[T]/(TP) on the algebra of ap and (a ,_. ca) : ap(R') � ap(R') on its points. (e) If O � N' � N � N" � 0 is exact then order(N) = order(N')order(N"). Let A be an abelian variety over k. For each n, Apn � Ker(pn : A � A) is a finite group scheme of order (pn)Zdim cAJ , i . e., the order is the same as when p # characteristic(k). The system A p 4 A p2 4 · · · is called the p-divisible (or Barsotti­ Tate) group A(p) of A. More generally, a p-divisible group of height h is a system of finite group schemes and maps N = (N1 i!., N2 � N3 !2, · · ·) such that Nn has order pnh and in- ! identifies Nn-l with the kernel of (Nn � Nn). For example Qp/Zp = ( Z/pZ � Z/p2Z � · · · ) andppoo = (pp � f.lp2 � f.lp � · · · ) are p-divisible groups of height one. A(p) is of height 2 dim(A). A homomorphism ¢ : N � N' of p-divisible groups is a family of maps ¢n : Nn � N� commuting with the maps in and i�. Exercise 3.3. (k algebraically closed.) For any abelian variety A there are maps ¢n : A pn � Apn such that Ker(¢n) = Ker(¢n+ l ) for all sufficiently large n. Deduce that A has � pdi m A points of order p, and that when equality holds A(p) = (Qp/ Zp) d im CAJ x (ppoo)dim CAl . (Such abelian variety is said to be ordinary.) Let W = Wk be the ring of Witt vectors over k ; it is a complete discrete valua­ tion ring of characteristic zero whose maximal ideal is generated by p and which has residue field k. There is a unique automorphism a ,_. aCPl of W which induces the pth power map on k. If k = Fp then W is the completion of the ring of integers in the maximal unramified extension Qpn of Qp and a ,_. aCPl is induced by the usual Frobenius automorphism of Qj,n over Qp. Let W[F, V] be the ring of noncommuta­ tive polynomials over W in which the relations FV = p = VF and Fa = a c p J F, a V = Va CP l , hold for all a E W. There is a contravariant functor, N ,_. DN = Dieudonne module of N, associating to each p-power order finite group scheme a W[F, V)-module which is of finite length as a W-module ; D defines an antiequiv­ alence of categories. The length of DN as a W-module is equal to the order of N. Thus manipulations with finite group schemes correspond exactly to manipulations with modules over the noncommutative ring W[F, V]. Examples : D(pp) = Wjp W = k ; F acts as 0, V acts as I ;

POINTS ON SHIMURA VARIETIES

mod p

175

D(ap) = k ; F = 0, V = 0 ; D(ZfpZ) = k ; F = I , V = 0. If N is unipotent and pN = 0, then DN = Lie(N) ; the bracket operation on Lie(N} is zero but it has the structure of a p-Lie-algebra and F acts as the "p-power" operation and V acts as zero. More generally, if N is unipotent and killed by p" , then DN = Hom(N, Wn) where WI = Ga = the additive group and wn = the Witt vectors of length n regarded as an algebraic group. There are canonical, nondegenerate, W-bilinear pairings ( , ) : DN x DN � W ® Qp/Zp such that (Fm, n) = (m, Vn)

, ( Vm, n)

= (m, Fn). The notion of Dieudonne module can be extended to p-divisible groups. On applying D to N = (N1 � N2 � · · ·) we obtain a sequence of modules and maps (DN1 � DN2 � ·), and we define DN = proj lim DNn . This is a W[F, V]-module which is free of finite rank equal to height(N) as a W-module. In classifying p-divisible groups one begins by considering them up to isogeny : N and N' are isogenous if there is a surjective homomorphism N � N' with finite kernel or, equivalently, if there exists an injective homomorphism DN' � DN whose cokernel has finite length over W. If we write W' = W[Ifp] = W ®zp Qp , W'[F, F- 1 ] = W'[F, V] = W[F, V] ®zp Qp, and D'N for DN ®zp Q p regarded as a W'[F]-module, then we see that N and N' are isogenous if and only if D' N � D'N'. Let .A be the category of W'[F]-modules whose objects occur as D' N for some p-divisible group N. When k is algebraically closed one knows that .A has exactly one simple object Dl = W'[F]f(Fr - ps) for each rational number A , 0 � A � I , A = s/r, (r, s) = 1. Dl has dimension r over W', End(Dl) is the unique division algebra over Qp of degree r 2 , and any D E .A can be written uniquely as a finite direct sum D = (Dl1) m i E£> EB (Dl•) m • with distinct A;. Then A� o · · · , A1 are the slopes of D and m,.r,., where A; = s;/r,., is the multiplicity of A;. We sometimes write (Ds l r)m as flsm lrm. Thus fls lr may now be a multipl,e of a simple module ; it has slope sfr with multiplicity r and has dimension r over W'. When k is algebraically closed and N is a p-divisible group over k, the slopes of D' N are called the slopes of N. Clearly N is uniquely determined up to isogeny by its slopes and their multiplicities. For example, all p-divisible groups of height one are isogenous (in fact, isomorphic) to /lpoo or Qp/Zp because D'(ppoo) = D I and D'(Qp/Zp) = no are the only Dl of dimension one over W'. There is only one simple Dl of dimension two over W' ; it is fll / 2 = D'(A(p)) where A is a supersingu­ lar elliptic curve (cf. §5). Let k have algebraic closure k =f. k. Any p-divisible group N over k defines a p-divisible group N;; over k and it is known that DN;; � DN ® wk W;;. If k = Fq with q = p a then £a : DN � DN is W-linear and so its characteristic polynomial det(T - PIDN) = f Ht
• • •

• •

• •

•

•

1 76

J. S.

MILNE

(as before) as a free Zrmodule of rank 2 dim(A). We write T1A TljA x TpA. Finally we note that to classify p-divisible groups up to isomorphism, it is neces­ sary to classify the (F, V)-stable lattices in the objects of vtt . References. The best introduction to the subject matter of this section is [6]. 4. S(Fp) as a family of abelian varieties. Fix a prime p not dividing LJ. From §2 we know that points of S( Fp) correspond to isomorphism classes of triples (A, i, (/j) where A is an abelian variety of dimension 2d over Fp, i is an action of 08 on A, and (/j is a KP-equivalence class of isomorphisms cp: TljA ---> V(Zlj) where TljA proj lim ptn A nCFp). (Recall that cpp : TpA ---> V(Zp) is defined only over the ground ring with p inverted, and Fp [p- 1 ] is the zero ring.) The 08 ® Pp-module tA satisfies the following condition : corresponding t o the idempotents ( A 8) and (8 V i n 08 ® Fp (4. 1 ) the subspaces � M2(Fp ) are free OF ® Fp-modules of rank 1 . If A i s defined by equations Za ( i ) TCi l , let A c p J be the abelian variety over Fp defined by the equations Zafil Ti . There is a Frobenius map F FA : A ---> A c pJ which takes a point with coordinates (tl > · · · tJ to (t1, · · · t£) . The map FA is a purely inseparable isogeny of degree p2a , which means that AF, the kernel of F, is a finite group scheme of order p2a with only one point in any field (so that only ap and p.p occur in any composition series for it). As groups, D(A) D(A C P l), but the identity map DA ---> DA c p > is (p)-linear, i.e., am ...... a CP> m for a E W, m E DA. The composite DA J. · · · a,) of P as being the coefficients of the equations defining A. Thus Frob(P) corresponds to (A C P> , ;c p> , cjJ C P l ) where ;c pJ and (/j C P J are such that FA defines a map of triples (A, i, (/j) ---> (A c p > , ;cp > , (/j C P> ). Finally we observe that there are "Heeke operators" acting. Let g E G(A1) and suppose that K' is an open subgroup of G(Z1) such that g- 1 K'g c K; then x >--+ xg : G(A) ---> G(A) induces a map G(Q)\G(A)/Koo K' ---> G(Q)\G(A)/KooK which arises from a map of varieties .9""(g) : K ' Sc ---> KSc. If P corresponds to (A', i', (/j') then .9""(g)P corresponds to (A, i, (/j) if there is an 08-isogeny a : A ---> A' such that =

=

=

,

,

=

=

,

commutes with m some positive integer. When we pass to Sp , only G(Alj) continues to act : if g E G(Alj) and P E K ' S(Fp) and §"(g)P E KS(Fp) correspond respectively to (A', i', (/j') and (A, i, (/j) then there is an isogeny a : A ---> A' whose kernel has order prime to p and a commutative diagram

POINTS ON SHIMURA VARIETIES

mod p

1 77

with m a positive integer prime to p. This definition is compatible with that over C in the sense that both mappings ff(g) come by base change from a mapping ff(g) : K ' S x spec Zc p l � K S x spec Zc p l where Zc p l = { m/n E Ql(n, p) = 1 } . (Mo �e concretely, this means that if (A, i, �) in characteristic zero specializes to (A, l, (/}) in characteristic p then ff(g)(A, i, �) specializes to ff(g)(A , l, (j).) 5. The isogeny classes. Fix a prime p not dividing L1 and consider pairs (A, i) where A is an abelian variety of dimension 2d over Fp and i is a homomorphism B 4 End0(A) such that i(l) = 1 . We write A - A' if A and A' are isogenous, and (A, i) - (A', i') if the pairs are B-isogenous in an obvious sense . .FP denotes the set of all B-isogeny classes and (A, i) ® Q the class containing (A, i). It will turn out (last paragraph below) that the map (A, i, �) ....... (A, i) ® Q : S(Fp) � .FP is surjec­ tive and so, to describe S(Fp), it suffices to describe JP and the fibres of the map. The first is done in this section and the second in the next. Note that Frob (and ff(g)) preserves the fibres. We first remark that, as in characteristic zero, there is a unique weak polarization on A inducing the given involution on B, and that it gives an F-equivalence class of pairings A n x A n � Gm for all n (cf. [17, §23]). In turn these pairings give an equivalence class of skew-symmetric pairings ¢1 : TIA X TIA � TIGm � zl with nonzero discriminant for each I :I- p, and a similar pairing cp p : DA x DA � W ; this last pairing satisfies the conditions cp p(Fm, n) = cpp(m, Vn) a2 , • • · ) >-+ (ar, a� · · · ) . ) THEOREM 5. 1 . (a) Let A be a simple abelian variety over Fq and let E = End0(A). Then E is a division algebra with centre Q[1r:], 1r: is an algebraic integer with absolute value q l 1 2 under any embedding Q[1r: ] 4 C, and for any prime v of Q[1r:] the invariant of E at v is given by

inv.(E)

=

1-

=

0

_

if v is real, if v j l, I :I- p,

ordv(1r:) - ord.(q) [Q [1r:] • .. Qp] if v I p. Moreover 2 dim(A) = [Q[1r:] : Q] [E : Q[1r:1 ] 1 1 2 and e = [E : Q[7r:]] I I2 is the least common denominator of the inv .(E). The characteristic polynomial PA (T) of 1r: : A � A is m(T)• where m(T) is the minimal polynomial of 1r: over Q. (b) The simple abelian varieties A and A' over Fq are isogenous if and only if there is an isomorphism Q[1r:A] � Q[1r:A '] such that 1r:A >-+ 7r:A ' · (c) Every algebraic integer 1r: which has absolute value q l i Z under any embedding Q[1r:] 4 C arises as the Frobenius endomorphism of a simple abelian variety A" over Fq . (d) For any abelian varieties A and B over Fq and any prime I (including I = p) the canonical map

178

J. S.

MILNE

Hom(A, B) ® Z1 --> Hom(A(l), B(l)) = Hom( T1A, T1B) is an isomorphism. (If l # p then A(l) and B(l) can be regarded as Gal(Fq (Fq) ­ modules.)

PROOF. The first part of (a) (the Riemann hypothesis) is due to Weil, part (c) to Honda, and the remainder to Tate ; see [17], [21], [7], [22l [23]. For example, if in (c) we take n: p a , q = p2a then we obtain an elliptic curve A !P such that End0(Atp) is a quaternion algebra over Q which is split everywhere except at p and the real prime. Any such elliptic curve is said to be supersingular. It follows easily from (a) that if Q[n:] has a real prime then either A is a super­ singular elliptic curve or becomes isogenous to a product of two such curves over =

Fq2·

From now on we. let p factor as (p) = Vr · · Vm in OF, where the }1; are distinct prime ideals, and we let d,. be the residue class degree of }1; over p ; thus d = I; d;. 5.2. Let (A, i) be as above. The centralizer of B in End0(A) is either : (a) a quaternion algebra B' over F which splits except at the infinite primes, the

PROPOSITION

primes where B is not split, and the pJor which d; is odd, and there does not split ; or (b) a totally imaginary quadraticfield extension F' ofF which splits B. In thefirst case A ,...., Acrd where A0 is a supersingular elliptic curve and in the second A ,...., A5 where A0 is an abelian variety such that F' c End0(A0).

PROOF. Suppose A ,...., A 0 x AI> r ;?: I , where A0 is a supersingular elliptic curve and Hom(A0, A 1 ) = 0. Then Endo(A) :;:::; M,(E) x End0(A 1 ), where E = Endo(A0), and B embeds into M,(E). Consider F 4 M,(E) ; we must have d/2r, but d = 2r is impossible because F does not split E [19, Theorem 10], and so r = d or 2d. The Skolem-Noether theorem shows that, when composed with an inner automor­ phism, the map F --> Mr(E) factors through M,(Q). Thus the centralizer C(F) of F in M,(E) is isomorphic to M, 1 a(F) ® E = M,1 a(E ® F). Let C be the centralizer of B in M,(E). Then B ® F C :;:::; C(F) because C(F) and B are central simple alge­ bras over F [19, §8]. It follows that either rjd = I , C = F, and B = E ® F, or rjd = 2 and C is a quaternion algebra over F such that, in the Brauer group of F, [B] + [ C] = [E ® F]. The first is impossible because B splits at infinite primes while E does not; thus the second holds, and this proves that case (a) of the proposition holds. Next assume that Hom(A0, A) = 0 when A0 is a supersingular elliptic curve, and fix a large subfield Fq of Fp such that A and all its endomorphisms are defined over Fq. From considering A/Fq we get a Frobenius endomorphism n: E Endo(A), and the assumption implies that there is no homomorphism Q[n:] --> R. Consider

B -B [n]-B ®F C - E I I I F- F [n:]- C I I Q - Q [n:]

where E is Endo(A) and C is the centralizer of B in E. Clearly, F[n:] = F would contradict our assumption. On the other hand we must have [F[n:] : F] ;;£ 2 and F [n:] = C for otherwise E would contain a commutative subring of dimension

POINTS ON SIDMURA VARIETIES

mod p

1 79

> 4d = 2 dim( A) over Q, which is impossible by 5. 1(a). Let F' = C = F[11:] ; it is a quadratic extension of F and can have no real prime because that would contradict our assumption. It splits B because, for any finite prime l :1- p, ( T1A) ®z1 Q 1 is free of rank 2 over F; = F' ® Q1 , from which it follows that B ® F; � M2(F;), and we are assuming that B splits at any infinite prime or prime dividing p. Let e be an idempotent :1- 0, 1 , in (B ® F') n End(A). Then A0 = eA is an abelian variety such that A "' A0 x A0 • Since elements of F' commute with e, F' c End0(A0). REMARK 5.3. In case (b) of 5.2, A0 is isogenous to a power of a simple abelian variety, A0 "' Ar, because the centre of E = End0(A) is a subfield of the field F'. It follows that, for any pair (A, i) as above, A is isogenous to a power of a simple abelian variety and hence End0(A) is a central simple algebra over the field Q[11:]. Let (A, i) and (A ', i') be such that there exists an isogeny a: A -+ A'. The Skolem­ Noether theorem shows that the map B _!_. End0(A) � End0(A'), where a * (r) = ara-1, differs from i' : B -+ End0(A') by an inner automorphism (r ...... f3 r f3 - 1 ) of End0(A'). Thus {3a is a B-isogeny A -+ A' , and we have shown that A "' A' im­ plies(A, i) "' (A', i'). We now consider in more detail the situation in 5.2(b). Let ll h · · · , lJ1, 0 � t � m, be the primes of F dividing p which split in F' and write lJ; = q,q; for i � t. Since OF n End(A0) and Zp both act on A0(p), their tensor product does, and the splitting F ® Qp � Fp , x · · · x FPm induces an isogeny A0(p) "' A0(lJ 1 ) x · · · x Ao(llm) and an isomorphism D 'A0 � D'Ao(ll 1 ) x · · · x D'Ao(llm). Clearly Ao(ll;) has height 2d, and so D' (Ao(lJ,)) has dimension 2d, over W'. Since cjJp(am , n) = cjJp(m , an) for a e F the decomposition of D'A0 is orthogonal for cf;p, and cjJp restricts to a non­ degenerate form on each D' (Ao(ll;)) . This implies that the set of slopes {A h il2 , · · · } of D' (Ao(ll;)) is invariant under il �-+ 1 - il. Fix an i � t. As F;; � F�i x F�:' acts on D'Ao(ll;), Ao(ll;) splits further : A0(ll;) "' A0(q,) x A0(q ;) , D'Ao(lJ,) � D' (A 0(q ,) ) x D ' ( A0 (q ;)) . Since F�; c End0(A(q,)) has degree d; = height(A0(q,)) over Qp, A(q,) is isogenous to a power of a simple p-divisible group : we may write D' A(q,) = Dk;ld;, 0 � k, � d,. Correspondingly, D' A(q ;) Dk �ld; with k; + k; = d,. Fix an i > t. The [F�; : Qp] 2d, = height Ao(ll;) and so, as above, Ao(ll;) is isogenous to a power of a simple p-divisible group and we may write DA0(lJ;) = D• 1r. Since sfr 1 - sfr we must have sfr = d,f2d,. We write k, d,/2. Note that for some i, 1 � i � m, we must have k, :1- d;/2 for otherwise all slopes of A(p) would equal f. Then (see §3 and 5 . 1 (a)) l 1t'fq 1 12 1 1 for all primes v of Q[11:] and so some power of it would equal one. On replacing Fq by a larger finite field we would have 11: = q 1 12, and this would imply that A is isogenous to a power of a supersingular elliptic curve, i.e., we would be in case (a). This means that t � 1 -at least one prime lJ; splits in F'. =

=

=

=

=

THEOREM 5.4. fp contains one element for each pair (F ' , (k;)v;;, ; �m ) where F' is a totally imaginary quadratic extension of F which splits B and is such that at least one lJ; splits in it ; if p, splits in F' then k, is an integer with 0 � k, � d, and otherwise k, = d,/2 ; /or at least one i, k, :1- d; - k,. When p, splits in F' we regard k; and k; as being associated to q, and q;, and we do not distinguish between two pairs (F', (k;)) and (F', (k;)) which are conjugate over F. There is one additional "supersingular" element.

1 80

J. S.

MILNE

For example, if F = Q then there is the supersingular isogeny class and one class for each quadratic imaginary number field F' which splits B and in which p splits. If p splits completely in F then there is the supersingular class and one class for each totally imaginary quadratic extension F' of F of the right type and choice of one out of each pair of primes dividing a lJ; which splits in F' ; one family of choices is not distinguished from the opposite family. PROOF OF 5.4. We first construct an isogeny class (A, i) ® Q corresponding to (F', (kh . · km)) . As before we let lJ;, 1 � i � t, be the primes dividing p which split in F'. Consider the ideal in OF'• .

,

where f = 2d1 dm . For some h, ah is principal, say ah for the nontrivial F-automorphism ofF' then 12:1t e F and . . •

= (12:) .

If we write a f-+

ii

Thus 12:1t = upfh with u a unit in OF . If u is a square in F then we may replace 12: by 12:/u 1 1 2 and obtain an equation 12:1t = q with q = pfh. If u is not a square then we replace 12: by 12:2/u and obtain a similar equation with q = p2fh. Note that the condi­ tion k ; #- k; for some ; implies that 12: ¢ F and hence that F' = F[12:]. Under any embedding F' 4 C, F maps into R. Thus complex conjugation on C induces a f-+ ii on F'. In particular 1t is the complex conjugate of the complex number 12: and so 12:1t = q implies that l1rl = q 1 1 2. Let A" be the abelian variety corresponding, as in 5. 1(c) to 12:, and let E = End0(A"). For any prime v of Q[1r] invv(E)

= = =

if vlp, (k,-/d;) [ Q[7r] 0 : Qp] if v IP and q;Jv, (k;/d;) [Q[7r]0 : Qp] if v jp and q;J v. 0

Let e0 be the denominator of invv(E) (when it is expressed in its lowest terms) and let e be the least common multiple of the e0 • Then 2 dim(A") = re where r = [Q[1r] : Q] . Clearly evlfF; : Q[7r]0] for any q j v, vjp, which implies (by class field theory) that F' splits E and (trivially) that e divides [F' : Q[1r]] = 2dfr. As [M2d ; re(E) : Q[1r]] = (2dfre) 2e2 = [F' : Q[1r1] 2, F' embeds into M2d l re(E) [19, Theorem 10]. Let A0 = Aidlre . The characteristic polynomial PA.(T) of 12: on A,. is c,.(T)• where c,.(T) is the minimal polynomial of 12: e Q[1r] over Q (5. 1(a)). Thus PAo(T) is c,.(T)2d l r which equals the characteristic polynomial of 12: e F' over Q. Corresponding to the splitting F; = F;1 x F�; x . . , we have A0(p) "' A0(q1) x A0(qi) x . . . and PAo(T) = P1(T)P{(T) . . . where P,(T) (resp. P;(T)) is the char­ acteristic polynomial of the image 12:; of 12: in F;; (resp. 12:; of 12: in F�'.) over Qp. Thus (see §3) A(q;) has slopes equal to ordq{7r;) = (fh/d;)k,/fh = k;d; ana A(q;) has slopes equal to k;/d,. Thus A = A0 x A0, regarded as an abelian variety over Fp. and the map i induced by B 4 M2(F') represent an isogeny class corresponding to (F', (kl > . . . , km)) . (A �d, i), where A 0 is a supersingular elliptic curve, represents the supersingular class. .

POINTS

ON

SHIMURA VARIETIES

mod p

181

Obviously if (A, i) (A', i') then both represent the supersingular class or correspond to the same pair (F', (kl> km)). It remains to show that if (A, i) and (A', i') both correspond to (F', (k l > . . . , km)) then (A, i) - (A', i'). By considering A and A' to be defined over some finite subfield of Fp we get elements n = nA e F' and n' = nA' e F'. The assumption implies that ordq(tr:) = ordq(n') for all q l p, q a prime ofF', and 5. 1 (a) then shows that l n/n' lv = 1 for all primes of F'. Thus n and n' differ by a root of 1 and so, after extending the finite field, we may take them to be equal. It follows that A and A', being isogenous to powers of the same abelian variety A,., are themselves isogenous, and 5.3 com­ pletes the proof. The proof that any class in ..Fp is represented by an element of S(Fp) requires the following lemma. "'

· · · ,

5.5. Let T c T1A be such that T1A/T is finite ; then there exists an isogeny A such that T1a maps T1A' isomorphically onto T.

LEMMA

a : A'

�

PROOF. The finiteness of T1A/T means that, for all n :> 0, the cokernel N of TfnT � T1A/nT1A is independent of n. Thus there is a map A n = T1A!nT1A .1� N. Define A' to be the co kernel of a ...... (ifJ (a), a) : A n � N x A, and a : A' � A to be (b, a) ...... na ; then (T1a)(T1A') = T. Let (A, i) represent a class in ..Fp and let 0' = B n End(A) ; it is an order in B. Regard (A, i) as being defined over a large finite field Fq ; then End(A) ® Z1 � Endp .(T1A) and 0' ® Z1 = (B ® Z1) n Endp 0(T1A). For almost all l, 0' ® Z1 will equal 08 ® Z1 and we take T1 = T1A ; for the remaining l we may choose a Tz of finite index in TzA which is stable under OB, i.e., such that Endp .(Tz) n (B ® Z1) = 08 ® Z1• Note that D(Tp/PTp) = M is a W[F, V]-module of finite length over W. We may choose Tp such that MfFM satisfies (4. 1). Let A' correspond to T = fi 1T1 as in the lemma. Then A' together with the obvious i and some (fi lies in S(Fp) and represents (A, i) ® Q. 6. An isogeny class. It remains to describe the set Z = Z(A, i, ifJA) of elements (A', i', (fi') of S(Fp) such that (A', i') is isogenous to a given pair (A, i). An 08-isogeny A' � A determines an injective map T1a : T1A' � T1A whose image A satisfies the following conditions : (a) A is 08-stable. (b) T1A/A is a finite group scheme. (More precisely, Coker(A/nA � T1AjnT1A) = Coker(A� � A n) = Ker(A' � A) for n :> 0.) (c) D(A/pA)/FD(A/pA) satisfies (4. 1). (For A/pA = Af, and so D(A/pA)/FD(A /pA) =

DA/F(DA).)

Consider all subobjects A of T1A satisfying (a), (b), (c). Any such A may be written A = AP x Ap with AP a Z�-lattice in T�A (in the usual sense of modules over Z�) and Ap c TpA. We let Y be the set of pairs (A, (/J) with A as above and (fi a K-equivalence class of isomorphisms AP � V(Z�). Since (A, (/J) is determined by a pair ((AP, (/J), Ap), we may write Y = YP x Yp . By 5.5, every (A, (/J) e Y arises from a triple (A', i', (fi') e S(Fp) equipped with an isogeny a ; A' � A. Thus we have a surjective map Y -� Z c S(Fp). For n a positive integer we set n(A, (/J) = (nA, n(/J), and we define Y ® Q to be the set of pairs (y, n) with y e Y and n e Z>o• where (y, n) and (y', n') are identi-

1 82

J. S. MILNE

fled if n'y = ny'. Then we write Y ® Q = ( YP ® Q) x ( Yp ® Q) where YP ® Q may be identified with the set of 08-stable lattices A in ( T1A) ® Q equipped with a K-equivalence class of isomorphisms A � T1A. There is an action of H(Q) = End0iA)x on Y ® Q: for a e H(Q) we choose a positive integer m such that rna is an isogeny of A and define a(A , qi , n) = ( T1(ma),

ifJ T1(ma)-l, mn). z.

LEMMA

6. 1 . The map Y -+ Z described above induces a bijection H(Q)\ Y ® Q �

PROOF. (A, qi , n) and (A', qi ', n') map to the same element of S(Fp) if and only if there exist D8-isogenies a

A' ==:' A and an DB-isomorphism f/Jo : T1A '

-+

a

V(Z1) such that

n (( T1a) T1A ' , ifJ0(T1a)) = (A, qi) and n' (( T1a' ) T1A ' , ifJ0( T1a ')) = (A', qi').

Then a ' a-1 makes sense as an element of End0(A) and a ' a-1 (A, qi , n) = (A', qi' , n'). LEMMA 6.2. The map G(A,) -+ YP ® Q, g �--+ (g ( T1A ) , ifJAg- 1 ), induces a bijection G(A ,)/K -+ YP ® Q. PROOF. Obvious. LEMMA 6.3. There is a one-one correspondence between Yp ® Q and the set X of W[F, V]-submodules M of D' A which are free of rank 4d over W, DB-stable, and

such that M/FM satisfies (4. 1).

PROOF. p: A -+ A induces maps in : AjpnA 4 Afpn+lA which define a p-divisible group A(p) = (AfpnA, in). The exact sequence 0 -+ A -+ TpA -+ N -+ 0 (N finite) gives rise to 0 -+ N -+ A(p) -+ A(p) -+ 0. On applying D we get 0 -+ DA -+ DA(p) -+ DN -+ 0. Since DN is torsion, we may identify D'A(p) with D'A. To (A, n) e YP we associate

n- 1 (DA(p)) e X. THEOREM

6.4. With the above notations,

Z(A, i, qi) � H(Q)\G(A,)

x

X/KP .

Frob acts by sending M e X to FM; the Heeke operator .r(g), g e G(A,), "acts" by multiplication on the right on G(A,). This simply summarizes the above. It remains to give a more explicit description of X. Note that, corresponding to the splitting D'A � D'A(p 1) X X D 'A ( P m) , we have X � x X Xm . 1 X It is convenient to write G;(Zp) = Aut08(A(p;)) and G;(Qp) = End08(A(p;)) x = End08(D' A(p;)) x . In the simplest cases G;(Q p) acts transitively on the lattices M c D' A(p;) which belong to X;, and in this case X; � G;(Qp)/G;(Zp). (To say that G;(Qp) acts transitively means that each A'(p;) is isomorphic to A(p;) and not merely isogenous ; cf. [6, p. 93].) PROOF.

. . •

• • •

POINTS ON SIDMURA VARIETIES

mod p

183

EXAMPLES 6.5. (a) F = Q , F ' is a quadratic extension o f Q , (p) = qq ' i n F', and (A, i, �) is in the isogeny class corresponding to (F', (0)). Then A(p) � (Qp/Zp)2 x (f.'poo)2 with 08 ® Zp = M2(Zp) acting in the obvious way on each factor. Thus G(Zp) = {(3 g) Ia, e Zp } and G(Qp) {(3 g) Ia, b e Q; } . In this case X = G(Qp)/G(Zp). Frob acts as (ij �). (b) As above, except (A, i, �) corresponds to (F', (1)). Then A(p) � (f.'poo)2 x (Qp/Zp)2 (i.e., in the splitting Fp = F� x F�,, F� now corresponds to the f.'poo factor). G(Zp), G(Qp) and X are as before but Frob acts as (� �). (c) F = Q, (A, i, �) is in the supersingular class. Then D'A(p) � D l / 2 x D l / 2 and End(D l 1 2) = Bp, the unique division quater­ nion algebra over Qp. B acts through the embedding B ® Qp � M2(Qp) � M2(Bp). Thus G(Qp), the centralizer of B ® Qp in M2(Bp) is (B�)x . Moreover G(Zp) may be taken to be o x where 0 is the maximal order in Bp. In this case X � G(Qp)/G(Zp) · Frob acts as multiplication by w, a generator of the maximal ideal of 0. (d) F arbitrary, p splits completely in F, (p) = lh · · · lJd , (A, i, �) corresponds to

b

.

(F', (k l > · · · , kd)) . Then X � xl

X • • • X xd where X; is as in case (a) if lJ; splits in F' and k; = 0, as in case (b) if lJ; splits and k; = 1 , and as in case (c) otherwise. (e) The general case. For a statement of the result, see [14]. (This case is treated in detail in : J. Milne, Etude d'une classe d'isogenie, Seminaire sur les groupes reductifs et les formes automorphes, Universite Paris VII ( 1977-1978).) Added in proof (November 1 978). The outline of a proof in [12] of the conjec­ ture for those Shimura varieties which are moduli varieties is less complete than appeared at the time of the conference. The above proof (completed in the report referred to in 6.5(e)) for the case of the multiplicative group of a quaternion algebra differs a little from the outline in that it depends more heavily on the Honda-Tate classification of isogeny classes of abelian varieties over finite fields. The complete seminar referred to in 6.5(e), which redoes in greater detail much of the material in this article and [3], will be published in the series Publications

Mathematiques de l'Universite Paris 7.

BIBLIOGRAPY 1. M. Artin, The implicit function theorem in algebraic geometry, Colloq. in Algebraic Geometry, Bombay, 1 969, pp. 1 3-34. 2. , Theoremes de representabilite pour les espaces algebriques, Presses de l'Universite de Montreal, Montreal, 1973. 3. W. Casselman, The Hasse- Wei/ �-function of some moduli varieties of dimension greater than one, these PROCEEDINGS, part 2, pp. 141-163. 4. P. Deligne, Travaux de Shimura, Seminaire Bourbaki, 1 970/71 , no. 389. 5. P. Deligne and M. Rapoport, Les schemas de modules de courbes elliptiques, Antwerp II, Lecture Notes in Math. , vol. 349, Springer, New York, pp. 143-3 1 6. 6. M. Demazure, Lectures on p-divisible groups, Lecture Notes in Math., vol. 302, Springer, New York, 1972. 7. T. Honda, Isogeny classes of abelian varieties over finite fields, J. Math. Soc. Japan 20 (1 968), 83-95. --

1 84

J. S.

MILNE

8. Y. Ihara, The congruence monodromy problems, J. Math. Soc. Japan 20 (1 968), 107-121. 9. On congruence monodromy problems, Lecture Notes, vols. 1 , 2, Univ. Tokyo, 1 968, �� ,

1969. 10. Non-abelian class fields over function fields in special cases, Actes Congr. Internat. Math., vol. 1, Nice, 1 970, pp. 381-389. 11. Some fundamental groups in the arithmetic of algebraic curves over finite fields, Proc. Nat. Acad. Sci . U.S.A. 72 (1 975), 3281 -3284. 12. R. Langlands, Letter to Rapoport (Dated June 1 2, 1 974-Sept. 2, 1 974). 13. Some contemporary problems with origins in the Jugendtraum, Proc. Sympos. Pure Math. , vol. 28, Amer. Math. Soc., Providence, R. I., 1 976, pp. 401 -418. 14. Shimura varieties and the Selberg trace formula, Canad. J. Math. 29 (1 977), 1 2921 299. 15. Y. Morita, Ihara's conjectures and moduli space of abelian varieties, Thesis, Univ. Tokyo (1 970). 16. D. Mumford, Geometric invariant theory, Springer, 1 965. 17. Abelian varieties, Oxford, 1 970. 18. M. Rapoport, Compactifications de l 'espace de modules de Hilbert-Blumenthal, Compositio Math. (to appear). 19. J. P. Serre, Expose 7, Seminaire Cartan 1 950/5 1 . 20. H . Swinnerton-Dyer, Analytic theory of abelian varieties, L.M.S. lecture notes 14 (1 974). 21. J. Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1 966), 1 341 44. 22. Classes d'isogenie des varietes abeliennes sur un corps fini, Seminaire Bourbaki, 1968-1 969, no. 352. 23. W. Waterhouse and J. Milne, Abelian varieties over finite fields, Proc. Sympos. Pure Math. , vol. 20, Amer. Math. Soc., Providence, R.I., 1 971 , pp. 53-64. ��,

�� ,

�� ,

�� ,

��,

�� ,

UNIVERSITY OF MICill GAN

Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 2, pp. 1 85-1 92

COMBINATORICS AND SHIMURA VARIETIES

mod p (BASED ON LECTURES BY LANGLANDS)

R. E. KOTTWITZ One of the major problems in the study of Shimura varieties is that of expressing their Hasse-Weil zeta-functions as products of L-functions associated to auto­ morphic forms. In [1] this problem has been solved for a certain class of Shimura varieties : those associated to an algebraic group G over Q which is the inverse image under ResnQ(B*) � ResF1Q(F*) of some connected Q-subgroup of ResF1Q(F*), where B is a totally indefinite quaternion algebra over a totally real number field F. In this paper we will discuss only the case G ResF1Q(B*). In this special case, the zeta-functions of the corresponding Shimura varieties can be expressed as products of automorphic L-functions associated to the group G itself (in general, groups besides G are needed as well). This case has already been discussed in [2] and the purpose of the present discussion is to provide further details on the com­ binatorial exercise referred to on the last page of that paper. This is worked out in detail in §4 of [1] for all of the subgroups of ResF1Q(B*) mentioned previously, but even so it is interesting to carry out the exercise in our situation (with further sim­ plifying assumptions added later) so that the main ideas can be understood more easily. First we sketch the arguments of [2] which lead to the combinatorial exercise. Let K be a compact open subgroup of G(A1) (where A1 is the ring of adeles of Q having component 0 at oo ). Let SK be the Shimura variety obtained from G and K as in [2]. The variety SK is defined over Q since B is totally indefinite, so its zeta­ function Z(s, Sx) is a product over the places v of Q of local factors Zv(s, Sx) . Let F' be a finite extension field of F which is Galois over Q. For a E Gal(F'/Q)/Gal(F'/F), let H" GL2(C) and let V" be the standard representa­ tion of H(f on C 2 . Then LGO = n (f H(f has a natural representation r on v 0" V(f , which may be extended to a representation of the L-group LG Gal(F'JQ) 1>< LG o by putting r (r) (@" v") = ®" w" for 1: E Gal(F'/Q) where.J:V" v,-1 " . The result of [2] is that Z(s, Sx) = fLr L(s - d/2, n, r)m C1r, K l up to a finite number of local factors, where d [F: Q], n runs over the representations of G(A)/Z(R) (where Z is the center of G) which occur in V ( G(Q)Z(R)\G(A)), and m(n, K) is an integer which is associated to n and K in a way that is described in [2]. The product over n is actually finite since m(n, K) turns out to be 0 for all but a finite number of n. The precise result is =

=

=

=

=

=

AMS (MOS) subject classifications (1 970). Primary 05C05, 14Gl0, 22E50, 22E55. © 1 9 7 9 , American Mathematical Society

1 85

1 86

R. E. KOTIWITZ

Zp (s, SK)

(1)

= n "'

L(s - d/2, 1Cp , r)m C n , K l

where 11:P is the local factor of 11: at a finite rational prime p which satisfies : (A) F is unramified at p ; (B) B splits at every prime ofF above p ; (C) K = KpKP where Kp i s a maximal compact subgroup of G(Q p) and KP i s a compact open subgroup of G(A�), A� being the ring of adeles of Q with component 0 at p and oo . For the rest o f this paper p will denote a fixed rational prime satisfying (A)-(C). Because of these assumptions about p, SK has good reduction at p, and m(11:, K) = 0 unless 1T:p is unramified. Take the logarithm of each side of (1), and use the power series expansion of log(l - X). Comparing the coefficients of the powers of p-s in the two sides, we see that (1) is equivalent to (2) Card SK (Fpi) = � m(11:, K)pid 1 2 tr(r(g")i) "

for all positive integers j, where 11: runs over all the representations of G(A)/Z(R) which are unramified at p and which occur in V(G(Q)Z(R)\G(A)), and where g"P is an element of the semisimple conjugacy class of LG associated to 11:p · For every place v of Q choose a Haar measure dg. on G(Q.) such that the measure of G(Z.) is 1 for almost all finite places v . Use dg. to define the action of L l (G(Q.)) on representations of G(Q.). Choose a Haar measure dzoo on Z(R), and use the quotient measure dgoo/dzoo to define the action of V(G(R)/Z(R)) on representations of G(R)/Z(R). LetfP be the characteristic function of the subset KP of G(A�) divided by the measure of KP (with respect to 11 "*P· dg.). It is pointed out in [2] that there exists f E ceca( G(R)/Z(R)) such that m(1C, K) = m(1C)tr 1CP{fPfoo) where 1f:P = ®v,p1T:v and m(11:) is the multiplicity of 11: in V(G(Q)Z(R)\G(A)) (it is known that this multiplicity is I , but we will not need to use this fact). By the theory of Heeke algebras, there exists a unique function fPl in the Heeke algebra .?t'(G(Qp). Kp) such that tr 1T:p(fjil) = pid / 2 tr(r(g"P)i) if 1T:p is unramified, = 0 if 11:p is ramified. co

00

So the right side of (2) becomes :E n m (11:) tr 1T:(fPfPfoo), which can be evaluated in terms of orbital integrals by means of the trace formula. For r E G(Q), let dg be the quotient of 11 .dg. on G(A) by some Haar measure dg7 on Gr(A) (where as usual G7 denotes the centralizer of r in G), and let m7 = meas(GiQ)Z(R)\G7(A)) where the measure used is the quotient of dg7 by the measure on Gr(Q)Z(R) � G7(Q) x Z(R) which is the product of the Haar measure on G7(Q) which gives points measure 1 with dzw Applying the trace formula, we find that (2) is equivalent to (3)

where r runs over a set of representatives of the conjugacy classes in G(Q). It turns out that the integral fcrCR) \G (R) foo(r 1 rg) dgoo is 0 unless r is central or elliptic at oo , and so the right side of (3) is

1 87

COMBINATORICS

(4)

where a7 = m7 fGr (R) \G(R) foo(g-1 rg) dg"" and r runs over a set of representatives for the conjugacy classes in G(Q) which are elliptic or central at oo . To g o further we must know more about Card SK(Fpi) · To know this number for all j it is enough (in principle anyway) to know explicitly the set SK(Fp) and the action of the Frobenius (/) on this set. Assumption. To make this description simpler, we assume from now on that p remains prime in F. First of all, for every triple p = (L, m', m") consisting of a totally imaginary quadratic extension L of F which is split at p and can be embedded in B, and in­ tegers m', m" such that m" > m' ;;; 0 and m' + m" = d, there is an associated

f/J-stable subset Yp of SK(Fp), and SK(Fp) is the disjoint union of the sets Yp and another (/)-stable subset Y0 • Let IP be L * regarded as an algebraic group over Q. It turns out that YP is isomorphic as a Gal{Fp/Fp)-set to lp(Q)\(G(A9) x Xp)/KP for some set Xp on which lp(Qp) and (/) act, the two actions commuting with each other. An embedding L � B gives us a map IP � G, and lp(Q) acts on G(A9) via Jp(Q) � Ip(A9) � G(A9) ; also Ip(Q) acts on Xp via Ip(Q) � lp(Qp), and KP acts on the pro­ duct G(A9) x XP by acting on G(A9) alone. For any positive integer j and for x E XP , let Tt = {g E lp(Qp) : f/Jix = gx}, and let ot be the characteristic function of Tt. Choose a Haar measure p. on lp(Qp), and for r E IP(Q) let (5)

rp (r) =

1-

-

p.(l,.) ot(r) where I,. is the stabilizer of x in lp(Qp). Choose a Haar measure v on G7(A9). Let v' = TI # p, oo dgv where dgv is the Haar measure on G(Q0) chosen before, and recall that we have defined fP to be the characteristic function of KP divided by v'(KP). It is easy to show that the contribution of YP to Card(SK(Fpi)) is 1:

xEXp mod lp (Qp)

J

-1

"' bT rp (T) G (A� \G(Af} fP(g T g) dv' d"u T f f where b7 = meas(Ip(Q)\G7(A�) x lp(Q p)). The measure used is the quotient of v x p. by the measure on lp(Q) which gives every point measure I . We see that (4) and (6) have roughly the same form, but we must sum (6) over all triples p = (L, m', m"), and it appears at first that there is nothing in (4) which corresponds to the sum over m', m". But in fact we will now see that fPl can be written in a natural way as a sum of functions, and it is this sum which corresponds to the sum over (6)

.f...J

T E [p (Q)

m', m".

