Collected Papers: Volume 2 (1951-1964)

(t~x-~)ts is defined

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211 ON ALGEBRAIC GROUPS OF TRA~qSFOR)/[ATIONS.

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at s~; this means that 2-1 is defined and has a representative on G~. Similarly, if we write t ~ t ~ t ~ -1, the representative of 29 on G-t is t ~ x y ~ ((tx~)t~-l)y~; let ~, ~ be two points of G with representatives r~, s~ on G~, G~ respectively; by Prop. 4 of the Appendix, we can choose 7 so that t.~ is generic on V over l~(r~,s~,t~,t~); the same will then be true of t, and also of tr~ and of (tr~)t~ -1 since V is a group-chunk; for a similar reason, this implies that ( x , y ) - - > ((tx)t~-l)y is defined at (r~,s~), and this completes the proof that G is a group. Now, going back to the space S constructed before, we transfer to G, S, by means of the birationa] correspondences 4)o, r the normal law given for V, W; in other words, for x, u generic and independent over K on V, W, and for 2 ~ o ( X ) , ~t~(u), we define ~ ( x u ) , and prove that this makes S into a transformation-space with respect to G. I n fact, the representative of 2~ on S~ is ((tx~)t~-~)u~, where t ~ t.~t~-~ as before; the rest of the proof is then quite similar to the proof given above. Naturally, if W is non-singular, S is non-singular; if W is everywhere normal, S is everywhere normal. Finally, if W is a homogeneous chunk, S is a homogeneous space. I n fact, in that case, let ~, ~ be any two points of S, with representatives as, b~ in S~, S~ respectively. Take x generic over K ( ~ , ~ ) on V; put 2'~2Ct, ~ " ~ 2 ~ . For u generic over K(x) on W, we have ~ I , ( 2 ~ ) ~ (xt~-!)u~; as W is a homogeneous chunk, x ' ~ (xt~-Z)a~ is defined and generic over K(~t, ~) on W, and therefore we have x ' ~ ( 2 ' ) ; similarly we have x" ~ ~I,(2") with x" ~ (xt~ -~) b~ generic over K(~,, 5) on W. That being so, there is an isomorphism of K(~t, b,x p) onto K(d, D, x") over K ( g , 5) which maps x' onto x " ; this can be extended to an isomorphism ~ of K((t, b, ~) onto some extension of K(~, b,x"). Then we have ~ ( ~ 2 " ~ b , and so 5 ~ ~-~2~. 7. From now on, it will be assumed that W and consequently S are everywhere normal. With this assumption, we shall construct an abstract variety S t, defined over k, and a birational correspondence F between S' and W, also defined over ~, so that the birational correspondence r o F, defined over K. between S' and S is an everywhere biregular mapping of S' onto S. This construction can then be applied to V itself, giving a variety G' and a birational correspondence Fo between G' and V, both defined over /~, such that r o Fo is biregular between G' and G. Transferring the normal laws for V, W to G', S" by means of F, Fo, we see that we have thus constructed a group G ' and a transformation-space S', birationally equivalent to V, W over k ; if W is pre-homogeneous and we have constructed S as a homogeneous space, S' will be a homogeneous space.

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I n constructing S', we may assmne that V operates faithfully on W; in fact, if this were not so, one could replace V by another pre-group ? satisfying this condition, according to Prop. 2 of w I, no. 3. Notations will now be the same as in no. 6, with the additional assumptions that W and consequently S are everywhere normal, and that V acts faithfully on W, so that G acts faithfully on S. Let /C' be any field containing /C. Let ~2 (s~) be a cycle of dimension 0 4=1

on V, rational over /C', and assume that s~ =2~s~ whenever i J : j . Then, if we put /C"=/c'(sl,. ' ' , s~), /C'" is a Galois extension of/C', i.e. separably algebraic and normal over/C'. Call K " the compositum of K and/C"; let u be a generic point of W over K " , and put w~ ~ s#. I f m is the dimension of the ambient affine space to W, we write w ~ = (W~l, 9 ',w~m). P u t now

~=i

/~=I

where T, U1,'" ', U m are indeterminates; let y be the point, in an afflne space of suitable dimension, whose coordinates are all the coefficients of the homogeneous polynomial y(T,U) except that of Tr; this is the so-called " C h o w point" of the cycle ~ (w~), and y(T, U) is its " C h o w form." As V acts faithfully on W, and the s~ have been assumed to be distinct, the eorollary of Prop. 5, w II, no. 5, shows that the w~ are all distinct. W e can therefore apply to them the following~ general result: LEM~A. I f in (1) we ta/ce the w~ to be any set of distinct points, and ko is the prime field, then the w~ are separably algebraic over ~co(y). By F-I.~, Th. 1, we need only show that a derivation D of the field /co(W~,-" ",wr) over /co(y) must be trivial. I n fact, applying D to (1), we get : r

as the w~ are all distinct, this cannot be an identity in T, U~,- 9 ", U,, unless all the Dwi, are 0. PROPOSITION 7. Notations being as defined above, we have/c'(y) ~ / c ' ( u ) provided the s~ are all distinct and satisfy the following condition: (S) The set of points ~ = ~ o ( s ~ ) any right-translation.

on G is not mapped onto itself by

The cycle ~ (wi) is the image of the cycle ~ (s~) by the mapping

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213 ON A L G E B R A I C GROUPS OF T R A N S F O R ) s

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x-,xu

of V into W ; it is therefore rational over U(u). By the main theorem on symmetric functions (VA, no. 7, Th. 1), this implies that y is rational over U ( u ) , i.e. that U ( y ) C U(u). On the other hand, the lemma shows that the w, are separably algebraic over U'(y); as we have u=s,-lw, by Prop. 5 of w I I , no. 5, u is therefore separably algebraic over U'(y), hence also over U ( y ) . Let (r be any automorphism over U ( y ) of the algebraic closure of U ( y ) ; as it induces an isomorphism of U(u) onto k ' ( u ~) over U, u ~ is generic on W over k', so that s~u~ is defined by Prop. 5. This gives (s,u)r ~, i.e. w~r162 ~. But the decomposition of the homogeneous polynomial y(T, U) into linear factors is uniquely determined; applying to (1), we see thus that the w,~ must be the same as the w, except for a permutation, i.e. that there is a permutation i--->r such that w ( = w o ( o . This can be written as s,r162 sr as the s.f are the same as the s, except for a permutation, we can write them as sinks,(o, where i--->r(i) is a permutation. Then we have r ~) =r which can be written as ,~(~)q)(u~) =2~(,)~, i.e. (~(u ~) ~ ( ~ ) - ~ g ~ ( o ~ . As G acts faithfully on S, the corollary of Prop. 5 shows that all the elements ~(~)-~2~(~) of G, for 1 ~ i ~ _ r, must coincide; if ~ is their common value, we have 2~(o ~ ( ~ ) T , which shows that the right-translation [ maps the set 2~ onto itself. By (S), this implies that [ is the neutral element of G, so that (~ (u ~) ~ ~, and therefore u ~ u . As u is separably algebraic over U(y), this shows that

~'(u) c k'(y). 8. Proposition 7 shows that we may write y ~ f ( u ) , where f is a birational correspondence, defined over ~,, between W and the locus Y of y over /cp in affine space. I f k'[y] is the ring generated over U by the coordinates of y, it is well-known that the integral closure of k'[y] in ~'(y) is a finitely generated ring over U, i.e. that it can be written as U [ y * ] , where y* is a point in a suitable a:ifine space; call Y* the locus of y* over /d in that Mfine space. As we have /c'(y*) ~ U ( y ) ~ U ( u ) , we may write y * ~ f * ( u ) , f* being a birational correspondence between W and Y*, defined over U. I t is usual to say that Y* is derived from Y by " n o r m a l i z a t i o n " over ~'. By Prop. 14 of the Appendix, since U ' is separably algebraic over k', U'[y*] is integrally closed in Jc"(y*). PROPOSITION 8. With the notations explained above, y* and ~t are: corresponding generic points over K" on Y* and S in a birational correspondence between Y* and S which maps Y* biregularly onto the {("-open set i

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In the first place, we prove that the coordinates wiz of the w, are all in U ' [ y * ] ; as they are in U ' ( y ) because Qf the relations w,=s~u and k'(u) = U ( y ) , it will be enough to show that they are integral over the ring k"[y], or in other words (e.g. by F-App. II, Prop. 6) that they are everywhere finite on W. I n fact, let ~r be any place of k"(y) such that y0r) is finite. Take r independent variables A1," " ",% over ]d'(y), and extend ~r to a place ~r' of U'(y, hl," 9 ",s at which every one of the r points (Xi, M w , , ' ' ",Mw,,~) is finite and =/=(0,.- . , 0 ) . The relation (1), by which y was defined, can be written

~,~ 9 ~ v (T, U) = II (~,T-- X (~,~,~) U~). i:1

#

Taking the values of both sides at ~', we see that the right-hand side does not become identically 0 at that place; as y(~) is finite, this implies that no ~ can become 0 at v ' ; but then wi~(~r) can be written as (~w~)0r')/X~(v') and is finite. This proves the assertion about the w~. We have thus shown that the mappings y*--)w~ of Y* into W are everywhere defined on Y*; as we have ~ t ~ A - ~ ( w ~ ) , this implies that y*--->~ is everywhere defined and maps Y* into the set ~ defined in Prop. 8. Conversely, the definition of y can be written

y(T, U) = H ( T - - X,I,,,(~,a) U~,) {=I

/z

if we call $~(~) the coordinates of ~(~). As 9 is everywhere defined on 9 (W), this shows that the mapping ~-->y is defined at every point of the set ~. As U[y*] is the integral closure of lc'[y] in U(y), it is therefore contained in the integral closure of the specialization-ring of every point of ~ on 8. But we have assumed that W and consequently S are normal, i. e. that the specialization-ring of every point of S (over any field of definition for 8) is integrally closed. This proves that ~--->y* is everywhere defined on the set ~. In view of what we have proved above, ~t is therefore the set of points of S where this mapping is defined, and is K"-open by Prop. 8 of the Appendix; more precisely, it is K'-open if K ' is the compositum of K and It'. This completes the proof. 9.

Denote now by S any cycle ~ (s~) on V, rational over the ground-

field to, consisting of distinct points st and satisfying condition (S). From such a set S, and taking U ~ k, we can derive as above a point y, which we now write as ys, and furthermore a point ys* such that k [ys*] is the integral

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215 ON A L G E B R A I C GROUPS OF T R A N S F O R ~ s

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closure of /c[ys] in /c(ys) ; as above, we call Ys* the locus of ys* over /C; we write gts for the open subset of S denoted by ~ in Prop. 8. I f we allow S to run through any finite set of cycles with the properties stated above, then all the varieties Ys* will be birational]y equivalent to W and to each other, and we can take the points ys* to be corresponding generic points of these varieties over /C. It is then an immediate consequence of Prop. 8 that the affine varieties Ys* (with empty " f r o n t i e r s " ) , and the birational correspondences between them for which the ys* are corresponding generic points of the Ys* over/C, determine an abstract variety S', and that this is biregularly equivalent over a suitable field (as a matter of fact, over K itself) with the union of the open sets ~s on 8. In order to prove that S' will be biregularly equivalent to S itself for a suitable choice of the cycles S, it is therefore enough, in view of the well-known " c o m p a c t o i d " property of open sets in the Zariski topology, to show that the family of all open sets ~s is a covering of 8. I n other words, we have to prove the following: PRoPosITIo~r 9.

Given any point ~t on S, there is a cycle S ~ ~, (s~)

on V, rational over/C, consisting of distinct points s~ and satisfying condition (S), and such that 2 i a e ~ ( W ) for all i. Assume that ~ has a representative as on S~; take x generic over K ( ~ ) ~ K ( a ~ ) on V, and put u ~ (xt~-~)a~, this being defined because W is a chunk. If we put, as usual, ~ o ( x ) and ( ~ a ~ ( u ) , we have then ~2g, so that u ~ , ~ ( 2 ~ ) . As the mapping x--->2a is everywhere defined on 11, this shows that the mapping x--->u of V into W is defined at the points s of V such that 2aE(P(W), and at those points only. Let F be the closed subset of V where the mapping x--->u is not defined; by Prop. 12 of the Appendix, there is a maximal ~-c]osed subset Fo of V contained in F ; then an algebraic point of V over /C is in F if and only if it is in Fo. Call F~ the union of the conjugates over /C of all the components of Fo; this is a /c-closed set on V, and its definition shows that the cycle S on V will satisfy the last one of the conditions stated in Prop. 9 if and only if it lies in V - - F , Now assume first that the field /C is infinite. Applying Prop. 13 of the Appendix to the variety V ~ ~ V - - F , and to the empty subset of V' X V', we obtain a separably algebraic point Sl over /C on V'; call s ~ , . . . , s~ all the distinct conjugates of s~ over /C; if this set satisfies condition (S), which will be the case in particular if d ~ 1, then it solves our problem. Suppose that this is not so, and therefore that d ~ l . For any r ~ d , let sa+~," " ",Sr be any set of r - - d points on V - - F ~ , distinct from one another and from s~," 9 .,sa; put S ' ~ { g ~ , ' 9 ",,~a} and S"~{2a+~," 9 ",~}. I f the set 12

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S ' U S " is mapped into itself by a right-translation r other than the identity, one of the following circumstances must occur: (i) r maps each one of the sets S', S " onto itself; then r is of the form s':lt ", with s', t" in S', and there must be two elements s", t" of S" such that t " ~ s " r ; (it) r maps S' into S " ; as d ~ 1, we can choose two distinct elements s', t' in S', and then s " ~ s%, t " ~ t ' r are in S", so that we have t " ~ (t's'-~)s"; (iii) r maps some s'~S' onto some # ' ~ S" and some t" e S' ofito some tlP~ S'; then s " ~ s't'-Itl ". Thus, in order to satisfy the requirements of Prop. 9, it is enough to take as sa+l," 9 ", sr the conjugates over k of a point s ~ sa+l of V - - F ~ , separably algebraic over/c, satisfying the following conditions : (a) no 3z, for d -~ 1 <-- l <-- r, coincides with any of the points .~ or 3jf13~ for 1 ~ i,j, h ~ d; (b) no pair of distinct conjugates of s over k lies on the graph of any of the birational correspondences x--->~(22~-~j), x-->~(&gj-~2) for l<=i,j<__d. As to (a), it will be satisfied provided we take s on V - - F ~ , where F2 is the union of F~, of the set Sl," 9 9 sa, and of the set of all conjugates over k of those algebraic points on V whose image on G coincides with one of the points .~jj-~a. Then our result follows at once by applying Prop. 13 of the Appendix to the variety V - - F 2 and to the union of the graphs of the birational correspondences in (b). I f ~ is finite, we have to proceed differently. Take any algebraic point s~ over /~ on V - - F ~ ; call s~,. 9 .,s~ its distinct conjugates over /~; if this set satisfies condition (S), it solves our problem. I f not, we use a result of Lang-Weil (this Jovl~N~tT,, vol. 76 (1954), p. 819) which says that, if 1 is sufficiently large, there must be a point s on V - - F 1 which is rational over the (unique) extension of k of degree 1. We take 1 prime and ~ d. I f s is rational over k, the cycle (s) solves our problem; if not, it is of degree 1 over ~; call sa+~,. --,sa+z its distinct conjugates over k; they are distinct from sl,. 9 sa, since the latter are of degree d over k. The set sa+~," 9 ", sa+z may solve our problem. I f it does not, the group g of right-translations mapping the set (2~+1,' 9 ',2d+~} onto itself is of order v ~ 1; as that set must be the union of cosets with respect to ~, v must divide l, and so g is cyclic of order l; call T a generator of g. Let r' be a right-translation mapping onto itself the set (g~,- 9 .,2~+~}. I f r' is not the identity and maps some element of the set {2a+1," 9 ',2a+~) into an element of the same set, it must be of the form r i, and therefore of order l; but this cannot be, since d ~ 1 is not a multiple of I. Therefore r' must map the set (2a+~,- 9 ",3a+;} into the set ( g , ' ' ' , 2 a } . As d ~ l , this is also impossible. Therefore the set s~,...,sa+~ solves our problem. This completes the proof of the results announced at the beginning of

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no. 7. Writing now G, S instead of G', S', we may restate them in a somewhat more complete form as follows: THEOrEm. (i) To every pre-group V, defined over a field k, there is a birationally equivalent group G, also defined over lc ; this is uniquely determined up to an isomorphism. (ii) To every pre-homogeneous space W with respect to V, defined over k, there is a birationally equivalent homogeneous space with respect to G, also defined over k; this is uniquely determined up to an isomorphism. (Hi) Let W be a pre-transformation space with respect to V, defined over ]c; let a be a point of W such that W is normal at a and that, if x is generic over ~(a) on V, xa and x-l(xa) are defined. Then there is a transformation-space S with respect to G, birationally equivalent to W over tr in such a way that the birational correspondence between them is biregular at a; S may be taken everywhere normal, and it may be taken to be non-singular if a is simple on W. Moreover, S is uniquely determined up to a birational correspondence which is biregular at every point of the form ,~, where ~ is the point corresponding to a on S and ~ is any point of G. Except for the statements about unicity, all this has been proved above. As to unicity, the statements in (i) and (ii) are special cases of the statement in ( i i i ) ; and the latter is an immediate consequence of the fact that the operations o~ G are everywhere biregular mappings of S onto S.

Appendix. I f X is any cycle, we denote by I X I the support of X , i.e. the closed set which is the set-theoretic union of the components of X. PROPOSITION 1. Let ~(x) be a regular extension of a field k, and/c(x,y) a regular extension of k ( x ) . Then ~(x,y) is a regular extension of k. This is an immediate consequence of F-IT, Th. 5. PROPOSITmN 2. Let ~(x) be a regular extension of a field k; let K be an overfield of ~, linearly disjoint from ~(x) over Ic; let 7c" be the algebraic closure of k in K. Then ~'(x) is the algebraic closure of k(x) in K ( x ) . Let y be an element of K ( x ) , algebraic over /c(x) ; we may take x to be

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a generic point over K of a variety V, defined over k, in an affine space; and then we m a y write y ~ F ( x ) , where F is a function on V, defined over K ; call F the g r a p h of F. As y is algebraic over k ( x ) , there is a polynomial P ~ k [ X , Y ] such that P ( x , Y ) ~=0 and P ( x , y ) - - 0 ; then P induces on the product V X D of V and of the afflne space D of dimension 1 a function which is not 0 on Y X D and is 0 on F. As I ~ has the same dimension as V, it m u s t be a component of the divisor ( P ) of P, and is therefore algebraic over /c. The smallest field of definition of 1~ containing k must ~hen be contained in U, so t h a t F is defined over U ; this implies t h a t y is in k ' ( x ) . C0~OL~A~Y.

I f K is primary over k, K ( x ) is primary over k ( x ) .

I n fact, the assumption means that U is pure]y inseparable over ~; this implies t h a t U ( x ) is purely inseparable over ~ ( x ) . PROPOSITION 3. Let ~(x) be a finitely generated extension of a field ~; then every field K such that ]c C K C l~(x) is finitely generated over k. Let t ~ ( t l , ' ' ' , t , ) be a m a x i m a l set of algebraically independent elements of K over /~; then K is algebraic over k ( t ) . Replacing k by k ( t ) , we see t h a t it is enough to prove our proposition in the case when K is algebraic over k. This being assumed, call /c' the smallest :field of definition containing k for the locus of x over the algebraic closure ~, of /~; then /c' is a finite algebraic extension of k and is algebraically closed in lc'(x) since k' (z) is regular over U. But then /d is the algebraic closure of k in /c'(x) and therefore contains the algebraic closure of k in /c(x), so t h a t K is contained in /c'. C01~0LLAI~Y. I f k ( x ) is regular over lc, so is K. PI~OI'OSlTIOX 4. Let t be a point, k a field, and let t l , " " ", tN be N independent generic specializations of t over lc. Let x be a point of dimension d < N over lc and such that to(x), k ( t ) are linearly disjoint over lc. Then there is an ~ such that t= is a generic specialization of t over l~(x). Call n the dimension of /~(t) over /~. By F-I6, Th. 3, every t= is a specialization of t over k ( x ) ; if none is generic, every t= m u s t have over /~(,) a dimension ~ n - - 1 ; but then ( x , t ~ , ' . . , t . ~ ) has over /~ a dimension <=d+N(n--1) < N n , which is impossibl% since (t~," 9 ",tN) has the dimension N n over lc. PROPOSITIONr 5. Let V be a variety, defined over a field ~; let K be an overfield of ]c and x a point of V. Let A and A' be the prime rational cycles,

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219 ON A L G E B R A I C GROUPS OF

TRANSFOR]~ATIONS.

377

over ~ and over K respectively, with the generic point x. Then A is the same as A' if and only if K and ~(x) are linearly disjoint over lc. We m a y replace V by any representative of V on which x has a representative, so that it is enough to prove our result for cycles in the affme n-space. F o r A to be the same as A', it is at any rate necessary t h a t they should have the same dimension, so t h a t K and k ( x ) m u s t be independent over /~; assume from now on t h a t this is so. A m o n g the coordinates of x, let (xl," 9 ", x~) be a m a x i m a l set of independent variables over /~ and therefore also over K ; write y for the point (xl,. 9 -,x~) and z for (X~+x,- 9 . , x ~ ) . By F - V I I 6 , Th. 12, A ~ A " if and only if A . ( y X S ~-~) is the same as A ' - ( y X S~-r) ; by F-VI3, Th. 12, this is so if and only if z has the same complete set of conjugates over K ( y ) as over /c(y), and therefore, by F-I4, Prop. 12 and F-I~, Prop. 6, if and only if K ( y ) and k ( y , z ) ~ T c ( x ) are linearly disjoint over k ( y ) . The l a t t e r condition means t h a t there is no relation ~ u ( ~ ( y ) ~ 0 in which the u~ are linearly independent elements of ]~(x) over /c(y) and the 4p~(y) are in K [ y ] and not all 0. Assume t h a t there is such a r e l a t i o n ; we may write O ~ ( y ) ~ ~,P~J(Y)~s, where the ~j are J

linearly independent elements of K over ~ and the P~s(Y) are in /c[y] and not all 0. Then we have F . v ~ 0 with v ~ F . u ~ P . ~ i ( y ) ; as the vj are in J

k ( x ) and not all 0 because of the assumptions on the u~ and P~s(Y), this shows that, when t h a t is so, K and /c(x) are not linearly disjoint over k. Conversely, assume t h a t there is a relation F. v~j ~ 0 in which the ~j are J

linearly independent elements of K over /~ and the v3. are in k ( x ) and not all 0; as the ~i are then also linearly independent elements of K ( y ) over k ( y ) , this implies t h a t K ( y ) and /c(x) are not linearly disjoint over 7c(y). C0nOLLA~Y. Let V be a variety, deft~ned over a field lc. Let A be a prime rational cycle on V over an overfield K of It. Then, if K' is any field such that tc C K' C K over which A is rational, A i~s prime rational over K' ; of all such fields K', there ~ one smallest one Ko; and an automorphism z of K over ~ transforms A into itself if and only if it induces the identity 0~

K o.

As in the proof of Prop. 5, it is enough to consider cycles in an affme space. Assume t h a t A is prime rational over K and rational over K ' C K, and write it as A ~ ~ n~A~, where the Ai are distinct prime rational cycles over K ' . L e t Z be a component of A1; it is algebraic over K', and so every conjugate of Z over K is a fortiori such over K% so t h a t every component of

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A is a component of A1 ; therefore we must have A ~ nlA1,. By F-Is, Prop. 26, the coefficient of Z in A is at most equal to its coefficient in A1 ; therefore we have A ~ A 1 . That being so, it follows from Prop. 5 and from F-I6, Th. 3 and F-IT, Lemma 2, that there is a smallest field Ko with the properties stated in our corollary; in fact, if x is a generic point of A over K, and if is the prime ideal in K [ X ] consisting of all polynomials in K [ X ] which are 0 at x, Ko is the smallest subfield of K such that ~ has a set of generators in Ko[X]. As ~ is also the ideal in K [ X ] whose set of zeros is the support I A ] of A, the last assertion follows from F-IT, Lemma 2. PROPOSITION 6. Let V be a variety, defined over a field k, and A a cycle on V ; assume either that A is a divisor on V or that the coefficients in A of all the components of A are ~ 0 modp, p being the characteristic. Then, of all the overfields of lc over which A is rational, there is one smallest one ko, k o is finitely generated over k; and an isomorphism a of leo over k onto some extension of lc leaves A invariant if and only if it leaves every element of ko invariant. Except for the last statement, this result is due to Chow. Let A be any cycle on Vi for every representative V, of V, call A, the sum of the terms in the reduced expression for A which pertain to components with representatives in V. ; then A is rational over an overfield K of k if and only if every A. is rational over K ; and an isomorphism of K which leaves A invariant must leave all the As invariant. Therefore it is enough to deal with cycles on an affine variety V. For such a cycle A, put A ~ ~ nA,~, where A, is the sum of the terms with the coefficient n in the reduced expression for A; then A is rational if and only if every cycle n A , is rational; and an isomorphism which leaves A invariant must leave all the A, invariant. Finally, if n = p~n' with n' prime to p, nA is rational if and only if p~A is rational. Therefore it will be enough to deal with the following two cases: (i) A is a cycl~ in affine space, consisting of a sum of distinct components; (it) A is a divisor on an affine variety V and of the form A ~ qAo, where q is a power of p and Ao is a sum of distinct components. (i) Let ~ be the ideal of all polynomials (with coefficients in the universal domain) which are 0 on the support ] A I of A ; this is the intersection of the prime ideals determined similarly by the components of A. The first assertion in our proposition will then be a consequence of F-IT, Lemma 2, if we prove that A is rational over a field K if and only if ~ has a set of generators in K [ X ] , i.e. if it is the extension to the universal domain

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221 ON A L G E B R A I C GROUPS OF T R A N S F O R I ~ A T I O N S .

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of the ideal 0i N K [ X ] ; the second assertion in our proposition also follows from the same lemma, provided one observes that, if /Co is the smallest field such that 9~ has a set of generators in /Co[X], an isomorphism which leaves A invariant must map /Co onto /Co, i.e. it must induce an automorphism in /%, so that the lemma in question is applicable. I f 9~ has a set of generators (P~) in K[X], the support I AI of A is the set of zeros of the P~ and is therefore K-closed. On the other hand, I A ] must also be K-closed if A is rational over K. In order to prove the equivalence of those two properties, one may then begin by assuming that [A I is K-closed. Consider first the case in which all the components of A are the conjugates of one of them, say Z, over K ; let x be a generic point of Z over ~ ; then A is rational over K if and only if K(x) is separable over K. Put K ' ~ K p-~, this being the smallest "perfect" field containing K. P u t : ~=~nK[X],

~'=~

n K'[X],

and call ~ ' the extension of ~3 to K'[X]. By F-IV2, Th. 4, and F-II1, Prop. 3, !13 and ~ ' consist of the polynomials, in K[X] and in KP[X] respectively, which are 0 at x; they are prime ideals; moreover, if P ' ~ r , some power prn of P ' is in ~3' and hence in ~ ' ; as ~ ' c ~ ' , this implies that ~ ' is primary and belongs to the prime ideal !13'. By F-Is, Th. 3, and F-I~, Prop. 19, we see that ~ 3 ' = ~ ' if and only if K(x) is separable over K, and therefore, as we have shown, if and only if A is rational over K. But, if 9~ is the extension of ~ to the universal domain, ~r must a f o r t i o r i be the extension of !13 to K'[X]. Conversely, if ~ ' = ~ ' , the extension of ~ to the universal domain is the same as that of ~ ' ; but it is well-known and easily verified that the latter must be a "radical" ideal, i.e. one consisting of all the polynomials which are 0 on a closed set; then one sees at once that it must be the same as 9~. This completes the proof in the special case we were considering. Now assume that I A 1 is any K-closed set; then we can write A as the sum of cycles A~ such that the components of each A~ are mutually conjugate over K, and ~ is the intersection of the ideals 9~ similarly determined b~y the A~. P u t : !13,= 9i, fq K [X],

~; = 9~,A g'[x],

and call ~ ' the extension of ~ to K'[X]. I f A is rational over K, all the A~ must be so, so that, as shown above, the 9I~ must be the extensions of the !13~ to the universal domain. I t is then easily seen that 9/ is the extension of the intersection of the ~ , i.e. of 9~ fq K[X]. Assume, on the other hand, that A is not rational over K ; then we have ~ ' - - ~ { for at least one i; from

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the unicity of the decomposition of an ideal into an intersection of primary ideals, it follows then that the intersection of the ~,', which is the extension of ~f n K [ X ] to K ' [ X ] , cannot be the same as the intersection of the ~ ( , which is 91 N K' I X ] . A f o r t i o r i , 9d cannot then be the extension of 91 N K [X] to the universal domain. This completes the proof for ease (i). (it) Let V be a variety, defined over It, in an aMne space; let Ao be a divisor on V and the sum of distinct components; let q be a power of the characteristic p 5/= 0; put A ~ qAo. I f P is any polynomial which is not 0 on V, denote by ( P ) v the divisor of the function induced by P on V. Call 91 the ideal of all the polynomials P, with coefficients in the universal domain, such that either P = 0 on V or ( P ) v F A . I f A is rational over an overfield K of /c, 91 is then the extension of 91 N K IX] to the universal domain, as follows at once from F - V I I I , , Th. 10. Conversely, assume that 91 is the extension of 91 N K [ X ] to the universal domain; we will prove that A is then rational over K ; our proposition will then follow from this as in ease (i). As a polynomial P is 0 on ]A ] if and only if some power P~ of P is in 91, our assumption on A implies that A is K-closed, and therefore that Ao is rational over K' = Kp -~. Let Z be a component of A. As well-known, there is a polynomial P such that ( P ) v = A o + B, where B has no component in common with Ao ; write P as P = Y. ~,P~, where the ~, are linearly indeper~dent over K' and the P, are in K ' [ X ] ; by F-VIII~, Th. 10, we have (Pi)v } A o for all i; and Z must have the coefficient 1 in at least one of the P,, since otherwise it would occur in B ; if we call that polynomial P', P' is then in K ' [ X ] , Z has the coefficient 1 in (P')v, and we have (P')v }Ao. But then P'q is in 91, and therefore, by hypothesis, may be written as Y~,j.Q~, where ihe y Oj are in 91 N K [ X ] . The latter fact implies that Z has at least the coefficient q in all the ( @ ) v ; as it has the coefficient q in P'q, it must have the eoeMeient q in one at least of the divisors ( Q j ) v ; as these divisors are rational over K, this implies that, if A~ is the sum of Z and its conjugates over K, qA~ is rational over K. As this is so for every component Z of A, A is therefore rational over K. PROPOSITION 7. Let U, V be two varieties, defined over a field k; let F be a l~-closed subset of U X V. Then the set A of the points a on U such that a )< V c F is k-closed. Let W~,. 9 -, Wm be those components of F which have the "projection" V on V (in the sense of F-IV3, F - V I I 3 ) ; if v is a generic point of V over /~, W~ has a generic point over ~ of the form (u~,v) ; and a e A if and only if,

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for v' generic over k(a) on V, (a, v') is a specialization of some (u~, v) over k. Let V~ be any representative of the abstract variety V; let vl be the representative of v on V1; the ambient affine space for F~ being embedded in a projective space, let Vo be the locus of v~ over k in that projective space. Let Fo be the union of the loci of the points (u~,vl) over ~ in U X Vo; Fo is k-closed on U X V o . Then A is the set of the points a on U such that Fo A (a X Vo) has a component of dimension ~ dim (Vo). As Vo is complete, our conclusion is now contained in Lemma 7 of my paper in Math. Ann., vol. 128 (1954), p. 104. PRoPosITIO~ 8. Let ~ be a mapping of a variety U into a variety V; let k be a field of definition for U, V and ~. Then the set of points of U where ~ is defined is k-open. (i) Assume first that U is an affine variety and V is the affine space of dimension 1. Let x be a generic point of U over k; put y ~ ( x ) . Let be the set of all polynomials P in /c[X] such that P ( x ) y is in k[x] ; this is an ideal in k [ X ] , containing the ideal ~ of those polynomials which are 0 at x and therefore on ~/. Since y may be written as Q ( x ) / P ( x ) , with P, Q in k [ X ] and P ( x ) ~ = 0 , we have ~[:/=~. As the points where ~ is not defined are the zeros of ~, the set of such points is k-closed. (ii) Take F as in (i), and assume that U is an abstract variety, with the representatives U~, on each of which a " f r o n t i e r " F~ (i. e. a k-closed set) is given, according to the definitions in F-VIIi. Call F~' the k-closed subset of U~ where ~ is not defined; the set F of the points of U where ~ is not defined is then the union of the images of the sets F~" N (U~--F~) by the canonical birational mappings of the U~ into U. I t is easily seen that F must be k-closed provided the following assertion is true: if x is a point of U with a representative x~ on some U~ which is a generic point over ~ of a component of F~', then every specialization zJ of x over k is in F. I n fact, let fl be such that x' has a representative x f on U~ ; then x must also have a representative x~ on U~, and, from the biregularity of the correspondence between U~, U~ at (x~,x~), it follows that x~ must be in F f ; as x f is a specialization of x~ over k, and as F f is ~-closed, x~' must then be in F~', so that x' is in F. This proves our result for this case. It follows trivially from this that our result remains true when U is an abstract variety and V is an affine space or more generally an affine variety. (iii) Let U be an abstract variety and let V be a k-open subset of an affine variety V~ ; let Vo be the projective variety whose part " a t finite dis-

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tance" is V1; then V o - - V is a L-closed subset Fo of Vo. Call r the graph of ~ on U X V o ; the set F of the points of U where ~, considered as a mapping of U into V, is not defined, is then the union of the set F1 of the points of U where r is not defined as a mapping of U into V1 and of the set-theoretic projection of r N (U X Fo) on U. As Vo is complete, the latter set coincides with the "projection" in the sense of F-IV3 and F-VII~ and is k-closed (e.g. by F-VIIi, Prop. 10 and 11); and F1 is to-closed, as shown in (it). Therefore F is k-closed. (iv) Let U, V be arbitrary abstract varieties; let x be a generic point of U over k; let the Vs be those representatives of V on which ~b(x) has a representative r and let Fs be the "frontiers" on the Vs. Then 4) is defined at a point of U if there is an ~ such that ~bs, considered as a mapping of U into V s - - F s , is defined there. Therefore the set where r is not defined is the intersection of the sets where the Cs are not defined; as the latter sets are k-closed by (iii), this completes the proof. COROLLARY 1. Let V be an abstract variety, defined over k, with the representatives Vs. Then, for each ~, the set ~s of the points of V which have a representative on Vs is k-open; and the canonical correspondence between V and Vs is an everywhere biregular mapping of ~ts onto V s - - F a if Fa is the frontier for V~. Let x be a generic point of V over k, and let xs be its representative on Vs; if we put xs ~ Ca(x), r is the "canonical correspondence" between V and Vs. Then ~s is the set of points where Ca, considered as a mapping of V into V s - - F ~ , is defined; it is k-open by Prop. 8. The rest is obvious. COROLLARY 2. Let V be an abstract variety, defined over a field k. Then there is a finite covering of V by k-open subsets of V, each of which is biregularly equivalent to an a~ne variety. Corollary 1 says that V has a covering by the k-open sets ~ , each of which is biregularly equivalent to the k-open subset V a - - F s of the aitine variety Vs. I t is therefore enough to prove our assertion for a k-open subset V - - F of an affine variety V defined over k. Let ~[ be the set of all polynomials in k[X] which are 0 on F ; it is an ideal in k[X], and, as F is /~-closed, it is the set of zeros of ~[. Let P~," 9 ",P,~ be a set of generators for ~ ; as F :/: V, they are not all 0 on V, and we may assume that P~,. 9 P~ are not 0 on V while P ~ + ~ , ' ' ' , P ~ are 0 on V, with l ~ r ~ m . For 1 "< p ~ r, call ~p the k-open subset of V consisting of the points where Pp is not 0 ; t h e ~ a r e a c o v e r i n g o f V---F. L e t x ~ ( x l , ' - .,x~) b e a g e n e r i c

[1955a]

225 ON ALGEBRAIC GROUPS OF TRANSFORIVIATIONS.

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point of V over k; let Vp be the locus of the point (z,.

9

z,, 1/Pp ( x , . 9 -, xn) )

in the affine space of dimension n ~-1.

Then Vp is biregularly equivalent

to tip. COROLLARY 3. Let V be a variety, defined over a field k. The set of points of V where V is normal (resp. relatively normal with respect to k) is a k-open subset of V. Let V* be the variety derived from V by normalization with reference to the smallest perfect field k ' ~ kp-~ containing k (resp. with reference to k ) ; let x be a generic point of V over k ; let x* be the corresponding point of V*, which is generic over k' (resp. over k) on V*. We may then write x * ~ @(x), where @ is a mapping of V into V*, defined over k' (resp. over k), Then the points where V is normal (resp. relatively normal) are those where is defined. As any k'-open set is also k-open, this proves the corollary. PROPOSITIO~ 9. Let V be a variety, defined over a field k; let F be a closed subset of V. For F to be k-closed, it is necessary that it should contain all the specializations over k of all its points; it is sufficient that it should contain all the generic specializations over k of all its points, or also that it should be invariant under all isomorphisms over k of a common field of definition K D k for its components. The necessity of the first condition follows from F-IVz, Th. 4; we first prove that this condition is sufficient. I n fact, it implies that, if z is a generic point over K of a component Z of F, the locus Z" of z over ~ is contained in F ; as Z is the locus of z over/~, Z' contains Z ; as z cannot be in any other component of F than Z, we get Z ' ~ Z ; thus all components of F are algebraic over k, and then F-IVz, Th. 4, shows that all the conjugates of Z over k must be contained in F. Now we show that the second condition implies the first one. Let x be any point of F and let x' be a specialization of x over k. Then if Y is the locus of x over ~, F-IV2, Th. 4, shows that x' must be on a conjugate V' of V over k. Let x 'p be a generic point of V' over /~; then x~' is a generic specialization of x over k by F-IVz, Th. 4, and is therefore in F by hypothesis, and x' is a specialization of x" over /7" and a f o r t i o r i over K and so is in F since F is K-closed. Finally the last condition implies the second one; for let x' be a generic specialization over k of a point x in a component Z of F ; then the isomorphism of k(x) onto k ( x ' ) over k which maps x onto x' can be extended to an isomorphism a of K ( x )

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onto a field Kr and then x' is on Z r by F-IV2, Th. 3, Coroll. 2, and is therefore in F if F is invariant under ~.

Let W be a subvariety of a product U X V, with the "'projection'" U on U; let k be a field of definition for U, V, W. Then the set-theoretic projection of W on U contains an open subset of U; and the union of all such sets is k-open. PROPOSITION 10.

The assumption means that, if (u,v) is a generic point of W over k, u is generic over k on U. Let V1 be a representative of V on which v has a representative v l ; F~ being the corresponding frontier, p u t V I ' = V~--F1, so that vl is in V~'; let W1 be the locus of (u, vl) over k on U X VI'. Let Vo be the projective variety whose p a r t " a t finite d i s t a n c e " is V~; p u t F o = V 0 - - V I ' ; fhis is a k-closed set on Vo. The set WI (1 (u X VI') can be writteIl as u X X, where X is either V~r (in the trivial case W = U X V) or else a k ( u ) - c l o s e d subset of V~'; as v~ is in X, X is not empty, so t h a t we can choose in it a point w which is algebraic over k ( u ) . Let W r be the locus of (u,w) over ~ on U X Vo, which has the same dimension as U ; call n t h a t dimension. Then the set C = W ' A (U X Fo) is a ~-closed subset of W', so t h a t all its components are of dimension ~ n - - 1 . As Vo is complete, the set-theoretic projection C' of C on U is then a ~-elosed subset of U. Let a be any point in U - - C ' ; as Vo is complete, there is a point ( a , b ) on W' with the projection a on U ; as ~ is not in C', b cannot be iI1 Fo and is therefore in V~', so that (a,b) is in WI. Therefore the ~-open set U - - C ' on U is contained in the set-theoretic pr6jection of W~ and a f o r t i o r i in that of W. The last assertion in our proposition is then an immediate consequence of the sufficiency of the last condition in Prop. 9. PROPOSITION 11. Let U, V, W be three varieties and U X V into W, all defined over a field k. Assume that, f is defined at (a,x) for x generic on V over k(a). Let those a e U such that, for x generic over k(a) orv V, f ( a , x ) k(a) on W. Then ~ is either empty or k-open on U.

f a m g p i n g of for every acU, ~ be the set of is generic over

Let r be the dimension of W ; for z generic over k on W, let z~,. 9 .,z~ be r algebraically independent elements of k ( z ) over k ; p u t z , ~ $ , ( z ) , where ~b* is a ~unction on W, defined over k. I t is clear that a point z' of W is generic over an overfield K of /r if and only if the ~, are all defined at z' and their values r are independent over K. L e t u , x be independent generic points of U, V over k; we m a y assume t h a t f ( u , x ) is generic over k on W, since otherwise ~ is empty. P u t f , = $ ~ o f ; ~ is then the set of

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those points a on U such that, for x generic over ]c(a) on V, the f i ( a , x ) are all defined and are algebraically independent over ]~(a). Take u, x as above; assume t h a t u is not in e ; we prove t h a t e m u s t then be empty. I n fact, the assumption on u means t h a t there is a polynomial 2 with coefficients in ~ ( u ) such that p(fl(~,x),.

- ",fr(~,x))

=o.

Write P ~ Y. t,M~(Z), where the My(Z) are monomials (with coefficient 1) y

in the indeterminates ZI," 9 ",Zr, and the t, are in k(u) and not all 0. Let a be any point of U, and take J generic over K ~ k (a) on V. Take a variable quantity X over ]c(u, z) ; extend the specialization u---~ a over k to a /~-valued place ~- of the field ]c(u, ,~) such t h a t the elements ;~t~ of ]~(u, X) are all finite and not all 0 at ~r; call t / t i l e value of ~t~ at ~r. As k ( u , ; Q and ]c(x) are independent regular extensions of k, the place ~ of k(u,X) and the isom o r p h i s m of k ( x ) onto k ( x ' ) over k which maps x onto x' make up a specialization of the set of quantities k(u,;~) U k ( x ) , which can be extended to a place ~r' of k ( u , A , x ) at which u, x and the ;~t, have respectively the values a, x" and t / . I f the f i ( a , x ' ) are not all defined, a is not in fl; if they are all defined, they are the values at ~' of the e l e m e n t s / ~ ( u , x ) of /~(u, ;~, x). I n the l a t t e r case, the relation

~ P ( f ~ ( ~ , ~ ) , . 9 . , f , ( ~ , ~) ) = o, taken at ~', gives an algebraic relation between the f~(a,~) whose coefficients t / are in /~ and are not all 0; this implies t h a t the f~(a,J) are not independent over K ~ ( a ) , so t h a t a is again not in ~t. This shows that, for ur a m u s t be empty. F r o m now on, therefore, we may assume t h a t n is in ft. We prove now t h a t a m u s t contain a /~-open set. Since the assumptions and the conclusion of our proposition are not affected if V is replaced by any birationally equivalent variety to V over ~ (the m a p p i n g f being transferred to the l a t t e r in an obvious m a n n e r ) , we may take for V an afflne v a r i e t y ; p u t x = (Xl,' ' ",x,~). Then we can write the f~ as f~(~, ~) = e ~ ( ~ ) / f o ( ~ ) , where Po, P~,'" ",P~ are polynomials in the indeterminates X ~ , ' ' - , X , ~ with coefficients in ~ ( u ) , and P , (x) ~ 0 for 0 G i _< r. Call M , ( X ) , with 0 ~ v G N , all the monomials in Xz,- 9 .,X~ which either are of degree 0 or 1 (i.e. equal to one of the monomials 1, X~," . .,X~) or occur in one at least of the P~; call 2 the point in the projective space p N with the homo-

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geneous coordinates (My(x)), and call Y- the locus of ~ over k. We maY' replace V by the birationally equivalent F ; then, writing V, x instead of ~, 5, and calling (xo,' 9 ",xiv) the homogeneous coordinates for x, we see that the f~(u,x) are expressed as z~/zo, with N

z~~ F~t~x~ ~:o

(0 ~-- i ~ r ) ,

where the t~ are elements of k (u). I f V is contained in any linear subvariety of pN, then the smallest linear subvariety of pY which contains V is defined over k; if this is of dimension N', we can express N - - N ' of the coordinates x~ linearly in terms of the others, with coefficients in k; thus we may assume that V is not contained in any linear subvariety of pN. We may write t ~ , = r where the r are functions on U, defined over k; as zo is not 0, we may assume that too ~ 1. By Prop. 8, the subset U" of U where all the ~ are defined and finite is k-open. Call n the dimension of V; as n is then the dimension of k ( u , x ) over k ( u ) , and the ft(u,x) are independent over k(u), we have r ~ n . P u t z j ~ , w ~ x ~ for r + 1 ~-- j ~ n, where the %.~, for r + 1 ~ j ~ n, 0 --~ v --~ N~ are (n - - r ) ( N + 1) independent variables over k ( u , x ) ; call S the affine space of dimension ( n - - r ) ( N - I - 1 ) . By F-II~, Prop. 24, the n - - r quantities z~+~/Zo,'' ",z,/Zo are algebraically independent over the field K ~ k ( U , W, Z1/ZO," " ", Zr/Zo).

Now take any a~ U'; take 2, v~ generic and independent over k(a) on V, S ; put ~ ~ r (a), 5~~ Z i ~ p for 0 ~ i _~ r, and ~ ~ Z w-j~ for r + 1 --__j ~ n. Y

As too ~ 1, and V is not contained in any linear variety, 2o is not 0; therefore, if we put f j ~ Z J Z o for r + l ~ ] _ ~ n , the functions f~,' 9 ' , f ~ on U" X S are defined at (a, ~) and have the values 2~/~o," 9", 2~/~o respectively. I f one assumes that ~/5o," 9 ", ~/2o are algebraically independent over k(a, ~) this implies a f o r t i o r i that ~/2o," 9 ",~/2o are so over k(a), i.e. that a ~ . Therefore, if we prove that there is a k-closed subset C of U' }( S such that, with the notations just introduced, the quantities ~/2o," 9",~n/2o are algebraically independent over k ( a , ~ ) whenever ( a , ~ ) is not in C, it will follow that ~ contains the set of those points a on U' such that a ~ S is not contained in C; and this set wilt be k-open by Prop. 7. I n other words, as long as we merely wish to prove that ~ contains a k-open set, it is enough to prove it for U ' / S and the functions f ~ z~/zo for 1 <: i ~ n instead of for U and for f , . 9 .,f,.. This means that, writing U instead of U ' X S,

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ON ALGEBRAIC GROUPS OF T R A N S F O R M A T I O N S .

it is enough to prove our assertion under the additional assumption r ~ n, the ~ being now everywhere defined on U, with boo ~ 1. This being now assumed, put Z~+l~ ~ w~x~, where the w, are N ~ 1 independent variables over ~ ( u , x ) . As l r is of dimension n over ~ ( u , w ) , there is a homogeneous polynomial P, with coefficients in k ( u , w ) and not all 0, such t h a t P(zo," 9 ", z~, zn+l) = O, and P is uniquely determined up to a factor in ~:(u,w). As zl/zo," " ",zn/zo are algebraically independent, there is at least one t e r m in P where z~§ occurs with a non-zero exponent; after m u l t i p l y i n g P with a suitable element of ~(u, w), we may assume t h a t the coefficient of this term is 1. Write all the other coefficients in P as C p ( u , w ) , where the Cp are functions on U X S n§ defined over It. We now prove our assertion about ~ by showing t h a t ~ contains the set of all points a on U such t h a t all the Cp are defined at (a, ~ ) for ~ generic on S n§ over ( a ) ; this is a k-open subset of U by Prop. 8 and 7. I n fact, let a be a point with t h a t p r o p e r t y ; take ~ generic over ~(a, ~ ) on V. P u t t~v ~ ~b~(a), these being all defined, according to our present assumptions ; p u t 2t = ~ ~ for 0 --< i <: n, and ~§ ~ ~ ~p2~. If we specialize the relation P(zo,"

9

",z.+~)

=0

over the specialization (~,~,~2) of ( u , w , x ) with respect to k, since the Cp are all defined at ( a , ~ ) , a homogeneous relation 5o," 9 ",5~+~ with coefficients in ] c ( a , ~ ) , containing 5~+~ with a exponent in a term of coefficient 1. This shows that 5~+~/5o is then over the field L ( ~ ) , where we have p u t L =

~ ( ~ , ~1/~o," 9 ,

we get, between non-zero algebraic

~,/~o).

Now take n ( N + l ) independent variables w~ over /c(a), for l _ < i < : n , 0 --< v --< N ; p u t y~ = ~ w~2~ for 1 --< i --< n ; what we have proved above shows that, for each i, YJZo is algebraic over L ( w ~ o , ' ' ' , w i z r fortiori over the field

and therefore a

L ' = L ( w~o, " 9 ", w~N ) = ~ ( a, W~o, " 9 ", w,~r ~/5o, " 9 ", 5~/5o ).

On the other hand, one sees j u s t as before, using F-II~, Prop. 24, t h a t the y~/5o, for l < - - i < _ n , are algebraically independent over the field ~(a,w~o," 9 " , w , ~ ) ; as they are algebraic over L', this implies that L ' has at ]east the dimension n over the l a t t e r field, so that the 2J2o, for 1 <--i <--n, m u s t be algebraically independent over it. But then they m u s t a f o r t i o r i be so over ~ ( a ) , which means t h a t a is in ft.

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This completes the proof of the following statement: the assumptions being again those of Prop. 11, ~ must either be empty or contain a 7~-open subset of U. Now we prove Prop. 11 by induction on the dimension of U, the conclusion being trivially true if that dimension is 0. Assume that is not empty; put X = U - - a ; we have proved that X is contained in a k-closed subset C of U. Call U~ the components of C; they are algebraic over k, and of dimension < dim(U). By the induction assumption, ~ A U~ is either empty or a ~-open subset of ~f~; in both cases its complement C~ on U,~ is a ~-closed subset of U. As X is the union of the C~, this shows that X is ~-closed. As it is obviously invariant by all automorphisms of ~ over ~, it must then be k-closed. PR0rOSlTIO~ 12. Let U be a variety defined over a field ~ ; let F be a closed subset of U. Then, among all the ~-closed subsets of U contained in F, there is one maximal set Fo. Let K be the smallest common field of definition containing k for all the components of F ; let ~ run through all the isonmrphisms of K over into the universal domain. As such an isomorphism ~ ]eaves all k-closed sets invariant, every k-closed subset of U which is contained in F is contained in all the sets F r and therefore in their intersection Fo; Fo is closed, since it is the intersection of closed sets; and it is k-closed, by Prop. 9. This proves the proposition. PROPOSITION 13. Let U be a variety defined over an infinite field k; let F be a closed subset of U X U. Then there is a point a on U, separably algebraic over k and such that no pair (a',a") of distinct conjugates of a over k is in F. Applying Prop. 12 to U X U, F and the algebraic closure ~ of k, we see that there is a ~-closed subset Fo of U }( U such that a subvariety of U }( U which is algebraic over k is contained in F if and only if it is contained in Fo ; this applies in particular to algebraic points over k oi1 U X U. By replacing F by the union of all conjugates over ]~ of all the components of Fo, we see that it is enough to prove our result in the case in which F is k-closed. We may assume that no component o[ F is contained in the diagonal oi U }( U, since the omission of such components does not affect the content of our proposition. Furthermore, we may, in order to prove our proposition, replace U by any k-open subset of U; in view of this, we first replace U by the set of its simple points, and then use Corollary 2 of Prop. 8 to replace U by an affine variety. Thus we may assume that U is a non-singular affin~

[1955a]

231 ON A L G E B R A I C GROUPS 0 E

389

TRANSPOR]VIATIO1WS.

variety, defined over ]C, and that F is a k-closed subset of U X U, no component of which is contained in the diagonal of U X U. Let n and N be the dimensions of U and of t h e ambient affine space, respectively. The case n ~ N is trivial, since in that case any rational point of U over k, e.g. 0, would solve our problem; therefore we assume n < N. Consider all sets of n linear equations: N

(1)

(1_<: i_<,~),

Xt,~x~=t,o V=I

and identify the set (1) with the point t = (t~o,t~) in the afl]ne space T of dimension n ( N + 1). In the space T, we consider the following sets: (a) Call A the set of those points t for which the left-hand sides of (1) are not linearly independent; as A can be described as the set of zeros of certain determinants, it is ko-closed, ]co being the prime field (one could easily see that A is in fact a variety, defined over ko). Put T ' = T - - A ; for

t~T', (1) defines a linear variety L(t) of dimension N

n.

(b) Take t generic over k on T ; by F-V1, Th. 1, U A L ( t ) is not empty, and, if u is a point in it, u is algebraic over k ( t ) and is generic on U over the field K=]C(t11," 9 .,t,,~). As the t~o are then in K(u), we have k(u,t) = K ( u ) , so that k(u,t) is a regular extension of k. Let W be the locus of (u,t) over ]C on U }< T ; by F-u Prop, 4, if t'eT', a point u' is in U A L ( t ' ) if and only if (u',t') is in W. By Prop. 10, there is a It-closed subset B of T such that T - - B is contained in the set-theoretic projection of W o n T ; then, if t ' ~ T - - (A U B), UN L ( t ' ) is not empty. ( c ) Let P p ( X ) ~ 0 , for l ~ p ~ r , be a set of equations for U with coefficients in k ; put Pp~ = OPp/OX~. Let D be the subset of U X T consisting of the points where the matrix

t~ P ~ , ( u )

,

(l<-i~n,l
=

l
--

is of rank < N ; since this can he expressed by the vanishing of determinants, D is a ]c-closed subset of U X T (as U is non-singular, it could be shown that D is actually a variety, defined over k). As W is not contained in D, D N W is a k-closed subset of W (also, in ~act, a variety), so that its cornportents have a dimension < n ( N + l ) . Let D' be the "projection" of D N W on T (i.e. the closure of the set-theoretic projection); this is a k-closed subset of T. Let u' be a point in U N L(t'), for t ' c T ' ; then, if L(t') is not transversal to U at u', (u', t') must be in D and therefore in

13

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[1955a] 390

DNW,

ANDRE WEIL.

and t' must be in D'. Therefore, if t ' r transversal to U at every point of U A L(t').