The function fPl in the Heeke algebra .Yf = .Yf( G(Qp). Kp) is characterized by the equation tr 1rp(Jp>) = pid / 2 tr(r (g,.P)i) for all unramified representations 1rp of G(Qp). Since p remains prime in F, Fp is a field (of degree d over Qp) and G(Qp) � GL2(Fp ) (since we assumed that B split at every prime of F above p). Regarding GL2(Fp) as the Fp points of GL2 , we get an isomorphism f �--+ JV from .Yf to the algebra of polynomials in a, b, a- 1 , which are symmetric in a, b (a and b are two indeterminates). Regarding GL2(Fp) as the Qp points of ResFJQ GL2, we get an iso­ morphism f �--+ 1- from .Yf to the algebra of functions on LG o x { (/)} which are

b- 1

188

R . E. KOTTWITZ

obtained by restricting linear combinations of characters of finite dimensional complex analytic representations of the complex Lie group L G. The relation between fV .and /- is · a = !v a 1 0 d x � x ... x J- g1 gd b 1 .� b . We want tofindjpl e Yf such that (Jpl)- (x) = pjd t 2 tr(r(xj)) forallx e L Go x {�} . Let I be the greatest common divisor ofj and d. Then for

(( gJ

( gJ )

((

) (

J)

((

we have tr r(xJ) = tr r a1 ' () aj b .?. b x a2 ' () aj+ l b � b + x 2 1 j l = ((a 1 . . . adV 1 1 + (b 1 . . . bd)j 1 1) 1 . It follows that up))Y((8 g)) = pjd /2 (aj / 1 + bj l l)l. For k' E z with 0 � k' let .fi. k' be the function in the Heeke algebra such that flk' = pjd 12 (ajk' ll bjk"ll + ajk"ll bjk'll) .

)

<

1/2,

(k•)

where k" = I - k'. If I is even, then for k' = k" = 1/2, let.fi. k' be the function in the Heeke algebra such that flk' = pjd 12(L ,)ajk' llbjk'll . By the binomial theorem we � k[1'=/Zlf. have Jl'p(j) - ,i,.,j O J, k' · Now consider a triple p = (L, m', m"), and let r e L - F. Recall that m" is determined by m' (since m' + m" = d). Choose an embedding L --+ B. Then r gives us an element of lp(Q) and of G(Q). We will show that the contribution of r to (6) is 0 unless m' is divisible by djl, in which case it is equal to the contribution of k' = lm' fd and r to the following rewritten version of (4) : wz1 dv' dgp � k'fO aT Gr (A�) \G(A�JP(g-1 r g) dv Gr (Qp) \G(Qp) ./j, k' (g-1 r g) dgT where dgr is the Haar measure on Gr(Qp) corresponding to the Haar measure f-t on Ip(Qp) under the isomorphism lp(Qp) ...::::. Gr(Qp) induced by the embedding L --+ B. It can easily be shown that ar = br with the choice of measures that we have made, so it is enough to show the following : THEOREM. With notations as above, rp < jl (r) = 0 unless dfl divides m', in which case it is Scr (Qp) \G(Qp) .fi. k' (g- 1 rg) (dgpfdgr) where k' = lm' fd. We begin the proof of this theorem by evaluating Scr (Qp) \G(Qp) fj, k' (g- 1 r g) . (dgp/dgr) · This is easy since L splits at p and the hyperbolic orbital integrals of any f e Yf can be read off from JV. Let a , (3 e F! be the two eigenvalues of r and assume without loss of generality that the valuation v(a) of a is less than or equal to the valuation v((3) of (3. Then the integral ofjj, k' over the orbit of r is equal to 0 unless v(a) = jk'/1 and v((3) = jk"/1, in which case it is

J

J

la f3/(a - (3)2 1

�>jd 2 (k)meas(/ )-1) /

'

o

COMBINATORICS

1 89

where /0 is the subgroup of lp(Qp) corresponding to @pP x @pP under Ip(Qp) ....::. x FJ . The final result is that the integral is equal to O unless v(a) = jk'/1 and v(,B) = jk"jl, in which case it is (k, )pik'dlt f.J.(/0)- 1 . We must now evaluate rpi(r). This is the combinatorial exercise referred to in the beginning of this paper. First we will describe XP and the actions of lp(Qp) and r!J on XP . Let Qpn be a maximal unramified extension of Qp containing Fp, let cgpn be its valuation ring, let BIJ' be the set of @'j,"-lattices in the two dimensional vector space Qpn EB Qj," over Qj,", and let BiJ be the set of classes of lattices in Qpn EB Qpn (two lattices Mt. M2 are said to be in the same class if there exists a nonzero element c E Qj," such that M1 = cM2). For any lattice M, let M denote the class of M. The set BiJ is the set of vertices of the Bruhat-Tits building of SL2(Qj,"). This building is a tree. The groups GL2(Qpn) and Gal (Qj,"/Qp) act on BIJ' and BIJ. Let FJ

B = (b

m'

)

0 " pm '

and let (J be the Frobenius element of Gal (Qj,"/Qp). Let XP be the set of sequences {M,.} iEZ of elements M,. of BIJ' satisfying : (A) M; � M,._1 � pM,. for all i E Z; (B) B(JdM; = Mi-d for all i E Z. The action of r!J on XP is given by r!J : {M,.} ...... {M;,} where M; = M,._ 1 . Identify lp(Qp) with A(Fp), where A = m m . The action of an element r E lp(Qp) on XP is given by r{M,.} = {rM,.}. Recall that we are trying to calculate rpi(r) = L: x f.J.(lx)- 1 ot(r), where the sum is taken over x E XP mod lp(Qp)· Let /1 be the subgroup of lp(Qp) = A(Fp) consisting of all matrices of the form where n E Z and c E FJ . As before let /0 = A (@pP) . Then lp(Qp) = /0 x It. Furthermore f.J-(10) = f.J.(lx) [10 : lx] = f.J.(lx) [Ip(Qp) : /1 /x], so rpi(r) is equal to f.J.(I0)- 1 Card S(r, j) where S(r, .i) = {x e XP mod /1 : r!Jix = rx} . Let � be the apartment in BiJ corresponding to the split torus A. Choose a point p0 in �, and let o1t = { x E BiJ : p0 is the point of � nearest x} . Let o!t' be a set of representatives for the lattice classes in o!f, or in other words, let o!t' be a subset of BIJ' such that no two elements of o!t' are in the same class and such that { M: M E A'} = o!t. Then the set of {M,.} E XP which satisfy (C) M0 E o!t' is a set of representatives of XP mod It. The condition r!Ji{ M,.} = r{ M,.} just says (D) r M,. = Mi-i · We can summarize this discussion by saying that S(r,.i) is the set of all sequences of elements of BIJ' satisfying (A)-(D). Any sequence {M,.} in S(r, j) determines a sequence { x,.} = { M,.} of elements of BiJ satisfying : (A') x,. is a neighbor of x,._1 ; (B') B(JdX; = X i-d ; (C') xo E A ; (D') rx,. = x,._i · Let S' (r, j) be the set of sequences { x,.} satisfying (A')-(D'). If the valuation of

1 90

R.

E.

KOTTWITZ

det(r) is not equal toj, then conditions (A) and (D) are incompatible, and S(r, j) is empty. So from now on we assume v(det r) = j. In this case, any element of S'(r, j) can be uniquely lifted to an element of S(r,j) (to check this one needs to use the fact that v(det B) = d) . So Card S(r, j) = Card S'(r, j) if v(det r) = j. Let r, s e Z be such that rj + sd = I. Conditions (B') and (D') together are equivalent to the two conditions : (E') qdi/1 x; = B-j / 1 rd/1 x; ; (F') r' B•q•dx; = X i-1 · We will now show that S'(r, j) is empty unless B-i 1 1rd 1 1 e /0• Suppose S'(r, j) is nonempty, and let {x;} e S'(r, j). Then p0 is the point of � closest to x0, so ?:Po is the point of 1:� closest to ?:Xo, where 1: = q-dj / 1 B-i l l rd l l, But ?:Xo = Xo by (E'), and 1:� = � since q� = �. B � = � and r� = �. So 1: fixes p0• Since q-di 1 1 fixes p0, we conclude that B -i l lrd l l fixes p0. But B-i l lrd l t is diagonal and its determinant is a unit (since we are assuming that v(det r) = j), so that the fact it fixes Po im­ plies it belongs to /0• So we have shown that S'{r, j) is empty unless B-i 1 1ra 11 e /0. Let a, f3 e FJ be the diagonal entries of r. so that r = {3 Z). Then B-i11rd1 1 e lo if and only if v(a) = jm' Id and v(f3) = jm"Id. In particular we see that q;i(r) = 0 for all r unless jm' ld and jm"ld are integers. Recalling that I = (j, d) and that m' + m" = d, we see thatjm'fd, jm"/d e Z if and only if dll divides m', andthat if dll does divide m', then rpi(r) is still zero unless v(a) = jm'ld = jk'/1 and v(f3) = jm"/d = jk"I I. Comparing this with the orbital integrals we have computed, we find that all we have left to show is the following lemma. LEMMA. Suppose B-i 1 1 rd l l e /0• Then Card S'(r, j) = (L,)pik' dll where k' = lm'ld. To prove this we first look more closely at conditions (E') and (F'). Let PJ� = {x e PJ : 1:(x) = x}, where 1: is the automorphism of PJ defined previously. Then (E') simply says that X; e (14� for all i e Z. Let Qj, be the completion of Q[,n. There exists a unique continuous action of GL2(Qj,) on PJ which extends the action of GL2{Q'f,n) on PJ (PJ is given the discrete topology). Each element of GL2(Qp) acts by a simplicial automorphism of the tree PJ. Using the fact that B-i 1 1rd 11 e /0, it is easy to show that there exists a diagonal matrix o with diagonal entries in Qj, - {0} such that o"jd/l ()- 1 = B-;ttrdtl. A simple calculation shows that PJ� = oPJ"djll where P4"di/l denotes the set of fixed points of qdi 1 1 in PJ. The subtree PJ"dj/l of PJ can be canonically identified with the Bruhat-Tits building of SL2 over the fixed field of Q[,n under qdi l t. So PJ"di/l is a tree in which every vertex has exactly pdi 1 1 + 1 neigh­ bors. Since (14� is simply the translate of PJudill by o, it too is a tree in which every vertex has exactly pdi l l + 1 neighbors. A further consequence of B-i 1 1rd l l e /0 is that PJ� contains �. The reason for this is that qdi 1 1 fixes � pointwise, as does every element of /0. In (F') the automorphism r' B•qsd of PJ appears. We will need to know the effect of this automorphism on �. The automorphism qsd acts trivially on �. The automorphism B acts on � by translation by m" - m'. The automorphism ra acts on � in the same way that Bi does, since B-i l lrd l l e /0, and therefore r acts on � by translation by (m" - m')jld. So r'Bsqsd acts on � by translation by I - 2k'. This discussion has reduced our problem to that of proving the following lemma. LEMMA. Let T be a tree in which every vertex has q + 1 neighbors, and let � be a

191

COMBINATORICS

subtree of T in which every vertex has 2 neighbors. Let p0 e �- Let I and k be nonnega­ tive integers such that I - 2k is positive. Let � be an automorphism of T which acts on � by translation by I - 2k. Let N(l, k) be the set of sequences of length 1 + 1 ofpoints x0, · · · , x1 in Tsatisfying : (i) X; is a neighbor of X;+l for i = 0, · · · , I - 1 ; (ii) Po is the point of � closest to xo ; (iii) XI = exo. Then N(l, k) = (i)q k .

Let us agree to call a sequence of points x0 ,

I joining x0 to x1• Then

· · ·,

x1 satisfying (i) a path of length

N(l, k) = .E Card {paths oflength I joining Xo to exo}, XQE./1

where A is the set of x0 e T which satisfy (ii). Let a, b e Z with a � 2b � 0, and let x, y be two vertices of T at distance a 2b apart. Then the number of paths of length a joining x to y is independent of the choice of x and y, and will be denoted by M(a, b). The figure below shows that the distance between Xo and exo is equal to I - 2k + 2n where n is the distance from x0 to p0. Xo

n

So N(l, k) = .E �=o Card {x0 e A : distance from x0 to Po is n} M(l, k - n). The number of points x0 at distance n from p0 such that p0 is the point of � closest to x0 is (1 - q-1)q n if n > 0 and is 1 if n = 0. Therefore N(l, k)

=

M(l, k) + ( 1

- q-1) .E �=l q n M(l, k - n).

The function M(l, k) on the set of pairs of integers (/, k) such that l � 2k � 0 is characterized by the three properties : (i) M(l, 0) = 1 for 1 � 0 ; (ii) M(l, k) = (q + I)M(l - 1 , k - 1) for l = 2k > 0 ; (iii) M(l, k) = M(l - 1 , k) + qM(l - 1 , k - 1) for I > 2k > 0. Condition (i) and the fact that {i), (ii), (iii) determine M uniquely are both obvious, and (ii), (iii) become obvious if it is observed that to give a path of length l from x to y is the same as to give a neighbor x' of x and a path of length l - 1 from x' to y. For l, k e Z, let m(l, k) be the coefficient of XI Yk in the formal power series ex­ pansion of f(X, Y) = (I - q Y) (1 - Y)-1 (I - X - qXY)-1 . We will show that M(l, k) = m(l, k) for I � 2k � 0. It is enough to show that conditions (i)-(iii) hold with M replaced by m. From the definition off(X, Y) we get (1 - X - qXY) · f (X, Y) = (1 - q Y) (I - Y)-1 . Comparing coefficients of X1 Y k on the two sides we see that m(l, k) - m(l - 1, k) - qm(l - 1 , k - 1) = 0 if / > 0, which shows

1 92

R.

E.

KOTTWITZ

that m satisfies (iii). Setting Y = 0 in the definition off(X, Y) we get !: l=o m(l, O)X1 = (I - X)-1 = 1 + X + X2 + · · . , which shows that m satisfies (i). We still have to show that m satisfies (ii). Let g(X, Y) = ( I - X - qXY)-1 and let n(l, k) be the coefficient of Xt Yk in the formal power series expansion of g(X, Y). We have ( 1 - X - qXY)-1 =

� (X + qXY)I = 1� to (k)xl-k(qXY)k = I; (kl )qkX I Yk, t ;;,O; ;;,O

k and therefore n(l, k) = (f.)qk. In particular for k > 0 we have n(2k - 1 , k) = k k 1 q k = 2 � qk-1 q = qn(2k - 1 , k - 1).

( f l)

)

e

This shows that the coefficient of X2k-l Yk in (1 - q Y)g(X, Y) is zero. But (I - q Y) · g(X, Y) is equal to (l - Y)f(X, Y), so the coefficient of X2k 1 P in (l - Y) f(X, Y) is zero, which means that m(2k - 1 , k) = m(2k - 1 , k - 1) for all k > 0. (7) We have already seen that if I > 0, then m(l, k) = m(l - 1 , k) + qm(l - 1 , k - 1). Take I = 2k with k > 0. Then m(2k, k) = m(2k - 1 , k) + qm(2k - 1 . k 1). Combining this with equation (7) we see that m satisfies condition (ii). Going back to our discusion of N(l, k), we see that -

-

00

N(l, k) = m(l, k) + (1 - q-1) I; q nm(l, k - n) n=1 since m agrees with M whenever M is defined and m(l, k - n) is 0 for n > k. We may use the equation above to extend the definition of N(l, k) to all integers I and k. With this convention we find that ,E N(!, k)X1 Yk is equal to 00

I; m(l, k)X I Yk + (1 - q-1) I; q n I; m(l, k - n)X I Yk l, k n =l l, k = f(X, Y) + - q-1) I; qn yn I; m(l, k - n)Xt p-n l, k n =l = f(X, + (1 - q-1) q n yn = f(X, Y)(1 - Y)(l - q Y)-1 = g(X, Y). 1 We have already computed the formal power series expansion of g(X, Y) . Looking back at the answer, we find that N(l, k) = (f.) q k. This proves the lemma.

Y) [1

(1

00

�

]

REFERENCES 1. R. P. Langlands, On the zeta-functions of some simple Shimura varieties, Notes, Institute for Advanced Study, Princeton, N. J., 1 977. 2. Shimura varieties and the Selberg trace formula, Canad J. Math. (to appear). -- ,

UNIVERSITY OF WASIDNGTON, SEATTLE

Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 193-203

NOTES ON L-INDISTINGUISHABILITY

(BASED ON A LECTURE OF R. P. LANGLANDS) *

D . S HELS TAD These notes are intended as a brief discussion of the results of [4]. Although we consider essentially just groups G which are inner forms of SL2, we emphasize for­ mulations (cf. [7]) which suggest possible generalizations. We assume that F is a field of characteristic zero. In the case that F is local there are only finitely many irreducible admissible representations of G(F) which are "L-indistinguishable" from a given representation. We structure this set, an L-packet, by considering not the characters of the members but rather sufficiently many linear combinations of these characters. In the case that F is global we consider certain L-packets of representations of G(A) and describe the multiplicity in the space of cusp forms of a representation 1C = ®v1Cv in terms of the position of the local representations 1Cv in their respective L-packets. 1. �(T), 'll( T) and C(T). Suppose that G is a connected reductive group defined over F, any field of characteristic zero, and that T is a maximal torus in G, also defined over F. Fix an algebraic closure F of F. Then we set �{T) = {g e G(F) : ad g- 1 /Tis defined over F} and 'll( T) = T(F)\�(T)fG(F). If g E �(T) then O" � g" = O"(g)g- 1 is a continuous 1-cocycle of @ = Gal(F/F) in T(F). The map g � (O" � g") induces an injection of 'll( T)into a subgroup C(T) of H 1 {®, T(F)) defined as follows. Let T.c be the preimage of T in the simply-connected covering group Gsc of the derived group of G. Then C(T) is the image of the natural homomorphism of H l (@, T.c(F)) into H l (@, T(F)). If H 1 {®, G.c(F)) = 1 and so, in particular, if F is local and nonarchimedean then 'll{ T) coincides with C{T). L-indistinguishability appears when G contains a torus T such that 'll ( T) is non­ trivial. 2. Groups attached to G (local case). Assume now that F is local. Fix a finite Galois extension K of F over which T splits. We replace F by K and @ by @KIF = Gal(K/F) in the definitions of the last section. An application of Tate-Naka­ yama duality then allows us to identify C(T) with the quotient of {A. e X* (T.c) : .E a eQJK/F O"A = 0 } by

{A

E

X* (T.c) : A = 1:: O"fl-11 aE I/!JK /F

- fJ-11 , fl-11 E

}

X* (T) ,

AMS (MOS) subject classifications (1 970). Primary 22E50, 22E55.

*This work has been partially supported by the National Science Foundation under Grant MCS76-0821 8 . © 1 979, American Mathematical Society

1 93

1 94

D. SHELSTAD

X* ( ) denoting Hom(GLr. ). A quasi-character " on X* (Tsc) trivial on this latter module defines, by restriction, a character on &( T). In [7] there is attached to each triple (G, T, K) a quasi-split group over F (there denoted H). We will pursue this just in the case that G is an inner form of SL2 • 3. Groups attached to an inner form of SL2 (local case). Suppose that G is an inner form of SL2 and that F is local ; we will continue with this assumption until § 14. We have then two groups to consider : S4 and the group of elements of norm one in a quaternion algebra over F. We may take PGL2(e) as LGo (notation as in [1]) and the diagonal subgroup as distinguished maximal torus LT o . Fix a maximal torus T in G, defined over F. A quasi-character " from the last section is just a @K 1 rinvariant quasi-character on X* (T). We fix an isomorphism between X* (T) and X * (LT0) = Hom(LT0, ex) as follows. If G is SL2 and T the diagonal subgroup we use the map defined by the pairing between X * (T) and X * (Lro) ; if T is arbitrary in SL2 we choose a diagonalization and compose the in­ duced map on X * (T) with that already prescribed. If G is the anisotropic form we may still regard T as a torus in SL2 and proceed in the same way. Using this isomorphism between X* (T) and X* (LT0) we transfer " to a quasi­ character on X * (Lro) ; using the canonical isomorphism between Hom(X * (LT0), ex) and LTo we then regard ,. as an element of LT0• At the same time we transfer the action of @K I F on X* (T) to X * (Lr o) and LT0, writing O"r for the new action of O" E @KIF · Here are the possibilities. If T is split then @K 1 F acts trivially and " is an arbitrary element of LT0• If T is anisotropic, suppose that T is defined by the quadratic ex­ tension E of F. We shall assume that K is some fixed large but finite Galois exten­ sion of F containing, in particular, E; @K 1 F acts on T through @E 1 F · Let 0"0 be the nontrivial element of @E I F and aV be a coroot for Tin G. Then 0"0av = - av so that (�>(aV)) 2 = 1 . Since aV generates X* (T) there are then just two possibilities for " · The nontrivial " defines the element c-o �) * of LTO ; here, and throughout these notes, we use (� �) * to denote the image in PG4(C) of the matrix (� �) in GL2 (e). The action of O"r on L To is described as follows : if O" e @K I F maps to the trivial element in @E l F under @K I F -+ @ElF then O"r acts trivially and if O" maps to 0"0 then O"r acts by

As an element of LT0, " is O"rinvariant, O" e @K I F · We define LHo to be the con­ nected component of the identity in the centralizer of " in LGo. Whatever T, if K is trivial then LH o = LG o and if " is nontrivial then LH o = LT0• Let O" e @K I F · Then since both LH o and LTo are invariant under O"r we may multiply O"r by an inner automorphism of LH o to obtain an automorphism O"n stabilizing LTo and fixing each root, if any, of LTo in LH0• The collection {O"n, O" E @K 1 F} defines a semidirect product LH = LH o XI WK I F where WK I F• the Weil group of K/F, acts through @K 1 F · In duality, we obtain a quasi-split group H over F. Specifically : (a) . If ,. is trivial (whatever T) then LH = LG = LGo x WKI F and SL2. (b) If T is split and " nontrivial then L H = LT o x WK l F and H = T. PROPOSITION.

H

=

195

L-INDISTINGUISHABILITY

(c) lf T is anisotropic and " nontrivial then L H = LTo ><J WK 1 F where w E WKl F acts trivially on L To if w maps to 1 under WK I F ---> @K I F ---> @E1F, and w acts by (g g)* ---> CS �)* if w maps to a o (E, (J 0 as before) ; and H = T.

To indicate that H is defined by (T, tc) we write H = H(T, K). Note that the choice (of diagonalization) made in defining the isomorphism between X*(T) and X*(L To) does not affect H(T, tc) . 4. Embedding L-groups. For later use we specify an embedding cH of L fl in LG. We refer to the proposition above. In (a) cH is to be the identity, in (b) cH is the inclusion and in (c) cH extends the inclusion of L ro in LGo by : if w E WKIF

(� �t maps to 1 under

X

W ---4

(� �t

X

WKI F -> @3KIF -> @E l F '

W

and

if w maps to a 0 • As remarked earlier there are analogues of these groups "H" for any connected reductive group G over F. In general, it need not be that L H embeds in LG; [7] indicates how to accommodate this (see Lemma I and the last several paragraphs). 5. Orbital integrals (normalization). We will transfer certain integrals from G to H. For this, normalizations are required. Fix a pair (T, tc) . We choose Haar measures dt and dg on T(F) and G(F) respectively. If /E C�(G(F) ) , o E 'l)(T), and r E T(F) is regular in G then we set r/J"Cr. .f) =

_ Jh-IT(F)h\G (F) f(g-1h-1rhg) ____!jg_ (dt)h

where h E %f(T) represents o and (dt)h is the measure on h-IT(F)h obtained from dt by means of ad h. For o E �(T) - 'l)(T) we set (/)0( ,f) = 0. Recall that %f(T), �(T), �(T) were defined in § 1 . In the case that " i s trivial (whatever T) we define (/)T 1 •( , f) = f/JT I I( , f) = c(G) 1::; (/Jii( , f) /jEC (T)

where s(G) = I if G = SL2 and s(G) = - I if G is anisotropic. We call r/JT II ( , ) a "stable" orbital integral. If T is split and " nontrivial then we define (/)T I •( r ' f)

=

l r 1 - rz l I; tc(o)r/Jii(r ) ,/ l n rzl 1 1 2 oEc crJ

where r �> rz are the eigenvalues of T · Suppose that T is anisotropic and that ,. is nontrivial. Then our normalization depends on two choices. First choose a nontrivial additive character cf;F on F. If E is the quadratic extension of F attached to T then we define J...(E/F, r/JF) as in [5]. Also choose a regular element r o of T(F). Let tc ' denote the quadratic character of px attached to E. Then we define

196

D.

rpr l •(r .J)

=

SHELSTAD

(

).(E/F, i/JF)tc' r; - r� 7 1 - rz

) Ilrn1 r-z l 1r1z2l

.r; tc(o)f/la(r . J)

i}E.f (T)

where the order on the eigenvalues T I > rz of r and r�. rg of r o is prescribed by fixing a diagonalization of T. A different choice of 1/JF or r o causes at most a sign change in the normalizing factor. 6. Transferring orbital integrals. The integrals rpr 1 •( , f) can be transferred to stable orbital integrals on H = H(T, tc) in the following sense : LEMMA. If f E C�(G(F)) then there existsfH E C�(H(F)) such that

rpr 1 1(r , fH) = rpr l •(r , f)

and

(i) ifG is anisotropic and tc trivial (so that H is SL2) then (/J T'Il(

, JH)

=

0 for T' split ;

(ii) if G is SL2 and K trivial (so that H is SL2 again) then for all T', provided that the Haar measures dt' are chosen consistently.

This is proved (in [4]) by a case-by-case argument. Note that if H is T then

rpr 1 1( , JH) is justfH itself.

For general (real) groups a formalism for transferring the (/JT 1 •( , f) to H is devel­ oped in [10]. Some progress towards obtaining a result as above for any real group is made in [9] and [10].

7. Stable distributions and a map. We define the space of stable distributions on G(F) to be the closed subspace, with respect to simple convergence, of the space of all distributions on G(F) generated by stable orbital integrals, that is, by the distri­ butions f f/JT 1 1(r , f), where r is a regular semisimple element of G(F) and T denotes the torus containing T · The transfer of the rpr 1 •( , f) to H establishes a correspondence (f, fH) between C�(G(F)) and C�(H(F)). Dual to this correspondence there is a well-defined map from the space of stable distributions on H(F) to the space of invariant distributions on G(F) ; that is, if 6lH is a stable distribution on H(F) then we may define a (con­ jugation-) invariant distribution 6l on G(F) by the formula 6l (f) = eH(fH), f E C� ( G(F) ) This map, 6lH ..... 6l, will be central to our study of L-indistinguish­ ability. We denote by X" the character of (the infinitesimal equivalence class of) an irre­ ducible admissible representation of G(F) ; we regard X" as a function on the regular semisimple elements of G(F), using the same normalization of Haar measure as in §5. There is a simple way to determine whether X"' as distribution, is stable. Let G be GL2 in the case that G is SLz , or the full multiplicative group of the underlying quaternion algebra in the case that G is anisotropic. Then a linear combination X of characters is stable if and only if X is invariant under G(F) . . . or (of relevance for generalizations (cf. [9])) if and only if X is invariant under &(T), T c G.

.....

.

L-INDISTINGUISHABILITY

197

8. Local L-packets and some stable characters. We assume that the Langlands correspondence has been proved for G ; in fact enough has been proved in each of the cases being considered. In the following definitions we also allow G to be a torus. We denote by f/J(G) the set of equivalence classes of admissible homomor­ phisms of W];. into LG o XI W];. (notation and definitions as in [1]). To each {rp} e f/J(G) there is attached a finite collection Drrp> of irreducible admissible represen­ tations of G(F), an L-packet. Two irreducible admissible representations of G(F) are said to be L-indistinguishable if they belong to the same L-packet. In the case that G is an inner from of SL2 there is a simpler definition : tr 1 and tr2 are L-indis­ tinguishable if and only if there exists g e G(F) such that trz is equivalent to tr1 o adg. We set X rrpl = .E "Enr.> X " · Clearly : PROPOSITION.

X frpl is stab/e.

Since the Langlands correspondence has been proved for any (connected reduc­ tive) real group [6] we can define characters Xrrpl in that case. In general X rrpl need not be stable ; however if the constituents of Dr"'> are tempered (cf. [3]) then Xr"'> is stable [9]. From § 12 on we will not need to distinguish in notation between an admissible homomorphism rp : W];. --+ LGo XI Wf. and its equivalence class ; we will then denote both by rp and write U"' , X"'' etc. 9. Character identities (introduction). We return to the map of stable distributions on H(F) to invariant distributions on G(F). If {rpH} e f/J(H) then, as we have observed, X rrpnl is stable. Its image in invariant distributions on G(F) is repre­ sented by some function X on the regular semisimple elements of G(F). This func­ tion X is computed by the Weyl integration formula. We write : X rrpnl � X · Recall the embedding cH of LH in LG (§4). We could have used W];. in place of WK l F in defining LG, LH, and tH · With these modifications, suppose that 'PH is an admissible homomorphism of w;. into LH. Then rp = tH o 'fJH maps Wf. to LG. Suppose that rp is admissible ; recall that (local} admissibility imposes the condition that the image of rp lie only in parabolic subgroups of LG which are "relevant to G" (cf. [1]). Then we say that {rp} e f/J(G) factors through {rpH} e f/J(H). Suppose that { rp} factors through { rpH }. Then linear combinations of the charac­ ters of the representations in Dr"'> make natural candidates for X ( � X rrpn> ).

10. "S0\S". We introduce a useful group. It is easier to work with a homomor­ phism rp : Wf. --+ LG rather than an equivalence class {rp}. We exclude the case of sp(2) [1] and corresponding special representation of G(F), and consider just an admissible homomorphism rp : WKIF --+ LG where K, LG (and LH, tH) are as earlier. We define S"' to be the centralizer in LGo of the image of rp ; s; will be the con­ nected component of the identity in S"' . If rp' is equivalent to rp then there exists g E LGO such that s"', = gS"'g- 1 and s;, = gs;g- 1 . Suppose that rp = tH o 'PH where 'PH is an admissible homomorphism of WK I F into LH and H = H(T, ti:). We regard " as an element of LT0• Recall that L H o is the connected component of the identity in the centralizer of " in L G o . By definition (§3), (JH fixes " ' (J e @K 1 F · It follows then that " lies in the center of the image of LH in LG. Therefore " centralizes the image of 'PH( WK 1 F) in LG; that is, " centralizes

I 98

e

D. SHELSTAD

q>( WK; F) · We have then that " Srp. We define srp(�>), or just s(�>) when qJ is under­ stood, to be the coset of " in s;\srp. Recall that this quotient appeared in [3]. Suppose that qJ' is equivalent to qJ and that qJ' = &H o f/J'H where ({Ju : WKIF � L H and H = H(T', �>'). Then ,., E srp' · Write (/)1 as ad g 0 (/), g E LG0• Then g-l ,.'g E Srp. We define srp(�>') to be the coset of g- 1 ,.'g in S;\Srp. We will see that s;\Srp is abelian. Therefore srp(�>') is independent of the choice for g. We continue with the same qJ, qJ' . The conjugation ad g induces an isomorphism behyeen s;\Srp and s;.\Srp' which carries srp(�>) to srp,(�>). Because both groups are abelian this isomorphism is independent of the choice of g in the equivalence qJ ' = ad g o f/J · We may therefore regard S;\Srp and the elements srp(�>) as attached to {qJ}. 11. Calculations. We compute explicitly s;\Srp and the elements s(�>) = srp(�>). We need take just one homomorphism qJ : WK l F � LG from each equivalence class. If we write as x WKIF • then the homomorphism qJ1 : WKIF � LG o = PGLz(C) lifts to a two dimensional representation ip 1 of WK l F [7, Lemma 3]. (i) Suppose that ip 1 is reducible. Then ;p 1 factors through ..- : WKIF � F x and is defined by a pair (fl. , lJ) of quasi­ characters on Fx . We take

qJ(w) qJ1(w) w, w e

iflt(w) = (fl.(z-6w)) lJ (z-�w))) '

If (fJ./lJ) 2 # I then srp = s; = LTO and if fl. = )) then srp = s; = LG0• However, if fl.flJ has order two then Srp = L To ><1 ( l ,(n) * ) ; recall that ( ) * denotes the image of( ) in PGL2(C). Hence S;\Srp = Z/(2). Whatever fi./lJ, qJ factors through L T, T a split torus. The corresponding " ((6 �) * , some z # I) lies in s; and s(�>) is trivial. Note that (this or any) qJ factors through H when H = H( , 1) ; s(l) is trivial also. Suppose that fl.flJ has order two. In (ii) we will show that a homomorphism equivalent to qJ factors through LT, where T is a torus defined by the quadratic extension E of F attached to fl.flJ, and that the associated s(�>) = srp(K) is the coset of (� 6) * . (ii) Suppose that ip 1 factors through a representation ip2 = Ind( WE l F • Ex, ()) of WEIF where E c K is a quadratic extension of F and () is a quasi-character on Ex . Then f/JI factors through qJ2, the projective representation defined by ip2. Let (1 ° be the n ontrivial element of @EI F and define O(x) = 0(Xa0), X E Ex . We realize WEIF explicitly as {x x p ; x Ex, p e { I , q0 } }, with multiplication rule (x x p)(x' x p' ) = x(x') P ap p' x pp ' where ap, p' = I unless p = p ' = (1 ° and aaa, aa is some (chosen) element a of Fx NmE 1 FEx . Then we may assume that ()(x) 0 qJ2(x x I) = 0 O(x) , and

e

,

_

(

)

-

199

L-INDISTINGUISHABILITY

where z is a c�osen square root of 8(a). Thus _

cp2(x

x

1)

=

(8(x) 0 ) O (x)

0

and cpz(l

x

*

,

x e EX,

(� �) . .

C1°) =

Let T be a torus attached t o E. Then cp factors through LT; indeed we chose the embedding c T of L T in LG (§4) so as to insure this. Recall that T = H(T, .t) where .t, as element of L T0, is (-A �) * ; s(.t) is the coset of (-A �) * in s;1s"'. To compute S"' , suppose first that (8/0) 2 #- 1 . Then

so that s;\S"' = Z/(2) ; we have already recovered the nontrivial element of s;\S"' as an s(.t) . Suppose now that 8/0 has order two. Then S"' coincides with cp 1 ( WKIF) ; that is, Therefore s;\S"' = Z/(2) (£) Z/(2). We have recovered only (-A �) * (and I) as an ( ) To recover the remaining elements of s;\S"' we recall homomorphisms cp', cp" : WKIF -+ LG which are equivalent to cp and factor through other LH. Let E0 be the quadratic extension of E defined by 8/0. Then pxfNmEoJFEt = Z/(2) (£) Z/(2) so that there are two distinct quadratic extensions E' and E" of F which are distinct from E and contained in E0• We pick a quasi-character 8' on (E'Y such that 8' o NmEotE' = 8 o NmEotE and 8' /0' is the quadratic character attached to E0/E' (O'(x) = 8'(x"'), 1 #- a ' e @E'tF) ; we pick 8" similarly. Then iiJ2 = Ind( WE'tF• (E'Y , 8') and ip� = Ind ( WE"tF• (E") x , 8") define representations fiJi, ip'{ of WKIF• each equivalent to iph and hence homomorphisms cp', cp" : WKI F -+ LG, each equivalent to cp. We define cpi, cp2 and cp'{, cp� as we did CfJh cp2 ; we realize cp2 and cp� as we did cp2. Then cp 1 ( WK 1 F), cpi( WK1F), cp'{( WKIF), S"' , S"', and S"'n all coincide and equal

s .t .

As with cp earlier, cp' ( . . . cp") factors through L(T') ( . . . L(T")), where T' ( . . . T") is some torus attached to E' ( . . . E"). Suppose T' = H(T', .t ') and T" = H(T", .t " ) . Then, writing cp = ad g ' o cp' = ad g " o cp", g ' , g " e LGo , we have ( ')

s .t

=

( ') =

s"' .t

g

and ( ")

s .t

=

( ")

s"' .t

=

g

"

'

( - 1 0) 0 1

( - � �)*

* g

g

' -1

"-1 .

200

D. SHELSTAD

An elementary argument shows that none of the conjugations ad g', ad g", ad(g ' )- 1 g " may fix (-0 �) * and so we conclude that {s(�e'), s(�e")} =

{ (� �t · ( _� �t } ·

Thus we have recovered each element of s;\s'P as an "s(�e)". We have one further case to consider : 0 = 0. We pick a quasi-character p. on px such that 0 = p. NmEI F · Then cp is equivalent to the homomorphism of type (i) above attached to the pair (p., 'p.) where ' is the quadratic character of px defined by the extension E/F. We now call that homomorphsim cp' . We had that s;,\s'P, has two elements and that (� ij)* is a representative for the nontrivial coset. We now recover this coset as an s'P,(�e). Let T be a torus attached to E. Then cp factors through LT and the associated ,. is (-0 �) * If cp ' = ad g cp, g E LG0, then -1 0 - - 0 1 g 0 l *g 1 = 1 0 * and we are done. (iii) Suppose that ip 1 is of tetrahedral or octahedral type. Then S'P = s; = 1 . This is a straightforward exercise. In summary : PROPOSITION. (1) s;\S'P is one of I , Z/(2) or Z/(2) Ef) Z/(2). (2) For each element s of s;\s'P there is a pair (T, �e) such that s = s(�e). 12. Character identities (continued). From now on we will not distinguish in notation between an admissible homomorphism cp: WK 1 F --+ LG and its equivalence class ; that is, we denote both by cp . . as we have noted, s;\S'P can be regarded as attached to the equivalence class. Suppose that cp factors through CfJH• where H = H(T, �e). Let s = s(�e). Then, in the notation of §§8, 9 : a

(

.