L(t') is

(d) Let X be any component of F ; let (u,v) be a generic point of X over ~. As X is not in the diagonal of U X U, we have u :f: v and may assume for instance that Ul-7(=Vl. Take the t,~ independent over k(u,v) for 1--
[1955a]

233 ON ALGEBRAIC GROUPS OF T R A N S F O R M A T I O N S .

391

P u t n ~ [U :/el ; as U is separably algebraic over lc, there are n distinct isomorphisms a of U into the algebraic closure of k. Each a can then be extended uniquely to an isomorphism, which we also denote by a, of U(x) onto U~(x) over ~ ( x ) . Let z be an element of U ( x ) , integral over U [ x ] and therefore also over k [ x ] ; then all the z~ are also integral over /~[x]. I f ~1," ' " , ~ are n linearly independent elements of to' over ~, it is wellknown that d e t ( ~ ~) is not 0; therefore all the z% and z among them, can be expressed as linear combinations of the n elements w ~ = ~ ( z ~ ; but these are integral over ~[xJ and are traces o v e r / c ( x ) of elements o f / d ( x ) , so that they are in ]~(x) ; as ~ [ x ] is integrally closed, the w~ are therefore in k [ x ] , so that z is in U [ x ] . T H E UNIVERSITY OF CHICAGO.

[ 1955b] On algebraic groups and homogeneous spaces I n a recent paper in the same JOURNAL ( [4] 0J~ the bibliography ; quoted hereafter as A G ) , I gave some results oil algebraic groups and transformationspaces, which supplement those in my Varidtds abdliennes ( [ 3 ] ; quoted as V A ) . Applications will now be made of that theory to somewhat more specific questions. I n no. 1, a rather general procedure is described for obtaining, from a given transformation-space S with respect to a group G and from a suitable cycle on S, another transformation-space with respect to the same group. As shown in no. 2, this includes as a special case the construction of coset-spaces and of factor-groups; thanks to the main theorem in AG, these can now be defined without enlarging the groundfield, whereas such an enlargement was required in their construction as previously given by S. Nakano ( [ 2 ] ) ; except for this, we have substantially followed his method. The rest of this paper is chiefly devoted to "principal homogeneous spaces," i.e. to those homogeneous spaces on which the group operates in a simply transitive manner. The pair consisting of such a space and of one point on it does not differ materially from a group ; thus there is little incentive for studying those spaces as long as one is not paying any attention to the groundfield or if the groundfield is algebraically closed. But it can happen that a principal homogeneous space contains no rational point over the groundfield over which it is defined; an example of this is given by the plane curve Xa-] - p Y Z ~ p 2 over the rational number-field, where p is a rational prime; this may be considered as a principal homogeneous space with respect to its jacobian variety, which is the plane curve Y Y ~ 4 X ~ - - 2 7 . More generally, Chow's work (cf. [1]) has shown that it is not always possible to map a curve " c a n o n i c a l l y " into its jacobian variety by a mapping defined over the groundfield, but that the curve can always be so mapped into a suitable principal homogeneous space with respect to its jacobian variety This, among other results, will be proved again here by a different method, which can be extended at once to a variety V of higher dimension and to its Albanese variety, provided the groundfield is one over which the latter is defined. I t will also be shown that the classes of principal homogeneous Received December 27, 1954. 493 Reprinted from the Am. J. of Math. 77, 1955, pp. 493-512, by permission of the editors, 9 1955The Johns Hopkins University Press. 235

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spaces with respect to a commutative group can be arranged into a torsiongroup, i.e. a group whose elements are all of finite order; and it follows at once from the results of no. 5 t h a t this group must be countable if the groundfield is finitely generated over the prime field. There seems to be no reason why it should be finite, even if the groundfield is the field of rational n u m b e r s ; a more detailed investigation of its structure, e.g. for the case of an elliptic curve over the field of ratiofial numbers, would be of considerable interest from the point of view of the theory of diophantine equations. 1. Let G be a group and S a transformation-space with respect to G, both defined over a field /r Let Z be either a divisor on S or a cycle on S whose components have coefficients which are prime to the characteristic of the universal domain. We denote by sZ, for any s e G, the transform of Z by the m a p p i n g u-->su of S onto itself. Let K be an overfield of k over which Z is rational. Let x be generic over K on G; by Prop. 6 of the Appendix of AG, there is a finitely generated extension k ( t ) of k which is the smallest overfield of k over which xZ is rational ; as xZ is rational over K(x), we have k(t) C K(x). I f x' is also a generic point of G over K , and ~ is the isomorphism of K(x) onto K(x') over K which maps x onto x', a will transform xZ into x'Z; if t' is the image of t under a, k ( t ' ) will then be the smallest overfield of k over which x'Z is rational, and, by Prop. 6 of the A p p e n d i x of AG, the cycle x'Z depends only upon t ' ; in other words, if a~ is an isomorphism of K(x) onto a field K(xl'), m a p p i n g x onto x l ' and t onto tl', we have x'Z~x~'Z if and only i f = t ' ~ t l ' . I n particular, take x ' = y x , with y generic over K(x) on G; then k ( t ' ) is the smallest overfield of k over which yzZ is r a t i o n a l ; as yxZ can be written as y(xZ), it is rational over /c(y, t ) , so that /c(t') C k(y, t) ; similarly, xZ can be written as y~(yxZ) and is therefore rational over k(y, t ' ) , so that k ( t ) C k ( y , t ' ) . This shows that lc(y, t ) = k ( y , t ' ) . P u t now z==yx, so that we have k ( t ' ) C K ( z ) ; take y' generic over K(x,y) on G, and call r the isomorphism of K(z) onto K(y'z) over K which maps z onto y'z=y'yx; let t" be the image of t' under r. Then t " is the image of t under the isomorphism r O a of K(x) onto K(y'yx) over K which maps x onto y'yx. I f Z is such that k ( t ) is a regular extension of k, then we may call T the locus of t over/c and, with the above notations, we may write t'-~-g(y, t), where g is a m a p p i n g of G X T into T, defined over /~. Then the results we have just proved mean that g satisfies ( T G 1, 2) of AG, no. 2, i.e. t h a t it is a normal law between G and T. A p p l y i n g now the main theorem of AG, we .e'er the following result:

[1955b]

237 ON A L G E B R A I C GROUPS A N D H 0 ~ I O G E N E O U S

SPACES.

495

PROPOSITION 1. Let G be a group and S a transformation-space with respect to G, both defined over a field It. Let Z be either a divisor on S or a cycle on S whose components have coeficients which are prime to the characteristic. Let K be an overfield of k over which Z ks rational; and assume that, if x is generic over K on G, the smallest extension U of k over which xZ is rational ks a regular extension of k. Then there is a transformationspace T with respect to G, defined over ~, and an everywhere defined mapping F of G into T, defined over K, such that the point t ~ F ( x ) is generic over k on T, that l c ' ~ ( t ) , and that F ( s s ' ) ~ s F ( s ' ) for all s, s' on G. For any s, s" in G, we have F ( s ) ~ F ( s ' ) if and only if sZ ~ gZ. If Z ks algebraic over ]c, one can talce for T a homogeneous space with respect to G. The existence of a transformation-space T with a generic point t over such that ] c ' ~ / c ( t ) has been proved above; moreover, with the same notations as above, we have It(t) C K ( x ) , k ( t ' ) C K ( y x ) , t ' ~ y t ; as K ( x ) , K ( y x ) are independent extensions of K , this shows, if K is algebraic over k, t h a t k ( t ) , k ( t ' ) are then independent extensions of k, i.e. t h a t T is pre-homogeneous, so that, by the m a i n theorem of AG, we m a y replace it by a birationally equivalent homogeneous space. As ]c(t) C K ( x ) , we m a y write t ~ F(x), with F defined over K ; then we have yt ~ F(yx), i. e. F(yx) ~ yF(x), for y generic over K ( x ) on G. This m a y be written as F ( x ) = y - ~ F ( y x ) , which shows, if s is any point of G and y is taken generic over K ( s ) on G, that F is defined at s. As F is everywhere defined, the relation F(yx) ~ y F ( x ) implies F(ss') = s F ( s ' ) for all s, s' on G. Now, for any s, s" on G, take x generic over K ( s , s p) on G; then xs, xs' are both generic over K on G, and therefore, if a is the isomorphism of K ( x s ) onto K ( x s ' ) over K which maps xs onto xs', ~ will map F ( x s ) onto F ( x s ' ) and the cycle xsZ onto xs'Z; then, as we have seen above, we have x s Z ~ x / Z if and only if F ( x s ) ~ F ( x s ' ) ; as the latter relation can be written x F ( s ) ~ x F ( s ' ) , so that these two relations are respectively equivalent to s Z = s ' Z and to F ( s ) ~ F ( s P ) , this completes our proof 2. We first apply Prop. 1 to the construction of the homogeneous space defined by a group G and a subgroup of G. PROPOSITION 2. Let G be a group, defined over a field It; let Z be a rational cycle over k on G, consisting of components with coe~cient 1, and such that its support I Z I is a subgroup of G. Then there ks a homogeneous space H with respect to G, defined over t~, and a rational point a over lc on H, with the following properties: (i) if we put, for a generic x over t~ on G,

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n~DR~ WEIL.

F ( x ) ~ xa, the mapping F of G onto H determines a one-to-one mapping of the cosets of I Z] in G onto the points of H ; (it) l~(x) is separable over k ( F ( x ) ) ; (iii) if ~ is a mapping of G into a variety V, deflated over an overfield K of te, and such that ~ ( x s ) ~ r whenever s ~ l Z [ and x is generic over K ( s ) on G, there is a mapping ~ of ~t into V, defined over K, such that ~ t p o F . I f ] Z J is a normal subgroup of G, then one can define on H a group-law, defined over ~, such that F is a homo~norphism of G onto H. The " s u p p o r t " of a cycle was defined at the beginning of the Appendix of AG. The assumption on ]Z I implies that Z has one and only one component Zo containing e, that this is a subgroup of G, and that all the components of Z are cosets of Zo in G. We apply Prop. 1 to the cycle Z on S ~ G, G acting on itself by the left-translations, and to the field le; if x is generic over 1~ on G, the smallest extension of /c over which xZ is rational is contained in k ( x ) , and hence, by AG-App., Prop. 3, Coroll., it is regular over /e. Therefore, by Prop. 1, there is a homogeneous space with respect to G, which we now call H, defined over k, and a mapping F of G into H, defined over le, with the properties stated in Prop. 1; in particular, F is everywhere defined, t ~ F ( x ) is generic over k on H, and k ( t ) is the smallest extension of k over which xZ is rational. I f we put a ~ F ( e ) , a is rational over k, and we have, for all s~G, F ( s ) ~ s a . I f s, s' are any two points on G, we have sa ~ s'a if and only if sZ ~ s'Z, i.e. if and only if s-~s'Z ~ Z; by the assumption on Z, the latter relation is equivalent to s-~s'~ J Z 1. Thus the points of H are in a one-to-one correspondence with the cosets of t Z] in G. Call F the graph of F on G X H . For any b r F N ( G X b ) is the set of those points (s, b) which are such that s a = b; in particular, if x is generic over k on G and if we put t = F ( x ) = x a , FN (GXt) is the set of the points (s, t) such that sa ~ xa, i.e. x - ~ s ~ l Z I ; this set can be written as I x Z l X t . As F . ( G X t ) is the prime rational cycle overTc(t) on G X H with the generic point (x, t) over k ( t ) , this shows that the prime rational cycle Z' over /e(t) on G with the generic point x has the same components as the cycle x Z ; as the latter is rational over k ( t ) and its components have the coefficient 1, this implies that Z ' ~ x Z . As the components of the prime rational cycle with the generic point x over k(t) have the coefficient 1, te(x) must be separable (i. e. "separably generated") over /~(t). As to (iii), let x be generic over K on G; put t = F ( x ) and w = r ; let w' be any generic specialization of w over K ( t ) ; this can be extended to a generic specialization x" of x over K ( t ) ; we have then w ' ~ r

[1955b]

239 ON ALGEBRAIC GROUPS AND :[tOMOGENEOUS SPACES.

497

and t ~ F ( x ' ) , and the latter relation implies that x' is on I xZ I, i.e. that it is of the form xs with s e l Z I . Let ~ be generic on G over K ( s ) ; we have (~(~s) ~ r specializing ~ to z over K ( s ) , we get d p ( x s ) ~ r since both sides are defined, i.e. w ' ~ w. This shows that w is purely inseparable over K ( t ) ; as it is at the same time rational over K ( x ) which is separable over K ( t ) , it is therefore rational over K ( t ) , and we may write w ~ r where r is a mapping of H into V, defined over K. This proves (iii). Finally, assume that I z t is a normal subgroup of G; let x, y be indeWe have pendent generic points of G over /~; put t = F ( x ) , u ~ F ( y ) . F ( x y ) ~ xF (y) ~ xu; this is a function of x, defined over I Z I, we have x s y = x y s ' with s ' = y - l s y e f Z l , and therefore F(xsy) ~ F ( x y ) ; by (iii) applied to the mapping x--->xu of G into H and to K = k ( u ) , this implies that F ( x y ) is rational over k ( u , t ) . Therefore the mapping u-->xu of H into H is defined over k ( t ) ; on the other hand, if U is any field of definition for that mapping, containing k, the image x a ~ t of a by it is rational over U, so that k ( t ) C U ; thus k ( t ) is the smallest field of definition for the mapping u ~ xu. This shows that G is not operating faithfully on H ; applying Prop. 2 of AG, no. 3, to G and H, we see that we can define on H a normal law of composition f such that F ( x y ) ~ f ( F ( x ) , F ( y ) ) . By the main theorem of AG, we can then replace H by a birationally equivalent group H', defined over k, with a mapping F' of G into H', also defined over k, such that F ' ( x y ) ~ F ' ( x ) F ' ( y ) , F ( x ) and F'(x) being corresponding generic points of H and H' over ]c when x is generic over k on G. As usual, from the relation F ' ( x ) = F ' ( y ) - l F ' ( y x ) which holds for x, y generic and independent over k, we deduce that F' is everywhere defined 1; therefore, if G is made to operate on [I" by the law (x,w) --->F'(x)w for x, w generic and independent over k on G and H', H ' is a homogeneous space with respect to G. But then the unicity assertion in the main theorem of AG can be applied to H and H ' and shows that they are biregularly equivalent; in other words, H itself, with the law f, is a group. This completes the proof. I t is easily seen that the pair (H,a) is uniquely determined, up to an isonlorphism, by the conditions (i), (it), (iii) in Prop. 2 ; in other words, if H ' and a' have similar properties, there is an everywhere biregular birational correspondence between H and H ' which maps a onto a' and transforms the law of composition between G and H into the law between G and H'. The space H may be called the coset-space determined by G and Z, and may be denoted by G/Z; if ]Z 1 is a normal subgroup of G, the space H, with the group-law 1 This is Theorem 1 of Nakano ( [2] ).

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determined by Prop. 2, is called the quotient-group (or factor-group) of G by Z, and is denoted by G/Z. 3. Before making another application of Prop. 1, we will introduce a new condition which a law of composition may satisfy. Let V, W be two varieties, g a mapping of V X W into W, and ~ a field of definition for V, W and g; consider the following condition: (TGI')

v~g(x,u),

If x, u are independent generic points of V, W over ~, and then k ( x , u ) ~ k ( x , v ) ~ k ( u , v ) .

The condition k ( x , u ) ~ ( x , v ) is equivalent to ( T G 1 ) of AG, no. 2. The condition k ( x , u ) - - k ( u , v ) implies that the dimension of V, which is the dimension of x over k ( u ) , is the same as that of v over ~(u), and therefore at most that of W; if the dimensions of V and of W are the same, this implies that v is generic over ~(u) on W, which is condition ( I t ) of AG, no. 2. Let k' be any field of definition containing ~ for the birational correspondence u--> v ~ g(x, u) between W and itself; if u is taken generic over lc'(x) on W, we have M(u) ~ / d ( v ) ; since ~(x) C ~'(u,v) by ( T G I ' ) , we have k(x) C ~'(u). Taking u' generic on W over k ' ( x , u ) , we get in the same manner k(x) C ~ ' ( u ' ) . As k ' ( u ) , k'(u') are independent regular extensions of k', their intersection is ~', so that k(x) C k'. This shows that ( T G I ' ) implies ( T G 3 ) . In view of the results of AG, end of no. 3, this shows that, if g satisfies ( T G I ' ) and ( T G 2 ' ) , or if two mappings f, g of V X V into V and of V X W into W are given and satisfy ( T G 1 r,2), then V is a pre-group and W a pre-transformation space, and V operates faithfully o n W.

I f a pre-group V and a pre-transformation space W satisfy ( T G Y ) , we say that W is a pre-principal space with respect to V; if at the same time V and W have the same dimension, so that, as we have shown, W is prehomogeneous with respect to V, we also say that V is simply pre-transitive o n W.

Let IV be a pre-principal space with respect to a pre-group V; by the main theorem of AG, we can construct a group G and a transformation-space S, birationally equivalent to V, W and defined over the same field k; then S is also pre-prineipal with respect to G. Let T be the locus of (u, xu) over ~ on S X S, x and u being independent generic points of G, S over k; put t ~ (u, xu); then ( T G I ' ) implies that /r C ~ ( t ) , i.e. that we may write x ~ ~b(t), where ~b is a mapping of T into G, defined over/~; conversely, if this is so for a transformation-space S with respect to G, S is pre-prineipal.

[1955b]

241 ON A L G E B R A I C GROUPS A N D ]-IOMOGENEOUS SPACES.

499

The space S will be called a principal space with respect to G if, for x, u generic and independent over/r on G, S and for t ~ (u, xu), we have x ~ ( t ) where ~b is an everywhere defined mapping, defined over k, of the locus T of t over k into the group G. I f at the same time S is homogeneous, it will be called a principal homogeneous space with respect to G. PROrOSITION 3. Let S be a pre-principal transformation-space with respect to a group G, both being defined over a field k. Then there is a l~-open subset P of S which is a princ@al transformation-space with respect to G; if G and S have the same dimension, P is uniquely determined and is homogeneous. Let T and ~ be defined as above; call F the It-closed subset of T where q~ is not defined. We first show that, if (a, b) is in F, (sa, s'b) is in F for all s, s' in G. I n fact, take x, u generic and independent over k(s,s') on G, S ; p u t v ~ xu, ul ~ su, Vl ~ s'v, x~ ~ s'xs-1; then we have vl ~ x~ul, and xl, ul are generic and independent over k(s,s') on G, S, so that (u,v) and ( u , Vl) are generic points of T over k(s, s'), and that x ~ ~(u, v), xl ~ ~(u~, v~) by the definition of ~ ; this gives

~(u,v) --s'-l~(su, s'v)s. I f (a,b) is in T, it is a specialization of (u,v) over lc(s,s'), and therefore (sa, s'b) is also in T ; then the above relation shows that r is defined at (a, b) if it is defined at (sa, s'b), i.e. that (a, b ) c F implies (sa, s'b) 9 F. As (e, u) is a specialization of (x, u) over k, e being the neutral element of G, T contains the diagonal A of S X S. As the projection of A on either factor of S X S is everywhere biregular, the projection of the /~-closed subset F A A of A onto S is a /~-closed subset F ' of S, consisting of the points a 9S such that ~ is not defined at (a,a). From what we have proved above, it follows that, if a~ F', sa ~F" for all s ~ G. For the same reason, if a is in S - - F ' , then ~b(a, sa) is defined for all s e G; as (a, sa, s) is then a specialization of (u, xu, x) over k, and x ~ ( u , xu), this shows that ~(a, s a ) - - s for all a e S - - F ' and all s e G, and therefore ~ ( a ~ s ) ~ l~(a, sa); in particular, if x is generic over/c(a) on G, the locus of xa over k ( a ) has a dimension equal to that of G. I f G is complete, every specialization (a, b) of (u, xu) over k can be extended to a specialization (a,b,s) of (u, xu, x) over k, so that b ~ s a ; in other words, every point of T must be of the form (a, sa), with a~S and s c G; then it follows from what we have proved above that such a point cannot be in F unless a and sa are in F p. Without attempting to decide whether this is still so in the general case, we shall merely show that, if u

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[1955b] 500

ANDRE WEIL.

is generic over k on S and (u,u') is in F, then u' must be in F'. In fact, suppose that this is not so; take x generic over Ir on G; call X, X' the loci of xu, xu' over ~(u,u') on S; by what we have proved above, they have the same dimension, which is that of G. By F-VI3, Th. 11, we have T N (u X S) ~ u X X ; at the same time, since we have shown that (u, xu') is in T, u X X ' is contained in T N (u X S ) ; as X and X ' have the same dimension, this implies that X ~ X : . But, as we have shown, since (u, u') is in F, (u, xu') must be in F, and therefore, since F is k-closed, u X X' must be contained in F ; as X ~ X ' , this implies that F contains (u, xu), which is generic on T over ~, and contradicts the definition of F. One may observe that, if S is pre-homogeneous, this again Shows that (a, b) cannot be in F unless a, b are in F ' ; for, if a~F" and x is generic on G over tc(a,b), xa has then over k(a) a dimension equal to that of G, and therefore is generic on S over/c(a) since in the present case the dimensions of S and G are equal; then if (a, b) is in F, so is (xa, b), and so b must be in F'. Now replace first S by S - - F ' ; as F ' is mapped onto itself by all operations of G, S - - F ' is again a transformation-space with respect to G, defined over k, and satisfies our other assumptions. Writing again S instead of S - - F ' , we see that it is enough to prove our result under the additional assumption that / 7 ' ~ . If G is complete or S is prchomogeneous, this already implies that S is principal. Otherwise we observe that, since T is also the locus of (z-lu, u) over /r and since we have ~(x-lu, u) --x, T and F are mapped onto themselves by the mapping (u,v)---. (v,u) of S X S onto itself. Call now F " the "projection" of F on either factor of S X S (in the sense of F-IV3 and F-VIIi, i.e. the closure of the set-theoretic projection); this will be the same, whether we project F onto the first or the second factor, and it is not S by what we have proved above, since F ' is empty; it is therefore a ~-closed subset of S. By what we have proved, F " is mapped onto itself by all operations of G. Then S - - F " is the principal space whose existence was to be proved. Finally, assume that the space S from which we first started was prehomogeneous; this means that T ~ S }(S. Let a, b be any two points in S - - F ' ; then, if x is generic on G over to(a, b), xa, xb are generic on S over k(a, b), and so there is an isomorphism r of to(a, b, xa) onto k(a, b, xb) over ]c(a,b), mapping xa onto xb; then we have x~a--xb, i.e. b=x-~x~a. This shows that S - - F " is homogeneous, and also that an open subset of S which is a transformation-space for G cannot contain a point of S - - F " without containing S - - F ' . Therefore S - - F ' is the only open subset of S which is a principal space with respect to G.

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243 ON A L G E B R A I C GROUPS A N D H O M O G E N E O U S SPACES.

501

I f N is a principal homogeneous space, the mapping $ of ~ r , S X S into G which has been defined above will be called the canonical mapping of S X S into G. For any a, b on S and s on G, the relations b=sa, s=ck(a, b) are equivalent; in particular, for any a on S and for x generic over /c(a) on G, the mapping x--->xa~v of G into S has the inverse v--->x= $(a,v); as both are everywhere defined, this is therefore an everywhere biregular mapping of G onto S, defined over /c(a). I n particular, if there is at least one rational point a over k on S, S is biregularly equivalent to G over k. 4. Let G be a group, V and W two varieties, and F a mapping of V ; K W into G, all defined over a field k. We may consider W ? < G as a transformation-space with respect to G, the law of composition between them being ( x , ( N , y ) ) ~ ( N , xy) for any N ill W and any x, y in G. We now apply Prop. 1 of no. 1 to the ease when we take for S this transformationspace W X G and for Z the graph of the mapping N-.-->F(M,N) of W into G, where M, N are independent generic points of V, W over /c. We must then consider the smallest field of definition /e' containing ~ for the mapping N-->xF(M,N) of W into G, where x is generic over lc(M,N) on G. As this mapping is defined over /c(x,M), U is a regular extension of /c, contained in/c(x, M). Then Prop. 1 shows that we may write / c ' = It(u), where u is a generic point over /c of a transformation-space U with respect to G; as /~(u) c / c ( x , M ) , we may write u = f ( x , M ) , where f is a mapping of G X V into U, defined over /~; moreover, as the mapping x.--->f(x,M) of G into U is no other than the mapping F defined in Prop. 1, we see that f is defined at every point (s, M) of G ;4 M, and that f(ss', M) = sf(s', 3f) ; taking s ' = e , and writing f(M) instead of f(e,M), this gives f ( s , M ) = s f ( M ) , and in particular u ~ xf(M). As the mapping N--~ xF(3i, N) is defined over lc(u), xF(M,N) is rational over / c ( u , N ) ; similarly, if y is generic over lc(x,M,N) on G, the mapping N..->yxF(M,N) is defined over l~(yu), and so yxF(M,N) is rational over lc(yu, N). As we can write

y = (yxF(M, N) ) (xF(M, N) )-1, this shows that y is rational over k(u, yu, N). If N' is generic on W over lc(x,y,M,N), y must then also be rational over /c(u, yu, N'); thus /c(y) is contained in lc(u, yu, N) and in lc(u, yu, N'); as these are independent regular extensions of tc(u, yu), their intersection is ]c(u, yu), and so we have ]c(y) C lc(u, yu). This means that U is a pre-principal space and may therefore, by Prop. 3, be replaced by a principal space, birationally equivalent to it.

244

[1955b] 502

ANDRI~ WELL.

The most interesting case is that in which there are two mappings F1, F2 of V, W into G, defined over some overfield K of k, such t h a t F ( M , N ) ~ F I ( M ) F 2 ( N ) for M, N generic and independent over K on V, W ; by the corollary of Th. 7, VA-18, this is always so whenever G is an abelian variety. Take x generic over K(M, N) on G, and put z ~ xF~(M) ; then the m a p p i n g N - - > x F ( M , N . ) ~ z F 2 ( N ) is defined over K(z), so that k(u) C K(z). As x, M are generic and independent over K ( N ) on' G, V, u is then generic over K ( N ) on U, and the dimension of U is t h a t of u over K ; the relation K(u) C K(z) shows t h a t this is at most the dimension of G. Therefore U is pre-homogeneous and may be taken to be a principal homogeneous space with respect to G. Moreover, we may write u ~ O ( z ) , where is a m a p p i n g of G into U, defined over K. I f we substitute yx for x, with y generic over K ( M , N , x ) on G, z is replaced by yz, and u by yu; this gives r ~ y ~ ( z ) , which may be written as r and thus shows t h a t ~ is everywhere defined. P u t t i n g now a ~ , ( e ) , we see t h a t a is rational over K and that u ~ z a , i.e. f(M) ~ F I ( M ) a . P u t now g(N) ~ F 2 ( N ) - l a ; g is a m a p p i n g of W into U, defined over K. As we have also g(N) ~ F ( M , N ) - I f ( M ) , g is also defined over the field k(M), and therefore also over k(M') if M ' is another generic point of V over K ; if we take M, M ' generic and independent over K on V, k(M) and k(M') are independent r e g u l a r extensions of k, so t h a t their intersection is k ; hence g is defined over k. Thus we have proved the following r e s u l t : PROPOSlTIO~ 4. Let G be a group, V and W two varieties and F a mapping of V X W into G, all defined over k. Assume that there are two mappings F~, F2 of V, W into G, defined over some overfield K of k, such that F ( M , N ) ~ F ~ ( M ) F 2 ( N ) for M, N generic and independent over K on V, W. Then there is a princ@al homogeneous space U with respect to G, and two mappings f, g of V, W into U, all defined over ~, such that f ( M ) = F ( M , N ) g ( N ) , i.e. F ( M , N ) ~ r where ~ is the canonical mapping of U X U into G. COROLLARY. Notations being as in Prop. 4, U, f and g are uniquely determined by G, V, W and F, up to an isomorphism. I n fact, assume t h a t Up, f', g" have similar properties ; then we have xf'(M)), where r is the canonical m a p p i n g for U'. This shows that the m a p p i n g N---->xF(M,N) is defined over k ( u t) with u ' ~ x f ' ( M ) ; thus, if we put u ~ x f ( M ) as before, we have k ( u ) C k ( u ' ) and m a y write u ~ r where r is a m a p p i n g of U" into U, defined over ~.

xF(M,N) ~ r

[195561

245 ON ALGEBRAIC GROUPS A]qD H O M O G E N E O U S SPACES.

503

Replacing x by yx, with y generic on G over k (M, N, x), we get yu ~ ~ (yu p) ; from this we conclude, in the usual manner, t h a t ~ is everywhere defined. Take any point a' on U', and put a ~ ( a ' ) ; as we have x a ~ r and as the mappings x--->xa, x - o x a ' are everywhere biregular mappings of G onto U and U', defined over k(a'), we see t h a t ~ is an everywhere biregular m a p p i n g of U' onto U. Moreover, we have ~ ( u ' ) ~ xtp(f'(M)), and therefore f ~ ~ o f'; from this one easily concludes t h a t g ~ ~ o g'. This proves our assertion. 5. Let G be a group, defined over a field k. We will now prove t h a t the classes of principal homogeneous spaces with respect to G, for birational equivalence over k, form a set. I n fact, let x, y be independent generic points of G over k; let (~ be the isomorphism of ~ ( x ) onto ~(yx) over ~ which maps x onto yx. Let H be any principal homogeneous space with respect to G, defined over k ; let a be an algebraic point over k on H , and p u t u ~ xa, so that u is generic over 7~ on H. Then ]c(u) is a regular extension of 7c contained in ~ ( x ) and such t h a t ~ ( u ) = ~ ( x ) ; moreover, we have

~(y,u) =k(y,u~)

= ~ ( u , u~)

since u ~ y u . Conversely, let ~ ( u ) be any such extension of k, and call U the locus of u over k; then we may write u r g (y, u), where g is a m a p p i n g of G X U into U, defined over ~; and one verifies at once t h a t this makes U into a pre-principal pre-homogeneous space with respect to G, and thus determines uniquely a class of birationally equivalent principal homogeneous spaces with respect to G. As every such class is determined by at least one such extension, this shows that these classes form a set. I f G is commutative, one can define canonically a commutative groupstructure on the set of classes of principal homogeneous spaces with respect to G. I n order to do this, we first observe that, if H is any transformationspace over a commutative group G, then the law (x,u)-->x-lu, for x~G, u e H , defines on H a structure of transformation-space with respect to G; this will be called the opposite transformation-space to H and will be denoted by H - ; it is a principal homogeneous space if H is such. PROPOSITION 5. Let G be a commutative group, defined over a field k. Let H~, for 1 ~-- i X n, be principal homogeneous spaces with respect to G, defined over k. Then there is a principal homogeneous space H with respect to G, defined over tr and an everywhere defined mapping f of HI }( H2 X " 99X H,~ into H, defined over ~, such that f(s~a,.

9 .,sna~)

=

s~"

. . s,,f(a,.

9

a~)

246

[1955b] 504

ANDRe] ~,VEIL.

for all s~e G and a~eH~. Moreover, H and f are uniquely determined up to an isomorphism of H. Put V ~ W ~ HI X H2 X " " " X 1t,,; call ~ the canonical mapping of H, X H~ into G, so that b~~ sa, is equivalent to s ~ q~,(04, b~) for a~, b~ in H~ and s in G. Let u = ( u ~ , . - .,u~) and v ~ ( v ~ , . 9 .,v,,) be two points of V; put

F(u, v) = fI +,(u,, ~,), i=I

where the right-hand side has a meaning since G is commutative. H~, choose a point 04, and put a ~ ( a , . 9 .,a~). We have

r

vd ~ r

vdr

On each

ud -~

for all i, as one verifies at once, and therefore, again because of the commutativity of G :

F ( u , v ) = F ( a , v ) F ( a , u ) -1. Thus the assumptions of Prop. 4 are satisfied, so that there is a principal homogeneous space U and two mappings f, g of V into U, all defined over /c, such that (1)

f(u) =F(u,v)g(v),

F ( u , v ) = , ~ ( g ( v ) , f ( u ) ),

where ~ is the canonical mapping of U )< U into G. Take any point b on V, and take v generic over k(b) on V; as F is defined at (b,v), the relation (1) shows that f is defined at b. Thus f is everywhere defined. As F ( u , u ) ~ e, the relation (1) gives g ( u ) = f ( u ) , i.e. f ~ g . If s l , ' ' ' , s ~ are any elements of G, and we put s ~ s l " 9 .s.~ and u ' ~ (slu~,. 9 ",s,u,,), we have F ( u ' , v ) ~ s - I F ( u , v ) and therefore, by (1), f ( u ' ) ~ s - l f ( u ) . If we now put H ~ U-, i.e. if we take for H the opposite space to U, H and f will have the properties stated in Prop. 5. Let us now assume that ~ and ~ have similar properties; put F ~ / ~ - . Put 2 = F ( u , v ) - ~ ] ' ( u ) , the multiplication in the right-hand side being that of ~7. I f the s~, s and u" have the same meaning as above, we have ]'(u')=s-~]'(u), so that ~ does not change if one replaces u,v by u',v. Therefore /c(2) is contained both in k ( u , v ) and in k(u',v). If the s~ have been taken generic and independent over lc(u,v) on G, t~(u,v) and lc(u',v) will be independent regular extensions of k ( v ) ; this gives k ( 5 ) C k(v), so that we may write 2 = t T ( v ) , with j defined over k. Then we have ]'(u) = F ( u , v ) g ( v ) ; by the corollary of Prop. 4, ~7, ~ and j must then be

[1955b]

247 ON

ALGEBRAIC GROUPS AND I~O~IOGENEOUS SPACES.

505

the same as U, f and f, respectively, except for an isomorphism of U onto U. This proves the assertion about unicity in Prop. 5. In Prop. 5, take n ~ 2; call ~1, ~ 2 the classes of H1, H2, and denote by t~l ~ ~ 2 the class of H. This defines on the set of classes of principal homogeneous spaces with respect to G a commutative group-structure. In fact, commutativity is obvious. Call ~ o the class of G, and therefore of all principal homogeneous spaces with respect to G which have a rational point over k. For any principal homogeneous space H with respect to G, the mapping f ( x , u ) ~ x u of G X H into H satisfies the condition of Prop. 5; there-fore we have ~ o ~-t~ ~ 9/ for all classes ~ . If ~ is the canonical mapping of H X H into G, then ~, considered as a mapping of H X H- into G, satisfies the condition of Prop. 5 ; therefore, if ~ - is the class of H-, we have 5~ -~ &t- ~ 9/0. Finally, let H1, H2, H~ be three principal homogeneous spaces with respect to G ; apply Prop. 5 successively to the following spaces: (a) to H~, H2, obtaining a space H12 and a mapping f12; (b) to H12, H3, obtaining H', f'; (c) to H2, H3, obtaining H2~, f2a; (d) to H~, H~3, obtaining H", f"; (e) to H1, H2, H3, obtaining H, f. Then the two mappings

f'(f~,~(u,u~),u,),

f"(u,f~(u~,m))

of H1 }( H2 X Ha into H', H'" satisfy the same condition as the mapping f. By the unicity assertion of Prop. 5, this shows that H', H" are isomorphic to H. This means that the addition &t~ + ~2 is associative. One proves quite similarly, by induction on n, that if :~ and ~ , are the classes of the spaces H, H, in Prop. 5, then ~ ~ ~ . ~ . In fact, let H', f' be the space and the mapping obtained by applying Prop. 5 to H~,. 9 -, H,~, so that ~ ' ~ & q + . . . + 5 ~ _ ~ by the induction assumption; and let H", f" be the space and the mapping obtained by applying Prop. 5 to H', H,, so that ~ " ~ ~ ' - ~ ~ by definition. Then the mapping

(ui,

.,u.) ~f"(f'(~.

,~_~),u,,)

of H~ X" 9 ' } ( H ~ into H " has the properties stated for f in Prop. 5~ so that, by the unicity assertion in Prop. 5, H" is isomorphic to H. From this one deduces that every element ~ of the group we have just described is of finite order. In fact, take on a space H of class ~t any positive cycle ~ a~ of dimension 0, rational over k.

Call H , any space of

class nt~; then there is a mapping f(u~," . ",u,) of the product of n factor~ equal to H into H , with the properties stated in Prop. 5. From the unicity assertion in Prop. 5, it follows that any permutation of the u, will change f

248

[1955b] 506

ANDRI~ W E I L .

into sf, with s~ G; as f is everywhere defined, we see that s ~ e by t a k i n g ul . . . . . u~; therefore f is a symmetric function, so t h a t f ( a l , - - -,a~) is rationM by the m a i n theorem on symmetric functions (VA-7, Th. 1). So H~ has a rational point over k, and is therefore isomorphic to G. Now, ~ being as before, put Ho ~ G and take, for each integer n ~ 0, a space H~ of class n ~ so that all the H . are disjoint. On the set | ~ U H~ yt

(which is of course not an algebraic v a r i e t y ) , we will define a commutative group-law f (in the sense of group-theory, not of algebraic geometry) such that Ho ~ G will be a subgroup of @ and that f induces on H ~ X H~, for all m, n, a m a p p i n g fm.~ of H,~ X H~ into H ~ satisfying the conditions in Prop. 5. As there is such a m a p p i n g f~.~ for each m, n, and as it is uniquely determined up to an automorphism of H ~ (i. e. up to left-multiplication by a rational point of G), we merely have to choose the f,~.~ so that the m a p p i n g f of (~ X ~ into @ which coincides with f~,~ on Hm X H , for all m, n satisfies the axioms for groups ; we do this as follows. F o r any n, we take fo,~(x, u) ~ xu for x c G , u~H~. We choose f-l.~ and, for all n ~ 0, f~,~ and f_~_~ so as to satisfy the conditions in Prop. 5. Now, for elements Ul," " ",u~+l of H1 in any number, we define ul" 9 .u~+~ inductively as being equal to ul for n ~ 0 and to f~,~(u~" 9 - u~,u.+~) for n ~ 1, similarly, for elements v~," 9 ',v,+~ of H_~, we define v~. 9 9v~+l as equal to vl for n ~ 0 and to f-~,-l(vl" " "v., v~+~) for n ~ 1. I t is then easily seen that, whenever m, n are both ~ 0, there is one and only one way of choosing fro,, so t h a t it satisfies the condition f ~ , , ( u ~ ' 9 -u~, u~+l" - 9u~+,) ~ ul" - 9u ~ when the u~ are in I-I1, we determine f_~_~ similarly, using H_~ instead of H , Finally, for m => n ~ 0, we choose f,~_~ and f_~,, so as to satisfy the conditions n

f~,~(ul"

9 .u~, vl"

9 .v,)

~

I I f - l . ~ ( v , , ~ , ) 9 u~+~. 9 . u ~

I-~,-(~,"

9 '~m,~"

" .u.)

=fIf-~,l(~,,~)

'v.+l"

9 .~

4=1

respectively, the u~ being any elements of H1 and the v~ any elements of H _ , I t is then a trivial m a t t e r to verify t h a t these choices of the f~., satisfy all the requirements for a commutative group-law on | The points on the H~ which are rational over k form a subgroup g of | As we have shown t h a t there are such points for some n ~ 0 , there is a smallest n ~ 0 for which there is such a point a ~ H ~ , this n is the order of ~ in the group of classes of principal homogeneous spaces with respect to G. Then ~ is the direct product of the group g o n g A G of rational

[1955b]

249 507

ON ALGEBRAIC GROUPS AND ~IO~MCOGENEOUS SPACES.

points over k on G and of the infinite cyclic group ), generated by a. The quotient-group | may be described as an algebraic group consisting of n components respectively isomorphic to Ho ~ G, H ~ , - - - , H , _ , 6. PROPOSITION 6. Let A be an abelian variety and H a principal homogeneous space with respect to A, both being defined over a field k. Let V1," 9 ", V~ be varieties, and F a mapping of VI X 9 " " X V~ into H, all these being defined over k. Then there is for each i a principal homogeneous space H~ with respect to A and a mapping F~ of V~ into H~, Ht and F~ being defined over k, and there is a mapping f of H1 X " 9 " X H , into H with the properties stated in Prop. 5, such that, for (M1," 9 ",M~) generic over k on V 1 X " 9 " X Vn, we have F ( M I , . 9 .,Mn) ~ f ( F l ( M ~ ) , "

9 ",F,(M~) ).

Moreover, all these are uniquely determined up to isomorph@ms. For n ~ 1, there is nothing to prove. I f the assertion is proved for a product of two factors, then this can be applied to the product V I X ( V 2 X " ' " X V ~ ) of V1 and V2X" 9 " X V ~ , so that the general case follows by induction on n. Thus it is enough to treat the case of two factors V, W and of a mapping F of V X W into H. Call ~ the canonical mapping of H X H into A; let ( M , N ) and ( M ' , N ' ) be two independent generic points of V X W over k; and put x~(F(M,N'),F(M',N')

),

y~(F(M,N),F(M,N')

).

so that we have xy = ~,(F(M, N ) , F ( M ' , N') ). As the mapping ( (M, M'), N p) --> x of (V X V) X W into A has the constant value e on the variety (M, M) X W, Th. 7 of VA-18 shows that x is rational over k ( M , M ' ) ; for a similar reason, y must be rational over k ( N , N ' ) ; in other words, there are mappings (I), $ of V X V, W X W into A, both defined over k, such that x ~ r and y ~ r By the corollary of Th. 7' of VA-18, 9 and q, satisfy the assumptions of Prop. 4, so that there are two. principal homogeneous spaces U~, U.~ with respect to A, mappings F~, G~ of V into U~ and mappings F2, G2 of W into U2, all defined over k, such that F~(M) ~ x G I ( M ' ) , F 2 ( N ) ~ y G 2 ( N P ) ; moreover, as r ~ ( N , N ) are defined and equal to e, we have G1 ~ F 1 , G~ ~ F 2 . Now call H~, H2 the spaces respectively opposite to U1, U2, and apply Prop. 5 to H~, H2: let / t be the principal homogeneous space and ~ the mapping of H~ X H~ into /~ with

250

[1955b] 508

ANDRI~ WEIL.

the properties stated in that proposition. As H1, H2 are opposite to U~, U2, w e have

F~(M') ~ x F I ( M ) ,

Put P(M,N)=](FI(M),F2(N)).

F2(N') ~ y F 2 ( N ) ,

multiplication in the right-hand sides being understood in the sense of H1, H2. By the definition of ~, we have then:

P(M',N') = (xy)#(M,N), while, as we have seen above, the same relation holds if F is substituted for iV. But then, as the corollary of Th. 7, VA-18, shows that the mapping

( (M',N'), (M,N) ) ---->xy of (V X W) X (V X W) into A satifies the condition of Prop. 4, the corollary of Prop. 4 shows that H, F must be the same as H, F except for an isomorphism of H onto H. Then, replacing ]~ by a mapping f of:H1 X H2 into H by means of that isomorphism, we have the spaces HI, H2 and the mappings F1, F2, f whose existence was asserted in our proposition. As to unicity, assume that there are spaces Hi*, H~* and mappings F~*, F2*, f* with the same properties. Then, x being defined as above, or equivalently by F(M', N') ~ xF(M, N'), we have

f*(F~*(M'), F2*(N')) ~ xf*(F~*(M), F~*(N')) = f*(xF~*(M), F2*(N')) and therefore F~*(M')= xFI*(M) since the mapping u---->f*(u,v) of H~* into H is, as easily seen. an everywhere biregular mapping of H~* onto H. But then the corollary of Prop. 4, applied to the mapping (M',M)-->x of V X V into A, shows that H~*, FI* are the same as H~, F~ except for an isomorphism. The same argument applied to y instead of x shows that H2, F~ are uniquely determined up to an isomorphism. Then Prop. 5 shows that f is uniquely determined. This completes the proof. 7. The foregoing results will now be applied to the theory of Jacobian varieties. As in VA-35, we consider a complete non-singular curve F of genus g > 0, defined over a field ~. I f a is any divisor on F, Prop. 6 of the Appendix of AG shows that there is a smallest field containing k over which a is rational; this field will be denoted by/r (a). In particular, if M1," 9", M~ are independent generic points of F over k and if we put m ~ ~ M~, then, i

by VA-4, Lemma 1, to(m) is the field k(M1,- 9 .,Mg)~ of symmetric functions of M~,.. ",M~, defined over k, i.e. the subfield of to(M1,. 9 .,Mg) consisting of those elements which are invariant under all permutations of

[1955b1

251 ON

ALGEBRAIC

GROUPS

AND

ts

SPACES.

509

M1," 9 ",Mg; such a divisor m will be called generic over k. As lz(m) is a regular extension of k, we may write it as k(u), where u is a generic point of a variety W over k, and we may write u ~ F ( M 1 , ' 9 .,Mg), with F defined over k ; as F is symmetric in the M~, this may also be written as u ~ F ( m ) . Now let the N~, P~, for 1 ~ i <_ g, be 2g independent generic points of Y over k ( m ) ; and p u t : x = ( N , . - . , K s , P1,. 9 ",Pg), this being a generic point over k of the product V ~ F X ' " X r of 2g factors equal to F. By VA-35, Lemma 11, there is a positive divisor m' g

g

on r linearly equivMent to m + ~. N ~ - - ~ P~, and it is uniquely determined i=1

~=I

and such that k (x, m ) ~ ~(x, m ' ) ; this implies that it is generic over k (x). Then, if we write u ' = F ( m ' ) , we have k ( x , u ) ~ I c ( x , u ' ) ; we may thus write u ' = g ( x , u ) , where g is a mapping of V X W into W which satisfies (TG 1). We now show that this mapping satisfies the condition ( T G 2 ' ) of AG, no. 3, Prop. 2, so that this proposition may be applied to it. I n fact, let y be a generic point of V over k(x,u); we may write

y = (Q_I," " ", Qg, RI," 9 ",Rg). Then the point u " ~ g ( y , u ' ) will be determined by u " ~ F ( m ' ) , where m" is the positive divisor linearly equivalent to m ' - f - ~ . Q ~ - - ~ . R~. Applying again VA-35, Lemma 11, we see that there is a positive divisor ~ S~ linearly equivalent to ~..N~--~P~+Y.Q~, and that it is generic over k(y,u) and rational over k(x,y).

But then In" is linearly equivalent to m ~ - ~ S ~ - - ~ , R ~ ,

which shows that, if we put z~

($1,. 9 .,Sg, R , .

9 .,Rg)

z is generic on V over k ( u ) and that we have u " ~ g ( z , u ) . This shows that g satisfies (TG 2'). Applying Prop. 2 of AG, no. 3 and the main theorem of AG, we see that there is a group J, a normal law g between J and W, and a mapping r of V into J such that g ( x , u ) ~ j ( ~ ( x ) , u ) for x, u generic and independent over k on V, W. P u t now K ~ k ' ( P ~ , . . . , P ~ ) , n ~ N ~ , and i

w =

(s.-

9 -, s ~ , 0 . "

9 ", Q ~ ) .

Since the P~, O~, N~ are generic and independent over k, Lemma 11 of VA-35

252

[1955b] 510

ANDR~ WELL.

shows that w is generic over K on V. At the same time, the linear equivalence by which the S~ were defined shows at once that g ( x , u ) ~ g ( w , u ) for u generic over k(x,w), and therefore g(c~(x),u)~g(~(w),u). Since J, by definition, operates faithfully on W, this implies r 1 6 2 as .w is generic over K on V, this shows that ~ ( x ) is generic over K on J. I t will now be shown that K(~(x)) ~ K 0 r ). I n fact, if u and u ' ~ g ( x , u ) are as before, u" is rational over K ( m , n ) ~ K ( u , n ) since the divisor m' is so by Lemma 11 of VA-35; therefore the mapping u-->u' is defined over K ( n ) , so that K(~(x)) C K ( n ) . P u t now K ' ~ K ( ~ ( x ) ) , so that u' is rational over K'(u) ; then m and m' are both rational over Kt(u). But Lemma 11 of VA-35 shows that n is rational over K ( m , m') and therefore over K'(u). I f ul is a generic point of W over K'(u), n will also be rational over K'(ul) ; as K'(u), K'(ux) are independent regular extensions of K', this implies that n is rational over K'. But now a comparison with the construction of the jacobian variety given in VA-36 shows that the latter coincides with our J over a suitably extended groundfield; more precisely, substituting K for k, ~ P~ for a, n for m and ~(x) for z in the treatment given in VA-36, we get the same law of composition for the field K(~(x)) as has been defined above. Alternatively, one may also reason as follows. Let Jx be the jacobian variety as defined in V A ; let ~bl be the "canonical m a p p i n g " of F into J1, also according to the definition of VA-37 (which will soon be replaced by a more appropriate one) ; let K1 be an overfield of the field K defined above, over which J~ and r are defined; take n generic over K1; put t ~ ( x ) , x being as above, and z ~S[~l(n)]. As we have then K~(x) ~K~(z) ~ K l ( n ) , the mapping x---->z defines a birational correspondence between J and J~, defined over KI. I f we write it as z ~ f ( x ) , f is everywhere de:fined by VA-15, Th. 6; and this, by the results at the beginning of VA-19, must then be of the form f(x) ~fo(X) + a, where a ~ f ( e ) and where fo is a homomorphism, so that (using the additive notation on J~ and the multiplicative notation on J ) we have f o ( x x ' ) ~ f o ( x ) + fo(x'). But then fo is again a birational correspondence, and, if g is the inverse mapping to fo, we have g(z + z') ~g(z)g(z') for z, z' generic and independent over K~ on J , This can be written as g (z) ~ g(z + z')g(z')-~; if then z~ is any point of Jx, and we take z" generic on J t over K~(zt), this shows that g is defined at zl. As fo, g are everywhere defined, they determine an isomorphism between J and J~. One could also, without making use of the results of VA, verify directly (for instance by making use of the criterion for the completeness of a group

[1955b ]

253 ON A L G E B R A I C GROUPS A N D HO2YIOGENEOUS SPACES.

511

given by VA-33, Th. 16) that the group J we have constructed here is complete and is therefore an abelian variety. Then the results we have proved above, combined with the corollary of Th. 7, VA-18, show at once that J has the properties stated in VA-36, Th. 18; since the whole theory of the jacobian variety depends upon nothing else, and these properties (as proved in VA-37) are characteristic of the jacobian variety, this would suffice for a complete treatment. From this discussion, we conclude that J is an abelian variety. Now apply Prop. 6 to the mapping ~ of V into J ; this defines 2g mappings of r into principal homogeneous spaces with respect to J, all defined over ~. As ~b is symmetric in the N~ and also in the P~, the unicity assertion in Prop. 6 shows at once that the first g mappings must coincide, and that the last g mappings must coincide; call F, F ' these mappings, and H, H ' the spaces into which they map r. Now, notations being the same as above in no. 6, put z' ~

(PI,"

" ", P g , N1," " ', N g ) .

Then we have, always with the same notations as before, u ~ g ( x ' , u ' ) , and therefore ~ b ( x ' ) ~ r -1. This, combined with the unicity assertion in Prop. 6, shows at once that H ' is the opposite space to H while F ' must be the same as F. We now embed J and H, in the manner explained at the end of no. 5, into a commutative group | consisting of principal homogeneous spaces H , with respect to J, all defined over ~, with Ho ~ J, H1 ~ H, in such a way that | is an infinite cyclic group, that the H . are the cosets of J in (~ and that the group-law in (~ induces on Hm X H., for all m, n, a mapping of Hm X H , into Hm+, defined over k and satisfying the conditions in Prop. 5. At the same time, we change from the multiplicative to the additive notation, not only in J but also in @. With this notation, we have, if x, 4~(x) and F have the same meaning as before,

4)(x) ~ ~ F(N~) - - ~ F(P,). Let us now extend the mapping F into a homomorphism of the group of divisors on F into (~, by putting F(a) ~ ~, n.~F(A~) for any divisor a ~ ~ n~A~, so that F ( a ) e l l , if n is the degree of a; in particular, F ( a ) is in J if and only if a is of degree 0. I f a is any point of H and M is a generic point of F over /c(a), the mapping M . - - > F ( M ) - - a of r into J, which is defined over ~(a), is a "canonical m a p p i n g " in the sense of VA-37; naturally it is only defined up to an additive constant; and, by the unicity assertion in Prop. 6,

254

[1955b1

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ANDR~ WEIL.

no such m a p p i n g can be defined over ~ unless H is isomorphic to G, i. e. unless H has a rational point over It. F r o m Th. 19 of VA-38, one deduces immediately that a divisor a on r is linearly equivalent to 0 if and only if

F(a)

=o.

In other words, the homomorphism a--) F(a) determines an isomorphism of the g r o u p of all divisor-classes (of any degree) on I' onto the g r o u p (~.

From the foregoing results, one concludes easily that these properties are characteristic for (~ and F, up to isomorphisms on J and its cosets H~ in (~. One may call (~ the J a c o b i a n g r o u p of F, and F the canonical m a p p i n g of F, and of the group of divisors on F, into the Jacobian group. I n substance, the construction of the varieties H~ has already been given by Chow (in [ 1 ] ) by a method belonging to projective geometry. TI{E UNIVIgIr

OF CHICAGO.

BIBLIOGRAPHY.

[1] W. L. Chow, " The Jacobian variety of an algebraic curve," American Journal of Mathematics, vol. 76 (1954), pp. 453-476. [2] S. Nakano, " Note on group varieties," Memoirs of the College of Science, University of Kyoto, Series A, vol. 27 (1952), Math. no. 1, pp. 55-66. [3] A. ~Veil, Vari~t~s abe;liennes et courbes algdbriques, Paris, ttermann et Cie, ]948. [4] - - , " On algebraic groups of transformations," American Jour~*al of Mathematics, vol. 77 (1955), pp. 355-391.