)

( )

a

.

LEMMA .

There exist integers (s, n), 7r: E ll'P, such that X'Pn � I: (s, n) Xn· 1CE U�

The proof [4] is again a case-by-case argument. Here is a summary of the ex­ plicit results. (A) G = S4. We consider cp as in (i), (ii), (iii) ofthe last section. (i) If (p./v) 2 ¥- I or if p. = v then U'P contains one element which we denote by n ; s;\S'P = 1 and ( 1 , n ) = 1 . I f p./v has order two then U'P has two elements which we denote by n1 and n2 . Recall that S;\ S'P = Z/(2) ; (s, n1) and (s, n2 ) are the two characters in s. These characters depend on the choices we made in normaliz­ ing orbital integrals (§5) ; that is, for some choices < , n1 ) is the trivial character and for others < n2 ) is the trivial one. (ii) We have already considered the case 0 = 0. If 0/0 is not of order two then U'P has two elements and s;\S'P = Z/(2) ; the result is as in (i). Suppose that 0/0 has order two. Then we had that s;\S'P is Z/(2) ® Z(2) ; U'P has four elements, too. Suppose U'P = {n,, i = I, . . . , 4} . Then we may (and do) normalize orbital integrals so that the ( , n;) are the four characters on s;\s'P. ,

20 1

L-INDISTINGUISHABILITY

(iii) Here Drp has one element, say n ; s;\Srp = 1 and ( 1 , n) = 1 . (B) G anisotropic. We have only to consider qJ as in (ii) with () #- 0 , and (iii). Suppose F = R. Then s;\Srp = Z/(2) since we exclude (iii) also. However Drp has only one element, say n; (s, n) is a character in s, trivial or nontrivial according to our choice of normalizing factors for orbital integrals. Suppose that F is nonarchimedean. Then Drp has the same number of elements as the corresponding L-packet in SL2, and the result is as there, except when qJ is of type (ii) with ()j[J of order two. Then the corresponding packet in SL2 has four elements ; Drp has only one element, say n. We must set ( 1 , n) 2 and (s, n) = 0, s #- 1 ; in particular, ( , n) is not a character. As for generalizing these identities we will report some progress for real groups in a forthcoming paper ; in the case that K is trivial, so that H is a quasi-split inner form of G, an appropriate identity is known provided that the constituents of nrp are tempered [9]. 13. Structure of local £-packets. From the results of the last section we conclude : PROPOSITION. If G is the quasi-split form (SL2) then the pairing ( , ) identifies nrp (noncanonically) as the dual group of s;\srp. If G is anisotropic we remark only the existence of the functions ( , n), n E Urp. With some qualifications, we can expect an analogue of the proposition for a general quasi-split real group (cf. [3]). 14. A global application. Suppose now that F is a global field and that G is SL2 . There are analogues for the groups H of the local case ; again H may be a maximal torus in G, defined over F, or G itself. We assume that K is some large but finite Galois extension of F and consider just homomorphisms qJ : WK 1 F ---+ LG defined by representations of the form Ind( WK 1 F• WK 1 E • ()) , where E is a quadratic extension of F contained in K and () is a grossencharacter for E not factoring through NmE I F· If Tis a torus attached to E then (/J factors through L T (as earlier, provided that L T is correctly embedded in L G). We define nrp to be the set of representations n = ®vnv of G(A) which are irreducible, admissible (as in [1]) and such that """ E nrp. for each place v . The set Hrp is an L-packet in the following sense. We define two irreducible admissible representations n1 ®vn� and n2 = ®vn� of G(A) to be £-indistinguishable if for all places v, n� and n� are £-indis­ tinguishable and for almost all v, n� and n� are equivalent. An L-packet is an equivalence class for this relation. We have singled out the packets Hrp for the following reason. As is proved in [4], two representations from such a packet may appear with different multiplicities in the space of cusp forms for G. One may appear, with multiplicity one, and the other not appear. For the remaining £-packets the members do appear with equal multiplicity (conjectured to be either zero or one). In [4] there is a formula for the multiplicity with which a representation from nrp appears in cusp forms ; it is this we wish to discuss. As in the local case, we define Srp to be the centralizer of qJ( WK l F) in LG o and s; to be the connected component of the identity in Srp. As in (ii) of § 1 1 we have that either s;\srp = Z/(2) or s;\Srp = Z/(2) EB Z/(2). For each place v, Srp embeds =

=

202

D. SHELSTAD

naturally in S'P. and s; in s;, There is then a natural map of s;\S'P into s;.\s'P, In notation we will not distinguish between an element of S�\S'P and its image in s;.\s'P, Recall that the local characters < , �v> depend on the choices we made when normalizing orbital integrals (§§5, 1 2). We chose a nontrivial additive character cf; F. on Fv and regular element r o = r� of T(Fv ) for each torus T anisotropic over Fv . Instead, we now choose a nontrivial additive character cj; on F\A and for each torus T anisotropic over F a regular element r o in T(F). At each place v where T does not split we use cf; v and r o to specify (])T 1 K( , ), " nontrivial. Fix s E s;\s'P and � E n'P. Then we have (s, �v> = 1 for almost all v. Therefore we may define (s, �> = Dv(s, �v> · Then [4] :

PROPOSITION. (i) (s, �) is independent of the choices madefor cj; and r0• (ii) < ' > induces a (canonical) surjection of n'P onto the dual group of s;\s'P. Also : THEOREM. The multiplicity with which � E n'P appears in the space of cusp forms is : 1 (s, �> . [s'Po\S'P] SESI;;\s. That is, those � for which < , �) is the trivial character appear with multiplicity one and those � for which < , �> is nontrivial do not appear. 15. Afterword. More generally, and of relevance also for Shimura varieties (cf. [8]), we can consider inner forms of a group G such that Res£ 1 FSL2 c G c ResE;FGL2 with E some finite Galois extension of F, and develop a local and a global theory in the same way. The analogous multiplicity formula is [4] :

__5p___

I; < s' � > 'P where d'P is defined as follows. If q; , q;' : WF -+ LG are admissible homomorphisms (cf. [1, § 1 6]) call q; and q;' locally equivalent everywhere if (/Jv is equivalent to q;� for each place v. Call q; and q;' weakly globally equivalent if q;' is equivalent to wq; where w is a continuous 1 -cocycle of WF with values in the center of LG o such that the restriction of w to each local group is trivial. Then d'P is the number of weak global equivalence classes in the everywhere local equivalence class of q;. In [8] L-indistinguishability plays a role in relating the zeta-functions of certain Shimura varieties to automorphic L-functions. Briefly, in the case discussed in [2], where G is the full multiplicative group of some quaternion algebra over a totally real field and there is no L-indistinguishability, functions L(s, �. p) appear (� an automorphic representation of G(A), p a certain representation of LG, fixed as in that lecture). In the general case of [8], where G is a subgroup of the full multi­ plicative group, we must consider at least some of the L-packets n'P where different multiplicities in cusp ( . . . automorphic) forms occur. Then q; : WF -+ LG factors through ty : LT ..... LG, with T a nonsplit torus of G. If q; = Cy 0 qJy, �T E n'PT and Pr = p · tr then L(s, �. p) = L(s, �r. py). There is a natural decomposition Pr = p} EfJ p�, where p}, p� may be reducible, and it is the functions L(s, �r. p�) which are relevant.

[S;\S'P]

s E S 0 \S0

L-INDISTINGUISHABILITY

203

REFERENCES 1. A. Borel, Automorphic L-functions, these PROCEEDINGS, part 2, pp. 27-61 . 2. W. Casselman, The H asse-Wif!il �-function 'of some moduli Ivarieties ofdimension greater than one, these PROCEEDINGS, part 2, pp. 141-163. 3. A. W. Knapp and G. Zuckerman, Normalizing factors, tempered representations and L-groups, these PROCEEDINGS, part 1 , pp. 93-105 . 4. J-P. Labesse and R. P. Langlands, L-indistinguishability for SL(2) (to appear). 5. R. P. Langlands, On Artin's L-functions, Rice Univ. Studies, vol. 56, no. 2, pp. 23-28, 1 970. 6. , On the classification of irreducible representations of real algebraic groups (to appear). 7. , Stable conjugacy ; definitions and lemmas (to appear). 8. , On the zeta functions ofsome simple Shimura varieties (to appear). 9. D . Shelstad, Characters and inner forms of a quasi-split group over R, Compositio. Math. (to appear). 10. , Orbital integrals and a family of groups attached to a real reductive group, Ann. Sci . Ecole Norm. Sup. (to appear). ---

---

---

--

COLUMBIA UNIVERSITY

Proceedings of Symposia in Pure Mathematics Vol . 33 (1 979), part 2, pp. 205-246

AUTOMORPIDC REPR'ESENTATIONS, SmMURA VARIETIES, AND MOTIVES. EIN MARCHEN

R.

P.

LANGLANDS

1. Introduction. It had been my intention to survey the problems posed by the study of zeta-functions of Shimura varieties. But I was too sanguine. This would be a mammoth task, and limitations of time and energy have considerably reduced the compass of this report. I consider only two problems, one on the conjugation of Shimura varieties, and one in the domain of continuous cohomology. At first glance, it appears incongruous to couple them, for one is arithmetic, and the other representation-theoretic, but they both arise in the study of the zeta-function at the infinite places. The problem of conjugation is formulated in the sixth section as a conjecture, which was arrived at only after a long sequence of revisions. My earlier attempts were all submitted to Rapoport for approval, and found lacking. They were too im­ precise, and were not even in principle amenable to proof by Shimura's methods of descent. The conjecture as it stands is the only statement I could discover that meets his criticism and is compatible with Shimura's conjecture. The statement of the conjecture must be preceded by some constructions, which have implications that had escaped me. When combined with Deligne's conception of Shimura varieties as parameter varieties for families of motives they suggest the introduction of a group, here called the Taniyama group, which may be of impor­ tance for the study of motives of CM-type. It is defined in the fifth section, where its hypothetical properties are rehearsed. With the introduction of motives and the Taniyama group, the report takes on a tone it was not originally intended to have. No longer is it simply a matter of for­ mulating one or two specific conjectures, but we begin to weave a tissue of surmise and hypothesis, and curiosity drives us on. Deligne's ideas are reviewed in the fourth section, but to understand them one must be familiar at least with the elements of the formalism of tannakian categories underlying the conjectural theory of motives, say, with the main results of Chapter II of [40]. The present Summer Institute is predicated on the belief that there is a close re­ lation between automorphic representations and motives. The relation is usually AMS (MOS) subject classifications (1 970). Primary 14010, 14F99. © 1979, American Mathematical Society

205

206

R.

P. LANGLANDS

couched in terms of L-functions, and no one has suggested a direct connection. It may be provided by the principle of functoriality and the formalism of tannakian categories. This possibility is discussed in the highly speculative second section. However there is a small class of automorphic representations which are certainly not amenable to this formalism. I have called them anomalous, and in order to make their significance clearer I have discussed an example of Kurokawa at length in the third section. Although the anomalous representations form only a small part of the collection of automorphic representations, they are frequently encoun­ tered, especially in the study of continuous cohomology, and so we have come full circle. Our long divagation has not been in vain, for we have acquired concepts that enable us to appreciate the global significance of the local examples described in the seventh section, which deals with the second of our original two problems. At all events, I have exceeded my commission and been seduced into describing things as they may be and, as seems to me at present, are likely to be. They could be otherwise. Nonetheless it is useful to have a conception of the whole to which one can refer during the daily, close work with technical difficulties, provided one does not become too attached to it, but takes pains to ensure that it continues to con­ form to the facts, and is. prepared to abandon it when that is called for. The views of this report are in any case not peculiarly mine. I have simply fused my own observations and reflections with ideas of others and with commonly accepted tenets. I have also wanted to draw attention to the specific problems, on which expert advice would be of great help, and I hope that the report is sufficiently loosely written that someone familiar with continuous cohomology but not with arithmetic can turn to the seventh section, overlooking the first few pages, and see what the study of Shimura varieties needs from that theory, or that someone familiar with Shimura varieties but not automorphic representations or motives will be able to find the definition of the Serre group in the fourth section and the Taniyama group in the fifth, and then turn to read the sixth section. Finally a word about what is not discussed in this report. The investigations of Kazhdan [26] and Shih [46] on conjugation of Shimura varieties are not reviewed ; their bearing on the problem formulated here is not yet clear. L-indistinguishabil­ ity is not discussed. Little is known, and that is described in another lecture [44]. Problems caused by noncompactness are ignored. There is a tremendous amount of material on compactification and on the cohomology of noncompact quotients, but no one has yet tried to bring it to bear on the study of the zeta-functions. The omission I regret most is that of a discussion of the reduction of the Shimura varieties modulo a prime [36]. Here there is a great deal to be said, especially about the structure of the set of geometric points in the algebraic closure of a finite field, starting with the work of Ihara on curves. I hope to report on this topic on another occasion. 2. Automorphic representations. Our present knowledge does not justify an attempt to fix a language in which the relations of automorphic representations with motives are to be expressed, Nonetheless that of tannakian categories [40] appears promising and it might be worthwhile to take a few pages to draw attention to the

AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES

207

problems to be solved before it can be applied in the study of automorphic repre­ sentations. We first recall the rough classification of irreducible admissible representations of GL{n, F), F being a local field, and of automorphic representations of GL(n, AF), F being a global field, reviewed in some of the other lectures ([3], [5], [35], [48], cf. also [7], [31]). In either case the representation n has a central character w and z � l w(z) l may be extended uniquely from Z(F), or Z(AF), to a positive character v of GL(n, F), or of GL(n, AF). If F is a local field then n is said to be cuspidal if for any K-finite vectors u and v, in the space of n and its dual, v- l (g) (n (g)u, v) is square-integrable on the quotient Z(F)\GL(n, F). To construct an arbitrary irreducible admissible repre­ sentation one starts from a partition {nh · · ·, n,} of n and cuspidal represen­ tations n 1 o · · ·, n, of GL(n;, F). If w; is the central character of n;, there is a real number s; such that lw;(z) l = lzl•; if z lies in the centre of GL{n;, F), a group isomor­ phic to Fx . Changing the order of the partition, one supposes that s 1 � • • • � s,. The partition defines a standard parabolic subgroup P of GL{n) and o = ®n; a representation of M(F), because the Levi factor M of P is isomorphic to GL(n 1) x · · · x GL(n,). The representation o yields in the usual way an induced represen­ tation /" of G(F). /" may not be irreducible, but it has a unique irreducible quotient, which we denote n 1 E8 · · ·EEl n,. Every representation is of this form and n 1 E8 · · ·EEl n, � n :i E8 · · · E8 n; if and only if r = s and after renumbering n; � n;. Thus every representation can be represented uniquely as a formal sum, in the sense of this notation, of cuspidal representations. The representation n 1 E8 · · · E8 n, is said to be tempered if all the S; are 0. We can clearly define, in a formal manner, the sum of any finite number of representations. We can in fact formally define an abelian catefory H (F) whose collection of objects is the union over n of the irreducible, admissible representations of GL(n, F). If n = n 1 E8 · · · E8 n" n' = n:i E8 · · · E8 n;, with n; and nj cuspidal, we set Hom(n, n')

=

{i, jl1t:; -"jl

EB

C.

The composition is obvious. The tempered representations form a subcategory H0 (F) . If F is a global field then n is said to be cuspidal if n is a constituent of the representation of GL(n, AF) on the space of measurable cusp forms rp satisfying (a) rp(zg) = w(z) rp(g), z E Z(AF), (b) Jz (AF)G (F)\G (AF) V -2(g) j rp(g) j 2 dg < 00 . I f n 1 o · · , n, i s a partition of n and n 1 o · · · , n, cuspidal representations of GL{n;, AF) we may again change the order so that s 1 � · • � s, and then construct the induced representation /" . Every automorphic representation is a constituent of some /" . For an adequate classification, one needs more. The following statement may eventually result from the investigations of Jacquet, Shalika, and Piatetskii­ Shapiro, but has not yet been proved in general. A. If n is a constituent of /" and of lu' then the partitions {nh · · · , n,} and {n:i, - · · , n;} have the same number of elements, and, after a renumbering, n; = n; and n; n;. •

•

�

208

R . P. LANGLANDS

If 7C; = (8).7C;(v) then one constituent of I" is the representation 7C = ®v 7C(v) with local components 1e(v) = 1e 1 (v) [±) · · · [±) 1e,(v). This representation will be denoted 1e = 1e 1 [±) · [±) 7C" but the notation is not justified until statement A is proved. The representations of this form will be called isobaric and can again be used to define an abelian category D(F). We agree to call 7C = 1e 1 [±) [±) 7C, tempered if each of the cuspidal re­ presentations 7C; has a unitary central character, that is, if each of the s; is 0. How­ ever the language is only justified if we can prove the following statement, the strongest form of the conjecture of Ramanujan to which the examples of Howe­ Piatetskii-Shapiro and Kurokawa allow us to cling. B. If 7C = (8)1e(v) is a cuspidal representation with unitary central character then each �{the factors 1e(v) is tempered. The tempered representations form a subcategory 0°(F) of D(F). It is clear that we have tried to define the categories H(F) and H0(F) in such a way that if v is a place of F there are functors H(F) --+ H(F.) and Do(F) --+ 0°(F.) taking 7C to its factor 1e(v). However without a natural definition of the arrows, there is no unique way to define Hom (7C, 1e') --+ Hom (1e(v), 1e'(v)) . If I" is not irreducible it will have other constituents in addition to 1e 1 [±) · · [±) 1e,. These automorphic representations will be called anomalous. Although the principle of functoriality may apply to them, there is considerable doubt that they can be fitted into a tannakian framework. Observe that statement A and the strong form of multiplicity one imply that if 1e is any automorphic representation there is a unique isobaric representation 1e ' such that 1e(v) - 1e'(v) for almost all v. If F is a global field the purpose of the tannakian formalism would be to provide us with a reductive group over C whose n-dimensional representations, or rather their equivalence classes, are to correspond bijectively to the isobaric automorphic representations of GL(n, Ap). This hypothetical group will have to be very large, a projective limit of finite-dimensional groups. We denote it by GucFJ · The category Rep(GocFJ ) of the finite-dimensional representations over C of the algebraic group GucFJ would certainly be abelian, but in addition it is a category in which tensor products can be defined. Moreover there is a functor to the category of finite-dimensional vector spaces over C. If (rp, X), consisting of the space X and the representation rp of GucFJ on it, belongs to Rep (GocFJ) one sim­ ply ignores rp. The tensor product satisfies certain conditions of associativity, com­ mutativity, and so on, and the functor, called a fibre functor, is compatible with tensor products and other operations of the two categories. A theorem of [40], but not the principal one, asserts that, conversely, an abelian category with tensor products and a fibre functor is equivalent to the category of representations of a reductive group, provided certain natural axioms are satisfied. Thus it appears that if we are to be able to introduce GocFJ we will have to as­ sociate to each pair consisting of a cuspidal representation 7C of GL(n, A) and a cuspidal representation 1e' of GL(n', A) an isobaric representation 7C (g) 1e' of GL(nn', A). In general, if 7C = 1e 1 [±) · · · [±] 1e, and 1e' = 1ei [±] · · · [±) 1e; we would set 7C (g) 7C' = ffi (7C; (g) 7Cj). · ·

· · ·

·

{,

j

AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES

209

In addition, we will have to associate to each isobaric representation 7T: of GL(n, A), n = I , 2, , a complex vector space X('IT:) of dimension n, together with isomorphisms . . ·

There are a large number of conditions to be satisfied, among them one which is perhaps worth mentioning explicitly. Suppose 7T: and 'IT:' are cuspidal and 7T: � 'IT:' = ffi�=1 'JT:; with 'IT:; cuspidal. Then the set {'IT:;, · · · ,'IT:,} contains the trivial representation of GL(I, A) if and only if 'IT:' is the contragredient of 'IT:, when it contains this repre­ sentation exactly once. At the moment I have no idea how to define the spaces X(7T:) ; indeed, no solid reason for believing that the functor 7T: � X(7T:) exists. Even though the attempt to introduce the groups Gn(F) may turn out to be vain the prize to be won is so great that one cannot refuse to hazard it. One would like to show, in addition, that if 7T: and 'IT:' are tempered then 7T: � 'IT:' is also, and thus be able to introduce a group Gna
X

Dp.

One will also wish to introduce, by a similar process, groups Gn
)

l wl- 1 1 2

X

W.

210

R.

P . LANGLANDS

Notice that according to these classifications there are homomorphisms of algebraic groups Gn cFJ --)- Gal(F/F) and G0ocFJ --)- Gal(F/F), the group on the right being a projective limit of finite groups. The principle of functoriality cannot be valid unless there are similar homomorphisms when F is a global field. Let Dp , Dt, D�, be the groups of multiplicative type whose modules of rational characters are, respectively, the module of all characters of px , or Fx\IF if F is global, of all positive characters, or of all unitary characters. Then DF = Dt x D� and there will be homomorphisms Gn cFJ

--)-

Dp,

Gno cFJ -)- D � .

If F is nonarchimedean and local we may also define Dun• D;;n , and D�n' by re­ placing the modules of characters of various types by the modules of unramified characters of the same type. The groups Dum Dtm and D�n contain a distinguished point over C, the Frobenius f/J, which is simply the image of a uniformizing parameter in px . In any case the formalism will certainly allow us to introduce for any representation a of the algebraic group Gn cFJ over C an L-function L(s, a) and if a corresponds to the representation n of GL(n, Ap) then L(s, a) = L(s, n) . But the reasons for wishing to introduce the groups GncFJ and Gno cFJ and the associated formalism are not simply, or even primarily, aesthetic. There are prob­ lems which will be difficult to formulate exactly without them. Suppose, for example, that n = ®vnv is an isobaric representation of GL(n, A) and each of the factors nv is tempered. For almost all v, nv is unramified and associated to a conjugacy class {gv} {g(nv)} in GL(n, C). Since nv is supposed tempered this class meets the unitary group U(n) and I may, as I prefer, regard it as a conjugacy class , in U(n). The general analytic analogue of the Tchebotarev theorem or the Sato­ Tate conjecture would be a theorem or conjecture describing the asymptotic distribution of the classes {gv} · Suppose the formalism existed and n were associated to a representation a of GnocFJ · The image a(GnocFJ (C)) would be a reductive subgroup H(C) of GL(n, C) with a maximal compact subgroup KH. For almost all v, av would factor through Gno (F,) -> D�n and av(f/Jv) would be defined. Since its conjugacy class in H(C) would meet KH, it would define a conjugacy class in KH, which we denote {av(f/Jv) } . Of course {avCf/Jv)} � {gv} and the asymptotic distribution of the classes {gv} can be inferred from that of the classes {avCf/Jv)} . There is a natural probability measure on the space of conjugacy classes in KH. If X is a set of conjugacy classes and if the set Y = U x E X x is measurable in KH, one takes meas X = meas Y. It is natural to suppose that it is this measure which defines the asymptotic distribution of the classes {avCtPv)} . To verify the supposition, it will be necessary to establish that if p is any representation of G0ocFJ over C then the order of the pole of L(s, p) at s = 1 is equal to the multiplicity with which the trivial representa­ tion of Gno cFJ occurs in p . If the existence of G0o cFJ were established, it would be easy enough to deduce this from the recent results of Jacquet and Shalika =

[25] .

AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES

21 1

Within this formalism, the principle of functoriality asserts that if F is a local field and G a reductive group over F then any L-packet of representations of G(F) is associated to a homomorphism q> : GucF) --+ LG of algebraic groups over C for which GucFl .'!.... LG '\. _ /

Gal(F/F) is commutative. If GucFl is replaced by Guo cFl • the L-packet should be tempered. If F is global, some caution will have to be exercised. If the L-packet H consists of 1C = (8)1e. for which the 1Cv are always tempered, it should correspond to a q> : Guo (F) --+ LG. Otherwise this may not be so, for a reason which will perhaps be clearer after an example of Kurokawa [29] is discussed in the next section. If there is a representation cj; : LG --+ GL(n) x Gal(F/F) and if the image cf;ill) of n given by the principle of functoriality is not isobaric then n can be associated to no q>. One may nonetheless hope to prove, both locally and globally, that to each q> : GucFl --+ LG is associated an L-packet, provided q> commutes with the homo­ morphisms to the Galois group. If G is not quasi-split the local behaviour of q> with respect to parabolic subgroups will also have to be taken into account [3]. For archimedean fields one recovers the usual classification. The few examples studied [44] suggest that questions about the multiplicity with which the elements of n occur in the space of automorphic forms will have to be answered in terms of q>. The principal reason for wishing to define the group Gn cFl is that it provides the only way visible at present to express completely the relation between auto­ morphic forms and the conjectural theory of motives [40]. The category of motives over F, a local or a global field, is Q-linear and tannakian, but it does not always possess a fibre functor over Q and seldom a single naturally defined one. Thus tan­ nakian duality associates to it not a group over Q but a group-like object, a "gerbe" in the rustic terminology which has become so popular in recent years. Over C this object becomes a group GMotCFl • and the relations between motives and automorphic representations will probably be adequately expressed by the ex­ istence of a homomorphism pF : Go cFl --+ GMot CFl defined over C. The field F can be local or global. The local and global homomorphisms are to be compatible with each other and with the formation of £-functions. Both the image and the kernel of PF will probably be rather large when F is a number field (cf. C.6.2 of [43]) but rather small when Fis local. 3. Anomalous representations. Since the anomalous representations cause some difficulty in the stu dy of the zeta-functions of Shimura varieties, it will be useful to acquire some feeling for them before going on. The brief remarks of the previous section suggest that an automorphic representation 1C of the reductive group G, or rather the L-packet H containing it, should be called anomalous if for some homomorphism cf; : LG --+ GL(n, C) x Gal(F/F) the principle of functoriality takes

212

R. P . LANGLANDS

1C or U to an anomalous representation of GL(n, A). It may be that the counter­ examples to the Ramanujan conjecture of Howe-Piatetskii-Shapiro [19] are anom­ alous in this sense. Since their paper is not available to me as I write, I have to test this suggestion on another example, discovered by Kurokawa and quite explicit. The group G is to be the projective symplectic group in four variables over Q. The L-group is then the direct product of LG0, the symplectic group in four variables and the galois group Gal(Q/Q). Since all the groups with which we shall deal in this section will be split, we may ignore the factor Gal (Q/Q). Let G1 be the product of PGL(2) over Q with itself, so that LG� = SL(2, C) x SL(2, C) and let G2 be GL(4) over Q. Define cp1 : LG� --+ LG to be the homomor­ phism

(;: ��) (;: �:) �1 X

-->

0

and let cp 2 be the standard imbedding of LGo in LG� which is GL(4, C). In order to analyze Kurokawa's example one must formulate his statements representation-theoretically. I state the facts necessary to this purpose, but have to ask that the reader understand the discrete series sufficiently well to verify them for himself. There is nothing to prove. It is simply a question of writing down explicitly for the special case of concern to us here some of the results of [17] and and [18], and some definitions from [31]. An automorphic representation 1C = ®1Cv of G(A) is associated to a holo­ morphic form of weight k in the classical sense of [39] if and only if 7C"' is a member of the holomorphic discrete series and lies in an L-packet Dq,ckl • where the restric­ tion of if;(k) to e x s;; WeiR has the form Z -->

with A = (2k - 3)/2, p. = t . Since G is the projective group, k must be even. Notice that zlz-l is to be calculated as z2l(zz)-l = z- 2.1 (zz)l. We will eventually take k = 10. On the other hand an automorphic representation 1C = ®1Cv of PGL(2, A) is associated to a holomorphic form of weight 2k - 2 if and only if 7C "' belongs to the discrete series and to an L-packet Drp w , where cp(k) : z -->

(

zz .l

--

.l

z- · z-·

)

with A as above. In particular cp1 takes the £-packet Drp Ckl ® Drpc2l , consisting in fact of a single representation, to Uq, Ckl . We want to apply the principle of functoriality to cp 1 and a very special auto-

AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES

213

morphic representation "" of G 1 (A). "" must be a tensor product 1r' ® 1r" of two representations of PGL(2, A). "";, and ""� will both be members of the discrete series, the first in Drp
lxl-1 / Z

of GL(l, A), and then construct the induced representation of GL(2, A) as in the preceding section. Any constituent it" of the induced representation factors through a representation it" of PGL(2, A). We so choose it" that ""� E Drp
f>

=

{Ad g h l g E G(R)} 0

and it will be best simply to let h denote an arbitrary element of f>. Recall that the pair (G, h) is subject to three conditions [10, § 1 .5) : (a) If w is the diagonal map GL(l) -+ GL(l) x GL(l) then the homomorphism h o w is central. (b) The Lie algebra @ of G(C) is a direct sum @ = p + f + p and if(zh z2) E �(C) then

214

R . P. LANGLANDS

X E p, = z11z2 X, = X, X E r, XE p. = zl zil X, In fact the summands p, f , and p vary with h, and when it is useful to make the depend­ ence on h explicit we write ph, th, and Ph · (c) The adjoint action of h(i, - i) on the adjoint group is a Cartan involution. Since G(R} acts on the real manifold .p by conjugation every element of @ defines a complex vector field on .p. Let Xh be the value of the vector field associated to X at h E .p. The complex structure on .p is so defined that the holomorphic tangent space at h is {Xhl X E p} and the antiholomorphic tangent space is {Xhl X E p}. If A1 is the ring of adeles over Q whose component at infinity is 0 and K is an open compact subgroup of G(A1) then G(A1)/K is discrete and Xx = .p x G(A1)/K is a complex manifold on which G(Q) acts to the left. If K is sufficiently small then any r E G(Q} with a fixed point lies in Z(Q) n K and thus fixes the whole manifold. We shall always assume that K is sufficiently small and then ad h(z)(X)

is a complex manifold, proved by Baily-Borel to be the set of complex points on an algebraic variety Shx = Shx(G, h) = Shx(G, ,P) over C. Deligne anticipates that Shx will often be a moduli space for a family of motives over C. This is sometimes so, the motives then being those attached to abelian varieties, but can certainly not yet be proved in general. Nonetheless there is a good deal to be learned from a rehearsal of the considerations that suggest such an interpretation of Shx. In essence one observes that Shx(C) is the parameter space for a family of polarized Hodge structures ; the difficulty is to show that these Hodge structures all arise from motives. A real Hodge structure V is a finite-dimensional vector space VR over R together with a decomposition of its complexification Vc = E9 p, q EZ vP• q , satisfying yq, p = VP, q . The collection of real Hodge structures forms a tannakian category over R, whose associated group is /?4. Indeed to a real Hodge structure V, one associates the representation rJ of 1?4 defined by (4. 1) The relations yq ,p = VP . q imply that· rJ is defined over R. Conversely each represen­ tation of 1?4 that is defined over R yields a Hodge structure, the elements of yp , q being defined by (4. 1 ) . The real Hodge structure Vis said to be of weight n if VP ,q = 0 whenever p + q #- n. Certainly any Hodge structure is a direct sum V = ffi n Vn with vn of weight n. We are however interested in the category of polarized rational Hodge structures. A rational Hodge structure V is formed by a finite-dimensional vector space VQ over Q, a direct sum decomposition VQ = E9 V(j, and real Hodge structures of weight n on Vfi = VJ ® Q R. There is a distinguished object of weight - 2, the Tate object Q( l ) , in the category of rational Hodge structures. The underlying rational vector space is Q(l)Q = 2�iQ 5:::; C and, by definition, Q( l) - 1 .- 1 = Q( l )c .

AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES , AND MOTIVES

21 5

It seems to be customary to identify the underlying vector space of Q(n) = Q(l)®" with (2n-i)"Q and Q(n)8 with (2n-i)"R !;;;;; C. The factors 2n-i have been chosen for reasons which need not concern us . It is no trouble to carry them along. If V is a rational Hodge structure and u the associated representation of f!lt let C be u( - i, i) acting on V8. If V is of weight n, a polarization of V is a bilinear form P : V x V -+ Q ( - n) satisfying : (a) For all u and v in Vc and all r e R(C) P(u(r)u, u(r)v) = u(r)P(u, v). Thus the form is compatible with the Hodge structures. (b) P(v, u) = ( - 1)"P(u, v). (c) The real-valued form (2n-i)"P(u, Cv) on V8 is symmetric and positive-definite. A rational Hodge structure is said to be polarizable if each of its homogeneous components admits a polarization, a polarization of the full structure being defined by polarizations of the homogeneous components. The category Jf(!}!!&(Q) of polarizable Hodge structures is tannakian, with a natural fibre functor WHod : V -+ V0 and an associated group GHod • reductive but overwhelmingly large. It does have factor groups of manageable size. If V is a polarizable rational Hodge structure, one may take the tannakian cate­ gory generated by V and Q(l) and the repeated formation of duals, sums, tensor products, and subobjects. The associated group is called the Mumford-Tate group of V and denoted by .Itff( V). It is finite-dimensional and reductive, and there is a surjection GHod -+ .Hff(V) defined over Q. If u is the representation of f!lt attached to V then .Hff( V) is simply the smallest subgroup of the group of automorphisms of the rational vector space underlying V which contains u(f!lt) and is defined over Q [37]. It is consequently connected. The polarizable rational Hodge structures for which .ltff(V) is abelian play a particularly important role in the study of Shimura varieties. They are said to be of CM type. The second description of the groups .ltff( V) shows that the category of such Hodge structures is closed under sums and tensor products and thus is a tan­ nakian category. The associated group has been studied at length in [41] and is often called the Serre group. At the risk of making a comparison with [41] difficult, for Serre himself employs a different notation, we shall denote the group by !7'. It is not difficult to describe !7'. Let Q be the algebraic closure of Q in C and let t e Gal(Q/Q) be complex conjugation. To construct X*(Y), the module of rational characters of !7', we start with. the module M of locally constant integral-valued functions on Gal(Q/Q). The Galois group acts by right translation and X*(Y) = {A e Mi (u l)(t + 1 )). = (t + 1)(u - 1)). = 0 'flu e Gal(Q/Q)} . In particular if ). e X*(Y), -

(u

1)).( 1)

+

(u

l) ).(c ) = 0 because the left side is (t + l)(u - 1)).(1). The lattice of rational characters of f!lt is canonically isomorphic to Z E9 Z and the homomorphism ho : f!lt -+ [!' dual to the homomorphism X*(Y) -+ X*(f!lt) which sends ). to ().(1 ) , ).(c)) is defined over R. The composite h o w : GL(l) -+ !7' is dual to the homomorphism X*(Y) -+ Z taking ). to ).(1) + A(t) and is defined over Q. -

-

216

R . P . LANGLANDS

To verify that the group !I' just defined in terms of its module of characters is the Serre group defined in terms of Hodge structures is easy enough. The existence of the two homomorphisms h0 and h0 o w implies that every representation of !I' defined over Q defines a rational Hodge structure. It is enough to show that these are polarizable when the representation is irreducible. To obtain the irreducible representations, one takes a A E X*(!I') and defines the field F by Gal(Q/F) = {a E Gal(Q/Q) I aA = A}. The underlying space of the representation is the vector space over Q defined by F, and the representation r = r1 is that defined symbolically by r1(s) : x E F -+ A(s)x. The weight of the associated Hodge structure is - (A( I) + A(c)) = n. There is an a E F such that c(a) = ( - I) na and ( - I)l <el j na is totally positive. A possible polarization is P(u, v) = (2n:i)-n Trnguac(v).

Conversely suppose one has a rational Hodge structure V whose Mumford-Tate group vltff(V) is abelian. Since there is a homomorphism � -+ vltff( V), the coweight GL(I) -+ � of the group � defined by z -+ (z, 1) also defines a coweight v V of Jtff( V). The lattice Y* of coweights of Jtff( V) is a Gal(Q/Q) module gen­ erated by v v . We define an injective homomorphism of its lattice of rational charac­ ters V* into X*(!l') by sending v E Y* to the element A given by a E Gal(Q/Q). The dual homomorphism !I' -+ vHff( V) is surjective and V is defined by a rational representation of !1'. We return to the Shimura varieties ShK(C) and show how one attaches families of rational Hodge structures to ShK(C). Let G0 be the largest quotient of G such that (a - l)(c + l)v V = (c + l)(a - l)v V = 0 for every coweight of the centre of G0. Let � be a rational representation of G on V which factors through G0• To each x = (h, g) in XK we associate a triple ( Vx , t x , q;"). vx is a rational Hodge structure whose underlying space is V� = Vg, the Hodge structure being defined by the representation � o h of �. The third term q;" is an isomorphism Vj1 -+ VA1 and is given by v -+ �(g)-l v. It is only defined up to composition with an element of �(K). The homogeneous components of V� are independent of x and may be written V0. It follows from Lemma 2.8 of [11] that if x E XK there is at least one collection p x of bilinear forms P� : V0 x V0 -+ Q( - n)g which is invariant under the derived group of G and is a polarization of vx . Let \13" denote the collection of all such polarizations. If g E G(Q), px E \13", and x' = rx then the collection px' given by P�' : (u, v) -+ P�(�(r-l)u, �(r-l)v) lies in \13"'. We must also verify that the family { V"} over XK is a family of rational Hodge structures in the sense of [9]. Otherwise it could not possibly be attached to a family of motives. There are two points to be verified. Let v;, q be the subspace of Ve of type p, q and set Vp = E{Jp';;;p Vp', q ' · The space v; £; Vc must be shown to vary holomorphically with x. In other words if v(x) is any local section of v; and Y any antiholomorphic vector field then Yv(x) also takes values in v;. The condition of transversality must also be established, to the effect that for the same v(x) and any holomorphic vector field Y the values of Yv(x) lie in V�1 •

AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES

217

There is certainly no harm in supposing that v(x) takes values in VP, q· Then one has to show that if X0 (h0, g'j) is fixed and Y is the vector field defined by XE @ then Yv(x0) lies in Vp when X E lJho and in v;_1 when X E f:iho · Let Kho be the stabi­ lizer of h o in G(R). We represent XK as the quotient G(A)/Kho K and lift v(x) to a function on G(A) which we write as =

The function u takes values in the constant space Vfq· Moreover Yv(xo ) �(X) u( I , g/) + Xu( l , gf) . The second term lies in Vfq for all X E @. If X E lJ then .;(X) Vfq � V�1, q_1 and if X E fj then .;(X) vrq � Vt1 . q+l · If r E G(Q) and x' = rx then v ---+ .;(r) v provides an isomorphism between ( Vx .� .., rpx) and ( Vx', �x', rpx'). Thus, if s E SK(C), any two elements of {(Vx, � .., rp"') I x ---+ s } are canonically isomorphic, and we may take ( V•, �s, rps) to be any one of them, and redefine our family as a family of rational Hodge struc­ tures, with supplementary data, over the base SK(C). The locally constant sheaf Fe(Q) of rational vector spaces underlying this family is the quotient of VQ x XK by the action r : (v, x) ---+ (.;(r)v, rx) of G(Q). For this quotient to be well defined, the group K must be sufficiently small, for G(Q) n KhK is then contained in the kernel of .; for all h (cf. II.A.2 of [41]). It is here that the condition that .; factors through G0 intervenes. When dealing with motives, one does not need to introduce the polarizations explicitly as part of the moduli problem, but in order to introduce a Hodge struc­ ture on the cohomology groups of the sheaves Fe(Q) one must verify that on each connected component of SK(C) a locally constant section s ---+ p s can be defined. Let G0(R) be the connected component of G(R) and choose X0 (h, g1) in XK. If =

=

X�

=

{(Ad g o h, gfk) j g E G0(R), k E K}

then the image of X� in SK(C) is open. If x and x' lie in X� and x' rx then r lies in (4.2) All we need do is find a collection P = { Pn } such that P E � .. for all x E X� and Pn (�(r)u, .;(r)v) Pn (u, v) if r lies in the group (4.2). Choose any P in � ..0• Then P E � .. for all x E X�. There are certainly homomorphisms An of G into the general linear group of vn such that =

=

The eigenvalues of A(r) are positive if r lies in G0(R)Kh. Moreover A is trivial on the derived group of G and factors through G0• It therefore follows from the results of II.A.2 of [41] that each An is the identity on all of (4.2). Although we have defined the families ( Vs, �s, rps) for any G, it is clear that SK(C) is not going to appear as a moduli space unless G is equal to G0, and so for the rest of this section we assume this. The moduli problem is best formulated completely in the language of tannakian categories. We can drop the polarizations and retain only the pairs ( Vs, rp5), or ( Vx, rpx), but we now have to emphasize that ( VX, rpx) is

21 8

R.

P . LANGLANDS

defined for every (finite-dimensional) representation � of G over Q, and so we write (�, V(�)) for the representation and ( V"(�), «p"(�)) for the pair ( Vx , «p"). On the category f]£gfJ'(G) of finite-dimensional representations of G we have the natural fibre functor (L)Rep (G) : (�, V(�)) --> V(�)Q and rt : (�. V(�)) --> Vx(�) is a ®-functor from :JllgfJ'(G) to .Yf(!!Ef?(Q) which satisfies (L) Hod o r;x = (L)Rep (Gl · Since .Yf(!!Ef?(Q) and f]£g£37!(GHod) are the same categories, r; x defines a homomorphism [40, 11.3.3.1] «px : GHod --> G and r;x may be defined by (�, V(�)) --> (� o «pX, V(�)). When we emphasize r;X. «p x appears as an isomorphism of two fibre functors However, when we emphasize «pX, as we shall, then these two fibre functors are the same, for they are both obtained from (L)Hod o r;" = (L)Rep (Gl by tensoring with A 1 , and «p x may be interpreted as an isomorphism of (L)�[p CGl · Such an isomorphism is given by a g- 1 E G(A1) [40, 11.3]. This is the g appearing in x = (h, g). Only the coset gK !";:::; G(A1) is well defined. We have arrived, by a rather circuitous route, at the conclusion that XK parame­ trizes pairs («p, g), «p being a homomorphism from GH od to G defined over Q, and g in G(A1) being specified only up to right multiplication by an element of K. In addition, «p is subject to the following constraint : H. The composition of «p with the canonical homomorphism :Jll --> GH od lies in �­ If r E G(Q) the pairs («p, g) and (ad r o «p, rg) will be called equivalent. The variety ShK(C) parametrizes equivalence classes of these pairs. One of the important tannakian categories is the category vlf(!! f7(k) of motives over a field k. It cannot be constructed at present unless one assumes certain con­ jectures in algebraic geometry, referred to as the standard conjectures [40]. It is covariant in k, and rational cohomology together with its Hodge structure yields a ®-functor h8 H : vlf{!!f7(C) --> .Yf(!!Ef?(Q). vlf(!! .oT(C) together with the fibre functor (L)Mot (c) of rational cohomology also defines a group GMot (c) over Q and h8 H is dual to a homomorphism hJH : GHod --> GMot (c) defined over Q. Implicit in Deligne's construction is the hope that any homomorphism «p' : GH od --> G satisfying H is a composite «p' = «p o h�H · According to the Hodge con­ jecture, «p would be uniquely determined [40, VI.4.5] and ShK(C) would appear as the moduli space for pairs («p, g), with g as before, but where «p is now a homo­ morphism from GMot Ccl to G defined over Q and satisfying : H'. The composition of «p with the canonical homomorphism fJ£ --> GMot CCJ lies in �­ This may be so but it will not be a panacea for all the problems with which the study of Shimura varieties is beset. So far as I can see, we do not yet have a moduli problem in the usual algebraic sense, and, in particular, no way of deciding over which field the moduli problem is defined. We can be more specific about this difficulty. Suppose 1: is an automorphism of C. Then 7:-1 defines a ®-functor r;(1:) : vlf{!! .oT(C) --> vlf{!! f7(C). Let GfJot CCJ be the group defined by vlf{!! .oT(C) and the fibre functor (L)Mot CcJ o 7](7:). The dual of 7](7:) is then an isomorphism over Q : «p(7:) : GMot (C) --> GfJot (C) · The homomorphism «p has a dual, a ®-functor r; : fl£ vlf(!! .oT(C) and the fibre functor (L)Mot CcJ o 7](7:) o r; defines a group GT ,'fi over Q. The ®-functor r; then defines a dual

AUTOMORPHIC REPRESENTATIONS , SHIMURA VARIETIES , AND MOTIVES

rp• :

G' · "' , and rp ' = rp' 0 rp(z-) :

GK!o t (C) -->

GMot (C)

219

--> G'· "' ·

Moreover the two fibre functors (l)d61 ccJ and (l)d61 ccl o r;(z-) are canonically isomor­ phic. As a consequence, there is a canonical isomorphism G(A1) --> G•· "'(A1). Let g' be the image of g. The pair (rp ', g') seems once again to define a solution to our moduli problem. The difficulty is that G'· "' may not be the group G or, even if it is, the composition of rp ' with the canonical homomorphism may not lie in .p. One of the purposes of the next two sections is to discover what G' · "' is likely to be. 5. The Taniyama group. There is one type of Shimura variety which is very easy to study, that obtained when G is a torus T. Then the set .p reduces to a single point {h}. For each open compact subgroup U of T(A1) the manifold Shu(C) con­ sists of a finite set and Shu = Shu(T, h) is zero-dimensional. In general a special point of (G, ,P) will be a pair (T, h) with T c:;; G and h E .p. If U = K n T(A 1) then Shu(C) is a subset of ShK(C), the points of which have traditional!y been referred to as special points, and I shall continue this usage. But it is best to give priority to the pair (T, h) rather than to the points of ShK(C) it defines. There are a number of un­ solved problems about Shimura varieties and their special points that I want to describe in the next section. To formulate them some Galois cocycles have to be defined. Deligne has shown me that my original construction gave, in particular, a specific extension of Gal(Q/Q) by the Serre group, Y', an extension I venture to call the Taniyama group and denote by .r. Since the cocycles needed are, as Rapoport observed, often easily defined in terms of .r, I begin by constructing it. The group Y' is an algebraic group over Q, and .r will also be defined over Q. Thus we will have an exact sequence Y' --> .r --> Gal(Q/Q) --> 1 . Recall that X*(Y') i s a module of functions on Gal(Q /Q), and that the Galois action on X*(Y') giving the structure of Y' as a group over Q is defined by right translation. We are still free to use left translation to define an algebraic action of Gal( Q/Q) on Y', and it is this action which is implicit in (5. 1). The extension will not split over Q but it will be provided with canonical splittings over each l-adic field Q�> Gal(Q /Q) --> .r(Q1), which will fit together to give Gal{Q /Q) --> .r(A1). Rather than attempting to work directly with Y', I choose a finite Galois exten­ sion L of Q, let ff' L be the quotient group of Y' whose lattice of rational characters consists of all functions in X*(.9") invariant under G(Q/L), and define extensions (5. 1)

(5. 2)

I

I

__.

__.

ff'L

__.

.r L

__.

Gal(£ab fQ)

-->

I,

afterwards lifting to Gal(Q /Q), and then passing to the limit. To motivate the construction we suppose that the extension is defined and that there is a section r --> a{r) of _r L --> Gal(Labf Q) with a(z-) E _r L(L). Let a(z- 1 )a(z- 2) = d,1, ,2a(z-1r2) with d,1, ,2 E ff'L(L), and with (5. 3)

Observe that the elements of the Galois group play two different roles. They are first of all elements of a quotient group of _rL , and secondly they are automorph­ isms of Lab and thus act on _r L(£ab) , since _r L is defined over Q. In the first role

220

R . P. LANGLANDS

they will be denoted by

p or a.

We have p(a (1:))

(5.4)

1:,

perhaps with a subscript added, and in the second by

= cp (1:) a(1:) with cp (7:) cp .- (a(1:))

=

E f/'L(L). Certainly p(c.-(7:)) cp (7:) .

In addition (5.5)

Conversely if we have collections {cp(7:)} and {d�1. <'2} satisfying (5. 3), (5.4), and (5.5), we can construct f7L over Q, together with the section a . Any splitting Gal(LabJQ) -+ f7(A1) will be of the form 1: -+ b(1:) a(1:) with b(1:) E f/'L ( A1 (L)). In order that it be a splitting, we must have b(7: 1 )7: 1 (b(7:2))d�!o <'2 = b(1:1 1:2). (5.6) If the b(1:) a(1:) are to lie in f7L(A 1) we must have p(b(1:)) cp (1:) = b(1:). (5.7) Again any collection {b(1:)} satisfying (5.6) and (5.7) defines a splitting, and it is our task to construct {b(1:)}, { cp (7:)}, and {d��o �J · The group !/' is a quotient of GHod and thus is provided with a canonical homo­ morphism h : � -+ !/'. Over C the group fJ.f is canonically isomorphic to GL(l) x GL(l). Restricting h to the first factor we obtain a coweight fl. of !/', the canonical coweight. If ). E X*(!/') then ()., fl. ) = ).(1). Since f/'L is a quotient of !/', fl. also defines a coweight of f/'L, which for convenience will also be denoted by fl. · If v is any coweight of f/'L and x any invertible element of L or of Aj(L) then x" will be the element of f/'L(L), or f/'L ( A1(L)) , satisfying ).(x") = x for all ). E X*(f/'L). Recall that we have two actions of Gal(L/Q), or Gai(LabfQ), on X*(f/'L) or on X* (f/'L) = Hom (X * (f/'L), Z), the lattice of coweights. That defined by right translation we write v -+ av, and that defined by left translation we write v -+ v1:. Thus ()., afl. ) = ).(a- 1 ) while ()., f1.7:) = ).(1:) , because an inverse intervenes in the action by left translation, and f1.7:- 1 = 7:f1.. Moreover u(x�<) = u (x)u" while 1:(x�<) = x�<� . These aspects of the notation have to be emphasized because at some points our convention of distin­ guishing between p , a on the one hand, and 1: on the other, fails us. For the study of Shimura varieties, it is best to take Q to be the algebraic closure of Q in C, and we shall do this. Thus we provide ourselves with an extension of the infinite valuation on Q to Q. There is a property of the Weil groups that will play a prominent role in our discussion. Let v be a valuation of Q and hence of Q, and let Q. and Q. be the completions of Q and Q with respect to v . Eventually v will be defined by the inclusion Q s;:; C, and Q. will be R and Q. will be C. In any case, the data provide us with imbeddings (5.8)

Gal(FvfQ.) 4 Gal(F/Q), if F is any finite Galois extension of Q in Q. The local and global Weil groups WF,to, and WFI" are defined as extensions (5.9)

AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES

and 1

� CF � WFtQ � Gal(F/Q) �

221

1.

We may imbed the arrows (5. 8) and (5.9) in a commutative diagram 1 1

� �

Fvx

l

CF

�

WF,tQ.

� Gal(F./Q.) � 1

l � � WFtQ IF

l

Gal(F/Q)

�1

Moreover we may so choose the central arrows that they are compatible with field extensions and upon passage to the limit yield I: WQ. � WQ. It is the image of WQ. in WQ that will be fixed, and IF may be changed to w � xiF(w)x-1 where x E CF and xa(x)-1 E F{' for all a E Gal(F./Q.). N ow let v be the valuation given by Q � C. Let F/:, = IT wl vF� . The natural map F/:, � CF is an imbedding, and we sometimes regard F/:, as a subgroup of CF · If we take an element 1: of Gal(Q /Q), lift to WQ, and then project to WuQ we obtain an element w = w(1:) of WuQ which is well defined modulo the connected component, and in particular modulo the closure of L/:,. We choose a set of representatives w,., a E Gal(L/Q), for the cosets of CL in WuQ in such a way that the following conditions are satisfied : (a) w 1 = 1 . (b) If a E Gal(LvfQ.) then w,. E wL.tQ. · (c) If p E Gal(L/Q) and a E Gal(LvfQ.) then WpWu = ap, 11 Wpu with ap.u E L;,. To arrange the final condition we may choose a collection f of representatives r; for the cosets Gal(L/Q)/Gal(L./Q.) and set w'l ,. = w'l w,. if a E Gal(Lv fQ.). We suppose that f contains 1 . With this choice we also have : (d) If {ap.u} is the cocycle defined by wpwu = ap.uWpu then a71,p = 1 for 7J E f and a E Ga1(L.IQ.). If w E WLIQ• let w,.w = c,.(w) w,. , cu ( w) E CL . If w = w(1:) we set bo(7:)

=

n uEGai (L/Q)

c,.(w)U." .

It lies in CL ® X* (9'L), but is not well defined, because w is not. However we can show that it is well defined if taken modulo L(;, ® X* (9'L), and that, in addition, it pehaves properly under extensions of the field L. The ambiguity in w now has no effect, for we are only free to replace w by uw where u = limnun and un lies in the image of L;, U, where U is a subgroup of the group of units of L defined by a strong congruence condition. But [41, II.A.2], n Q a(v)111' = 1 uEGai (Q/ )

for all v E U. Consequently I] cu Cuw)111'

=

(l)a(u)111')(l) cu Cw)111')

is congruent modulo L;, ® X* (9'L) to IT ,. cu(w)111'. Suppose the representatives w,. are replaced by e,.w,., and, hence, ap.u by

222

R. P. LANGLANDS

a�, a = epp(ea)e";"Jap.a· If (1 e Gal(L./Q.) then e17 e L'{; and p(e11) e L;,. Since ap,a and a;, a must then both be in L�, we infer that eP = epa (mod L�) when (1 e Gal(L.JQ.). Moreover ca(w) is replaced by c;(w) = ene;;-1c(w) and b0(1:) by

b(kt:)

=

{ l}e:�e;;-17�'}bo(7:) .

The factor may be written Since this change has no effect on b0(1:). In this argument we have denoted the image in Gal(L/Q) of 1: e Gal(Q/Q) by the same symbol, a practice we shall continue to indulge in. If we modify I then w is replaced by xwx- 1 with x e CL and xa(x)- 1 e L: for all a e Gal(L./Q). Then ca(xw.x-l) = a(x)a7:(x-l)ca(w) and ITa(x)-a�'a7:(x)a" = fl a(x)t7C.-'- 1J" .

17

17

Since a(x) = x (mod L;,) if a e Gal(LvfQ.), the same argument as before shows that b0(1:) is unchanged. Finally suppose that L £ L'. Then b (1:) e CL ® X*(,SP L ') and b0(1:) e CL ® X*(,SPL) are both defined, and we must verify that is taken to b0(1:) by the canonical mapping of the first group to the second. Either Lv = R or L. = C, and the two cases must be treated separately. Suppose first that L. = C and hence that Gal(L'/L) n Gal(L�/Q.) = { I }. Since both b0(1:) and b0(1:) are independent of the choices of coset representatives, we may choose those which make it easiest to verify that b0(1:) is the image of b0(1:). Let e be a set of representatives for the cosets Gal(L'/L)\Gal(L'/Q)/Gal(L�/Q.) containing 1 . Every element a of Gal(L'/Q) may be written uniquely as a product (1 = '7Jp, ' e Gal(L'JL), 7J e e , p e Gal(L�/Q.). We may suppose that w; = w�w�w� with w� = 1 and w� e wL�IQv " Let w' be w(1:) with respect to L', and w be w(7:) with respect to L. Then under the canonical map 1r : Wu1g -> Wug the element w' maps to w. If a e Gal(L/Q) it lifts to a unique element of Gal(L' /Q) of the form 7JP · We suppose that W17 is the image of w�w;. Thus if

0

with d7J.p(w') e WL 'IL then ca(w) Gal(L' /Q) then

=

b0(7:)

7r(d7J.p(w')) . On the other hand if a 1

wcd7].p(w')

=

ca,(w')wc.

Consequently, by the very definition of 1r, ca(w)

=

IT C171(w') .

0'!-fl

=

'7JP lies in

AUTOMORPHIC REPRESENTATIONS , SHIMURA VARIETIES , AND MOTIVES

223

It follows immediately that b0(i) is the image of b�('r). If L. = R then X * (sPL) � Z and the Galois group acts trivially. Suppose we replace W11 by e11W11 with e11 E CL . Then cu(w) is replaced by e11e-;;/. Since I1(e11e;,.1 ) 11P u

=

I1(e11e;,.1 )P u

=

I,

this has no effect on b0{'r), and when defining b0(-r) we need not suppose that the collection { w11} is subject to the constraints (a), (b), and (c). If we want to define b�(-r), we may still need to be careful about the choice of the coset representatives w�, a E Gal(L'/Q). However, since we are only interested in the image of b�(-r) in g>L, we may again ignore (a), (b), and (c). We choose a set e of representatives for the cosets Gal(L'/L)\Gal{L'/Q), write a = p'Y), p E Gal(L' /L), 'YJ E e, and take w� = w�w�. If a E Gal(L/Q) is the image of 'YJ• we take W11 = n-(w�). The argument can now proceed as before. Let b(-r) be a lift of b0(-r) to h ® XAYL) = g>L (A(L)) and let b(-r) be the projec­ tion of b (-r) on g>L ( A 1(L) ) . The element b (-r) is well defined modulo g>L(L) and, as we shall see, this bit of ambiguity will cause us no difficulty. But we have to fix one choice. The first point to verify is that d�1 o �2

=

b (-rl )-r l (b (-rz) ) b(-rl -rz) - 1

lies in g>L(L). When verifying this, we may choose the liftings b(-r1), b(-r2), and b(-r1-r2) in any way we like. We choose liftings cu(w1) and c11(w2) of cu(w1) and C11(w2) to h and take b(-r1)

Since cu{w1w2)

=

=

IT cu (w l ) uP , u

b(-rz)

=

ITciT(wz) uP. u

cu (w1ku�1 (w2), we may take cu {w1w2) to be cu(w1)cu�Jw2). Because -r:t1(b(-rz)) = ITcu (wz)u';- 1�' = ITcuq(wz) uP, u u

the element dq, ,2 will then be I . Finally we have to establish that the elements cp (-r) defined by equation (5.7) lie in g>L(L), for equations (5.4) and (5. 5) will then follow immediately. It will suffice to show that for any w E WuQ and any p E Gal(L/Q) ( 5 . 1 0)

lies in L(;, ® X* (YL). Suppose w = w 1 w2 and w1 projects to -r1 E Gal(L/Q). Then C11(w) = cu(w1)cu�1(w2) and ( 5. 1 0) is equal to

{ I] cu(wl)uPp (cu(wl))-pup}-r1 { I] c11(Wz)uPp (cu(wz))-pup}·

Consequently we need only verify that (5 . 1 0) lies in L/;, ® X* (YL) for w in CL and for w = w�. If w lies in CL then cu(w) = a(w) and Il ua(w)up = Il upa(w)Pul'. The expression ( 5. 1 0) is therefore equal to I . If w = w� then cu (w) = aM and (5. 1 1) However it follows from condition (c) that ap . u = ap, 11, (mod L�). Since (1 + c) · (I - c 1 )p = 0, the right side of (5. 1 1) lies in L;;, ® X (sPL). *

224

R. P. LANGLANDS

Since b(r), although defined for -r E Gal(Q/Q), depends only on the image of -r in Gal(La bfQ), the groups f7 L and f7 are now completely defined. The ambiguity in the b(-r) is easily seen to correspond to the ambiguity in the choice of the section a(-r). If E £ Q is any finite extension of Q, we let f7E be the inverse image of Gal(Q/E) in !7. If E £ La b we may also introduce f7k. The group !7f has been introduced by Serre [41], who uses it to formulate some ideas of Taniyama. He makes its arith­ metic significance quite clear, but his definition is sufficiently different from that given here that an explanation of the reasons for their equivalence is in order. If X is an idele of L then cp(x) = rr Gai (L/Q) a(x)ap is an element of 9"L (A(L)) . By 11.4 of [41] there is an open subgroup U of the group of ideles h such that cp(x) = I if X E u n LX The standard map of IL onto Gal(LabfL) restricts to u and if 'C = "C(x) is the image of x we may take w = w('C) to be the image of x in CL . Then cp(x) is a lifting of b0('C) to 9"L (A (L) ) . However cp(x) depends only on '"· and thus we may take b('C) = cp(x). Then d,h '<2 = I and cp('C) = I if 'C, '"h "C2 lie in the image Gal(LabIF) of U. Here F is the finite extension of L defined by U. The elements cp('C) are in fact 1 for all '" E Gal(LabfL). If we choose a set of representatives e for Gal(LabfF)\Gal(LabfL) and set b(n;) = b(-r)b(r;), '" E Gal(£ab/F), 7J E e, then, in general, d•1. •z will depend only on the images of 'Cb -r2 in Gal(F/L), and !7f may be ob tained by pulling back an extension 0

1

->

9"L

->

f7u

->

Gal(F/L)

->

I

to Gal(LabfL). Here f7 u is the quotient of !7f by the normal subgroup {a("C) i-r E Gal (La b{ F)}. It is the extension f7u that Serre defines directly. He denotes it by the symbol Sm . The map ¢ defines a homomorphism of LX/LX n u � LX U/LX into 9"L(L) and to verify that f7u is the group studied by Serre, we have only to verify that it can be imbedded in a commutative diagram

The right-hand arrow is x -> "C(x)-1. We do not have to pass to the quotient but may define the homomorphism h/LX -> !7 f(L) (5. I 2) directly. The central arrow is then obtained by composing with the projection !7f(L) -> f7u(L) . The homomorphism (5. 1 2) is x

->

{ I] a(x)ap} b('C)-1a('C)-1

ih is the image of x in Gal(LabfL). The composition of our splitting Gal(La bf£) -> !7 f(Q1) with !7f(Q1) -> f7 u (Q1) is either Serre's e1 or its inverse, presumably its inverse, for we are so arranging matters that the eigenvalues of the Frobenius ele­ ments acting on the cohomology of algebraic varieties are greater than or equal to 1 . The cocycle p -> cp('C) certainly becomes trivial at every finite place, but i s not necessarily trivial at the infinite place, v . Indeed under the isomorphism

AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES

225

H l (Gal(L.JQ.), ffL(L.)) � H- l (Gal(L./Q.), X* (ffL)) given by the Tate-Nakayama duality it corresponds to the element of the group on the right represented by(l - 7:-l)p. If, as has been our custom, we denote the image of 1: in Gal(L/Q) again by 1: we may suppose that w(1:) = w., for the class of { cp{1:)} depends only on thi� image. According to the discussion of formula (5. 1 1) IT ap,puu<�- 1-l) p = IT (aP• 7J' a-l p, 7J) fl11 (.- 1-l) p = eP (7:) U 7) E f

lies in L;, ® X* (ffL) = ffL(L00). By definition, the classes of {cp(7:)} and {ep(7:)} are inverse to one another in Hl(Gal(L/Q), ffL(A(L)) ). Thus all we need do is calculate the projection of ep(7:) on ffL(L.) for p E Gal(L./Q.). Observe first of all that if we agree to choose coset representatives satisfying (d), then a P • 7J' ap,� = p(a;,�) a P7J · ' = ap 71, , . Again if P'YJ = 1J1p1 then a P 7J • ' = ClqiPb t = 1J l (a Pb ,) a 7JI, Pit a;.� PI = 'YJ l (a Pb ,).

Since aPb , E L�, the term on the right has a projection on L� different from 1 only if 'YJ l = 1 . Then 1J too equals 1 , and so the projection of ep(1:) on ffL(L.) is

in conformity with our assertion. Suppose T is a torus over Q, provided with a coweight It such that (5. 1 3)

(1 + c) (1: - l)p

=

(1: - 1)(1 + c)p

= 0

for all 1: E Gal(Q/Q). Then there exists a unique homomorphism cjJ : !7 --. T such that the composition of cjJ with the canonical coweight of !7 is It · We can transport the cocycles p --. cp(1:) from !7 to T, obtaining cocycles {cp{7:, p)} as well as b(1:, p) E T(A1(L)) . However if T and h : � --. T define a Shimura variety and It is the restric­ tion of h to the first factor of � the condition (5. 1 3) will not necessarily be satisfied. Nonetheless, we may repeat the previous construction and define b(1:, p) and {cp(7:, p)} for all 1: such that (1 + c)(1:- 1 - l)p = 0. This generalization is necessary for the treatment of those Shimura varieties Sh{G, h) for which G is not equal to G0• It should perhaps be observed that b(1:, p) is not insensitive to the ambiguity in the choice of w = w(1:), although { cp{7:, p)} is. However, if Z is the centre of G the ambiguity all lies in Z(A1) n K, and may be ignored. If E £ C the motives over E whose associated Hodge structure is of CM type are themselves said to be of CM type. They form a tannakian category CM( E) with a natural fibre functor wcM cE» given by rational cohomology. Since one expects that the natural functor CM(Q) --. CM(C) is an equivalence, we may as well sup­ pose that E £ Q. According to the hopes expressed at the end of the previous section there should be an equivalence 1J : CM(Q) --. �tff[l}J(ff) and an isomorphism WRep (9') o 7J --. WcM (Ql • which would enable us to identify !7 with the group GcMCQl • defined by the category CM(Q} and the functor c.ocMCQl . If E £ Q, one hopes that in the same way it will be possible to identify ffE with GcM(El · There are properties which this identification should have, and it is necessary to describe them explicitly. First of all, ifF £ E the diagram

226

R. [I'

I

--->

P. LANGLANDS §"

1

E

--->

§" F

1

GcM CQJ ---> GcMCEJ ---> GcMCFJ

should be commutative. The other properties are more complicated to describe. Suppose -r E Gal(Q/Q) and -r takes E to E'. lts inverse then naturally defines a ®-functor 7](-r) from CM(E') to CM(E). Let GtMCE' ) be the group defined by the category CM(E') and the functor WcMCEJ o 7](-r). The dual of 7](-r) is then an isomorphism �(-r) : GcMCEJ � GtMCE') · In terms of representations, 7](-r) associates to every representation (.;', V(.;')) of GcM CE'J a representation (�, V(�)) of GcMCEJ · Since WcMCE'J and wcM CEJ o 7](-r) become isomorphic over Q there is a family of homomorphisms, one for each e, ¢(�') : V(e)Q --> V(�)(l, compatible with sums and tensor products. If ¢'(�) is another possible family then there is a t E !YE'(Q), such that cjJ'(e) = cjJ(e)�'(t). Finally since the two functors we"� CE ' ) and we"� CEJ o 7](-r) are canonically isomorphic, arising as they do from the /-adic cohomology, there is a canonical family of isomorphisms ¢4�') : V(�')At --> V(�)Ar On the other hand, suppose a(-r) E !Y(Q) maps to -.. Let a(a(-r)) = ca(-r)a(-r). We may use a(-r) to associate to every representation (.;', V(�')) of !YE' a representation (�, V(�)) of !YE· The space V(�) is obtained by twisting V(�') by the cocycle {�'(cu{-r)- 1 )}. The representation � is t --> e(a(-r)ta(-r)-1 ) . This functor (�', V(�')) --> (�, V(�)) is to be (isomorphic to) that obtained from 7](-r) by identifying !Y E and GcMCEJ and !Y E ' and GcMCE'J · Moreover one possible choice for ¢(�') is to be the isomorphism ¢0(�') : V(e)Q --> V(�)Q implicit in the definition of V(�). If b(-r)a(-r) is the image of -r under the canonical splitting Gal(Q/Q) __:. !Y(A1), then ¢A/e) is to be ¢0(�') o �'(b(-r)) - 1 . If -r E Gal(Q/E) then E' = E and 7](-r) : CM(E') --> CM(E) is the identity functor. Thus the functor (.;', V(e)) --> (�, V(�)) from Rep(TE,) = Rep(TE) to Rep(TE) must be canonically isomorphic to the identity. A final property, which seems to be independent of the preceding ones, is that this isomorphism should be given by �, --> � and ¢0(�') o �'(a(-r)) : V(e) --> V(�). Observe that a(-r) is now in !YE and these transformations are defined over Q. I assume that all this is so, just to see where it leads, and especially to see what it suggests about the groups Gr,rp introduced in the previous section. But there are some lemmas to be verified first. I conclude the present section by describing a property of the Taniyama group whose significance was pointed out to me by Casselman. It will be needed to show that the zeta-functions of motives, and especially abelian varieties, of CM type can be expressed as products of the L­ functions associated to representations of the Weil group. The point is that there is a natural homomorphism � of the Weil group WQ of Q into !Y(C) and thus for any finite extension F of Q a homomorphism �F : WF --> !YF(C). To define it we work at a finite level, defining WuQ --> ;YL(C), and after­ wards passing to the limit. Fix for now a set of coset representatives w,. which satisfies (a), (b), and (c). If w E WuQ we define b0(w) = fluE Gai CL!Q) c,.(w) 11�'. If -r is the image of w in Gal(LabjQ) then b0(w) = b0(-r) (mod L{;, ® X *(Y'L)), and we may lift b0(w) to b(w) in Y'L(A(L)) in such a way that the projection of b(w) in Y'L(A1(L)) is b(-r). However a simple

AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES

227

calculation shows that if..- I> z-2 are the images of wi > w2 then b(w1 )-r 1 (b(w2))b(w1 w2)- 1 E f/L(L). Since its projection on SL(Aj(L)) is equal to dr�> rz• it is itself equal to dr�o rz· If b.(w) is the projection of b(w) in f/L(L.) = f/L(C), we may define rp by rp : w � b.(w)a(z-) . If the coset representatives W17 are changed then rp is replaced by rp ' = ad a o rp, a E f/L(C), but this is of no importance. If GwF is the group over C defined by the tannakian category of contiauous, finite-dimensional, complex, semisimple representations of WF then lfJF is the com­ posite of the imbedding Wp � GwiC) and an algebraic homomorphism cpp : GwF � !TF· Moreover if the principle of functoriality is valid, there is a surjection GucFJ � GwF and we can expect to have a diagram GU(F)

1

PF

__,

GwF ---;;;-;-'

whose two composite arrows differ by ad s, s e f/(C) s;;; !Tp(C). 6. Conjugation of Shimura varieties. The principal purpose of this section is to formulate a conjecture about the conjugation of Shimura varieties, a conjecture whose first justification is that it is a simple statement which implies what we need for the study of the zeta-functions at archimedean places and is compatible with all that we know. Some lemmas are necessary before it can be stated, and we shall see that these lemmas together with the hypothetical properties of the Taniyama group suggest an answer to the question that arose at the end of the fourth section. This answer in its turn throws new light on the conjecture, so that we can weave a consistent pattern of hypotheses, and our task will be ultimately to show that it has some real validity. We need a construction, which we make in sufficient generality that it applies to all Shimura varieties and not just those associated to motives. Suppose the pair (G, ,))) defines a Shimura variety, T and f' are two Cartan subgroups of G defined over Q, and h : f!ll � T, h � f!ll � f' both lie in .)). Let f-t and f1 be the coweights of T and f' obtained by restricting h and h to the first factor of f!/l, and choose a finite Galois extension L which splits T and f'. Let -r E Gal(Q/Q) ; then the coweights (1 + c )(z-- 1 - 1)p and (1 + c ) (z-- 1 - 1)(1 are both central and they are equal. Choose w = w(z-) as before, and set b o(Z", p) = n c�(w)""�' , bo(z- , fl ) = n cu(w)"".U. 17

17

Let b (z- , p) and b(-r, fl ) be liftings to T(A(L)) and T(A(L)) and let B(-r) = B(z- , f-t• fl) be the projection of b(-r, fl)- 1 b(z- , p) on G(Aj(L)) . Although we may not be able to define the cocycle {cp (-r , p) } , we can define { cp(-r, /-tad)} if /-tad is the composition of f-t with the projection to the adjoint group. It may be as well to check that B(z-) is indeed independent of the choice of w and of the coset representatives w.,., provided the usual conditions (a), (b), and (c) are satisfied. If w.,. is replaced by e.,.w.,. then, apart from a factor in L'{;, ® X*(T), b0(-r, p) is multiplied by n 1)E f e�Cl+c) (r-1-ll l', and bo(Z", fl) by n 1)E f e�Cl+c) (r- 1-l)Ji. Since these two terms are central and equal by assumption, the change has no effect on B(z-). If w is replaced by xwx- 1 with xa(x- 1 ) e L{; for a e Gal(L.IQ.), then b0(-r, p) is modified

228

R. P. LANGLANDS

by the product of an element in L/:o @ X* ( T) and TI 'I x'� CHtl C<- 1-lJ ,u r: . Since b0(r:, fl) undergoes a similar modification, B(r:) is not affected. One can also show easily that B(r:) is not changed when L is enlarged ; the argument is once again basically the same as that used to treat b(r:). B(r: ) does depend on the choice of b(r:, f.J.) and b(r: , f1). These choices made, we will use them consistently to define cp(7:, f.l.ad), cp(r:, /1ad), and, when (1 + c)(r:- 1 - l ) f.J. = (1 + c)(c 1 - I)fl = 0, cp(r:, f.J.), cp(r:, p ). Thus cp(r: , f.l.ad) is to be the proj ection of p (b(r:, f.J.)-1) b (r:, f.J.) in G.d (Aj(L)) . With these conventions, the ambiguity in B(r:) will cause no harm in the construction to be given next. Let G< . .u and G'· ii be the groups obtained from G by twisting with the cocycles { cp(r:, f.l.ad )-1 } and { cp(r: , flad)-1 } . We are going to verify the following : FIRST LEMMA OF COMPARISON. (i) If Cp = cp(7:, f.i.ad ) then rP = B(r:) ad cp-1 (p(B(r:)- 1 ))

lies in G< . .u(L). (ii) The cocycle {rp} in G< . .u(L) bounds. (iii) If ( I + c)(r:- 1 - l )f.J. = ( I + c)(r;- 1 - l)p = 0 then 7P =

cp(r:, p)-1 cp(r:, f.J.).

The third assertion is clear ; it is the other two with which we must deal. The element r p can be obtained by projecting (6. 1) on G(A1(L)) . This makes it perfectly clear that rP is not affected if w = w("t"j-is--re· placed by xw(r:) with x E CL . Thus we may assume that w = wn where 1: is here also used to denote the image of 1: in Gai(L/Q). Then we have to show that b(r: , p )- 1 p (b(r:, p ))

=

b(r:, f.J.)- 1 p (b(r: , f.J.)) (mod G(L"') G(L)) .

It follows easily from (5. 1 1) that if iiP· 'I is a lift of aP·'I to congruent to

h,

then both sides are

The desired equality follows. To prove the second assertion we shall apply Hasse's principle, but for this we need a group G whose derived group Gde r is simply connected. Let Gsc be the simply connected covering of Gde r · The Cartan subgroup T defines Tder and Tsc · We have an imbedding X* (T.J -> X* (T) and Gde r = G.c if and only if the quotient is torsion­ free. If we can construct a diagram of Gal(L/Q)-modules

in which P is torsion-free and P -> M is a surjection whose kernel is a free

AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES

229

Gal(L/Q)-module, then we can use it to define a central extension G' of G with --> G(R) will be surjective. To construct the diagram, we choose an exact sequence of Gal(L/Q)-modules X*(T') = Q. We will have Gde r = G�c = G,0 and G'(R)

0 -> N -> P -> M -> 0,

with P torsion-free and N free. Then Q is the set of all (x, p) in X*(T) EEl P for which x and p have the same image in M. We lift ,u to ,u' = (,u, v) with v E P. If f1 = ad g o ,u and g is the image of g ' , we set p ' = ad g' o ,u'. If the assertion is valid for ,u', p ' , and G', it is valid for ,u, p, and G. Consequently we may suppose that Gde r is simply-connected. There are two types of ambiguity in B('r). lt can be changed to tB('r) with t E t(L). Then {rp} is replaced by {(trp ad cp-1 (p(t)- 1 )}, and its cohomology class is not affected. We can also change B('r) to B('r)t with t E T(L) . Then { cp} is replaced by { c�} with c� = p(t;l)cptad • tad being the image of t in Gad • and {rp} is replaced by {r�}, with r� = rp ad cp-1 (p(t- 1))t. o ad c;1 (p(o- 1 )) then r� = (ot)ad- 1 c� (p(ot)- 1 ) . Consequently the ambiguity i n B(z) has n o effect o n the assertion (ii). Rather than prove (ii) directly for a given choice of the pair ,u. p, we want to prove it for a succession of pairs. For this one should first check that the validity of (ii) defines an equivalence relation. If ,u and f1 are interchanged then {r p} is replaced by {r;1 }, and if rp = o ad cp-1 (p(o- 1 )) then r;1 = o- 1 ad cp-1 (p(o)) . To show the transitivity, we introduce a new notation, denoting rP by rp(fl , ,u), and cP by cp(,t.t). Suppose rp(,uz , ,t.t3) = t ad c;1 (,t.t3) (p(r - 1 )), rp(,t.tr . ,t.tz) = s ad cp-1 (,u z)(p(s - 1 )) .