[1955c] On a certain type of characters of the id6le-class group of an algebraic number-field

Notations will be the same as in my previous work on class-field theory (Sur la th~orie du corps de classes, J. Math. Soc. Japan, 3 (1951), pp. 1-35 ; cf. also Sur les " f o r m u l e s explicites" de la th~orie des hombres premiers, Comm. Lund (M. Riesz jubilee volume), 1952, pp. 252-265). If K is any field, K* denotes the multiplicative group of non-zero elements of K. We consider an algebraic number-field k; k, means its completion with respect to a valuation v; in particular, k~, k~(1~ p ~ r~), k~(r, + 1 <~~<_rl + rf) denote the completions of k with respect to the prime ideal ~, to the real archimedian valuation vp and to the imaginary archimedian valuation v~, respectively; k~ may be identified (canonically) with the real number-field R, and k, may be identified (non-canonically) with the complex number-field C; put ~ = [k~: RJ. The id~le group Ik is the subgroup of ]-[ k* consisting of the a = ( a , ) s u c h that almost all a, (i.e., all except a finite number) are units. We denote by Pk the group of principal id~les, and by Ck=Ik/Pk the group of id~le-classes. Each id~le a = ( a , ) determines in an obvious manner an ideal a - ( a ) of k; we put: [] a ]]=N(a)-' ~ [a~ I'z. Then a ~ ] ] a [] is a representation of Ik into R* (in fact, into R*), taking the value 1 on Pk. Group characters will be understood in the extended sense, i.e. as continuous representations into C* (not necessarily of absolute value 1). The groups Ik, C~ will be topologized in the usual manner. A character )~ of Ck may also be regarded as a character of Ik, taking the value 1 on Pk; because of the known structure of Ck, such a character can always be written as )~,(a) ]] a ][~, where a e R and )~1 is a character of absolute value 1. The Hecke L-series attached to a character X of Ck can be constructed as follows. Let f be the conductor of )~; ff a = ( a , ) is an id~le such that a z = l for l ~ 2 ~ r ~ + r ~ and a ~ = l for every prime divisor ~ of f, )/(a) depends only upon the ideal a=(a); under those Reprinted from Proc. Intern. Symp. on Algebraic Number Theory, Tokyo-Nikko, 1955, pp. 1-7.

255

256

[1955cj 2

A. WEIL

circumstances, we p u t ~ ( a ) = x ( a ) . Then the L-series attached to x is ~ , ~ ( a ) N a .3, the sum being extended to all integral ideals r p r i m e to f. We shall denote by G(f) the group of the fractional non-zero ideals in k whose expression in t e r m s of p r i m e ideals does not involve any p r i m e divisor of f. We have t h u s attached, to e v e r y character )~ of Cx with the conductor f, a c h a r a c t e r % of G(f). Clearly ~ is completely d e t e r m i n e d by its values at the p r i m e ideals which do not divide f. A t the same time, X induces on the subgroup ]~k~ of Ix a character X

of t h a t g r o u p ; if we make use of the fact t h a t X m u s t be the product of a character of absolute value 1 and of a character I[ a I]~ we see t h a t )/, on t h a t group, can be w r i t t e n as:

X((a~))=F[(az/] az [)-s~, I a~ [~x<~+~~

(1)

w h e r e the f~, are integers and a and the ~o~ are real n u m b e r s . Now denote by k*(f) the subgroup of k* consisting of all elements a/a', w h e r e ~, ~' are integers in k such t h a t a------d----1 mod. f. Then X((a)) is a character of k*(f), which coincides on k*(f) w i t h the character • of k* given by the formula

=

I

I

(2 )

in which a~ denotes the image of ~ in kz (the l a t t e r being identified w i t h R or with C, as the case m a y be). Conversely, assume that for some integral ideal m of k we have a character ~ of the group G(m), and that t h e r e are integers f~ and real n u m b e r s a, ~o~ such t h a t ~ ( ( a ) ) = X ( a ) for a e k*(m), X being defined by (2). L e t a be an id~le; t h e r e is a S e k * such that, if we p u t b--~a, then, for e v e r y p r i m e divisor p of m, b~ is a u n i t in k~ and is ------1modulo the highest power of p dividing m; and $ is u n i q u e l y d e t e r m i n e d in k* modulo the subgroup k*(m) of k*. Now put: ~(a) =~((b)) 1-[ (b~/l bz I)-~l bz I~x(~+~x~. X

Our assumption on ~ implies t h a t t h e right-hand side does not depend upon the choice of ~ when a is given; and one sees at once that X is a c h a r a c t e r of Ix, taking the value 1 on Px and satisfying (1), t h a t its conductor f divides m, and t h a t the c h a r a c t e r ~ of G (f) associated w i t h X coincides w i t h @ on G(m). I t is clear t h a t a-->(a) defines a homomorphism of k*(m) into G(m) whose kernel is the group E(m) of all units e in k such t h a t ,.~-1 mod. m; E(m) is of finite index in the g r o u p E of all units in k. Notations being as above, we see that • takes the value 1 on

[1955c]

257 On a Certain Type of Characters of the Id~le-Class Group

E(f), so that, if m is the index of E(f) in E, • takes the value 1 on E . Conversely, let the f~, a, cpz be given; let X be defined by (2); and assume that t h e r e is an i n t e g e r m > 0 such t h a t • is 1 on E . Then • is 1 on a subgroup E ' of E of finite index. By a t h e o r e m of Chevalley, this implies t h a t t h e r e is an ideal m such t h a t E ' ~ E ( m ) ; t h e n • is I on E(m) and t h e r e f o r e d e t e r m i n e s a character of the image of k*(m) in G(m), which can "then be extended to a c h a r a c t e r of G(m), hence, for a suitable divisor f of m, to a c h a r a c t e r ~ of G(f) associated with a c h a r a c t e r 1~ of Ck with the conductor f. A character x of C~ is of finite order (in the group of all characters of CD if and only if it is 1 on the connected component of 1 in [k, i.e. if and only if f , = 0 for all ~, ~ = 0 for all ~, and a = 0 ; by class-field theory, such characters are those associated with the cyclic extensions of k; for such a )r all values of ~, are roots of u n i t y . Our purpose is now to show t h a t all the values of ~ m a y be algebraic for certain characters )r which are not of finite order. In fact, assume t h a t all the ~ , are 0 and that ~ is rational; t h e n ~((a)) has algebraic values on k*(f), i.e. ~ has algebraic values on t h e imuge of k*(~) in G(f); as t h a t image is of finite index in G(f), all the values of ~ m u s t be algebraic. The f~ and a being given, a necessary and sufficient condition for the existence of such a c h a r a c t e r % is, as we have seen, t h a t t h e r e should be an i n t e g e r m such t h a t ~ ( ~ / I r I)~z~ :=1 for all ~ e E ; replacing m b y 2 m , ~his can also b y w r i t t e n as

I I (~,/~,Y, = 1.

(3)

We shall say that )r is of type (A) if all the q~ are 0 and a is rational; for such a character, the integers f, will be such t h a t (3) holds, for a suitable m, for all ~ e E . Conversely, if the f , are given integers, and if t h e r e is an m such that (3) holds for all e e E , t h e n t h e r e will be a character )r of t y p e (A) belonging to the f~; and all such characters will be of the f o r m )~(a):~o(a)lta]I ~, w h e r e X0 is a c h a r a c t e r of finite order and p is rational. In partiaular, if k is a totally i m a g i n a r y quadratic extension of a totally real number-field k0, then, by Dirichlet's theorem, the group E0 of the units in k0 is of finite index in E ; if m is that index, e ~ m u s t then be totally real for e v e r y ~ e E , so t h a t (3) holds on E , for t h a t value of m and for a r b i t r a r y values of the f , . More generally (as A r t i n pointed out to me d u r i n g the symposium), Minkowski's t h e o r e m on units in absolutely normal number-fields makes it possible to reduce the problem of finding all characters of t y p e (A) of a field k to an exercise in Galois theory, and it will be enough

258

[1955c] 4

A. WEIL

here to state the result. Let /Co be the maximal totally real subfield of k; then k contains at most one totally imaginary quadratic extension of /Co; for two such extensions could be written as k o ( l / - a ) , ko(V'-/~), with a, fl totally positive in ko; then # contains the totally real field ko (V'a~), which must be the same as ko, so that the two extensions must be the same. Now let us call trivial those characters of type (A) which are of the form Xo(a)I[ all ~, with Xo of finite order and p rational. In order for a field k to have non-trivial characters of type (A), it is necessary and sdfficient that it should contain a totally imaginary quadratic extension kl of its maximal totally real subfield k0; and then all such characters are of the form

x(a) = x~(Nk/k,(a))Xo(a)

(4)

where Xo is of finite order, X~ is a character of type (A) of #~, and Nk/,, denotes the relative norm from I, to I,,, which extends the relative norm of elements of Iv over #~ in the obvious manner. Thus, in a certain sense, all non-trivial characters of type (A) come from totally imaginary quadratic extensions of totally real fields. We shall say that a character X is of type (A0) if the character • of k* associated with it according to (2) is of the form where the rz, sx are integers, and the sign may depend on a; such a character is called trivial if it is of the form Xo(a) Ilall ~, with Xo of finite order and m an integer. Non-trivial characters of type (Ao) are those non-trivial characters of type (A) for which 2o is an integer andf,~--2a mod. 2 for all b. It is easily seen that the character X of C, defined by (4) is of type (Ao) if and only if the character )/1 of C,, which appears in (4) is of type (Ao). I f x is of type (Ao), the values taken by ~ on the image of k*(f) in G(f), which are the values taken by • on k*(f), are all contained in the compositum of k and of its conjugates over Q (the rational number-field). As that image is of finite index in G(f), the values of on G(f) must all lie in a finite extension of this field. Thus: I f a character X of the id~le-class group C, of the field k is of type (A), the coe~cients of the Hec#e L-series associated with X are algebraic numbers; i f X is of type (Ao), these coefficients all lie in a finite algebraic extension K of Q. It is tempting to conjecture that the converse statements are also true; but I have not examined this question. In the second statement, it would be of interest to determine the smallest field K containing all the coefficients of the L-series, i.e. containing all the

[1955c]

259 On a Certain Type of Characters of the Id~te-Class Group

values taken by ~ on G(f). If N is the index in G(f) of the image of k*(f) in G(f), then it is clear at any rate that all the values taken by x ~ on G(f) lie in the field Ko generated by the values taken by • on #*. The determination os K0 amounts to an exercise in Galois theory; one should observe that Ko need not contain k. We now come back to the construction given above for X when the values of X are given on G(m), m being a multiple of f. It obviously depends upon the following fact (which is equivalent to the theorem on the independence os valuations on k): Is I(m) is the group of the id~les a=(a~) such that a ~ = l for all 2, and a,~-I for every prime divisor P of m, then the group Pk/(m) is everywhere dense in Ik. It amounts to the same to say that the image of I(m) in C~ is everywhere dense in Ck. This implies that a character os Ck is completely determined by its values on I(m). We shall denote by I'(m) the compact subgroup of I(m) consisting of the id~les a e I(m) such that ( a ) : l ; then I(m)/I'(m) is discrete and may be identified with G(m). Let ~o be any representation os Ck into a complete topological group /'; as usual, we make no distinction between ~o and the corresponding representation of Ik into F. Assume that there is an m such that ~ : 1 on I'(m). Then ~ determines a representation ~ of G(m) into F, and ~ is uniquely determined by ~ since the image of I(m) in Ck is everywhere dense. We may now ask, conversely, whether, if a representation ~ of G(m) into F is given, it determines a representation ~ os Ck into F. This will be so if and only if the representation into F of the image of I0n) in Ck which is determined by ~5 is continuous for the topology induced on that image by the topology os Ck; for then i~ will be uniformly continuous, and can be extended by continuity. This is easily seen to amount to the following condition. To every neighborhood V of the neutral element in F, there must be an integer N and an ~ > 0 such that we have ~((a)) e V for every a e k*(m ~) which satisfies the conditions I a ~ - i I ~ for all 2. Now let z be a character of Ck os type (A0); then ~ takes its values in a subfield K of C, of finite deawee over Q. If m is any multiple of the conductor of X, we have, for ~ e k*(m)and a~>0 for all p:

~((,~))= x(,~) = T[,~I~,~;,':,. 4,.

Let w be any valuation of K, and let K~ be the completion os K with respect to w; the above criterion shows that ~ determines a representation ~ of C, into K~*, satisfying )~w(a)=~((a)) for a e I(m),

260

[1955c] 6

A, WElL

provided e i t h e r w is a valuation at infinity o r w is attached to an ideal ~ and we take m =pf, where p is the rational prime which is a multiple of ~. As K is embedded in C, we may of course take for w the valuation Wo induced by the ordinary absolute value on C; then ~,Oo=)~. Other valuations of K at infinity determine characters of Ck in the usual sense, i.e. representations of Ck into C; the corresponding L-series are the conjugates over Q of the series attached to the given X. On the other hand, for each prime ideal ~ in K, we get a representation ~ of Ck into K~, invariantly associated with )~. As the connected component of 1 in the group K~ is {1}, )r takes the value 1 on the connected component of 1 in Ck. As C~ is the direct product of its maximal compact subgroup and of a group isomorphic to R, and as ) ~ takes the value 1 on the latter group, ) ~ must map C, onto a compact subgroup of K~ and therefore onto a subgroup of the group U~ of units in K~. Now let co be any character of the compact group U~; as U~ is the projective limit of finite groups, ~o must be of finite order; therefore o2oX~ is a character of finite order of C,, which, by class-field theory, determines a cyclic extension k' of k. If, for a given )~ and ~, we make all possible choices of o,, these cyclic extensions will generate a certain abelian extension k()~, ~) of k; the compositum of these for all ~ will be an abelian extension k(x) of k which is thus invariantly attached to x. If X is of finite order n, its values on Ik (not merely those on some I(m)) are n-th roots of unity; then, for every w, ~ . is the transform of ~ by an isomorphism into K,* of the multiplicative group of the n-th roots of unity; in that case, k(%) is the cyclic extension attached to )~ by class-field theory. In all other cases k()~) will be an infinite extension of k. If X is the trivial character :g(a)=Ha]], k()~) is the maximal cyclotomic extension of k; more generally, if )~ is any trivial character of type (A0), k()~)will be contained in the maximal cyclotomic extension of a cyclic extension of /c of finite degree. As to the non-trivial characters of type (Ao), some of them arise in connection with the theory of abelian varieties with complex multiplication; in fact, all the characters of type (Ao) can be expressed in terms of those which arise in that manner and of the trivial ones. Taniyama has'proved that the L-series attached to the characters of type (A0) belonging to abelian varieties with complex multiplication are precisely those which occur in the zeta-functions of such varieties; and his recent work (done since the symposium) has shown that the fields generated by the points of finite order on these varieties are

[1955c]

261 On a Certain Type of Characters of the Id~le-Class Group

closely related to the fields k(z) defined above. For more general results, including these as rather special cases, the reader must be referred to his forthcoming publications; all that can be said here is that they tend to emphasize the imDortance of the characters which we have discussed and of their remarkable properties. UNIVERSITY OF CHICAGO

[ 1955d] On the theory of complex multiplication I shall concentrate chiefly on those aspects of m y work which have not been duplicated by the parallel and independent investigations of Shimura and of Taniyama. A preliminary account of their results, which are more complete than m y own in several important respects, appears in this same volume; it is understood that t hey will later give a full exposition of the whole theory. We need the concept of polarized variety; the word "polarization" is chosen so as to suggest an analogy with the concept of "oriented manifold " in topology. Let V be a complete non-singular variety; X being a divisor on V, denote by C(X) the class of all the divisors X ' such that there are two integers m, m', both >0, for which m'X' is algebraically equivalent to reX. We say that the class C(X) determines a polarization of V if it contains at least one ample complete linear system, or in other words if there exists a projective embedding of V for which the hyperplane sections belong to C(X). Thus a polarized variety may be regarded as a variety with a distinguished class of projective embeddings. The class C(X) is uniquely determined by any divisor in it;'every divisor in C(X) will be called a polar divisor of V for the polarization determined by that class. It is clearly the same to say that a variety is polarizable or that it is projectively embeddable. Let V be a variety, defined over a field k. Let X he a divisor on V, defining a polarization of V. If the smallest field containing k, over which X is rational, is not algebraic over k, then X belongs to an algebraic family, defined over an algebraic extension of k, and may be replaced by a member of that family, algebraically equivalent to X and algebraic over k. Thus we may assume that X itself is algebraic over k. Call Y the sum of all conjugates of X over k; "if p is the characteristic, then, for a suitable m, p~"Y will be rational over k; and one sees immediately that it determines a polarization of V, although not necessarily the same as the original one. We say that a polarized variety V is defined over k if V is defined over k and if there is on V a polar divisor which is rational over k; this Reprinted from Proc. Intern. Symp. on Algebraic Number Theory, Tokyo-Nikko, 1955, pp. 9-22.

263

264

[1955d] lO

A. WEIL

amounts to saying that V has a projective embedding which is defined over k. As an important example, we mention the case of the jacobian variety J of a curve F ; the canonical divisor 0 on J (canonical, that is to say, up to a translation) determines a polarization of J which will be called its canonical polarization. A classical result, due to Torelli, and for which it would be .worth while to give a modernized proof covering the abstract case, asserts that two curves are isomorphic if and only if their canonically polarized jacobians are isomorphic. Let A be an abelian variety; we denote by A* its dual, and by C1 the canonical homomorphism of ~(A) onto A*, with the kernel Q~(A). Every divisor X on A determines a homomorphism ~Ox of A into A*, defined by cpxu=C1 ( X ~ - X ) . If p = 0 , the degree v(~ox) of ~ox is always the square of an integer. If X>-0, ,(~x) is >0, i.e. 9x is surjective, if and only if there is an m > 0 such that m X determines an ample complete linear system on A, i.e. if and only if X determines a polarization of A. Conversely, let A be polarized; then every polar divisor X on A determines a homomorphism ~Ox of A onto A*; in the extension ~ ( A , A * ) ~ ) Q by Q of the group of homomorphisms of A into A*, q~x is uniquely determined by the polarization of A up to a positive rational factor. If ~ is a homomorphism of A* onto A such that ~q~x is of the form m ~ , then ~-I(X) determines a polarization of A* which is canonically associated with that of A. In the case p = 0 , there will be a polar divisor X on A such that every polar divisor is algebraically equivalent to a multiple m X of X; such a divisor will be called basic; if, for such a divisor X, we put ,(~o~)=r ~, r is called the rank of the polarized variety A. As usual, if A, B are abelian varieties, J((A, B) will denote the additive group of homomorphisms of A iuto B, ~r B) its extension by Q (i.e. the vector-space ~ ( A , B ) ~ Q over Q), ~ ( A ) the ring of endomorphisms of A, ~0(A) its extension by Q. If i e J/0(A, B ) a n d ,(i)-~0 (which implies that A, B have the same dimension, since the " d e g r e e " ,(~) of i is not defined otherwise), then i-~ is defined and is in ~o(B, A). If ~ is a homomorphism of A into B, its transpose ~2 is the homomorphism of B* into A* defined by ti(C1Z)=C1 (~-~(Z)) for every Z e Q~,(B); this extends to an isomorphism of J(o(A, B) onto

J(o(B*, A *). If A is a polarized abelian variety, and X is a polar divisor of A, put, for every a ~ ~o(A), d=~o~t'%'~ox; then ~ - ~ a ' is an involutory antiautomorphism of the algebra (Ao(A), canonically attached to the polarization of A. The trace a being defined on ~o(A) as

[1955d]

265 On the Theory of Complex Multiplication

11

usual, we have a ( ~ a ' ) > 0 for every a-7~=0 in ~go(A). If A is the (canonically polarized) jacobian of a curve, then ~ - a ' is no other than the so-called "Rosati antiautomorphism" Let a be a homomorphism of an abelian variety A onto an abelian variety B of the same dimension; if Y is a divisor on B, and if we put X=a-I(Y), we have ~ox=~a.~o~.a; in particular, if Y determines a polarization on B, so does X on A. If A is polarized and X is a polar divisor of A, and if ~ is an automorphism of the non-polarized A, then it will be an automorphism of the polarized A if and only if there are integers m, m', both >0, such that m~x=m'%.~Ox.Ot; taking degrees on both sides, we get m = m ' . But this may be written as d~=SA and implies a ( ~' ) : a ~ A ) . As a(~'~) is a positive nondegenerate quadratic form on ~Zo(A), and the additive group of ~ ( A ) is finitely generated, this shows that the group of automorphisms of a polarized abelian variety is finite (a result originally due to Matsusaka, whose proof, based on a different idea, is to appear shortly). From now on, A will be a polarized abelian variety of dimension n; we usually write ~ , ~0 instead of J ( A ) , ~o(A); on ~o, we have the trace a and the antiautomorphism a ~ d . For every prime l, not equal to the characteristic, ~ has a faithful representation R, of trace a by endomorphisms of a free module of rank 2n over the l-adic integers; this can be extended to a representation R, of (~o by endomorphisms of a vector-space of dimension 2n over l-adic numbers. If the characteristic is 0, J / has a faithful representation R of trace a by endomorphisms of a free abelian group of rank 2n (viz., the fundamental group of the complex torus defined by A under any embedding of its field of definition into C); this can be extended to a representation R of ~Zo in a vector-space of dimension 2n over Q; and the representations R~ can be derived from R by extending the group (resp. the vector-space) on which R operates by the ring of /-adic integers (resp. by the /-adic number-field). Moreover, ~//may also be considered as operating on the Lie algebra of A, i.e. on the tangent vector-space to A at 0; if p = 0 , this can be extended to a representation Ro of ~o by endomorphisms of a vector-space of dimension n over any common field of definition for A and its endomorphisms. By embedding such a field into C, one finds that R decomposes over C into Ro and the imaginary conjugate representation Ro; if we call 60 the trace of R0, we have a=~o+~o. Let ~ , . . . , e~ be orthogonal idempotents in ~o, i.e. elements such that ~ = ~ for all i and ~ j = 0 for i=/==j; put ~ o = ~ A - ~ , ~ ; we can write -z~=o~,/m, where m is an integer and d0,'" ", al, are in ~ . Call

266

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A~ the image of A by a~; then it is easy to see that A is isogenous to Ao • "" 9• A~, and that ~(~)--2 dim (A~). Let 0 be a semi-simple commutative subalgebra of ~o; in terms of suitable orthogonal idempotents r ~ , it can be written as O=~,K~r where the K, are fields. As ~o has faithful representations with the rational-valued trace o, C has representations of the same type; this implies that, if ~---~$,~, is in C, with $, e K~ for l _ ~ i ~ h , we have ~($)=~,,~ Tr(~,), where Tr is the ordinary trace (taken in K, over Q for each i) and the ,, are integers > 0. If the A~ are defined as above, we have 2dim(A~)=a(r QJ, hence ~, ,,EK~ : Q~ ~ 2n. Assume now that ~. ~K, : Q~ ~ 2n ; then the latter inequality must be an equality, and we must have v , = l for all i. That being so, a representation of ~o of trace a is equivalent (over an algebraically closed field) to one in which all elements of C appear as diagonal matrices and in which the diagonal elements corresponding to some element of C are all distinct; then the commuter C' of 0 in ~0 i~ also represented by diagonal matrices, which implies that it is commutative and semi-simple; what we have said about 0 can now also be applied to 0', and it easily follows from this that C'=C. In particular (as Shimura also proved), if U/o contains a field K of degree ~ 2 n , K must be of degree 2n, must contain ~A and the center of ~-~o, and is a maximal commutative subalgebra of ~o. When that is so, A must be isogenous to a product B • . - . • B, where B is simple; in fact, if this were not so, ~ o would be the direct sum of algebras ~o(A,), the A, being proper subvarieties of A, at least one of which would have to contain a field isomorphic to K, while we have just shown that ~o(A,) cannot contain a field of degree > 2 d i m (A~). Assume now that A is isogenous to a product /~• . . . • of r factors B of dimension m, so that n=rm; then ~o is the ring of matrices of order r over the division-algebra ~o=~_~o(B). Call /c the center of ~0, which we identify with the center of ~o, so tha~ K ~ k; call , the degree of k, p~ the dimension of -~o as a vectorspace over k. As K is of degree 2n/, over k and is maximally commutative in ~4o, it is known that ~o, as a vector-space over k, must be of dimension (2n/~)~; this gives rg=2n/,, hence 2 r e = g , ; therefore a maximal subfield of -~0, containing k, is of degree 2m. If now we assume that p = 0 , ~o must have a faithful representation by rational matrices of order 2m; as it is known that the order of such a representation must be a multiple of ,p~, this gives p = l , ~ o = k . Moreover, any polarization of B determines an automorphism ~-->U of k, of order 1 or 2, such that Tr ($$')~ 0; if ko consists of the elements

[1955d]

267 On the Theory of Complex Multiplication

13

of k invariant under that automorphism, this implies that ko must be a totally real field, and that k is either ko or a totally imaginary quadratic extension of ko. As before, call R0 the representation of ~o determined by the Lie algebra of A; call So the representation of -~0 which is similarly defined; then the representation of k of trace Tr~/Q decomposes into So and 20; this implies that k=/=ko and that So is the direct sum of m one-dimensional representations of ~o, i.e. of m isomorpbisms q~, of k into the universal domain, inducing on k0 all its distinct isomorphisms into the algebraic closure Q of Q. Moreover, Ro induces on k the representation (n/m)So; this implies that Ro induces on K the sum of the one-dimensional representations ~ ( 1 ~ ~ ~ m, 1 ~ i ~ n/m), where, for each i, the e ~ are all the isomorphisms of K into Q which induce q~ on k. Still assuming p = 0 , consider now any field K of degree 2n containing a totally imaginary qua~lratic extension k of a totally real field ko, the latter being of degree m. Let the ~ , be all the isomorphisms of ko into Q; for each ~, let ~ be an isomorphism of k into Q, inducing -#~, on ko, and let the ~oz~,for 1 ~ i ~ n/m, be all the isomorphisms of ~" into Q which induce e~ on k. We ask for the abelian varieties A of dimension n such that ~o(A) contains a field isomorphic to K and that Ro induces on K a representation which is the sum of the ~o~. Taking C as universal domain, it is easily seen that A is uniquely defined by these conditions up to an isogeny over C and that it can be constructed as follows. Consider the mapping $-4(~0~(~)) of K into C~; let M be the image under that mapping of a " m o d u l e " m in K, i.e. of a free abelian subgroup of rank 2n of the additive group of K; then the complex torus C~/M defines an abelian variety A with the required properties. If ( $ , , . . - , $./~) is a basis for K considered as a vector-space over k, we may in particular take m = ~ , ~ n , where n is a module in k; then one finds that A is the product of n/m varieties B of dimension m. This shows that A cannot be simple unless n==m. By a CM-extension of a totally, real field Ko of degre n over Q, we shall understand a system (K; {~}) consisting of a totally imaginary quadratic extension K of Ko and of n isomorphisms ~oz of K into Q, inducing on K0 all the isomorphisms of Ko into Q.

If we

consider Q as embedded in C, K can then be written as Ko(Q, where is such that -{2 is a totally positive element of _~ and that all the ~o~({) have a positive imaginary part; $ is uniquely determined by that condition up to a totally positive factor in Ko; conversely,

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[1955d] 14

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the CM-extension (K; [~ox}) is uniquely determined by I(o and $ and will also be written as/(o((5)). The CM-extension Ko((4)) will be called primitive if it cannot be written as Ko((~)) with 4~ lying in a proper subfield of K0; Ko((4)) is primitive if and only if there is no conjugate ~' of ~ over Q, other than ~, such that C/~ is a totally positive algebraic number. The proof given above shows that every CMextension of a totally real field of degree n determines a " c a t e g o r y " of mutually isogenous abelian varieties of dimension n, and that the latter are simple if and only if the former is primitive. In consequence, it seems reasonable to deal first with the simple abelian varieties belonging to primitive CM-extensions, even though some important results have already been obtained by Taniyama for more general cases. From now on, let (k; [~o~}) be a primitive CMextension, given once for all, of a totally real field ko of degree n; we consider the abelian varieties A of dimension n which belong to it in the sense described above. This means that there is an isomorphism ~o of k onto JT0(A) such than Roo~O decomposes into the sum of the ~o~. As (k; [~o~}) is primitive, it is easily seen that ~ is uniquely determined by this condition, so that it may be used to identify k with ~0(A); this identification will be made from now on. Then the ring ~4(A) is identified with a subring r of the ring o of all integers in k. If K is a field of definition for A and for all the endomorphisms of A, k will have a representation of trace ~_] ~o~ by matrices of order n over the field K. One finds that, for k to have such a representation, it is necessary and sufficient that K should contain the field k, generated over Q by the values taken by that trace on k. Conversely, if K is a field of definition for A, containing kt, it must be a field of definition for all the endomorphisms of A. One should observe that k~ need not contain k. We now consider polarized abelian varieties belonging to a given CM-extension. The rank of such a variety, for p = 0 , has been defined above as the integer r=,(~Ox) ~/2 if X is a basic polar divisor. By using the representation of our varieties as complex toruses when C is taken as universal domain, one finds that, f o r a given CM-extension (k; [~o~}), a given ring of endomorphisms r, and a given value of the rank r, there is at most a finite number of distinct types of polarized abelian varieties with respect to isomorphism over the universal domain. If A is such a variety, its group of automorphisms is the multiplicative group of the roots of unity in r. Call o~ a generator of that group, and N its order. Let K be a field of definition for the polar-

[1955d]

269 15

On the Theory of Complex Multiplication

ized variety A and for its automorphism co; let X be a positive polar divisor on A, of which we may assume that it is rational over K and that it determines an ample complete linear system; after replacing X, if necessary, by the sum of its transforms by the N automorphisms of A, we may also assume that it is invariant by (o. Now identify A with its image under the projective embedding of A defined by the complete linear system determined by X; then ~ is induced on A by an automorphism ;2 of the ambient projective space which leaves invariant the hyperplane Ho such that Ho.A=X. If we take the homogeneous coordinates (X0,. 9-, X~) in that space so that H0 is defined by 2(o=0, $2 will appear as a linear substitution: m

(Xo, xl,

. . .,

(Xo,

clix

, . . .,

cmiX

).

For any r ~ m, let the Pr~ be a base for the space of homogeneous polynomials of degree r N in the X~ which are invariant under that substitution; let U~ be the locus of the point r with the homogeneous coordinates Pr~(x), in a projective space of suitable dimension, when x is a generic point of the ambient space of A. By adjoining the N-th roots of unity, if necessary, to the groundfield, and writing the substitution ~9 in diagonal form, one shows that all the varieties Ur are isomorphic to one another. Call U any one of them; call V the image of A in U b y r and call F the mapping of A onto V induced by ~ ; we say that 11, together with the mapping F, is the quotient of A by the group generated by ~o. Now, for each one of the finitely many types of varieties belonging to given data (k; [~o~}), r, r, we can construct a representative A by means of a complex torus. A variety A obtained by this method need of course not be defined over an algebraic number-field. However, I have given (in a paper j u s t published in the Amer. J. of Math".) a criterion for a variety, defined over a field/(1, to be isomorphic to a variety defined over a subfield /Co of /(1; by using this criterion, it is easily seen that, for each type of varieties belonging to the given data, there is a representative which is defined over an algebraic number-field. As this is only a special case of some important unpublished results of T. Matsusaka on the field of moduli of a polarized abelian variety, I need not give more details here; however, it will be worthwhile to consider more closely the case in which A is defined over an algebraic number-field, even though Matsusaka's results could also be applied to that case. Let therefore A be a 1) A. Weil, The field of definition of a variety, Amer. J. of Math., 78 (1956), pp. 509-524.

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[1955d1 16

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polarized abelian variety belonging to the given data and defined over an algebraic number-field which we may assume to be a finite Galois extension K of kt. Let Ko be the field of the elements of K which are invariant under all those automorphisms of K over ks which transform A into a variety isomorphic to A; the degree of Ko over kt is at most equal to the number of possible types of varieties belonging to the given data. Let a be an automorphism of K over /(o; there is an isomorphism ao of A onto A ~ uniquely determined up to an automorphism of A and algebraic over K; therefore every conjugate of a~ over K i s of the form a.o~~. Call V t h e quotient of A by its group of automorphisms, and F the canonical mapping of A onto V; then there is an isomorphism /~ of V onto V ~ uniquely determined by the condition / ~ o F = F % a ~ ; it must be the same as its conjugates over K, and is therefore defined over K; and we have p~o=fl~o/~, for any two automorphisms r, a of K over K0. Applying the results of the paper quoted above, we conclude from this that there is a variety V0 defined over I(o and an isomorphism ~o of V0 onto V, defined over K, such that ~o=q~%~o-1. Let now A1 be any variety, isomorphic to A, defined over an algebraic number-field /(1 containing kt. If K~ does not contain K0, there must be an automorphism r of the field of all algebraic numbers over /s which does not leave invariant all elements of Ko; then, if a~ is an isomorphism of A onto A1, its transform by ~ is an isomorphism of A ~ onto A , so that A and A ~ must be isomorphic; but this contradicts the definition of K0. Therefore we have /(1 ~ Ko. If now V~ is the quotient of A~ by its group of automorphisms, F~ the canonical mapping of A1 onto V , P~ the isomorphism of V onto V~ such that / ~ o F = F ~ o ~ , and a any automorphism of KK~ over /s we have/~1-/~o/~o , hence ( ~ o ) ~ which shows that /3~q~ is an isomorphism of V0 onto V , defined over K , Call z a generic point of A over K, and w the corresponding point on V0, i.e. w=~o-~(F(z)). To each primitive N-th root of unity ~, we can associate the set of those functions 0 on A, defined over Q, which satisfy 0(o~z)=r for each such function, there is a funct i o n f on Vo such that f(w)=O(z)~; call ~ the set consisting of those functions f on Vo. If f ~ ~ , and ~ is another primitive N-th root of unity, , being an integer prime to N, then ~ consists of the functions f~h N, where h runs through the set of all functions on V0, defined over Q; also, if an automorphism of Q over /40 maps ~ onto r it will transform the functions in ~ into the functions in ~ . We say that V0, together with the sets of functions !~, is the Kummer wriety attached to the given type of abelian varieties (for a more

[1955d]

271 On the Theory of Complex Multiplication

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general definition, valid for arbitrary polarized abelian varieties, we refer the reader to a forthcoming publication by Matsusaka); and we say that this Kummer variety is defined over K0. It is clear that a type of abelian varieties is completely determined by its Kummer variety. We can now formulate the basic problems of complex multiplication for simple abelian varieties: I. Characterize the fields Ko for the types of abelian varieties belonging to given data (k; [qozt), l" and r. II. For each such type, characterize the fields generated over Ko by the images on Vo of the points of finite order on a variety A of that type. III. Determine the zeta-function of any abelian variety of the given type, over a field of definition of that variety containing kt and therefore Ko. For n - - l , the complete solution of problems (I) and (II) is given by the classical theory of complex multiplication, and problem (III) was solved recently by Deuring. For arbitrary n, Taniyama has now solved a problem which includes the general case of (III) as a special case; the independent and overlapping investigations of Shimura, Taniyama and myself give the solution of (I) and (II) in the case r=L~; and one may hope that the general case will not offer insurmountable difficulties any more. The basic tool here is Shimura's theory of reduction modulo a prime ideal, by means of which our problems can be reduced to problems on abelian varieties over finite fields. I shall sketch briefly the main ideas involved here. As above, let A be a variety of one of the given types, defined over a field K containing kt. Shimura's theory shows that, for almost all prime ideals ~ in /( (i.e., for all except a finite number), one can reduce A and its endomorphisms modulo ~, obtaining an abelian variety A(~3) of dimension n defined over the finite field with N(~) elements and an isomorphism of r = , d ( A ) into ~J4(A(~)). Then the Frobenius endomorphism of A(~) (induced by the automorphism of the universal domain which raises every element to its N(~)-th power) commutes with every element of the image of r in ~?/(A(~3)), since such an element is an endomorphism of A(~) which is defined over the field with N(~3) elements. By the results proved above, this implies that the Frobenius endomorphism can be identified with an element 7r of the field k:=~ ~o(A), and more precisely with an integer in k (not necessarily in r). The mapping ~-~Tr determines the zeta-function of A over K; and Taniyama has shown that a more detailed study of

2 72

[1955d] 18

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the properties of this mapping leads directly to an expression of the zeta-function in terms of Hecke L-functions attached to characters of type (Ao) of the field K (cf. p. 4 of this volume); as could be expected, these are characters which come from the quadratic extension k of the totally real /co (in the sense of formula (4), p. 4). In fact, the connection between characters " o f type (Ao)" and abelian varieties with complex multiplication appears to be so close that it can hardly be accidental; and any f u t u r e arithmetical interpretation of the characters of type (Ao), corresponding to the interpretation given by class-field theory for the characters of finite order of the id~le-class group, ought to take complex multiplication into account. As to problems (I) and (II), I will consider only the case ~=-o. The method sketched below could perhaps be applied without substantial changes to a ring r such that r=~ and that the classes of ideals in r which belong properly to r (i.e. which consist of ideals m such that, in Dedekind's notation, m : m = r ) form a group. If n = l , all the subrings of o have these properties; for n > l , it does not seem to be known whether any proper subring of o has them; in order to treat the general case of problems (I) and (II), one will presumably have to rely more heavily upon the /-adic representations R~ than is done here. We first have to look more closely into the relation between k and k,. Taking C as universal domain, and taking k to be embedded in it, call k' the compositum of k and all its conjugates over Q; call G the Galois group of k' over Q; call H, H, the subgroups of G corresponding respectively to the subfields k, kt of M; call ~ the automorphism $-->~ of k'. Call S the set of those automorphisms of k' over Q which induce on k one of the isomorphisms ~p~. Thus S is the union of cosets with respect to H, i.e. we have HS=H: we have G=S""S,:,; more generally, if o' is any transform of 6 by an inner automorphism of G, we have G - S = S a ' = a ' S . The assumption that (k; [~o~}) is primitive amounts to saying that H consists of all the elements 7 of G such that 7 S = S . On the other hand, H, consists of Che elements 7' of G such that S,y'=S. The subgroup of G corresponding to ko is H"-"Ha; and one finds that Ht'-%Ht is a group, corresponding to a totally real subfield of k' of which k~ is a totally imaginary quadratic extension. Write S as the union of distinct cosets t~-lH, with respect to H~; for each t~, let ~ be the isomorphism of ks into k' induced by ~ on k~. Then (k,; [~,}) is a primitive CM-extension, and the relation between (k; [~o~}) and (kt; [~,}) is symmetric. This suggests that one should look for a relation between the categories

[1955d]

2 73 On the Theory of Complex Multiplication

19

of abelian varieties belonging to these CM-extensions; as to what this may be, I have no conjecture to offer. Before coming back to our problems, we must also observe that, for any abelian varieties A and B, J((A, B) is a right J(A)-module and a left ~(B)-module. If cp is an isomorphism of a commutative subring C of ~yT(A) onto a subring of ~ ( B ) , and if one considers only those e cJ((A, B) for which s for all 7 e C, the distinction between right and left is not necessary. In particular, consider two abelian varieties A, A' of dimension n, belonging as above to the primitive CM-extension (k; {q~}). Then they are isogenous; and, by considering the operation of J((A, A') on the Lie algebra of A, one sees that a S = Sa for all ~ e J/o(A, A') and all $ e k. Thus J(o(A, A') is a vectorspace over k; as such, it is clearly of dimension 1; and J/[A, A') is a module over the ring generated in k by .Y/(A) and ~ ( A ' ) . If now we assume that J ~ ( A ) = ~ ( A ' ) = o , then J((A, A') is an o-module, isomorphic to an o-ideal whose class is uniquely determined; if a is a nonzero element of ~o(A, A'), and if ~t is the set of the $ e k such that $~ e J((A, A'), ~ is an ideal in that class. If we take a e SEt(A, A'), we have 1 e a, so that a-~ is an ideal in o. In particular, the dual A* of A is isogenous to A; and, if A is polarized and Y is a basic divisor on A, ~o~ is in ~a((A, A*). If we assume A to have o as its ring of endomorphisms, the same will be true of A*, and the set of the S e k such that Sq~ye ~r(A, A*) will be an o-ideal in k. One finds, in fact, that it can be written as f;~0, where fo is an ideal in the ring of integers of /co, and that the rank r of A is r =N(fo). When that is so, we say that A belongs to fo; it is clear that all the conjugates of A over kt will belong to fo. Thus, in discussing our problems (I) and (II) for r=o, we may confine our attention to those types which belong to (k; {~o~}), r = o and a fixed fo. Let A, A' be two such varieties; let Y, Y' be basic divisors on A, A'; if ~ e J((A, A'), and if we put Z = ~ - ' ( Y ' ) , q~;~o~ will be in k; one finds that in fact it must be a totally positive integer; call it f(~). Take an a-J=0 in J/o(A, A'), so that we can write J((A, A')=a,~, where a is an o-ideal in k; then one finds that there is a totally positive peko such that pr

and that f(Sa)=p$~ for all Sea.

One may

call this a positive hermitian form on t~. The form psi, defined on ~, and the form p~$~, defined on an idea] a~, will be called equivalent if there is a ~ e k such that a~=~-~a and g~=p~; the class determined for this equivalence relation by the form pS~ on (~ will be denoted by (a ; p). That being so, the class of the form f($a) on the ideal a deter-

274

[1955d] 20

A. WEIL

mined by ~ ( A , A')=ae~ is independent of the choice of ~ and will be denoted by {A': A}; A and A' are isomorphic if and only if this class is (o ; 1). On the classes of forms, we define a group law by putting

(a ; p). (a'; p') = (~';

pp').

Then, if A, A', A" all belong to the same type, we have: {A": A} = {A": A'} 9[A': A}; and, if r is any automorphism of f~ over kt, we have {A'~: A *} = {A': A}. It immediately follows from this that every field Ko occurring in problem (I) for r = o is abelian over ks, with a Galois group which is isomorphic to a subgroup of the group of classes of forms (a; g). Take a field of definition K for A, A' and their endomorphisms and homomorphisms; again by Shimura's theory, we can reduce all of these modulo almost all prime ideals in K. For sueh a prime ~, d((A, A') is mapped isomorphieally onto its image in d f ( ~ ) = ~ ' ( A ( ~ ) , A'(~)) and may be identified with that image; similarly we can identify d/o(A, A') with its image in the extension of d((~) by Q. One sees at once that an element of the latter set is in d(o(A, A') if and only if it commutes with all elements of k. Now put ~ " = J t ( ~ J ) ~ o ( A , A'); this is clearly an o-module containing d((A, A'); we show that JC'=~C((A, A'). In fact, assume that this is not so; as both are o-modules, there will be a $ in k and not in such that ~r A')cd('. But (e.g. by using a representation of A, A' as complex toruses over C) one can see that there are elements a~ of ~'(A, A ' ) a n d elements a'~ of d((A', A) such that 3 ~ = ~ , a ~ (this is a speeial ease of the faet that, if A, A', A" are three varieties of the given type, d((A, A") is no other than the tensor-produet, taken over o, of the o-modules d((A, A') and df(A', A")). This gives $=~,a~.(&r~), so that ~ must be an endomorphism of A(~), which is absurd. Now let Pc be a prime ideal in k,; we assume that it is not ramified in K0 and also that it has in a suitable field K a non-exceptional prime divisor ~, i.e. one modulo which one can reduce A, all its conjugates over k~, and the endomorphisms and homomorphisms of these varieties. Put q=N0~t). Take for A' the transform of A by an automorphism T of Q over k~ which induces on K0 the Frobenius substitution for p,. By what we have seen above, the Frobenius homomorphism of A(~) onto A'(~), induced by the automorphism x ~ x q of the universal domain, will be the image of an element ~ of J((A, A'); and we have f(w)--q. Then, if we put 3'(A, A ' ) = q - ~ , q is an ideal in o, such that q~=q% and we have {A': A}--(q-~; q).

[1955d]

275 On the Theory of Complex Multiplication

21

By class-field theory, an abelian extension is completely determined by the knowledge of the Frobenius substitution for almost all prime ideals; therefore (I) will be solved if we determine the correspondence p~-> c~. Let m be a multiple of the order of the Frobenius substitution for ~ in Ko; then r~ transforms A into a variety A1 isomorphic to A. Call al an isomorphism of A onto A~; this is uniquely determined up to an automorphism, i.e. up to a root of unity. Then the automorphism x - ~ x ~'~ of the universal domain induces a homomorphism of A(~) onto AI(~), which, as above, may be identified with an element of ~ ( A , A1); this can be written as Tral with 7r e o; ~r is uniquely determined up to a root of unity. Proceeding as above, we find that 7r~=q ~ and that 7r~=q% One should observe that, if N ( ~ ) = q ~ and m is a multiple of h, then A I ( ~ ) - - A ( ~ ) , so that in that case a~ can be determined uniquely by prescribing that it should reduce to the identity mapping on A(~); then ~r also is completely determined. Now, in order to find q, it is enough to determine the prime ideal decomposition of v in a suitable field for some suitable choice of m, e.g. for m = h. B u t this has been done by T a n i y a m a (cf. w3 of his contribution to this volume). The conclusion is that, for almost all t~, we have q=l~I ~ ( ~ ) provided of course ideals in subfields of U (the smallest Galois extension of Q containing k) are identified in the customary way with the ideals they generate in U. This formula contains the solution of problem (I) for r=~. One should observe that, while the prime ideal decomposition of ~r, together with the relation ~r~=q *'~, determines 7r up to a root of unity, this is not enough for the calculation of the zeta-function, where a more vrecise result (also contained in Taniyama's work) is required. For r=~, problem (II) can be treated by an entirely similar method. We consider the pairs (A, a), where A is an abelian variety of one of the types discussed above, and a is a point of finite order on A. If ~ is the ideal in % consisting of those ~ for which Sa=0, we say that a belongs to ~. The type of the pair (A, a) will be considered as given by the type of A, i.e. by the data (k; [~o~}), ~--o and f0, and by the ideal zT in o. Consider two such pairs (A, a) and (A', a'). If we write, as above, J((A, A ' ) = a a and f ( $ ~ ) - p ~ , an element ~o of a such that ~0~ maps a onto a'; it uniquely modulo a~ and is such that $00+~m=a. That define an equivalence relation between triplets a, p,

there will be is determined being so, we $0, where a,

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are as before and $o is such that $oO+an=a, by defining two triplets a, g, $o and a', p', $~ to be equivalent if there is a ~.ek such that a'=~-'% g ' = p ~ and ~ - ' ~ o mod. a'n; and we denote by (a; p; ~o) the class of such a triplet. Then the class of the triplet a, p, $o attached as above to the two pairs (A, a) and (A', a') is independent of the choice of ~; it will be denoted by {(A', a'} : (A, a)} ; the two pairs are isomorphic if and only if the class is (o; 1 ; 1). A group law between equivalence classes is defined by putting (a ; p; -:o)"(a'; p'; $'o)= (aa'; pp'; ~o~). Proceeding as above, one finds that the Frobenius substitution for Pt, in the field generated over Ko by the image of the point a on Vo, is (q_t; q; 1). This solves problem (II).

UNIVERSITY OF CHICAGO

[1955e] Science Franqaise?

L ' a n dernier, un h o m m e politique assez c o n n u d6jA, et qui l'est encore plus A pr6sent, s'6tonnait que, depuis fort longtemps, a u c u n s a v a n t fran?ais n'efit re?u de prix Nobel. L'occasion 6tait solennelle ; il exposait son p r o g r a m m e de g o u v e r n e m e n t . S'il faisait cette constaration, ce n'6tait pas seulement, sans doute, p o u r s'attfister d ' u n e situation si humiliante p o u r notre a m o u r propre national. C'est qu'il e n t e n d a i t que la prise du pouvoir lui d o n n e r a i t la ,facult6 d ' y p o r t e r remade. Or: sont-ils, ces rem6des ? Sont-ils fort cach6s ? E t ce qui est p o u r nos h o m m e s politiques u n sujet d'6tonn e m e n t en est-il un pour les initi6s ? Ici, je d e m a n d e la permission de raconter m o n histoire, ou p l u t 6 t de copier quelques passages d ' u n article que j'6crivis, fort jeune encore, A m o n retour d ' u n v o y a g e en Am6rique, il y a pros de vingt ans. Ce v o y a g e faisait suite A b e a u c o u p d'autres, en Allemagne, en Angleterre, en Italie, en Russie mSme (est-il p r u d e n t de l'avouer ?), et j u s q u ' e n Asie. Mon article fur soumis ~ quelques revues, qui le jug~rent impubliable ; il y a des v6rit6s qui ne sont pas bonnes A dire ; on ne se priva pas de me le faire savoir ; je ne profitai gu~re de la lemon... ~ J ' e n ai assez, 6cfivais-je. J ' a i m e v o y a g e r A l'6trang e r ; mes amis s a v e n t que m o n a m o u r - p r o p r e national n'est pas chatouilleux A l'exc~s, et j'ai pfis d6s longt e m p s l'habitude d'entendre, sans trop m'6mouvoir, qu'on discute, parfois sans bienveillance, de m o n pays, de ses h6tels, de ses femmes, de ses politiciens. Q u ' y Reprinted from La Nouvelle N.R.F., Paris, Imp. Crrtr, 3e annre, n~ 25, pp. 97-109.