If rP

=

Observing that rp(,t.tz , ,t.t3) ad cp-1 (,t.t3)(p(s- 1 ))rp(,t.tz, ,t.t3)- 1 = ad cp-1 (,u2)(p(s- 1 )) and that rp(,u r . ,u3) = rp(,t.t r . ,u2)rp(,u2, ,u3), one deduces with little effort that r/,u r . ,t.t3) = st ad cp(,u3)(p(st)- 1 ) . Transitivity established, we return to the original notation. If f1 i s conjugate to ,u under G(Q), say f1 = ad x o ,u then rP = x ad cp- 1 (x- 1 ) = x ad c;1 (p(x- 1 )),

and certainly bounds. In general f1 and ,u are not conjugate under G(Q), but they are conjugate under G(R). Since G(Q) is dense in G(R) we may take advantage of the transitivity and assume that they are conjugate under Gde r(R). It is now that the assumption that Gde r is simply connected intervenes. If we are

230

R. P. LANGLANDS

careful in our choice of b(z-, p.) and b(z-, p.), defining them by liftings of cu(w) to h, then B(z-) and the TP will lie in GJ.fr· Moreover the cocycle { rp} in GJ.!r(L) certainly bounds at every finite place. Since we are applying Hasse's principle, we need only verify that it bounds at infinity as well. One begins with a calculation similar to the one made while studying the cocycle {cp(t·)}. If p E Gal(L/Q), set and define ep(-c, p.) in a similar fashion. The projection of { rp} on G�·�'(L,.,} is co­ homologous to ep(z-, p.)ep(z-, p.)-1. If p E Gal(Lv/Qv) the projection of ep(z-, p.) on G�·�'(Lv) is Thus all we need do is show that/p('t", p.)/p('t", p.)-1 bounds in Gr·P(Lv)· Recall that the cocycle defining G�·P(Lv) is For brevity, we write oio. ar,�} = {ap(v)}. If X E G(R) we write x(p.) = ad X

x (p.) = p. then

0

P. ·

If

x/p(z-, p.) f,(z-, p.)-1 ad hp(p(x-1)) = ap((xz-- 1 ,x- 1 - z- -1)p.), because p(x) = x and fp(z-, p.)-1 ad hp(x-1) = x-1 /p(z-, p.)-1. If w lies in the normalizer of Tde r in Gder(Lv) then (6.2) However {wp(w-1)} is a cohomology class in Tde r(Lv) or TJ
(6.3) wp. = z--1p.. Then (1 + e)(z--1 - l)p. = 0 and {c;1(z-)} = {c;1(z-, p.)} is defined. It bounds in G(L). Once again it is enough to verify this when Gder is simply connected, although this time the modifications made to arrive at a simply-connected derived group would be different . It is no longer important that P � M be surjective, or that its kernel be free, but there must be a v e P which maps to p. and is fixed by Gal(Q/E). The set of all z- which satisfy (6.3) for at least one w form a group. Let E = E(G, h) = E(G, ,P) be its fixed field. Then E £ L. In order to apply Hasse's principle we have to arrange that {cp(z-)} lies in Gde r(L) and that it obviously bounds in Gde r (A1(L)) . Since cp(z-) only depends on the image of z- in Gal(L/Q), we may cal­ culate it when w = w� . If @i is a set of coset representatives for Gal(L/Q)/Gal(L/E), then

AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES

231

We write ava. r = v(au. �)av.a � a;, � and vap. = vp. + v(a - l)p.. Now for a e Gal(L/E), v(a - l)p. is a weight of the derived group and f L Ti a a��;-I J .u may be lifted to Tde r (A(L)) . On the other hand Ti v Ti a a�;,� a;,"f = 1 . If a = Ti a aa.r then a lies in CE and lifts to a' in IE. Moreover Ti v v(a)v.u lifts to Ti v ))(a') v.u which lies in T(A). If iivtM is a lifting of ava . r to IL then we so choose cp(-c) that (6.4) modulo T(L"'). It remains to verify that the {cp-1('c)} so defined bounds at the infinite place. The projection of the right side of (6.4) on CL ® X*(Tde r) is IT pvacr -1 -I J .u IT aiXICD'� -1 -IJ .u v, D' ap, va • aEGa!(LIQ) p, Thus, restricting {cp-1 (z-)} to Gal(L/Q.) and projecting on TdelL.) , we obtain a class cohomologous to {ap ((z--1 - I)p.)} = {ap ((c.o - l)p.) } . We have observed already that it is shown in [45] that the right side bounds in GdelL.) . Although there is one more consequence to be derived from (6.3), there are some things that must first be said about the general (T, h), or (T, p.). We drop the assumption (6.3) for a while, and return to it later. We have as­ sociated to the pair (T, p.) and z- a twisted form a� ... of G. The twisting of T in G is trivial, and T� = T is a Cartan subgroup of a� . .u. Let p.� be z-- lp.. There is a unique homomorphism h� : fJll --+ T� whose restriction on the first factor is p.� and which is defined over R. The pair (G� . .u, h�) defines a Shimura variety. The roots { r } of T in G are the same as the roots of P in a� ... . However the classification into compact and noncompact differs for the two pairs. The root r of Tin G is compact or noncompact according as ( - 1 ) is 1 or - 1 . In order for (G• . .u, h•) to define a Shimura variety, r must be compact or noncompact as a root of T� according as ( - I ) is 1 or - 1 . However the ideas used in the proof of Lemma A.8 of [34] show that the type of r is changed on passing from T, G to P, a� . .u if and only if ( - I) = - 1 . We are now almost ready to formulate a conjecture about the conjugation of Shimura varieties. After discussing the conjecture and its consequences, we shall show how it can be heuristically justified in terms of the Taniyama group and motives. Recall that ShK(C) = G(Q)\.f) x G(A1)JK. Thus if g e G(A1) and K1 = g-1Kg, then right multiplication by g defines a morphism _

•

�(g) : ShK(C) --+ ShK1 (C). It is algebraic and is called a Heeke correspondence. If z- is an automorphism of C, we set Shk(G, h) = Shk = ShK ®.- 1 C, the Shimura varieties being at the moment only defined over C. Then �·(g) : Shk --+ Shh If G = T is a torus and K = U then Shu is 0-dimensional. Let z- also denote the element of Gal(Q/Q) defined by z-. If we set U• = U then

232

R. P. LANGLANDS

Shu(C) = T(A1)/ U = T<(A1)j U< = Shu, · This gives us an isomorphism Shu = Shu(T, h) ---> Shu = Shu,(T<, h<). In addition there is the natural map r from complex points of Sh&(T, h) to complex points of Shu(T, h). Define rpr = rpr(U, T, h) by the commutativity of the diagram Shu(T, h)

,

Shb(T, h) / �� r Shu,(T , h<)

In general Gr . p and G are different. But G< . p is defined by the cocycle {c;1(r, ,U ad) } and in G.d(A1(L)) Thus

g -> gr = ad b(r, ,Uad)-1(g) defines an isomorphism of G(A1) with G<·P(A1). Conjecture. There is a family of biregular maps rpr = rpr (K, G, h) , K s; G(A1) defined over C, taking Shk- (G, h) to Shx,(G<·P, h<), and rendering the following diagrams commutative : Shu(T, h) 4 Shx(G, h) r rr r (a) Sh[r(T, h) 4 Shk(G, h)

�' 1

1�,

Shu,(T<, h<) 4 Shx,(G<·P,h<) Here U = T(A1) n K and the rpr in the left column is rpr(U, T, h). (b) The conjecture in this form refers to a specific T and a specific h factoring through T. If we choose another pair (f, h) we obtain another group G<· P. and an­ other collection {tl>r = rpr (K; G, h)} . The conjecture is inadequate as it stands, and must be supplemented by a statement relating rpr and tP r · Observe that there can be at most one family {rpr} satisfying the conditions (a) and (b). We have already associated to the two pairs (T, h) and (f, h) a cocycle { rp } which bounds in G<·P(L). Let rP = u ad c;1 (p(u- 1 )). Then g ---> ugu- l defines an isomorphism of Gr ·�'(Q) with G<· P.(Q) or of G< . P(A1) with G<· P.(A1). Set uK r = uK
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES

233

If this statement is true when u is replaced by a v in G(L.) = G(C) which also trivializes { rp} restricted to Gal(L./Q.) then it is true for u. An examination of the proof of the second property of {rp} shows that we can take v to lie in x- 1 wT(L.), x and w being as in that proof. Since x- 1 w(..-- 1 ,u) = ..-- 1 p, we have ad v o h• = lz•. We infer that ad u carries �· to .f)• and hence defines a bijection G• . .u(Q)\�· X G•·I'(At)f K• ----> G•· P(Q)\,P• X G•· P(Af)/"K• and an isomorphism ¢ : ShK.( G• · .u, h•) ----> Sh.uK< (G•· P, h•).

Since u and B(..-) both trivialize { rp} in G• . .u (AJ
----+ ShKr( G•· P' h•) rp

are commutative. The conjecture as it stands certainly implies that the conjugate of a Shimura variety is again a Shimura variety. Together with its supplement, it implies the usual form of Shimura's conjecture [10]. To verify this one applies the Weil criterion [49] for descent of the field of definition. For this we need families of isomorphisms /p : ShHG, h) ----> ShK(G, h) defined for automorphisms p of C over E(G, h) and satisfying f"P = /p/C. Choose a Cartan subgroup T and an h which factors through it. We know that when ..- fixes E(G, h) the cocycle {c;1(..-, ,u)} is defined and bounds in G(L). Let c;1(..-, ,u) = vp(v- 1 ). Then g � vgv- 1 is an isomorphism of G(Q) with G• . .u(Q). Methods which we have already used show easily that The composite ad v o h is conjugate under G•·.u(R) to h•. Consequently ad v defines an isomorphism � � �· and then, as before, we obtain from an isomorphism On the other hand mutativity of

zv

sK(G, h) ____. s.xeo• . .u, h•). = b(..-, ,u)- 1 with z E G• . .u(A1). We define f. by the com­

234

R. P. LANGLANDS

I omit the calculations, lengthy but routine, by which it is deduced from the con­ jecture and its supplement that.f.. does not depend on the choice of T and h and that the cocycle conditionf"P = /p iC is satisfied. Up to now the SK have been taken as varieties over C, but by the criterion for descent we may now define them over E(G, h) in such a way that the .f.. are simply the identity maps. It has to be verified that the models thus obtained are canonical, but the construction is clearly such that only the case that G is a torus T need be considered. Let a be the transfer of w = w, to CE and a' a lifting of a to IE. The proof that in this case cp(r, p.) is trivial shows in fact that we may take it to be 1 and b = b('r, f-l) to be n Gal {L/Q) /Gal {L/E) l.l(a')"l'. We take V to be 1 and Z to be b- 1 = b('r, p.)- 1 . The composition .- 1

Shu(T, h) ----> Shu(T, h)

ff (b)

---->

Shu(T, h)

is then the identity for all 1: fixing E(T, h), and this is just the condition that Sh{T, h) be the canonical model. Suppose E(G, h) � R. If we take the canonical model for ShK then the complex conjugation defines an involution 8 of the complex manifold ShK(C). It is necessary to have a concrete description of this involution in terms of the representation ShK(C)

= G(Q)\.P

X

G(At)f K,

and one purpose of the conjecture is to provide it. Choose some special point (T, h). If E(G, h) � R then we may define b(c, f-1.) and {cp(c, p.)} . However the condition (c) on the coset representatives w" used to define b(c, p.) implies that b(c, p.) = 1 . We may also take cp(c, p.) to be 1 , and then v may be taken to be 1 as well. It follows that h and h' are conjugate in G(R). Since Kh, = Kh, r; : ad g o h' --> ad g o h is a well-defined map of .p to itself. Consequence of the conjecture. The involution 8 may be realized concretely as the mapping (h, g) --> (r;(h), g) of G(Q)\.P x G(A1) /K to itself. Since we are comparing two continuous mappings which commute with the lJ(g), g E G(A1), it is enough to see that they coincide on the point in Su(T', h') represented by (h', 1). Since /, is the identity and v = z = 1 , q;, takes this point to the point in ShK,(G'• I', h') = ShK(G, h) represented again by (h', 1). It follows immediately from condition (a) of the conjecture that c applied to the point re­ presented by (h', 1) is (h, 1). For each T and f-1. let l.l(f-1.) be the fibre functor from f?lUff!JJ(G) to �CfJ'(G'·�') which takes (�', V(�')) to (�, V(�)) , with � = �', and with V(�) being the space obtained from V(�') by changing the Galois action to (J : X --> e(ca(7:, p.)- 1) fl(X). lf (t, h) is another special point, and if l.l (p., p) is defined by the diagram

AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES

235

then, according to the first lemma of comparison, the two fibre functors WRep(G<·PJ and WRep (G•·Pl o v{,u, fi) are isomorphic. On the other hand, we associated, at the end of the fourth section, to each pair (q/, g) a group G�· "' and a pair (�', g'). The groups G and G�· "' are associated to the same tannakian category, and there is thus an equivalence of categories v(�) : BU&J(G) --+ BU&J(G�·'�'), determined up to isomorphism. If � factors through the Serre group, then it can be factored through a Cartan subgroup T of G and defines a coweight p of T. The hypothetical properties of the Taniyama group imply that a�, rp may be taken to be a�. ;; with v(�) being v(p). Thus we have a diagram fl,ftff&J (G�·P) • < 11. rpl

� / fJfC&J(G) / " (rp)

fJftff&J(G�. "') which is commutative up to isomorphism of functors. Moreover the two fibre functors WRep (Gr·P) and WRep (G•·�J o v(,u, �) are isomorphic. If we choose an isomor­ phism between them, We obtain [40, 11.3.3] an isomorphism over Q, G� .rp --+ G�•P, Composing with �· we obtain a homomorphism �" : GMot (C) ---+ a� . P. Since there are so many special points, it is not unreasonable to hope that v(,u, �) exists for all �' and that WRep(G•·P) and wRep (G•·�J o v(,u, �) are always isomorphic. The second lemma of comparison in conjunction with the hypothetical properties of the Taniyama group implies that the composition of �" with the canonical homomorphism fJf --+ GMot(c) lies in s;,� when � factors through the Serre group, and once again we may surmise or hope that this will be so in general. If � � is the biregular map appearing in the conjecture then the composition � � o -.-1 defines a map from the set of complex points on ShK(G, h) to the set of com­ plex points on ShK.(G� . ,.. , h�). The idea is that �� o -.-1 will take (�, g) to a pair (�", g"), by a process which can be defined within the moduli problem. We have just seen how to obtain �", at least at the hypothetical level at which we are working. To obtain g" we observe that we have two homomorphisms (6.5) One is obtained from the chosen isomorphism over Q by extending scalars to A1. The other is obtained by a lengthy composition. The g from which we start pro­ vides an isomorphismwtlct
x --+ f (b(-r, ,u)-1)x

of

V(e')

with V(.;). Composing

236

R . P. LANGLANDS

(6.6) and (6.7) we obtain a second isomorphism between the two fibre functors figuring in (6.5). According to general principles, it can be obtained by composing the first with an element (g")- 1 in G� ·�-'(A1) [40, §II]. If one can establish, in some way or another, that the map ¢� : (q>, g) -+ (q>", g") is really defined, then to prove the conjecture and its supplement one will only need to verify that the composite ¢ � o 1: is complex analytic. However our purpose here has been to see how the wheels mesh, not to find the mainspring. 7. Continuous cohomology. If G = G0 then, according to the principles of the fourth section, we should be able to attach to each point of ShK(C) an equivalence class of pairs (q>, g). Here q> is a homomorphism from GMot(c) to G defined over Q and if r; is the associated 0-functor fJU&J(G) -+ .A(!} !7( C), then g defines an isomorphism (J)AJ M

ot (C) 0

'Yl -+ "I

(J)Af

Rep (G) •

In general we have mappings ShK(G, h) -+ ShK 0(G0, h0), and by pulling back we can associate to each point of ShK(C) a pair (q>, g) where g is again in G(A1), but q> now takes GMot (C) to Go. If (.;, V(.;)) is a representation of G0 over Q or, what is the same, a representation of G factoring through G0, then to each point s of ShK(C) we may associate the motive M(s, .;) defined by .; o ¢, together with the isomorphism wt!c, t (c) (M(s, �))

�

V(.;)A1

defined by g. The variety ShK(G, h) should be defined over E = E(G, h) and so should this family of motives. Suppose now that the variety ShK(G, h) is proper. In the best of all possible worlds, one might be able to form the cohomology groups M', 0 � i � 2 dim ShK, of the family M( · , .; ) , which would again be motives, now over E, and thus correspond to representations ai of GMo t (E) · If the formalism of the second section were established, one could compose ai with PF to obtain represen­ tations pi = ai ® PF of Gn<El · Then the basic problem would simply be to describe the image pi(Gn(E)) . But we do not have all this formalism, and one of the principal reasons for studying Shimura varieties is the hope that by grappling with the specific arithmetic problems they pose we will obtain an insight that will help us with its construction. Informed by the general principles and hypotheses we are attempting to establish, we can try to formulate questions that are, at least in part, tractable and which if answered will confirm or, if the answer is other than expected, perhaps refute these principles. In the present context we can first observe that even if the Mi remained unde­ fined, the zeta-function Z(z, Mi) can be defined directly in terms of the data at our disposal. It is a product over the places of E, II vZvCz, Mi) . At a nonarchimedean place it can be defined by the /-adic representation of the Galois group on the ith cohomology group of the /-adic sheaf F�(Q1) associated to .;, as in the papers [30] and [34]. Since our principal concern now is with the factors for the archimedean places, we need not enter into details. The field E is contained in C, and the archimedean places are obtained by ap­ plying automorphisms of C, or of Q, to E. We first define the factor Z0(z, Mi) for the place v given by E � C. We have seen that we can associate to .; a locally con-

AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES

237

stant sheaf Fe(Q) over ShK(C). Moreover, we have an analytic family of polarized Hodge structures on Fe(C) = Fe(Q) ® C. By a construction of Deligne [12], [50] this defines a Hodge structure on the cohomology groups Hi = Hi (ShK (C) , Fe(C)). In accordance with the ideas of Serre [42] the factor Zv(z, M•) will be defined as L(s, pi) where pi is a representation of the Weil group Wc!Ev on Hi . The Hodge structure on Hi defines a representation of �(R). Since ex = �(R), this can be used to define pi on ex � WctEv If Ev is equal to C, this defines pi completely. If E. = R then to define pi completely we also have to define pi(w), if w is the element of the Weil group which projects to c and has square - 1 . Since ShK is defined over E, c also defines an involution on ShK(C} which we denoted by (). What we need is a map of order two ¢ : (J *F ---+ F,

F = Fe(C),

such that on each fibre .q (J *FsP, q = pPO(s)

---+

psq, p

•

The associated map c* on cohomology takes H P.q to H q , p and we set pi(w) equal to ( - l )Pc*.

To define ¢ we have to assume that the consequence of the conjecture which was described in the previous section is valid. It can be proved directly in several cases. Then () can be obtained by taking the map (h, g) -+ (r;(h), g) and passing to the quotient. Given (h, g), the fibres at the image of (h, g) and (r;(h), g) may both be identified with V(�)c and ¢ is simply the identity map. If we replace the imbedding E � C by z--1 : E -+ C, -r being an automorphism, then the complex manifold ShK(C} is replaced by Shk(C), which we may identitfy by means of 'P r with ShK,(C), the manifold associated to ShK,(Gr ,p, hr). Thus the factor of the zeta-function defined by the place associated to z--1 : E = E(G, h) -+ C can be calculated by replacing G by G� ·�', h by h�, and E by z--1(£) = E(G� ·�', h) with the place defined by its inclusion in C. The space V(�) has also to be twisted by the cocycle {�(c11(-r, ,u)-1)}. The function Z(z, Mi) defined, the immediate problem is to show that it can be expressed as a product of L-functions associated to automorphic representations (7 . 1 )

Z(z, Mi)

= ij L(z - ai, J

ni, ri) .

Here ai e C is a translation, ni is an automorphic representation of some group Hi, and ri is a representation of the L-group LHi. The first step is to decide which Hi, which ni, and which ri intervene in the product. The first step is to use the theory of continuous cohomology to compute the cohomology groups of the sheaves Fe(C) together with their Hodge structure, and thus to compute Zv(z, Mi), v being again defined by E � C. Using this together with an analysis of the L-packets of automorphic representations of G [44], one searches for an identity (7. 1) which is at least valid when both sides are replaced by their factors at v. An example is discussed in detail in [34]. The identity found, it must be verified for the local factors at the other places v' . If v' is an archimedean place, then the theory of continuous cohomology will allow us to compute Zv,(z,Mi) in terms of the automorphic representations of a G� ·P, a group which

238

R. P. LANGLANDS

differs from G by an inner twisting. To make the comparison it will be necessary to have established the principle of functoriality for the pair G ano G � . ,., and to have understood in detail how it manifests itself. This is bound up with the study of L­ packets and is primarily an analytic problem, for we expect that the trace formula will give us a good purchase on it [23]. At the finite places the identity (7. 1) is difficult to treat as it stands, and for reasons familiar from topology one replaces the left side by (7.2)

Z(z)

=

IT Z(z, Mi) C-I l ' i

modifying the right side accordingly. The right side is then analyzed by the trace formula, at least if there is no ramification or, at worst, a mild sort [8], [14], [30] . I do not see at the moment any general way of dealing with a truly nonabelian situation, although a rather curious method has been discovered by Deligne for treating the group GL(2) rt3]. If there is no ramification, the factor of (7.2) at a finite place can be analyzed by the fixed point formulae of /-adic cohomology. Apart from combinatorial difficul­ ties [28] the critical factor is to have a reasonably explicit description of the set of geometric points on ShK(G, h) in itp, the algebraic closure of the residue field at a prime p of E, together with the action of the Frobenius on it [32]. This is an idea first applied by Ihara [20], who has since intensively studied the structure of this set for Shimura curves [21], [22]. Not much has been done when there is ramification. The first thing is to analyze in reasonably simple cases the manner in which the variety reduces badly. Some interesting discoveries have been made for curves [8], [14], [15], but higher dimen­ sional varieties behave in a more complicated manner. However tools are available for studying their reduction, and it is time to begin. None of these steps will be easy to carry out. The study of L-packets is in an em­ bryonic stage, and even the combinatorial problems will demand considerable ingenuity in their solution [27]. There is still a great deal to be learned from the study of specific examples. The theory of continuous cohomology is itself in its infancy, and my purpose in this section is to draw attention to some problems which arise in the study of Shimura varieties and which the mature theory should resolve. I begin by introducing a representation r of the L-group LG which will play a fundamental role in the discussion. The group LG is a semidirect product LGo x Gal(Q/Q). If T is a Cartan subgroup of G over Q then there is an isomorphism of XiT), the lattice of coweights of T, with X * (Lro), the lattice of weights of Lro, defined up to an element of the Weyl group. In particular if {T, h) is a special point then fl. defines an orbit e in X*(LT0), and e is independent of{T, h) . Let r 0 be the representation of LGo whose set of extreme weights is e. The group Gal(Q/Q) acts on the weights of LTo and preserves the set of dominant weights. The group Gal(Q/E) fixes the set e. Theus Gal(Q/E) fixes the dominant element p.v in e, and we may extend r 0 to LGo x Gal(Q/E) in such a way that Gal{Q/E) acts trivially on the weight space of p.v . The extended representation will also be called r 0 , and we define r to be the induced representation

AUTOMORPHIC REPRESENTATIONS , SHIMURA VARIETIES , AND MOTIVES

r = Ind(LG, LG o

x

239

Gal(Q/E), r 0) .

Led d be the dimension of ShK(G, h). One expects that the function (7.2) will be equal to a product of functions (7.3) L(z - d/2, n, p). Here n is an automorphic representation of one of the groups H attached to G in [33]. There is, in general, an imbedding (/) : LH c... LG and p is a subrepresentation of r o f/J· If w is any place of Q, let rw be the restriction of r to the L-group LGw, which equals LGo x Gal(QwfQw). Implicit in this notation is an imbedding Q !;;;;; Qw. Then the double cosets in Gal(Q/E)\Gal(Q/Q)/Gal(Qw/Qw) parametrize the places of E dividing w, and rw = EBvlw r0, with rv = Ind(LGw, LG o x Gal(Qw/E.), r�) and r�(-r) = r0(u.-ru;1) if O"v is some element in the coset defining v. The repre­ sentation p will also be a direct sum p = EBvlwPv • and the function (7.3) will be a product IT w iT vlw L(z - d/2, nw, p0). The factor corresponding to the place v is L(z - d/2, 1Cw , p0). Since we shall only be interested in the place v given by E !;;;;; C, we shall write r and p instead of r. and Pv · Moreover we shall write an automorphic representation of G(A) (or of H(A)) as n ® n1, n being a representation of G(R) and n1 of G(A1). Any irreducible representation n of G(R) lies in some L-packet Hrp where f/J is a homomorphism from WetR to LG. If a is the character of We1R obtained by com­ posing WetR -+ R with the absolute value, let c/JI(n) = a-d 1 2 ® (r o (/) ). Then L(s - d/2, n, r) = L(s, c/JI(n)) . On the other hand, suppose n ® n1 is an automorphic representation of G(A). Let it act on the subspace U ® U1 of the space of automorphic forms. If h e .f) then, according to the principles of continuous cohomology [4], its contribution to the cohomology of ShK(e) with values in F�(C) in dimension i is (7.4) HomKiAigft ® V, U) ® UK1 . Here t is the Lie algebra of Kh and V the dual of V(�). The space U� is the space of vectors in U1 fixed by K. The action of z e ex = �(R) defining the Hodge structure sends f/J ® u to f/J ' ® u with qJ ' (X ® �)

= ({J(r;(z)X ® � (h(z-l)) v) .

Here X -+ r;(z)X is the action which multiplies the exterior product of p holomor­ phic and q antiholomorphic vectors by z-P :z-q . Thus r;(z) = ad h(z- 1 ) o r;(z) and qJ ' (X® v) = n(h(z-l))({J(r;(z)X ® v) . If E !;;;;; R we may extend this action of ex to an action of WeiR = We/E.. Let n e G(R) be such that ad n o h = h'. Then the element of w which projects to t and has square - 1 sends f/J ® u to f/J ' ® u with qJ ' (X ® v)

= n(n)qJ (ad n-l(X) ® �(n- l)v) .

240

R. P. LANGLANDS

In either case the representation of Wc;E. on (7.4) factors as cji(n:) ® I , where cji(n:) acts on HomKh(Ai@jf ® V, U). Let ¢2(n:) be the element in the representa­ tion ring of Wcm defined by ¢ z(n;) = Efl( - l ) i

Ind( Wc/R • Wc!Ev• cf;•"(n:)) . Let m(n:1 , K) be the dimension o f Ulf. If for all n: we had ¢ z( n:) = m(n:)cf; 1(n:) (7.5) we could expect a relation K (7.6) Z(z) = n L(z - d/2, n:, r ) m (1Coo) m (1Cj, ) . "

Here, as a single exception, we have taken n: to be a representation of G(A), its com­ ponent at infinity being denoted n:"', and r to be the representation of LG. However (7.5) is not always valid, and it is the true form of the relation between ¢1(n:) and ¢2(n:) that we must discover, for it is the clue to the correct expression of(7.2) as a product of £-functions associated to automorphic representations. Let U(t;) = {n:h · · · , n:r} be the set of discrete series representations with the same central and infinitesimal characters as �- Then U(t;) = n is an £-packet U'P and the representations ¢1 (n:,) are all equal. We denote them by ¢1(0) . The continuous cohomology of the representations n:; is completely understood [4], and it is a simple exercise to prove the following lemma. LEMMA. EBi=1 ¢ z (n:j) = ( - I ) d¢1(0) .

Thus, in this case, the relation (7.5) fails when the £-packet has more than one element. In order to correct (7.6) one has to replace r by a subrepresentation. However r is in general irreducible as a representation of LG, and so we have to introduce the groups L H of [33], and begin the study of £-indistinguishability. If we accept £-indistinguishability, but expect no other difficulties with the cor­ rection of (7.6), then we have to be prepared to prove that every irreducible com­ ponent of ¢2(n:) is a component of ¢1(n:) . But we will again be deceived. There is another difficulty. It appears already in the simplest of the examples considered by Casselman [6] and Milne [36], although they had no occasion to draw attention to it. Suppose G is the group associated to a quaternion algebra over Q which is split at infinity but not at p . Let n:1 = n: P ® n;P, and suppose n: ® n:1 is trivial on the centre, n:P is one­ dimensional and trivial on the maximal compact subgroup Kp of G(Qp), and n: is either one-dimensional or the first element of the discrete series. r; is taken to be trivial. If K = KPKp then L(s - 1-, n: ® n:1 , r) should appear in the zeta-function Z(s, ShK) with the exponent ± m(n:P, KP) . Here m(n:P, KP) is the multiplicity with which the trivial representation of KP occurs in n:�, and the sign is positive if n: is one-dimensional and negative if it is the first element of the discrete series. As Casselman and Milne show in their lectures, this is so locally almost everywhere. One can probably show without great difficulty that the local statement is correct at p as well when n: belongs to the discrete series, for n: then contributes to the co­ homology in dimension one and n:P = n;((Jp) where (JP is a special representation of the thickened Weil group. In particular

AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIFS, AND MOTIVES

L(z - ! ,

n"p ,

r) =

241

I _I e/p• ' l e i = 1 .

However if n- is one-dimensional then n- contributes to the cohomology in dimen­ sions zero and two and the corresponding local contribution to the zeta-function should be

{ (1 - e/p•)(II - epfp•) }

m (1rP, KP)

·

The factor inside the brackets is not L(z - t. n-p , r). The difficulty is resolved if we realize that when n-, and hence n- ® 1CJ > is one-dimensional we should not be using L(z - f, n- ® n-1, r) at all but rather L(z - t, n-' ® n-j, r) where n-' ® n-j is the one-dimensional representation of G'(A) = GL(2, A) defined by the same character of the idele-class group as n- ® n-1. Since G'(Qv) - G(Qv) and n-� - n-v for almost all places v, the error of using n- ® n-1 instead of n-' ® n-{ is not detected when one only considers the local zeta­ function almost everywhere. The significance of the considerations of the second and third sections begins to appear. The representation n- ® n-1 and the representation n-" ® n-j of G'(A) associated to it by the principle of functoriality are anomalous, because n-; is one­ dimensional for almost all places v while n-; is infinite-dimensional. The isobaric representation equivalent to n-" ® n-j almost everywhere is n-' ® n-/. It was implicit in the discussion of the second section that anomalous representations would have nothing to do with motives, and so it should come as no surprise now that we must discard n- ® n-1 and replace it by n-' ® n- / . In this example n- itself was not changed for G(R) - G'(R) and n- - n-'. However in general we must expect that n- itself will have to be modified. Thus the proper factor will not be L(z - d/2, n- ® n-1, r) but L(z - d/2, n-' ® n-/, r), where n-' ® n-/ is an automorphic representation of a group G' obtained from G by an inner twisting. Again n-� will have to be equivalent to n-v almost everywhere. Since at the moment we are primarily interested in the infinite place. we simply ask whether it is possible to find a candidate for n-' or, rather, for an L-packet {n-'} = U'. There are apparently two conditions to be satisfied, the first arising from the compatibility of functional equations. (a) Let n- E U"' and let {n-'} = Hrp' · For any additive character ¢ of R and any re­ presentation q of LG = LG', is equal to

e ' (z, q o cp, ¢) = e(z, q o cp, ¢) L(l - z, q o cp) L(z, q o cp)

�

e ' (z, q o cp' , ¢) = e (z, q o cp', ¢) L( - z, q (z, (J cp

0,

0

j')

.

(b) It is possible to find a summand ¢0(n-') of ¢ 1 (n-') which is such that ¢2(n-)

a¢o (n-'), a E Z.

=

These conditions are only tentative, and may have to be modified in the course of time, but they will serve for the explanation of our problem.

242

R . P . LANGLANDS

The first condition involves only rp and rp' and we begin by constructing some pairs that satisfy it. Fix an element w of Wc18 that projects to c and satisfies w2 = - 1 . We may suppose that rp(w) = a x c with a in the normalizer of LTo in LG0• Then rp(w) also normalizes LT0• Let rp(c) denote the transformation of X * (LT0) or of X* (LT0) defined by rp(w). We may also suppose that rp takes ex to L TO and that rp(z) = zA z"'CeJ A with A e X* (LT0) ® C and A - rp(c)A e XiL ro). The representation rp' will be defined in a similar way. Thus rp'(w) = a' x c with a' in the normalizer of LT0, and rp'(z) = zA z'I'' Ce l A . Notice that A is to be the same for rp' as for rp. However a, which is given, is replaced by a', which we must now define. We suppose that rp{c) sends every root to its negative, and choose A in X* (LT0) such that AV(a) = e2ni<J., J.v> for any weight Av of LTo which is orthogonal to all roots. We shall take a' to lie in L To and to be such that AV(a') = e2ni<J.'J.v> when Av is orthogonal to all roots. Here ).' is still to be defined. If we also denote by a the operator on X* (LT0) ® C defined by a and if we let q be one-half the sum of the positive roots then ).' is to be given by the equation ).' = 1 ; a A (1 - a)( + rp'(c)) A + i .

�

_

Observe that the action of rp'{c) is the same as that of c . We are assuming that rp is a given, well-defined homomorphism, and hence [31] that

In order to show that rp' is also well defined we must verify that (7.7) We begin with the equations a rp'(c) = rp'(c)a = rp(c) and a {l + rp(c)) = 1 + rp(c), remarking also that the square of both rp(c) and rp'(c) is the identity. We infer that the left side of (7. 7) is equal to {1 + rp'(c)) ( l + rp(c)) A 2

_

( 1 - a) ( l + rp'(c)) A + 4

�

= q. The sum is in turn congruent to - rp(c) A _ (1 - a) {l + rp'(c)) A = 1 - rp'(c) A . 2 4 2

because {1 + rp'(c)) q/ 2 1

Consequently the homomorphsim rp' can be constructed whenever rp is defined and rp(c) sends every root to its negative. The following lemma is valid in this generality. LEMMA. For any representation u of the Wei/ form of LG and any nontrivial char­ acter ¢ ofR, e ' (r , u o rp , ¢) = e ' (-r, u o rp', ¢ ) .

243

AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES

The proof is a computation based on the proof of Lemma 3.2 of [31] and on Chapters 5 and 6 of [24], but it is rather lengthy, and not worth including here. Observe that we could have started with rp', defined by an a ' in LT o , and, reversing the process, passed to ). and a . More generally if LM and LM' are two parabolic subgroups of LG, and rp : We1R --+ LM has an image which lies in no proper par­ abolic subgroup of LM, then we can use the process to pass to a rp" whose image lies in the minimal parabolic ofLM, and thus of LG or LM', and afterwards reverse it to pass from rp" to a rp' : We1R --+ LM' whose image lies in no proper parabolic subgroup of LM'. My intention now is simply to show, by means of a few examples, how for a given 11: in some n"' one can choose one of the rp' just described so that the condition (b) is satisfied for the elements 11:' of n"'. . Of course the problem is to decide if such a choice is always possible. Without more examples or a general theorem, we cannot be at all confident that this is so. If all the continuous cohomology of ' ® 11: is zero there is no difficulty satisfying (b). We take rp' = rp and a = 0. The simplest nontrivial example is obtained by taking ' trivial and 71: trivial. Let 71: E n"'. Then rp(z) = zqztp C tl q , with q equal again to one-half the sum of the positive roots. If LM is the parabolic subgroup of LG correspondi ng to the minimal parabolic of G over R, then the image of rp lies in LM and rp(t) takes every root of L To in LM to its negative. Define rp ' as above, with a ' e LT0• G' can be taken to be the quasi-split form of G over R. The continuous cohomology of 11: is all in even dimensions and all of type (p, p) for some p. To compute it one observes that it is the same as the cohomology of the compactdual, which can be computed by using Schubert cells. One verifies without difficulty that for 11:' e n"', the representati on (/JI(11:'), which depends in reality only on rp ' , is equivalent to (/J2(7C). If G is not quasi-split over R then rp' is different from rp. If G is not quasi-split over R then it is certainly not quasi-split over Q, and the trivial representation of G(A) is anomalous. Once again we see that the passage from 11: to 11: ' is the local expression of the passage from an anomalous representation to one which is not anomalous. Other interesting examples are the representations 11: = J;, j of PSU(n, 1) dis­ cussed in Chapter XI of the notes of Borel-Wallach [4]. Take � trivial. In this case ¢z(7C) is ( - I )i+i times a representation induced from ex ' the representation of ex used having the weights z-i z-i, z-:-i-1 z -i-1 , . . . , z- Cn-j l z - Cn -i l . (7.8) Here 0 ;;;;; i + j ;;;;; n - I and 0 ;;;;; i, j. Borel and Wallach lapse into vagueness at one point, and it may be that the roles of i and j should here be reversed, but that is of little consequence. The group LGo is SL(n + 1 , C) and L T0 may be taken to be the group of diagonal matrices. The representation r o is the standard representation of SL(n + 1 , C). It is easy enough to deduce from [4] that if 11: e n"' then

with A being equal to (n/2

-

i, n/2, n/2

-

1,

· ··,

n/2

-

i + 1 , n/2

-

i

-

1,

244

R. P. LANGLANDS

- n/2 + j + I , - n/2 + j - I , · · · , - n/2, - n/2 + j). The numbers occurring here are n/2, n/2 - 1 , · · · , - n/2, but the order is somewhat unusual. The transfor­ mation g>(t) is given by

We are of course using the obvious representation of the elements of X* (LT0) ® C as sequences of n + I complex numbers whose sum is 0. Suppose, to be definite, that i � j. We will choose g>' to be such that the trans­ formation g>'(t) takes (xi> · · · , Xn+I ) to ( - Xn+ l > - Xm · · · , - Xn-i+ l > - Xn-i • · · · , - xi+Z• - Xn-i+ I > · · · , - Xn-i • - xi + I > • · · , - x 1 ). The indices within the gaps decrease or increase regularly by one. If rc' E n'P' then the representation (h(rc') is induced from a representation of ex with weights

(7.9)

z -i- I z-i-1 , . . . , z - ; z- (n-j+ l ) =- (j-1 ) '

• • .

z-
.•

- (n-i) z-i ; 'z

· , z-nz-n , z- (n-j) =- (n-i l .

Happily the set (7.8) i s a subset of (7.9) and the condition (b) i s satisfied. It should be observed that the representation ¢0(rc') that is chosen to satisfy (b) will have to be, except for some degenerate values of i and j, a proper subrepresen­ tation of ¢ 1 (rc'). This phenomenon will, I hope, be taken into account by L-in­ distinguishability. For example if e is the element of LTo with diagonal entries _______. ..___

j

�

I, - 1, - 1,· · ·, - 1, I, · · ·, I, - 1, · · ·, - 1, I

then e commutes with g>'( Wc1R) and ¢0(rc') may be taken to be the restriction of ¢ 1 (rc') to the + I eigenspace of r(e). REFERENCES 1. J. Arthur, Eisenstein series and the trace formula, these PROCEEDINGS, part I , pp. 253-274. 2. W. Baily and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (I 966), 442-528. 3. A. Borel, Automorphic £-functions, these PROCEEDINGS, part 2, pp. 27-6 I . 4 . A . Borel and N . Wallach, Seminar o n the cohomology of discrete subgroups of semi-simple groups, Institute for Advanced Study (1 976/77). 5. P. Cartier, Representations ofp-adic groups : A survey, these PROCEEDINGS, part I , pp. 1 1 I-I 55. 6. W. Casselman, The Hasse- Wei/ �-function of some moduli varieties of dimension greater than one, these PROCEEDINGS, part 2, pp. I4I-I63. 7. --, GL(n), Algebraic Number Fields (£-Functions and Galois Properties), A. Frohlich , ed., Academic Press, New York, I 977. 8. I. V. Cherednik, Algebraic curves uniformized by discrete arithmetical subgroups, Uspehi Mat. Nauk 30 (1 975), 1 83-1 84. (Russian)

AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES

245

P. Deligne, Travaux de Griffiths, Sem. Bourbaki (1 970). Travaux de Shimura, Sem. Bourbaki (1971). 1 1 . -- , La conjecture de Wei/ pour les surfaces K3, Invent. Math. IS (1 972), 206-226. 12. -- , Theorie de Hodge. III, Inst. Hautes Etudes Sci. Pub!. Math. 44 (1 974), 5-78. 13. -- , Letter to Piatetski-Shapiro. 14. P. Deligne and M. Rapoport, Les schemas de modules de courbes elliptiques, Modular Functions of One Variable. II, Lecture Notes in Math., vol. 349, Springer, New York, 1 972, pp. 1 43-3 1 6. 15. V. G. Drinfeld, Coverings of p-adic symmetric spaces, Functional Anal. Appl. 10 (1 976), 29-40. (Russian) 16. J. Giraud, Cohomologie non-abe/ienne, Springer-Verlag, Berlin and New York, 1 97 1 . 17. Harish-Chandra, Representations of semi-simple Lie groups. VI, Amer. Math. J . 7 8 (1 956), 564-628. 18. --, Discrete series for semi-simple Lie groups. I, Acta Math. 113 (1 965), 241-3 1 8 ; II, 1 1 6 (1 966), 1-1 1 1 . 19. R . Howe and I . Piatetski-Shapiro, A counterexample to the "generalized Ramanujan coniec­ ture " for (quasi-) split groups, these PROCEEDINGS, part 1 , pp. 3 1 5-322. 20. Y. Ihara, Heeke polynomials as congruence (,-functions in elliptic modular case, Ann. of Math. (2) 85 (1 967), 267-295. 21. --- , On congruence monodromy problems, vols. 1, 2, Lecture notes from Tokyo Univ. , 1 968, 1 969. 22. --- , Some fundamental groups in the arithmetic of algebraic curves over finite fields, Proc. Nat. Acad. Sci . 72 (1 975). 23. S. Gelbart and H. Jacquet, Forms of GL(2) from the analytic point of view, these PROCEED­ INGS, part 1 , pp. 21 3-25 1 . 24. H . Jacquet and R . P . Langlands, Automorphic forms on GL(2), Lecture Notes in Math., vol. 1 1 4, Springer, New York, 1 970. 25. H. Jacquet and J. Shalika, A non-vanishing theorem for zeta-functions of GL(n), Invent. Math. 38 (1 976). 26. D. Kazhdan, On arithmetic varieties, Lie Groups and Their Representations, Halsted, New York, 1 975. 27. R. Kottwitz, Thesis, Harvard Univ., 1 977. 28. -- , Combinatorics and Shimura varieties mod p, these PROCEEDINGS, part 2, pp. 1 85-192. 29. N. Kurokawa, Examples of eigenvalues of Heeke operators in Siegel modular forms, 1 977 (preprint). 30. R. P. Langlands, Modular forms and 1-adic representations, Modular Forms of One Variable. II, Lecture Notes in Math., vol . 349, Springer, New York, 1 973, pp. 361-500. 31. -- , On the classification of irreducible representations of real algebraic groups, Institute for Advanced Study, Princeton, N. J., 1 973. 32. ---, Some contemporary problems with origins in the Jugendtraum, Proc. Sympos. Pure Math., vol. 28, Amer. Math. Soc., Providence, R. I., 1 976, pp. 401-4 1 8 . 33. --- , Stable conjugacy : definitions and lemmas, Canad. J. Math. (to appear). 34. -- , On the zeta-functions of some simple Shimura varieties, Canad. J. Math. (to appear). 35. --- , On the notion of automorphic representation, these PROCEEDINGS, part 1, pp. 203-207. 36. J. Milne, Points on Shimura varieties m od p , these PROCEEDINGS, part 2, pp. 1 65-1 84. 37. D. Mumford, Families of abelian varieties, Proc. Sympos. Pure Math., vol. 9., Amer. Math. Soc., Providence, R. I . , 1 966, pp. 347-35 1 . 38. I . Piatetski-Shapiro, Multiplicity one theorems, these PROCEEDINGS, part 1 , pp. 209-212. 39. H. Resnikoff and R. L. Saldana, Some properties of Fourier coefficients of Eisenstein series of degree two, Crelle 265 (1974), 90-109. 40. N. Saavedra Rivano, Categories tannakiennes, Lecture Notes in Math., vol. 265, Springer, New York, 1 972. 41. J.-P. Serre, Abelian 1-adic representations and elliptic curves, Benj amin, New York, 1 968. 42. --- , Facteurs locaux des fonctions zeta des varietes algebriques (definitions et conjectures), Sem. Delange-Pisot-Poitou, 1 970. 43. -- , Representations 1-adiques, Algebraic Number Theory, Tokyo, 1 977. 9.

10. -- ,

246

R.

P. LANGLANDS

44. D. Shelstad, Notes on £-indistinguishability, these PROCEEDINGS, part 2, pp. 1 93-203 . 45. , Orbital integrals and a family of groups attached to a real reductive group, 1 977 (preprint). 46. K.-Y. Shih, Conjugations of arithmetic automorphic function fields, 1 977 (preprint) . 47. J . Tate, Number theoretic background, these PROCEEDINGS, part 2, pp. 3-26. --

48. N. Wallach, Representations of reductive Lie groups, these PROCEEDINGS, part 1 , pp. 71-86. J. Math. 78 (1 956), 509-524. 50. S. Zucker, Hodge theory with degenerating coefficients. I, 1 977 (preprint).

49. A. Wei!, The field of definition of a variety, Amer. THE INSTITUTE FOR ADVANCED STUDY

Proceedings of Symposia in Pure Mathematics Vol. 33 ( 1 979), part 2, pp. 247-290 ,

,

,

VARIETES DE SHIMURA : INTERPRETATION MODULAIRE, ET TECHNIQUES DE CONSTRUCTION DE MODELES CANONIQUES

PIERRE DELIGNE SOMMAIRE Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Rappels, terminologie et notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 1. Domaines hermitiens symetriques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 1 . 1 Espaces de modules de structures de Hodge . . . . . . . . . . . . . . . . . . . . . . . . 25 1 1 .2 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 1 . 3 Plongements symplectiques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 2. Varietes de Shimura . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 2.0 Preliminaires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 2.1 Varietes de Shimura . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 2.2 Modeles canoniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 2. 3 Construction de modeles canoniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 2.4 Loi de reciprocite : preliminaires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 2. 5 Application : une extension canonique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 2.6 La loi de reciprocite des modeles canoniques . . . . . . . . . . . . . . . . . . . . . . 284 2.7 Reduction au groupe derive, et theoreme d'existence . . . . . . . . . . . . . . . . 285 Bibliographie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Introduction. Cet article fait suite a [5], dont nous utiliserons les resultats essen­ tiels (ceux des paragraphes 4 et 5). Dans une premiere partie, nous tentons de motiver les axiomes imposes aux systemes (G, X) (2. 1 . 1) a partir desquels sont definies les varietes de Shimura. On montre que, grosso modo, ils correspondent aux espaces de modules de structures de Hodge x+ du type suivant. (a) x+ est une composante connexe de l'espace de toutes les structures de Hodge sur un espace vectoriel fixe V relativement auxquelles certains tenseurs t 1 . . tn sont de type (0, 0). Le groupe algebrique G est le sous-groupe de GL(V) qui fixe les t; et X est l'orbite G(R) · x+ de x+ sous G(R). ·

4 WS (MOS) subject classifications (1 970). Primary 10D20, 14099 ; Secondary 200 1 5 , 20030. © 1 979, American Mathematical Society

247

248

PIERRE DELIGNE

(b) La famille de structure de Hodge sur V parametree par x+ verifie certaines conditions, verifiees par les families de structures de Hodge qui apparaissent na­ turellement en geometrie algebrique : pour une structure complexe convenable (et uniquement determinee) sur x+, c'est une variation de structure de Hodge polarisable. L'espace x+ est automatiquement un domaine hermitien symetrique ( = espace hermitien symetrique a courbure < 0). Les domaines hermitiens symetriques peuvent tous etre ainsi decrits comme des espaces de modules de structures de Hodge ( 1 . 1 . 1 7 ) , et je crois cette description tres utile. Par exemple : le plongement d'un domaine hermitien symetrique D dans son,dual i5 (une variete de drapeaux) correspond a !'application (une structure de Hodge) >-+ (la filtration de Hodge correspondante). Les descriptions comme "domaine de Siegel de 3e espece" s'inter­ pretent en disant que, sous certaines hypotheses, si on superimpose a une structure de Hodge une filtration par le poids, on obtient une structure de Hodge mixte, d'ou une application de D dans un espace de modules de structures de Hodge mixtes (cf. les constructions de [1, III, 4. 1]). Ce dernier point ne sera ni mentionne, ni utilise dans l'article. Ce point de vue, et la description de certaines varietes de Shimura comme es­ paces de modules de varietes abeliennes, sont lies par le dictionnaire : il r evient au meme (equivalence de categorie A >-+ H1(A, Z) ) de se donner une variete abe­ lienne ou une structure de Hodge polarisable de type {( - 1 , 0), (0, - 1 )} (il s'agit ici de Z-structures de Hodge sans torsion ; par passage au dual (A >-+ Hl(A, Z)) , on peut remplacer {( - 1 , 0), (0, - 1 ) } par {(1 , 0), (0, 1) } ) . Polariser la variete abelienne revient a polariser son H1 . Avec parametres, de meme, il revient au meme de se donner un schema abelien polarise sur une variete complexe lisse S, ou une varia­ tion de structures de Hodge polarisee de type {( - 1 , 0), (0, - 1)} sur l'espace analy­ tique s•n. Une famille analytique de varietes abeliennes, parametree par s•n, est automatiquement algebrique (ceci resulte de [3]). Pour interpreter des structures de Hodge de type plus complique, on aimerait remplacer les varietes abeliennes par des "motifs" convenables, mais il ne s'agit encore que d'un reve. Au numero 1 .2, nous donnons une description commode basee sur le formalisme precedent de la classification des domaines hermitiens symetriques, en terme de diagrammes de Dynkin et de leurs sommets speciaux. Au numero 1 .3, nous clas­ sifions un certain type de plongements de domaines hermitiens symetriques dans le demi-espace de Siegel. Les resultats sont paralleles a ceux de Satake [11]. Une application de l'astuce unitaire de Weyl, pour laquelle nous renvoyons a [7], ra­ mene la classification a la connaissance d'un fragment de la table, donnee par exemple dans Bourbaki [4], donnant !'expression des poids fondamentaux comme combinaison lineaire de racines simples. Le lecteur ctesireux d'en savoir plus sur les variations de structure de Hodge, et la far;on dont elles apparaissent en geometrie algebrique, pourra consulter [6] ( dont no us ne suivons pas les conventions de signes) ; certains faits, enonces dans [6], sont demontres dans [7]. Aux numeros 2. 1 et 2.2 nous definissons, dans un langage adelique, les varietes de Shimura KMc(G, X) (notees KMc(G, h) dans [5], pour h un quelconque element de X), leur limite projective Mc(G, X) et la notion de modele canonique. Je renvoie au texte pour ces definitions, et dirai seulement qu'un modele canonique de

VARIETES DE SHIMURA

249

M0(G, X) est un modele de M0(G, X) sur le corps dual (2.2. 1) E(G, X), i.e. un schema M(G, X) sur E(G, X) muni d'un isomorphisme M(G, X) ® E < G .Xl C .:::... Mc(G, X), cette fayon de definir Me(G, X) sur E(G, X) ayant des proprietes convenables ( G(Af)­ equivariance, et proprietes galoisiennes des points speciaux (2.2.4)). On definit

aussi la notion de modele faiblement canonique (meme definition que les modeles canoniques, avec E(G, X) remplace par une extension finie E c C). Ils jouent un role technique dans la construction de modeles canoniques. Les differences ap­ parentes entre les definitions de 2. 1 , 2 . 2 et celles de [5] proviennent d'un autre choix de conventions de signes (action a droite contre action a gauche, loi de reciprocite en theorie du corps de classes global. . .). Pour une description heuristique, je renvoie a }'introduction de [5]. Pour une breve description, sur des exemples, de comment on passe du langage adelique a un langage plus classique, je renvoie a [5, 1 .6-1 . 1 1 , 3 . 1 4--3 . 1 6, 4. 1 1-4. 16]. Dans [5], nous avions systematise les methodes introduites par Shimura pour construire des modeles canoniques. Dans Ia seconde partie du present article, nous perfectionnons les resultats de [5]. Au numero 2.6, nous determinons l'action du groupe de Galois Gal(Q/E) sur }'ensemble des composantes connexes geometri­ ques d'un modele faiblement canonique (suppose exister) de M0(G, X) sur E, sans supposer, comme dans [5], que le groupe derive de G est simplement connexe. Le point essentiel est la construction, donnee au numero 2.4, d'un morphisme du type suivant. Soient G un groupe reductif (connexe) sur Q, p : G --+ G le revetement universe} de son groupe derive Gd•r et M une classe de conjugaison, definie sur un corps de nombres E, de morphismes de Gm dans G. On construit un morphisme qM du groupe des classes d'ideles de E dans le quotient abelien G(A) /pG(A) · G(Q) de G(A). Ce morphisme est fonctoriel en {G, M), et, si F est une extension de E, le diagramme C(F)

l�

G(A) /pG(A) · G(Q)

C(E)

est commutatif. Si G n'a pas de facteur G' sur Q tel que G'(R) soit compact, on dectuit du theoreme d'approximation forte que

n0 ( G(A) /pG(A)G(Q)) = n0 ( G(A) / G(Q)) , et qM fournit une action sur n0 (G(A)/ G(Q)) de n0 C(E), le groupe de Galois rendu

abelien Gal(Q/E)•h d'apres la theorie du corps de classes global. La seconde idee nouvelle-en fait un retour au point de vue de Shimura-est !'observation suivante : les resultats de 2.6 permettent de reconstruire un modele faiblement canonique ME(G, X) de M0(G, X) a partir de sa composante neutre M8(G, X+) (une composante connexe geometrique, dependant du choix d'une com­ posante connexe x+ de X), munie de l'action (semi-lineaire) du sous-groupe H de G(A!) x Gal(Q/E) qui Ia stabilise. Soient Z le centre de G, G•d le groupe- adjoint, G•d(R)+ Ia composante neutre topologique de G•d(R), et G•d(Q)+ = G•d(Q) n G•d(R)+. L'adherence Z(Q)- de Z(Q) dans G(Af) agissant trivialement sur M0(G, X), l'ac-

250

PIERRE DELIGNE

tion de H sur M8(G, X+) se factorise par H/Z(Q)-. On peut en fait faire agir un groupe un peu plus gros, extension de Gal(Q/E) par le complete de G•d(Q)+ pour Ia topologie des images des sous-groupes de congruence de Gd•r(Q) . A isomorphisme unique pres, cette extension ne depend que de G•d , Gd er et de Ia projection x+•d de x+ dans G•d (2. 5). On Ia note tfE(G•d , Gder , x+•d) . La composante neutre M8(G, X+) est Ia limite projective des quotients de x+•d par les sous-groupes arithmetiques de G•d (Q)+, images de sous-groupes de congruence de Gd•r(Q). On verifie enfin que les conditions que doivent verifier son modele M8(G, X+) sur Q, et I' action de tfE(G"d , Gd•r, x+•d), pour correspondre a un modele faiblement canoni­ que, ne dependent que du groupe adjoint G•d , de x+•d , du revetement Gd•r de G•d et de I' extension finie E (contenue dans C) de E(G•d , x+•d). Ces conditions definis­ sent les modeles faiblement canoniques (resp. canoniques, pour E = E(G•d , x+•d)) connexes (2.7. 10). Le probleme de !'existence d'un modele canonique ne depend done, en gros, que du groupe derive. Cette reduction au groupe derive est une version beaucoup plus commode de Ia maladroite methode de modification centrale de h dans [5, 5. 1 1 ] .

En 2 . 3 , nous construisons une provision de modeles canoniques a I' aide de plon­ gements symplectiques, en invoquant [5, 4.21 et 5.7]. Les resultats de 1 . 3 nous per­ mettent d'obtenir les plongements symplectiques desires avec tres peu de calculs. En 2.7, nous expliquons Ia reduction au groupe derive esquissee ci-dessus et nous deduisons de 2.3 un critere d'existence de modeles canoniques qui couvre tous les cas connus {Shimura, Miyake et Shih). Dans l'article, nous utilisons }'equivalence entre modeles faiblement canoniques et modeles faiblement canoniques connexes pour transporter a ces derniers des resultats de [5) (unicite, construction d'un modele canonique a partir d'une famille de modeles faiblement canoniques). II eut ete plus naturel de transposer les demon­ strations, et de transposer de meme Ia fonctorialite [5, 5.4], et le passage a un sous­ groupe [5, 5.7] (evitant par la la sybilline proposition [5, 1 . 1 5]). Le manque de temps, et la lassitude, m'en ont empech6. J'ai recemment demontre qu'on pouvait donner un sens purement algebrique a Ia notion de cycle rationnel de type (p, p), sur une variete abelienne A (sur un corps de caracteristique 0). On peut retrouver a partir de Ia le critere d'existence 2. 3 . 1 d e modeles canoniques, e t donner une description modulaire des modeles obtenus avec son aide (cf. [9]). Cette description ne se prete malheureusement pas a la reduc­ tion modulo p. Cette methode evite le recours a [5, 5.7], (et par la a [5, 1 . 1 5]) et fournit des renseignements partiels sur Ia conjugaison des varietes de Shimura. 0. Rappels, terminologie et notations. 0. 1 . Nous aurons a faire usage du theoreme d'approximation forte, du theoreme d'approximation reelle, du principe de Hasse et de Ia nullite de H 1 (K, G) pour G

semi-simple simplement connexe sur un corps local non archimedien. Des indica­ tions bibliographiques sur ces theoremes sont donnees dans [5, (0. 1 ) a (0.4)]. Signa­ Ions en outre I' article de G. Prasad (Strong approximation for semi-simple groups over function fields, Ann. of Math. (2) 105 (1 977), 553 - 572) prouvant le theoreme d'approximation forte sur un corps global quelconque. Soit G un groupe semi­ simple simplement connexe, de centre Z, sur un corps global K. Nous n'utiliserons

VARilhES DE SHIMURA

251

le principe de Hasse pour H l (K, G) que pour les classes dans l'image de H l (K; Z). En particulier, les facteurs £8 nous indifferent. 0.2. Groupe reductif signifiera toujours groupe reductif cormexe. Un revetement d'un groupe reductif est un revetement connexe. Groupe adjoint signifiera groupe reductif adjoint. Si G est un groupe reductif, nous noterons G•d son groupe adjoint, Gder son groupe derive, et p : G -> Gd er le revetement universe! de Gd er . Nous noterons souvent Z (ou Z(G)) le centre de G, et (conflit de notation) Z celui de G. 0.3. Nous noterons par un exposant 0 une composante connexe algebrique (par exemple : zo est Ia composante neutre du centre Z de G). L'exposant + designera une composante connexe topologique (par exemple : G(R)+ est Ia composante neutre topologique du groupe des points reels d'un groupe G). On notera aussi G(Q)+ Ia trace de G(R)+ sur G(Q). Pour G reductif reel, no us noterons par un indice + I' image inverse de G•d(R)+ dans G(R). Meme notation + pour la trace sur un groupe de points rationnels. Pour X un espace topologique, nous notons 1r0(X) !'ensemble de ses composantes connexes, muni de Ia topologie quotient de celle de X. Dans I' article, l'espace 7ro(X) sera toujours discret, ou compact totalement discontinu. 0.4. Un domaine hermitien symetrique est un espace hermitien symetrique a cour­ bure < 0 (i.e. sans facteur euclidien ou compact). 0.5. Sauf mention expresse du contraire, un espace vectoriel est suppose de di­ mension finie, et un corps de nombres est suppose de degre fini sur Q. Les corps de nombres que nous aurons a considerer seront le plus souvent contenus dans C; Q designe la cloture algebrique de Q dans C. 0.6. On pose i = proj lim Z/nZ = Ti p ZP • A f = Q ® i = TI P Qp (produit restreint) et on note A = R x A f l'anneau des adeles de Q . On designera parfois encore par A l'anneau des adeles d'un corps global quelconque. 0.7. G(K), G ®F K, GK : pour G un schema sur F (par exemple un groupe alge­ brique sur F) et K une F-algebre, on designe par G(K) I'ensemble des points de G a valeur dans K, et par GK ou par G ® F K Ie schema sur K deduit de G par extension des scalaires. 0.8. Nous normaliserons l'isomorphisme de reciprocite de Ia theorie du corps de classes global ( = choisirons lui ou son inverse) 1roA"t:/E* ___::::__. Gal(Q/E)•b de telle sorte que Ia classe de l'idele egal a une uniformisante en v et a 1 aux autres places corresponde a un Frobenius geometrique (l'inverse d'une substitution de Frobenius) (cf. 1 . 1 .6 et Ia justification loc. cit. 3.6, 8 . 1 2). ·

1. Domaines hermitiens symetriques. 1 . 1 . Espaces de modules de structures de Hodge. 1 . 1 . 1 . Rappelons qu'une structure de Hodge sur un espace vectoriel reel V est une bigraduation Vc = ffi Vpq du complexifie de V, telle que VPq soit le complexe conjugue de VqP . Definissons une action h de C* sur Vc par la formule (1 . 1 . 1 . 1) h(z)v = z-p:z-qv pour V E Vpq . Les h(z) commutent a Ia conjugaison complexe de V0, done se deduisent par ex-

252

PIERRE DELIGNE

tension des scalaires d'une action, encore notee h, de C* sur V. Regardons C comme une extension de R, et soit S son groupe multiplicatif, considere comme groupe algebrique reel (autrement dit, S = RciR Gm (restriction des scalaires a Ia Weil)) ; on a S(R) = C*, et h est une action du groupe algebrique S. On verifie que cette construction definit une equivalence de categories : (espaces vectoriels reels munis d'une structure de Hodge) ..... (espaces vectoriels reels munis d'une action du groupe algebrique reel S). A !'inclusion R* c: C* correspond une inclusion de groupes algebriques reels Gm c: S. No us noterons wh ( ou simplement w) la restriction de h- 1 a Gm , et l'appel­ lerons le poids w : Gm ..... GL( V). On dit que V est homogene de poids n si Vpq = 0 pour p + q f; n, i.e. w(i\) est l'homothetie de rapport i(n . Nous noterons /-lh (ou simplement f-t) !'action de Gm sur Vc definie par f-t(z) v = z-Pv pour v E Vpq . C'est un compose Gm ..... Sc ..... h GL( Vc). La filtration de Hodge Fh (ou simplement F) est definie par FP = EB r ":; p v rs . On dit que V est de type rff c: Z x Z si Vpq = 0 pour (p, q) ¢ rff . Plus generalement, si A est un sous-anneau de R tel que A ® Q soit un corps (en pratique, A = Z, Q ou R), une A-structure de Hodge est A-module de type fini V, muni d'une structure de Hodge sur V ®A R. EXEMPLE 1 . 1 .2. L'exemple fondamental est celui oil V = Hn (X, R), pour X une variete kahlerienne compacte, et oil VPq c: Hn (X, C) est l'espace des classes de cohomologie representees par une forme fermee de type (p, q). D'autres exemples utiles s'en deduisent par des operations de produits tensoriels, passage a une struc­ ture de Hodge facteur direct, ou passage au dual. Ainsi, le dual Hn(X, R) de Hn (X, R) est muni d'une structure de Hodge de poids - n. L'homologie, ou Ia coho­ mo1ogie, entieres fournissent des Z-structures de Hodge. ExEMPLE 1 . 1 .3. Les structures de Hodge de type {( - 1 , 0), (0, - 1)} sont celles pour lesquelles !'action h de C* = S(R) est induite par une structure complexe sur V; pour V de ce type, Ia projection pr de V sur v- 1 • 0 c: Vc est bijective, et verifie pr (h(z) v) = z pr( v) . EXEMPLE 1 . 1 . 4. Soit A un tore complexe ; c'est le quotient L/F de son algebre de Lie L par un reseau F. On a F ® R ....::::. L, d'oil une structure complexe sur F ® R. Considerons celle-ci comme une structure de Hodge (1 . 1 .3). Via l'isomorphisme r = H1 (A , Z), c'est celle de 1 . 1 .2. EXEMPLE 1 . 1 . 5. La structure de Hodge de Tate Z(l) est Ia Z-structure de Hodge de type ( - 1 , - 1) de reseau entier 211: iZ c: C. L'exponentielle identifie C* a CjZ(l), d'oil un isomorphisme Z(l) = H1 (C * ). La structure de Hodge Z(n) = dfn Z(l)®n (n E Z) est Ia Z-structure de Hodge de type ( - n, - n) de reseau entier (271: i) nZ. On note . . . (n) le produit tensoriel de . . . par Z(n) (twist a Ia Tate). 1 . 1 .6. REMARQUE. La regie h(z)v = z -Pz -q v pour v E vpq est celle que j'utilise dans (Les constantes des equations fonctionnelles des fonctions L, Anvers II, Lecture Notes in Math., vol. 349, pp. 501-597) et !'inverse de celle de [6]. Elle est justifiee d'une part par l'exemple 1 . 1 .4 ci-dessus, d'autre part par le desir que C * agisse sur R(1 ) par multiplication par Ia norme (cf. Joe. cit., fin de 8 . 1 2). 1 . 1 .7. Une variation de structure de Hodge sur une variete analytique complexe S consiste en (a) un systeme local V d'espaces vectoriels reels ;

VARilhES DE SHIMURA

253

(b) en chaque point de S, une structure de Hodge sur la fibre de V en s, variant continument avec s. On exige que la filtration de Hodge varie holomorphiquemen.t avec s, et verifie l'axiome dit de transversalite : la derivee d'une section de FP est dans FP- 1 • II sera souvent donne un systeme local Vz de Z-modules de type fini, tel que V = Vz ® R. On parlera alors de variation de Z-structure de Hodge. De meme pour Z remplace par un anneau A comme en 1 . 1 . 1 . REMARQUE 1 . 1 . 8. Regardons S comme etant une variete reelle, dont l'espace tangent en chaque point est muni d'une structure complexe, i.e. d'une structure de Hodge de type {( - 1 , 0), (0, - 1)}. L'integrabilite de la structure presque complexe de S s'exprime en disant que le crochet des champs de vecteurs est compatible a la filtration de Hodge du complexifie du fibre tangent : [T0 .- 1 , ro.- 1 ] c ro.- 1 . De meme, l'axiome des variations de structure de Hodge exprime que la derivation (fibre tangent) ®R (sections coo de V) --> (sections coo de V) ( ou plut6t le complexifie de cette application) est compatible aux filtrations de Hodge : ovFP c FP pour D dans T0 ,- 1 (holomorphie) et ovFP c FP- 1 pour D arbitraire (transversalite). PRINCIPE 1 . 1 .9. En geometrie algebrique, chaque fois qu'apparait une structure de Hodge dependant de parametres complexes, c'est une variation de structure de Hodge sur l'espace des parametres. L'exemple fondamental est 1 . 1 .2, avec para­ metres : si f : X --> S est un morphisme pro pre et lisse, de fibres Xs kahleriennes, les Hn(X., Z) forment un systeme local sur S, et la filtration de Hodge sur le com­ plexifte Hn(X., C) varie holomorphiquement avec s et verifie l'axiome de transver­ salite. 1 . 1 . 10. Une polarisation d'une structure de Hodge reelle, de poids n, V est un morphisme If! : V ® V --> R( - n) tel que la forme (2n i)n 1/f(x, h(i)y) soit symetrique et definie positive. De meme pour les Z-structures de Hodge, en rempla<;ant R( - n) par Z( - n), . . . . Puisque 1/f(h(i)x, y) = 1/f(x, h( - i)y) (h(i) est trivial sur R( - n) ) , et que h( - i)y = ( - I) nh(i)y, la condition de symetrie revient a : If! symetrique pour n pair, alterne pour n impair. Les structures de Hodge qui apparaissent en geometrie algebrique sont des Z­ structures de Hodge homogenes et polarisables. Exemple fondamental : les theo­ remes de positivite de Hodge assurent que fin(X, Z), pour X une variete projective et lisse, est polarisable (noter que h(i) est !'operation notee C par Weil dans son livre sur les varietes kahleriennes) . 1 . 1 . 1 1 . Soient des espaces vectoriels reels ( V;);"' 1 et une famille de tenseurs (si)J EJ dans des produits tensoriels de puissances tensorielles des V; et de leurs duaux. On s'interesse aux families de structures de Hodge sur les V;, pour lesquelles les s1 sont de type (0, 0). Pour interpreter cette condition "type (0, 0)" dans des cas particu­ liers, noter que f : V --> W est un morphisme si et seulement si, en tant qu'element de Hom( V, W) = V* ® W, il est de type (0, 0). Soit G le sous-groupe algebrique de T1 GL( V;) qui fixe les s1. D'apres 1 . 1 . 1 , une famille de structures de Hodge sur les V; s'identifie a un morphisme h : S --> T1 GL(V;). Pour que les si soient de type (0, 0), il faut et il suffit que h se factorise par G : il s'agit de considerer les morphismes algebriques h : S --> G. On peut regarder G, plutot que le systeme des V; et s1, comme 1' objet primordial : si G est un groupe algebrique lineaire reel, il revient au meme de se donner h : S -->

254

PIERRE DELIGNE

G, ou de se donner sur chaque representation V de G une structure de Hodge, fonctorielle pour les G-morphismes et compatible aux produits tensoriels (cf. Saavedra, [10, VI, §2]). Les morphismes wh et fi.h de 1 . 1 . 1 deviennent des mor­ phismes de Gm dans G, et de Gm dans Gc respectivement. 1 . 1 . 12. La construction 1 . 1 . 1 1 amene a considerer les espaces de modules de structures de Hodge du type suivant : on fixe un groupe algebrique lineaire reel G, et on considere une composante connexe (topologique) X de l'espace des mor­ phismes ( = homomorphismes de groupes algebriques sur R) de S dans G. Soit G1 le plus petit sous-groupe algebrique de G par lequel se factorisent les h E X : X est encore une composante connexe de l'espace des morphismes de S dans G 1 . Puisque S est de type multiplicatif, deux elements quelconques de X sont conjugues : l'espace X est une classe de G1 (R)+-conjugaison de morphismes de S dans G. C'est aussi une classe de G(R)+-conjugaison, et G 1 est un sous-groupe invariant de la composante neutre de G. 1 . 1 . 1 3. Vu 1 . 1 .9 et 1 . 1 . 10, nous ne considerons que les X tels que, pour une famille fidele V,. de representations de G, on ait (a) Pour tout i, la graduation par le poids de v,. (de complexifiee la graduation de V;c par les V,1- = EB fi+q =n vr) est independante de h E X. Conditions equiva­ lentes : h(R*) est central dans G(R)O ; Ia representation adjointe est de poids 0. (f3) Pour une structure complexe convenable sur X, et tout i, la famille de struc­ tures de Hodge definie par les h E X est une variation de structure de Hodge sur X.

(r) Si V est la composante homogene d'un poids n d'un V,., il existe 1/f : V ® V --+ R( - n) qui, pour tout h E X, soit une polarisation de V. PROPOSITION 1 . 1 . 14. Supposons w!rifie (a) ci-dessus. (i) II existe une et une seule structure complexe sur X telle que les filtrations de Hodge des V,. varient holomorphiquement avec h E X. (ii) La condition 1 . 1 . 1 3(f3) est verifiee si et seulement si Ia representation adjointe est de type {( - 1 , 1), (0, 0), (1, - 1)}. (iii) La condition 1 . 1 . 1 3(r) est w!rifiee si et seulement si G 1 (defini en 1 . 1 . 12) est reductif et que, pour h E X, l'automorphisme interieur int h(i) induit une involution de Cartan de son groupe adjoint. (i) Soit V Ia somme des V,.. C'est une representation fidele de G. Une structure de Hodge est determinee par Ia filtration de Hodge correspondante (plus la graduation par le poids si on n'est pas dans le cas homogene) : en poids n = p + q, on a ypq = FP n f'q. L'application cp de X dans la grassmannienne de Vc : h r-+ la filtra­ tion de Hodge correspondante, est done injective. Nous allons verifier qu'elle identifie X a une sous-variete complexe de cette grassmannienne ; ceci prouvera (i) : Ia structure complexe sur X induite de celle de la grassmannienne est la seule pour laquelle cp soit holomorphe. Soient L l'algebre de Lie de G et p : L --+ End( V) son action sur V. L'action p est un morphisme de G-modules, injectif par hypothese. Pour tout h E X, c'est aussi un morphisme de structures de Hodge. L'espace tangent a X en h est le quotient de L par l'algebre de Lie du stabilisateur de h-a savoir le sous-espace LOO de L pour la structure de Hodge de L definie par h. L'espace tangent a la grassmannienne en cp(h) est End( Vc)/F0 (End( Vc)) . Enfin, dcp est le compose

VARIETES DE SHIMURA

255

L/£00 __!____. End( V)/End( V)OO

]'

]'

Lc/FlLc__i__. End( Vc)/FlEnd( Vc) Puisque p est un morphisme injectif de structures de Hodge, drp est injectif; son image est celle de LcfFOLc, un sous-espace complexe, d'oil l'assertion. (ii) L'axiome de transversalite signifie que l'image de drp est dans p- 1 End( Vc)/£0End(Vc), i.e. que Lc = F- 1 Lc. Pour prouver (iii), nous ferons usage de [7, 2.8], rappele ci-dessous. Rappelons qu'une involution de Cartan d'un groupe algebrique lineaire reel (non necessaire­ ment connexe) G est une involution u de G telle que Ia forme reelle G11 de G (de conjugaison complexe g --+ u(g)) soit compacte : G11(R) est compact et rencontre toutes Ies composantes connexes de G11(C) = G(C). Pour C E G(R) de carre central, une C-po/arisation d'une representation V de G est une forme bilineaire ?f! G-invari­ ante, telle que ?f!(x, Cy) soit symetrique et defini > 0. Pour tout g E G(R), on a alors ?f!(x, gCg-ly) = ?f!(g-lx, Cg-ly) : Ia notion de C-polarisation ne depend que de la class de G(R)-conjugaison de C. Rappel. 1 . 1 . 1 5 [7, 2.8]. Soient G a/gebrique reel et C E G(R) de carre central. Les conditions suivantes sont equivalentes : (i) Int C est une involution de Cartan de G ; (ii) toute representation reel/e de G est C-po/arisable ; (iii) G admet une representation reel/e C-polarisable fide/e. On notera que la co ndition 1 . 1 . 1 5(i) entraine que GO est reductif-pour avoir une forme compacte. Elle ne depend que de Ia classe de conjugaison de C. Prouvons 1 . 1 . 14(iii). Soit G2 le plus petit sous-groupe algebrique de G par lequel se factorisent les restrictions des h E X a Ul c: C*. Pour qu'une forme bilineaire ?f! : V ® V --+ R( - n) verifie 1 . 1 . 1 3(r ), il faut et il suffit que (277:i) n ?J! : V ® V --+ R soit invariant par les h(U 1 )-donc par G2-(ceci exprime que ?f! est un morphisme), et une h(i)-polarisation. D'apres 1 . 1 . 1 5, 1 . 1 . 1 3(r) equivaut a : int h(i) est une involu­ tion de Cartan de G2• On en deduit d'abord que G1 est reductif: G2 l'est, pour avoir une forme com­ pacte, et G 1 est un quotient du produit Gm x G2. Puisque G2 est engendre par des sous-groupes compacts, son centre connexe est compact : il est isogene au quotient de G2 par son groupe derive. L'involution 0 = int h(i) est done une involution de Cartan de G2 si et seulement si e'en est une de son groupe adjoint, et on conclut en notant que G1 et G2 ont meme groupe adjoint. Les conditions en 1 . 1 . 14 ne dependant que de (G, X), on a le COROLLAIRE 1 . 1 . 1 6. Les conditions 1 . 1 . 1 3(a), (�), (r) ne dependent pas de lafamil/e fidele choisie de representations V;. CoROLLAIRE 1 . 1 . 17. Les espaces X de 1 . 1 . 1 3 sont les domaines hermitiens symetriques. A. Prouvons X de ce type. On se ramene successivement a supposer : (1) Que G = G1 : remplacer G par G1 ne modifie ni X, ni les conditions 1 . 1 . 1 3. (2) Que G est adjoint : par(l ), G est reductif, et son quotient par un sous-groupe

256

PIERRE DELIGNE

central fini est le produit d'un tore T par son groupe adjoint G•d. L'espace X s'identifie encore a une composante connexe de l'espace des morphismes de SfGm dans G•d : si un tel morphisme se releve en un morphisme de S dans G, avec une projection donnee dans T, le relevement est unique. Les conditions enoncees en 1 . 1 . 14 restent par ailleurs verifiees. (3) Que G est simple : decomposer G en produit de groupes simples G; ; ceci decompose X en un produits d'espaces X; relatifs aux G;. So it done G un groupe simple adjoint, et X une G(R)+ -classe de conjugaison de morphismes non triviaux h : SfGm --+ G, verifiant les conditions de 1 . 1 . 14(ii), (iii). Le groupe G est non compact : sinon, int h(i) serait trivial (par (iii)), Lie G serait de type (0, 0) (par (ii)) et h serait trivial. Soit h E X. D'apres (iii), son centralisateur est compact; il existe done sur X une structure riemannienne G(R)+-equivariante. D'apres (ii), h(i) agit sur l'espace tangent Lie(G)/Lie(G)OO de X en h par - 1 : l'espace X est riemannien symetrique. On verifie enfin qu'il est hermitien symetrique pour Ia structure complexe 1 . 1 . 14(i). II est du type non compact (courbure < 0) car G est non compact. B. Reciproquement, si X est un espace hermitien symetrique, et que x E X, on sait que Ia multiplication par u (lui = 1) sur l'espace tangent T:x a X en x se prolonge en un automorphisme m:x(u) de X. Soient A le groupe des automorphismes de X, et h(z) = m(z/ t) pour z E C*. Le centralisateur A x de x commute a h, et Ia condition de 1 . 1 . 14(ii) est done verifiee : Lie(A:x) est de type {0, 0), et T:x = Lie(A)/Lie(A:x) de type {( - 1 , 1), (1, - 1)}. Enfin, on sait que A est Ia composante neutre de G(R), pour G adjoint, et que l'espace riemannien symetrique X est a courbure < 0 si et seulement si Ia symetrie h(i) fournit une involution de Cartan de G (voir Helgason [8]). 1 . 1 . 1 8 . Indiquons deux variantes de 1 . 1 . 1 5 (cf. [7, 2. 1 1]). (a) On donne un groupe algebrique reel reductif (0.2) G, et une classe de G(R)­ conjugaison de morphismes h : S --+ G. On suppose que wh-note w-est central, done independant de h (condition 1 . 1 . 1 3(a)), et que int h(i) est une involution de Cartan de Gfw(Gm ). Puisque G est reductif, w(Gm) admet un supplement G2 : un sons-groupe in­ variant connexe tel que G soit quotient de w(Gm) x G2 par un sous-groupe central fini. II est unique : engendre par le groupe derive et le plus grand sous-tore compact du centre. II contient les h(U1 ) (h E X), et int h(i) en est une involution de Cartan. Si V est une representation de G, sa restriction a G2 admet done une h(i)-polarisa­ tion f/J. Si V est de poids n, w(Gm ) agit par similitudes, done G de meme : pour une representation convenable de G sur R, f/J est covariant. Pour cette representation, R est de type (n, n) ; ceci permet de faire agir G sur R(n), de fa<;:on compatible a sa structure de Hodge, et de voir Iff = (27Ci)-"f/J comme une forme de polarisation G­ invariante : V ® V --+ R( - n) (b) Supposons que G se deduise par extension des scalaires a R de G0 sur Q, et que w soit defini sur Q. Le groupe G2 est alors defini sur Q, car c'est l'unique supplement de w(Gm ), et tout caractere de G/G2 est defini sur Q, car ce groupe est trivial ou isomorphe, sur Q, a Gm . Si une representation (rationnelle) V de G0 est de poids n, les formes bilineaires G-invariantes V ® V --+ Q( - n) forment un espace vectoriel F sur Q. L'ensemble de celles qui sont de polarisation (rei. h E X) est Ia trace sur F d'un ouvert de FR, et cet ouvert est non vide d'apres (a). II existe done des formes de polarisation 7/f : V ® V --+ Q(n) G-invariantes. .

VARIETES DE SHIMURA

257

On prendra garde que les formes en (a) et (b) ne sont pas toujours de polariza­ tion pour tout h' E X: si h' = int(g)(h), la formule 1/f(x, h'(i)y) = gl/f (g-lx, h(i)g-ly) montre que la forme (271:i) nlf! (x, h'(i)y) est symetrique et definie-mais definie posi­ tive ou negative selon I' action de g sur R( - n) . 1 .2. Classification. Dans la suite de ce paragraphe, nous utilisons la relation 1 . 1 . 1 7 entre do­ maines hermitiens symetriques et espaces de modules de structures de Hodge, pour reformuler certains resultats de [1] et [8], et donner quelques complements. 1 .2. 1 . Considerons les systemes (G, X) formes d'un groupe algebrique reel simple adjoint G, et d'une classe de G(R)-conjugaison X de morphismes de groupes al­ gebriques reels h : S ---> G, verifiant (les notations sont celles de 1 . 1 . 1 , 1 . 1 . 1 1 ). (i) La representation adjointe Lie(G) est de type {( - 1 , 1), (0, 0), (1, - 1) } (en particulier, h est trivial sur Gm c S) ; (ii) int h(i) est une involution de Cartan ; (iii) h est non trivial ou-ce qui revient au meme (cf. 1 . 1 . 17)-G est non compact. D'apres 1 . 1 . 1 7, les composantes connexes des espaces X ainsi obtenus sont les domaines hermitiens symetriques irreductibles. L'hypothese (ii) assure que les involutions de Cartan de G sont des automor­ phismes interieurs, done que G est une forme interieure de sa forme compacte (cf. 1 .2.3). En particulier, G, etant simple, est absolument simple. La classe de G(C)-conjugaison de f.l-h : Gm ---> Gc ne depend pas du choix de h E X. Nous Ia noterons Mx. PROPOSITION 1 .2.2. Soit Gc un groupe algebrique complexe simple adjoint. A chaque systeme (G, X) forme d'une forme reelle G de Gc, et de X verifiant 1 .2. 1 (i), (ii), (iii) associons Mx · On obtient ainsi une bijection entre classes de Gc(C)-con­ jugaison de systemes (G, X), et classes de Gc(C)-conjugaison de morphismes non­ triviaux fl. : Gm ---> Gc verijiant Ia condition suivante. ( * ) Dans Ia representation ad fl. de Gm sur Lie(Gc), seuls apparaissent les caracteres z, 1 et z-1 .

Pour verifier 1 .2.2, nous utiliserons la dualite entre domaines hermitiens syme­ triques, et espaces hermitiens symetriques compacts : 1 .2.3. Soient G une forme reelle de Gc, X une classe de G(R)-conjugaison de mor­ phismes de S/Gm dans G, et h E X. La forme reelle G correspond a une conjugaison complexe (J sur Gc ; definissons G* comme la forme reelle de conjugaison complexe int (h(i)) (J : G*(R) = {g E (C) I g = int(h(i) ) (J(g) } .