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ferais-je, si t o u t cela est vrai ? que m ' i m p o r t e , si t o u t cela est faux ? Mais j'en ai assez, q u a n d je rencontre un ehimiste, qu'il me d e m a n d e i n v a r i a b l e m e n t : ~ Pourquoi la chimie fran~aise est-elle tombfie si bus ? ~; si c'est u n biologiste : ~cPourquoi la biologic fran~aise va-t-elle si mal ? ~; si c'est un physicien : ~ Pourquoi la physique fran~aise... ~; mais je n'ach~ve pas, c'est toujours la mfime question d o n t on me rebat les oreilles, et j'en suis encore ~ chercher la r@onse. Bien stir, q u a n d je d e m a n d e des pr~eisions, il arrive qu'on reconnaisse qu'il existe encore chez nous, duns tel ou tel domaine, des s a v a n t s fort distingu~s. Q u a n t ~ moi, m a t h ~ m a t i c i e n t o u t ~ fait ignorant de t o u t e science sinon de la mienne, je ne puis discuter ; souvent je me hasarde, en r@onse l'~ternelle question, ~ suggfirer : ccMais un tel... ? ~ et je cite un nom, illustre chez nous ; mais j'ai fini par y renoncer, car pour une lois qu'on m ' a v o u e : c~E n effet, il y a t o u t de mfime un tel ~, trop souvent, l'illustre coll~gne est assomm~ aussit6t d ' u n m o t dfidaigneux, d ' n n sourire, on s i m p l e m e n t d ' u n h a n s s e m e n t d'@aules... (( E n t e n d o n s - n o u s . Les mceurs de la gent universitaire, depuis qnelque douze ans que je la frfiquente, me sont un peu connues ; qu'on ne vienne pas me parler ici de jalousie, d'ignorance ou de pr~jugfi : on n'expliquera pas ainsi que tous ces coll~gues fitrangers, et s u r t o u t les jeunes, me posent toujours, ~ pen pros dans les m~mes termes, la m~me question. Ils reconnaissent sans se gfiner, de quelque pays qu'ils soient, l'importance des centres scientifiques anglais, am~ricains, russes, allemands ; ils s a v e n t apprficier aussi, parfois avec beaucoup de chaleur, les mfirites de tel s a v a n t fran~ais. Ce ne p e u t fitre la jalousie qui les fair tous parler ; il y a a u t r e chose ; il y a, faut-il le dire, un fait : ils doivent avoir raison. Cela est fficheux; expliquons-le c o m m e nous pouvons ; mais m i e u x v a u t le reconnaitre ; m i e u x v a u t mfime, c o m m e je le fais ici ~ dessein, s'exagfirer pent-~tre l%tendue du mal que de s o t t e m e n t fermer les y e u x . Assez parlfi (avec des majuscules) de Science F r a n ~ a i s e ; assez invoqufis les m~nes de Pasteur, de Poincar~, de Lavoisier : qu'ils se reposent en paix, car ils l'ont bien m~rit~, ce repos qu'on ne v e u t pas accorder leurs ombres ; la Science Fran~aise, apr~s tout, c'est nous, c'est les vivants, et leurs noms ne sont pas une

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m i n e d o n t on nous ait octroy6 la concession ~ perp6tuit6 ; si nous ne savons pas nous e x a m i n e r avee s~v6rit6, sans complaisance facile, d ' a u t r e s le font p o u r nous. Quelques-uns diront : c
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esdme, parfois avec respect. Je sais qu'on les c o m p t e sur les doigts, et que leur 6minence ne rend que plus sensible la platitude de la contr6e environnante. Pourquoi n'y en a-t-il pas plus ? D~s 1937, j'avais c m en apercevoir les raisons : ~cSupposons que, dans tel ou tel domaine, disons la th6orie des hombres (il ne me co~te rien d'en parler, elle n'est pas enseign6e dans les universit6s fran~aises), les maitres v~ritables soient venus ~ faire d6faut ; que les chaires les plus en vue et les positions dominantes se trouvent occup6es par des hommes, non pas ignorants ou sans comp6tence, mais sans 6clat, ou, chose peut-~tre plus grave encore, par de ces savants (ilssont nombreux, et, pour des raisons qu'il faudrait bien examiner, ils le sont tout particuli6rement dans les universit6s fran~aises) ~ qui quelques travaux brillantsont valu au d6but de leur carri6re une r6putation qu'ils n'ont pu on ne se sont pas souci6s de soutenir. Que va-t-il se passer, si de tels h o m m e s (charg6s d'honneurs, sans doute, et de titres) sont install6s an pouvoir ? Car, reconnaissons-le, c'est un pouvoir v6ritable qu'ils d6tiennent ; pouvoir de distribuer les places; pouvoir, plus important encore lorsqu'il s'agit de science exp6rimentale (et c'est pourquoi chaque jour en m e levant je remercie Dieu de m'avoir fair math6maticien), d'allouer les cr6dits de laboratoire et les moyens de recherche; pouvoir, de par les positions qu'ils occupent, d'attirer~ soi les jeunes, et de conserver pour soi les collaborateurs qui ~ d'autres sont refus6s. De ces jeunes, que va-t-il arriver ? quel est l'avenir d'une science dont l'enseignement est une lois tomb6 entre les mains de pontifes de cette esp~ce ? Maints exemples, que j'ai pu 6tudier (et non pas seulement en France, qu'on le croie bien; je ne crois pas tout parfait ailleurs,et j'ai observ6 en d'autres pays des ph6nom~nes tout semblables), permettent de donner de ce qui doit se passer une description assez pr6cise : le tableau clinique de la maladie (comme disent, je crois, les m6decins) est bien connu. De tels h o m m e s ne tardent pas ~ tomber en dehors des grands courants de la science : non pas de la Science Fran~aise, mais de la science (sans majuscule) qui est universelle; ils travaillent, souvent honn~tement, de tr6s bonne foi et non sans talent, ou, d'autres fois,ils font semblant de

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travailler, mais en t o u t cas ils sont 6trangers aux grands probl~mes, aux id6es vivantes de la science de leur 6poque ; et A leur suite, c'est t o u t e lear 6cole qui se t r o u v e 6gar6e dans des eaux stagnantes (parfois bourbeuses, mais, cela, e'est une autre histoire) : des jeunes gens bien dou6s passent les ann~es les plus importantes de lear carri~re scientifique, les premieres, ~ travailler des probl~mes sans port@ et dans des voies sans issue. I1 faudrait les envoyer ~ l'6tranger, ces jeunes gens, les initier ~ toutes les m6thodes, ~ toutes les id6es ; car, q u a n d bien m~me il s'agirait du maitre le plus 6minent, qu'est-ce qae l'61~ve d ' u n seal maitre ? Mais quoi ? L'on a trop peur de perdre des collaborateurs et des disciples, et, ~ leur place, de voir revenir des juges, des juges s6v~res. Qu'il est pr6f6rable de les garder aupr~s de soi, de s'en faire aider, de les maintenir, a u t a n t qu'il se peut, dans des voies trac6es! Qu'ils aient du talent, c'est b i e n ; qu'ils soient sages de plus, et (sans nuire la hi6rarchie ni ~ l'ordre d'aneiennet6) routes les voies leur sont ouvertes ; et, s'ils sont sages, le talent m~me apr~s t o u t n'est pas indispensable, une bonne petite chaire les r6compensera. ~> J'ai copi6 m o t pour mot, et m o n manuscrit de 1937 est 1~ pour le prouver. J'avais bien 6crit <; je n'avais pour cela d ' a u t r e raison que celle que j'en donnais, c'est-~-dire qu'il n'existait alors aueune chaire de ee titre. J'6tais assez naif pour esp6rer ainsi ne blesser personne, dans eet article or1 je blessais t o u t le monde. Je ne pr6voyais pas q u ' u n jour serait cr66e ~ la Sorbonne une chaire de th6orie des nombres, ni que ce serait au profit d ' u n administrateur chevronn6, ancien reeteur et directeur de ministate ; encore moins pouvais-je pr6voir que la n o m i n a t i o n de son successeur serait l'occasion d ' u n 6pisode qu'il v a u t la peine de raeonter iei, car il complete, fort h e u r e u s e m e n t du point de r u e du clinieien, fort f~cheusement de t o u t autre point de rue, le <
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mand, l'autre fran~ais. Nous les appellerons A e t ]3. Tous deux se sont partieuli+rement distingu6s en th6orie des hombres. A 6tait rest6 eu Allemagne jusqu'au d6but de 194o. I1 n'6tait pas juif. Ses sentiments d'hostilit6 au r6gime 6talent bien connus de ses coll~gues et n'6taient pas ignor6s des autorit6s universitaires; mais il n'avait jamais eu aucune activit6 politique et n'avait pas 6t6 inqui6t6. Peut&tre aurait-il quitt6 son pays plus t6t s'il n'avait pas pens6, par sa pr6sence, renforcer ce qui restait alors en Allemagne de pens6e libre ; il est vrai aussi qu'un voyage en Am6rique l'avait convaincu que le elimat intellectuel de ce pays lui convenait real. S'il se d6cida ~ 6migrer en 194o, ce qui n'alla pas sans difficult6s ni risques, ce fut sans doute qu'alors il d6sesp6ra de l'avenir de l'Europe. P e n d a n t la guerre, il fur trait6 par les Am6ricains en r6fugi6, c'est-~-dire assez real. Du moins y trouva-t-il de quoi vivre et poursuivre ses travaux, tandis que la plupart et les meilleurs des savants allemands qui avaient cherch6 un asile en France ~ la suite des premieres pers6cutions hitl6riennes avaient dfi en repartir faute de possibilit6s de travail. Vers la fin de la guerre, l'une des chaires si envi6es de l'Institute for Advanced Study, de Princeton, lui rut offerte ; il l'aecepta, et se fit citoyen am6ricain. Mais l'Allemagne se relevait de ses ruines mat6rielles et intellectuelles plus vite qu'on n'avait pu s'y attendre ; le travail seientifique y redevenait possible. Malgr6 la longueur de son s6jour en Am6rique, A n'avait pu s'aecoutumer ~ bien des aspects de la vie am6ricaine qui lui avaient d6plu d~s l'abord ; quelques-uns de ses amis rest6s en Allemagne s'en aper~urent. I1 n'en fallut pas plus. Bient6t, la plus c61~bre des universit6s d'Allemagne occidentale lui offrit une chaire. I1 ne pouvait ~tre question 1~ d'une situation mat6rielle comparable celle qu'il avait ~ P r i n c e t o n ; mais on salt que les universit6s allemandes peuvent, dans une certaine mesure, proportionner le traitement ~ la valeur et ~ la r6putation seientifiques ; on lui en offrit un fort sup6rieur ~ celui de la plupart de ses coll~gues allemands ; pour l'attirer, on lui offrit le remboursement de tous ses frais de d6m6nagement. Suivant la loi allemande, la nomination d'un 6tranger ~ une chaire universitaire

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lui conf6re de plein droit la naturalisation ; on offrit au professeur A, ~ son choix, de reprendre la nationalit~ allemande ou de conserver la nationalitfi am~ricaine. E t t o u t cela fur fair sans qu'il efit rien demandS, sans qu'il efit eu ~ rien demander. I1 accepta ; et sa pr~sence et son enseignement n ' o n t pas peu contribu~ ~ rendre l'Universit~ dont il s'agit une pattie du lustre qu'elle avait eu autrefois. Quant au Fran~ais, la d6claration de g u e r r e l ' a v a i t surpris 5 P r i n c e t o n ; mobilisable, il w i t l'avis de l'ambassade de France, qui lui r e c o m m a n d a de rester o~ il ~tait ; un professeur fran~ais ~ l%tranger, pensait-on alors assez raisonnablement, ~tait plus utile 5 la France q u ' u n soldat de plus sons l'uniforme. I1 passa done la guerre aux l~tats-Unis. Celle-ci finie, les postes les plus brillants lui furent bient6t offerts ; Princeton, Harvard, Columbia se le disputbrent. C'est dans cette derni6re universit~ qu'il se fixa, dans l'une des meilleures chaires qu'elle efit ~ offrir ; quelque t e m p s auparavant, il s%tait fait naturaliser am~rieain. MMs lui aussi se lassa des ~;tats-Unis, et dfisira rentrer en France. C'est ici que son histoire cesse de ressembler 5 la prficfidente. T o u t d'abord, en France, on n'offre pas une chaire nn savant, si distingufi soit-il ; il faut qu'il fasse acte de candidature ; il faut le plus souvent qu'il fasse ses visites de candidature, formalitfi destin~e principalement p e r m e t t r e ~ ceux d o n t il postule les suffrages de juger de la souplesse de son ~chine. Les amis du professeur B, mis au courant de ses intentions, a t t e n d i r e n t longtemps une occasion favorable. Plusienrs chaires devinrent vaeantes, mais chaque fois les jeux ~taient faits. Enfin le titulaire de la chaire de th~orie des nombres prit sa retraite ; les amis de B pens6rent qu'il ne convenait pas de difffirer plus longtemps. Pour eette chaire, B ~tait si fiminemment qualififi qu'il ne semblait pas que quieonque, en France ou ailleurs, pfit la lui disputer sans ridicule. Mais il fallait d'abord que cette candidature ffit recevable. La loi fran~aise n ' a d m e t plus, depuis longtemps d~j~, que nos universitfis puissent s'enrichir par des nominations de savants ~trangers, comme c'est l'usage dans presque t o u s l e s autres pays ; il e n e s t ainsi, q u a n d bien m~me l%tranger ne serait tel que par naturali-

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sation. Mais, par hasard, B n'avait pas perdu sa nationalit6 frangaise en acqu6rant la nationalit6 am6ricaine. Force fur ~ la Sorbonne d'enregistrer sa candidature. Alors se d6clencha une campagne d'une violence extraordinaire. On vit un membre de l'Institut monter en personne sur la br~che pour d6fendre la citadelle menac6e. On feignit de mettre en doute la valeur math6matique du candidat. Parmi les 6loges que la critique avait d6cern6s ~ ses ouvrages, on rechercha les r6serves et les objections de d6tail ; par un montage habile de citations tronqu6es, on composa un texte qui pflt impressionner d6favorablement les incomp6tents; or, comme c'est l'ensemble d'une Facult6 qui vote sur chaque nomination, routes sp6cialit6s r6unies (depuis les math6matiques jusqu'~ la botanique), c'est n6cessairement, en chaque cas, une majorit6 d'incomp6tents qui d6cide. On reprocha ~ B de n'~tre pas rentr6 endosser un uniforme en 1939; on mobilisa contre lui les (~anciens c o m b a t t a n t s ~ et (( anciens r6sistants ~ professionnels ; il n'en manque pas dans l'Universit6, dont toute la carri~re ne se fonde que l~-dessus ; et je ne parle pas de ces patriotes tardifs, toujours cherchant ~ faire oublier qu'ils se sont d6shonor6s, et obligeant par 1~ m~me, quoi qu'on en air, ~ s'en souvenir toujours. On gonfla les m6rites du suppl6ant du pr6c6dent titulaire de la chaire. On fit si bien que ce suppl6ant l'emporta. La seule consolation de B fur que son 61oignement l'avait pr6serv6 de participer ~ cette m~16e sordide. Ses amis s%taient charg6s pour lui des visites de c a n d i d a t u r e ; c'6taient des savants fort distingu6s, eux aussi; le r6sultat a prouv6 qu'ils auraient pu mieux employer leur temps. Mais l'histoire ne finit pas 1A. A tort ou ~ raison, le bruit se r6pandit que la direction de l'enseignement sup~rieur d~sirait r~cup~rer pour la France un math6maticien si 6minent et ne se refuserait pas ~ une cr6ation de chaire en sa faveur pour peu que la Sorbonne la demand~t. N'6tait-ce pas l'occasion pour ces Messieurs de r6parer leur erreur sans chagriner personne ? Si le minist~re n'avait pas les intentions qu'on lui pr~tait, du moins l'honneur de la Sorbonne serait sauf, ou presque. Mais non : c'6tait bien le talent trop distingu6 du candidat qui l'excluait. Nouveau vote, nouvel 6chec.

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E t le professeur B est toujours ~ l'Universit6 Columbia, qui s'eu f61icite et efit 6t6 bien en peine de le remplacer. Ainsi joue la loi de la cooptation des m6diocres, que je m'imaginais d6couvrir en 1937; loi d ' a u t a n t plus fatale qu'il faut ~ un h o m m e des qualit6s de premier ordre pour qu'il d6sire attirer aupr~s de lui ses 6gaux, au risque qu'ils lui soient sup6rieurs. U n h o m m e m6diocre, au contraire, cherchera toujours ~ s'entourer non pas seulement de m6diocres, rnais de plus m6diocres que lui ; il le faut bien, pour faire briller ses minces m6rites. Ce n'est pas d'hier que la plupart de IIos institutions scientifiques sont prises dans les rouages de ce m6canisme inexorable. La situation est-elle sans remade ? Peut-on imagiiier des r6formes qui en am~neraient le redressement ? Ce n'est pas douteux. I1 reste assez d'616ments sains dans le m o n d e scientifique fran~ais, il y a assez de talent parmi les jeunes pour permettre les plus s6rieux espoirs si on se d6cidait ~ faire le n6cessaire. J'eii ai assez dit pour faire comprendre qu'une telle r6forme ne p e u t partir que d'en haut. I1 y faudrait un acte d'autorit6 ; et elle se heurterait A la plus violente r6sistance de la part de la majorit6 des universitaires fran~ais, de l'Institut, du Coll~ge de France, de corps constitu6s et de personIIalit6s d o n t il est d'usage de ne parler en public que sur un t o n de profond respect. Peut-~tre, apr~s tout, n'y faudrait-il qu'encore un peu plus de courage que pour s'attaquer aux int6r~ts des viticulteurs ou au privilege des bouilleurs de cru. La politique, dit-on, est l'art du possible; oh, en pareille mati~re, est le possible ? Je ne suis pas politicien ; ce n'est pas m o n m6tier de le savoir. Rien de ce que je vais dire n'est impossible en soi, puisque t o u t cela se pratique sous nos yeux dans les pays qui sont ~ la t~te du m o u v e m e n t scientifique moderne. Sommes-iious encore capables de nous instruire leur exemple ? Je n'eii sais rien ; si nous ne le sommes pas, t a n t pis pour nous. Quelles sont donc ces r6formes qui pourraient nous tirer de la profonde orni~re oh nous sornmes ? I1 n'y a pas l~ grand myst~re ; tous ceux qui y ont quelque peu r6fl6chi sans pr6jug6 et de bonne foi saveiit bien ~ quoi s'en tenir l~-dessus. I1 suffira ici d'indiquer bri~vement quelques points essentiels.

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D'abord, il faut s'attaquer ~ une organisation vicieuse, qui fait de l'Universit6 de France un monstre hydroc6phale, dont la Sorbonne est la t~te difforme et les universit6s de province sont les membres exsangues. Lors de la r6forme de Liard, il est notoire que celui-ci c6da ~ des pressions 61ectorales en acceptant beaucoup plus d'universit6s qu'il ne le jugeait souhaitable. I1 disait, paralt-il, que cela n'avait pas d'importance, parce que la p l u p a r t mourraient d'elles-m6mes. I1 n'avait pas pr6vu que les autorit6s locales, municipalit6s, chambres de commerce, fi~res du prestige qui en rejaillissait sur elles, leur accorderaient t o u t juste le soutien n6cessaire pour les faire subsister et en tirer quelques menus services, sans bien e n t e n d u leur donner les ressources qui en auraient fait de vrais centres intellectuels. LA oil par hasard se forme en province un noyau scientifique int6ressant, il v6g6te faute d'6tudiants ; les bons 6tudiants se dirigent sur Paris ot~ ils se t r o u v e n t noy6s dans la foule et ne p e u v e n t que rarement tirer profit de l'enseignement de maitres d6bord6s de tous c6t6s. M~me dans les quelques domaines off la France tient encore son rang, il ne saurait ~tre question de trouver assez de maitres et de chercheurs pour m o n t e r plus de quatre ou cinq grands centres. Done, la premiere r6forme dolt consister ~ rabaisser la p l u p a r t de nos universit6s au rang de centres prop6deutiques interm6diaires entre le secondaire et le sup6fieur; ~ cr6er, en province, environ quatre grands centres scientifiques bien dot6s en h o m m e s et en moyens, dans des localit6s bien choisies qui ne seraient pas n6cessairement des grandes villes ; ~ d6charget Paris de son trop-plein sur ces centres par des mesures appropri6es, dont le d6tail ne serait pas difficile formuler. E n second lieu, il faut changer radicalement le mode de nomination des professeurs. Le mieux serait de s'inspirer du syst~me anglais et de m e t t r e toutes les nominations importantes entre les mains de comit6s restreints offrant un m i n i m u m de garanties d'impartialit6 et de comp6tence, comit6s qm devraient obligatoirement (comme il se fait en Angleterre, avee d ' a u t a n t plus de soin qu'il s'agit d'une chaire plus importante) consulter largement l'opinion scientifique internationale et en tenir le plus grand compte. Dans ces comit6s devraient

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SCIENCE FRAN~AISE ?

13

entrer pour une large p a r t des savants d6sign6s par le ministre et choisis eux-m~mes en t e n a n t compte de l'opinion internationale. Notons en passant qu'en Angleterre une visite de candidature serait suffisante pour disqualifier aussit6t un candidat. E n troisi~me lieu, mais en troisi6me lieu seulement, il faudrait donner, non aux universit6s actuelles, mais aux quatre ou cinq grands centres qu'il s'agit de constituer, non seumement des ressources, mais aussi une autonomie financi~re qui les m l t sur le m6me plan que les grandes universit6s anglaises, allemandes, am6ricaines, et que notre H a u t Commissariat de l'~nergie Atomique. Je ne veux pas me donner le ridicule ici de r6p6ter, apr~s t a n t d'autres qui le crient bien fort depuis vingt ans, que la science cofite chef. C'est vrai, encore qu'on ait pu assez souvent autrefois, et qu'on puisse peut-~tre encore (mais exceptionnellement) aujourd'hui faire avec des moyens modestes d ' i m p o r t a n t e s d6couvertes. Je ne veux pas rappeler les statistiques humiliantes qu'on a publi6es maintes reprises sur le b u d g e t de la recherche scientifique en France compar6 ~ celui qu'on y consacre ailleurs. ~cJe vous ferai de bonne chore, disait Maitre Jacques, si vous me donnez bien de l'argent. ~ I1 avait raison; et, q u a n d m~me il aurait 6t6 un fripon, cela n'aurait pas suffi ~ lui donner tort sur ce point. I1 faudra donc de l'argent, bieu de l'argent, pour les laboratoires, les biblioth~ques, le personnel subalterne. I1 faudra bien aussi se d6cider ~ payer d 6 c e m m e n t le personnel scientifique p r o p r e m e n t dit. I1 faudra permettre ~ nos grands centres scientifiques de recruter celui-ci par contrats individuels, comme le fait le H a u t Commissariat de l']~nergie Atomique et comme le font les grandes Universit6s 6trang~res. I1 faudra qu'on puisse, le cas 6ch6ant, n o m m e r ~ nos grandes chaires et ~ la direction de nos grands laboratoires des 6trangers qualifi6s. Les Anglais, d o n t les traditions scientifiques valent bien les n6tres, le font parfois et s'en t r o u v e n t bien ; la France l'a fait autrefois; pourquoi faut-il que notre amour-propre national en soit arriv6 ~ l'emporter sur notre int6r~t bien compris ? I1 faudra que les contrats que nos grands centres seront en mesure d'offrir leur p e r m e t t e n t d'entrer en concurrence, avec quelque chance de succ~s, avec les institutions similaires ~ l'6tranger.

288

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LA NOUVELLE N.R.F.

Bien entendu, lorsqu'on se sera r6solu ~ traiter conven a b l e m e n t nos savants, on sera en droit d ' a t t e n d r e d'eux qu'ils se consacrent honn~tement ~ leur enseig n e m e n t et ~ leurs recherches. Mais il ne sera pas besoin pour cela de r~glements draconiens. Ce n'est pas de gaiet6 de cceur que t a n t d'universitaires, chez nous, cherchent un suppl6ment h de maigres ressources dans des pratiques vari6es otl se consume leur t e m p s et leur 6nergie : cumul d'enseignements de bas 6tage, trust des examens et concours, et trop souvent raise ~ la disposition de l'ilidustrie priv6e de laboratoires officiellement consacr6s la science pure. E n Angleterre, en Am6riqlie, en Allemagne, l'industrie priv6e a ses propres laboratoires de recherche, souvent si largement con~us qu'il s'y fait de n o m b r e u x t r a v a u x scientifiques de grande valeur. E n France, les industriels, quand ils lie travaillent pas sur licences 6trang~res, t r o u v e n t trop souvent plus 6conomique de faire travailler ~ leur compte uli laboratoire universitaire en 6change d ' u n suppl6melit de t r a i t e m e n t d6risoire accord6 au professeur qui le dirige. Bien entendu, la liaison entre science pure et science appliqu6e est chose h a u t e m e n t souhaitable ou plut6t indispensable, mais qui ne s'obtient pas en 6touffant celle-l~ au profit de celle-ci par des arrangements qui constituent de v6ritables escroqueries aux d6pens de l']~tat. Assur6ment, bien d'autres questions se posent : recrut e m e n t des jeulies, r61e des c~grandes 6coles ,, liaison entre l'enseignement et la recherche. Je ne crois pas utile d'en discuter ici. Je ne pense pas qu'aucune d'elles puisse offrir de difficult6 s6rieuse dans un climat redeveliu favorable. Le redeviendra-t-il ? Se trouvera-t-il un chirurgieli pour m e t t r e sur la table d'op6ration un malade qui pr6t e n d qu'il ne s'est jamais mieux port6 ? S'il lie s'en rencontre pas, faut-il d6sesp6rer ? ou attendre le salut de l'ilitervention miraculeuse du g6nie ? ~Bien stir, disait m o n article de 193 7, le g6liie perce q u a n d m~me ; le g6nie se fait toujours sa place, ~ travers t o u s l e s obstacles; bien stir... (je n'en suis pas si stir que ~a). Oui, mais pour le g6nie m~me, que d'alin6es perdues ; quels retards, quelles sordides difficult6s ; et tous les autres, ceux qui auraient pu faire oeuvre utile, maintenir,

[1955e ]

289 SCIENCE

FRAN~AISE

15

?

en a t t e n d a n t la venue d u g6nie, une tradition honorable et parfois glorieuse, tous ces autres, quoi d ' e u x ? Souvent, ils s'aperqoivent des ann6es perdues ; un peu trop tard, ils se r e m e t t e n t ~ l'6cole ; ils t e n t e n t de se refaire une place dans la eolonne en marehe, q u a n d leur esprit a perdu sa souplesse et sa plastieit6 ; fls se hissent avec difficult6 ~ u n 6chelon off d ' a u t r e s a v a n t eux parvinrent, puis, l'effort fourni, ils y restent, ils sont d6pass6s. Ils y restent, et l'histoire recommence. Car voil~ Oil l'on se sent d6sesp6rer, le tragique cerele vicieux : l'histoire recommence... U n e fois provincialis6, une fois tomb6 darts l'orni6re, on y reste. Sauf miracle, bien stir ; ear l'esprit, c'est le m i r a c l e ; mais n ' y eomptons pas trop, ou plut6t, le miracle arrive ~ qui a u r a su le m6riter. ~ J e n'6tais gu6re optimiste en I937. J e ne le suis pas plus ~ pr6sent. ANDR]~ W E I L ,

Pro]esseur ~ l' Universitd de Chicago.

6159-x-I955. - - Imp. CR]~Tt~, C or b eil- Essonnes (S.-et-O.}.

[ 1956] The field of definition of a variety

Let V be a variety, defined over an overfield K of a groundfield k. Consider the following problems: ( P ) Among the varieties, birationally equivalent to V over K, find one which is defined over t~. ( P ' ) Among the varieties, birationally and biregularly equivalent to V over K, find one which is defined over ~. Froblems o~ these types arise, for instance, in Chow's recent work on abelian varieties over function-fields ( [ 1 ] ) , in my work on algebraic groups ( [ 4 ] ) , and also in the unpublished work of Shimura and of Taniyama on complex multiplication. Criteria for those problems to have a solution are implicitly contained in Chow's paper ( [ 1 ] ) and in Lang's subsequent note on a related subject ( [ 2 ] ) ; the purpose of the present paper is to develop them more explicitly. Without restricting the generality of the problem, we may assume that K is finitely generated over /c; we shall make the restrictive assumption that it is separable over /c. Then it is a regular extension of the algebraic closure /c' of k in K ; and ]c' is a separably algebraic extension of /~ of finite degree. Thus it will be enough for our purpose to discuss ( P ) and ( P ' ) , firstly when K is separably algebraic over k, and secondly when it is regular over k. As usual, we do not distinguish between mappings and their graphs. I n particular, we do not distinguish between a birational correspondence T between two varieties V and W (this being defined as a subvariety of V X W which satisfies certain conditions) and the m a p p i n g of V into W determined by T. The inverse mapping, T -~, is then a m a p p i n g of W into V or also a birational correspondence between W and V. To prevent misunderstandings, I take this opportunity for pointing out that (by abuse of language) I call T everywhere bireguMr only when T is biregular at every point of V and T -~ is so st every point of W; when that is so, T might more suitably be ~*Received January 16, 1956. 509 Reprinted from the Am. J. of Math. 78, 1956, pp. 509-524, by permission of the editors, 9 1956 The Johns Hopkins University Press.

291

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[1956] 510

A N DR 1:~' W E L L .

called an i.somorphi~m between V and IV, and a ~C-isomorphism if /C is a ticld of definition for V, IV and T.

Section I. Separably Algebraic Extensions of the Groundfield. 1. Let k be a separably algebraic extension of a groundfield he, of tlnite degree n. Call ~ the set of all distinct isomorphisms of /C (over /Co, i.e., leaving all elements of tee invariant) into the algebraic closure go of Ice; ~t consists of n distinct isomorphisms, including the identity automorphisnl of /c. I f a c ~ , and if oJ is any isomorphism over leo of an overfield of /C~, we denote by ~o~ the isomorphism of k defined by putting ~ ( ~ ) ~ for every ~ c k. Let V be a variety, defined over /C; assume that there is a variety Vo, defined over he, and a birational correspondence f, defined over k, between Vo and V. Then, for a e ~ t , r c ~ , the mapping f ~ . ~ f f o ( f r -1 is a birational correspondence between V r and V ~. We now modify our problem ( P ) as follows : (A) Let k be a separably algebraic extension of /co of finite degree; let ~ be the set of all distinct isomorphisms of k into ~o. Let V be a variety,

defined over k; for each pair (a,'r) of elements of ,~, let f~.~ be a birational correspondence between V ~ and V ~. Find a variety Vo, defined over he, and a birational correspondence f, defined over k, between Vo and V, such that f ~ . a ~ f~o (f~)-~ for all ac ~, r c ~. Tm~oREM 1. Problem (A) has a solution if and only if the f ~ are defined over a separably algebraic extensiol~ of /co atgd satisfy the following conditions : (i)

f , , p = f ~ . , , o f , , p for all p, ~, r i~t ~ ;

(ii)

f ...... ~ (f,.~)~ for all a, r in ~ and all automorphisms r of ko

over ~o.

Moreover, wheat that is so, the solution is unique, up ~o a birational transformation oft Vo, defined over ~co. The conditions are obviously necessary. I f the problem has two solutions this is a birational correspondence between Vo and Vo', defined over /C. Writing that (Vo, f) and (Vo',f') are solutions of (A), we find F ~ F ~ for all ~, r; thus, F is invariant under all automorphisms of ~o over /Co; therefore it is defined over /Co; this proves the unicity assertion in Theorem 1.

(Vo, f) and (Vo',f'), put F ~ f ' - ~ o f ;

[19561

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OF

A VARIETY.

511

Now assume that (i), (if) are fulfilled; then, if r r are in 3 and is an automorphism of ~o over the compositum of k ~ and k ~, (if) shows that f,.r is invariant under ~; if f~.r is defined over a separably algebraic extension of ko, this implies that it is defined over the compositum of k ~ and k ~. Let x be a generic point of V over k ; for each ~ e $t, put xo f,~.,(x), c being the identity automorphism of k. If we put p ~ r in (i), we see that fr162is the identity mapping of Vr therefore we have x~ ~ x. Let K be the compositum of the fields/cG for all a e ,~ ; this is a Galois extension of ko; call P its Galois group. Take any ~ e P ; as x and xe~ are generic points of V and V% respectively, over K, there is one and only one isomorphism o)* of K(x) onto K(x~) which induces ~o on K and maps x onto x~. As f .... is a birational correspondence, defined over K, we have K(x) ~K(x~,~), so that ~* is an automorphism of K(x). Applying (if) to anv extension of ~ to an automorphism of ~o, we get:

putting p : r and a : e o ~ in (i), we find that the right-hand side of this relation is x ~ ; therefore ~* maps x~ onto x ~ for all ~. From this it immediately follows that the mapping o~-->o~* is a homomorphism of P into the group of all automorphisms of K(x), and more precisely an isomorphism of P onto a group F* of automorphisms of K(x). Call ko(y) the field consisting of those elements of K(x) which are invariant under r * ; it is finitely generated over/~o ( [ 4 ] , App., Prop. 3). As K(x) is regular, hence separable, over K, and K is separable over ko, K(x) is separable over ko (Bourbaki, Alg., Chap. V, w 7, no. 4, Prop. 7) ; hence ko(y) is separable over ko. Any element of the algebraic closure of ko in ko(y) must be in the algebraic closure of K in K(x), which is K since K(x) is regular over K ; as such an element is invariant under P*, it must then be invariant under P, and so it must be in ko. Thus we have proved that k,)(y) is regular over ko. Call Vo the locus of y over ko. I f an element ~ of F induces the identity on k, o~* leaves x invariant; as P* is the Galois group of K(x) over ko(y), this implies that k(x) C k(y), so that we may write x ~ f ( y ) , where f is a mapping of Vo into V, defined over k. We have K(x) CK(y), hence K ( x ) ~ K ( y ) since K(y) is contained in K(x) by definition. This shows that f is a birational correspondence between Vo and V. Transforming the relation x ~ f ( y ) by any o,* in P*, and calling a the isomorphism of k induced on k by ~*, we get x,,~f~(y), hence f,,,=fr by (i), this shows that (Vo, f) is a solution of our problem.

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2. I n the introduction, we formulated, in addition to the problem ( P ) , a more precise problem ( P ' ) . We may modify the problem (A) similarly, by requiring f to be everywhere biregular; call (A p) this modified problem. For (A') to have a solution, it is obviously necessary that the f , ~ should be everywhere biregular and satisfy the conditions in Theorem 1. Assume that this is so; let (Vo, f) be a solution of (A). Then, if (Vo',f') is a solution of (A'), the unicity of'the solution of (A) shows that we must have fr ~ f o F -1, where F is a birational correspondence between Vo and Vop, defined over/co. Thus problem (A') may be reformulated as follows. (B) Let k and ko be as in problem ( A ) ; let V and Vo be varieties, respectively defined over k and over ko; let f be a birational correspondence, defined over k, between Vo and V. Find a variety Vo' and a birational correspondence F between Vo and Vo', both defined over ko, such that the birational correspondence f o F ~ between Vo' and V ks everywhere biregular. I t is obvious that, if (B) has a solution, this is unique up to a koisomorphism; therefore the'same is true for (A'). I f (B) has a solution, then (,~ being defined as before) the birational correspondence f - o (f~)-i between V ~ and V ~ must be everywhere biregular for all a, r in ~. We will prove that this condition is also sufficient, at any rate if V is a k-open subset of a projective variety, defined over k. This will be an immediate consequence of the following result. PR01"OSlTIO.'r 1. Let k, ko, ,.~ be as in (A). Let Vo be a variety, defined over ko; let V be a projective (resp. a~ne) variety, defined over Is; let f be a birational correspondence, defined over Is, between Vo and V. Then there is a projective (resp. affine) variety W and a birational correspondence F between Vo and W, both defined over 1% such that Fof-~ is biregular at every point of V where the mappings f~ o f-~ are defined for all a e ,~. Let S be the ambient space of V, projective or affine; f may be regarded asia mapping of Vo into S. Call a l ~ , a 2 , " 9 ",an the elements of ,~, and put F~ ~ (f%. 9 -, f~,,) ; this is a mapping of Vo into the product S )< 999)< S of n factors equal to S, and is defined over the compositum K of the fields k ~. It is clear that F~ o f-~ is defined wherever all the f~ o f-~ are defined. Let x be a generic point of Vo over /Co; let W1 be the locus of Fz(x) over K ; put u = F I ( ~ ?) = (Xl,"" ',X~Z). AS ffl=tE, we have x ~ f ( x ) , so that the image of u by the mapping f o F~ -~ is x~; this shows that f o F~ -~ is the mapping induced on W~ by the projection of the product S X" " ")< S onto its first factor, and is therefore everywhere defined. Thus the birational correspon-

[1956]

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OF

DEFINITION

OF

A

VARIETY.

513

dence F1 o f-~ between V and W1 is biregular wherever all the f~of-~ are defined. Let zl," ' ' , z ~ be ~ points of S ; if S is the projective m-space, p u t z ~ (z~o,- 9 ", z~,~) ; and let z' be the point, in a projective space of suitable dimension, whose homogeneous coordinates are all the monomials z~,lz2~,. 99 z,,~, with 0 ~ m ~ m for every i. I f 8 is the affine m-space, put zt ~ (z~l,. - . , z ~ ) , put Z~o~ 1 for 1 --~ i _< n, and let z' be the point, in an affme space of suitable dimension, whose coordinates are the same monomials as before. I n either case, put z ' ~ , t , ( z , . . . , z ~ ) ; it is well-known t h a t ~ is an everywhere biregular m a p p i n g of S X " ' " X S onto its image in projective (resp. affine) space. P u t now F2 ~ o F~; then F_. is a birational correspondence between Vo and W2 ~ r and F 2 o f - 1 is biregular wherever all the f,rof-~ are defined. i f S is projective, let (1, f ~ ( x ) , . . . , f , , ( x ) ) be a set of homogeneous coordinates for f ( x ) ; the f~ are functions on Vo~ defined over b. P u t fo ~ 1. Then we have F , ~ (go," 9 ",g~), where the gp are all the monomials

I f ~ is an automorphism of K over ~, gp~ is again one of the gp, which we m a y write as g~(p); the m a p p i n g ~ - - > ~ ( p ) determines a representation of r (the Galois group of K over k) as a group of permutations on the gp. F o r a given p, let -/p be the subgroup of r determined by co(p) ~ p ; then, for o e F, o)(p) takes a number of distinct values equal to the index dp of 7p in r . I f Kp is the subfield of K consisting of the elements of K invariant under ~,p, gp is defined over Kp ; therefore, if ( ~ , " 9 ", ~ p ) is a basis of Kp over bo, we may write gp ~ ~ ~hp~, where the hp~, for 1 ~ v <--dp, are functions on Vo, 1s

defined over /co . Then we have, for all r e P : y

If, in this relation, we take for (o a set of representatives of the dp cosets of ),p in F, we get a linear substitution expressing the dp distinct functions g~(p) in terms of the dp functions h ~ ; and, since K~ is separable over /co, t h a t substitution is invertible. F r o m this it follows immediately that, if we call F ( x ) the point whose homogeneous coordinates are all the functions h ~ (where p runs through a set of representatives for the classes of equivalence determined by the p e r m u t a t i o n group r on the set {0, 1," 9 ",r}, and where, for each p, we take l ~ - - ~ d p ) , F is of the form ~I, oF2, where 9 is an automorphism of the ambient projective space of W2. I f S is affine, we p u t

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f ~ ( f ~ , ' ' ",f,~), f o ~ l , and we define F by the same formulas as in the projective case, but regard it as a mapping of Vo into an affine space; then we have again F ~ ~I, oF2, ~I, being now an automorphism of the ambient afflne space of W2. In either case, the mapping F is defined over ko; if W is the locus of F ( x ) over ko, W and F have the properties required by our proposition. S. Before applying this to the problems {A') and (B), we need a general lemma : LEMMA ]. Let f be a birational correspondence between two varieties U and V ; let k be a field of definition for U, V, f. Then the sets of points where f and f-1 are respectively biregular are k-open, and f determines a kisomorphism between them. Call U' the set of points of U where f is defined, U'" the set of points of U where it is biregular; call V" the set of points of V where f-1 is defined, W' the set where it is biregular. By [4], App., Prop. 8, U' and V" are/c-open. Call f' the restriction of f to U ' X V' (i.e. the birational correspondence between U' and V' whose graph is the set-theoretic intersection of the graph of f with U ' } < V ' ) . If f is biregular at a point a of U, it is defined at a, so that a e U ' ; and, if we put b = f ( a ) , f-~ is defined at b, so that b e V ' ; therefore (a, b) is on the graph of f', and f' is defined at a. Conversely, let a be a point of U' where f' is defined; put b = f ' ( a ) ; then b is in V', so that f-~ is defined at b; as U' and V' are open, f is then defined at a, and we have b = f ( a ) ; thus f is biregular at a. This shows that U" is the set of points of U' where f' is defined; similarly V " is the set of points of V' where f'-~ is defined; this implies that they are/c-open. If f " is the restriction of f to U" }( V", it is everywhere biregular by definition (i. e., f " is biregular at every point of U", and f"-~ is so at every point of V"). In order to formulate our results on problems (A') and (B), we will say that a variety U, defined over a field k, is projectively (resp. a~nely) embeddable over k if it is k-isomorphic to a k-open subset of a projective (resp. affine) variety, defined over /C. TtlEOREM 2. Problem (B) has a solution (Vo',F') provided V i.~ projectively embeddable over/c and the birational correspondence f~ o f ~ between V and V ~ is everywhere biregular for every isomorphism cr of k over ~'o into #o. When that is so, Vo' is projectively embeddable over It(;; it is a~n~ly embeddable over k o if V i~v so over k.

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FIELD

OF D E F I N I T I O N

OF A V A R I E T Y .

515

We may assume V to be a /c-open subset of a projective (resp. affine) variety, defined over /C. Take W and F as in Proposition 1; then F o f - ~ is biregular at every point of V; therefore, by Lemma 1, it is a /c-isomorphism between V and the /c-open subset Vo' of W where f o F -~ is biregular. As the f~ o f-~ are everywhere biregular Vo' is also the subset of W where f~ o F -~ is biregular, for every (r; therefore it is invariant under all automorphisms of ~o over /co, so that it is /co-open, by [4], App., Prop. 9. Then, if F ' is the restriction of F to Vo )< Vo', (Vo',F') is a solution of (B). TI~EORE~ 3. Problem (A') has a solution, i.e., problem (A) has a ,~'olution (Vo, f) for which f is everywhere biregula~, provided V is pro]eetively ernbeddable over /C and the f~., are everywhere biregular and satisfy the conditions in Theorem 1. When that is so, Vo is projectively embeddable over/co; it is a]~nely ernbeddable over /co if V is so over /C. The solution is unique up to a ~co-iSomorphism.

Section II. Regular Extensions of the Groundfield. 4. Let l~ow /,~ denote the groundfield. Let T be a variety, defined over /C; let t be a generic point of T over k. When we denote by Vt a variety, defined over k ( t ) , we will agree, whenever t' is also a generic point of T over /c, to denote by Vt, the transform of Vt by the isomorphism of k ( t ) onto k(t') over /C which maps t onto t'. Similarly, if a mapping, defined over k(t)~ is denoted by re, ft' will denote its transform by the same isomorphism; if t, t', t" are three independent generic points of T over k, and ft',t is a mapping, defined over /c(t, t'), we denote by ft",e' the transform of ft'.t by the isomorphism of /c(t,t') onto /c(t',t") over /C which maps ( t , t ' ) onto

(t', t") ; etc. Let Vt be a variety, defined over k ( t ) ; assume that there is a variety V, defined over /C, and a birational correspondence ft, defined over k ( t ) , between V and Vt; then ft' oft -1 is a birational correspondence between Vt and Vv. We therefore modify problem ( P ) of the introduction as follows: (C) Let T be a variety, defined over cb field k , let t, t' be independent generic points of T over k. Let Vt be a variety, defined over k ( t ) ; let ft',t be a birational correspondence, defined over /c(t,t'), between Vt and Vt,. Find a variety V, defined over /C, and a birational correspondence ft, defined over /c(t), between V and Vt, such that ft, t ~ f t , oft -1. THEOREM 4.

the condition :

Problem (C) has a solution if and only if ft'.t satisfies

298

[19561 516

(i)

ANDRI%' WEIL.

:,,,.,~f,,,,,,oft,.,

where t'" is a generic point of T over l~(t,t'). When that is so, the solution is unique, up to a birational transformation on V, defined over to. The condition is obviously necessary. The proof for the unicity of the solution, when one exists, is quite similar to the proof of the corresponding statement in Theorem 1. Now, assuming (i) to be fulfilled, we shall construct a solution of ( C ) . We may replace T by any birational transform of T over k, and so we may assume that T is an affine variety. S i m i l a r l y we m a y assume that Vt is an aftine variety; and, taking x to be a generic point of Vt over k ( t ) , we may replace x by ( x , t ) and Vt by the locus of (x,t) over / c ( t ) ; after that is done, Vt is still an affine variety, and we have k(t) C / r from now on, assume that this is so, and assume that x has been taken generic oi1 Vt over k ( t , t', t"). By [4], App., Prop. 1, /c(x) is a regular extension of /~; call X the locus of x over k. P u t X'~ft, t(x) ; this is a generic point of Vt, over k(t, t', t " ) ; by the definition of Vt,, this implies that there is an isomorphism of k ( t , x) onto k(t',x') over /c, m a p p i n g t onto t' and x onto x ' ; therefore we have lc(t') C lc(x'), hence k(x,t') C k(x,x'). As the definition of x' shows k(x,x') to he contained in k(t',t,x), i.e. in k(x, t'), it follows that we have k(x,x') ~ k(x, t') ; therefore x' has a locus W, over k ( x ) . Let k ( v ) be the smallest field of definition containing/~ for W , ; as k(v) C k(x), tc(v) is a regular extension of k. Call V the locus of v over k; we may write v ~ G(x), where G is a m a p p i n g of X into V, defined over /c. I f we put x"=ft.,.t(x), W, is also the locus of x" over /c(x) ; as the fields lr x') and k,(x,x") are respectively the same as lc(x, t') and k(x, t") and are therefore algebraically i n d e p e n d e n t over k(x), W, is also the locus of ~J' over k(x,x'). But (i) may be written XtP~ft,.t'(XI); therefore W,. is the same as W~. This implies that the isomorphism of It(x) onto k(x') over k which maps x onto x' leaves invariant all the elements of the smallest field of definition of W~, hence also all the elements of k ( v ) , so that we have

G(x) = G ( x ' ) . On the other hand, let K be an overfield of t% algebraically independent from k(x,x') over /c; if q~ is any function on X, defined over K, it will induce on W~ a function which is defined over K ( v ) ; if O(x) ~ q ~ ( x ' ) , that function is a constant, so that its constant value must be in K(v). This shows that K ( v ) is the subfield of K ( x ) consisting of the elements of K ( x ) which are invariant under the isomorphism of K ( x ) onto K(x') over K m a p p i n g x onto x'. Now the relation S ' ~ ft",t (X) shows that x" is rational over ~:(t", t, x),

[1956]

299 THE

FIELD

OF I ) E F 1 N I T I O I ' q OF A V A R I E T Y .

517

i. e. over k ( t " , x), so t h a t we may write x " ~ 4t,,(x), where (bt" is a m a p p i n g of X into Vt,,, defined over /c(t"). The relation x " ~ f t , , t , ( x p) m a y then be written as x " ~ S~t,,(x'). A p p l y i n g to the field K ~ lc(t") and to the function 4~r' what we have proved above, we conclude from this t h a t /c(x") C k(t", v). As we have G ( x ) = G(x'), hence also G ( x ) ~ G ( x " ) , the isomorphism of k ( x ) onto /c(x") over /c which maps x onto x" leaves v ~ G(x) i n v a r i a n t ; applying the inverse of that isomorphism to the relation ~ ( x " ) C k ( t " , v ) , we get lc(x) C k ( t , v ) , hence k ( x ) ~ l c ( t , v ) since lc(t) and lc(v) are both contained in /c(x). Also, since k ( x ) and k ( t ' ) are algebraically independent over lc, the same is true of /c(v) and k ( t ' ) ; as the isomorphism of ts(x') onto k ( x ) over k which maps x' onto x maps t' onto t and v onto itself, this implies that /c(v) and /c(t) are algebraically independent over k. As the relation lc (x) ~ / s (t, v) can also be written lc (t, x) ~ lc (t, v), we conclude t h a t Vt and V are birationally equivalent over /c(t), so t h a t we may write x ~ f t ( v ) , where ft is a birational correspondence between V and Vt, defined over /e(t). Then we have x ' ~ f t , ( v ) . Therefore (V, ft) is a solution of our problem. We also see that X is birationa]ly equivalent to T }( V over /c. 5. J u s t as in Section 1, we consider the problem (C') which consists in finding a solution (V, ft) of (C) such t h a t ft is everywhere biregular. F o r such a solution to exist, it is necessary that ft'.t should be everywhere biregular; it will be shown that this is sufficient. As in Section I, if we make use of Theorem 4, we see t h a t (C') may be reformulated as follows:

Let ]c, T and t be as in (C) ; let V and Vt be varieties, respectively defined over k and over /c(t) ; let ft be a birational correspondence, defined over k ( t ) , between V and Vt. Find a variety V" and a birational correspondet~ce F between V and V', both defined over t~, such that the birational correspondence ft o F 1 between V" and Vt is everywhere biregular. (D)

I n order to solve ( D ) , we need some p r e l i m i n a r y results. LEMMA 2. Let F and H be mappings of a variety X into two varieties W, T, all the~'e bei~tg defined over a field /c; x being a generic point of X over Is, assume that t ~ H ( x ) is generic over 7c on 7' and float x has a locus Vt over k ( t ) . Let Ft be the mapping of Vt into W induced by F on Vr. Then F is defined (~t every point of Vt where Ft and H are both defined. I t is clearly enough to treat the case in which X is an affiue variety and W i-s the affine line. Then Ft is the function on Vt, defined over ~ ( t ) ,

300

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such that F t ( x ) ~ F ( x ) . I f F t is defined at a point a of Vt, we can write it as F t ( x ) ~ P ( x ) / Q ( x ) , where P, Q are polynomials with coefficients in k ( t ) , such that Q(a)~=0. More explicitly, we have

P ( X ) = 2~ A , ( t ) M , ( X ) ,

Q ( X ) -~- ] ~ t q ( t ) N j ( X ) ,

where tile h,, /~1 are functions on T, defined over k, and the Mi, N i are monomials in the indeterminates ( X ) ; and we have

(1)

Xt~(t)Nj(a) # o. J

Then we have F ( x ) = , ~ ( x ) / q z ( x ) ,

(2)

o(x) ~XXdH(x))M,(x), *

with

*(x) = X , j ( H ( x ) ) N s ( x ) . f

As a is on Vt, (t,a) is a specialization of ( t , x ) over k ; if H is defined at a, we must have H ( a ) ~ t. As t is generic on T over k, the functions A,, t~j are defined at t ; therefore the functions ~,, o H , / ~ j o H are defined at a on X, with the values ) q ( t ) , /~j(t). That being so, the relations (1), (2) show that F is defined at a on V. PROPOSITION 2. Let k, T, t, t" be as in (C) ; let V be a variety, defined over k; let Vt be a variety, defined and projectively (resp. affinely) embeddable over k ( t ) ; let ft be a birational correspondence, defined "over k ( t ) , between, V and Vt. Then: (i) if a is a point of Vt where ft, oft -1 is biregular, there is an affine variety W and a birational correspondence F between V and W, both defined over k, such that F oft -1 is biregular at a; (ii) if ft, oft -~ is everywhere biregular, there is a variety W, defined and projectively (resp. affinely) embeddable over k, and a birational correspondence F between V and W, defined over k, such that F o ft -1 is everywhere biregular. We may assume that Vt is a k ( t ) - o p e n subset of a variety, defined over k ( t ) , in a projective (resp. affine) space S. We m a y also assume that T is a projective (resp. affine) v a r i e t y ; let S ' be its ambient space. I f S, S' are affiue, S X S' is an affme space ; if they are projective, call 9 the well-known biregular embedding of S X S ' into a projective space S " of suitable dimension. L e t v be generic on V over k(t, t'), and p u t x ~ f t ( v ) . We may replace Vt by a suitable k ( t ) - o p e n subset of the locus of ( x , t ) over k ( t ) in the affme case, of r over k ( t ) in the projective case; after t h a t is done,

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301 TIfE

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OF DEFINITION

OF A VARIETY.

519

we have /c(t) C k ( x ) , and therefore k(x) ~ k ( t , x ) ~ k ( t , v ) , so that x has a locus X over k, birationally equivalent to T X V, and that we may write t ~ H ( x ) , where H is a mapping of X into T, defined over /~; moreover, the mapping H is everywhere defined on X. Now, since X is birationally equivalent to T X V over lc, and V is birationally equivalent to Vt over lc(t), X is birationally equivalent to T X Vt over k(t). More explicitly, if we put x ' ~ f t , ( v ) , x' is generic over k ( t ) on X, and we have k ( x ' ) = l c ( t ' , v ) , hence k ( t , x ' ) ~ l c ( t , t ' , x ) , so that we may write x r ~ gt (t', x), where g t is a birational correspondence, defined over k ( l ) , between T X Vt and X. We have t ' ~ H ( x ' ) , and we may write x ~ c k t ( x ' ) , where qSt is a mapping of X into Vt, defined over k ( t ) ; then (H, q6t) is the mapping of X into T X Vt, inverse to gt. The mapping gt induces on the subvariety t ' X Vt of T X Vt the mapping (t',x)----> x ' ~ f t , ( f t - l ( x ) ) ; and qst induces on Vt, the mapping x'---> x, i. e. the mapping ft o fr -1. Applying L e m m a 2, we see that gt is defined at ( t ' , a ) whenever a is a point of Vt where ft' oft -~ is defined, and that ~bt is defined at every point of Vt, where ft oft, -~ is defined. Therefore gt is biregular at (if, a) whenever a is a point of Vt where fr oft -~ is biregular. Now let Ao be the k(t)-closed subset of T X Vt where gt is not biregular ; and assume first that a is a point of Vt with the property stated in (i). Then (t'~ a) is not in Ao, so that T >( a is not contained in Ao ; let Ax be the (non-dense) k(t, a)-closed subset of T consisting of those points t~ such that ( t , a ) eAo. By [4], App., Prop. 12, there is a ~-closed subset A2 of T containing all ~-closed subsets of T contained in A1; in particular, every point of A~ which is algebraic over k must be in A~. Let As be the union of the components of A2 and of their conjugates over /r put T ' = T - - A 3 ; this is a k-open subset of T such that, if tl is any algebraic point over k in T', gt is biregular at (tx, a). On the other hand, assume, as in (ii), that ft, oft -~ is everywhere biregular. Then gt is biregular at every point of t' X Vt, so that Ao has no point in common with t' X Vt. This implies that the projection of Ao on T is non-dense in T, so that, if we call A~' the closure of that projection, it is a (non-dense) /c(t)-closed subset of T. Let A2' be the maximal J~-closed subset of T contained in A~'; let Aa' be the union of the components of A~' and of their conjugates over k ; put T " ~ T - - AJ. Then T " is/c-open on T ; and, if t~ is any algebraic point over /~ in T", gt is biregular at every point of t~ X Vt. Now let t~ be a separably algebraic point over k in T' (resp. T " ) ; if k is finite, we may take for t~ any algebraic point over 1c in T ' (resp. T " ) ,

302

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WEIL.

since in that case every algebraic extension of k is separable; if k is infinite, we apply [4], App., Prop. 13. Let ti," 9 ", t~ be the distinct conjugates of tl over k. As they are in T' (resp. T ' ) , gt is biregular at (h,a) (resp. at every point of h X Vt), and a fortiori at (h, x), for 1--~ i -~- ~; therefore it induces on t~ X Vt a birational correspondence g~ between Vt and the locus V~ of the point g ~ ( x ) ~ g t ( h , x ) over l~(t,t~) in the projective (resp. affine) ambient space of X ; and g~ is biregular at a (resp. at every point of Vt). But, as we have already observed, the relation/~ (x) ~ k (t, v) shows that X is birationally equivalent to T X V; we may write x ~ f ( t , v), where f is a birational eorrespondence between T X V and X, defined over ]c; then we have x ' ~ f ( t ' , v ) ; and f is the product of Yt and of the birational correspondence (t', v) --> (t', x) between T X V and T X Vt. As the latter correspondence is biregular at (h, v), and gt is biregular at (t~, x), for I ~ i ~ n, we see that f is biregular at (h, v), and that we have

g~(~) = g, ( t,, ~) = i ( h, ~). As the point f ( G v ) has the same locus over ]~(h) as over k(t, h), this shows that Vt is defined over k ( h ) . As every automorphism of ~ over k can be extended to an automorphism of ~(v) over k ( v ) , this also shows that V~ is the transform of V~ by the isomorphism of ]a(t~) onto ]~(h) over k which maps t~ onto h. Also, if f~ is the mapping of V into V~, defined over k ( h ) , which is such that f ~ ( v ) ~ f ( h , v ) , we have f ~ g ~ o f t ; and f~ is the tran~form of f~ by the isomorphism of ]~(t~) onto ~ ( h ) over ]a which maps t~ onto t~. Now apply Proposition 1 to the variety V, defined over the groundfield k, to the variety V~, defined over k(t~), and to the birational correspondence f~ ; this gives a projective (resp. afflne) variety W and u birational eorrespondence F between V and IF, both defined over k, such that F o f~-~ is biregular wherever all the f~ o f - 1 are defined, i.e. wherever all the gi o gt -~ are defined, Now, in ease (i), all the g, are biregular at a, so that all the g~og~-~ are biregular at the point gJa) ; therefore F o ft-% which is the same as (F o f~-~) o gl, is biregular at a; as this involves merely a local property of W at the image of a by that mapping, we may replace W, in the projective ease, by one of its afflne representatives. Thus we have solved our problem in ease (i). I n ease (ii), g, is biregular at every point of Vt; as we have just shown, this implies that F o f t -1 is biregular at every point of Vt, so that it determines an isomorphism of Vt onto a k(t)-open subset W' of W. The assumption in (ii) implies that 1'11' is invariant under the isomorphism of ]~(t) onto k ( t ' ) over /~ which maps t onto t'. From this and from [4], App., Prop. 9, it follows easily that W' is k-open; thus (IV', F) is a solution of our problem.

[19561

303 T:HE F I E U D OF D E F I N I T I O N

OF A V A R I E T Y .

5~1

COROLLARY. Let t~, T, t and t' be as in (C) ; let V be a variety, defined over lc; let Vt be a variety, defined over k ( t ) ; let ft be a birational correspondence between V and Vt, defined over k ( t ) and such that ft, oft -~ is everywhere biregular . Then, if a is any point of Vt, there is an alfine variety W and a birational correspondence F between V and W, both defined over k, such that F o f t -~ is biregular at a. We may assume that t" has been taken generic on T over k ( t , a ) ; take t" generic on T over k ( t , t ' , a ) . Call a', a" the images of a by ft, oft -~ and by ft" o ft-% respectively. The isomorphism of /~(t, a, t') onto k(t, a, t") over /~,(t,l~) which maps t' onto t" maps a" onto a"; therefore, if Vt,~ is a representative of the (abstract) variety Vt, on which a' has a representative a,', the point a" of Vt,, has a representative a,~" on Vt,,a. Let ft', be the hirational correspondence between V and Vt,~ which is determined by ft'. As ft,, o ft, -~ is everywhere biregular and maps a' onto a", ft"a oft,, -~ is biregular at a~'. Applying Proposition 2(i) to V, Vt,~ and ft'~, we get a solution ( W , F ) of our problem. 6.

~ow we can deal with problems (D) and (C').