Le morphisme h est encore defini sur R, de SfGm dans G* : on a h(C*JR*) c G *(R) ; definissons X* comme la classe de G*(R)-conjugaison de h. La construction (G, X) ---> (G*, X*) est une involution sur !'ensemble des classes de Gc(C)-conjugai­ son des systemes (G, X) formes d'une forme reelle G de Gc, et d'une classe de G(R)­ conjugaison de morphismes non triviaux de S/Gm dans G. Elle echange les (G, X) comme en 1 .2.2, et les (G, X) tels que G soit compact et que X verifie 1 .2. 1(i). On sait que les formes reelles compactes de Gc sont toutes conjuguees entre elles. Puisque si g E Gc normalise une forme reelle G, on a g E G(R) (ceci parce que G est adjoint), la dualite ramene 1 .2.2 a I' en once suivant :

258

PIERRE DELIGNE

LEMME 1 .2.4. Soit G une forme compacte de Gc. La construction h --+ /-lh induit une bijection entre (a) classes de G(R)-conjugaison de morphismes h : S/Gm --+ G, verifiant 1 .2. l(i), et (b) classes de Gc(C)-conjugaison de morphismes p : Gm --+ Gc. verifiant 1 .2.2(•). Soient T un tore maximal de G, et Tc son complexifie. On verifie d'abord que !'application h --+ ph : Hom(S/Gm , T) --+ Hom(Gm , Tc) est bijective. Si W est le groupe de Weil de T, on sait que Hom(U l , T)/ W ____:____. Hom(U l , G)/G(R) et L'application h --+ /-lh induit done une bijection Hom(S/Gm , G)/G(R) ____:____. Hom(Gm , G)/GJC), et, pour que h verifie 1 .2.l(i), il faut et il suffit que p,. verifie 1 .2.2(•). 1 .2.5. Soit G un groupe algebrique complexe simple adjoint. Nous allons enume­ rer les classes de conjugaison de morphismes non triviaux p : Gm --+ G verifiant 1 .2.2(•), en terme du diagramme de Dynkin D de G. Rappelons que ce dernier est canoniquement attache a G en particulier, les automorphismes de G agissent sur D-On peut identifier ses sommets aux classes de conjugaison de sous-groupes paraboliques maximaux. Soient T un tore maximal, X(T) = Hom(T, Gm), Y(T) = Hom(Gm , T) (le dual de X(T) pour l'accouplement X(T) x Y(T) --+ o Hom(Gm , Gm) = Z), R c X(T) !'ensemble des racines, B un systeme de racines simples, a0 l ' oppose de la plus grande racine et B+ = B U {a0} . Les sommets de D sont parametres par B, et ceux du diagramme de Dynkin etendu n+ par B+. Une classe de conjugaison de morphismes de Gm dans G a un unique representant p E Y(T) dans la chambre fondamentale (a, p) � 0 pour a E B. Il est uniquement determine par les entiers positifs (a, p) (a E B) et, G etant adjoint, ceux-ci peuvent etre prescrits arbitrairement. La condition 1 .2.2( * ) , pour p non trivial, se recrit (ao , p) = 1 . -

-

Ecrivons la plus grande racine comme combinaison lineaire de racines simples, = 0, avec n(a0) = 1, et appelons speciaux les sommets de n+ tels que, pour la racine correspondante a E B+, on ait n(a) = 1 . On sait que le quotient du groupe des copoids par celui des coracines agit sur n+, et de fa<;on simplement transitive sur ]'ensemble des sommets speciaux. Les sommets speciaux soot done les conjugues sous Aut(D+) du sommet correspondant a a0, et leur nombre est l ' in­ dice de connexion l n- 1 (G) I de G (cf. Bourbaki [4, VI, 2 ex 2 et 5a)]). La condition ( *) se recrit (•)" Pour une racine simple a E B correspondant a un sommet special de D, on a (a, p) = 1 . Pour les autres racines simples, (a, p) = 0. 1 .2.6. Au total, les classes de Gc(C)-conjugaison de systemes (G, X) comme en 1 .2.2 soot parametrees par les sommets speciaux du diagramme de Dynkin D de Gc. En particulier, pour G une forme reelle donnee de Gc, X est determine par le I:aEB+n(a)a

'

VARIETES

DE

SHIMURA

259

sommet special s(X) correspondant ( G(R) c Gc(C) est en effet son propre normali­ sateur). Le sommet correspondant a x- 1 = {h- 1 1h E X} est le transforme de s(X) par !'involution d'opposition. Dans 1 .2.3, G et G* sont des formes interieures l'un de l'autre. S'il existe X verifiant 1 .2. 1 (i), (ii), (iii), G est done une forme interieure de sa forme compacte. En d'autres termes, Ia conjugaison complexe agit sur le diagramme de Dynkin de Gc par !'involution d'opposition. PROPOSITION 1 .2.7. Soit G un groupe algebrique reel simple adjoint, et supposons qu'il existe des morphismes h : C*/R* -+ G verifiant 1 .2. l {i), (ii), (iii). L'ensemble de ces morphismes a alors deux composantes connexes, echangees par h �--+ h- 1 . Chacune a pour stabilisateur Ia composante neutre G(R)+ de G. L'hypothese (ii) assure que le centralisateur K de h(i) est un sous-groupe compact maximal de G(R). En particulier, 7Co(K) ..::. 1CoG(R). II a meme algebre de Lie que le centralisateur de h. Ce dernier est un groupe algebrique connexe-en tant que centralisateur d'un tore-et compact-en tant que sous-groupe du centralisateur de h(i). II est done topologiquement connexe et Centr(h) = K+ = K n G(R)+. Le centre de K+ est de dimension 1 : le complexifie de K+ est le centralisateur de /1h done, d'apres ( * )", un sous-groupe de Levi d'un sous-groupe parabolique maximal. On peut aussi le deduire de ce que Ia representation de K+ sur Lie(G)/Lie(K+) est ir­ reductible (cf. [8, preuve de V, 1 . 1]). Le morphisme h est done un isomorphisme de SfGm avec le centre connexe de K+, et, K+ determine h au signe pres. A fortiori, h(i) determine h au signe pres. Des lors (a) L'application h �--+ h(i) est 2 : 1 . (b) Elle envoie isomorphiquement l'orbite G(R)+jK+ de h sous G(R)+ sur !'en­ semble G(R)/K de toutes les involutions de Cartan dans G(R). La proposition en resulte. CoROLLAIRE 1 .2. 8 . Soit (G, X) comme en 1 .2. 1 , et s le sommet correspondant du diagramme de Dynkin de Gc. (i) Si s n'est pas .fixe par /'involution d'opposition, G(R) et X sont connexes. (ii) Si s est fixe par /'involution d'opposition, G(R) et X ont deux composantes connexes; les composantes de X sont echangees par h �--+ h- 1 , et par les g E G(R) - G(R)+. Signalons que le cas (i) est encore caracterise par les conditions equivalentes (i') le systeme de racines relatif de G est de type C (plutot que BC) ; (i") X est un domaine tube. 1 .3 . Plongements symplectiques. 1 . 3. 1 . Soit V un espace vectoriel reel, muni d'une forme alternee non degeneree 1/f. Le demi-espace de Siegel s+ correspondent admet la description suivante : c'est l'espace des structures complexes h sur V, telles que 1/f soit de type ( 1 , 1 ) (pour !'identification (1 . 1 . 3) entre structures complexes et structures de Hodge de type {( - 1 , 0), (0, - 1)}) et que la forme 1/f(x, h(i)x) soit symetrique et definie positive. Si on remplace "defini positif" par "defini", le double demi-espace de Siegel S± obtenu est une classe de conjugaison de morphismes h : S -+ CSp(V) (CSp = similitudes symplectiques ; dans [5], ce groupe est note Gp). 1 .3.2. Soient G un groupe algebrique reel adjoint (0.2) et X une classe de conjugai­ son de morphismes h : S -+ G. On suppose verifiees les conditions (i), (ii) de 1 .2. 1 , et o n remplace (iii) par

260

PIERRE DELIGNE

(iii') G est sans facteur compact. Le systeme (G, X) est done un produit de systemes (G,, X,) comme en 1 .2. 1 , et X, correspond a un sommet special du diagramme de Dynkin de G,c (1 .2.6). Considerons les diagrammes oil G est le groupe adjoint du groupe reductif Gh et oil X1 est une classe de G 1 (R)-conjugaison de morphismes de S dans G1 • On dispose d'une section G � Gh de sorte que V est une representation de G . Notre but est Ia determina­ tion 1 .3 . 8 des representations complexes irreductibles nontriviales W de G , qui est essentiellement equivalent a figurent dans Ia complexifiee d'une representation ainsi obtenue. Ce probleme celui resolu par Satake dans [11]. LEMME 1 . 3 . 3 . II suffit qu 'existe (Gh X1 ) � (G, X), comme ci-dessus, et une repre­ sentation lineaire ( V, p) de type {( - 1 , 0), (0, - 1)} de Gh telle que W figure dans Vc. Remplacant G1 par le sous-groupe engendre par le groupe derive G;_ et par I' image de h, on se ramene a supposer que int h(i) est une involution de Cartan de G 1 fw(Gm)· II existe alors sur V une forme de polarisation W ( 1 . 1 8(a)) , telle que p soit un mor­ phisme de (Gh X1 ) dans ( CSp(V), S±) . 1 .3 .4. Considerons le systeme projectif {H,.)n E N suivant : N est ordonne par divisibilite, H,. = Gm • et le morphisme de transition de H,.d dans Hn est x � xd (lim proj H,. est le revetement universel-au sens algebrique-de Gm). Un morphisme fractionnaire de Gm dans un groupe H est un element de lim inj Hom(H,., H). De meme pour le groupe S. Pour p. : Gm � H fractionnaire, defini par p." : H,. = Gm � H, et V une representation lineaire de H, V est somme des sous-espaces V11 (a E ( 1/ n)Z) tels que, via p. " , Gm agisse sur Va par multiplication par x" 11 • Les a tels que V11 =1- 0 sont les poids de p. dans V. De meme, un morphisme fractionnaire h : S � H determine une decomposition de Hodge fractionnaire p, s de V (r, s E Q). LEMME 1 . 3.5. Pour h E X, soit ilh le relevement fractionnaire de fl. h a Gc. Les re­ presentations W de 1 . 3.2 sont celles telles que p.h n 'ait que deux poids a et a + 1 . La condition est necessaire : Relevant h en h 1 E Xh on a fl.h, = p.h · v, avec v cen­ tral. Sur V, f1.h1 a les poids 0 et 1 . Si - a est !'unique poids de v sur W irreductible dans Vc, les seuls poids de P.h sur W sont a et a + 1 . Pour W non trivial, I' action de Gm via p.Hn assez divisible) est non triviale (car Gc est simple), done non centrale, et les deux poids a et a + 1 apparaissent. La condition est suffisante : Prenons pour V l'espace vectoriel reel sous-jacent a W, et pour groupe G 1 le groupe engendre par !'image de G , et par le groupe des homotheties. Pour h E X, de relevement fractionnaire ii a G, soit h l (z) = h(z)z-az l -a. Si Wa et Wa+l sont les sous-espaces de poids a et a + 1 de W, iz agit sur W11 (resp. Wa+ l ) par (z/z)a (resp. (zfz) l+ a), et h 1 par z (resp. z) : h 1 est un vrai mor­ phisme de S dans Gh de projection h dans G, et V est de type {( - 1 , 0) (0, - 1)} rei. h 1 . II ne reste qu'a appliquer 1 .3.3. 1 .3.6. Traduisons la condition 1 . 3.5 en terme de racines. Soient T un tore maximal de Gc, t son image inverse dans Gc, B un systeme de racines simples de T, et p. E Y(T) le representant dans Ia chambre fondamentale de Ia classe de conjugaison de fl.h (h E X). Si a est le poids dominant de W, le plus petit poids est - -r (a), pour ..-

26 1

VARilhES DE SHIMURA

!'involution d'opposition. II s'agit d'exprimer que (f-t, {)) ne prend que deux valeurs a et a + 1 , pour {3 un poids de W. Ces poids etant tous de la forme (a + une com­ binaison Z-lineaire de racine), et les (f-t, z ) , pour r une racine, etant entiers, la con­ dition s'exprime par (f-t, - z"(a)) = (f-t, a) - 1 , soit (f-t, a + -.(a)) = I . Determinons les solutions de (1 .3.6. 1). Pour tout poids dominant a, (f-t, a + -.(a)) est un entier, car a + -.(a) est combinaison Z-lineaire de racines. Si a =1- 0, il est > 0, sans quoi f-t annulerait tous les poids de la representation correspondante. Un poids dominant a verifiant (1 .3.6. 1) ne peut done etre somme de deux poids : (1 .3.6. 1)

LEMME 1 .3.7.

Seuls les poids fondamentaux peuvent verifier 1 .3.6. 1 .

1 .3.8. D'apres 1 .3. 7 , les representations W cherchees se factorisent par un facteur simple G, de G, et leur poids dominant est un poids fondamental ; il correspond a un sommet du diagramme de Dynkin D, de G,c. La condition necessaire et suf­ fisante (1 .3.6. 1) ne depend que de la projection de f-t dans G,c ; celle-ci correspond a un sommet special s de D, ( 1 . 2.6), et s a racine simple a5• Le nombre (f-t, m), pour m un poids, est le coefficient de as dans !'expression de m comme combinaison Q­ lineaire de racines simples. Pour m fondamental, ces coefficients sont donnes dans les tables de Bourbaki [4]. lis sont donnes par la table suivante, oil sont enumeres les diagrammes de Dynkin munis d'un sommet special (entoure). Chaque sommet correspond a un poids fondamental m , et on l'a affecte du nombre (f-t, m). Les sommets correspondant aux poids qui verifient (1 .3.6. 1) sont soulignes. TABLE 1 . 3.9. Dans la table, . . . indique une progression arithmetique. A p+ q- l (sommet special en pieme position)

.

. .

+ q) j(p + q) pqj(p + q) qj(p 1-- . .. . .... . . . ... . .... . .... -B1- . . . . . .. . . ......... . ...p. .. --:2._ •

•

•

1 /2 1 1 � ··················································· � . . . l/2 1 /2 1--- ··················································· �

i- . . . ..

D{!+ 2

:. :. .:. . . . . . . . . . . -<::: k ( : � r: : � k

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

�� ·�;, 2

·

5

T3 13

/2

;, �

5 /?

4/2 I

+ 1 /2

3 /2 -G)

+2

�

5)

262

PIERRE DELIGNE

REMARQUES 1 .3. 1 0. (i) Pour G simple exceptionnel, aucune representation W ne verifie 1 .3.2. (ii) Pour G simple classique, sauf le cas lYf (I � 5), les representations W de 1 .3.2 forment un systeme fidele de representations de G. Pour IJll , on obtient seulement une representation fidele d'un revetement double de G (a savoir, Ia composante connexe algebrique du groupe des automorphismes d'un espace vectoriel sur H muni d'une forme antihermitienne non degeneree-une forme interieure de S0(2n)) .

2. Varietes de Shimura. 2.0. Preliminaires. 2.0. 1 . Soient G un groupe, r un sous-groupe et q; : r --+ L1 un morphisme. Sup­

posons donnee une action r de L1 sur G, qui stabilise r, et telle que (a) r(q;(r)) est l'automorphisme interieur int7 de G ; (b) q; est compatible aux actions de L1 , sur F par r, e t sur lui-meme par automor­ phismes interieurs : q;(r(o)(r)) = int,;(q;(r)). Formons le produit semi-direct G ><1 L1. Les conditions (a), (b) reviennent a dire que !'ensemble des r · q;(r)- 1 est un sous-groupe distingue, et on definit G * rL1 comme Ie quotient de G ><1 L1 par ce sous-groupe. On notera que les hypotheses entrainent que Z = Ker(q;) est central dans G, et que Im(q;) est un sous-groupe distingue de L1. Les !ignes du diagramme o ___. z ___. r --- L1 ___. L1/r --- o (2.0. 1 . 1)

II

!'

!'

II

0 ---> Z ---> G --> G * r L1--+ L1/F --> 0

soot exactes, d'ou un isomorphisme (2.0. 1 .2) F\ G -"'---. L1\ G * r L1 et, mise en evidence, une action a droite de G * rL1 sur F\G. Pour cette action, G agit par translations a droite, et L1 par I' action a droite r - 1 . Si G est un groupe topologique, que L1 est discret, et que !'action r est continue, la groupe G * r L1, muni de la topologie quotient de celle de G ><1 L1, est un groupe topologique, G/Ker(q;) en est un sous-groupe ouvert, et !'application (2.0. 1 .2) est un homeomorphisme. La construction 2.0. 1 garde un sens dans la categorie des groupes algebriques sur un corps. Si G est un groupe reductif sur k, on a un isomorphisme canonique G = G *z eal Z(G) (pour l'action triviale de Z(G) sur G). 2.0.2. Soient G un groupe algebrique sur un corps k, et G•d le quotient de G par son centre Z. L'action par automorphismes interieurs de G sur lui-meme (x, y) --+ xyx- 1 : G x G --+ G est invariante par Z x { e} agissant par translation, done se factorise par une action de G•d sur G. Prendre garde que !'action de r E G•d(k) sur G(k) n'est pas necessairement un automorphisme interieur de G(k) (Ia projection de G(k) dans G•d(k) n'est pas toujours surjective). Un exemple typique est !'action de PGL(n, k) sur SL(n, k). De meme, !'application "commutateur" (x, y) = xyx-1y -1 : G X G --+ G est invariante par Z x Z agissant par translation, et se factorise par une application "commutateur" ( , ) : G•d x G•d --+ G.

VARIETES DE SHIMURA

263

Tout ceci, et le fait que ces "commutateurs" et "automorphismes interieurs" verifient les identites usuelles se voit au mieux par descente, i.e. en interpretant G comme un faisceau en groupes sur un site convenable, et G•d comme le quotient de ce faisceau en groupes par son centre. En caracteristique 0, si on s'interesse seule­ ment aux points de G sur des extensions de k, il suffit d'utiliser Ia descente galoi­ sienne-cf. 2.4. 1 , 2.4.2. Variante. Pour G reductif sur k, les groupes G et G ont meme groupe adjoint, et les constructions precedentes pour G et G sont compatibles. En particulier, !'ap­ plication commutateur ( , ) : G x G -+ G a une factorisation canonique ( ' ) : G X G ---+ G•d X G•d ---+ (} ---+ G. On en deduit que le quotient de G(k) par le so us-groupe distingue pG(k) est abelien. 2.0.3. Soient k un corps global de caracteristique 0, A l'anneau de ses adeles, G un groupe semi-simple sur k et N = Ker(p : G -+ G). Soient S un ensemble fini de places de k, As l'anneau des S-adeles (produit restreint etendu aux v ¢ S) et posons Fs = pG(A5) n G(k) (intersection dans G(A5)). C'est le groupe des ele­ ments de G(k) qui, en toute place v ¢ S, peuvent se relever dans G(kv) (se rappeler que p : G(A) -+ G(A) est propre). La suite exacte longue de cohomologie identifie G(k)/pG(k) a un sous-groupe de H 1 (Gal(k/k), N(k)), et F5/pG(k) aux elements localement nul, en les places v ¢ S, de ce sous-groupe. En particulier, F5/pG(k) est contenu dans le sous-groupe H�(Gai(k/k), N(k)) des classes dont Ia restriction a tout sous-groupe monogene _est triviale (argument et notations de [12]). Si Im Gal(k/k) est l'image de Galois dans Aut N(k), on a. H�(Gal(k/k), N(k)) = H�(Im Gal(k/k), N(k)) (loc. cit.) ; en particulier, F5/pG(k) est fini. PROPOSITION 2.0.4. (i) Fs ne depend que de /'ensemble des places v E S oil le groupe de decomposition Dv c Im Gal(k/k) est non cyclique. En particulier, il ne change pas si on ajoute a

s les places a l'infini.

(ii) F5/pG(k) s 'identifie au sous-groupe du groupe fini H�(Im Gal(k/k), N(k)) forme des classes de restriction nulle a tout sous-groupe de decomposition Dv, v ¢ S. En particu/ier, pour S grand, on a F5/pG(k) = H�(lm Gal(k/k), N(k)). La restriction d'un element de H�(Im Gal(k/k), N(k)) a un groupe de decom­ position cyclique est automatiquement nulle, d'oi:l (i). Pour (ii), on peut supposer que S contient les places a l'infini. Le principe de Hasse pour G (pour des classes venant du centre) assure alors que tous les elements du groupe (ii) sont effective­ ment realises comme classe d'obstruction. COROLLAIRE 2.0.5. Tout sous-groupe de S-congruence assez petit de G(k) est dans Fs .

Si U est un sous-groupe de S-congruence, UfU n pG(k) est fini : !'obstruction a relever dans G(k) meurt dans une extension galoisienne de degre et de ramification bornees, done est dans H 1 (Gal, N(k)) pour Gal un quotient fini de Gal(k/k). Des conditions de S-congruence permettent alors de passer de ce H 1 a F5/pG(k), cf.

[12].

REMARQUE

2.0.6. On notera par ailleurs que si G verifie le theoreme d'approxi-

264

PIERRE DELIGNE

mation forte rei. S, tout sous-groupe de S-congruence sur F8/pG(k).

U

c Fs

de G(k) s'envoie

CoROLLAIRE 2.0.7. Pour toute place archimedienne v, un sous-groupe de S-con­ gruence assez petit U de G(k) est dans Ia composante connexe topologique G(kv)+ de G(kv).

Puisque G(k.) est connexe, on a G(k")+ = pG(k.) et (2.0.4 et 2.0.5).

U

c Fs = Fsu tvl c G(k.)+

CoROLLAIRE 2.0.8. Le sous-groupe G(k)pG(A8) de G(A8) est ferme, topo/ogique­ ment isomorphe a pG(As) *rs G(k) (i.e. pG(A8) en est un sous-groupe ouvert).

C'est un sous-groupe parce que, vu 2.0.2, pG(A8) est distingue dans G(A8), avec un quotient commutatif. Soit T :=> S assez grand pour que G(k) soit dense dans G(Ar) (approximation forte). Notons kr- s le produit des kv pour v E T - S. Pour K un sous-groupe compact ouvert de G(Ar) , on a G(k)pG(A8) = G(k)p(G(k) · G(kr-s)

x

p- 1K) c G(k)(pG(kr- s) x K) .

D'apn!s 2.0. 5, pour K assez petit, on a dans G(Ar) : G(k) n K c Fr , d'ou dans G(As) : G(k) n (pG(kr-s) X K) c rs c pG(As)· L'intersection de G(k)pG(As) avec le sous-groupe ouvert pG(kr-s) x K est done contenue dans pG(A8), et le corollaire en resulte. COROLLAIRE 2.0.9. Si (} verifie /e theoreme d'approximation forte rei. S, /'adherence de G(k) dans G(A8) est G(k) · pG(A8).

2.0. 1 0. So it T un tore sur k et S un ensemble fini de places contenant les places archimediennes. Soit U c T(k) le groupe de S-unites. D'apres un theoreme de Chevaelly, tout sous-groupe d'indice fini de U est un sous-groupe de congruence (voir [12] pour une demonstration elegante). II en resulte que si T' --> T est une iso­ genie, l'image d'un sous-groupe de congruence pour T' est un sous-groupe de con­ gruence pour T. 2.0. 1 1 . Soient G reductif sur k, p : (} --> Gder le revetement universe I de son groupe derive, et zo Ia composante neutre de son centre Z . Voici quelques corollaires de 2.0. 1 0 (on suppose que !'ensemble fini S de places contient les places archimedien­ nes). CoROLLAIRE 2.0. 1 2. Pour U d'indice fini dans le groupe des S-unites de Z(k) il existe un sous-groupe compact ouvert K de G(As) te/que

G(k) n (K . Gder( As) ) c Gder (k) . U. Appliquons 2.0. 10 a l'isogenie zo --> GJGder : pour K petit, un element r de G(k) dans K · Gd•r(As) a dans (GJGd•r)(k) une image petite, pour Ia topologie des sous­ groupes de S-congruence, done peut se relever un petit element z de Z(k), et r =
z.

CoROLLAIRE 2.0. 1 3 .

Le produit d'un sous-groupe de congruence de Gder et d'un sous-groupe d'indice fini du groupe des S-unites de Z0(k) est un sous-groupe de S­ congruence de G(k).

VARIETES DE SHIMURA

265

COROLLAIRE 2.0. 1 4. Tout sous-groupe de S-congruence assez petit de G(k) est contenu dans Ia composante neutre topologique G(R)+ de G(R). Appliquer 2.0. 1 3, 2.0.7 a Gd•r , et 2.0. 10 a zo. 2.0. 1 5 . On sait que Gder(k)pG(A) est ouvert dans G(k)pG(A) (car image inverse de {e} c le sous-groupe discret GfGder(k) de (G/Gd•r)(A)) . D'apres 2.0.8, G(k)pG(A) est done un sous-groupe ferme de G(A). On pose

(2.0. 1 5 . 1 )

n-(G) = G(A)/G(k)pG(A).

L'existence de commutateurs 2.0.2 montre que !'action de G•d (k) sur n-(G), deduite de !'action 2.0.2 de G•d sur G, est triviale. 2. 1 . Varietes de Shimura. 2. 1 . 1 . Soient G un groupe reductif, defini sur Q, et X une classe de G(R)-conjugai­ son de morphismes de groupes algebriques reels de s dans GR . On suppose verifies les axiomes suivant (les notations sont celles de 1 . 1 . 1 et 1 . 1 . 1 1 ) : (2. 1 . 1 . 1) Pour h e X, Lie(GR) est de type {( - 1 , 1), (0, 0), ( 1 , - 1) } . (2. 1 . 1 .2) L'involution int h(i) est une involution de Cartan du groupe adjoint GRad · (2. 1 . 1 . 3) Le groupe adjoint n'admet pas de facteur G' defini sur Q sur lequel la projection de h soit triviale. L'axiome 2. 1 . 1 assure que le morphisme wh (h e X) est a valeurs dans le centre de G, done est independant de h. On le note Wx, ou simplement w. Quelques simpli­ fications apparaissent lorsqu'on suppose que : (2. 1 . 1 .4) Le morphisme w : Gm --+ GR est defini sur Q. (2. 1 . 1 . 5) int h(i) est une involution de Cartan du groupe (G/w(Gm)) R· D'apres 1 . 1 . 14(i), X admet une unique structure complexe telle que, pour toute representation V de GR , Ia filtration de Hodge Fh de V varie holomorphiquement avec h. Pour cette structure complexe, les composantes connexes de X sont des domaines hermitiens symetriques. La preuve de 1 . 1 . 17 montre aussi que si l'on decompose G'Ji en facteurs simples, h se projette trivialement sur les facteurs com­ pacts, et que chaque composante connexe de X est le produit d'espaces hermitiens symetriques correspondant aux facteurs non compacts. L'axiome 2. 1 . 1 . 3 peut en­ core s'exprimer en disant que G•d (resp. G, cela revient au meme) n'a pas de facteur G' {defini sur Q) tel que G'(R) soit compact, et le theoreme d'approximation forte assure que G(Q) est dense dans G(Af). 2. 1 .2. Les varietes de Shimura KMc (G, X)-ou simplement KMc-sont les quo­ tients K Mc(G, X) = G(Q)\X x (G(Af)/K) pour K un sous-groupe compact ouvert de G(Af). D'apres 1 .2.7, et avec les notations de 0.3, 1 'action de G(R) sur X fait de n-0 (X) un espace principal homogene sous G(R)/G(R)+ . Puisque G(Q) est dense dans G(R) (theoreme d'approximation reel), on a G(Q)/G(Q)+ ...::. G(R)/G(R)+ , et, si x+ est une composante connexe de X, on a K Mc(G, X) = G(Q) +\X+ x (G(Af)/K) .

Ce quotient est Ia somme disjointe, indexee par !'ensemble fini G(Q)+\G(Af)/K de doubles classes, des quotients Fg\X+ du domaine hermitien symetrique x+ par les images Fg C:: G•d(R)+ des sous-groupes r; = gKg- l n G(Q)+ de G(Q)+ . Les Fg sont des groupes arithmetiques, d'ou une structure d'espace analytique sur Fg\X+ .

266

PIERRE DELIGNE

L'article [2] fournit une structure naturelle de variete algebrique quasi-projective sur ces quotients, done sur KMe(G, X). Si rg est sans torsion (tel est le cas pour K assez petit) ; il resulte de [3] que cette structure est unique. Plus precisement, p our tout schema reduit Z, un morphisme analytique de Z dans Fg\ X+ est automatique­ ment algebrique. 2. 1 .3. On a

n-0 KMe = G(Q) \ n-0(X)

(G(Af)/K) = G(Q)\G(A)/G(R)+

K = G(Q)+\G(A!)fK. Puisque G(Af)/K est discret, on peut remplacer G(Q)+ par son adherence dans G(Af). La connexite de G(R) assure que pG(Q) c G(Q)+ . Par le the o reme d'approxi­ mation forte pour G, pG(Q) est dense dans pG(Af), et G(Q)+ ::::> pG(Af). Des lo rs, x

?ro KMe = G(Q)+pG(Af)\G(Af)/K = G(A!)fpG(Af) · G(Q)+ . K

(2. 1 .3.1)

x

= G(Af)/G(Q)+ . K,

puisque pG(Af) est u n sous-groupe distingue, avec u n quotient abelien. De meme, posant x 0n-(G) = n-0n-(G)/n-0 G(R)+ , on a ?ro K Me = G(A)/pG(A) = n-(G)/G(R)+

(2. 1 .3.2)

G(Q) · G(R)+ x K x K x0n-(G)/K. ·

=

En particulier, n-0 KMe ne depend que de l'image de K dans G(A)/pG(A). 2. 1 .4. Pour K variable (de plus en plus petit), les KMe forment un systeme pro­ jectif. II est muni d'une action a droite de G(Af) : un systeme d'isomorphismes g : KMe � g -!Kg Me. II est commode de considerer p1utot le schema Mc(G, X) -ou simplement Me-limite projective des KMe. La limite projective existe parce que les morphismes de transition sont finis. Ce schema est muni d'une action

a droite de G(Af), et il redonne les KMe : KMe = Me iK.

Nous nous proposons de determiner Me et sa decomposition en composantes connexes. DEFINITION 2. 1 .5. Fixons une composante connexe x+ de X. La composante neutre M8 de Me est Ia composante connexe qui contient /'image de x+ x {e} c X X G(Af). DEFINITION 2. 1 .6. Soient G0 un groupe adjoint sur Q, sans facteur G� defini sur Q tel que G�(R) soil compact et G1 un revetement de G0 . La topologie -r(G1) sur G0(Q) est cel/e admettant pour systeme fondamental de voisinages de /'origine /es images des sous-groupes de congruence de G1(Q). Nous noterons 11 (rei. G1), ou simplement 11, Ia completion pour cette topologie. Soit p : G0 --> G1 ]'application naturelle, notons - ]'adherence dans G1(Af), et posons r pG0(A) n G1(Q). Puisque G0(R) est connexe, r c G1(Q) + . On a

=

(2.0.9, 2.0. 14)

Go(Q)11 (rel. G1) = G1(Q) - *c1 cQl Go (Q) pGo(A!) *r Go( Q), Go (Q)+II(rel. G1) = GI (Q)+ *c1 CQl + Go(Q) + pGo (Af) *r Go( Q) + . PROPOSITION 2. 1 .7. La composante neutre M8 est Ia limite projective des quotients F\X+ , pour r un sous-groupe arithmetique de G•d(Q) + , ouvert pour Ia topo/ogie -r(Gd•r).

(2. 1 .6. 1 ) (2. 1 .6.2)

=

=

267

VARIETES DE SHIMURA

D'apn!s 2. 1 .2, c'est Ia limite des F\X 0 , pour F l'image d'un sous-groupe de con­ gruence de G(Q)+ · Le Corollaire 2.0. 1 3 permet de remplacer G par Gd•r. 2. 1 .8. La projection de G dans G•d induit un isomorphisme de X+ avec une classe de G(R)+-conjugaison de morphismes de S dans G'k_d et, d'apres 2. 1 .7, M8(G, X) ne depend que de G•d, Gd•r et de cette classe. Formalisons cette remarque. Soient G un groupe adjoint, x+ une classe de G(R)+-conjugaison de morphismes de S dans G(R) qui verifie (2. 1 . 1), (2. 1 .2), (2. 1 .3) et G1 un revetement de G. Les varietes con­ nexes de Shimura (rei. G, G1, X+) sont les quotients F\X+, pour F un sous-groupe arithmetique de G(Q)+ , ouvert pour Ia topologie -r(G1) . On note M8(G, Gh x+) leur limite projective, pour r de plus en plus petit. On notera que !'action par transport de structure de G(Q)+ sur M8(G, Gh x+) se prolonge par continuite en une action du complete G(Q)+A (rei. G1). Avec les notations de 2. 1 .7, et !'identification ci-dessus de x+ avec une classe de G(R)+-conjugaison de morphismes de S dans G'k_d , on a M8(G, X)

= M8(G•d, Gd•r, x+).

2. 1 .9. Soient Z le centre de G, et Z(Q)- !'adherence de Z(Q) dans Z(A f). D'apres Chevalley (2.0. 10), c'est le complete de Z(Q) pour Ia topologie des sous-groupes d'indice fini du groupe des unites ; il re<;oit isomorphiquement !'adherence de Z(Q) dans n:0Z(R) x Z(Af). Pour K c G(A f) compact ouvert, on a Z(Q) · K Z(Q)- · K (dans Z(A!)), et K Mc = G(Q)\X

=

G(Q) Z(Q)

=

x

(G(A!) fK) =

\x

x

CoROLLAIRE

x

=

G(Q) Z(Q)

\X

x

-

(G(A!) fZ(Q) ) .

2. 1 . 1 1 . Si les conditions 2. 1 .4 e t 2. 1 . 5 sont verifiees, o n a Mc(G, X)

G(Q)\X X G(A!).

Dans ce cas, Z(Q) est discret-dans Z(A!) et Z(Q)-

COROLLAIRE

(G(A!) fZ(Q) · K)

( G(A!)fZ(Q)-) est propre. Ceci permet le pas­

2. 1 . 10. On a Mc(G, X)

x

( G(At) f Z(Q)- . K) .

L'action de G(Q)/ Z(Q) sur X sage a Ia limite sur K: PROPOSITION

���j \X

2. 1 . 12. L'action

a droite de

=

= Z(Q) .

G(A!) se factorise par G(A!)JZ(Q)-.

=

2. 1 . 1 3 . Soient G•d(R)1 !'image de G(R) dans G•d (R), et G•d(Q)1 G•d(Q) n G•d(Rk L'action 2.0.2 de G•d sur G induit une action (a gauche) de G•d(Q)1 sur le systeme des K Mc i nt(r) : K Mc � rKr-1 Mc,

et a la limite sur Me. Pour r E G•d(Q)+, cette action stabilise la composante neutre (done toutes les composantes, cf. ci-apn!s) et y induit !'action 2. 1 .6. Convertissons cette action en une action a droite, notee · r · Si r est !'image de

268

PIERRE DELIGNE

o e G(Q), l'action · r coincide avec l'action de o , vu comme element de G(Af) : pour u E Me image de (x , g) E X X G(AI), u . r est image de (r- l (x) , int71 (g)) = (o-t(x) , o- lgo) ,..., (x , go) mod G(Q) a gauche.

Au total, nous obtenons ainsi une action a droite sur Me du groupe

(2. 1 . 1 3. 1)

G(A ! ) G•d(Q) 1 - G(Af) * G•d(Q)+ . Z(Q) - G (Q)+IZ(Q) Z(Q) - * G (Q)IZ(Q) _

PROPOSITION 2. 1 . 14. L'action a droite de G(Af) sur 7eoMefait de 7eoMc un espace principal homogene sous son quotient abelien G(A!)JG(Q)+ = 1r o7e( G). Cela resulte aussitot par passage a Ia limite des formules 2. 1 .3. 2. 1 . 1 5. Puisque G•d(Q) agit trivialement sur 1e(G) (2.0. 1 5), et que G•d(Q) + sta­ bilise au moins une composante connexe (2. 1 . 1 3), le groupe G•d(Q) + les stabilise toutes. Pour !'action 2. 1 . 1 3 du groupe (2. 1 . 1 3 . 1 ) sur Me. le stabilisateur de chaque composante connexe est done

(2. 1 . 1 5. 1)

��S��

* G (Q)+IZ(Q) G•d(Q)+ =

2 . o. 1 3

G•d(Q) +A(rel. Gd•r).

Resume 2. 1 . 16. Le groupe G(AI)jZ(Q)- * G CQ) IZ CQ) G"d(Q)1 agit a droite sur Me. L 'ensemble profini 7eoMc est un espace principal homogime sous taction du quotient abelien G(AfjG(Q)+ = 1r o7e(G) de ce groupe par /'adherence de G•d(Q)+ . Cette ad­ herence est le complete de G•d(Q)+ pour Ia topologie des images des sous-groupes de congruence de Gder(Q). L 'action de ce complete sur Ia composante neutre, une fois convertie en une action a gauche, est /'action 2. 1 .8. 2.2. Mode/es canoniques. 2.2. 1 . Soient G et X comme en 2. 1 . 1 . Pour h E X, le morphisme /-th (1 . 1 . 1 , com­ plete par 1 . 1 . 1 1) est un morphisme sur C de groupes algebriques definis sur Q : /-th : Gm � Gc. Le corps dual ( = reflex field) E(G, X) c C de (G, X) est le corps de definition de sa classe de conjugaison. Si x+ est une composante connexe de X, on le notera parfois E(G, X+). Soient (G', X') et (G", X") comme en 2. 1 . 1 . Si un morphisme f : G' � G" envoie X' dans X", on a E(G', X') => E(G", X"). 2.2.2. Soient T un tore, E un corps de nombres, et f-t un morphisme, defini sur E, de Gm dans TE. Le groupe E * , vu comme groupe algebrique sur Q, est Ia restric­ tion des scalaires a Ia Weil RE10(Gm). Appliquant .RE1o ·a f-t• on obtient RE1o(f-t) : E * � REIQ TE . On dispose aussi du morphisme norme NEto : RE1oTE � T (sur les points ra­ tionnels, c'est Ia norme, T(E) � T(Q)) . D'ou par composition un morphisme NE1o o RE1o(f-t) : E * � T. Nous le noterons simplement NRE(f-t), ou meme NR(p,). Si E' est une extension de E, f-t est encore defini sur E', et (2.2.2. 1)

2.2.3. Soient en particulier T un tore, h : S � TR et X = {h } . Si E c C contient E(T, X), le morphisme /-th est defini sur E, d'ou un morphisme NR(f-th) : E * � T. Passant aux points adeliques modulo les points rationnels, on en deduit un homo­ morphisme du groupe des classes d'ideles C(E) de E dans T(Q)/T(A), et, par pas­ sage aux ensembles de composantes connexes, un morphisme

VARIETES DE SHIMURA

269

noNR(p.h) : no C(E) -+ n0(T(Q)/T(A)). La theorie du corps de classe global identifie n0 C(E) au groupe de Galois rendu abelien de E. Le groupe n0 (T(Q)\ T(A)) est un groupe profini, limite projective des groupes finis T(Q)\T(A)/T(R)+ x K pour K compact ouvert dans T(Af). C'est n0T(R) x T(Af)/T(Q)-. Les varietes de Shimura KMc(T, X) sont les ensembles finis T(Q)\{h} x T(Af)/K = T(Q)\ T(A!)fK. Leur limite projective T(A!)fT(Q)-, calculee en 2. 1 . 10 est le quotient de n0(T(Q)\T(A)) par n0 T(R). Nous appellerons morphisme de reciprocite le morphisme rE(T, X) : Gal(Q/E) a b -+ T(Af)/T(Q)- inverse du compose de l'isomorphisme de la theorie du corps de classe global (0.8), de n0 NR(p.h) et de Ia projection de n0 T(A)/T(Q) sur T(A!)fT(Q)-. II definit une action rE de Gai(Q/E)ab sur les KMc(T, X) : Q ..... Ia translation a droite par re (T, X) (Q). Le cas universe! (en E) est celui oil E = E(T, X) : il resulte de (2.2.2. 1) que !'ac­ tion rE de Gal(Q/E) est Ia restriction a Gal(Q/E) c Gal(Q/E(T, X)) de rE c r . x l . 2.2.4. Soient G et X comme en 2. 1 . 1 . Un point h e X est dit special, ou de type CM, si h : S -+ G(R) se facto rise par un tore T c G defini sur Q, On notera que si T est un tel tore, !'involution de Cartan int h(i) est triviale sur !'image de T(R) dans le groupe adjoint, et que cette image est done compacte. Le corps E(T, {h}) ne de­ pend que de h. C'est le corps dual E(h) de h. No us transporterons cette terminologie aux points de K M0( G, X) et de Mc(G, X) : pour x e KMc(G, X) (resp. Mc(G, X)), classe de (h, g) e X x G(A!), Ia classe de G(Q)-conjugaison de h ne depend que de x. Nous dirons que x est special si h l'est, que E(h) est Ie corps dual E(x) de x, et que Ia classe de G(Q)-conjugaison de h est le type de x. Sur !'ensemble des points speciaux de KMc(G, X) (resp. de M0(G, X)) de type donne, correspondant a un corps dual E, nous allons definir une action r de Gai(Q/E). Soient done x e KMc(G, X) (resp. Mc(G, X)), classe de (h, g) e X x G(Af), T c G un tore defini sur Q par lequel se factorise h, Q e Gai(Q/E), et r(Q) un representant dans T(Af) de rE(T, {h})(Q) e T(Af)/T(Q)-. On pose r(Q)x = classe de (h, r(Q)g) . Le lecteur verifiera que cette classe ne depend que de X et de (1 . L'action ainsi definie commute a !'action a droite de G(A!) sur M0(G, X). 2.2. 5. Un modele canonique M(G, X) de M0(G, X) est une forme sur E(G, X) de M0(G, X), muni de !'action a droite de G(Af), telle que (a) les points speciaux sont algebriques ; (b) sur !'ensemble des points speciaux de type -. donne, correspondant a un corps dual E(-r), le groupe de Galois Gal(Q/E(-r)) c Gal(Q/E(G, X)) agit par !'action 2.2.4. Par "forme" nous entendons : un schema M sur E(G, X), muni d'une action a droite de G(Af), et d'un isomorphisme equivariant M ® E cc . x l C ...::::. M0(G, X). Soit E c C un corps de nombres qui contient E(G, X) . Un modele faiblement canonique de M0(G, X) sur E est une forme sur E de M0(G, X), muni de !'action a droite de G(At), qui verifie (a) et (b * ) meme condition que (b), avec Gal(Q/E(-r)) remplace par Gal(Q/E(-r)) n Gal(Q/E). 2.2.6. Dans [5, 5.4, 5. 5], nous inspirant de methodes de Shimura, nous avons

270

PIERRE DELIGNE

montre que Mc(G, X) admet au plus un modele faiblement canonique sur E (pour E(G, X) c E c C), et que, lorsqu'il existe, il est fonctoriel en (G, X). 2.3. Construction de modeles canoniques. Dans ce numero, nous determinons des cas oil s'applique le critere suivant, demontre dans [5, 4.2 1 , 5.7], pour construire des modeles canoniques. Critere 2.3. 1 . Soient (G, X) comme en 2. 1 . 1 , V un espace vectoriel rationnel, muni d'une forme alternee non degeneree l/f, et S± le double demi-espace de Siegel cor­ respondant (if 1 .3.1). S'i/ existe un p/ongement G 4 CSp( V), qui envoie X dans S±, alors Mc(G, X) admet un modele canonique M(G, X).

PROPOSITION 2.3.2. Soient (G, X) comme en 2. 1 . 1 , w = wh (h e xr et ( V, p) une representation fidele de type {( - 1 , 0), (0, - 1)} de G. Si int h(i) est une involution de Cartan de GRfw(Gm), il existe une forme alternee l/f sur V, te/le que p induise (G, X) 4 {CSp( V), S±) .

Par hypothese, Ia representation fidele V est homogene de poids 1 . Le poids w est done defini sur Q, et on prend pour l/f une forme de polarisation comme en 1 . 1 . 1 8(b). -

COROLLAIRE 2.3.3. Sojent (G, X) comme en 2. 1 . 1 , W = Wh (h E X), et ( V, p) une representationfidele de type {( - 1 , 0), (0, - 1)} de G. Si le centre zo de G se dep/oye sur un corps de type CM, il existe un sous-groupe G2 de G, de meme groupe derive et par lequel sefactorise X, et uneforme alternee l/f sur V, telle que p induise (G2, X) 4 {CSp( V), S±)) .

L'hypothese sur zo revient a dire que le plus grand sous-tore compact de ZB est defini sur Q. On prend G2 engendre par le groupe derive, ce tore, et l'image de w, et on applique 2.3.2. 2.3.4. Soit (G, X) comme en 2. 1 . 1 , avec G Q-simple adjoint. L'axiome (2. 1 . 1 .2) assure que GR est forme interieure de sa forme compacte. Exploitons ce fait. (a) Les composantes simples de GR sont absolument simples. Si on ecrit G comme obtenu par restriction des scalaires a Ia Weil : G = RF!0G• avec G• absolument simple sur F, cela signifie que F est totalement reel. Posons les notations : I = }'ensemble des plongements reels de F, et, pour v E I, G. = G•®F. v R, D. = dia­ gramme de Dynkin de Gvc · On a GR = TI G., Gc = IJ G.c, et le diagramme de Dynkin D de Gc est Ia somme disjointe des D.. Le groupe de Galois Gal(Q/Q) agit sur D et I, de facon compatible a la projection de D sur /. (b) La conjugaison complexe agit sur D par l'involution d'opposition. Celle-ci est centrale dans Aut(D). Des lors, Gal(Q/Q) agit sur D via une action fidele de Gal(Kn/Q), avec Kn totalement reel si l'involution d'opposition est triviale, quad­ ratique totalement imaginaire sur un corps totalement reel sinon. 2.3.5. On a X = IJ x., pour X. une classe de G.(R)-conjugaison de morphismes de S dans G• . Pour G. compact, x. est trivial. Pour G. noncompact, x. est decrit par un sommet s. du diagramme de Dynkin D. de G.c (1 .2.6). Quelques notations : Ic = I' ensemble des v E I tels que G. so it compact, Inc = I- I., De (resp. Dnc) = Ia reunion des D. pour v E Ic (resp. v E Inc), Gc (resp. Gnc) = le produit des G. pour v E Ic (resp. v E Inc) ; de meme pour les revetements univeG­ sels ; enfin, 2(X) = I' ensemble des s. pour v E Inc · La definition 2.2. 1 donne : ·

VARIET!�S

DE

SHIMURA

27 1

PROPOSITION 2.3.6. Le corps dual de (G, X) est le sous-corps de KD fixe par le sous-groupe de Gal(KD/Q) qui stabilise l,'(X).

2.3.7. Supposons qu'il existe un diagramme

(2.3.7. 1) Le revetement universe! G de G se releve dans Gh ce qui permet de restreindre Ia representation V a G. Le quotient de G qui agit fidelement est par hypothese le groupe derive de G1 . Appliquons 1 .3.2, 1 .3.8 au diagramme On trouve que les composantes irreductibles non triviales de Ia representation Vc de Gncc se factorisent par l'un des G.c (v e Inc), et que leur poids dominant est fondamental, de l'un des types permis par Ia Table 1 .3.9. L'ensemble des poids dominants des composantes irreductibles de Ia representation Vc de Gc est stable par Gal(Q/Q). Puisque Gal(Q/Q) agit transitivement sur I, et que Inc -:f. 0 , on trouve que (a) Toute composante irreductible W de Vc est de Ia forme ®vET w., avec w. une representation fondamentale de Gvc ( v e T c I), correspondant a un sommet 't"(v) de D •. Nous noterons 9'( V) !'ensemble des 't"( T) c D pour W c Vc irreducti­ ble. (b) Si S E 9'( V), S n Dnc est vide ou reduit a un seul point ss E Dv (v E Inc), et, dans Ia Table 1 .3.9 pour (D., s.), s8 est l'un des sommets soulignes. (c) 9' est stable par Gal(Q/Q). On a 9' rt. { 0 } . S i un ensemble de parties 9' de D verifie (b) et (c), nous noterons G(S")c le quotient de Gc qui agit fidelement dans Ia representation correspondante de Gc. La condition (c) assure qu'il est defini sur Q. Le cas le plus interessant est celui oil (d) 9' est forme de parties a un element. Si 9' verifie (b), (c), !'ensemble 9'' des {s} pour s E s E 9' verifie (b), (c), (d), et G(S"') domine G(S"). Dans Ia table ci-dessous-deduite de 1 . 3.9-nous donnons La liste des cas oil il existe 9' verifiant (b), (c). D'apres 1 .3. 10, ce ne peut etre le cas que si G est de l'un des types A, B, C, D, et ces types seront successivement passes en revue. L'ensemble 9' verifiant (b), (c), (d) maximal, et le groupe G(S") correspondant (il domine tous Ies G(S"), pour 9' verifiant (b), (c)). TABLE 2 . 3.8 . Types A, B, C. Le seul S" verifiant (b), (c), (d) est !'ensemble des {s}, pour s une extremite-correspondant a une racine courte pour les types B, C-d'un dia­ gramme D. (v E I). Le revetement G(S") est le revetement universe!. Type D1 (I � 5). Pour qu'il existe 9' verifiant (b), (c), il faut et il suffit que les (G., X.) (v e Inc) soient ou bien tous de type Df, ou bien tous de type Dfl. Dis­ tinguons ces cas : Sous-cas Df. Le 9' verifiant (b), (c), (d) maximal est !'ensemble des {s} pour s a l'extremite "droite" d'un D •. Le revetement G(S") est le revetement universe!. Sous-cas Df. L'unique 9' verifiant (b), (c), (d) est !'ensemble des {s} pour s

272

PIERRE DELIGNE

l'extremite "gauche" d'un D • . Le revetement G(9') de G est de la forme RngG*, pour G * le revetement double de G forme de S0(2l), cf. 1 .3 . 1 0. Type D4• Rempla�ons 9', verifiant (d), par S = {s l {s} e 9'} . La condition (b) sur 9' devient : S est contenu dans l'ensemble E des extremites de D, et S n I(X) = 0 . Pour Ia definition de I(X) voir 2.3.5. La partie de E stable par Gal(Q/Q) maximale pour cette propriete est Ie complement de Gal(Q/Q) · .2'( X). Bile rencontre chaque D. en 0, 1 ou 2 points. Dans Ie premier cas, il n'existe pas 9' verifiant (b), (c). Dans Ie second (resp. 3e), elle (resp. son complement) est l'image d'une section -r de X -+ I, invariante par Galois ; -r(J) est disjoint de (resp. contient) I( X). Ap pelant -r( v) le sommet "gauche" de D., on retrouve Ia situation de D1 (I ;;;;; 5) : Sous-cas D�. II existe une section -r de X -+ I avec -r(I) => .Z(X). Cette section est alors unique, et Ia situation est Ia meme qu'en Df, l � 5. Sous-cas Df. On donne une section -r de X -+ I avec -r(J) n .2'(X) = 0 . Si on n'est pas dans le cas D�. cette section est unique, et l'unique 9' verifiant (b), (c), (d) est l'ensemble des { s } pour s l'extremite "gauche" d'un D• . Le revetement G(9') de G est de Ia forme RngG•, pour G• un revetement double de G• qui se decrit en terme de -.. Pour Ia suite de ce travail, il nous sera commode de redefinir le cas Df comme excluant D�. Avec cette terminologie, il existe 9' verifiant (b), (c) si et seulement si (G, X) est de l'un des types A, B, C, DR, DH et, sauf pour Ie type DH, il existe 9' verifiant (b), (c), (d) tel que G(9') soit le revetement universel de G. 2.3.9. Nous aurons a considerer des extensions quadratiques totalement imagi­ naires K de F, munies d'un ensemble T de plongements complexes : un au-dessus de chaque plongement reel v E I• . Un tel T definit une structure de Hodge hT : S -+ K3 sur K (considere comme un espace vectoriel rationnel, et sur leque1 K * agit par multiplication) : si J est !'ensemble des plongements complexes de K, on a K ® C = Cl, et on definit hT en exigeant que le facteur d'indice ff E J soit de type ( 1 , 0) pour ff E T, (0, - 1) pour if E T, et (0, 0) si ff est au-dessus de Inc · Le resultat principal de ce numero est la ­

-

PROPOSITION 2.3. 10. Soit (G, X) comme en 2. 1 . 1 , avec G Q-simple adjtJint, et de l'un des types A, B, C, DR, DH. Pour toute extension quadratique totalement imagi­ naire K de F, munie de T comme en 2.3.9, il existe un diagramme

pour lequel (i) E(G1, X1) est le compose de E(G, X) et de E(K * , hT). (ii) Le groupe derive G� est simplement connexe pour G de type A, B, C, DR , et le revetement de G decrit en 2.3.8 pour le type DH.

Soit S le plus grand ensemble de sommets du diagramme de Dynkin D de Gc tel que { {s} 1 s E S} verifie 2.3.7(b), (c). Nous l'avons determine en 2.3.8. Le groupe de Galois Gal(Q/Q) agit sur S, et on peut identifier S a Hom(K5, C), pour Ks un produit convenable d'extension de Q, isomorphes a des sous-corps de Kv puisque Gal(Q/K0) agit trivialement sur D, done sur S. En particulier, Ks est un produit de corps totalement reels ou CM. A la projection S -+ I correspond un morphisme F -+ K5• Pour s E S, soit V(s) la representation complexe de Gc de poids dominant le

273

VARIETES DE SHIMURA

poids fondamental correspondant a s. La classe d'isomorphie de la representation EB V(s) est definie sur Q. Ceci ne suffit pas a ce q u 'on puisse la definir sur Q ; l' obstruc­ tion est dans un groupe de Brauer convenable. Toutefois, un multiple de cette representation peut toujours etre defini sur Q. Soit done V une representation de G sur Q, avec Vc EB V(s) n , pour n convenable. No us noterons V5 !'unique facteur de Vc isomorphe a V(s) n . Ces facteurs sont permutes par Gal(Q/Q) de fa<;on com­ patible a !'action de Gal(Q/Q) sur S, et la decomposition Vc EB v. correspond done a une structure de K8-module sur V: sur v• . Ks agit par multiplication par l'homomorphisme correspondant de Ks dans C. Notons G ' le quotient de G qui agit fidelement sur V. C'est le revetement de G considere en (ii). Soit h E X, et relevons h en un morphisme fractionnaire (1 .3.4) de S dans G�. On en deduit une structure de Hodge fractionnaire sur V, de poids 0. Soit s E S, et v son image dans I. Le type de la decomposition de v. est donne par la Table 1 .3.9 : (a) si v E I" V, est de type (0, 0) ; (b) si v E Inc• V, est de type {(r, - r), (r - 1 , 1 - r)} oil r est donne par 1 .3.9 : c'est le nombre qui affecte le sommet s de n., muni du sommet special qui definit "'

"'

x• .

On definit une structure de Hodge h2 de V, en gardant V, de type (0, 0) pour Ic, et, pour v E Inc• en renommant la partie de type (r, - r) (resp. (r - 1, 1 - r)) de V, comme etant de type (0, - 1) (resp. ( - I , 0)) . Si G2 est le sous-groupe algebri­ que de GL( V) engendre par G ' et Kf, la structure de Hodge h2 est un morphisme S --+ G2R . Notant X2 sa classe de G2(R)-conjugaison, on dispose de (G2, X2) --+ (G, X), et E(G2, X2) = E(G, X). Munissons K ® F V de la structure de Hodge h produit tensoriel de celle de V et de celle de K (2.3.9). On a (K ®F V ) ® R = EevEI(K ®F.v R) ® R ( V ®F. v R). Cette decomposition est compatible a la structure de Hodge, et sur le facteur cor­ respondant a v E Ic (resp, v E Inc), la structure de Hodge est le produit tensoriel d'une structure de type {( - 1 , 0), (0, - 1)} sur K ®F.v R (resp. V ®F.v R) par une de type {(0, 0)} sur V ®F.v R (resp. K ®F.v R). Au total, h3 est de type {( - I , 0), (0, - I )} . Si G3 est le sous-groupe algebrique de GL(K ®F V) engendre par K* et G2, la structure de Hodge h3 est un morphisme S --+ G3R . Si X3 est la classe de conjugaison de h3 , on dispose de (G3, X3) --+ (G, X). Le groupe derive de G3 est G ', et E(G3, X3) est le compose de E(G2, X2) = E(G, X) et de E(K*, hT) · Pour obtenir (G 1 o X1 ) cherche, il ne reste plus qu'a appliquer 2.3.3 a ( G3, Xg) et a sa representation lineaire fidele K ® F v. REMARQUE 2. 3. I l . La construction donnee se generalise pour fournir un dia­ gramme (2.3.7. 1) oil 9'( V) est n'importe quel ensemble de parties de D verifiant 2.3.7(b), (c) . En gros : (a) si 9' verifie 2.3.7(b), (c), on definit Ky par Hom{Ky, C) = 9', on construit une representation V de G telle que 9'( V) = 9', et Ia decomposition Vc = EB Vs (S E 9') fournit sur V une structure de Ky-module ; (b) la structure de Hodge fractionnaire de Vs est de type (0, 0) pour S au-dessus de Ic, de type {(r, - r), (r - I, 1 - r)} avec r decrit---c omme ci-dessus-par le point de S au-dessus de Inc sinon ; (c) on convertit {(r, - r), (r - 1 , I - r)} en {(O, - 1), ( - I , O)} comme plus haut ; vE

274

PIERRE DELIGNE

(d) pour convertir le (0, 0) en {( - 1 , 0), (0, - 1 )}, on tensorise V, sur Kfl', avec K'g, de type CM muni d'une structure de Hodge convenable h. Par cette methode, on obtient pour (G�o X1) un groupe derive G(S"), et un corps dual compose de E(G, X) et E(K:S.* , h). Noter que, meme pour .9' verifiant (b), (c), (d), Ia conversion indiquee de (0 , 0) est plus generate que celle de 2.3. 10. REMARQUE 2.3. 1 2. Pour les types A, avec l,'(X) fixe par l'involution d'opposition, B, C et D R, le corps dual E( G, X) est le sous-corps (totalement reel) de Kv fixe par le sous-groupe de Gal(Kv/Q) qui stabilise Ic. Si Ic = 0 , c'est Q. Si Ic (resp. Inc) est reduit a un element v, c'est v(F) . Les E(K, hr) sont des extensions de E(G, X). REMARQUE 2.3. 1 3 . Pour ces types, et DH, les V, de 2.3. 1 0, pour v -e Inc• sont de type {( - t , !) , ( t , - !) } . Ceci permet, dans 2.3.10, de remplacer G2 par le sous­ groupe de GL( V) engendre par F* et G' . Si lc = 0 , on peut meme le remplacer par le sous-groupe de GL( V) engendre par Q * et G ', et le critere 2.3.2 s'applique direc­ tement a ce groupe, d'ou (Gt. X1 ) avec E(Gt. X1 ) = E(G, X) ( = Q hors le cas DB). 2.4. Lois de reciprocite : preliminaires. Les constructions de ce numero nous permettrons, au numero 2.6, de calculer Ia loi de reciprocite des modeles canoniques, i.e. l'action du groupe de Galois sur l'ensemble des composantes connexes geometriques. Bien qu'elles s'exprini.ent mieux dans le langage de la descente fppf, nous les avons exprimees dans celui de Ia descente galoisienne, le croyant plus familier aux non-geometres. Ceci expose a quelques redites et inconsequences, et introduit des hypotheses parasites de separabilite ou de caracteristique 0. Soit G un groupe reductif sur un corps global k. Avec la notation de 2.0. 1 5, notre but est de construire des morphismes canoniques des deux types suivant. a. Pour k' une extension finie (qu'on supposera separable) de k, et G' deduit de G par extension des scalaires a k', un morphisme norme

(2.4.0. 1) b. Pour T un tore, et M une classe de conjugaison, definie sur k, de morphisms de T dans G, un morphisme

(2.4.0.2)

qM

:

""(T) --+ ""(G).

Si m e M(k), qM sera Ie morphisme qm induit par m ; la difficulte est de montrer que ce morphisme ne depend pas du choix de m, et de construire qM meme si M n'a pas de representant defini sur k. Les proprietes de fonctorialite de ces morphismes seront evidentes sur leur definition. 2.4. 1 . Nous utiliserons systematiquement le langage des torseurs (que je prefere a celui des cocycles), et celui de la descente galoisienne, sous la forme que lui a donnee Grothendieck (cf. SGA 1, ou SGA 4112[Arcata]). Descente galoisienne : Soit K une extension separable finie d'un corps k. Pour construire un objet X sur k (par exemple un torseur), il suffit de construire (a) pour toute extension separable k' de k, telle qu'il existe un morphisme de k-algebre de K dans k', un objet Xk' sur k' ; (b) pour k" une extension de k', un isomorphisme 'Xk", k' : xk' ® k" � xk, ; et de verifier (c) une compatibilite 'Xk"k"'Xk"k' = 'Xk'"k' · En pratique, cela signifie que pour construire X, on peut .supposer }'existence d'objets auxiliaires qui n'existent que sur une extension separable K de k-a charge

VARIETES DE SHIMURA

275

de montrer Ie X construit ne depend pas-a isomorphisme unique pres-du choix d'un tel objet auxiliaire. REMARQUE 1 . La descente galoisienne est un cas particulier de Ia localisation en topologie etale ; une construction comme en (a), (b), (c) ci-dessus sera souvent introduite par l'adverbe "localement". EXEMPLE 2.4.2. Expliquons le relevement canonique utilise en 2.0.2 de I' applica­ tion commutateur. L'usage de Ia descente galoisienne-plutot que fppf-nous oblige a supposer que Ia projection de G sur G•d est lisse, et a ne considerer que ( , ) : G•d(k) x G•d(k) --+ G(k), plutot que le morphisme G•d x G•d --+ G. Si x�o x2 E G•d(k), on peut, localement, ecrire X; p(x;)z; avec z; dans le centre de G. L'element X; est unique, a multiplication par un element du centre de G pres. Le commutateur de x1 et x2 ne depend pas de l'arbitraire dans Ie choix des x;, et on pose (x� o x2) x 1 x2x11 x21 . 2.4.3. Pour G un groupe algebrique sur un corps k, un G-torseur est un schema P sur k, muni d'une action a droite de G qui en fasse un espace principal homo gene. Le G-torseur trivial Gd est G muni de l'action de G par translations a droite. On identifie les points x e P(k) aux trivialisations de P (isomorphismes cp : Gd � P) par cp(g) xg. Si f : G1 --+ G2 est un morphisme, et P un Grtorseur, il existe un G2-torseurf(P) muni de f : P --+ f(P) verifiant f(pg) f(p)f(g), et il est unique a isomorphisme unique pres. Nous nous interesserons a Ia categorie [G1 --+ G2] des G1-torseurs P munis d'une trivialisation de f(P). Pour morphismes, on prend les isomorphismes de Grtorseurs, compatibles a Ia G2-trivialisation. On note H0(Gl --+ G2) le groupe des automorphismes de (G1d, e) (c'est Ker(G1(k) --+ G2(k))) et H1 (G1 --+ G2) l'en­ semble (pointe par (G1d, e)) des classes d'isomorphie d'objets. Chaque x e G2(k) definit un objet [x] de [G1 --+ G2] : le G1-t'orseur trivial G1d, muni de Ia trivialisation x de f(G1d) G2d. Quand cela ne prete pas a confusion, nous le noterons simplement x. L'ensemble des morphismes de [x] dans [y] s'iden­ tifie a {g e G1(k) if(g)x y} : a g, associer u --+ gu : G1d --+ G l d· Un objet est de Ia forme [x] si et seulement si, en tant que G1-torseur, il est trivial-d'ou une suite exacte

=

=

=

=

=

(2.4.3.1)

1

=

H0(G1 -----+ G2) -----+ G t (k) -----+ G2(k) 1 1 1 -----+ H (G1 -----+ G2) -----+ H (G t ) -----+ H (G2) -----+

(ceci ne decrit pas l'image inverse de p e Hl(G1) ; pour Ia decrire, il faut proceder par torsion, comme dans [13]). 2.4.4. Si f est un epimorphisme, de noyau K, il revient au meme de se donner le G1-torseur P G2-trivialise par x e f(P)(k), ou le K-torseur J- 1 (x) c P : le foncteur nature) [K --+ {e}] --+ [G1 --+ G2] est une equivalence. Plus generalement, si g : G2 --+ H induit un epimorphisme de G1 sur H, et que K; = Ker(G; --+ H), le foncteur nature) est une equivalence [K1 --+ K2] --+ [G1 --+ G2] . 2.4. 5. Si G est commutatif, Ia somme s : G x G --+ G est un morphisme, et on ·definit Ia somme de deux G-torseurs par P + Q s(P x Q). Si G1 et G2 sont com­ mutatifs, on additionne de meme les objets de [G1 --+ G2], qui devient une categorie .de Picard (strictement commutative) (SGA 4, XVIII, 1 .4).

=

276

PIERRE DELIGNE

Tout ce qui precede vaut pour des faisceaux en groupes sur un topos quelconque. 2.4.6. Si k' est une extension finie de k (le cas oil k' fk est separable no us suffit) et G' un groupe algebrique sur k', le foncteur de restriction des scalaires a la Weil Rk'lk est une equivalence de la categorie des G'-torseurs avec celle des Rk'lkG'­ torseurs. Ceci correspond au lemme de Shapiro H 1 (k', G') Hl(k, Rk'lkG). Si G' se deduit par extension des scalaires de G-commutatif-sur k, on dispose d'un morphisme trace Rk'lkG' --+ G-d'ou un foncteur trace Trk'lk des G'-torseurs dans les G-torseurs. Plus generalement, pour G 1 --+ G2 un morphisme de groupes com­ mutatifs on trouve un foncteur additif

=

(2.4.6. 1 )

De tels foncteurs sont decrits avec une grande generalite dans [SGA 4 , XVII, 6.3]. Pour k'/k separable, on peut donner une definition simple par descente : locale­ ment, k' est somme [k' : k] copies de k, [G� --+ G2] s'identifie a la categorie des [k' : k] uples d'objets de [ G 1 --+ G2], et Trk'lk a la somme. Quand les groupes sont notes multiplicativement, on parlera plutot de foncteur norme Nk ' lk· 2.4.7. Soient G un groupe reductif (0.2) sur k, et p : G --+ G le revetement uni­ versel de son groupe derive. Le cas particulier de 2.4.3 qui nous importe est {G --+ G]. Soient G•d le groupe adjoint de G, Z le centre de G, et Z celui de G. Le morphisme G --+ G•d est un epimorphisme, d'ou une equivalence (2.4.4) (2.4.7. 1 )

[Z ----. Z] ----. [G ----. G].

Puisque Z et Z sont commutatifs, [Z --+ Z] est une categorie de Picard strictement commutative (2.4.5). Utilisant !'equivalence (2.4.7.1), on fait de [G --+ G] aussi une telle categorie. Nous allons calculer [xd + [x2] dans [G --+ G], et les donnees d'as­ sociativite et de commutativite. On suppose p : G --+ Gd•r separable pour pouvoir proceder par descente galoisienne. Ecrivant (localement) x,- p(g,.)z,- , avec z,­ central, on a des isomorphismes g,- : [z,-] --+ [x,-] et g1 g2 : [z 1 z2] --+ [x 1 x2], d'ou un isomorphisme

=

(2.4.7.2) Si on change de decomposition : x,.

= p (g;) ;., avec = g;.u,. (u,- E Z) , le diagramme z

g,-

est commutatif. L'isomorphisme (2.4.7.2). (2.4.7.3) est done independant des choix faits. Le lecteur verifiera facilement que, via cet

VARIETES DE SHIMURA

277

isomorphisme, Ia donnee d'associativite est deduite de l'associativite du produit, et que Ia donnee de commutativite est (x 1 o x2) : [x2x 1 ] --+ [x1 x2]. II verifiera aussi que si Y; = p(g,)x; (i = 1 , 2), Ia somme des g,. : [x;] --+ [y;] est

g l + gz = g l int.,1 (gz) : [x l xz] --+ [Y l Yz] , oil int designe I' action de G sur G (definie par transport de structure, ou via l'action 2.0.2 de G•d = (}•d). La categorie [G --+ G] etant de Picard, !'ensemble Hl(G --+ G) des classes d'iso­ morphie d'objets est un groupe abelien. Les formules ci-dessus montrent que l'injection G(k)/p(G(k)) --+ H 1 (G --+ G) est un homomorphisme. D'apn!s (2.4.3. 1), c'est un isomorphisme si H 1 (G) = 0. Soient k' une extension finie de k, et notons par un ' l'extension des scalaires a k'. L'equivalence (2.4.7. 1) permet de deduire de 2.4.6. 1 un foncteur trace (que nous baptiserons norme)

(2.4.7.4)

PROPOSITION 2.4.8. Si k est un corps local ou global, le morphisme deduit de (2.4.7.4) par passage a /'ensemble des classes d'isomorphie d'objets induit un mor­ phisme de G(k')fpG"(k') dans G(k)fpG(k) :

G(k)fpG(k)

G(k')/pG(k')

(2.4.8.1)

r

r

Hl(G'---+ G') � Hl (G ---+ G).

Si H l(G) = 0, Ia fleche verticale droite est un isomorphisme, et l'assertion est evidente. Cette nullite vaut pour k local non archimedien. Pour k local archimedien, le seul cas interessant est k = R, k' = C, et le diagramme commutatif Z(C) --+> H 1 (Gc---+ Gc)

lNc!R

(2.4.8.1)

Z(R)

1

Hl(G ---+ G)

montre que est encore defini-a valeurs dans l'image de Z(R) (et meme de sa composante neutre). Pour k global, et x E G(k')/p(G(k')), l'image de Nk'tix) E Hl(G --+ G) dans H 1 (G) est done localement nulle. D'apres le principe de Hasse, elle est nulle. Nous n'utili­ sons ici le principe de Hasse que pour les classes de cohomologie dans l'image de H l (Z), de sorte que les facteurs £8 ne creent aucun trouble. L'image Nk'tix) est done dans G(k)fpG(k), comme promis. 2.4.9. Pour k local non archimedien d'anneau des entiers V, k' non rami:fie sur k, d'anneau des entiers V', et G reductif sur V, le morphisme 2.4.8 induit un morphisme de G( V')/pG( V) : on le voit en repetant les arguments qui precedent sur V, Ia des­ cente galoisienne etant remplacee par Ia localisation etale (ici, formellement identi­ que a une descente galoisienne sur le corps residuel). On peut done adeliser 2.4.8 : pour k global, le produit restreint des morphismes 2.4.8, pour les completes de k, est un morphisme Nk ' tk : G(A')jpG(A') ---+ G(A) /pG(A).

278

PIERRE DELIGNE

Divisant par le rnorphisme trace global, on obtient enfin le rnorphisme (2.4.0. 1) Nk'tk : 11: (G ') ---> 11:(G).

De merne que Ia construction du morphisme (2.4.0. 1 ) repose sur celle du foncteur (2.4. 7 . I), celle de (2.4.0.2) reposera sur Ia Construction 2.4. 10. Soient G reductif connexe sur k, p : G � G le revetement

universe/ du groupe derive, T un tore sur k, et M une classe de conjugaison, definie sur k, de morphisme de T dans G . On definira unfoncteur additif qM : [{e} ----> T] ---> [G ----> G]. Nous donnerons de Ia construction deux variantes. I ere methode. Localement, il existe m dans M. Posons X(m) = Z m( T) c G et Y(m) = p- 1 X(m) = Z · {p- 1 m(T))O. Les groupes X(m) et Y(m), extensions de tores par un sous-groupe central de type multiplicatif, soot commutatifs. Ils donnent lieu a un diagramme z___, Y(m) ·

1

1

z___,X(m) <----- T Si g dans G conjugue m en m', il conjugue (l)m en (l)m' (on le fait agir sur G par l'action de (Jad = Gad) . De plus, l'isomorphisme int(g) de (l)m avec ( l )m' ne depend pas du choix de g: si g centralise m, il centralise Z, m(T), Z, ainsi que (p- 1 (T)) O (un tore isogene a un sous-tore de T), done X(m) et Y(m). Deux m dans M etant localement conjugues, ceci permet d'identifier entre eux les diagrammes (1 )m, et d'en deduire un diagramme unique z___, y (I) Z _____, X ,_____ T

1

1

defini sur k. On definit qM cornme etant le foncteur compose ---->

T] ----> [ Y ----> X] . : 4 [Z ----> Z] __::_, [G ----> G]. 2 2eme methode. On suppose p : G � Gde r separable, pour proceder par descente galoisienne. Les objets de [{e} � T] n'ayant pas d'autornorphismes, un foncteur de [{e} --+ T] dans [G � G] est simplement une loi qui a t e T(k) assigne un G­ torseur G-trivialise qM([t]). Procedons par descente galoisienne. Localement il existe m e M(k). Soit qm le foncteur [t] � [m(t)]. Nous allons definir un systeme transitif d'isomorphismes entre les qm. Ceci fait, nous pourrons definir qM comme etant l'un quelconque des qm. Pour definir tm' , m : qm ...::::. qm'• on choisit g tel que m' = gmg- 1 (nouvelle applica­ tion de Ia methode de descente) et on pose tm' , m([t]) : [m( t )] � [m'( t )] est (g, m(t)) e G(k). On a bien m'(t) = (p((g, m( t)))m (t ) , et il reste a verifier que (g, m( t )) est independant de g. L'espace C des g qui conjuguent m en m' est connexe et reduit, en tant que tor­ seur sous le centralisateur du tore m(T). La fonction (g, m(t)) de C dans G a une qM : [{ e }

279

VARIETES DE SHIMURA

projection p((m, m(t))) = m'(t)m(t)- 1 dans G constante. La fibre de crete, elle est constante. La construction est recapitulee dans le diagramme commutatif (2.4. 10. 1)

=

(qm([t])

[m(t)], m'

=

p

etant dis­

gmg- 1 ).

Construisons Ia donnee d'additivite de qM. C'est Ia donnee, pour chaque tl> t2 e T(k), d'un isomorphisme qM([t1 tz]) --+ qM([td) + qM([tz]), ces isomorphismes etant compatibles aux donnees d'associativite et de commutativite pour Ia somme dans [G --+ G]. On les definit par descente : pour m e M, et m' = gmg- 1 , le diagramme suivant est commutatif

(fleches obliques 2.4. 10. 1), et definit l'isomorphisme cherche independamment de m. Sa commutativite exprime une certaine identite, dans G , entre commutateurs (2.4.2), et Ia projection de cette identite dans G resulte de ce que les fleches ecrites ont un sens. Pour Ia prouver, on note que localement (descente) c'est Ia projection d'une identite analogue pour G x Z0-de projection vraie dans G x zo, done vraie dans G . La compatibilite a l'associativite et a Ia commutativite se voit par descente, en fixant m ; on utilise que le commutateur 2.4.2 est trivial sur m(T). 2.4. 1 1 . Complements. (i) La construction 2.4. 10 est compatible a I' extension des scalaires : notant par un ' l'extension des scalaires de k a une extension k' de k, on definit de fa<;on evidente un isomorphisme de foncteurs additifs rendant commutatif le diagramme [{e} � T] � [G � G]

1

1

[{e} � T'] � [G' � G']

(ii) On peut repeter Ia construction 2.4. 10 sur une base quelconque ; Ia descente galoisienne est a remplacer par Ia localisation etale. (iii) La construction 2.4. 10 est compatible aux foncteurs normes. Pour definir l'isomorphisme de foncteurs additifs rendant commutatif le diagramme [{e}

�

1

T'] ____!_!i__, [G ' � G']

12.

4. 7. 4

[{e} � T] � [G � G]

(notations de (i), avec k'/k fini separable), et verifier ses proprietes, le plus simple

280

PIERRE DELIGNE

est de proceder par descente : localement, k' devient une somme k1 de copies de k, -+ G'] devient [G' -+ G]I, Ia norme (trace) devient Ia somme, et tout est trivial. 2.4. 1 2. Pour k local non archimedien, Hl(G) = 0 et, passant aux ensembles de classes d'isomorphie d'objets, on deduit de 2.4. 10 un morphisme qM : T(k) -+ G(k) /pG(k). Pour G reductif sur l'anneau des entiers V de k, T un tore sur V et M sur V, il induit un morphisme de T(V) dans G( V)/pG( V) (2.4. 1 l(ii)) . Pour k archimedien, on peut rencontrer une obstruction dans Hl(G), mais elle disparait pour x dans Ia composante neutre (topologique) T(k)+ de T(k) : par 2.4. 1 1(ii), elle depend continument de x, et est nulle pour x = e. On a done encore un morphisme T(k)+ -+ G(k)fpG(k) . Pour k global, prenant le produit restreint de ces morphismes pour les completes de k, on trouve qM : T(A)+ -> G(A )/pG(A) . (2.4. 12. 1) Si T(k)+ = { x e T(k) I pour v reel, x est dans T(kv)+ } , le principe de Hasse, utilise comme en 2.4.8, fournit qM : T(k) + -> G(k)/pG(k) . · (2.4. 1 2.2) Puisque T(A )+jT(k) + ...:::::. T(A)/ T(k) (theoreme d'approximation reel pour les tores), on obtient finalement par passage au quotient le morphisme (2.4.0.2) promis : [G'

qM : 7t: ( T) -+ 7t: (G). 2.5. Application : une extension canonique.

2.5. 1 . Soit G un groupe reductif sur Q. On suppose que G n'a pas de facteur G' ( defini sur Q) tel que G'(R) soit compact. Le theoreme d'approximation forte assure des lors que G(Q) est dense dans G(Af). Le cas qui nous importe est celui d'un groupe comme en 2. 1 . 1 . Reprenons le calcul de 2. 1 .3. Pour K compact ouvert dans G(AI), (2.5. 1 . 1)

11:0 ( G(Q)\ G(A)/K) = G(Q) \11:0 ( G(R)) x ( G(AI)/K) = G(Q) \ G(A)/ G(R)+ x K = G(Q)+\ G(Af) /K

et on peut remplacer G(Q) (resp. G(Q)+) par son adherence dans G(AI). Celle-ci contient pG(Af), un sous-groupe distingue a quotient abelien de G(Af), et, avec les notations de 2.0. 1 5, (2.5. 1 .2) 7t:o ( G(Q)\ G(A)/K) = 11:(G)jG(R)+ x K = 7t:o7t: (G)/K = G(Af) / G(Q)+ K. Passant a Ia limite sur K, on en deduit que ·

11:0 ( G(Q)\ G(A)) = 7t:o7t: (G) = G(Af) / G(Q)+ -.

Soit n011: (G) le quotient G(Af) / G(Q)+ de n011:(G) par 11:0G(R) + (0 . 3 ). La suite 0 -> G(Q)+/Z(Q) - -> G(A f) /Z(Q) - -> n011:(G) -> 0 (2.5. 1 .3) est exacte. L'action de G•d sur G induit une action de G•d(Q ) sur cette suite exacte. L'existence du commutateur (2.0.2) montre que l'action de G•d(A) sur G(A)fpG(A) est triviale-a fortiori celle de G"d(Q) sur n01r(G). L'application du sous-groupe G(Q) / Z(Q) de G(Q)+/Z(Q)- dans G•d(Q)+ verifie les conditions de 2.0. 1 . Le groupe + G(Q)+/ Z(Q)- * G (Ol +tz