THEOREM 5. Problem (D) has a solution if and only if ft, oft -1 is everywhere biregular for t' generic over It(t) on T. The condition being obviously necessary, assume that it is fulfilled. By the corollary of Proposition 2, there is, to every point a of Vt, an affine variety W~ and a birational correspondence Fa between V and We, both defined over /c, such that F~oft-~ is biregular at a; call ~ the /~(t)-open subset of Vt where F~ oft -~ is biregular, and call Wa' its image on Wa by F~ o ft-% which is a /~(t)-open subset of Wa. Then WJ is the subset of W~ where ft o F~-~ is biregular ; as in the proof of Proposition 2, this implies that Wa" is invariant under the isomorphism of /~(t) onto /~(t') over ~ which maps t onto t', and we again conclude from this that W~' is /c-open. As we have a ~ ~2~ for every a e Vt, the open sets r form a covering of Vt; by the well-known " c o m p a c t o i d " property of open sets in the Zariski topology, there must be finitely many points as on V such that the sets ~a~ cover Vt. Then the k-open subsets W~' of the afflne varieties W~o, together with the birational correspondences Fap o F~. -~ between them, define an abstract variety, which, together with the obvious birational correspondence between it and V, solves our problem. THEOREI~[ 6.

Problem (C') has a solution, i.e., problem (C) has a

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ANDRs WEIL.

solution (V, ft) for which ft is e~,erywhere biregular, if and only if ft',t is everywhere biregular and satisfies eondHiott (i) in Theorem 4. The solution 'is unique up to a k-isomorpl~is.m. This is an immediate consequence of Theorems 4 and 5. 7.

As to the projective or affine embeddability of the solution of prob-

lems (D) and (C'), we have the following result. THEOREM 7.

Let V be a variety, defined over a field k, and projectively embeddable over an overfield K of k. Then V is projectively ( resp. a]finely ) (resp. a~nely) embeddable over lc provided (i) K is separable over k or (it) 1; is everywhere normal with reference to 1,'. The assumption means that there is a birational correspondence f, defined over K and biregular at every point of V, between V and a subvariety of a projective (resp. affine) space; if we regard f as a mapping of V into that space, it has a smallest field of definition ~' containing k; we may replace K by k ' ; after that is done, K is finitely generated over k. I f K is separable over k, it is a regular extension k l ( t ) of the algebraic closure kl of k in K, and kl is a separably algebraic extension of k of finite degree. Proposition 2(it) shows that V is then projectively (resp. affinely) embeddable over k~; by Theorem 2, this implies that the same is true over ~; this completes the proof in case (i). I f K is not separable over k, let ]c* be the union of the fields kP-'~, for n = 1 , 2 , . . . ; then the compositum K* of K and k* is separable over k*, so that, by what we have just proved, V is projectively (resp. affinely) embeddable over k*. In order to deal with case (it), it is therefore enough to prove our theorem in the case in which V is everywhere normal with reference to k, and K is purely inseparable over ]~; I owe the proof for this to T. Matsusaka; it is as follows. We may again assume that K is finitely generated over /c; as it is purely inseparable, it is contained in some field ] d ~ lWq, where q is a power of the characteristic. Then there is a mapping f' of V into a projective (resp. affine) space, defined over Z", such that f' determines a birational correspondence, biregular at every point of V, between V and the closure W' of its image by f'; then W' is a projective (resp. affine) variety, defined over .~', and f' determines a /d-isomorphism between V and a It'-open subset of W'. Call the automorphism ~---)~q of the mfiversal domain; put W ~ W'~; W is then a projective (resp. affine) variety, defined over k. Let x be a generic point of V over /c; then W' is the locus of the point y ' ~ f f ( x ) over /d, and W is

[1956]

305 TItE

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OF A V A R I E T Y .

523

the locus of the point y~y"~ over k. As y' is rational over k'(x), y is so over k ( x ' ) ; we may write y ~ g ( x ) , where g is a mapping of V into W, defined over k; as we have k ' ( y ' ) ~ / c ' ( x ) , we have k ( y ) ~ I c ( x ' ) , which implies that k(x) is purely inseparable over k(y). I n the projective case, let U be the projective variety derived from W by normalization in the field k ( x ) ~ ; U is birationally equivalent to V over k; let z be the point of U which corresponds t~ x on V. I n the affine case, we take for z a point in a suitable affinc space such that k[z] is the integral closure of the ring k[y] in the field k(x), and for U the locus of z over /C. I n either case we may write z ~ f ( x ) , where f is a birational correspondence between V and U, defined over k. By definition, U is everywhere normal with reference to k, and the mapping h ~ g o f-~ of U into W is everywhere defined and such that the (settheoretic) inverse image of every point of W for that mapping consists of finitely many points of U. Let a be any point of V; let (a, b) be a specialization of (x,z) over x---)a with reference to k,; then, as h is defined at b, (a,b,h(b)) is a specialization of (x,z,y) over k. As f' is defined at a, g is also defined there, so that we must have h ( b ) ~ g ( a ) ; therefore b is one of the finitely many points of U whose image by h is g(a). As V is normal at a by assumption, with reference to /c, this implies that f is defined at a, and that we have b = f ( a ) . We have g ( a ) ~ f ' ( a ) G hence f ' ( a ) = g ( a ) ~ - ' ; as g(a) = h ( b ) , this shows that f ' ( a ) is the unique specialization of y' over z---) b with reference to k"; as f' is biregular at a, f'-~ is defined at f'(a), and therefore x has no other specialization than a over z---)b with reference to U, hence also with reference to k by F-II~, Prop. 3. As U is normal at b, with reference to /C, this implies that f ~ is defined at b ~ f ( a ) . We have thus shown that f is biregular at every point of V, so that it is a k-isomorphiim between V and a /c-open subset of U. As a special ease (already contained in Proposition 2), we see that, in problem (D), V' is projectively (resp. affinely) embeddable over k if Vt is so over/c(t) ; similarly, in problem (C'), V is projectively (resp. affinely) embeddab]e over /C if Vt is so over /c(t). 8. In [4], the construction carried out in Nos. 7-9 can be advantageously replaced by the application of our Theorem 6 to the situation described in No. 6 of that paper. The application is entirely straightforward, so that no further details need he given; this shows that the recourse to the Lang-Weil 1U is the "derived normal mt~del of W in tile field k(x)" according to Zariski's delinition (!5], pp. 69-70); of. also [3].

306

[1956] 57)~4

ANDRt~ WEIL.

Theorem, J.e.. in substance, to the so-called "Riemann hypothesis" in the case of a finite groundfield (loc. cir., p. 374) was unnecessary; so is the assumption of normality in the final result (loc. cir., p. 375) ; normality had to be assumed there merely because of the use made of the Chow point in the construction on p. 370, whereas in the present paper a different device was adopted (in the proof of Proposition 1). Of course, in the main theorem of [4] (p. 375), parts (i) and (ii) remain unchanged. For the sake of completeness, we give here the improved result by which part (iii) of that theorem may now be replaced: PROPOSITION" 3. Let G be a group and W a chunk of transformation-space with respect to G, both defined over t~. Then there is a transformation-space S with respect to G, and a birational correspondence f between W and S, both defined over lc, with the following properties: (a) f is biregular at every point of W; (b) for every s e G and a e W such that sa is defined, we have f(sa) = s f ( a ) ; (c) every point of S can be written in the form sf(a), with s e G and ae W. Moreover, S is uniquely determined by these properties up to a k-isomorphism compatible with the operations of G. UNIVERSITY OF CHIOAG0.

REFERENCES.

[1] W. L. Chow, "Abelian varieties over function-fields," Transactions of the American M a t h e m a t i c a l Society, vol. 78 (1955), pp. 253-275. [2] S. Lang, "AbeIian varieties over finite fields," Proceedings of the National A c a d e m y of Sciences, vol. 41 (1955), pp. 174-176. [3] T. M a t s u s a k a , "A note on m y paper ' S o m e t h e o r e m s on abelian varieties '," N a t . Se. Rep. Ochanomizu Univ., vol. 5 ( ] 9 5 4 ) , pp. 21-23. [4] A. Weil, " On algebraic g r o u p s of t r a n s f o r m a t i o n s , " A m e r i c a n J o u r n a l of Mathematics., vol. 77 (1955), pp. 355-391. [5] O. Zariski, " Theory and a p p l i c a t i o n s of holomorphic f u n c t i o n s on algebraic varieties over a r b i t r a r y groundfields," M e m o i r s of the A m e r i c a n M a t h e m a t i c a l Society, no. 5 (1951), pp. 1-90.

[ 1957a] Zum Beweis des Torellischen Satzes

Vorgelegt v o n t t e r r n 1~I. D e u r i n g

in d e r S i t z u n g v o m 8. F e b r u a r 1957

Die yon Siegel geschaffene Theorie der hSheren Modulfunktionen kann als Theorie der Moduln ffir abelsche FunktionenkSrper angesehen werden; ihre Anwendung auf das klassische Problem der Moduln algebraischer Kurven beruht auf einem Satz yon 1~. Torelli, der ungefi~hr besagt, dab eine algebraische Kurve durch die Normalperioden der abelschen Integrale 1. Gattung auf der Kurve v611ig bestimmt ist. Der Beweis, der von Torelli selbst ffir diesen Satz gegeben wurde (Rend. Ace. Lincei [V] 22 [1914], p. 98), ist im wesentlichen algebraisch-geometrischer Natur. Eine moderne, vSllig einwandfreie und lfickenlose Darstellung dieses Beweises liegt nieht vor; dasselbe grit auch ffir den durchaus klassisch formulierten und an mehreren Stellen ziemlich skizzenhaften Beweis yon A. Andreotti (Mdm. Acad. Belg. 27 [1952], fase. 7). In der vorliegenden Arbeit soll der Torellische Satz in einer solehen Fassung formuliert und bewiesen werden, dab er auch im ,,abstrakten Fall" (beliebiger Charakteristik) seine Gfiltigkeit behalf. Die ganz einfache Beweisidee, die mit dem Andreottischen Ansatz eng verwandt ist, ist leider durch teehnische Schwierigkeiten etwas entstellt, welche es nStig machen, die F~lle niedrigeren Gesehlechtes besonders zu behandeln. I. Zur abstrakten Formulierung des Torellischen Satzes braucht man den Begriff einer ,,polarisierten" abelschen Mannigfaltigkeit (vgl. meinen Vortrag ,,On complex multiplication", Tokyo Symposium on Numbertheory, 1955, sowie eine demn~chst erscheinende Arbeit yon T. Matsusaka im American Journal o/ Mathematics). Zun~chst sei A eine abelsche Mannigfaltigkeit im klassischen Fall, d.h. fiber dem komplexen Zahlk6rper; der zugehSrige FunktionenkSrper sei der KSrper aller periodischen meromorphen Funktionen in einem n-dimensionalen Vektorraum R fiber dem komplexen Zahlk6rper mit einem vorgegebenen Periodengitter G vom Rang 2 n. Damit G tats~chlieh Periodengitter eines solchen KSrpers der Dimension n sei, ist bekanntlieh notwendig und hinreiehend, dal3 es eine auf G • G ganzzahlige alternierende Reprinted from G6ttingen Nachrichten 1, 1974, by permission of Akademie der Wissenschafien GOttingen.

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Bilinearform B gebe, die Imaginiirteil einer im Raum R positiv definiten Hermiteschen Bilinearform ist. Nun sagt man, dab jede solche Bilinearform B eine Polarisierung der abelsehen Mannigfaltigkeit A ----RIG bestimmt, wobei zwei Formen B, B' dann und nur dann dieselbe Polarisierung von A bestimmen, wenn sie sich nur um einen konstanten Faktor unterseheiden. Man sagt, dab A polarisiert ist, wenn man auf A eine bestimmte Polarisierung gew/~hlt hat. Wenn s/~mtliehe Elementartefler der Form B einander gleich sind, sagt man, dab die dureh B polarisierte Mannigfaltigkeit A zu der Hauptschar der polarisierten abelsehen !~annigfaltigkeiten der Dimension n geh6rt. Wenn eine abelsche Mannigfaltigkeit keine komplexen Multiplikationen (d. h. keine nieht-trivialen Endomorphismen) besitzt, dann ist sie nur auf eine Weise polarisierbar. Man kann sieh leieht iiberzeugen, dab es sieh in der klassischen Theorie der abelsehen Funktionen (z.B. in den Arbeiten von Hermite und Humbert) meistens um polarisierte abelsohe Mannigfaltigkeiten handelt; nur dann, wenn von solehen Mannigfaltigkeiten die Rede ist, durf man von ihren Moduln spreehen. Z.B. ist die Siegelsehe Theorie der hSheren Modulfunktionen niehts underes als eine Theorie der l~oduln ftir die Hauptschar der polarisierten abelschen l~annigfultigkeiten einer gegebenen Dimension, indem ein System soleher !~Ioduln dureh einen Punkt im Fundamentalbereieh der Modulgruppe bestimmt ist. Im ubstrukten Fall sei X ein nicht-ausgearteter positiver Divisor auf einer abelsehen Mannigfaltigkeit A ; darunter versteht man einen positiven Divisor X, der hSehstens durch endlieh viele Trunslutionen in einen dazu linear /~quivalenten Divisor verwandelt wird. Dann sagt man, dub X eine Polarisierung von A bestimmt, wobei zwei solehe Divisoren X, X' dann und nur dunn dieselbe Polurisierung von A bestimmen, wenn es zwei positive ganze Zahlen m, m' gibt, ftir welehe m X und m'X" algebraiseh /~quivalent sind; jeder positive Divisor X" mit dieser Eigensehaft heiBe ein Polardivisor ftir die dureh X polarislerte abelsehe Mannigfaltigkeit A. Aus dem Hauptsatz der Theorie der Thetufunktionen folgt sofort, daft im klassischen Fall dieser Polarisierungsbegriff mit dem oben definierten zusammenf/~llt. Insbesondere sei J die Jacobische Mannigfaltigkeit einer algebraischen Kurve C; die Bezeichnungen seien dieselben wie in meinem Bueh Varidtds abdliennes et courbes algdbriques, Paris 1948 (zitiert als VA). Es sei ~ die (bis auf eine Translation eindeutig bestimmte) kanonisehe Abbfldung von C in J. Wenn a----~aiMi ein Divisor auf O ist, werde ich zur Abkiirzung ~(a)---i

~ a ~ ( M ~ ) sehreiben (statt S[~(a)] wie in VA). Wie in VA wird dureh W~ diejenige Untermannigfaltigkeit yon J bezeiehnet, die aus allen Punkten der Form ~0(a) besteht, wenn a die Gesamtheit der positiven Divisoren vom Grad r auf C durehl/iuft. Start Wg-1 wird meistens 0 gesehrieben; werm f ein Divisor der kanonisehen Klasse auf C ist, besteht O ~ W~_x aueh aus den Punkten (~) - - ~0(m), wenn m die Gesamtheit der positiven Divisoren vom Grad g - - 1

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durchlauft; O wird also durch die Abbfldung u -+ ~ ( ~ ) - u auf sich selbst abgebildet. Start Wg_~ werden wir W schreiben; es sei W* das Bild von W bei der Abbildung u --> ~ (1) - - u yon J auf sich selbst. Wenn X ein Zyklus (und insbesondere eine Mannigfaltigkeit) auf J und a ein Punkt auf J ist, so wird (wie in VA) durch Xa der aus X durch die Translation a entstehende Zyklus bezeiehnet. Wir wollen unter der kanonischen Polarisierung von J diejenige verstehen, welche von O oder (was auf dasselbe hinausl~uft) yon einem beliebigen unter den Divisoren Oa bestimmt wird. Diese Polarisierung ist offenbar invariant bei jeder Abbfldung u --> • u + a v o n J auf sich selbst (dasselbe gilt iibrigens fiir jede Polarisierung einer abelschen Mannigfaltigkeit, wie z.B. sofort aus VA-73, prop. 31 folgt). Im klassisehen Fall gehSrt jede kanoniseh polarisierte Jacobische Mannigfaltigkeit zu der Hauptsehar der abelsehen Mannigfaltigkeiten derselben Dimension. 2. Der eigentliehe Inhalt des Torellischen Satzes besagt nun, daB eine polarisierte abelsche Mannigfa~tigkeit im wesentlichen h6chstens auf eine Weise die kanoniseh polarisierte Jacobisehe Mannigfaltigkeit einer Kurve sein kann. Genauer li~Bt er sich wie folgt formulieren: tIauptsatz. Es seien C, C' zwei Kurven vom gleichen Geschleoht g ~ 1 ; es sei (bzw. q~') die kanonische Abbildung yon C (bzw. C') in ihre Jacobische Mannig]altigkeit J (bzw. J'). Es gebe einen Isomorphismus ~ der kanonisch polarisierten Mannig/altig]ceit J au] die kanonisch polarisierte Mannig/altiglceit J'. Dann gibt es einen Isomorphismus ] yon C au/ C' mit der Eigenscha/t, daft ~ o q~ -~ • qJ o ] + a ist mit konstantem a; dabei sind ], a und das Vorzeichen + dutch ~ eindeutig bestimmt, wenn C und C" nicht hyperelliptisch sind; bei hyperelIiptischen C und C" sind / u n d a dutch ~ und das Vorzeichen -4- eindeutig bestimmt, wobei das letztere beliebig gewahlt werden kann. Der Beweis ergibt sieh dadurch, dab man eine explizite Konstruktion ftir C und ~ angibt, wenn J als gegeben gedacht wird. Zuni~chst wird gezeigt, dab die Sehar {O~} der aus O durch Translation entstehenden Mannigfaltigkeiten durch die Polarisierung yon J eindeutig bestimmt ist; das folgt im klassischen l~all aus bekannten S~tzen aus der Theorie der Thetafunktionen und im abstrakten l~all aus folgendem Satz: Satz 1. E i n positiver Divisor Z au/ J laflt sich dann und n u t dann dutch eine Translation in 0 aber/i~hren, wenn Z ~ 0 ist (wobei X ~ O wie in VA-57 dureh das Bestehen der linearen .~quivalenz X, ~ X fiir alle t definiert ist). Es ist ni~mlich O~ -~ O fiir alle a. Es sei Z ~ O m i t positivem Z; dann gibt es nach VA-62, th. 32, cor. 2 ein a, ftir welches Z ~ O~; indem wir Z durch Z_~ ersetzen, brauehen wir also nur zu zeigen, daB, wenn Z ~ O und Z positivist, Z mit O zusammenfallen muB. Es sei x ein generischer Punkt auf J (in bezug auf einen gemeinsamen Definitionsk5rper ftir alle in Betracht kommenden Mannigfaltigkeiten); dann ist naeh VA-41, th. 20 ~ - 1 ( O ~ ) ~ - ~ P ~ , wo die

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P~ (1 ~ i < g) P u n k t e auf C sind mit der Eigenschaft ~ ( P ~ )

~- ~(t) -t- x;

i

die P, mfissen also unabh~ngige generische P u n k t e auf C sein, woraus folgt, dab l ( ~ P,) ~- 1. A u s Z ~ O folgt, dab ~-I(Z~) ein mit ~ Pi linear ~quivalenter positiver Divisor auf C sein muB; wegen l ( ~ P,)--~ 1 fallt er also mit ~ Pi zusammen; demnaeh liegt jeder P u n k t T(Pi) auf einer Komponente yon Z~, und der P u n k t g

auf einer Komponente yon Z. Da dieser Punkt, wie letztere Gleichung zeigt, auf O generisch ist, so mul~ O Komponente yon Z sein; also ist Z - - O positiv und ~ 0, woraus folgt, dab Z - O ~ 0 ist, w.z.b.w. N u n wird jedem Divisor X auf J durch die in VA-45 und VA-48 definierte Abbildung X -~ 8~ ein Endomorphismus ~ yon J zugeordnet; dabei ist 8o der identische Automorphismus 6j yon J ; nach VA-57, th. 30 ist dann und nur dann 8x ~ Or, wenn X ~ Y. Wenn X Polardivisor auf der kanoniseh polarisierten Mannigfaltigkeit J ist, so gibt es nach der Definition zwei positive ganze Zahlen m, m', ffir welche m X mit m ' O algebraisch ~quivalent ist; um so mehr besteht dann die Relation m X ~- m ' O , also m(5'X ~- m'(~j; dann mul3 aber (z.B. nach VA-65, th. 33, cor. 1) n ---- m ' / m ganzzahlig sein, woraus folgt, dab 8x----nOj und X - ~ n O mit ganzzahligem n gilt. Sagen wir, daB ein Polardivisor X auf J minimal ist, wenn es zu jedem Polardivisor Y auf J eine ganze Zahl r gibt, fiir welche Y -~ r X . J e t z t folgt sofort aus Satz 1, dab ein Polardivisor dann und nur dann minimal ist, wenn er der Schar {O~} der aus 0 durch Translation entstehenden Divisoren angeh6rt. I)amit ist diese Schar in invarianter Weise a u f der polarisierten Mannigfaltigkeit J gekennzeichnet. 3. Wir sind jetzt schon imstande, den l~all g--~ 2 zu erledigen. In diesem Fall ist n~mlich O ~ W~ ~ ~ (C). I m Hauptsatz kSnnen sich dann die Kurven a(T(C)), ~'(C') auf J ' hSehstens dutch eine Translation unterscheiden. Da ~, ~' die K u r v e n C, C' isomorph auf ~0(C), ~'(C') abbilden, so muB es also / und a geben, ffir welche ~ o ~ --~ ?' o / + a; dab / und a dabei eindeutig bestimmt sind, ergibt sich daraus, dab (z.B. nach VA-62, th. 32, cor. 2) O durch keine Translation in sich selbst fibergeffihrt wird. Bekanntlich ist jede Kurve vom Geschlecht 2 hyperelliptisch; auf einer solchen Kurve C hat ni~mlich die kanonische Vollschar (bestehend aus allen positiven Divisoren, die mit einem beliebigen kanonischen Divisor ~ linear ~quivalent shad) den Grad 2 und die Dimension 1. Es gibt also einen Automorphismus h yon C derart, dab ftir jeden P u n k t M auf C der Divisor M + h ( M ) der kanonischen Vollschar angehSrt; dann ist ~ ( h ( M ) ) = ~(~) - - T ( M ) . Wenn nun ~, ~', a, ], a dieselbe Bedeutung haben wie oben und /1-----] o h , a l = ~ ( q ~ ( ~ ) ) - - a gesetzt wird, so ist ~ o ~ - - - - - - r o / ~ + a ~ ; dabei sind/~, a~ dureh diese Gleichung eindeutig bestimmt, da sonst / ----/~ o h t

t

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und a durch die Gleichung ~ o ~ : ~ ' o ] + a nicht eindeutig bestimmt w~ren. Damit ist der Hauptsatz im Fall g = 2 vollsti~ndig bewiesen. Die in der Literatur h~ufig vorkommende Behauptung, dab im klassischen l~all eine der Hauptsehar angehSrige abelsehe Mannigfaltigkeit der Dimension 2 ,,im allgemeinen" Jacobisehe Mannigfaltigkeit einer Kurve ist, l~Bt sich jetzt leicht auch im abstrakten Fall in folgender pr~ziseren Fassung formulieren und beweisen: Satz 2. E s sei A eine polarisierte abelsche Mannig/altigkeit der Dimension 2. E s gebe au] A einen Polardivisor X , derart, daft deg (X 9X~) ~- 2 sei ]i~r generisches u. D a n n ist entweder X eine Kurve vom Gesehlecht 2, A die kanonisch polarisierte Jacobische Mannig]altigkeit yon X und die kanonische Abbildung yon X in A die identische; oder A ist das Produlct 1" • 1" zweier elliptischer Kurven 1", 1", und X ist von der F o r m 1" • a' + a • I ~, wo a, a' Punlcte au] 1" bzw. 1" sind. Zur bequemeren Formulierung des Beweises wollen wir (in Erweiterung einer Definition in VA-22) folgendes verabreden: Wenn A eine beliebige abelsche Mannigfaltigkeit und X eine Untermannigfaltigkeit von A ist, so werden wir unter der durch X erzeugten abelsehen Untermannigfaltigkeit yon A die kleinste abelsehe Untermannigfaltigkeit B von A verstehen, die si~mtliche Punkte x - - x" mit x und x' in X enth~lt. Aus VA-21 folgt, dab B die dureh diese Punkte erzeugte Untergruppe von A ist; wenn x ein beliebiger Punkt yon X ist, so ist auch B die im Sinne von VA-22 durch X_~ erzeugte abelsche Untermannigfaltigkeit von A. Z.B. wird die Jacobische Mannigfaltigkeit J einer Kurve C durch die Kurve ~ (C) erzeugt, wenn ~ die kanonische Abbildung yon C in J ist; also wird auch J durch jede der Mannigfaltigkeiten W~ erzeugt. Wegen VA-20, th. 9 folgt daraus, dal3 W~ ftir r ~ g mit keiner abelschen Mannigfaltigkeit birational ~quivalent sein kann. Nach dieser Bemerkung werden wir Satz 2 zun~chst in dem Fall beweisen, wo der Polardivisor X eine Kurve ist, d. h. wo X aus einer einzigen Komponente mit dem Koeffizienten 1 besteht. Es sei i die Injektion, d . h . die identische Abbildung, von X in A. Das Geschlecht g yon X kaim weder 0 noeh 1 sein; im ersteren Fall mfiBte n~mlieh i nach VA-19, th. 8 konstant sein; im letzteren Fall w~re i naeh VA-20, th. 9 bis auf eine Translation ein Homomorphismus, also w~re X bis auf eine Trans]ation eine abelsche Untermannigfaltigkeit von A, und es w~re deg (X. Xu) ~ 0. Es sei J die Jacobische Mannigfaltigkeit yon X; es sei ~ die kanonisehe Abbildung yon X in J ; k sei ein Definitionsk6rper ffir A, X, J, ~. Setzen wir w : ~ (M) + ~ (N) und x ~- i(M) + i(N), wo M, N zwei unabh~ngige generische Punkte yon X fiber k sind. Der Oft (,,locus") von w fiber k ist dann nach unseren iiblichen Bezeiehnungen W~; nach VA-39, prop. 15 ist k(w) der KSrper k(M, N)8 der symmetrischen Funktionen von M, N fiber k; also ist k ( x ) in k(w) enthalten, so dab wir x ~ F ( w ) sehreiben k6nnen, wo F eine Abbildung yon W2 in A ist. Es sei X" der aus X

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dureh die Abbfldung u - - > - - u laervorgehende Divisor; wegen VA-61, th. 31, eor. 2 folgt aus der Voraussetzung tiber X, dab deg (X. X~) ~ 2; nach VA-13, th. 4, eor. 2 mul~ also x generiseh tiber lc auf A und k (M, N) algebraisch vom Grad 2 tiber k ( x ) sein; wegen k ( x ) ~ k(M, N)s ist demnaeh k ( x ) -~ k ( M , N ) , ~- k (w). Damit ist gezeigt, dal3 F eine birationale Abbfldung yon W~ auf A ist. Wie oben bemerkt, kann W2 ftir g ~ 2 mit keiner abelschen Mannigfaltigkeit birational ~quivalent sein; es ist also g -~ 2 und W2 : J ; nach VA-20, th. 9 ist dann F bis auf eine Translation ein Isomorphismus von J auf A ; andererseits folgt aus VA-18, th. 7, cor., dab ~ und F o i sieh hSchstens durch eine Translation unterscheiden kSnnen. Damit ist Satz 2 in dem Fall, wo X eine Kurve ist, vollst~ndig bewiesen. Falls X keine Kurve ist, kSnnen wir X ~-- ~ X (~ schreiben, wobei X m . . . . , X (~) Kurven sind. Dann ist ftir generisches u: 2

deg (X. X,)

~ deg (X (~ 9X (~

Da jedes Glied auf der rechten SeRe _~ 0 ist, so folgt unmittelbar aus dieser Gleichung (unter Benutzung von VA-22, prop. 6) erstens, dab jede Kurve X (~ bis auf Translation eine abelsche Untermannigfaltigkeit von A sein muB, und zweitens, dab es nur zwei solche Kurven gibt; dann ist deg (X m .X(,2~) ---- 1. Wegen VA-13, th. 4, cor. 2 ist damit der Beweis beendet. I m klassischen Fall gentigt eine polarisierte abelsche Mannigfaltigkeit dann und nur dann der Voraussetzung yon Satz 2, wenn sie der Hauptschar angehSrt; also ist eine solche Mannigfaltigkeit entweder eine Jaeobische Mannigfaltigkeit oder ein Produkt zweier elliptischer Kurven. Es ist mir nicht gelungen, die entsprechende Frage ftir g -~ 3 zu entscheiden. 4. Zur Behandlung des allgemeinen Falles brauchen wir einige Hflfss~tze. Bekanntlich pflegt man zu sagen, dab ein posit~ver Divisor m a u f der Kurve C in der durch einen Divisor a bestimmten linearen Vollschar ~ (bestehend aus allen m i t a linear ~quivalenten positiven Divisoren auf C) enthalten ist, wenn es einen positiven Divisor m' gibt, ftir welchen m + m' der Schar angeh6rt. Unter Benutzung dieser Sprechweise gilt folgender Hflfssatz: Hflfssatz 1. E8 sei a ein Divisor vom Grad 0 au] der Kurve C; es sei a ----- q~(a) O; es sei ~ die dutch f + a bestimmte lineare Vollschar. D a n n besteht die Punktmenge 0 (-~ O~ aus den P u n k t e n ~(m), wenn m die in der Vollschar enthaltenen positiven Divisoren vom Grad g - 1 durchlau/t. Die Vollschar hat dann und n u t dann einen F i x p u n k t P, wenn a yon der F o r m a -~ q~(P - - Q) * ist, wo Q ein P u n k t yon C ist; in diesem Fall ist 0 ~ Oa = W~(p) ~ W_~(~). Die erste Behauptung folgt unmittelbar aus der Tatsache, dab O, O~ aus den Punkten ~(m) bzw. ~(f + a - m') bestehen, wobei m, m' die positiven Divisoren vom Grad g - 1 durchlaufen. Damit P Fixpunkt der Schar ~ sei, ist notwendig und hinreichend, dab l(I + a - - P) -- l(f + a) sei; nach dem Riemann-Rochsehen Satz l~Bt sich diese Gleichung in der Form / ( P - - a )

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-~ l ( - - a) + 1 sehreiben. D a n n ist 1( P - - a) ~ 1, also P - - a m i t einem positiven Divisor ~quivalent; da P - - a v o m Grad 1 ist, gibt es d a n n einen P u n k t Q derart, d a b P--a ~ Q, also a - ~ ~ ( P - - Q ) ist. U m g e k e h r t , w e n n letztere Gleichung b e s t e h t u n d a ~= 0 ist, ist l ( P - a) = l(Q) -~ 1, l ( - - a ) -~ o; wie oben gezeigt wurde, ist d a n n P F i x p u n k t v o n ~ . I n diesem Fall b e s t e h t ~ aus den Divisoren P + ~, wo ~ die durch ~ - - Q b e s t i m m t e Vollschar durchl~uft. Es sei der positive Divisor m v o m G r a d g - - 1 in dieser Schar ~ e n t h a l t e n ; d a n n gibt es einen posiriven Divisor m ' v o m G r a d g - - 1 derart, d a b m + m ' der Sehar ~ angehSrt; also ist P entweder K o m p o n e n t e v o n m oder K o m p o n e n t e von m ' . I m ersten Fall k a n n m a n m ~ P + n schreiben, wo n ein positiver Divisor v o m G r a d 9 - 2 ist, der in der durch ~ - Q b e s t i m m t e n Vollschar e n t h a l t e n ist. Aus d e m R i e m a n n - R o c h s e h e n Satz folgt aber sofort, d a f alle positiven Divisoren v o m G r a d g - 2 in dieser Vollschar e n t h a l t e n sind; jeder Divisor P + n m i t p o s i t i v e m n v o m G r a d g - - 2 ist also in der Sehar ~ enthalten. Zweitens sei m wie oben in ~ e n t h a l t e n u n d P K o m p o n e n t e v o n m ' . D a n n kSnnen wir m ' -~ P + n ' sehreiben m i t einem positiven n' v o m G r a d g - - 2; w e n a u m gekehrt n' ein beliebiger positiver Divisor v o m G r a d g - - 2 ist, so ist wie oben P + n' in ~ enthalten, so d a f es einen positiven Divisor m v o m G r a d g - 1 g i b t derart, d a f m + P + 11' der Schar ~ angeh6rt; d a n n ist m in ~ enthalten, u n d es ist m ~t+a--P--n" ~ f--rr D a m i t ist bewiesen, daft O r~ Oa aus den P u n k t e n ~0(P + n), ~ ( [ - 1 t ' - - Q ) besteht, wenn u, 11' die Menge aller positiven Divisoren v o m G r a d g - 2 durchlaufen. D a m i t ist der Hflfssatz vollst~ndig bewiesen. 5. Ftir g ~ 5 gilt weiter der folgende Hflfssatz: Hflfssatz 2. Die Bezeichnungen seien dieselben wie im Hil/ssatz 1; die Scha~ sei fixpunkt]rei, und das Geschlecht g yon C sei > 5. Welter ],abe 0 r~ Oa mehr als eine Komponente. D a n n gibt es eine Komponente dieser Menge, die dutch unendlich viele Translationen in sich selbst i~berge/ahrt wird. Es sei k ein algebraiseh abgeschlossener DefinitionskSrper ftir C, J , ~ u n d die K o m p o n e n t e n y o n a; M 1 , . . . , M~_~ seien g - - 2 unabh~ngige generische P u n k t e v o n C fiber/c; wir setzen m ----~ M~. Wegen a ~= 0 ist l(~ + a) ----g - 1, v

also l (~ + a - - m) ~-- 1 ; es ist also [ + a - - m ~ 11 m i t einem durch diese Relat i o n eindeutig b e s t i m m t e n positiven Divisor n v o m G r a d g; 11 ist d a a n rational fiber d e m K S r p e r k ( m ) ---- k(M1 . . . . . Mg_~),, z.B. n a e h VA-41, th. 20. J e d e K o m p o n e n t e y o n O ~ O~ h a t die Dimension g - - 2 u n d ist algebraiseh tiber k, also rational tiber k, weft k algebraisch abgesehlossen ist; folglieh h a t ein in Bezug a u f k generiseher P u n k t einer solchen K o m p o n e n t e die Dimension g - - 2 tiber k. N a e h Hflfssatz 1 li~ft sieh ein soleher P u n k t in der F o r m ~ (~) sehreiben, wo ~ ein in ~ e n t h a l t e n e r positiver Divisor v o m Grad g - 1 ist. E s sei ~ ~-- R1 + - . - -t- Rg-1; d a m R ~ 0:) tiber k die Dimension g - 2 habe, mfissen wenigstens g 2 der P u n k t e B~, z.B. R x , . . . , R~_~, fiber k u n a b h a n g i g

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[1957a] Andrg Weil

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sein; es gibt also einen Isomorphismus von k(R1 . . . . . Rg_2) auf k(M1 . . . . . Mg-2) fiber k, der Rt in M~ ffir i ---- 1 . . . . . g - - 2 fiberffihrt. Dieser Isomorphismus kann zu einem Isomorphismus von B ( R 1 , . . . , R~-I) in das Universalgebiet erweitert werden; dadureh wird Rg_l in einen P u n k t N von C fibergeffihrt, also ~ in den Divisor m -[- N ; dann ist ~0(m + N) generiseh fiber k auf derselben K o m p o n e n t e von O ('~ O~ wie ~0(~). D a m i t ist gezeigt, dab es zu jeder K o m p o nente y o n O r-~ O~ einen generischen P u n k t der F o r m ~0(m + N) gibt. Dabei muB der Divisor m + N in ~ enthalten sein; da aber die Relation m + n ,~ ~ + a den positiven Divisor rt eindeutig bestimmt, mug N eine der g K o m p o n e n t e n des Divisors n sein. Da II fiber k (m) rational ist, ist u entweder ein rationaler Primdivisor fiber k(m) oder S u m m e yon mehreren solehen Primdivisoren. I m ersten Fall sind samtliehe K o m p o n e n t e n von n zueinander fiber k(m) konjugiert; dasselbe gilt also ffir die P u n k t e ~(m + N), so dab je zwei dieser P u n k t e generische Spezialisierungen voneinander fiber k (m) und u m so mehr fiber k sind; folglieh haben sie denselben Ort fiber k, so dab O ~ O~ nieht mehr als eine K o m p o nente haben k a n n entgegen der Voraussetzung. Demnaeh ist n yon der F o r m rt-----rtl~...+nh, wo die nt rationale Primdivisoren fiber k(m) sind, mit h ~ 2. Es sei vt der Grad von rtt; dann ist ~ vt = g, also vt ~ g - - 1 ffir alle i wegen i

h ~_ 2. Ffir jedes i i s t ~0(ui) rational fiber k(m), also aueh fiber k ( M 1 . . . . . Ma_2) , u n d laBt sieh also (nach VA-18, th. 7, eor.) in der F o r m yJ(M1) + ~v'(M2) + 999 sehreiben, wo % ~v', . . . g - 2 Abbildungen von C in J sind, die (weil ihre Summe eine symmetrische F u n k t i o n von M1 . . . . . Mg_~ ist) sich voneinander h6ehstens durch Addition einer K o n s t a n t e n unterseheiden. Demnaeh k5nnen wir ~(rt~) in der F o r m g--2

(nt) = ~ ~o~(M~) + c~ = ~t (m) + c~ sehreiben, wo ffir jedes i ~ eine Abbfldung y o n C in J und c~ eine K o n s t a n t e bedeutet; dabei ist jedes ~t bis a u f eine K o n s t a n t e eindeutig bestimmt. Ffir jedes i i s t aber ~t bis auf eine K o n s t a n t e v o n d e r F o r m a~ o % wo at ein Endomorphismus yon J ist; wit dfirfen also annehmen, da$ ~Pt = at o ~v; da k algebraiseh abgesehlossen ist, sind dann die at u n d c~ fiber k definiert. D a / ( n ) --~ 1 ist, so ist u m so m e h r l(rtt) ---- 1 ffir jedes i; folglieh ist ffir j e d e s i n~ algebraiseh (und sogar, wie m a n sieh leieht fiberzeugt, rational) fiber k(~ (ut)). W a r e insbesondere ~ (nt) algebraisch, also rational, fiber k, so ware dasselbe ffir jede K o m p o n e n t e von ut der Fall; dann ware aber ein soleher P u n k t F i x p u n k t der Schar ~ entgegen der Voranssetzung, weil m -4- u die Dimension 9 - - 2 fiber k hat und folglieh ein generiseher Divisor der Vollschar ist. D e m n a e h ist at # 0 ffir jedes i. Wenn At die abelsehe Untermannigfaltigkeit von J ist, auf welehe J dureh ~t abgebildet wird, und dt die Dimension yon A~ bedeutet, so ist also d~ > 0 ffir jedes i.

[1957a]

315 41

Z u m Beweis des Torellischen Satzes

Setzen wir fl---- ~j + ~ ~i. D a n n ist: i

woraus folgt, dab fl(~(m)) fiber k rational ist. D a J dureh den O r t W v o n (m) fiber k erzeugt wird, so muB also fl k o n s t a n t sein. Weft fl ein E n d o m o r phismus y o n J ist, ist d e m n a e h fl = 0, d . h . - - x = ~ a i x ffir jedes x a u f J . i

W e n n x generiseh a u f J fiber k gew~hlt ist, h a t ai x die Dimension d~ fiber k, also ~ ~ x hSehstens die Dimension ~ di. D a m i t ist bewiesen, d a b ~ d~ > g. i

i

i

Andererseits wird J dureh ~v(C), also A t dureh ~ (C) erzeugt; d u r c h I n d u k tion n a c h r ffir 1 < r < 9 - - 2 sieht m a n d a n n leieht, dab die Dimension y o n ~ ( M 1 -t- 999 M,) fiber k gleieh r i s t , solange dieser P u n k t a u f A~ nicht fiber k generisch ist; insbesondere ist die Dimension v o n ~i(m) fiber k gleich v~ -~ inf (g - - 2, d~) ; sie ist d a n n u n d n u r d a n n gleieh di, wenn ~ (m) a u f A ~generiseh ist. N u n ist aber v~ ebenfalls die Dimension v o n q (n~) fiber k; d a dieser P u n k t a u f W~ liegt, so ist v'~ < v~; es ist d a n n und n u r d a n n v~ = v~, wenn ~ (n~) a u f W~ generiseh ist; d a J dureh W~ u n d A~ dureh den O r t von yJ~(m) fiber k erzeugt wird, u n d d a sieh ~i(m) und q(rti) n u r u m die K o n s t a n t e ci u n t e r scheiden, so folgt aus v: - - vi, dab J = Ai, also d~ ---- g u n d v~ -----g - 2 ist. N e h m e n wir an, es sei v'i < g - - 2 ffir jedes i; d a n n k a n n nieht v~ ~--vi sein; also ist v~ < v~; zugleieh m u g aueh v~ ---- di sein, also d~ < v~ ffir jedes i, was der Ungleichung ~ d~ > g = ~ vi widersprieht. Folglieh dfirfen wir z.B. i

i

a n n e h m e n , d a b v~ ---- g - - 2 ist; d a n n ist vl > g - - 2, also % < 2. W e n n v~ = 1 w~re, so w~re, wegen 0 < v~ < v~ ffir jedes i, v~ ~-- 1, also v~ ---- v~ -~ g - - 2 u n d g -~ 3 entgegen der Voraussetzung; also ist v~ = 2. W e n n v~ ~-- 2 w/~re, so w~re v~ = g - - 2 u n d g = 4, entgegen der Voraussetzung; also ist v~ = 1 u n d (wegen g > 5) v~ = d~. So ist gezeigt, dab h = 2, v~ = g - 2, v~ = 2, v~ = d~ ---- 1 ist; der Ort y o n ~%(m) ist A~ u n d h a t die Dimension 1; und der Ort B yon q(n~) e n t s t e h t aus A~ dureh die Translation cz. Setzen wir rt~ ~-- N + N ' ; weder N noeh N ' ist algebraisch fiber k; da aber v~ ~-- 1 ist, so ist N" algebraiseh fiber k (N). U n t e r den Mr muB es d a n n g - - 3 P u n k t e geben, z.B. M~, . . . , Mg_a, die fiber k(N, N') u~abh/~ngig sind. D e r Divisor M~ + 999 A- Mg_a + n~ v o m Grad g - 1 h a t d a n n fiber k die Dimension g - - 2 und ist in der Sehar ~ e n t h a l t e n ; also ist, wenn wit t = ~ ( n ~ ) , w=~(M a+...+M~_a) setzen, t + w generischer P u n k t einer K o m p o n e n t e Z v o n O r~ O~. Dabei shad t, w unabh~ngige generische P u n k t e v o n B bzw. Wg_a fiber k; Z wird also dureh jede Translation, die durch einen P u n k t y o n A~ b e s t i m m t ist, in sich selbst fibergeffihrt. D a m i t ist der Hflfssatz bewiesen. U m festzustellen, ob Hflfssatz 2 aueh ffir g ---- 3 u n d g ---- 4 gtiltig bleibt, w~ren ziemlich langwierige Fallunterseheidungen nStig; ich h a b e sie nieht bis zu E n d e durehgeffihrt.

316 42

[1957a] Andrd Well

Ein positiver Divisor c v o m Grad 2 auf einer K u r v e C vom Gesehlecht g ~_ 2 h e i f e hyperelliptisch, wenn l(r = 2 ist; in diesem Fall heiBe ebenfalls die dureh c bestimmte lineare Vollschar der Dimension 1 hyperelliptisch. Wie bekannt ist, gibt es auf einer K u r v e C vom Gesebleeht g ~ 2 h6ehstens eine hyperelliptische Vollschar, und C heil~t hyperelliptiseh, wenn es auf C eine solche Vollsehar gibt. Hflfssatz 3. Keine der Manniq]altiglceiten W1 = q~(C). . . . . W~_~ = W, Wg_l = O, W* wird in sich selbst durch eine Translation (aufler der identischen ) aberge/i~hrt. Es wird dann und n u t dann W dutch eine Translation in W* tlberge]ahrt, wenn C hyperelliptisch ist; in diesem Fall ist W* ~-- W~(c), wenn r ein hyperelliptischer Divisor au] C ist. DaB die erste B e h a u p t u n g ffir O gilt, ist z.B. in VA-62, th. 32, cor. 2 enthalten. Ffir r < g - - 1 ist O der Ort von v + w fiber k, wenn v, w unabhi~ngige generisehe P u n k t e y o n W~ bzw. W~-~-x fiber k sind; wenn also W~ dureh eine Translation in sieh selbst fibergeffihrt wfirde, w~re das aueh der Fall ffir O. Da W* naeh seiner Definition (in Nr. 1 oben) aus W ---- W~_~ dureh die Abbildung u -~ ~ (~) - - u entsteht, so gilt die erste B e h a u p t u n g aueh ffir W*. N e h m e n wir an, die Translation c ffihre W in W* fiber; es sei c ein Divisor v o m Grad 2 mit ~(r = c. Zu jedem positiven Divisor m v o m Grade g - - 2 m u f es dann einen positiven Divisor m ' desselben Grades geben, derart, dab f - - m ~ m' + r mit anderen Worten, es m u f l ( f - c - m) ~ 1 sein. Das Bestehen dieser Ungleiehung ffir generisches m fiber/c (c) ist aber mit der Ungleiehung 1( ~ - - c) g - - 1 gleiehwertig, und umgekehrt folgt aus der letzten Ungleiehung, dab l(f--c--m) _~ 1 ffir jedes positive m v o m Grad g - - 2 . Wegen des RiemannRoehsehen Satzes bedeutet aber l ( f - - r _~ g - - 1 niehts anderes als 1(r ~ 2. Hflfssatz 4. Es eei V der Oft des Punktes qJ( M - - N ) abet k, wenn M , N zwei unabhangige generische Punkte yon C ~ber k sin& Dann hat V die Dimension 2. Es sei ~ die Menge der P u n ~ e a ~= 0 au] J mit der Eigenscha/t, daft 0 r-~ O~ zwei Komponenten hat, wovon keins durch elne nicht-identische Translation in sich selbet abergeht. Dann ist 9~ = V - - {0}, wenn g ~_ 5 und C nicht hyperelliptisch ist; dagegen gibt es, wenn g ~ 5 und C hyperelliptisch ist, eins Kurve Vx au/ V derart, daft 9~ : V - - V1. Es seien P , Q, P ' , Q' irgend vier P u n k t e auf G; aus ~o(P - - Q ) : ~0(P' - - Q ' ) folgt, daft P - - Q ~ P ' - - Q ' , also P + Q ' ~ P'+Q, und welter, wenn C nieht hyperelliptiseh ist, dab entweder P =- P ' , Q : Q' oder P ---- Q, P ' =- Q'. Die Abbfldnng (P, Q) -> ~ 0 ( P - - Q ) yon G>~ C auf V ist also auBerhalb der Diagonale y o n C • G eineindeutig, falls C nieht hyperelliptisch ist; in diesem Fall hat also V die Dimension 2. Wenn C hyperelliptiseh ist, so gibt es einen Automorphismus h von G derart, dab ffir jeden P u n k t P y o n C der Divisor P + h ( P ) hyperelliptiseh ist; dann ist, wenn c irgendein hyperelliptiseher Divisor ist, ~ (h (P)) -~ r (c) - - ~0(P) ffir jeden P u n k t P ; indem dann N ' : h (N) gesetzt wird, sind M, N ' unabhgngige P u n k t e auf C fiber k, und ~o( M - N)

[1957a]

317 Z u m Beweis des Torellischen Satzes

43

nichts anderes als ~(M + N') --~0(c), also g die aus W2 durch die Translation --~(c) entstehende Fl~ehe. Nun sei a ein Punkt in der Menge 9~; nach den Hflfss~tzen 1 und 2 muB jedenfalls a auf V liegen; es ist also a = ~ ( P - Q) ~= 0, und O ~ Oa ist Vereinigungsmenge yon W~(p) und W*_~(Q). Da die letzteren Mannigfaltigkeiten nach Hilfssatz 3 keine Translation in sich selbst zulassen, so gehSrt umgekehrt dann und nur dann ein solcher Punkt der i~Ienge 9~ an, wenn W~(p) mit W*_~(Q)nicht zusammenf~llt, also (nach Hflfssatz 3) wenn P + Q nieht hyperelliptisch ist. Folglich ist wie behauptet 2 ---- V - - {0}, falls C nicht hypereUiptisch ist; im hyperelliptisehen Fall besteht dagegen 2[ aus allen Punkten yon V - - {0}, die sich nieht in der Form ~ (P - - Q) mit hyperelliptischem P + Q schreiben lassen. Wenn nun P - t - Q hypereUiptisch ist, so ist Q -~ h(P), wo h dieselbe Bedeutung hat wie oben; es sei Vt der Ort des Punktes qJ(M - - h ( M ) ) -~ 2~(M) - - ~(c) fiber/c bei generischem M fiber k auf C. Es gibt bekanntlich Punkte P a u f G derart, dab P ~- h (P); also liegt 0 auf der Kurve Vt. Es ist dann 2 ---- V - - V1, W.z.b.w. 6. I m Falle g > 5 ist nun der Hauptsatz eine unmittelbare Folge der oben bewiesenen Hilfss~tze. Oben ist schon die Sehar {Oa} in invarianter Weise auf der polarisierten Mannigfaltigkeit J gekennzeichnet worden. Es sei T = Ot eine beliebig gew~hlte Mannigfaltigkeit aus dieser Schar. Die Menge der Punkte a :4= 0 auf J mit der Eigensehaft, dab T ~ Ta zwei Komponenten hat, wovon keine dureh eine nieht-identische Translation in sich selbst fibergeht, ist dann nichts anderes als die im Hflfssatz 4 definierte Menge 9/; damit ist aueh die Fl~che V als abgeschlossene HiJlle yon 9~ (in der Zariskisehen Topologie anf J) in invarianter Weise definiert. Die Menge aller Komponenten der Durchsehnitte T ~ Ta, wenn a die Punktmenge V - {0} durehl~uft, besteht dann aus allen Mannigt-r wenn P, Q die Kurve C durchlaufen. Es sei faltigkeiten Wt +v(p), W* nun X eine beliebig gewfihlte Mannigfaltigkeit aus dieser Menge; es sei Y die Menge aller Translationen, die X in eine Mannigfaltigkeit derselben Menge fiberftihren. Aus dem Hflfssatz 3 folgt sofort, dab Y eine Kurve ist, die aus der Kurve ~0(C) dureh eine Abbfldung der Form u -~ + ( u - ~(P)) entsteht, wo P ein Punkt yon Gist. Wir haben also eine explizite Konstruktion angegeben, wodurch auf der polarisierten l~Iannigfaltigkeit J die Schar der aus ~ (C) dutch eine Abbfldung u --> • u + a entstehenden Kurven in invarianter Weise gekennzeiehnet ist. Damit ist der Hauptsatz ffir g > 5 vollst~ndig bewiesen. 7. Um die F~lle g --~ 3 und g ~-- 4 zu behandeln, werden wir zuerst die tangentialen linearen Mannigfaltigkeiten zu den Mannigfaltigkeiten W~ bestimmen. Hilfssatz 5. Es sei F eine Abbildun9 einer Mannig]altiglceit U der Dimension n in eine Mannig/altigkeit V; es sei ]c ein Definitionsk6rper /at U, V, F ; es sei u

318 44

[1957a] Andrd Weil

ein generischer Punier yon U abet k; der Punlct v ~ - F ( u ) sei ein]ach au/ V. Unter diesen Umstdnden ist k (u) dann und nur dann separabel algebraisch ~eber k (v), wenn die zu F i m Punkte u tangentiale lineare Abbildung vom Rang n ist. Bekanntlich ist k(u) dann und n u r dann separabel algebraiseh fiber k(v), wenn es keine niehtverschwindende Derivation von k(u) fiber k(v) gibt. Die Derivationen yon k(u) fiber ]c stehen aber in eineindeutiger Beziehung zu denjenigen Tangentialvektoren zu U im P u n k t u, die fiber k(u) rational sind; wenn ~ ein solcher Vektor ist, wird n~mlich durch die Gleichung D[/(u)] ---d / ( u ; t~), wo ] eine beliebige fiber k definierte numerische F u n k t i o n auf U ist u n d d] das Differential yon ] bedeutet, eine Derivation D yon k (u) definiert; und ]ede Derivation yon k(u) fiber k li~Bt sich so definieren. Die in dieser Weise d e m Vektor ~ zugeordnete Derivation D verschwindet dann und nur dann auf k (v), wenn D durch die tangentiale lineare Abbfldung zu F in u auf 0 abgebildet wird. Die B e h a u p t u n g des Hflfssatzes folgt u n m i t t e l b a r daraus. N u n sei A eine abelsehe Mannigfaltigkeit der Dimension n; es sei T der tangentiale V e k t o r r a u m zu A in O; im Siane der Theorie der algebraisehen Gruppen ist T nichts anderes als die Liesehe Algebra von A, oder genauer der dieser Algebra untergeordnete Vektorraum. Wenn a ein beliebiger P u n k t auf A ist, so wird dureh die Translation u -~ u + a der V e k t o r r a u m T auf den zu A in a tangentialen V e k t o r r a u m Ta isomorph abgebildet; wir werden meistens T~ mit T dureh diesen Isomorphismus identifizieren. Bekanntlich wird dadurch der duale V e k t o r r a u m zu T m i t dem Vektorraum der translationsinvarianten Differentiale (oder, was dasselbe ist, der Differentiale 1. Gattung) a u f A identifiziert. Es sei co ein solehes Differential; es sei F eine Abbildung einer Mannigfaltigkeit U in A. Dem Differential w @ird durch die zu F transponierte Abbfldung $'* ein Differential F ' c o auf U zugeordnet, das durch die Gleichung

F*o~(u; ~) = o~(F(u); F ' , ) definiert werden kann, wo u ein generischer P u n k t auf U, D ein Tangentialv e k t o r zu U in u, und F" die zu F in u tangentiale lineare Abbildung bedeuten. D a n n ist bekanatlich F * co ein Differential 1. G a t t u n g auf U; da wir hier diese Tatsache nur im Falle der kanonischen Abbildung einer K u r v e in ihre Jacobische Mannigfaltigkeit benutzen wollen, so brauehen wir auf ihre Begrfindung nieht n~her einzugehen; in diesem Fall ist sie unmittelbar klar. B e t r a c h t e n wir wieder eine K u r v e C v o m Geschlecht g, ihre Jaeobische Mannigfaltigkeit J und ihre kanonische Abbildung ~ in J ; es sei k ein DefinitionskSrper ffir C, J , ~. Es sei M irgendein P u n k t auf C; da ~ in M biregul~r ist, so wird durch die zu ~ in M tangentiale lineare Abbfldung die Tangente zu C in M auf eine Gerade im tangentialen V e k t o r r a u m zu J in ~0(M) abgebildet. Dieser V e k t o r r a u m ist oben mit dem tangentialen Vektorraum zu J in 0 identifiziert worden; damit ist der Tangente zu C in M eine Gerade durch 0 in T zugeordnet, die wir ira folgenden mit t (M) bezeichnen wollen. Es ist

[1957a]

319 Zum Beweis des Torellischen Satzes

45

leicht einzusehen (z.B. durch die Wahl geeigneter affiner Repr/~sentanten ffir die in B e t r a c h t k o m m e n d e n a b s t r a k t e n Mannigfaltigkeiten), daB, wenn M" eine Spezialisierung yon M fiber k ist, t (M') die eindeutig bestimmte Spezialisierung von t (M) fiber M --->M' in bezug auf k ist; also ist t eine fiber k defiuierte Abbfldung (im Sinne der algebraischen Geometrie) von C in die Malmigfaltigkeit der Geraden dureh 0 in T. Es sei P der (g - - 1)-dimensionale projektive R a u m der unendlichfernen P u n k t e zu T ; die P u n k t e yon P stehen in eineindeutiger Beziehung zu den Geraden dureh 0 in T ; wir wollen mit ~ (M) den der Gerade t (M) zugeordneten P u n k t von P bezeichnen; x ist also eine Abbfldung yon C in P. Satz 3. Der Vektorraum der Di~erentiale 1. Gattung au/ J wird dutch die zu transponierte Abbildung q~* au] den Raum der Di~erentiale 1. Gattung au] C isomorph abgebildet. Da beide R/~ume dieselbe Dimension g haben, so brauchen wir nur zu zeigen, dab ~* o~ ~= 0 ist, wenn das Differential 1. G a t t u n g w auf J nicht 0 ist. Es seien M 1 . . . . . Mg g unabh/~ngige generisehe P u n k t e y o n C fiber k; es sei u = (M1 -t- 999 q- Mg). Da k ( M 1. . . . . Ms) fiber k(u) separabel algebraiseh ist, kSnnen wir Hilfssatz 5 auf die Abbildung (M 1 . . . . , Ms) -~ u von G • 999• G auf J anwenden. Daraus ergibt sieh sofort, dab die Geraden t (M1) . . . . . t (Ms) nicht alle in einer echten linearen Untermannigfaltigkeit von T liegen kSnnen. W e n n n u n q0* eo ~- 0 w/~re, so mfiBte (naeh der Definition von ~* oJ) t (M)ffir j eden P u n k t M auf C in der durch o~ -~ 0 b e s t i m m t e n H y p e r e b e n e in T liegen (wobei wir den R a u m der Differentiale 1. G a t t u n g auf J mit dem zu T dualen R a u m identifiziert haben). Dureh eine /~hnliche B e t r a c h t u n g kann m a n folgenden allgemeineren Satz beweisen, den wir hier nur nebenbei erw/~hnen. Es sei F eine Abbildung einer K u r v e C in eine abelsche Mannigfaltigkeit A der Dimension n; es sei k ein DefinitionskSrper ffir C, A, F ; es seien M1 . . . . . M s n unabh~ngige generische P u n k t e von C fiber/c und u ---- _~ (M 1 ~- 999 -b M~). U n t e r diesen Umst/~nden ist ]c (M1 . . . . . Ms) dann und nur dann separabel algebraiseh fiber/r (u), wenn durch F * kein nichtverschwindendes Differential 1. Gattung auf A auf 0 abgebildet wird. J e d e r H y p e r e b e n e durch 0 in T entspricht im projektiven R a u m P ebenialls eine H y p e r e b e n e ; die H y p e r e b e n e in P, die in dieser Weise der durch o~ ----- 0 definierten H y p e r e b e n e in T entsprieht, werde durch H~ bezeichnet; dabei bedeutet oJ wie oben ein nichtverschwindendes Differential 1. Gattung a u f J (bei Identifizierung solcher Differentiale mit den Linearformen in T). Aus den Definitionen folgt sofort, dab H~ dam1 und nur dann den P u n k t ~(M) enth/~lt, wenn ~*eo im P u n k t M versehwindet. Es sei nun (~ol, . . . , ~og) eine Basis ffir den R a u m der Differentiale 1. G a t t u n g auf J, also ffir den zu T dualen R a u m ; die eo~ kSnnen dann als ein System (gew6hnlieher) K o o r d i n a t e n in T und als ein System homogener K o o r d i n a t e n in P b e t r a c h t e t werden.

320

46

[1957a] Angrd Weil

Es ist nach dem obigen klar, dab die Verh~ltnisse der Differentiale ~* w~ im Punkt M yon C, also die Werte ~*co~(M; ~) ffir einen beliebigen nichtverschwindenden Tangentialvektor ~ zu C in M, ein System homogener Koordinaten fiir den Punkt v(M) sind. Die Abbfldung ~ von C in P ist also diejenige, die zur kanonischen linearen Vollschar gehSrt. Bekanntlich folgt unmittelbar aus dem Riemann-Rochschen Satz, dab v ein Isomorphismus von C auf ~(C) ist, wenn C nicht hyperelliptisch ist. I m hyperelliptischen Fall ist, wenn M generisch auf C fiber k gew~hlt wird, k (M) separabel vom Grad 2 fiber k ( ~ ( M ) ) ; der nicht-identische Automorphismus von k(M) fiber k (3 (M)) ist dann derjenige, der oben in Nr. 5 (beim Beweise von Hflfssatz 4) mit h bezeichnet wurde, so dab M + h(M) ein hyperelliptischer Divisor ist. 8. Satz 4. E s 8el a ~ A1 + 999+ Ar ein positiver Divisor vom Grad r ~ g au] C. D a n n und nur dann ist a -~ ~ (a) ein ein/acher P u n k t yon Wr, wenn l (a) 1 ist; in diesem Fall ist die lineare Untermannig/altigkeit yon P, die in P der tangentialen linearen Mannig/altiglceit zu Wr in a entspricht, der Durchschnitt der Hyperebenen H ~ , wenn w die Gesamtheit der Di~erentiale 1. Gattung au] J durchl~u/t, ]t~r welche (q~*o~) ~ a. Zunachst sei a einfaeh auf W~; es sei Tr die tangentiale lineare Mannigfaltigkeit zu W~ in a und L die ( r - 1)-dimensionale lineare Untermannigfaltigkeit yon P, die T~ entsprieht. Wenn M ein generiseher Punkt von C tiber k(A1 . . . . , A~) ist, so enth~lt W~ den Ort yon ~(M + A S + 999+ A~) fiber diesem K6rper; die Tangente zu dieser Kurve im Punkte a ist aber nach dem oben Bewiesenen die Gerade t(A1); also enth~lt T~ diese Gerade, und L enth~lt den Punkt v(A1). Nehmen wir an, es sei l(a) ~_ 2; dann gibt es zu jedem Punkt BI auf C einen Divisor b in der durch a bestimmten linearen Vollsehar derart, da~ b ~ B1; b l~Bt sieh also in der Form B1 + 999+ B~ sehreiben, und es ist ~(l}) -----~(a) -----a. Aus dem oben Bewiesenen folgt nun, dab v(B~) in L liegen muB. Da B~ beliebig ist, ist also die ganze Kurve v(C) in L enthalten, was unmSglieh ist. Demnaeh ist 1(a) ---- 1, wenn a einfaeh ist auf Wr; das Umgekehrte wurde in VA-40, prop. 18 bewiesen. Nun sei l(a) ~-- 1, also a einfach auf W~; wir haben oben bewiesen, da~ L die Punkte v ( A 1 ) , . . . , v(A~) enth~lt. Andererseits ist l ( [ - a) ---- g - r wegen des Riemann-Roehsehen Satzes; m. a. W. bilden die Differentiale co, die der Bedingung (~*r genfigen, einen Vektorraum der Dimension g - - r ; der Durchschnitt L" der Hyperebenen H~, wenn eo diesen Vektorraum dnrehl~uft, hat also dieselbe Dimension r - 1 wie L u n d enth~lt die Punkte ~(Ax), . . . . v (A~). l~alls diese Punkte in keiner linearen Mannigfaltigkeit niedrigerer Dimension liegen, folgt sehon daraus, dab L" mit L zusammenf~Ut. Das ist aber jedenfalls dann der Fall, wenn A~ . . . . . Ar fiber k unabh~ngig sind, da wir beim Beweis von Satz 3 sogar gezeigt haben, dai3 die Geraden t(A~) . . . . . t(Au) in keiner eehten liaearen Untermannigfaltigkeit von T liegen kSnnen, falls A~ . . . . . Ag fiber k unabh~ngig sind.

[1957a]

321 Zum Beweis des ToreUischen Satzes

47

Der allgemeine Fall folgt daraus duroh Spezialisierung. Wenn u ein genefischer Punkt yon W~ fiber k ist, so ist ni~mlieh die tangentiale lineare Mannigfaltigkeit zu W~ in a die eindeutig bestimmte Spezialisierung derjenigen in u fiber u ~ a in bezug auf k. Man braueht also nur zu beweisen, daB, wenn die Behauptung yon Satz 4 ftir die tangentiale lineare Mannigfaltigkeit zu Wr in a = ~ (a) riehtig ist, sie auch ffir jede Spezialisierung a' yon a riehtig bleibt. Es seien nun bl . . . . . bg_~ g - - r fiber k (a, a') unabhi~ngige generisehe positive Divisoren vom Grad g - r - 1; es sei K tier dureh die Komponenten dieser Divisoren fiber k erzeugte KSrper; a' ist noeh Spezialisierung von a fiber K. ' Die Bedingungen (~* to~) ~ a + bQ, (~* we) ~ a ' + bQ, bestimmen dann niehtverschwindende Differentiale %, w~ eindeutig bis auf konstante Faktoren; es ist dana unmittelbar einzusehen, dab bei der Spezialisierung a ~ a' fiber K die Hyperebene H ~ sieh ftir jedes ~ auf H ~ spezialisiert. Wegen der Wahl der b~ sind sowohl die co~ wie die co~ linear unabhangig; der Durehsehnitt der Hyperebenen H ~ wird also auf den Durchsehnitt der H ~ spezialisiert. Damit ist alles bewiesen. 9. Naeh diesen Vorbereitungen kehren wir zum Beweis des Torellisehen Satzes in den F~llen g = 3, g ---- 4 zurfiek. Erstens sei g ---- 3. Naeh Satz 4 ist dann O = W2 im meht-hyperelliptischen Fall singulariti~tenfrei. I m hyperelliptisehen Fall hat dagegen O genau einen singuli~ren Punkt c = ~(c), w o r irgendeinen hyperelliptisehen Divisor auf C bedeutet. Im letzteren Fall ist dana O_r eindeutig dadureh bestimmt, dalt es die einzige Mannigfaltigkeit in tier Sehar {O~} ist, die in 0 einen singuli~ren Punkt hat. Da diese Mannigfaltigkeit aus allen Punkten ~0(M A- N)

--

c

= ~o(M)

--

~

(h (N))

besteht, so ist sie niehts anderes als die im Hilfssatz 4 mit V bezeiehnete F1/~ehe. Der Beweis des Itauptsatzes im hyperelliptisehen Fall g ---- 3 1/Liftsieh darm genauso wie in Nr. 6 dureh Betraehtung der Komponenten der Durehschnitte T c-~ T~ ffir a in V - - {0} zu Ende ffihren, wobei T irgendeine Marmigfaltigkeit aus der Sehar {O,} (z.B. V selbst) bedeutet. Im nicht-hyperelliptisehen Fall sei wieder T = Ot eine beliebig gewKhlte M'annigfaltigkeit aus der Sehar {O~}. Es sei k o ein algebraiseh abgesehlossener gemeinsamer DefinitionskSrper ffir J und T; es sei k ein k o enthaltender DefinitionskSrper ffir C, ~0 und t. Da O dureh die Abbildung u --> ~o(~)- - u anf sich selbst abgebildet wird, so wird T durch die Abbildung u -->F(u) : q~(~) + 2 t - - u auf sich selbst abgebildet. Da O, also aueh T keine nieht-identische Translation in sich selbst zulassen, so ist E die einzige unter den Abbfldungen u -+ c - - u mit konstantem c, die T auf sich selbst abbildet; dadureh ist F nach Wahl yon T in invarianter Weise gekennzeichnet. Es ist klar, dal3 die Tangentialebenen zu T in den Punkten u und F (u) zueinander parallel sind, d. h. dab sie

322

[1957a]

48

Andrg Weil

bei der vorgenommenen Identifizierung der tangentialen Vektorr~ume zu J in u u n d in F (u) zusammenfallen. I m vorliegenden Fall ist 9 ein Isomorphismus yon C auf eine singularit~tenfreie K u r v e 4. Grades in der projektiven Ebene. Es sei u ein generischer P u n k t yon T fiber k; m a n k a n n u in der F o r m u = ~(M1 q- M~) q- t sehreiben, wo M1, M2 zwei unabh~ngige generisehe P u n k t e von C fiber k sind. Die Gerade durch v(M1) und v(M~) ist dann generiseh fiber k, woraus folgt, dab sie zwei weitere S e h n i t t p u n k t e ~ (N1), ~ (N~) mit T (C) hat, und dal~ je zwei ihrer S e h n i t t p u n k t e mit v(C) fiber k unabh~ngig shad; also sind je zwei der P u n k t e M1, M2, N1, N~ 1/nabh~ngige generisehe P u n k t e yon G fiber k; und der Divisor M1 q - M 2 Jr N1 q - N z gehSrt zu der kanonisehen Vollsehar. Es folgt nun unmittelbar aus Satz 4, dab es auf T auBer u genau 5 P u n k t e gibt, in denen die Tangentialebene zu T zur Tangentialebene zu T in u parallel ist; das shad n~mlieh die P u n k t e u" -= q~(N~ -~- N2) + t,

v~j -~ ~ ( M i ~- Nj) -~- t

( i , j ---- 1,2).

Dabei ist u' nichts anderes als E (u). Wenn wir also ffir v ehaen von u und F (u) versehiedenen P u n k t y o n T wahlen, in dem die Tangentialebene zu T zur Tangentialebene zu T in u parallel ist, so kann das nur einer der vier P u n k t e vi~ sein; u n d es ist dann u - v y o n der F o r m ~ ( M - N), wo M, N zwei unabhi~ngige generische P l m k t e yon C fiber k sind. Es ist aber klar, dab ein solcher P u n k t v fiber/co(u ) algebraisch sein muB, weft es sonst unendlieh viele solche P u n k t e g~be; u - v hat also hSchstens die Dimension 2 fiber k0. Da aus Hilfssatz 5 folgt, da~ der Ort von u - v fiber/C die dort mit V bezeichnete Fl~ehe ist, so muB u - - v denselben Ort V fiber Ico haben wie fiber k. Wir haben also die Fl~ehe V als Ort des Punktes u - - v fiber/c o konstruiert; dabei wurde u als generischer P u n k t yon T fiber k gewii~hlt, und v wurde in der oben angegebenen Weise definiert. Es sei nun u 0 ein beliebiger generischer P u n k t y o n T fiber k0; der Isomorphismus yon k o (u) auf/c o (%) fiber/co, der u a u f u 0 abbildet, li~Bt sieh zu einem Isomorphismus der abgesehlossenen Hfille yon /co(u) auf diejenige yon/c0(u0) erweitern. Dadureh ist ersichtlieh, daB es wieder genau vier P u n k t e auBer u0 und F (u0) gibt, in denen die Tangentialebene zu T zur Tangentialebene zu T in u o parallel ist, und daB, wenn v 0 einer dieser vier P u n k t e ist, der Oft von u0 - - Vo fiber/co die Flaehe V ist. Da nun wieder V in vSllig invarianter Weise konstruiert ist, verl~uft der Beweis genauso wie frfiher weiter. 10. Endlieh sei g -~ 4. W e n a C hyperelliptisch ist, so besteht die kanonisehe lineare Vollschar auf C aus den Divisoren der F o r m c + c' -t- c", wo c, c', c" hyperelliptisehe Divisoren shad; folglieh gibt es zu ehaem positiven Divisor a ---- A -{- A' + A " vom Grad 3 nur ehaen zweiten solehen Divisor a' derart, dab

[1957a]

323 Zum Beweis des ToreUischen Satzes

49

a ~ a' kanonisch ist, falls keiner der Divisoren A ~- A ' , A ~ A " , A" ~- A " hyperelliptisch ist. U n t e r Benutzung des Riemann-Roehsehen Satzes folgt dann ans Satz 4, dab die singularen P u n k t e von O genau die P u n k t e ~(r ~ M) sind, wo c einen hyperelliptisehen Divisor und M einen P u n k t auf C bedeuten. Die singularen P u n k t e irgendeiner Mannigfaltigkeit aus der Schar {Oa} bflden also eine Kurve, die sich yon ~ (C) nur dureh eine Translation unterscheidet. D a m i t wird offenbar der hyperelliptische Fall g -= 4 erledigt sein, sobald wir zeigen, wie m a n diesen Fall v o m nicht-hyperelliptischen Fall in invarianter Weise unterscheiden kann. Das wird daraus folgen, da~ im letzteren Fall O n u r einen oder zwei singulare P u n k t e hat. Von nun an setzen wir voraus, dal] C nicht hyperelliptiseh sei. Da v ein Isomorphismus von C auf 9 (C) ist, wollen wir die Bezeiehnungen dadurch vereinfaehen, dal~ wit C mit ~ (C) durch v identifizieren. Es ist dann C eine singularitatenfreie K u r v e v o m Grad 6 im 3-dimensionalen projektiven R a u m P . Aus dem Riemann-Rochschen Satz ist sofort ersiehtlieh, dal~ es wenigstens eine Flache F , vom Grad 2 in P gibt, die C enthalt, sowieeine F l a c h e F a v o m Grad 3, die C aber nicht F , enthalt. Da C v o m Grad 6 ist, so muB dann $'~ die einzige Flaehe v o m Grad 2 sein, die C enthalt; und es ist C ~ F , . F3. W e n n eine Gerade D mit C drei P u n k t e gemeinsam hat, so mug sie in $'3 liegen. Die in F2 enthaltenen Geraden bflden aber eine oder zwei Scharen der Dimension 1, je n a c h d e m F2 ein Kegel ist oder nicht. Folglich kSnnen zwei S e h n i t t p u n k t e einer solehen Gerade mit C n i c h t fiber k unabhangig sein; dabei ist wie frfiher mit k ein Definitionsk6rper ffir C, J , ~ gemeint. Also hat die Verbindungsgerade zweier unabhangiger generischer P u n k t e yon C fiber k keinen dritten Punk~ mit C gemeinsam 1). Die B e s t i m m u n g der singularen P u n k t e von O wird dureh folgenden Hflfssatz geleistet: Hilfssatz 6. E i n positiver Divisor a yore Grad 3 au/ C genCegt dann und nur dann der Bedingung [(a) ~ 2, wenn er sich in der F o r m a : C 9D schreiben lgtflt, wo D eine au/ F2 liegende Gerade ist, und der Schnitt C 9D au/ F~ berechnet wird. Es sei zuerst bemerkt, da~, wenn F2 ein Kegel ist, C den Scheitel von F.~ nicht enthalten k a n n ; wegen C ---- F 2 9~'3 mfil3te namlich sonst dieser Scheitel a u f C singular sein. N u n ist nach dem Riemann-Rochschen Satz die Bedingung l(a) _~ 2 m i t der folgenden gleichwertig: es muB eine Gerade D im R a u m P geben, derart dab fiir jede D enthaltende E b e n e H die Relation H 9C ~ a besteht. D a n n muB offenbar jede K o m p o n e n t e yon a sowohl auf C wie auf D liegen. W e n n D genau zwei P u n k t e A, B m i t F , gemeinsam hatte, so ware D 1 Im Falle der Charakteristik 0 kann bekanntlich nur dann die Verbindungsgerade zweier unabh~ngiger Punkte auf einer Kurve C einen dritten Punkt mit C gemeinsam haben, wenn C in einer Ebene liegt. Im Falle der Charakteristik p ~ 0 gilt dieser Satz nicht mehr; ein Gegenbeispiel wird durch den Ort des Punktes mit den homogenen Koordinaten (1, t~, t~2, t~3) geliefert.

324

50

[1957a] Andrd Weil

zu/~2 in A und in B transversal; eine generische Ebene durch D hatte dann in A und in B hSchstens die Sehnittmultiplizitat 1 mit C; also hatte a hSehstens die zwei Komponenten A, B, jede hSchstens mit dem Koeffizienten 1, was der Voraussetzung widersprieht. Es habe D genau einen Punkt A mit F~ gemeinsam; dann muB A auf C liegen, weft sonst a =- 0 ware; naeh der oben gemachten Bemerkung mull also A ein einfaeher Punkt yon 2' 2 sein. Wenn D nieht die Tangente zu C in A ware, so h~tte wiederum eine generisehe Ebene dureh D die Schnittmultiplizit~t 1 mit C in A, was wie oben zu einem Widersprueh ffihrt. Werm sehlielllieh D die Tangente zu C in A ist, aber nieht in F~ liegt, so nehmen wir fiir H die Tangentialebene zu F~ in A ; dann ist H 9F 2 ~- D' + D " , wo D', D'" zwei Geraden sind; falls F2 ein Kegel ist, ist D' -----D". Nun ist die Sehnittmultiplizit/~t von H mit C in A dieselbe wie die Schnittmultiplizitat yon C mit dem Zyklus D" + D'" in A auf der F1/~ehe F~; sic ist also gleich 2, weft C die Tangente D hat und folglieh sowohl zu D' wie zu D'" in A transversal ist. Demnach kann A hSehstens den Koeffizienten 2 in a haben, was wiederum der Voraussetzung widersprieht. Damit ist bewiesen, dall die Gerade D auf F2 liegen mull. Wenn die Ebene H dureh D geht, so ist t l . F2 = D +4- D', wo D' eine Gerade ist; und H 9C, im Raum P berechnet, ist nichts anderes als der Zyklus (D + D') 9 C, berechnet auf F~. Wenn H generiseh gew~hlt ist, so geht D ' dureh keine Komponente yon a; es mull also D . C ~ a sein; und, wenn es so ist, ist H . C ~ a ftir jede Ebene H dureh D. Wegen C ---- F2 9$'3 ist aber D 9C, auf F~ genommen, nichts anderes als D 9_Fs, bereehnet im Raum P ; D 9C hat also den Grad 3, so dall aus D . C ~ a die Gleiehung D 9C ~-- a folgt. Bei Berfieksiehtigung von Satz 4 zeigt Hilfssatz 6, dall die singularen Punkte yon O die Punkte ~ (D 9C) sind, wenn D die Gesamtheit aller auf F~ liegenden Geraden durehli~uft, und D 9C auf $'2 bereehnet wird. Nun folgt aus dem oben bewiesenen, dall, wenn zwei verschiedene Geraden D, D" auf F2 in einer Ebene H liegen, der Zyklus (D + D ' ) . G, aufF~ bereehnet, derselbe ist wie der Zyklus H 9 C, bereehnet in P ; da die Zyklen der Form H 9C kanonische Divisoren auf C sind, so sieht man, dall unter diesen Umst~nden die Relation D 9C + D" 9C ~ f besteht. Daraus folgt sofort, dall, wenn F2 ein Kegel ist, samtliche Divisoren D 9C miteinander/~quivalent sind; in diesem Fall hat also O genau einen singularen Punkt. Wenn F~ kein Kegel ist, so gibt es auf F 2 zwei Seharen yon Geraden; wenn D und D" nieht derselben Schar angehSren, so liegen sie in einer Ebene. Daraus folgt, dall die Divisoren D . C, wo D eine dieser Scharen durehl~uft, miteinander ~quivalent sind. Also hat dann O entweder einen oder zwei singul/ire Punkte; eine genauere Betraehtung wiirde zeigen, dall es in diesem Fall tatsachlich zwei verschiedene singul/~re Punkte auf 0 gibt. Nun sei H e i n e generisehe Ebene in P in bezug auf k; es sei k (H) ihr kleinster (k enthaltender) DefinitionskSrper; k ( H ) hat dann die Dimension 3 fiber k. Da H generiseh ist, so hat sie bekanntlieh genau sechs versehiedene Schnittpunkte mit C. Es seien M, M ' zwei dieser Schnittpunkte; ihre Verbindungs-

[1957a1

325 Z u m Beweis des Torellischen Satzes

51

gerade ist definiert fiber k(M, M'). Da H dureh diese Gerade geht, so hat k(H) hSehstens die Dimension I fiber k (M, M'); folglieh hat k (M, M') wenigstens die Dimension 2 fiber k; M, M' mfissen also fiber k auf C unabhi~ngig sein. Je zwei der 6 Schnittpunkte yon H mit C sind also fiber k unabh~ngig, woraus folgt, dab keine drei dieser Punkte auf einer Geraden liegen kSnnen. Wenn M, M', M " drei dieser Punkte sind, so bestimmen sie also eine Ebene, die keine andere als H sein kann; demnach mull k(H) in k(M, M', M") enthalten sein, was nur dann mSglich ist, wenn M, M', M " fiber k auf C unabh/ingig sind. Damit ist gezeigt, dall je drei der Schnittpunkte yon H mit C fiber k auf C unabh~ngig sind. Setzen wir H . C ---- ~ M i , und ui~h : 9(Mi + Mj + Mh) ftir jedes i

Tripel (i, j, h). Nach Satz 4 sind die Tangentialhyperebenen zu O in den 20 Punkten ui~-h alle zueinander parallel. Wenn umgekehrt die Tangentialhyperebene zu O in einem Punkt v ---- ~(N + N' + N") zu den Tangentialhyperebenen zu 0 in den Punkten u~j-a parallel ist, so mull naeh Satz 4 H 9C ~ N + N' W N " sein, woraus folgt, d a l l v einer der Punkte ui~h ist. Die Punkte u ~ i h - ucj,h, shad nun die Punkte

O, ~o(M1--M~), q~(MI + M~--Ms--M4), ~o(MI + M2+ M 3 - - M 4 - - M s - - M s ) , sowie diejenigen, die daraus durch eine beliebige Permutation der M~ entstehen. Die Punkte ~ (M, - - M j ) haben die Dimension 2 fiber k; wir werden beweiseh, dall sie dadureh unter den Punkten u i ~ h - ue~,h, charakterisiert sind. Zungchst shad die Punkte uiCh diejenigen, wo die Tangentialhyperebene zn 0 zur Tangentialhyperebene zu k in u~2a parallel ist; da diese Bedingung nur endlieh viele Punkte bestimmt, so sind sie alle algebraiseh fiber k(Ulzs). Da u m gellerisch auf O ist, so haben alle Punkte uijh --ue~,~, h6chstens die Dimension 3 fiber k. Welter ist, weft ~ Mi ein kanonischer Divisor ist: i

~0(M1 ~- M2-~- Ms - - M 4 - M5 - - M s ) -= 2 ~P(M1 -~-M~ + M 3 ) - - q (f) = 2 ux2s--(f); also ist Ulna algebraiseh fiber k ( u l = - u45e); demnach hat der letztere KSrper die Dimension 3 tiber k; dasselbe gilt dann ffir alle daraus dureh Permutation der M, entstehenden Punkte. Der Beweis daftir, dall der Punkt q(M 1 + M ~ - M s - Ma) ebenfalls die Dimension 3 hat, ist etwas umstgndlieher. Er beruht auf folgendem Hilfssatz: Hilfssatz 7. Es gibt eine Spezialisierung (M~) yon (M~) abet k , / a t welch, M~, M'3 abet k unabhangig sind und M~ = M'~ ist. Es seien M~, M~ zwei unabhgngige generische Punkte yon C fiber k. Da Ma, Mi, Ma fiber k unabh/~ngig sind, so kann man sie jedenfalls auf M~, M~, M', spezialisieren und diese Spezialisierung zu einer Spezialisierung (H', M~) yon (H, M~) erweitern0 wo also M~ = M~ ist. Dann ist H' 9C = ~ M'~, also insbesondere H" 9C ~ M~ + M'~, woraus folgt, dall H" dureh die Verbindungs-

326 52

[1957a] Andrd Weil

gerade D yon M~ und M'8 geht, falls diese P u n k t e versehieden sind, und durch die T~ngente D zu C in M~, falls M~ ~ M~ ist; die dadurch definierte Gerade D ist in beiden F~llen rational fiber dem KSrper K = k (M~, M~). N e h m e n wir an, dab die P u n k t e M~, M '3 fiber/r nicht unabh/~ngig seien; dann hat K die Dimension 1 fiber k. Die P u n k t e M'~, und insbesondere M~ und M's, sind alle algebraisch fiber dem kleinsten DefinitionskSrper k (H') ffir H", da M~, M'2 fiber k unabh/~ngig sind, hat demnach k(H') mindestens die Dimension 2 fiber k; H" ist also nieht algebraisch fiber K und ist generisch fiber K in der Schar der durch D gehenden Ebenen. Es seien F ~ 0, F ' = 0 lineare Gleichungen ffir D mit Koeffizienten in K ; die Gleichung ffir H ' 1/~Bt sich in der F o r m F -k tF' = 0 sehreiben, wo t die Dimension 1 fiber K hat. Der P u n k t M~ kann nicht auf D liegen, weil sonst D die Verbindungsgerade der unabh/~ngigen P u n k t e M~, M~ w/~re und keinen dritten P u n k t M~ mit C gemeinsam h/~tte (bzw. nicht die Tangente zu C in M~ sein kSnnte); H" ist dann die durch D und M~ bestimmte Ebene, so dab K(t) in K(M~) enthalten und M' generischer P u n k t von C fiber K ist. Wegen der Gleichung H'.C--M1--M'3=2M~+M~JrM~ folgt daraus sofort, dab M~ fiber K(t) inseparabel algebraisch sein muB, und zwar hSehstens v o m Grad 4. Wenn K (M~) fiber K (t) rein inseparabel (vom Grad 2, 3 oder 4) w/~re, so h/~tten der K 6 r p e r K (M~) und die K u r v e C das Gesehleeht 0. Es muB also K (M~) rein inseparabel vom Grad 2 sein fiber einer separablen Erweiterung K ' yon K (t) vom Grad 2; dann ist aber K ' hyperelliptiseh, und K(M~) ist mit K" fiber K isomorph; also ist C hyperelliptisch, was unserer Voraussetzung widerspricht. Naeh dem Hflfssatz hat also ~ ( M I + M s - M s - M4) die Spezialisierung ~(M~ - - M~) fiber k, mit unabh/~ngigen M~, M 'a fiber/~; letzterer P u n k t hat die Dimension 2 fiber k; wenn der erstere keine gr6Bere Dimension fiber k h/~tte, so w~re die Spezialisierung eine generische; der erstere P u n k t lieBe sieh dann selbst in der F o r m ~ ( N - N') sehreiben mit unabh/~ngigen N, N'; und es w/~re M 1 + M s -k N'~-~ M8 + M4 q- N. Dabei kann nicht l(M 1 -k M2 -k N') > 1 sein; dann mfiBte n~mlich nach Hflfssatz 6 die Verbindungsgerade von M1 und M~ auf F~ liegen, was bei der Verbindungsgerade zweier unabh/~ngiger P u n k t e auf C nicht der Fall sein kann. Es muB also M~ q - M s + N ' - ~ Ms + M4 Jr N sein, was offenbar unm6glich ist. Daraus ergibt sieh folgende invariante Konstruktion ffir die Flgehe V. Man wahle eine beliebige Marmigfaltigkeit T---- Ot aus der Sehar {0~}; es sei ]co ein algebraiseh abgesehlossener Definitionsk6rper ffir J und T ; es sei u o ein generiseher P u n k t von T fiber k 0. U n t e r den 20 P u n k t e n von T, in denen die Tangentialhyperebene zu T zur Tangentialhyperebene zu T in u 0 parallel ist, gibt es 9 P u n k t e mit der Eigenschaft, daB, wenn Ul einer derselben ist, der P u n k t v ~- u~ - - u 0 die Dimension 2 fiber k 0 hat. D a n n ist g der Ort yon v fiber k 0. Das ist n mlich in dem oben Bewiesenen enthalten, falls k o zu

[1957a]

327

Z u m Beweis des Torellischen Satzes

53

gleicher Zeit ein DefinitionskSrper ffir C, q u n d t i s t . W e n n das nicht der Fall ist, w~hlen wir einen k 0 enthaltenden DefinitionskSrper ffir C, q und t. Es i~ndert sich nichts, wenn wir u o durch einen anderen generischen P u n k t yon T fiber k o ersetzen; wir dfirfen also annehmen, d a b uo sogar fiber k generisch ist; m i t denselben Bezeichnungen wie oben kSnnen wir dann u0 = ulna ~- t setzen. Die P u n k t e , wo die T a n g e n t i a l h y p e r b e n e zu T m i t derjenigen in u o parallel ist, sind d a n n die P u n k t e ui~a + t; wenn u~ irgendeiner dieser P u n k t e ist, so ist er algebraisch fiber ko(Uo) ; der P u n k t (uo, u~) h a t also dieselbe Dimension fiber k wie fiber ko, n~mlich 3; folglich h a t er einen Ort fiber k, u n d zwar denselben wie fiber k o; u n d die Dimension yon uo - - u~, sowohl fiber k wie fiber ko, ist nichts anderes als die Dimension des Brides dieses Ortes bei der Abbridung (x, y) -> x - - y y o n J • J a u f J . Oben ist bewiesen worden, dab es 9 P u n k t e u~ gibt, ffir welche dieses Bild die Dimension 2 h a t und dab es dann die Fl~che V ist. D a m i t ist unsere K o n s t r u k t i o n v611ig gerechtfertigt. Den Beweis des H a u p t s a t z e s im vorliegenden Fall k6nnen wir nun genau wie oben fortsetzen. Es b r a u e h t k a u m gesagt zu werden, dab ein einheitlicherer Beweis sehr zu wfinsehen w~re.

[1957b] Hermann Weyl (1885-1955) (avec C. Chevalley)

(~Quand Hermann Weyl et Hella annonc~rent leurs fian~ailles, l'6tonnement rut g~n~ral que ee jeune homme timide et peu loquace, ~tranger aux cliques qui faisaient la loi dans le monde math~matique de GSttingen, efit remport~ le prix convoit~ par tant d'autres. Ce n'est que peu h peu que l'on comprit ~ quel point Hella avait eu raison dans son choix... 1,~ Peut-Stre les v~rit~s math~matiques, comme les femmes, font-elles leur choix entre ceux qu'elles attirent. Est-ce le mieux dou~ qu'elles choisissent, ou le plus s~duisant ? celui qui les d~sire le plus ardemment, ou eelui qui les a l e mieux m~rit~es ? Elles semblent se tromper parfois; souvent il faut du temps pour s'apercevoir qu'elles ont eu raison. Timide, peu loquaee, ~tranger aux cliques, tel apparaissait donc Hermann W e y l / t ses d~buts; tel il devait rester au fond de lui-m~me, en d~pit des succ6s d'une brillante earri~re. Comme beaucoup de timides une fois rompues les barri~res de leur timiditY, il ~tait capable d'enthousiasme et d'~loquence: (~Ce soir-lh, dit-il en raeontant sa premiere rencontre avec celle qu'il devait 6pouser 2, je d~crivis l'incendie d'une grange auquel je venais d'assister; elle me dit plus tard qu'h m'~couter elle s'~tait ~prise de moi aussit6t. ~) Ses propres confidences nous le montrent profond~ment influen~able aussi, jusque dans sa pens~e la plus intime : (~Mon tranquille positivisme 1 E x t r a i t des paroles p r o n o n e e e s p a r C o u r a n t a u x obs~ques de Hella W e y l le 9 s e p t e m b r e i9t~8. Cette c i t a t i o n , e o m m e plusieurs a u t r e s p a r la suite, est tir6e d ' u n e notice inedite eonsaeree p a r H e r m a n n Weu ~t la m 6 m o i r e de Hella Weyl. Nos a u t r e s citations p r o v i e n n e n t des p u b l i c a t i o n s de W e y l . Reprinted from Enseign. Math. IIl, 1957, pp. 157-187, by permission o f the editor.

329

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[1957b] 158

C. C H E V A L L E Y

ET

A.

WEIL

fut 6bran]6 quand je m'6pris d'une jeune musicienne d'esprit trbs religieux, membre d'un groupe qui s'6tait form6 autour d'un h6g61ien connu... Peu apr6s, j'6pousai une 616re de Husserl; ainsi, ce fur Husserl qui, me d6gageant du positivisme, m'ouvrit l'esprit ~ une conception plus lil~re du monde. )) I1 avail alors vingt-sept ans. C'est ainsi qu'on volt se dessiner, vers l'6poque de son mariage, quelques-nns des prineipaux traits d'une des personnalit6s math6matiques les plus marqnantes et attachantes de la premi6re moiti6 de ce si6cle, mais aussi de l'une des plus difficiles h serrer de pr6s. (~A country lad of eighteen ~, un gars de eampagne de dix-huit ans, ainsi se d6crit-il lui-m~me h son arriv6e h G6ttingen. (~ J'avais choisi cette universit6, dit-il, principalement parce que le directeur de mon lye6e 6tail un cousin de Hilbert et re'avail donn6 pour eelui-ei une lettre de reeommandation. Mais il ne me fallut pas longtemps pour prendre la r6solution de lire et 6tudier tout ce que cet homme avail 6crit. D6s la fin de ma premi6re ann6e, j'emportai son Zahlbericht sous mort bras et passai les vaeances ~ le lire d'un bout ~ l'autre, sans aucune notion pr6alable de th6orie des nombres ni de th6orie de Galois. Ce fnrent les mois les plus heureux de ma vie... 3 )) Un peu plus lard, ce sont les joies de la d6converte: (~Un nouvel 6v6nement fur d6cisif pour moi: je fis une d6couverte math6matique importante. Elle concernait la loi de r6partition des fr6quences propres d'un syst6me continu, membrane, corps 61astique ou 6ther 61ectromagn6tique. Le r6sultat, conjectur6 depuis longtemps par les physiciens, semblait encore bien loin alors d'une d6monstration math6matique. Tandis que j'6tais fi6vreusement occup6 ~ mettre mon id6e au point, ma lampe h p6trole avait commene6 h fumer. Quand je terininai, une 6paisse pluie de flocons noirs s'6tait abattue sur mon papier, sur rues mains, sur mon visage.~) A c e moment, il est d6j& privatdozent /~ GOttingen. Bientbt c'est le mariage, la ehaire ~ Zurich, la 3 ,, De r o u t e rues e x p S r i e n c e s spiritnelles, ~crit-il ailleurs, celles qui m ' o n t c o m b l e de la p l u s g r a n d e joie f u r e n t , e n i 9 0 5 , q u a n d j ' ~ t a i s 5 t u d i a n t , l ' S t u d e d u Zahlberichl ct, en i 9 2 2 , la l e c t u r e de m a i t r e E c k h a r t q u i m e r e t i n t fasein6 p e n d a n t u n s p l e n d i d e h i v e r en E n g a d i n e .

[1957b]

331 HERMANN

WEYL

(1885-1955)

159

guerre. Au bout d'un an de garnison h Sarrebruck (comme simple soldat, pr6cise-t-il), le gouvernement suisse obtient qu'il soit rendu ~ son enseignement /~ l'Ecole polytechnique f6d6rale. <<Je ne puis gu6re me souvenir d'un instant de joie plus intense que le beau jour de printemps, en mai 1916, oh Hella et moi franchimes la fronti6re suisse, puis, arriv6s chez nous, descendimes de nouveau jusqu'au lac ~ t r a v e r s la belie ville paisible. ~ I1 reprend ses travaux. Un cours profess6 h Zurich sur la relativit6 parait en volume, en 1918; c'est 18 c616bre Raum, Zeit, Materie, qui connait cinq 6ditions en cinq ans et, profitant de la vogue extraordinaire du sujet jusque parmi les profanes, r6pand le nora de Weyl bien au-del~ du monde des math6maticiens o~ sa r6putation n'6tait plus h faire. Les ogres de chaires viennent d'un peu p a r t o u t ; celle de Gbttingen en 1922, off il s'agissait de la succession de Klein, fur l'occasion pour lui d'un d6bat de conscience particuli6rement dimcile. A y a n t retard6 sa d6cision t a n t qu'il pouvait, a y a n t encore au dernier m o m e n t pareouru avec sa femme les rues de Zurich en pesant sa r6ponse, il partit enfin au bureau de poste pour t616graphier son acceptation. Arriv6 devant le guiehet, ce fur un refus qu'il t616graphia; il n'avait pu se r6soudre ~ 6changer sa tranquillit6 zurichoise contre les incertitudes de l'Allemagne d'apr6s guerre. < Mais en 1929, quand GOttingen lui ogre la succession de Hilbert, il se laisse tenter. <~Les trois ann6es qui suivirent, dit-il, furent les plus p6nibles que Hella et moi ayons connues. ~>C'est le nazisme, d'abord imperceptible nuage ~ l'horizon, qui grandit r u e d'ceil, s'abat en trombe sur l'Allemagne en d6sarroi, y recouvre t o u t de bone sanglante. Par bonheur pour H e r m a n n Weyl, l'Institute for Advanced Study de Princeton, nouvellement cr66, ogre de le sauver du d6sastre. I1 h6site. I1 aceepte, il refuse. I1 aceepte de nouveau l'ann6e suivante; c'est de Zurich qu'i] envoie sa d6mission ~ G~ttingen en 1933 et qu'il part pour l'Am6rique. I1 n'eut done pas 5 subir ce stage souvent long, parfois humiliant et p6nible, que les circonstances ont impos6 h beaucoup de savants r6fugi6s aux Etats-Unis. La chaire de l'Institute

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A.

WElL

lui assura d'embl6e le confort mat6riel et la situation de premier plan darts le monde scientifique am6ricain auxquels tout, certes, lui donnait droit. Ce furent, dit-il, des ann6es heureuses que cel]es qu'il passa ~ Princeton. Sans doute ne s'aecoutuma-t-il jamais ~ porter ais6ment ce qu'il appelle <~le joug d'une ]angue 6trang6re ~>. Mais, grace au respect e t ~ l'affection qui l'entour6rent d~s l'abord, il se sentait enfin chez lui; et on sent percer nouveau le gars de campagne des premiers jours de GSttingen lorsqu'il d6peint le plaisir qu'il 6prouva, en 1938, ~ poss6der son lopin de terre et ~ y bgtir sa maison. Si la mort de sa femme, en i948, le d6chira cruellement, un second mariage, quelque temps apr~s, lui fit retrouver son 6quilibre. A y a n t pris sa retraite l'Institute, il partagea d6sormais son temps entre Princeton et Zurich. Une attaque cardiaque l'emporta h l'improviste, peu apr~s les f6tes de son soixante-dixi6me anniversaire. A son arriv6e en Am6rique, il avait d6jh donn6 en math6matique le meilleur de lui-m~me, et il le savait. Pour tout autre que lui, la tentation efit 6t6 grande de se reposer sur ses lauriers, de s'abandonner ~ un rSle de ~ pontife ~. Combien n'en est-il pas dont toute l'activit6, pass6 un certain Age, eonsiste g aller de commission en commission, pour y discuter gravement des m6rites de t r a v a u x de ~ jeunes ~> qu'ils n ' o n t pas lus, qu'ils ne connaissent que par our'-dire! Hermann W e y l se faisait une bien autre et bien plus haute id6e de son m6tier de professeur. I1 vit que Princeton seul, ~ notre 6poque, peut 6tre ce qu'ont 6t6 autrefois Paris, puis GSttingen: un centre d'6ehanges, un ~ clearing-house ~>des id6es math6matiques qui circulent de par le monde. Rappelant l'intense vie math6matique qui s'6tait d6velopp6e autrefois ~ GSttingen sous l'influence dominante de Hilbert, il a 6crit: <~Les id6es font boule de neige en un pareil point de condensation de la recherche ~>;et il ajoute: (~Nous avons assist6 h quelque chose de semblable ici ~ Princeton pendant les premi6res ann6es d'existence de l'Institute for Advanced Study. ~> S'il en a ~t6 ainsi, c'est en grande partie a lui qu'en revient le m6rite. I1 se donna pour tache prineipale de se maintenir au courant de l'actualit6, de renseigner et 6clairer les chercheurs, de leur servir d'interpr6te, de comprendre mieux qu'eux ce qu'ils faisaient ou essayaient de faire; il s'y consacra en route

[1957b]

333 HEttMANN

WEYL

(1885-1955)

i61

modestie, conscient de faire oeuvre utile, conscient d'y ~tre irrempla~able. Dans sa production, qui, pendant toute cette p~riode, reste abondante et d'une extraordinaire varietY, on retrouve la trace de ses lectures, des s6minaires et discussions auxquels il prenait part, des probl~mes sur lesquels de tous c5t6s on sollicitait ses avis. Parmi ces travaux, il n'en est gu~re qui n'61ucide un point difficile ou ne comble une lacune f~eheuse. Cette activit6 s'est poursuivie jusque dans ses derni~res ann6es. Par une supr6me coquetterie peut-~tre, sa dernibre publication aura 6t6 une ~dition rajeunie, compl~tement refondue, de son premier livre, ]ivre toujours utile, encore actuel, auquel par eette r6vision il a donn6 une vitalit6 nouvelle. Qui de nous ne serait satisfait de voir sa carriSre scientifique se terminer de m6me ?

Un Prot6e, qui se transforme sans eesse pour se d~rober aux prises de l'adversaire, et ne redevient lui-m~me qu'apr~s le triomphe final: telle est l'impression que nous laisse souvent H e r m a n n Weyl. N'est-il pas all6, pouss6 par le milieu sans doute, par l'occasion, mais aussi par (~l'inqui6tude de son g~nie ~), jusqu'~ se muer en logiclen, en physieien, en philosophe ? L'axiome Ne sutor ultra crepidam nous interdit de le suivre si loin en ses mStamorphoses. Mais, dans son oeuvre math6matique m~me, il n'est que trop fr6quent qu'il vous glisse entre les mains lorsqu'on eroit le mieux le saisir; et il faut avouer que la t~che de ses leeteurs n'en est pas faeilit~e. I1 est vrai qu'il appartient une p6riode de transition dans l'histoire des math~matiques et qu'il s'en est trouv6 profond~ment marqu6. Souvent il a pu prendre un plaisir grisant ~ se laisser entrainer ou ballotter par les eourants opposes qui ont agit6 eette 6poque, stir d'ailleurs au fond de lui-m~me (comme lorsqu'il s'abandonna un m o m e n t l'intuitionnisme brouw6rien) que son bon sens foncier le garantirait du naufrage. Son oeuvre a grandement eontribu6 ce changement de vision qui a fait passer de la math~matique elassique, fond6e sur le hombre r6el, ~ la math6matique moderne, fond6e sur la notion de structure. L'emploi syst6matique et tout abstrait du rev~tement universel, la notion de vari6t6 analytique

334

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C. C H E V A L L E Y

ET

A.

WElL

eomplexe, l'emploi courant et la popularisation, jusque parmi les physiciens, de l'alg6bre vectorielle et du concept d'espaee de repr6sentation d'un groupe, t o u t cela vient a v a n t tout de lui. Mais, s'il 6tait trop 616ve de Hilbert pour ne pas inelure parmi ses outils la m6thode axiomatique, s'il 6tait trop math6matieien aussi pour en d6daigner les sucd6s (son chaleureux 61oge de l'ceuvre d ' E m m y Noether serait lh, si besoin 6tait, pour en faire foi), ee n'6tait pas /~ elle qu'allaient ses sympathies. I1 y v o y a i t <~le filet dans lequel nous nous effor~ons d ' a t t r a p e r la simple, la grande, la divine Id6e ~; mais, darts ce filet, il semble avoir toujours craint que l'on n'attrapfit que des eadavres. A la dissection impitoyable sous le jour cru des projecteurs, il pr6f6fair, en bon romantique, le jeu troublant des analogies, auquel se prdte si bien le langage de la m6taphysique allemande qu'il affeetionnait. Plut6t que de saisir l'id6e brutalement au risque de }a meurtrir, il aimait bien mieux la guetter dans la p6nombre, l'accompagner dans ses 6volutions, la d6crire sous ses multiples aspects, dans sa vivante complexit6. Etait-ee de sa faute si ses lecteurs, moins agiles que lui, 6prouvaient parfois quelque peine le suivre ?

<~Le v6ritable principe de Dirichlet, a dit Minkowski dans un passage que Weyl citait volontiers, ce fur d ' a t t a q u e r les problbmes au moyen d'un m i n i m u m de calcul aveugle, d'un m a x i m u m de r~flexion lucide. ~ E t Weyl a 6crit de son maitre Hilbert : <~Un trait earact~ristique de son oeuvre, c'est sa m6thode d ' a t t a q u e directe; s'affranchissant de t o u t algorithme, il revient toujours au probl~me tel qu'il se pr6sente dans sa puret~ originelle. ~ En deux ou trois occasions, il a atteint pleinement luim~me ~ eet id6al de perfection classique, par exemple dans son travail de 1916 sur l'6gale r@artition modulo 1, et encore darts ses m~moires j u m e a u x sur les fonctions presque pgriodiques et sur les repr6sentations des groupes compacts. Comme il est naturel, ee sont lh, parmi ses travaux, eeux qu'on relit avee le plus de plaisir, ceux dont il est le plus facile aussi de rendre compte. Aussi est-ce par eux que nous commencerons, renongant /~ un ordre logique impossible h suivre lorsqu'il s'agit d'analyser

[1957b]

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(1885-1955)

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une oeuvre aussi riche. L'origine du premier, nous dit-il, se trouve dans un travail sur le phgnom~ne de Gibbs, ou s'~tait pr6sent~e incidemment une question d'approximation diophantienne; il s'6tait agi de faire voir que tout nombre irrationnel peut gtre approeh~ par une suite de fractions p~/q~ satisfaisant aux conditions qn = o (n), ] ~ - - pJq~ ] = o (l/n). Un peu plus tard l'attention de Weyl fut attir6e par F. Bernstein sur 18 probl6me du m o u v e m e n t moyen en m6eanique e61este, probl6me qui remontait /~ Lagrange, et dont Bohl s'oceupait alors; il s'agit 1/~ de d6terminer le comportement asymptotique, pour t ~ ~ , de l'argument d'une somme finie d'exponentielles Ea, e (X~ t), les X 6tant r6els ~. Ce fut l'occasion pour lui d'observet d'abord que son lemme diophantien entrainait ais6ment l'6gale r6partition modulo i de la suite (n~) pour a irrationnel, r6sultat qui fut obtenu en m6me temps par Bohl et par Sierpinski. Mais Weyl,/~ l'6eole de Hilbert et surtout par ses propres recherehes sur les valeurs et fonctions propres, avait aequis un sens trop juste de l'analyse harmonique pour s'en tenir 1/~, Convenons de d6signer par M (xn) , pour toute suite (x~), la limite pour n ~ ~ , si elle existe, de la moyenne des nombres x~, ..., x~. Dire que la suite (%~) est 6galement r6partie modulo i t

6quivaut /~ dire qu'on a M [ / ( ~ ) ] = f [ (x) dx pour certaines 0

fonetions p6riodiques partieuli6res, /~ savoir pour les fonctions de p6riode I qui coincident dans l'intervalle [0, 1] avec une fonetion caraet6ristique d'intervalle. Weyl s'aperr que, si eette propri6t6 est v6rifi6e pour les fonctions en question, elle l'est n6cessairement aussi pour route fonction p6riodique de p6riode 1, int6grable au sens de Riemann, et en partieulier pour les earact6res e (nx) du groupe additif des r6els modulo 1; r6eiproquement, si elle l'est pour ces derni6res fonctions, elle l'est aussi, en v e r t u des th6or~mes classiques sur la s6rie de Fourier, pour toute fonction p6riodique int6grable au sens de Riemann, de sorte que la suite (%~) est 6galement r6partie modulo 1; la d6monstration de ces assertions est imm6diate. Le r6sultat sur l'6gale r6partition modulo I de la suite (n~) pour e irrationnel Ici, comme

d a n s t o u t c e q u i s u i t , o n p o s e e (t) ~

e2rdt .

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[1957b1

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C. C H E V A L L E Y

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WElL

d~coule de 1~ aussit6t, sans aucun lemme diophantien; en rempla?ant le groupe des r6els modulo i par un tore de dimension queleonque, on obtient de m~me, et sans ealeul, la forme quantitative des c~l~bres th~or~mes d'approximation de Kroneeker. T o u t eela, si neuf ~ l'@oque du travail de Weyl, nous parait present bien simple, presque trivial. Mais aujourd'hui encore le lecteur reste ~tonn~ de voir eomme Weyl, sans reprendre haleine, passe de 1~ ~ l'~gale r@artition d'une suite (P (n)), off P est un polyn6me queleonque. Cela revient naturellement, d'apr~s ce qui precede, ~ l'~valuation des sommes d'exponentielles Ee(P(n)), probl~me qui avait ~t~ d~j~ l'objet des recherehes de H a r d y et Littlewood. Plus pr~eis~ment, il s'agit de d~montrer la relation N

y , e (P (n)) =

o (N)

n~0

lorsque P e s t un polyn6me off le coefficient du terme de plus haut degr6 est irrationnel. Pour donner une idle de la m6thode de W e y l , qui (avee les perfectionnements qu'y ont apport6s Vinogradov et son ~eole) est rest6e fondamentale en th6orie analytique des nombres, eonsid6rons le cas off P est du second degr~. Posons done: N /j n=0

e 4tant irrationnel. On 6erira alors, eomme dans l'~valuation elassique des sommes de Gauss: N

Isxp = sxs N =

~,

e ( e ( r n = - - n ') + ~ ( m - - n ) )

rrt,n=0 +N

=

Z r=-

e( r= + N

n~I r

off on a substitu5 n + r/~ m, et off I~ d~signe l'interseetion des deux intervalles [0, N] et [ - - r, N - - r]. Si on d6signe par ~ la derni~re somme (eelle qui est 5tendue/t l'intervalle I~), on a donc I srl 2 <~ E levi. Comme % est une somme de N ~- I termes

[1957b]

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au plus, on a [ % [ < N + I quel que soit r; comme d'autre part % est somme d'une progression g6om6trique de raison e (2~r), on a aussi: I%'1 ~< I sin ( 2 ~ r ) [

-~ 9

Soit 0 <~ z < 1/2; en vertu de l'6gale r6partition des nombres 2~r modulo 1, ]e nombre des entiers r de l'intervalle [ - - N, + N] qui sont tels que 2~r soit eongru modulo t ~ un nombre de Fintervalle [ - - ~, -k z] est de ]a forme 4zN ~- o (N), et est doric ~< 5zN dbs que N e s t assez grand. Pour chaeun de ces entiers, on a 1% ] ~ N + 1; pour tousles autres, on a ] % I ~ l/sin (~z). On a donc, pour N assez grand: 2N+I sin (= e)

[%[~ < 5~N(N + t) + - -

Pour N assez grand, le second membre sera ~< 6zN~; comme il en est ainsi quel que soit ~, on a bien s N = o (N). Si le degr6 du polyn6me P e s t d -? I avee d > l, la d6monstration se fait de m~me (et non par r@urrence sur d) au moyen d'un lemme sur l'6gale r @ a r t i t i o n modulo I d'une fonetion multilin6aire de d variables. Le r6sultat s'6tend aux fonctions de p variables par r6currence sur p. Avec cet admirable m6moire, Weyl 6tait d6j/~ tr~s prSs des fonctions presque p@iodiques. I1 s'y agissait, en effet, en premier lieu, des sous-groupes cycliques et des sous-groupes ~ un param~tre d'un tore de dimension finie, tandis que la th6orie des fonctions presque p@iodiques traite, dirions-nous, des sousgroupes h u n param6tre d'un tore de dimension infinie. On peut m~me dire que cette th6orie, qui suscita rant d'int@~t pendant une dizaine d'ann6es h la suite des publications de H. Bohr en 1924, eut pour principale utilit6 de m6nager la transition entre le point de vue classique et le point de r u e moderne au sujet des groupes compacts et localement compacts. Au temps m~me o/x Weyl s'occupait h GSttingen d'6gale r @ a r t i t i o n modulo 1, vers 1913, les premi@es id6es sur les fonctions presque p@iodiques y 6talent ((dans Fair ~>. Le probl6me du m o u v e m e n t m o y e n portait sur les sommes d'exponentielles, finies il est vrai, et Weyl en

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avait trait6 des cas particuliers, sans d'ailleurs approfondir la question, qu'il ne devait r6soudre compl~tement, toujours par la m6me m6thode, que lorsqu'il s'y t r o u v a ramen6 vingt-cinq ans plus tard. Mais H. Bohr, alors 61~ve de Landau, s'occupait de s6ries d'exponentielles en vue de l'6tude de ~ (s) dans le plan complexe, probl~me auquel Weyl va bient6t s'int6resser en passant, d6terminant m6me par sa m6thode le c o m p o r t e m e n t asymptotique de ~ (i -k it). D'autre part, les 6]~ves de Hilbert 6talent accoutum6s ~ consid6rer les termes de la s~rie de Fourier comme fonctions propres, et les coefficients de cette s6rie comme valeurs propres, d'op6rateurs convenablement d6finis. I1 semble donc que les voles fussent toutes pr6par6es dans l'esprit de Weyl, lorsque a p p a r u r e n t les premiere t r a v a u x de H. Bohr sur les fonctions presque p6riodiques, pour reprendre la question du point de vue des ~quations int~grales. Mais il est rare qu'un math6maticien, qu'il s'agisse du plus grand ou du plus humble, parcoure le plus court chemin d'un point ~ un autre de sa trajectoire. A v a n t de revenir aux fonctions presque p6riodiqnes ~ l'occasion d'une conf6rence de H. Bohr h Zurich, Weyl avait men6 ~ bien ses m~morables recherches sur les groupes de Lie et leurs representations, et avait congu l'id~e, d'une audace extraordinaire pour l'6poque, de ~ construire ~ les repr6sentations des groupes de Lie compacts par la complete d6composition d'une representation de degr6 infini. Blas6s que nous sommes par l'exp6rience des trente derni6res ann6es, cette id6e ne nous 6tonne plus; mais son succ~s semble avoir fait l'effet d'un vrai miracle ~t son auteur; ~ c'est lh, r6p~te-t-il ~ maintes reprises, l'une des plus surprenantes applications de la m6thode des 6quations int6grales ~>. D~jh I. Schur avait 6tendu au groupe orthogonal, au moyen de l'Ol~ment de volume invariant dans l'espace de groupe, les relations d'orthogonalit~ entre coefficients des repr6sentations que Frobenins avait d6couvertes pour les groupes finis ; mais il y avait loin de lh/~ un th6or6me d'existence. Weyl n'h6site pas h introduire, sur un groupe de Lie compact, l'alg~bre de groupe, toujours congue chez lui comme alg6bre des fonctions continues par rapport au produit de convolution h = f * g,

h(s) = f [ ( s t -1) g(t)dt,

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et dont il fait un espaee pr6hilbertien au m o y e n de la norme (f I] [~ ds)~; les int6grales, naturellement, sont prises au m o y e n de l'616ment de volume i n v a r i a n t que fournit la th6orie de Lie, et que Hurwitz avait sans doute ~t6 le premier h utiliser s y s t 6 m a t i q u e m e n t . Dans cet espace, l'op6rateur / - ~ ~ * ~ * /, off ~ d~signe la fonetion ?~ (s) = ~ (s-l), est hermitien et compl~t e m e n t continu; d'apr~s la th6orie de E. Schmidt, ses valeurs propres forment done un spectre discret, et h chacune correspond un espace de fonetions propres de dimension finie, dont on constate i m m 6 d i a t e m e n t qu'il est i n v a r i a n t par le groupe; c'est done un espaee de repr6sentation de celui-ci. Les th6or~mes de Schmidt fournissent a]ors le d6veloppement de 9 suivant les coefficients des repr6sentations ainsi obtenues, d6veloppement qui converge au sens de la norme. C'est lh une g6n~ralisation directe de la m6thode de Frobenius basSe sur la r~duction de la repr6sentation r6guli~re d ' u n groupe fini; la seule diff6renee, c o m m e l'observe Weyl, e'est l'absence d ' u n 61~ment unit6 dans l'alg~bre d ' u n groupe c o m p a c t ; W e y l y suppl6e par un artifice tir6 de la th6orie des s6ries de Fourier, h savoir l ' a p p r o x i m a t i o n de la masse unit6 plac6e h l'origine p a r une distribution de masses h densit6 continue, concentr6e dans un voisinage de l'origine; la convolution avec celle-ei eonstitue un (~op6rateur r6gularisant)), d'emploi courant aujourd'hui, mais dont c'6tait sans doute la premi6re apparition dans le cadre de la th6orie des groupes de Lie ; Weyl s'en sert pour d6montrer que toute fonction continue p e u t 6tre approch6e, non seulement au sens de la norme, mais m~me uniform6ment, par des combinaisons lin6aires de coefficients de repr6sentations. Bien que le m6moire de W e y l se limit~t n6cessairement aux groupes de Lie, il a v a i t atteint en r6alit~, du premier coup, des r6sultats d~finitifs sur les representations des groupes c o m p a c t s ; apr6s la d~couverte de la mesure de H a a r , il n ' y eut pas un m o t ~ changer ~ son expos6, et, chose rare en m a t h 6 m a tique, il ne vint m6me ~ personne l'id6e de le r6crire. Si, comme nous le raisons aujourd'hui, on consid~re une fonetion presque p6riodique c o m m e d~terminant une representation du groupe additif des r6els darts un groupe compact, et q u ' o n suppose acquise pour celui-ci ]a notion de mesure de H a a r , on d~duit

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imm6diatement des r6sultats de Weyl expos6s ci-dessus le d6veloppement de la fonetion en s6rie d'exponentielles. Les outils manquaient h Weyl, en i926, pour adopter ce point de r u e ; il y suppl6e en rempla~ant l'int6grale par une moyenne sur la droite, d6finie comme limite pour T-+ + zr de la valeur moyenne sur l'intervalle It, t ~ 3?] lorsque cette limite est atteinte uniform6ment par rapport au param6tre t. En 1926, il n'allait pas de soi que la th6orie des 6quations int6grales s'appliqu~t h cette moyenne; Weyl est oblig6 de consacrer une bonne partie de son travail A justifier cette application. I1 convient d'observer d'autre part que, sur un groupe compact, la mani6re la plus simple de construire la mesure de Haar consiste j u s t e m e n t h attacher ~ chaque fonction continue une valeur moyenne, par un proc6d6 directement inspir6 de la th6orie des fonctions presque p6riodiques. Que Weyl, en revanche, air cru voir dans la th6orie de Bohr (~le premier exemple d'une th6orie des repr6sentations d'un groupe vraiment non compact )> (par opposition a p p a r e m m e n t avec les groupes de Lie semi-simples dent les repr6sentations, dans son esprit, se ramenaient, par la (( restriction unitaire )), h celles de groupes compacts), cela montre qu'il se faisait encore quelque illusion sur le degr6 de difficult6 des probl~mes qui restaient ~ r6soudre. Ce n'en est pas moins lui qui a ouvert la voie h t o u s l e s progrbs ult6rieurs duns cette direction.

Sur le reste de son oeuvre d'analyste, nous serons beaucoup plus brefs, d ' a u t a n t plus qu'il a lui-m~me excellemment rendu compte d'une honne partie de cette oeuvre dans sa Gibbs Lecture de t948. D6butant, il partieipa activement au courant de recherches qui se proposait d'approfondir et d'appliquer A des probl6mes vari6s d'analyse la th6orie spectrale des op@ateurs sym6triques. Citons particu]igrement, dans cet ordre d'id6es, sa Habilitationsschri/t de 19i0, off il 6tudie un op@ateur diff6rentiel autoadjoint L sur la demi-droite [0, + ~ ]: d (p(t)d~) L (u) = ~t - - q (t) ~,

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Od p, q sont ~ valeurs r6elles et p (t) > O. Sur tout intervalle fini [0, 1], cet op6rateur, soumis aux conditions aux limites du t y p e habituel, (du/dt)o = hu (0), (du/dt)z = h' u (1), rel6ve de la th6orie de Sturm-Liouville ou, en termes modernes, de la th6orie des op6rateurs compl~tement continus; le spectre est r~el et discret et se compose des X pour lesquels l'~quation L u = Xu a u n e solution satisfaisant aux conditions aux limites impos6es. Le passage ~ la limite l ~ -1- w fair apparaitre, non seulement un spectre continu qui peut couvrir tout l'axe r6el, mais encore des ph6nom~nes impr~vus dont la d6couverte est due ~ Weyl. Les plus int6ressants concernent le comportement des solutions pour l-~-t- ~ lorsqu'on donne ~t X une valeur imaginaire fixe; chose remarquable, ils sont ind6pendants du choix de la valeur donn6e ~ X. C'est ainsi que Weyl est amen~ en particulier ~ la distinction fondamentale entre le cas du (~point limite ~ et le cas du <~cercle limite ~>: l'une des propri6t6s caract~ristiques du premier, c'est que l'~quation Lu = Xu y poss+de, quel que soit X imaginaire, une solution et une seule de carr~ sommable sur [0, + w ], tandis que toutes ses solutions le sont, pour X imaginaire, dans le eas du cerele limite. Weyl ~tudie aussi le passage ~ la limite 1-~ + c~ pour les d6veloppements de Sturm-Liouville sur [0, l]; il en tire des formules int6grales o~ apparaissent en g~n~ral des int6grales de Stieltjes, comme on pouvait s'y attendre. Le probl~me des moments de Stieltjes n'est d'ailleurs pas autre chose qne le probl+me aux diff6rences finies, analogue ~ l'6quation Lu ---- Xu sur la demidroite, et Hellinger fit voir par la suite que la m6thode de Weyl s'y transporte prescIue telle qnelle. Mais Weyl put aussi la transposer plus tard ~t un probl~me diff+rentiel o~ le param~tre spectral intervient non lin6airement, ainsi qu'au probl6me aux diff6renees finies correspondant (auqnel il a donn6 le nora de probl6me de Piek-Nevanlinna) ; il apporta m~me ~ cette occasion quelques am61iorations notables ~ son premier expos6. Si celui-ei a donn~ lieu depuis lors ~t des g~n6ralisations assez vari6es, il ne semble pas que la signification v~ritable des r~sultats de Weyl sur les probl6mes ~ param~tre non lin6aire ait jamais 6t6 tir6e au clair. Une autre s6rie de t r a v a u x traite de la r6partition des valeurs propres, dans divers probl~mes de t y p e elliptique. Ils

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reposent principalement sur un principe qui plus tard fut popularis6 par Courant sous la forme suivante: si A est un 0p6rateur sym6trique compl6tement eontinu dans un espaee de Hilbert H, sa n-i6me valeur propre est la plus petite des valeurs ((~minimum m a x i m o r u m ~) que peut prendre la norme de A, c'est-/~-dire le hombre max (Az, z ) / ( z , z), sur un sousespaee de H de codimension n - - 1. Une lois acquise la th6orie des op6rateurs eomplbtement continus, la v6rifieation de ce principe est d'ailleurs imm6diate. Mais Weyl l'adapte en virtuose routes sortes de situations de physique math6matique. Quant au e o m p o r t e m e n t asymptotique des fonetions propres, il avait, nous dit-il en 1948, eertaines conjectures: ~ mais, n ' a y a n t fair pendant plus de trente-einq ans aucune t e n t a t i v e s6rieuse pour les d6montrer, je pr6f6re, ajoute-t-il, les garder pour moi ~; il aura done laiss6 ee probl6me plus diffieile en h6ritage /~ ses suecesseurs.

C'est en 61~ve de Hilbert encore, et en analyste, que Weyl dut aborder le sujet d'un des premiers eours qu'il professa/~ GSttingen comme jeune privatdozent, la th6orie des fonctions selon Riemann. Le eours terrain6 et r6dig6, il se retrouva g6om6tre, et auteur d'un livre qui devait, exereer une profonde influence sur la pens6e math6matique de son si~cle. Peut-~tre s'6tait-il propos6 seulement de remettre au gofit du jour, en faisant usage des id6es de Hilbert sur le prineipe de Dirichlet, les expos6s traditionnels dont l'ouvrage classique de C. Neumann fournissait le modble. Mais il dut lui apparaitre bient6t que, pour substituer aux constants appels ~ l'intuition de ses pr6d6eesseurs des raisonnements corrects et, eomme on disait alors, ~ rigoureux ~ (et dans l'entourage de Hilbert on n'.admettait pas qu'on trich~t l~-dessus), c'6taient avant t o u t les fondements topologiques qu'il fallait renouveler. Weyl n'y semblait gubre pr6par6 par ses t r a v a u x ant6rieurs. I1 pouvait, dans cette tfiche, s'appuyer sur l'ceuvre de Poincar6, mais il en parle h peine. I1 mentionne, comme l ' a y a n t profond6ment influene6, les recherches de Brouwer, alors darts leur premi6re nouveaut6; en r6alit6, il n'en fait aucun usage. De fr6quents contacts avec Koebe, qui d6s lors

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s'6tait consacr6 tout entier h l'uniformisation des fonctions d'une variable eomplexe, durent lui ~tre d'une grande utilit6, particuli~rement dans la mise au point de ses propres id6es. La premibre 6dition du livre est d6di6e ~ F61ix Klein, qui bien entendu, comme Weyl le dit dans sa pr6face, ne pouvait manquer de s'int6resser h u n travail si voisin des pr6occupations de sa jeunesse ni de donner ~ l'auteur des conseils inspir6s de son temp6rament intuitif et de sa profonde connaissance de l'ceuvre de Riemann. Bien qu'il n'efit jamais connu celui-ci, c'6tait Klein qui, ~ GSttingen, incarnait la tradition riemannienne. Enfin, dans Fun de ses m6moires sur les fondements de la g6om6trie, Hilbert avait formul6 un syst6me d'axiomes fond6 sur la notion de voisinage, en soulignant qu'on trouverait lh le meilleur point de d6part pour <(un traitement axiomatique rigoureux de l'analysis situs ~). De tous ees 616ments si divers que lui fournissaient la tradition et le milieu, Weyl tira un livre profond6ment original et qui devait faire 6poque. Le livre est divis6 en deux chapitres, dont le premier contient la pattie qualitative de la th6orie. Les notions de (~surface ~) (vari~t~ topologique de dimension 2/~ base d~nombrable) et de ((surface de Riemann ~> (vari~t~ analytique complexe h base d~nombrable, de dimension complexe l) y sont d~finies au moyen de syst~mes d'axiomes, inspir6s naturellement de celui de Hilbert, mais qui cette lois (saul une l~g~re omission dans la premiere ~dition) ~taient destines ~ subsister sans retouches, et devaient servir de module ~ Hausdorff pour son axiomatisation de la topologie g~n~rale. Dans la premiere et la deuxi~me ~dition, la condition de base d~nombrable apparait sous forme de condition de triangulabilit~ ; et la triangulation joue un grand rSle dans la suite du volume; elle devait ~tre 61imin~e enti~rement de la troisi~me ~dition. Les questions touchant au groupe fondamental, au rev~tement universel, ~ l'orientation, sont 61ucid6es avec soin dans un esprit tout moderne, ainsi que les rapports entre propri6t6s homologiques et p6riodes des int6grales simples sur la surface. Dans la premi6re et la deuxi~me 6dition, l'auteur va jusqu'h la construction, pour les surfaces orientables compactes, d'un syst6me de <(r6trosections ~), c'est-~-dire essentiellement d'une base privil6gi6e pour le premier groupe d'homologie;

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comme il le dit lui-m&me, il aurait pu, au prix d'un 16ger effort suppl~mentaire, aller jusqu'/~ la repr6sentation de la surface au moyen d'un ~<polyg0ne canonique >>/~ 4g c6t6s (g d6signant le genre), et /t la d6termination explicite du groupe fondamental, et on peut regretter qu'il ne l'ait pas fait. Mais la construction mgme des rgtrosections, n~cessairement bas~e sur la triangulation, disparait dans la troisigme ~dition, au profit d'un traitemerit plus purement homol0gique o/1 n'interviennent que des recouvrements. En tout cas, pour tout l'essentiel, ee chapitre constitue une raise au point /~ peu prbs d6finitive des questions qu'il traite. Les th4orgmes d'existence font l'objet du deuxi~me chapitre. Weyl y donne du principe de Dirichlet une d~monstration sireplifi6e, bas6e naturellement sur l'id~e de Hilbert qui consiste, comme on sait, ~ op@er dans l'espace pr~hilbertien des fonetions diff@entiables a v e c l a norme de Diriehlet; mgme dans la troisigme 6dition, il n'a pas cru devoir suivre la variante qu'il avait pourtant contribu~/~ cr6er lui-mgme, et qui consiste R op6rer par projection orthogonale dans le compl6t6 de l'espaee en question, puis ~ montrer apr~s coup que la solution obtenue est diff@entiable. Une fois acquis le principe de Diriehlel,, l'auteur en tire les principales propri6t6s des int6grales ab61iennes et des fonctions multiplicatives, le thbor~me de I/iemann-I/och, puis le th6or~me de l'uniformisation, c'est-Mdire la representation conforme du rev~tement universel de la surface de t/iemann sur une sph@e, un plan ou un disque. Si on laisse de c6t6 les cas de genre 0 ou 1, le rgsultat peut s'exprimer en disant que route surface de t/iemann compacte, de genre > 1, peut se d~finir comme quotient du plan non-euclidien par un groupe discret de d~plaeements sans point fixe. <~Ainsi, dit Weyl dans la pr6face de la premi@e 6dition, ainsi nous pgngtrons dans le temple off la divinit6 est rendue ~ elle-m~me, d~livr~e de ses incarnations terrestres: le cristal non euclidien, oit l'arch6type de la surface de tliemann se laisse voir dans sa puret~ premiere...>> C'est en songeant sans doute /~ ce passage que Weyl dit plus tard de sa preface que (~plus encore que le livre lui-m~me, elle trahissait la jeunesse de son auteur >). Nous dirions aujonrd'hui qu'on a construit pour la surface de ttiemann un module qui est cano-

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nique h u n d6placement pr6s dans le plan non euclidien; autrement dit, on a associ6 canoniquement une structure h une autre. Mais qui saurait mauvais gr6 h Weyl, apr6s avoir achev6 un livre de cette valeur, d'avoir exprim6 d'une mani6re peut-Stre un pen trop romantique son enthousiasme juv6nile ?

C'est en 19t6, pendant la guerre, que Weyl fit paraitre en Suisse son premier m6moire de g6om6trie, sur le c616bre probl6me de la rigidit6 des surfaces convexes. Ici encore, GSttingen lui avait fourni son point de d6part. Sous la direction de Hilbert, Weyl avait collabor6 h la publication des oeuvres compl6tes de Minkowski, off la th6orie des corps convexes tient rant de place. D'autre part, Hilbert avait montr6 comment on peut faire d6pendre les in6galit6s de Brunn-Minkowski de la th6orie des op6rateurs diff6rentiels elliptiques. L'espace t/3 6tant consid6r6 comme espace euclidien, et (x, y ) d6signant le produit scalaire dans R a, soit V un corps eonvexe dans cet espace, d6fini au moyen de la fonction d'appui H; eela veut dire que H satisfait aux conditions II(x + x') ~< H(x) + tt(x'),

H(Xx) ~ ?,H(x)

pour ), >~ 0,

et que V e s t l'ensemble des points y satisfaisant h (x, y> ~< H (x) quel qne soit x. Si on suppose H diff6rentiable en dehors de 0, ]e volume de V est alors donn6 par une formule vol (V) = f H.Q (H) d ~ ,

o6 l'int6grale est 6tendue h la sph6re unit6 S o d6finie par (x, z} ~- i, oh do~ d6signe l'616ment d'aire sur So, et oh Q (H) est une forme quadratique par rapport aux d6riv6es partielles seeondes de H. Soient F, F' deux fonctions, diff6rentiables en dehors de 0, satisfaisant toutes deux h la condition d'homog6n6it6 F (;~x) = kF (x) pour X ) 0; soit B (F, F') la forme bilin6aire sym6trique par rapport aux d6riv6es partielles secondes de F et h celles de F' qui se d6duit de la forme quadratique Q (H) par lin6arisation, c'est-~-dire qni est telle que Q (H) = B (H, H);

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posons aussi LF ( F ' ) = B (F, F'). On v6rifie facilement, au moyen de la formule de Stokes, que l'int6grale I (F, F', F") = j" F".B (F, F')d co, So

o5 F " d6signe une troisi~me fonction satisfaisant aux m~mes conditions que F et F', d6pend sym~triquement de F, F' et F " Cela revient ~ dire que LF, consider6 comme op6rateur diff~rentiel sur les fonctions sur S OprolongSes/~ R 3 par homog6n6itS, est un op~rateur autoadjoint. Si V', V" sont deux corps convexes d~finis par des fonctions d'appui H', H", les formules ci-dessus m o n t r e n t que les <~volumes mixtes ~ associ~s par Minkowski V, V', V" ne sont autres que les nombres I (H, H, H') et I (H, H', H " ) ; de plus, nn calcul simple montre que L H est elliptique. Darts ces conditions, comme le fair voir Hilbert dans ses Grundzi~ge, l'application h L . de la thSorie des op6rateurs autoadjoints elliptiques conduit, pour le cas diffSrentiable, /~ l'inSgalit~ de Brunn-Minkowski. Mais il se trouve que L~ n'est autre qu'un op6rateur qui se pr~sente dans la thSorie de la dSformation infinitSsimale de la surface E fronti~re de V; jointe aux rSsultats de Hilbert, cette observation, due ~ Blaschke, entrainait l'impossibilit6 d'une telle d~formation pour E. Enfin, Hilbert, ~ propos des fondements de la g~om6trie, avait d6montr~ l'impossibilit6 d'appliquer isom~triquement une sphere sur une surface convexe non sph6rique. D'ailleurs, des rSsultats analogues sur les poly~dres convexes avaient 5t5 obtenus jadis par Cauch$: non seulement un poly~dre convexe n ' a d m e t aueune dSformation infinit~simale, mais encore, si P et P' sont deux poly~dres convexes a d m e t t a n t m~me schema combinatoire et a y a n t leurs c6t6s correspondants 5gaux, ils ne peuvent diff~rer Fun de l'autre que par un d~plaeement ou une sym6trie. Tout cela mettait ~ ]'ordre du jour l'extension anx surfaces convexes du second th~or~me de Cauchy. Mais Weyl ne s'arr~te pas 14. I1 consid~re en m~me temps un probl~me d'existence que, faute d'une conception claire de la notion de vari6tg riemannienne abstraite, personne n'avait encore m~me formnl6. I1 s'agit de savoir si <~toute surface

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couvexe ferm6e, donn6e in abstracto, est r6alisable )) on, c o m m e nous dirions m a i n t e n a n t , si route vari6t6 riemannienne compacte, s i m p l e m e u t connexe, de dimension 2, h courbure p a r t o u t positive, a d m e t un plongement isom6trique dans l'espace euelidien Ra; la question d'unicit6, pour ce probl~me d'existence, est alors celle m~me dont Weyl 6tait patti. I n t e r r o m p u dans son travail par sa mobilisation en 1915, il se contenta d'esquisser son id6e de d6monstration, et ne la m e n a jamais ~ terme. I1 p a r t du fair que toute <(surface convexe in abstracto ~) p e u t ~tre repr6sent6e c o n f o r m 6 m e n t sur la sph6re So, donc d6finie p a r un ds 2 donn6 sur S o sous la forme ds 2 = e 2r 2, off d~ est la longueur d'arc (( naturelle )) et (I) une fonction diff6rentiable sur So; soit Z ((I)) la (( surface abstraite )) ainsi d6finie. La condition que Z ((I)) soit h courbure p a r t o u t positive s'exprime par une in6galit6 diff6rentielle K (O) > 0; on constate aussitSt que Fensemble des q) qui y satisfont est convexe; il s'ensuit que Fensemble des surfaces convexes abstraites est connexe. L'id6e de Weyl est alors d ' a p p l i q u e r au probl~me une m6thode de coatinuit6. T o u t revient, I d6signant l'intervalle [0, 1], h d6terminer une application ~ de S o x I dans R 3 de telle sorte que l'application x -~ ~: (x) = ~ (x, v) de S o dans R a applique isom6triquem e n t Z (vq)) sur une surface convexe S~ = ~ (So) , et cela pour t o u t z C I. Pour cela, W e y l consid6re ~9/bv c o m m e une d6formarion infinit6simale de S~, dont la d6termination se ram~ne la solution d'une 6quation Au ~ f, off / est une fonction sur So, d6pendant de S~, et A cst essentiellement l'op6rateur elliptique L . relatif h S~. L'application de la m6thode de Hilbert h cette 6quation donne donc, en principe, une 6quation diff6rentielle fonctionnelle pour ~ ; il s'agit d'en t r o u v e r une solution sur l'intervalle I qui se r6duise pour z = 0 h l'application identique de S o dans R3; et on p e u t esp6rer y p a r v e n i r au m o y e n de l'une quelconque des m6thodes classiques de r6solution des 6quations diff6rentielles. Une d6monstration complSte a 6t6 obteuue r 6 c e m m e n t par Nirenberg en suivant cette vole; les br~ves indications donn6es ici sufilront t o u t au moins ~ faire a p p a r a i t r e l'extr6me hardiesse de l'id6e de H e r m a n n Weyl.

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Rentr6 ~ Zurich en 1916, Weyl eut, semble-t-il, quelque vell6it6 de revenir aux surfaces convexes; un m6moire oh il reprend les r6sultats de Cauchy sur les poly6dres pr6sageait peut6tre un mode d'attaque has6 sur des m6thodes moins infinit6simales et plus directes. Mais c'est bient6t la relativit6 qui attire et accapare son attention. Lh encore, il 6tait dans la tradition. Minkowski avait particip6 activement au courant de recherches qui s'6tait d6velopp6 a u t o u r de la relativit6 restreinte. Hilbert suivait de pros les t r a v a u x d'Einstein et cherchait, sans grand succ6s d'ailleurs, h 6claircir les probl~mes de la physique par la m6thode axiomatique. (( I1 faut en physique un autre t y p e d'imagination que celle du math6maticien ~), constate plus tard H e r m a n n Weyl, non sans quelque m61ancolie, dans sa notice sur Hilbert. Sans doute, en 6crivant ces roots, songeait-il aussi $ sa propre exp6rience et h cette (( th6orie de Weyl ~) ~ laquelle, disait-il vers la m~me 6poque, il ne croyait plus depuis longtemps. Mais h partir de 1917, et p e n d a n t plusieurs ann6es, son enthousiasme est d6bordant. E n 1918, il publie son cours de l'ann6e pr6c6dente sur la relativit6 sous le titre Baum, Zeit, Materie. (~A l'occasion de ce grand sujet, 6crit-il dans la pr6face de la premi6re 6dition, j'ai voulu donner un exemple de cette interp6n6tration, qui me tient t a n t h cceur, de la pens6e philosophique, de la pens6e math6matique, de la pens6e physique... )); mais, ajoute-t-il avee une modestie non exempte de naivet6, ((le math6maticien en moi a pris le pas sur le philosophe ~); et ce ne sont pas les math6maticiens qui s'en plaindront. Son ouvrage, dans ses cinq 6ditions successives, fit beaucoup pour r6pandre parmi les math6maticiens et les physiciens les connaissances g6om6triques et les notions essentielles de l'alg~bre et de l'analyse tensorielles. A partir de la troisi6me 6dition, on y trouve aussi un expos6 de la (~th6orie de Weyl )), premier essai d'une (~th6orie unitaire ~ englobant dans un m~me sch6ma g6om6trique les ph6nom~nes 61ectromagn6tiques et la gravitation. Elle 6tait fond6e, dirionsnous h pr6sent, sur une connexion li6e au groupe des similitudes (d6fini au m o y e n d'une forme quadratique de signature (1,3)), au lieu qu'Einstein s'6tait born6 h des connexions li6es au groupe de Lorentz (groupe orthogonal pour une forme de signature

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(1,3)), et plus pr~cis~ment ~ la connexion sans torsion d~duite canoniquement (par transport parall~le) d'un ds 2 de signature (1,3). Cette th~orie eut du moins le m~rite d'~largir le cadre de la g~om~trie riemannienne traditionnelle et de preparer les voies aux (~g~om~tries g~n~ralis~es ~) de Cartan, c'est-it-dire ~ la th~orie g~n~rale des connexions li~es ~ un groupe de Lie arbitraire. Qnant aux preoccupations philosophiques de Weyl pendant cette p~riode d'intense fermentation, elles ne tard~rent pas (heureusement, serions-nous tent~s de dire) ~ se couler dans un moule plus ~troitement math~matique, l'amenant /~ chercher une base axiomatique aussi simple que possible aux structures g~om~triques sous-jacentes h la th~orie d'Einstein et h la sienne; c'est lh c e qu'il appelle le (~ Raumproblem ~), le probl~me de l'espace; il y consacre plusieurs articles, un cours profess~ h Barcelone et h Madrid, et un opuscule qui reproduit ces lemons. I1 s'agit lfi en r~alit5 de caract~riser le groupe orthogonal (attach4 h une forme quadratique, soit complexe, soit r~elle et de signature quelconque) en rant que groupe lin~aire, par quelques conditions simples au moyen desquelles on puisse rendre plausible que la g~om~trie de l'(~univers ~) est d~finie localement par un tel groupe. Bien entendu, c'est la thdorie des groupes de Lie et de leurs representations qui domine la question; Weyl en donne une esquisse dans un appendice de son livre. De son c6t~, Cartan ne tarda pas h donner, du principal r~sultat math~matique de Weyl sur ce sujet, une d~monstration bas~e sur ses propres m~thodes. I] n'~tait pas dans le temperament de Hermann Weyl, une fois parvenu ainsi au seuil de l'ceuvre de Cartan, de se contenter d'y jeter un coup d'ceil rapide. D'autre part,/~ la suite peut-~tre d'une remarque de Study qui l'avait bless~ au vif, il avait commencd ~ s'intSresser aux invariants des groupes classiques. Study, dans une preface de 1923, iui avait reproeh~, ainsi qu'aux autres relativistes, d'avoir, par leur n~gligence /~ l'~gard de ce sujet, contribu~ ~ (~la mise en jach~re d'un riche domaine culturel ~); il entendait surtout par lh la thSorie des invariants du groupe projectif, dans laquelle il ~tait d'usage de faire rentrer t a n t bien que mal les autres groupes h l'occasion de l'~tude des

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covariants simultan6s de plusieurs formes. P a r une r6action bien caract6ristique, Weyl r6pondit ~ Study, avec une promptitude extraordinaire, par un m6moire off il reprend h la base la th6orie classique au moyen d'identit6s alg~briques dues ~ Capelli et indique aussi comment elle s'6tend aux groupes orthogonaux et symplectiques; ce qui ne l'emp~che pas de protester que, ((si m~me il avait connu aussi bien que S t u d y lui*m~me la th6orie des invariants, il n'aurait eu nulle occasion d'en faire usage dans son livre sur la relativit6: chaque chose en son lieu !)~. La synth~se entre ces deux courants de pens6e - - groupes de Lie et invariants - - s'op~re darts son grand m6moire de i926, m~moire divis~ en quatre parties, dont il dit lui-m~me vers la fin de sa vie qu'il repr6sente (( eu quelque sorte le sommet de sa production math6matique ~. L'6tude qu'avait faite Young, vers 1900, de la d~composition des tenseurs en tenseurs irr6ductibles d6finis par des conditions de sym6trie avait abouti en substance h la d6termination de toutes les representations (( simples ,, c'est-h-dire irr~ductibles, du groupe lin6aire sp6cial; mais, enferm~es qu'6~aient ces recherches dans le cadre de la th6orie traditionnelle, il leur ~tait impossible, par d6finition, d'obtenir ce r~sultat sous la forme que nous venons de lui donner. De son c6t6, Cartan, parti de la th6orie g6n6rale des groupes de Lie, avait d6termin6 toutes les representations en question, sans d'ailleurs, semble-t-il, faire le lien entre ses r6sultats et ceux d'Young. D6signons par G le groupe lin6aire sp6cial, et par g son alg~bre de Lie, qui se compose de toutes les matrices de trace 0; soit 4 l'ensemble des matrices diagonales contenues dans g. Une repr6sentation simple de G d6termine une representation simple p de q, donc une repr6sentation de 4. Cartan montre que l'espace V de la repr6sentation p e s t engendr6 par des vecteurs qui sont vecteurs propres de routes les operations p ( H ) , pour H C 4. Soit e l'un de ces vecteurs propres; on a p ( H ) . e = k ( H ) e, off ~ est une forme lin6aire sur ~, qu'on appelle le p o i d s de e; si H est la matrice diagonale de coefficients al, ..., an, il est facile de voir que k (H) est de la forme m I a ~ - ~ . . . - ~ m n an, Ofl les m i sont des entiers d6termin6s l'addition pros d'un m~me entier. Si on ordonne lexicographiquement l'ensemble des syst~mes (mj, ..., ran) de n entiers, on

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obtient donc une relation d'ordre dans l'ensemble des poids des repr6sentations de G. On appelle poids/ondamental d'une repr6sentation simple le plus grand des poids de cette repr6sentation pour la relation d'ordre qu'on vient de d~finir. Cartan avait montr~ que ce poids d6termine compl~tement la repr6sentation (h une ~quivalence pros), qu'il correspond/~ un syst~me d'entiers (mi) tel que m~ ~ ... ~ ran, et que r~ciproquement tout syst~me d'entiers satisfaisant ~ ces in~galit6s appartient au poids fondamental d'une repr6sentation simple de G. Soient de plus p, p' deux representations simples de G, operant respectivement sur des espaces vectoriels V, V'; soient )~, k' leurs poids fondamentaux; soient e, e' des vecteurs de V, V', de poids respectifs )~, k'. Le produit tensoriel p | ~' de p et p' (dit parfois encore (~produit kroneck~rien ~), et not6 le plus souvent p • ~' par Weyl) est une reprSsentation op6rant sur un espace V | V' de dimension ~gale au produit de celles de V e t V', qni est form~ de combinaisons lin6aires d'~l~ments se transformant par G comme les praduits formels xx', avec x ~ V, x'E V'; et, pour cette representation, le vecteur e • e' est de poids k ~- ),'. Soit W le sous-espace de V | V' engendrd par e | e' et ses transform6s par G; il dScoule facilement des r~sultats de Cartau que W ne peut pus se d4composer en somme directe de sous-espaces invariants par les operations de G; et Cartan avait cru pouvoir d~duire de lh que W fournit la repr6sentation simple de poids dominant k -[- )?. Weyl observa que cette conclusion est ill~gitime tant qu'on ne sait pas h priori que les representations de G sent toutes semi-simples (c'est-h-dire compl~tement r~ductibles). A vrai dire, ce dernier r~sultat n'~tait pas indispensable pour se convaincre du fait que la d6composition ~l'Young de l'espace des tenseurs fournit toutes les representations simples de G; Young avait en effet ~tabli l'irr~duetibilit~ des repr6sentations qu'il avait construites, et il suffisait d'~tablir par un calcul facile que leurs poids dominants sent tous ceux pr4vns par la th~orie de fiartan. Mais on n'efit obtenu ainsi que la classification des representations simples. Au contraire, en d~montrant la complete r6ductibilit8 de routes les representations de G, Weyl en obtint du mSme coup (compte tenu des r~sultats de Young et Cartan) la classification d~finitive, qui s'exprime par le fait que

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route <~grandeur ]in6aire }~, comme il dit, se d6compose en tenseurs irr6duetibles. On sait aujourd'hui d6montrer le th6or6me de compl6te r6ductibilit6 par des m6thodes alg6briques; c'est 1~ le point de d6part de la th6orie cohomologique des alg6bres de Lie. Mais c'est de consid6rations tout autres que Weyl tire sa d6monstration. I1 observe, comme l'avait d6j~ fait Hurwitz dans son m6moire sur la construction d'invariants par la m6thode d'int6gration, que la th6orie des repr6sentations du groupe lin6aire sp6cial complexe G est 6quivalente ~t celle des repr6sentations du groupe G u form6 des matrices unitaires a p p a r t e n a n t ~ G; en derni6re analyse, cela tient ~ c e que route identit6 alg6brique entre coefficients d'une matrice unitaire reste vraie pour une matrice quelconque. Or Gu poss6de une propri6t6 importante qui n ' a p p a r t i e n t pas a G: il est compact, ce qui permet, comme l'avait fait voir Hurwitz, de construire des invariants pour Gu et par suite pour G par int6gration dans l'espace du groupe Gu au moyen de l'616ment de volume invariant fourni par la th6orie de Lie. La m6thode classique qui permet d'6tablir la compl6te r6ductibilit6 des repr6sentations des groupes finis par construction d'une forme hermitienne, d6finie positive, invariante par les op6rations du groupe, s'6tend alors d'elle-m~me au groupe G~. Ce n'est pas seulement le th6or6me de compl6te r6duetibilit6 pour G que Weyl tire de la restriction au groupe unitaire Gu; il s'en sert aussi pour calculer explicitement les caract6res et les degr6s des repr6sentations simples de G. On voit tout de suite, en effet, que si Z e s t le caract6re d'une repr6sentation de Gu, et si s e s t une matrice diagonale unitaire de d6terminant I et de coefficients diagonaux e (xl) , ..., e (x~), la valeur de X (s) s'exprime comme somme de Fourier finie en xl, ..., x~ et ne change pas par une permutation quelconque des x i. Weyl montre que ces propri6t6s, jointes aux relations d'orthogonalit6 fournies, elles aussi, par la m6thode d'int6gration, suffisent d6jg ~ d6terminer compl6tement les caractbres et ~ en obtenir des expressions explicites. La suite du m6moire de Weyl est consacr6e ~ l'extension des m6thodes ci-dessus aux groupes orthogonaux et symplectiques, puis aux groupes semi-simples les plus g6n6raux. Soit cette

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fois .q une alg~bre de Lie semi-simple eomplexe; pour en ~tudier les repr6sentations, Weyl va appliquer la mSthode de restriction unitaire au groupe adjoint G de g, mis sous forme matrieielle relativement ~ une base eonvenable de g. Pour qu'il y air dans G (~assez ~>d'op~rations unitaires, il est nSeessaire que .~ admette ce qu'on appelle aujourd'hui une forme eompaete, ou pour mieux dire une base telle que ]es eombinaisons lin~aires r6elles des ~lSments de cette base forment l'alg~bre de Lie d'un groupe compact. En examinant ehaque groupe simple s@ar~ment, Cartan avait vSrifi~ dans chaque eas l'existence d'une forme compaete; Weyl en donne une d~monstration a priori basSe sur les propri~t~s des eonstantes de structure de g. Cela fait, il introduit ]e groupe G~ des operations unitaires de G, et son alg~bre de Lie g~. Le groupe G~ est compact, et la th~orie des representations de % est 5quivalente ~ celle des reprSsentations de g. Mais iei se pr~sente une difficult~ nouvelle; du fait que G~ peut n'~tre pas simplement connexe, la th~orie des reprSsentations de gu n'est plus entiSrement ~quivalente h celle des reprSsentations de G~. Si on eherehe h r~tablir l'~quivalence en remplagant G u par son rev~tement universel G~, qui, lui, est simplement connexe, il devient n~eessaire de s'assurer que eelui-ei est compact, et aussi d'en faire un groupe, loealement isomorphe G~,. Ce dernier point, qui devait peu apr~s 5tre ~lueid5 par Sehreier, est compl~tement laiss5 de c6t5 dans le mSmoire de Weyl. Mais c'est dans le premier que r~sidait la vSritable diffieult~. La question revient naturellement h faire voir que G~ a un groupe fondamental fini. Pour cela, Weyl introduit un sousgroupe A~ de G~, qui joue le m~me r61e que le groupe des matrices diagonales dans la th~orie du groupe unitaire special. T o u t ~l~m e n t s de G~ est eonjugu5 ~ un ~l~ment de A~; excluant certains 5lSments s, dits singuliers, qui forment un ensemble a y a n t trois dimensions de moins que Gu, s n'est eonjugu5 qu'/~ un hombre fini d'515ments de A u ; de plus, les 516ments de A u qui ne sont pas singuliers forment dans A~ un domaine simplement connexe A. Supposons que s d~erive dans G~ une eourbe fermSe F qui ne reneontre pas l'ensemble des ~lSments singuliers. Si s (t) est le point de param~tre t sur F, on peut dSterminer par eontinuit5 une courbe a (t) dans A~ telle que, pour t o u t t, a (t) soit eonjugu5 ~ s (t).

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Quand le point s (t) revient /~ sa position initiale s (1) = s (0), le point a (t) vient en un point a (1) qui est un 616ment de A~ conjugu6 de a (0), ce qui ne laisse pour ee point qu'un nombre fini de possibilit6s. Si on a a (1) = a (0), la courbe d6crite par a (t) est ferm6e, et par suite r6duetible a un point dans A; Weyl montre que P e s t alors elle-m~me r6ductible /~ un point. I1 en r6sulte facilement que le groupe fondamental de l'ensemble des 616ments non singuliers de (3~ est fini. De cela, et du fait que les 616ments singuliers se r6partissent sur des sous-vari6t6s a y a n t an moins trois dimensions de moins que On, Weyl conelut (/~ vrai dire sans d6monstration) que G~ lui-m~me a un groupe fondamental tlni. Ce point ~tabli, la voie est ouverte /~ la g~n~ralisation complete au cas semi-simple des r~sultats obtenus pour le groupe lin6aire sp6eial. Weyl d6montre la complete rdduetibilit6 des representations de g, et d~termine explicitement le caract~re et le degr~ d'nne repr6sentation simple de poids dominant donn~. Ici encore, cette d6termination r6sulte des relations d'orthogonalit~ entre caractbres et des propri6t~s formelles de la restriction Z d'un caract~re au groupe A~ qui reeouvre A~ dans le rev~tement simplement connexe G~ de G~. Ce groupe est un tore; Z e s t une combinaison lin~aire finie de caraetbres de ce tore, invariante par les op6rations d'uu certain groupe fini S d'automorphismes du tore qui g6n6ralise le groupe des permutations de z~, ..., z~ dont il a 6t6 question plus haut/~ propos du groupe unitaire special. Le groupe S, dont les dgveloppements ult6rieurs de la th~orie out montr6 qu'il y joue un r61e fondamental, s'appelle m a i n t e n a n t le groupe de Weyl. Enfin la th6orie s'ach~ve par la d6monstration de l'existence des repr6sentations simples de poids fondamental donn~. Pour les alg~bres simples, cette existence avait 6t~ 6tablie par Cartan par des constructions directes dans ehaque cas partieulier. Weyl, lui, applique au groupe compact G~ la m6thode de d6composition de la <~repr6sentation r6guli~re ~>, obtenue au moyen de la th~orie des 6quations int~grales suivant l'id~e que nous avons expos6e plus haut. Pour eonclure /~ partir de 1/~, il lui faut encore un lemme de nature plus technique, ~none6 seulement dans le m~moire de 1926, et dont Weyl n'a publi6 la d6monstration que

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dans son cours de 1934-35 (paru h Princeton sous forme de notes mim6ographi6es, The structure and representations o/ continuous groups).

Beaucoup plus tard, Weyl revint sur la d6termination des repr6sentations des groupes semi-simples dans son ouvrage The classical groups, their in~ariants and representations. L ' e s p r i t de ee livre est assez diff6rent de celui du m6moire de 1926. L ' o b j e t de l ' a u t e u r est m a i n t e n a n t d'une p a r t de d6montrer par des m6thodes p u r e m e n t alg6briques les r6sultats d6j/~ obtenus au sujet des repr6sentations des groupes classiques (groupe lin6aire g6n6ral, groupe lin6aire sp6cial, groupe orthogonal et groupe symplectiqne), et d ' a n t r e p a r t de faire la synthSse entre ces r6sultats et ]a th6orie formelle des invariants qui s'6tait d6velopp6e sons l'influenee de Cayley et Sylvester au cours du x~x e si6cle. Esp6rait-il cette lois se laver d6finitivement du reproehe de S t u d y en r a m e n a n t h la vie cette th6orie qui 6tait sur le point de sombrer dans l'oubli ? I1 nons dit lui-m~me que la d6monstration par Hilbert du th6or6me g6n@al de finitude a v a i t t~presque tu6 le sujet ~>; on peut se d e m a n d e r si Weyl ne lui a u r a pas, en r6alit6, port6 le coup de grace. La situation dans laquelle on se trouve en th6orie des invariants est la suivante. On a uric ou plusieurs repr6sentations lin6aires p, p', ..., d'un groupe G, op6rant sur des espaces r e c t o rids V, V', ... On consid@e des fonetions F (a, x', ...) d6pendant d'un a r g u m e n t a dans V, d'un a r g u m e n t a' dans V', etc. et s'exprim a n t c o m m e polynSmes par r a p p o r t aux eoordonn6es de ces arguments, homog6nes par r a p p o r t aux coordonn6es de chacun d'eux. Une telle fonction s'appelle un i n v a r i a n t si, pour t o u t s dans G, on a F (s.x, s.x', ...) ~- F (x, x', ...) .

Si J~, ..., Jh sour des invariants, t o u t polyn6me en J1, ..., Jh en est un aussi p o u r v u qu'il satisfasse aux conditions d'homog6n6it6 impos6es. Le premier probl6me de ]a th6orie est de t r o u v e r des invariants J~, ..., Jh tels que t o u t autre invariant puisse s'6crire comme polyn6me en les J~; eela fait, on se propose 6galement

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de d~terminer les relations alg~briques F (J1, ..., Jh) = 0, dites (~syzygies ~, qui lient entre eux les invariants q u ' o n a construits. Pla~ons-nous plus partieuli~rement dans le cas off G a ~t~ identifi0, au m o y e n d'une certaine representation ,ol, avee un sous-groupe du groupe lin~aire/~ n. variables, operant sur l'espace vectoriel V 1 ~- k n, off k est un corps de base q u ' o n suppose de caract~ristique 0. Consid0rons d ' a b o r d le cas off les repr~sentations ~, f , ..., coincident toutes avec 91; on dit alors q u ' o n eherche les invariants d'un certain n o m b r e de ~ vecteurs ~ (on entend par 1~ des vecteurs de V1). R e p r e n a n t sans grand changem e n t son travail de 1924 par lequel il a v a i t r~pondu ~ Study, Weyl montre alors, pour un groupe G unimodulaire, que la d~termination des invariants de vecteurs en n o m b r e quelconque peut se ramener, au m o y e n des identit~s de Capelli, au probl~me analogue pour n I vecteurs. Si G n ' e s t pas unimodulaire, ce r~sultat reste vrai pour les (~invariants relatifs ~ (polyn6mes se multipliant par une puissance du d~terminant de s quand on t r a n s f o r m e tons les vecteurs par s). Weyl d0duit de 1~ la solution des deux probl~mes ci-dessus pour le groupe unimodulaire et pour le groupe orthogonal; et il ~tend cette solution au cas des invariants d~pendant, non seulement d'un certain h o m b r e de vecteurs ~ cogr~dients ~ (se t r a n s f o r m a n t suivant p1), mais aussi d ' u n certain n o m b r e de veeteurs (~contragr~dients ~ (se transf o r m a n t c o m m e les formes lin~aires sur V1). Ensuite il passe aux invariants d~pendant de ~ quantit~s ~> x, x', ... a p p a r t e n a n t des espaces de representation quelconques du groupe ~tudi~; le cas off x, x', ... sont des formes homog~nes p a r r a p p o r t aux coordonn~es d ' u n vecteur ~contragr~dient ~ est celui dont t r a i t a i t plus particuli~rement la th~orie classique. Pour pouvoir aborder ]a question dans ce cadre g~n~ral, il faut a v a n t t o u t connaitre les representations simples du groupe; aussi une partie i m p o r t a n t e du livre est-elle consacr0e ~ la d~termination algabrique des representations (~tensorielles ~ des groupes classiques. Cela fair, Weyl m o n t r e que les i n v a r i a n t s d~pendant de plusieurs ~ quantit~s ~ d'esp~ee quelconque s ' e x p r i m e n t c o m m e polynSmes en un n o m b r e fini d'entre eux; il ~tend ce rSsultat, dans une certaine mesure, au groupe affine. Enfin, il emploie la m~thode d'int~gration pour d~montrer le r~sultat correspondant pour les

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repr6sentations quelconques d'un groupe compact, le corps de base 6rant cette lois le corps des r~els.

Pour f~ter son soixante-dixi~me anniversaire, les amis et 61~ves de Hermann Weyl publi@ent un volume de Selecta extraits de son oeuvre. I1 n'y a peut-6tre pas lieu de se f61iciter de cette mode des morceaux choisis destin6s '~ cS16brer la raise la retraite de math6maticiens 6minents. C'est trop pour les uns; ce n'est pas assez pour les autres. Du moins le volume en question contient-il une bibliographie complete de l'oeuvre de Hermann Weyl, ~tablie par ordre chronologique s, et dont nous avons naturellernent fair grand usage pour r6diger la pr6sente notice. Pour rem6dier en quelque mesure aux inSvitables lacunes de celle-ci, nous donnons ci-dessous une liste des mSmoires de Weyl, classSs par sujet; rien ne peut mieux, croyons-nous, en faire ressortir l'Stonnante vari6t6. Les num@os, bien entendu, renvoient h la liste des Selecta.

I. Analyse. a) Equations int6grales singuli@es: i, 3. b) Probl~mes de valeurs propres et dSveloppements fonctionnels associSs h des 6quations diff@entielles ou aux diff@ences finies: 6, 7, 8, 12, 103.

c) R6partition des va]eurs propres d'op@ateurs complStement continus en physique math6matique: 13, 16, 17, 18, 19, 22.

d) Espace de Hilbert: 4, 5. e)

Ph6nombne de Gibbs et analogues: 10, 11, 14.

/)

Equations diff@entielles li6es h des probl~mes physiques: 36-37 (d~veloppements asymptotiques, apparent6s au ph6nom~ne de Gibbs, au voisinage d'une discontinuit6 dans un

n convient de signaler qu'on n ' a pas fait figurer clans cette bibliographie les notes de cours, publi6es sous formc mim~ographi6e par l ' I n s t i t u t e for A d v a n c e d Study de Princeton, ct qui reproduisent plusieurs des cours qu'il y professa.

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probl~me d'61eetromagnOtisme), 123-t24-125 (6tude directe d'une Oquation diff6rentielle li6e /~ un probl6me de eouehe limite).

g) Problbmes elliptiques: 121 (principe de Dirichlet trait6 par la <
Ggomglrie.

a) Surfaces et poly6dres convexes: 25, 27, t06. b) Analysis situs: 24, 26, 57-58-59, 159. c) Connexions, g6om6trie diff6rentielle li6e ~ la relativit6: 30, 3i, 34, 43, 50, 82. d) Volume des tubes: 1t6 (eontient d~jfl, essentiellement, la formule de Gauss-Bonnet pour les vari~t~s plongSes darts un espace euelidien). III.

In~ariants et groupes de Lie.

a) ~ Raumproblem ~: 45, 49, 53, 54. b) Invariants: 60 (1 re partie), 63, 97, i17, 122. c) Groupes de Lie et leurs representations: 6t, 62, 68, 69, 70, 74, 79, 80, 81. IV. [telati~itg. 29, 33, 35, 39, 40, 46, 47, 48, 51, 52, 55, 56, 64, 65, 66, 89, 93, 134, i35.

[1957b]

359 HERMANN WEYL (1885-1955) V.

t87

Thgorie des quanta.

75, 83, 84, 85, 86, 87, 90, 9i, 100, 101, 140, 141. VI. Th~orie des algdbres. a) Matrices de Riemann: 99, t07, 108. b) Questions diverses: 96, 105 (spineurs, en commun avec

R. Brauer), 109, 110, 143. VII.

Th6orie g6om6trique des nombres

(d'apr~s Minkowski et Siegel). 120, 126, 127, 136. VIII. Logique. 9, 32, 41, 60 (2e pattie), 67, 77, 78. IX. Philosophie. t11, 118, 119, 138, 142, 156, 163. X.

Articles historiques et biographiques.

15, 88, 94, 95, 102, 131, 132, 137, 147, 149, 150, 152, 157, 160, 161, 162; et la con%renee <~Erkenntnis und Besinnung~>, Studia Philos., 15 (Basel, 1955) (traduction frangaise dans Rer de Thgol. et Philos., Lausanne, 1955). XI.

Varia.

2, 28, 38, 76, 92, 115, 128, 133, 139, 144, 146, 148, 151, 158.

[1957cl ] R6duction des formes quadratiques, d'apr6s Minkowski et Siegel La th6orie s'ins6re dans le sch6ma suivant, dont on rencontrera d'autres exemples par la suite. Soit G u n groupe de Lie semi-simple; soit F u n sous-groupe discret de G; classiquement, on se propose de trouver, pour F dans G, un " d o m a i n e fondamental", c'est-fi-dire un syst6me de reprdsentants dans G pour G/F possddant des propridt6s sympathiques. Si K est un sous-groupe compact de G, G operera sur l'espace homog6ne H = G / K (notation: on fera op6rer G ~ droite sur H; H est donc l'ensemble des classes/t gauche, K x , suivant K dans G); il est immddiat que, si F est discret darts G, F, en tant que sous-groupe de G, op6re sur H d'une mani6re " p r o p r e m e n t discontinue" (ce qui veut dire: quelle que soit la partie compacte X de H, les 7 ~ F tels que X c~ X7 r 0 sont en nombre fini). En fait, on prend toujours pour K un sous-groupe compact maximal de G; il r6sulte de la th6orie des groupes semi-simples que K est d6fini par l/t d'une mani6re unique, fi un automorphisme intdrieur pr6s de G; autrement dit, H est ddfini d'une mani6re unique fi un isomorphisme pr6s. Sans restreindre essentiellement la gdndralit6 des probl6mes qu'on se propose de traiter, on peut supposer que G n'admet pas de sous-groupe compact invariant; H = G / K est alors, au sens de Cartan, 1' "espace riemannien sym6trique" associ6/t G. Du point de vue thdorique, il est c o m m o d e de prendre G semi-simple au sens strict (non seulement au sens infinitdsimal, mais au sens global), c'est-/L-dire de centre r6duit/t l'616ment neutre (et non pas seulement de centre discret). Du point de vue des calculs explicites, il est souvent c o m m o d e de calculer avec des matrices, ce qui conduit /t 6crire des groupes, infinit6simalement simples ou semi-simples, ayant un centre discret (par exemple le groupe lindaire spdcial sur R, SL(R, n), dont le centre, sin est pair, est _+ 1,, off 1, est la matrice unit6). D'ofi maints abus de langage, dont on s'excuse d'avance. Dans la th6orie de la r6duction des formes quadratiques au sens classique, on part du groupe Go = PL+(R, n) (partie connexe du groupe projectif r6el fi n variables " h o m o g 6 n e s " - quotient du groupe lin6aire L+(R, n)/t n variables, de ddterminant > 0 , par son centre = quotient du groupe lin6aire sp6cial SL(R, n), it n variables, de d6terminant 1, par son centre), et du groupe discret F0, image dans G o du groupe multiplicatif F des matrices de d6terminant _+ 1 /l coefficients dans Z. Un sous-groupe compact maximal de Go est l'image K o dans Go du groupe orthogonal spdcial SO(R, n) (matrices orthogonales de ddterminant 1). Soit P l'espace des formes quadratiques positives non d6g6ndr6es fin variables: n

F(x) = ' x . A . x =

~

aijxixj,

'A = A.

i,j--I

(un vecteur x, en notation matricielle, sera toujours conqu c o m m e matrice ",i n lignes et 1 colonne). On 6crira A >> 0 pour exprimer que la matrice sym6trique A est 360

[1957c]

361

Reduction des formes quadratiques, d'apres Minkowski et Siegel celle d'une forme positive non degendrde. Le groupe L(R, n) opere sur P par la loi (A, X) -~ tX 9A - X, off A ~ P e t X est un element de L(R, n) ecrit comme matrice. Toute forme de P s'ecrivant comme somme de n carrds (de formes lineaires en les x;), le groupe opere transitivement dans P: l'element 1, de P (c'est-fi-dire la forme "standard" ~ x~) etant invariant par le groupe orthogonal O(R, n), P s'identifie l'espace homogene L(R, n)/O(R, n). Dans l'espace vectoriel (de dimension n(n + 1)/2) de routes les formes quadratiques dans R" (ou, ces formes 6rant exprimdes au moyen de la base canonique de R", dans l'espace des matrices symdtriques fi n lignes et n colonnes), Pest ddfini par les inegalites P(x) > 0 (x r 0) et forme donc un c6ne convexe; on verifie immediatement que ce c6ne est ouvert; sa frontiere est l'ensemble des formes positives ddgenerees. Par passage au quotient par la relation d'equivalence dont les classes d'equivalence sont les rayons ("demi-droites") issus de 0, P determine une partie convexe Po d'un espace projectif de dimension n(n + 1)/2 - 1. Par passage au quotient, le groupe projectif Go = P L ( R , n) opere dans Po; il resulte de ce qui precede qu'il y opere transitivement, et que P0 s'identifie fi Go/Ko, espace "riemannien symetrique" associe/t Go. La theorie de Minkowski conduit/t la determination, dans Po, d'un domaine fondamental pour le groupe discret F o, qui est un polyedre convexe (plus exactement, la reunion de l'intdrieur d'un polyedre convexe et d'une pattie convenable de sa frontiere). Soit d'abord n = 2 ("formes binaires"); on ecrit F ( x ) = ax 2 + 2 b x y + cy2;

P e s t le c6ne determine par a > 0, ac - b 2 > 0 (intdrieur de l'une des nappcs d'un cene du second degr6 dans R3); Po est, dans le plan projectif, l'intdrieur de la conique ac - b 2 = 0. C o m m e X --. t X 9A 9X est une representation du groupe lineaire dans l'espace des formes symdtriques A, il s'ensuit, par passage au quotient, que les operations de Go dans Po sont des homographies ou automorphismes du plan projectif ambiant, conservant la conique frontiere de Po; Po 6rant pris comme "modele cayleyien" de la geometrie plane hyperbolique, G induit donc, sur P0, un sous-groupe du groupe des automorphismes de cette geometrie; on verifie sans peine que c'est marne exactement la pattie connexe de ce dernier groupe (c'est-fidire le "groupe des ddplacements non-euclidiens"). La correspondance entre le "modele cayleyien" et le demi-plan de Poincar6 s'obtient comme suit :/t toute forme F e P on fait correspondre, d'une part le point f qu'elle determine dans Po, d'autre part celle des racines de l'equation a z 2 + 2bz + c = 0 dont la partie imaginaire est > 0; la correspondance f ~ z e s t une correspondance biunivoque entre P0 et le demi-plan supdrieur de la variable z; les operations de Go dans ce demi-plan sont 6videmment celles du groupe homographique reel de determinant > 0 (groupe des ddplacements non-euclidiens dans le modele de Poincare). Pour determiner un point de P, on peut, au lieu de se donner une forme quadratique dans R", se donner une forme quadratique F, positive non degenerde, dans un espace vectoriel E de dimension n sur R, et une base (el . . . . . e,) de cet espace. Si (x, y ) est le produit scalaire dans E, associe fi F de la maniere habituelle, les donnees

362

[1957c] Reduction des formes quadratiques, d'apres Minkowski et Siegel en question determinent la forme quadratique dans R" donnee par

Posons A = Hai~ll, air = (ei, ej~; soit X = ]lxijl] un 61ement du groupe lindaire L(R, n); par definition, la transformee de A par X est A' = tX. A - X , ce qui s'ecrit aussi n J A' = Ila'ijll avec aijt = (ei,/ ej), ei! =

xkiek.

2 k

1

Dire que X est ~ coefficients entiers de determinant _+ 1 6quivaut/t dire que les "lattices" (sous-groupes discrets de rang n de l'espace E) engendres par (el . . . . . e,) et par (e'l . . . . . e',) coincident. L'ensemble form6 par A et tous ses transformes par le groupe F est donc l'ensemble des matrices A' = (e~, e:i) lorsqu'on fair parcourir fi (e'1. . . . . e',) l'ensemble de t o u s l e s systemes de n gdndrateurs du lattice engendr6 par el . . . . . e,. Supposons qu'on ait choisi, dans P0, un systeme de reprdsentants Mo pour la relation d'dquivalence ddterminee par les operations du groupe F 0 sur Po; convenons momentandment de dire que la matrice A d'une forme quadratique dans R" est "rdduite" si le point qu'elle determine dans Po appartient /t M0, et aussi qu'une base (e~ . . . . , e,) de l'espace E est "reduite" pour une forme quadratique F dans E si la matrice A des (ei, e j) est reduite. I1 rdsulte de ce qui precede qu'dtant donnds une forme F et un lattice A darts E, il y a au moins un systeme de generateurs de A qui est une base reduite pour E, et que deux tels systemes determinent ndcessairement la m0me matrice A, donc ne different l'un de l'autre que par une transformation du groupe orthogonal de F. Choisir un systeme de representants M 0 revient donc "fi peu pres" fi enoncer une loi qui, /t tout couple form6 d'une forme quadratique F et d'un lattice A, permette d'associer, avec le moins d'ambigu'ite possible, un systeme de generateurs de A. On va, d'apres Minkowski, formuler une telle loi. Soit d'abord n = 2; on prendra pour el un vecteur va0 du lattice A dont la " l o n g u e u r " F(el) ~/2 soit la plus petite possible, puis pour ez un vecteur dont la longueur soit la plus petite possible parmi ceux qui ne sont pas de la forme le~ (t ~ R). I1 est clair qu'il n'y a qu'un hombre fini de vecteurs el satisfaisant/t la 1c condition, et qu'il y e n a au moins deux; des considerations geometriques 61ementaires (et evidentes) font voir qu'il y e n a exactement deux, saufdans les cas suivants : (a) lattice de Gauss (points fi coordonnees entieres dans le plan muni de la forme x2 ~_ y2); (b) lattice hexagonal (engendre par les vecteurs (1, 0) et (89 , / 3 / 2 ) d a r t s le plan muni de x 2 + y2). I1 s'ensuit que, dans tousles cas, les divers choix possibles de el se deduisent les uns des autres par une rotation laissant A invariant (rotation d'angle ~ dans le cas general, d'angle m~/2, resp. m7c/3, avec m entier, dans les cas (a), resp. (b)). Quant au second vecteur e2, on constate non moins aisement qu'il est determine d'une maniere unique au signe pres (une lois q u ' o n a choisi el) saul dans le cas d'un lattice engendre par deux vecteurs (1, 0) ct (2l, y), avec I):1 >x/3/2, dans le plan muni de x 2 + y2. O n levera l'indetermination de signe, autant

[1957c]

363

Rdduction des formes quadratiques, d'apr6s Minkowski et Siegel que possible, en convenant de prendre e 2 tel que ( e l , e2) > 0; il se trouve que, lorsque cette r6gle laisse subsister une ambiguit6, le lattice A admet la droite 0el pour axe de symdtrie, et les choix possibles de e 2 sont symdtriques l'un de l'autre par rapport 5- cet axe. Une base (el, e2) du plan muni d'une forme quadratique F sera dite rOduite si elle poss6de les propridtds 6nonc6es ci-dessus, par rapport au lattice qu'elle engendre et 5- la forme F. Une forme quadratique F = a x 2 "4- 2 b x y + cy 2 dans R 2 sera dite rOduite si la base canonique de R 2, form6e des vecteurs (1, 0) et (0, 1), est r6duite par rapport 5. cette forme. Si on 6crit les indgalitds F((1, 0)) _< F((0, 1)) < F((_+ 1, 1)) on obtient imm6diatement les conditions a < c, 12b[ _< a, qui sont donc ndcessaires pour que F soit r6duite (avec bien entendu a > 0 puisque F doit 6tre positive non d6g6n6r6e). La condition subsidiaire imposde 5. ez s'6crit ici b _> 0. Doric: 0
0<2b


Rdciproquement, ces indgalitds entrainent, d ' a b o r d que F est positive non d6gdn6r6e (c'est clair), puis que F est rdduite (c'est facile 5. v6rifier). Elles d6terminent, dans P0, c'est-5.-dire dans l'int6rieur de la conique ac - b 2 = 0, un triangle Mo, ayant un sommet sur la conique. Puisque, 6tant donn6s un lattice et une forme, la r6gle ci-dessus permet toujours de construire au moins une base rdduite, il s'ensuit que M o contient un syst6me complet de reprdsentants pour le groupe F o opdrant dans P0. Du fait que, pour F et A donnds, les divers choix possibles de la base r6duite se d6duisent tous les uns des autres par une rotation ou une sym6trie conservant A, on conclut que M o constitue marne un tel syst6me de repr6sentants. I1 s'ensuit que P0 est rdunion de M o et de tous ses transformds par Fo (" pavage" du plan non euclidien par les transformds du domaine fondamental). Naturellement, ces transform6s ne sont pas deux fi deux sans point c o m m u n : cela tient justement aux cas d'ambiguit6 possible dans le choix d'une base rdduite, ou, ce qui revient au marne, au fait que F o n'op6re pas sur Po "sans point fixe". L'analyse ddtaill6e du pavage en question est classique, n'offre pas de difficult& e t e s t sans intdr6t pour ce qui suit. O n notera que si, au lieu de partir du groupe F des matrices enti6res de d6terminant +_ 1, on 6tait parti du sous-groupe F + de F form6 des matrices de d6terminant 1, on aurait 6t6 conduit naturellement fi un domaine fondamental deux lois plus grand, correspondant 5. la notion d'"dquivalence propre" des formes quadratiques (au lieu de l'6quivalence " p r o p r e ou impropre" qui a servi ci-dessus); pour l'6quivalence propre, une forme sera dite rdduite si elle satisfait 5. 0 < a < c, 12bl < a; ce sont les conditions classiques de Gauss, correspondant, dans le demiplan de Poincar6, au domaine fondamental classique pour le groupe modulaire (d+termin+ par ]zl > 1, IR(z)[ _< 89 Passons au cas g6ndral. Dans un espace vectoriel E de dimension n sur R, une base (el . . . . . e,), engendrant un lattice A, sera dite r~duite par rapport 5. une forme quadratique F (positive non d6g6ndr6e) si elle satisfait aux deux conditions suivantes : (i) P o u r chaque i, soit E i l'ensemble des vecteurs e 6 A tels que (el . . . . . ei 1, e)

364

[1957c]

Rdduction des formes quadratiques, d'aprds Minkowski et Siegel fasse pattie d'un systeme de n gdndrateurs pour A; e~ est alors un vecteur de longueur minimum parmi ceux de E~. (ii) O n a ( e i , e i + l ) _ > 0 p o u r 1 _ < i _ < n - 1. C o m m e ci-dessus, la condition (ii) sert "a se ddbarrasser, dans la mesure du possible, de rambiguitd de signe inhdrente au choix de e~ quand F et A sont donnds. Si on s'dtait laissd guider par le casn = 2, on aurait, dans (i), pris pour E~ l'ensemble des vecteurs de A qui ne sont pas dans l'espace vectoriel engendrd par e , , . . . , e~ 1 ; avec cette condition plus forte, il n'aurait pas dtd vrai que tout lattice possdde une base rdduite par rapport ~ une forme F donnde, d'ofl ndcessitd de modifier assez fortement l'dnoncd des rdsultats ultdrieurs (il est/t noter que, dans les gdndralisations de la thdorie, par exemple aux corps de nombres algdbriques, on ne peut dviter ces dnoncds d'aspect plus compliqud). On notera que, pour que e e Ei, il faut et il suffit que l'image de e dans le quotient A/At_ ~ de A par le lattice engendr6 par e ~ , . . . , e~_ 1 soit un dldment "primitif" de ce quotient (c'est-~-dire non divisible par un entier > 1). I1 revient au marne de dire que E iest rensemble des vecteurs ~ i xjej Off X l , . . . , X n sont des entiers tels que le plus grand c o m m u n diviseur (x~ . . . . . x,) de x~, x~+ 1. . . . . x, est dgal ~ 1. Par suite, pour qu'une forme quadratique positive non ddgdndrde F ( x ) = Y',u a u x l x j soit rdduite (ce qui veut dire que la base canonique de R" est rdduite par rapport 5, la forme), il faut et il suffit qu'elle satisfasse aux conditions: (i) F ( x l . . . . . x , ) > a , pour tout systdme d'entiers (Xl . . . . , x , ) tels que (xi . . . . .

x.)

=

1;

(ii) ai, i+l -> 0 pour 1 < i _< n - 1. I1 est clair d'ailleurs que la condition (i), jointe /t al 1 > 0 (resp. all -> O) suffit fi entrainer que F est positive non ddgdndrhe (resp. positive). Si on remarque qu'avec les notations ci-dessus on a e~ c E~ et ej-4-_ e~ ~ Ej chaque fois que i < j, on en conclut que (i) entraine les indgalitds: aii <_ ajj

(i < j)

(I)

12aul _< a,,

(i < j).

(II)

et

De plus, un thdordme fondamental, dfi/t Minkowski, affirme qu",i tout n correspond un C, > 0 tel que (i) entraine aussi rindgalitd: alia22 . . . a , ,

<_ C n det(A),

A = I1%11-

(III)

Soit M la partie de P ddfinie par les conditions (i) et (ii); soit Mo son image dans Po- L'ensemble M 6tant ddfini par une infinitd d'indgalitds lindaires et homogdnes par rapport aux a u, il est 6vident que c'est un ensemble convexe "positivement homogene". C o m m e il est dvident que tout lattice A possdde au moins une base rdduite par rapport ~ une forme donnde, Mo contient un systeme de reprdsentants de Po par rapport au groupe F0 . On peut donner "ace sujet des 6noncds beaucoup plus prdcis (voir plus loin), mais ils ont peu d'importance du point de vue des applications. Les rdsultats vraiment importants sont lids ~ l'introduction de deux familles d'ouverts dans P (resp. P0) qu'on va ddfinir maintenant.

[1957c]

365 R6duction des formes quadratiques, d'apr+s Minkowski et Siegel

Pour chaque t > l, soit S(t) la partie de P d6finie par les in6galitds :

aii < t a i + l , i + 1

(1 _< i _< n -- 1) ]

12aijl < ta,

(1 _< i < j ___ n) /

(A)

a l i a 2 2 . . , ann < C,t" det(aij) Le th60r6me de Minkowski montre que M c S(t) pour t > 1 (c'est principalemerit sous cette forme qu'on a fi l'utiliser). 11est clair que les S(t) forment une famille, croissante avec t, de parties ouvertes de P, dont P e s t la r6union. On notera So(t ) l'image de S(t) dans P0. D'autre part, la r6duction de la forme quadratique F ~t une somme de carr6s par la m6thode des alg6bristes babyloniens (m6thode connue 6galement, dans la litt6rature, sous les noms de " m 6 t h o d e de Jacobi" et "m6thode d'orthogonalisation de Schmidt", /t moins qu'il ne convienne plut6t de l'attribuer ",i quelque savant russe ...) permet, d'une mani6re et d'une seule, d'6crire:

F(x) =

~ aiaxix j

i,j 1

= Zn (di xi q- =~i+llijXj)2 i=1

j

ou encore, en termes matriciels, A = tT- D 9 T, off D est la matrice diagonale ayant dl . . . . . d, pour coefficients diagonaux (et 0 partout ailleurs), et T la matrice triangulaire (au sens strict, c'est-fi-dire ayant 1 partout dans la diagonale principa!e) dont les coefficients sont les tii pour i < j, et les bii (1 si i = j, 0 sinon) pour j _< i. Par r6currence, on d6montre ais6ment que les di, tij sont des fonctions rationnelles des a~j, fi d6nominateurs r 0 dans P. Cela pos6, soit S'(u), pour tout u > 1, l'ensemble des points de P de la forme A = tT- D 9 T, off D est une matrice diagonale, fi coefficients diagonaux d l , . . . , d,, et T une matrice triangulaire au sens strict, fi coefficients tlj pour i < j, satisfaisant aux in6galit6s:

0 < di < udi+ 1

(1 _< i _< n -- 1);

]tii[ < u

(1 _< i < j _< n).

(B)

D'apr6s ce qu'on vient de dire, il est clair que les S'(u) forment, eux aussi, une famille croissante d'ouverts de P, dont la rdunion est P. Des calculs triviaux permettent de v6rifier que tout S(t) est contenu dans un S'(u), et r6ciproquement. I1 s'ensuit 6videmment que M est contenu dans S'(u) pour u assez grand. On notera S'o(U) l'image de S'(u) dans P0Orl doit ',i Siegel le tr6s important r6sultat suivant (pour l'6noncer commod6merit, on conviendra, pour toute partie S de P, et toute matrice inversible X, de noter S x le transform6 de S par X, i.e. l'ensemble des ' X . A 9X pour A ~ S): Quels que soient t > 1 et m entier :~0, rensemble des matrices X ?t coefficients entiers, de d~terminant m, telles que S(t) c~ S(t) x ~ O, est fini. I1 s'ensuit naturellement qu'on peut en dire autant pour S'(u). C o m m e M ~ S(t), on en conclut en particulier, en prenant m = + 1, que, pour tout t > 1, S(t) est

366

[1957c] Rdduction des formes quadratiques, d'apr6s Minkowski et Siegel

contenu dans la rdunion de M e t d'un n o m b r e fini de transformds de M p a r des 616ments du g r o u p e F. Enfin, on peut v6rifier, p a r exemple p a r calcul explicite, que, dans P0 consid6r6 c o m m e espace riemannien sym6trique Go/K o, chacun des ensembles (non compacts) So(t) est de volume fini, p o u r le volume " n a t u r e l " ddfini dans l'espace Po (volume qui est naturellement invariant p a r G o, et que cette condition d6termine d'une mani6re unique fi un facteur pr6s). C o m p t e tenu de ce qui pr6c6de, cela 6quivaut n a t u r e l l e m e n t / t dire que M 0 est de volume fini, ou encore que l'espace homog6ne Go/F o est de volume total fini. La d 6 t e r m i n a t i o n du volume de ce dernier espace (l'616ment de volume invariant dans Go 6tant choisi explicitement une fois p o u r routes) est d u e / t Minkowski. P o u r m6moire, on ajoute ce qui suit, d o n t l'intdrat est d ' o r d r e historique et esthdtique mais dorlt en r6alit6 on ne semble gu6re avoir fi faire usage. En premier lieu, l'ensemble convexe M, ddfini p a r les indgalitds (i) et ( i i ) j o i n t e s fi l'indgalit6 al 1 > 0, est en r6alit6 une pyramide convexe; a u t r e m e n t dit, il suffit p o u r le d6finir d ' u n nombrefini d'indgalitds prises p a r m i celles qu'on vient de mentionner. O n en a vu (sans d6monstration, mais la d 6 m o n s t r a t i o n dans ce cas est facile) un exemple, p o u r n = 2; p o u r n = 3 et n = 4, on conna~t un syst6me explicite d'indgalitds ddfinissant M ; rien de tel n'est connu p o u r n quelconque. De plus, M est l'adhdrence, darts P, de son intdrieur. Enfin, non seulement P e s t rdunion de M e t de tous ses transform6s M x p a r les 616ments X du g r o u p e F (ou, plus pr6cis6ment, p a r les transform6s de M p a r un syst6me de reprdsentants, darts F, des classes suivant le centre { + 1,} de F : car ce centre op6re trivialement sur P), mais encore ces transformds forment une triangulation ou un " p a v a g e " de P, au sens suivant: leurs int6rieurs sont disjoints; l'intersection de deux quelconques d'entre eux est une p y r a m i d e convexe de dimension plus petite: tout c o m p a c t n'a de points c o m m u n s qu'avec un n o m b r e fini de ces transform6s.

[1957c2] Groupes des formes quadratiques indefinies et des formes bilindaires alternees 1. Quelques concepts g~n~raux. C o m m e precddemment, soient G un groupe semi-simple non compact, K un sousgroupe maximal de G, F u n sous-groupe discret de G. L'espace homogene G/K est l'espace riemannien symetrique associe 5- G. O n est amene 5- considdrer les proprietds suivantes de F: (I) v(G/F) < + c~ (v designe bien entendu le volume invariant, ou mesure de Haar, sur G; il est invariant 5- droite et 5- gauche, parce que G est semi-simple, et se transporte 5. G/F d'une maniere 6vidente). (II) II existe dans G u n ouvert U de mesure finie (i.e. v(U) < + oc) tel que U F = G (autrement dit, l'image de U dans G/F est G/F), et que U - 1 U n F soit fini (autrement dit, il n'y a qu'un hombre fini d'61ements ~' e F tels que U7 rencontre U). (III) G/F est compact. I1 est clair que (III) ~ (II) ~ (I). Mais, entre (II) et (III), il y a lieu d'inserer une propriet6 de plus; pour l'enoncer, on introduit la notion suivante. Deux groupes F, F' sont dits commensurables si F c~ F' est d'indice fini dans F et dans F'. On demontre sans peine que c'est 15-,pour les sous-groupes d'un groupe G donnd, une relation d'equivalence. II s'ensuit que les x ~ G tels que xFx i soit commensurable "a F forment un groupe F (dit " g r o u p e des transformations" de G/F), et que, si F et F' sont commensurables, les "groupes de transformations" F, F' qui leur sont associes coincident, ce qui implique en particulier que F' c ~. Cela pos6, on s'interesse aussi 5- la propriete: (M) II existe darts G u n ouvert U de mesure finie, tel que KU = U (U est " s a t u r e " par rapport/t K, ou encore est l'image reciproque, par l'application canonique de G sur G/K, d'un ouvert de G/K), que U F = G, et que U - 1U c~ xF soit fini quel que soit x ~ ~. (On observera que, dans le groupe F, toute classe 5- gauche suivant F est contenue dans une reunion finie de classes "a droite, et reciproquement, de sorte que, dans la derniere condition de (M), on peut ecrire Fx au lieu de xF sans rien changer). I1 est clair que (III) ~ (M) ~ (II). Si (II) est satisfaite, Siegel dit que F est "de premiere espece." Si (M) est satisfaite, on propose de dire que F est " m i n k o w s k i e n " dans G. Les theoremes 6nonces dans le premier expose disent essentiellement que, dans le groupe Go = PL+(R, n), le groupe Fo = PL(Z, n) ("groupe modulaire") est minkowskien; dans ce cas, on verifie facilement que le groupe des transformations ['o associe 5-F 0 est P L + ( Q , n); on prend pour ensemble U Fun des ouverts So(t), S'o(u) introduits precedemment; le fait que (M) est verifi6 (et non seulement (II)) est precisement le theoreme de Siegel. Ce meme exemple montre que (M)

367

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[1957c] Groupes des formes quadratiques ind~finies et des formes bilindaires alterndes

n'entraine pas (III); en revanche, on ignore, (pour parler plus prudemment, le conf6rencier ignore) si (I), (II), (M) sont vraiment distinctes. En dehors des groupes fuchsiens, pour lesquels on poss6de des modes de d6finition g6om6triques (par un "polygone fondamental") et, comme dirait l'autre, fonction-th6or6tiques (rev6tement universel de surfaces de Riemann avec ramifications donn6es), il semble bien que tousles groupes connus satisfaisant 5, (I) soient des groupes/l d6finition arithm6tique, et qu'en vertu des travaux de Siegel on puisse affirmer que tous ces groupes sont minkowskiens. [N.B. On peut donner de tous ces groupes "arithm6tiques" une d6finition unique, comme suit: Soit A une alg6bre semi-simple sur Q, munie d'un antiautomorphisme involutif J; soit ~ un " m o d u l e " dans A (sous-groupe additif de type fini de A, tel que 9J~Q = A). Soit Ar l'extension de A ~ R, munie de l'antiautomorphisme qui 6tend J; soit G le groupe des automorphismes de Ar (munie de J, c'est-~-dire groupe des automorphismes de l'algbbre A r qui commutent avec J); G est semi-simple, et c'est m6me le groupe semi-simple "classique" le plus g6n6ral ("classique" signifiant que G n'a aucun facteur qui soit un des groupes simples exceptionnels). On prend pour F l e plus grand sous-groupe de G qui envoie ~ dans ~R.] I1 est facile de d6montrer que, si (II) est satisfaite, F est engendrO par les 616ments de U c~ U?, donc de typefini. On peut se demander si le 9roupe des relations entre ces g6n6rateurs de F est lui-m6me de type fini; on d6montre assez simplement qu'il en est ainsi, du moins, si KU = U [travailler dans G/K, et se servir du fait connu que G/K est simplement connexe; exercice recommand6 aux lecteurs]. Enfin, il est imm6diat que, si F est minkowskien, il en est de m6me de tout groupe commensurable /t F; on revanche, " o n " ignore s'il peut arriver que, de deux groupes commensurables, l'un soit "de premi6re esp6ce" et l'autre ne le soit pas. C'est marne 15, une des raisons pour lesquelles il est recommand6 de toujours travailler avec (M), plut6t qu'avec (II), malgr6 la complication apparente de la derni6re partie de la condition (M). 2. Formes ind~finies.

Soit, pour commencer, F une forme quadratique inddfinie non-dd9~n~rOe dans un vectoriel E de dimension n sur R Soit G le groupe orthogonal de F (groupe des automorphismes de E qui laissent F invariante). Soit U le dual de E; toute forme bilin6aire sur E d6termine canoniquement, comme on sait, une application lin6aire de E dans E'; en particulier, la forme bilin6aire F(x, y) associ6e/~ F, c'est-/tdire telle que F(x) = F(x, x), d6terminera une application lin6aire f de E sur E', qui est sym6trique (i.e. ~/ --- f ) et de rang n. Soit K un sous-groupe compact de G; il laisse invariante, comme chacun sait, au moins une forme positive non d6g6n6r6e qs; soit ~0 l'application de E sur E' associ6e fi ~P. On peut, par le choix d'une base convenable darts E (cf. Bourbaki, Alg., Chap. IX), mettre F, q5 sous la forme F = Z, d, x2, @ = Zi x~; la matrice de q~ 1.[, pour cette base, sera la matrice diagonale de coefficients dl, . . . , d,. Tout automorphisme de E qui laisse F et @ invariantes commute 6videmment ~i qo-lf ou autrement dit laisse invariants les sous-espaces Ev de E form6s respectivement des

[1957c]

369 Groupes des formes quadratiques inddfinies et des formes bilin6aires altern6cs

vecteurs propres de q~- if par rapport aux valeurs propres de ~0- if, valeurs propres qui ne sont autres que les di (ou plut6t les 616ments distincts parmi ceux-ci). Soit E+ (resp. E ) la somme directe de ceux des E,. qui sont associds fi des valeurs propres > 0 (resp. <0). Alors K est contenu dans le produit des groupes orthogonaux K + , K , des formes induites respectivement darts E+ et dans E par F; K+, K sont compacts, puisque F est positive (resp. n6gative) non ddgdn6r6e sur E+ (resp. E_). Par suite, pour que K soit sous-groupe compact maximal de G, il faut et il suffit qu'on ait K = K+ • K , ou autrement dit que K soit le groupe des automorphismes de E qui laissent invariantes F et une forme positive non d6gdn6rde do telle que q0- i f n'ait pas d'autres valeurs propres que _+ 1, ou, ce qui revient au m6me, telle que (q~ lf)2 = 1. I1 est clair alors que les points de G/K (le riemannien sym6trique associ6 5. G) sont en correspondance biunivoque avec les do possedant cette propridt6; il revient au m~me de dire qu'ils sont en correspondance biunivoque avec les couples (E+, E ) de sous-espaces compldmentaires de E tels que F induise sur E+ une forme positive non d6gdndr6e, et sur E_ une forme n6gative non d6g6n6rde et que de plus E+ et E soient orthogonaux l'un fi l'autre, par rapport F ; ceux-ci se d6terminent r6ciproquement. D'ailleurs, si (p, q) est la signature de F, E+ et E ont ndcessairement les dimensions p, q. Enfin, en vertu de la loi d'inertie, si E+ est un sous-espace de E de dimension p sur lequel F induise une forme positive non d6gdndrde, F induit ndcessairement sur l'orthogonal E_ de E+ une forme ndgative non d6g6n6r6e, et r6ciproquement. Cela permet de repr6senter, canoniquement, G/K comme partie ouverte d'une grassmannienne (fi savoir, soit la grassmannienne des sous-espaces de E de dimension p, soit celle des sous-espaces de E de dimension q); ces ouverts peuvent facilement 6tre ddfinis par des in6galit6s cxplicites. C o m m e prdcddemment, soit P l e c6ne convexe de toutes les formes positives non d6g6ndrdes dans E. A toute forme ind6finie non ddgdndrde F est associ6, d'apr6s ce qui pr6c6de, l'ensemble V(F) des dOe p telles que (~o lf)2 = 1. I1 est clair que les do ~ V(F) ont la propri6t6 do(x) > IF(x)] quel que soit x e E. Ces derni6res in6galit6s ddfinissent une partie convexe de P (ferm6e dans P), dont V(F) est la fronti6re (comme il r6sulte imm6diatement de la possibilit6 de r6duire simultan6ment F et une forme quelconque de P fi la forme diagonale). On a, pour tout x c E, IF(x) l = info~v~f~do(x), d'o/l s'ensuit ais6ment que F est d6termin6e, d'une mani6re unique au signe pr6s, par V(F). Tout cela subsiste d'ailleurs si F est "d6finie", mais devient sans intdr6t, V(F) 6tant alors r6duit/~ {F} ou bien b, { - F}. O n a l'habitude de dire, par abus de langage, que les do 6 V(F) sont les "majorantes" de F (ce sont en rdalit6 les 616ments fronti6res de l'ensemble des majorantes). Le groupe •(E, E) op6re transitivement dans P, comme on a vu, au moyen de do ---, do o X (pour l'application q~ de E sur E', canoniquement associ6e /t do, cela s'6crit ~0 --, tX. q~. X; ainsi en particulier si on 6crit en matrices, apr6s choix d'une base). Bien entendu, V(F o X) est l'ensemble transform6 de V(F) par do --* do ~ X; en particulier, V(F) est invariant par tout 616ment X du groupe orthogonal G de F; la mani6re dont G op6re sur V(F) est celle m~me qu'on obtient en transportant fi V(F), au moyen de la correspondance biunivoque entre G/K et V(F) ddfinie ci-dessus, les opdrations de G sur l'espace homog6ne G/K. Donc, pour 6tudier les propri6tds locales de la bijection G/K ~ V(F), il suffit de les 6tudier au voisinage du point d o = E x ~ 2 , p o u r F d o n n 6 e par F = x 2 + . . . + x p 2- x p +2 l _ _ , . . - x , . 2

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[1957c] Groupcs des formes quadratiques inddfinies et des formes bilin6aires altern6es

Cela ne fait pas de difficult6; on en conclut que V(F) est une vari6t6 analytique r6elle, plong6e dans P, et que G/K ~ V(F) est un isomorphisme au sens analytique r6el (en particulier, c'est une application diff6rentiable de rang 6gal 5, la dimension pq de G/K). Hermite introduisit les id6es exposdes ci-dessus (qu'on s'est content6 d'assaisonner de sauce bourbachique) en vue de la th6orie arithmdtique ("rdduction") des formes quadratiques ind6finies. La pyramide convexe M ("domaine fondamental de Minkowski") 6tant d6finie dans P c o m m e il a 6t6 dit dans l'expos6 pr6c6dent, on dira que la forme inddfinie F est r~;duite si V(F) rencontre M ; pour que cela air un sens, il faut naturellement q u ' o n soit dans R n, puisque la d6finition de M est relative "fi une base d6termin6e. On dira qu'une forme est ~ coefficients entiers s'il en est ainsi de la matrice de la forme bilindaire associde. I1 est ais6 de voir qu'il

n'y a qu'un nombre fini de Jormes r(duites gt coefficients entiers de dOterminant donnO ( ~ 0 ) . I1 revient au mSme de faire voir que l'ensemble des matrices B symdtriques, h coefficients entiers,de d6terminant donn6 b ~ 0,tellesque V(B) c~ S'(u) ~ 0, est fini, pour chaque valeur donn6e de u. Cela vient de la propri6t6 des S'(u) contenue dans le lemme trivial suivant: L E M M E . - - I I y a une matrice n unimodulaire fi coefficients entiers, telle que, quel que soit u > 1, il y ait un u' > 1 pour lequel 'n. S'(u)- 1. n (ensemble des matrices ~n- A 1. n, pour A ~ S'(u)), soit contenu dans S'(u'). [ O n prendra pour n la matrice de la permutation (1,2 . . . . . n)--*(n,n - 1. . . . . 1), ou autrement dit la matrice (6i,,+ a_ i); le lemme r6sulte alors de ce que, darts le groupe triangulaire, T ~ T - ~ transforme tout compact en un compact (propridt6 qui ne caract6rise nullement le groupe triangulaire).] Cela pos6, V(B) c~ S'(u) est l'ensemble des A ~ S'(u) tels que ( A - IB)2 = 1, i.e. : A = BA

1B = t(n

1B)-(tTc-A-I-n).(n IB).

D o n c la matrice enti6re u - 1B, de d6terminant b, transforme un point de S'(u') en un point de S'(u); si u et par suite u' sont fixds, il n'y a, d'apr6s le thdor6me de Siegel, qu'un n o m b r e fini de matrices susceptibles de faire une pareille chose. C.Q.F.D. [N.B. O n n'a pas suppos6 B ind6finie, donc le cas des formes positives est inclus; ce cas est d'ailleurs facile/t liquider directement.] Ce qui pr6c6de sert principalement b, d6montrer que, darts le groupe orthogonal G d'une forme quadratique ind6finie F, non d6gdn6rde, ',i coefficients entiers, le " g r o u p e des unit6s [arithm6tiques]" de F, intersection de G avec le groupe des matrices de d6terminant _+ 1 sur Z, est minkowskien. P o u r cela, soit B la matrice de F ; d'apr6s ce qui pr6c6de, les matrices Bi 6quivalentes h B (i.e., transform6es de B par des matrices sur Z de d6terminant _+ 1)), relies que V(BI) rencontre S(t), sont en n o m b r e fini (pour un choix, fix6 une lois pour toutes, de t > 1). Pour chacune, soit M~ une matrice sur Z, de ddterminant _+ 1, transformant B dans Bg, i.e. telle que Bg = tM i 9B . Mi. Alors A ~ ' M i ~- A-M~ - t e s t une bijection de V(Bi) sur V(B); soit Ui l'ouvert de V(B), image par cette bijection de S(t) n V(BI); soit U la rdunion des U~. O n montre que U (plus exactement, l'image rdciproque dans G de l'ouvert

[1957c]

371 Groupes des |brmes quadratiques ind6finies et des formes bilinaaires altern6es

de G/K, image de U p a r la bijection ddfinie prdcddemment entre G/K et V(B)) a l e s propridtds 6noncdes darts (M). Q u a n t 5- la premi6re, tout revient 6videmment 5. d d m o n t r e r que, quelle que soit la forme ind6finie F, l'ouvert S(t) r~ V(F) est de mesure finie au sens de l'unique mesure invariante (par r a p p o r t au groupe G) ddfinie dans V(F); cela se fait p a r des m a j o r a t i o n s explicites. P o u r m o n t r e r que U contient un syst6me c o m p l e t de repr6sentants p a r r a p p o r t au sous-groupe (c'est la deuxi6me propridt6 b, v6rifier), soit A e V(B); on peut transformer A en un point t M - A 9M de S(t) au m o y e n d ' u n M de d6terminant +_ 1 sur Z ; celui-ci est alors dans V(B') avec B' = t M . B - M , ce qui implique, p a r d6finition des B i, que B' est l'un des Bi, doric que M i M ~ est une " u n i t 6 " de B, et aussi que le transform6 ~(M. M i 1). A - ( M . M ~ 1) par M . M F 1 est dans Ui, d o n c dans U; autrement dit, A est darts le transform6 de U p a r Mi- M - 1. Q u a n t au dernier point 5- verifier, il r6sulte du thdor6me de Siegel. C o m m e dans la r6duction des formes positives, on peut se p r o p o s e r de calculer (et non pas seulement de majorer) le volume de G/F, l'unit6 de volume 6tant explicitement choisie. LS-, on ne s'en tire pas 5- si bon march6. Le lecteur est pri6 de se r e p o r t e r 5- Siegel. En revanche, on peut, darts ce qui prdc6de, r e m p l a c e r S(t) p a r la p y r a m i d e de M i n k o w s k i ; on obtient alors un " p a v a g e " de G/K p a r un " d o m a i n e f o n d a m e n t a l " et ses transform6s p a r le groupe des unites de B, ce pavage offrant aux a m a t e u r s de jouissances esth6tiques ",ipeu pr6s les m6mes agr6ments que celui dc M i n k o w s k i darts le c6ne P des formes positives. A rioter toutefois qu'fi cause des B~ en n o m b r e fini, le d o m a i n e f o n d a m e n t a l se compose, non pas d'un, mais de plusieurs morceaux, d o n t chacun est une esp6ce de polyadre convexe; en a t t r i b u a n t fi chacun de ces m o r c e a u x une couleur diff6rente, on obtient, p o u r l'ensemble du pavage, des r6sultats fort pittoresques (cf. H. Weyl). A u t a n t qu' " o n " peut savoir, cela ne sert strictement 5- rien. Enfin, ce qui prdc6de permet de decider dans quel cas G/F est c o m p a c t ; il faut et il sutfit p o u r cela, visiblement, que S(t) c~ V(B'), ou, ce qui revient au marne, S'(u) c~ V(B') soit d'adhdrence c o m p a c t e dans le c6ne P p o u r tout B' 6quivalent B. O r c'est l'ensemble des A eS'(u) tels que B' 1AB'-IA = 1,; p o u r qu'il soit compact, il faut et il suffit que l'adh6rence du c6ne de s o m m e t 0 qu'il engendre n'ait aucun p o i n t c o m m u n , autre que 0, avec la fronti6re du c6ne P (i.e., ne contienne aucune forme d6gdn6r6e # 0 ) . Mais ce c6ne est l'ensemble des A e S'(u) tels que B'-IAB' 1A = 2 . 1 , , avec 2 _> 0; si A a p p a r t i e n t 5- l'adhdrence de ce c6ne et cst d6g6n6r6e, on a u r a B'-IAB'-~A = 0. Ecrivant que A est dans l'adh6rence de S'(u) et est # 0, on trouve que A est de la forme

o) avec C non d6gdndrde; de B ' - I A B ' 1A = 0, on conclut alors que forme

(: 0)

B '-1

est de la

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[1957c] Groupes des formes quadratiques ind6finies et des formes bilin6aircs altern6es

d o n c a au moins un coefficient d i a g o n a l 0. P a r suite, B' 1, d o n c aussi B ' = tB' 9B ' - 1 9B', d o n c aussi B, " r e p r 6 s e n t e n t " 0 (ce qui veut dire que, si F est la forme de matrice B, F(x) = 0 a une solution rationnelle ~ 0). La r6ciproque s'ensuit de m6me. A u t r e m e n t dit, p o u r que G/F soit compact, il faut et il suffit que B ne repr6sente pas 0 (ce qui peut arriver p o u r n = 3 et p o u r n = 4; en revanche, un th6or6me classique de Meyer affirme que toute forme ind6finie fi n _> 5 variables, "fi coefficients entiers, " r e p r 6 s e n t e " 0).

3. Formes altern6es; groupe de Siegel. C'est le couplet suivant de la c h a n s o n ; il se chante sur le marne air. Soit F bilin6aire altern6e non d6g6n6r6e sur E; cela exige, bien entendu, que E soit de dimension paire 2n. Soit f l'application de E sur E' d6finie p a r F. Soit G le g r o u p e des a u t o m o r p h i s m e s de F ; soit K un sous-groupe c o m p a c t de G; il laissc invariante une q~ positive non d6g6n6r6e, "filaquelle a p p a r t i e n t une a p p l i c a t i o n q) de E sur E'. L'adjoint, p a r r a p p o r t fi ~, de l ' a u t o m o r p h i s m e i = q ~ - l f de E, est - ~ ; il s'ensuit que t est " s e m i - s i m p l e " (du p o i n t de vue matriciel, cela veut dire que z peut ~tre r6duit fi la forme diagonale, sinon sur R, en tout cas sur C), b, valeurs propres routes p u r e m e n t imaginaires. Si 1 a au moins deux valeurs propres distinctes et non imaginaires conjugu6es l'une de l'autre, E se d6compose en s o m m e directe de sous-espaces d o n t chacun est invariant p a r tout a u t o m o r p h i s m e de E qui c o m m u t e avec l; on en conclut,/t peu pr6s c o m m e au w qu'alors K ne peut 6tre maximal. P o u r que K soit maximal, il faut et il suttit que t n'ait que deux valeurs p r o p r e s distinctes, imaginaires conjugu6es l'une de l'autre; en multipliant qb p a r un facteur scalaire > 0, on peut supposer que ces valeurs propres sont + i , ce qui revient fi dire que t 2 = - 1. En ce cas, t d6termine une structure complexe sur E (on d6finira dans E la multiplication scalaire p a r les complexes au moyen de (~ + i~)x = :~x + [4lx); on 6crira E, p o u r E muni de cette structure; p o u r celle-ci, il est imm6diat que la forme bilin6aire fi valeurs complexes H = 9 + iF est hermitienne positive non d6g6n6r6e (N.B. lci, et dans ce qui suit, on note indiff6remment p a r ~ , p a r abus de langage, soit la forme q u a d r a t i q u e introduite ci-dessus, soit la forme bilin6aire associ6e.) C o m m e les 616ments de K c o m m u t e n t avec ~, ce sont des a u t o m o r p h i s m e s de E muni de sa structure complexe; il est clair alors que K est le g r o u p e unitaire d6termin6 par la forme hermitienne H. I1 contient donc toujours un centre non discret, form6 des multiples e"- 1 de l ' a u t o m o r p h i s m e identique (cela, au sens de la structure complexe). Dans le cas des formes q u a d r a t i q u e s ind6finies de signature (p, q), le centre du sous-groupe c o m p a c t m a x i m a l est non discret si p = 2 ou q = 2, et dans ce cas seulement. O n d6montre que l'existence d ' u n tel centre est n6cessaire et suffisante p o u r qu'il y air sur G/K une structure complexe invariante p a r G; on v a l e v6rifier dans le cas pr6sent. [N.B. La suftisance de la condition se justifie en g6n6ral c o m m e suit: K op6re darts G/K, avec un point fixe qui est le point de G/K qui c o r r e s p o n d fi K lui-m6me; il op6re donc sur l'espace des vecteurs tangents fi G/K en ce p o i n t ; dans cet espace, chacun des deux 616ments d ' o r d r e 4 du centre de K d6finit un a u t o m o r p h i s m e de cart6 - 1, et p e r m e t doric de d6finir une structure complexe, invariante p a r K. O n peut en faire a u t a n t en chaque point; on a ainsi une structure presque complexe;

[1957c]

373 Groupes des formcs quadratiques ind6finies et des formes bilineaires altern6es

reste ~, montrer qu'elle est int6grable. On peut le voir par exemple (d'apr6s Ehresmann) en remarquant qu'en g6n6ral, pour une structure presque complexe, le "d6faut d'int6grabilit6" s'exprime par un "tenseur mixte," celui qui donne les coefficients des co/~J~., dans l'expression des diff6rentielles d~)~ des formes ~i)~ de type (1, 0); en exprimant que ce tenseur est invariant par le centre de K, on trouve qu'il s'annule.] En d6finitive, on voit que G/K a 6t6 mis en correspondance biunivoque avec l'ensemble des structures complexes sur E pour lesquelles F est la partie imaginaire d'une forme hermitienne positive non d6gdndrde H = 4 ) + iF, et aussi avec l'ensemble V(F) des parties r6elles (I) de telles formes; comme au w V(F) est une sous-varidt6 du c6ne P des formes positives non ddgdn6rees sur E, et G/K V(F) est une bijection analytique rdelle de G/K sur V(F). Si on est dans R 2", et qu'on suppose F donn6e par une matrice fi coefficients entiers, on ddmontre, exactement c o m m e au w que le groupe des "unitds arithmdtiques" de F est minkowskien darts le groupe des automorphismes de F. Dans un expos6 ultdrieur, on ddfinira, d'une mani&e plus ou moins explicite, un ouvert U de G/K satisfaisant 5 la condition (M); ce sera fait, du moins, pour le " g r o u p e de Siegel" (ou " g r o u p e modulaire d'ordre n") proprement dit, qui est celui des unites si F est donnde dans R 2" par une matrice alternde de dOterminant 1. Si " o n " a du vice, " o n " ddfinira m~me, dans ce dernier cas, un " d o m a i n e fondamental" qui, avec ses transform6s, fournit un beau pavage de l'espace G/K. [N.B. I1 est connu que, par un choixtconvenable de 2n gdndrateurs pour le sous-groupe Z z" des vecteurs fi coordonndes enti6res dans R 2", toute formc alternde /t coefficients entiers peut s'6crire Y,i di(xiy,,+~ - x,+~yi), off les di sont des entiers, les "diviseurs 616mentaires", dont chacun est multiple du pr6c6dent. I1 n'y a donc pas besoin de la th6orie de la rdduction, dans ce cas, pour montrer qu'il n'y a, pour un d6terminant donn6, qu'un nombre fini de formes non 6quivalentes deux 5 deux. De plus, toutes ces formes sont 6quivalentes sur Q. Or il est facile de voir que les groupes d'unit6s arithm6tiques de deux formes 6quivalentes sur Q sont toujours commensurables; cela est vrai aussi, bien entendu, pour les formes quadratiques; mais ici on peut en conclure que les groupes de toutes les formes alterndes fi coefficients entiers sont commensurables au groupe de Siegel.] O n va s'occuper maintenant de structure complexe. Pour cela, on introduit le "complexifi6" de E, qu'on notera Ec (pour raison typographique, au lieu de la notation canonique Ec; c'est, c o m m e on sait, E | C muni de sa structure vectorielle sur C; on consid6re E c o m m e plong6 dedans de la mani6re 6vidente). Tout automorphisme t de E, de carr6 - 1 , se prolonge 5. Ec en un automorphisme analogue, qui d6termine une d6composition de Ec en somme directe des sousespaces Vi, V_~ formds des vecteurs propres relatifs aux valeurs propres i resp. - i de z; Vii, V_i sont sous-espaces de Ec sur C, donc sont espaces vectoriels sur C, de dimension n; on a d'ailleurs V_~ = ~, off, suivant l'usage, la barre denote l'imaginaire conjugu6 (d6fini dans Ee de la mani6re 6vidente). Si iE ddsigne l'ensemble des vecteurs "imaginaires purs" de E~. (image de E par x ~ ix), Ec = E @ iE est une somme directe; si ~ ("partie rdelle") est le projecteur de E~ sur E qu'elle d6termine, il est imm6diat que 9l induit sur Vii un isomorphisme de la structure complexe de Viisur la structure complexe de E qui est d6termin6e par l, c'est-',i-dire

374

[1957c] Groupes des formes quadratiques indefinies et des formes bilineaires alternees

sur celle de E,. D o n c ~est c o m p l e t e m e n t ddtermin6 p a r la d o n n e e de Vii.Rdciproquement, soit V~un sous-espace de Ec de d i m e n s i o n n sur C; p o u r que ~ induise sur V~une bijection de V, sur E, il faut et il suffit q u ' o n air l'une des relations 6quivalentes V~ r~ iE = {0}, Vi r~ E = {0}, Vi r~ V~ = {0}; lorsqu'il e n e s t ainsi, V~ p e r m e t doric de ddfinir sur E, p a r t r a n s p o r t de structure au moyen de N, une structure complexe, donc un a u t o m o r p h i s m e t de E de carr6 - 1 ; si alors on 6tend t it E,, V~sera l'espace des vecteurs p r o p r e s de l relatifs it la valeur p r o p r e i. Soit (5 la grassmannienne complexe des sous-espaces de Ec de dimension n sur C; d a n s (5, soit ,3 l'ensemble des sous-espaces d o n t l'intersection avec E se rdduit it {0} ; c'est un ouvert dans (5; d'apres ce qui precede, il s'identifie avec l'ensemble des structures complexes sur E. Si on s'est d o n n e c o m m e precedemment, sur E x E, une forme bilindaire alternde non degeneree F, celle-ci peut s'etendre it une forme Fc sur Ec • Ec; de m e m e p o u r l'extension q~c d'une forme symetrique qb. Si on a, sur E • E, qS(x, y) = F ( - tx, y) (ce qui equivaut it la relation z = q~- l f e c r i t e au debut de ce numero), la relation a n a l o g u e sera vraie p o u r les extensions de F, 0), t, it Ec; en particulier, sur V~, Fc et q5c induisent une forme alternee F ' et une forme symetrique qb' telles que l'on air @' = - iF', ce qui exige evidemment F ' = 0. A u t r e m e n t dit, V~est alors un espace isotrope maximal de F~ ( " i s o t r o p e " signifie j u s t e m e n t que F~ induit 0 sur Vii; on trouve alors, par exemple par le choix d ' u n e base convenable, que Vii, etant de d i m e n s i o n n, est isotrope maximal parce que F c e s t non ddgeneree). Ces espaces forment une sous-variet6 (analytique complexe) ~9 de la grassmannienne (g; on verifie sans difficulte que le g r o u p e des a u t o m o r p h i s m e s de E~ qui laissent F, invariante (le " c o m p l e x i f i e " du g r o u p e G) opere transitivement sur ~3. Reciproquement, soit V ~ r~ ~3; p u i s q u ' o n a E c = Vii| V i, on peut, quel que soit z~E~, 6crire z = u + v, u e Vii, v e V-i, et on a alors zz = iu - iv; si de meme on a z' = u' + v', avec u' ~ V/, v' e I/ i, on a u r a Fc(u, u') = 0 et Fc(v, v') = 0 parce que V/et p a r suite V~ = ~ sont isotropes p o u r F,.; cela p e r m e t de calculer F~(--lz, z') et de voir que cette forme bilineaire est symetrique. I1 faut exprimer de plus que @ est positive non degeneree sur E, ce qui equivaut b. F ( - tx, x) > 0 quel que soit x ~ 0 dans E. Or, l ' i s o m o r p h i s m e de E, sur V/, inverse de r i s o m o r p h i s m e de V/sur E, induit sur V~p a r 9~, s'ecrit x ~ z = x - i - t x (verification immediate); en tenant c o m p t e de ce que Vii, l / ~ = ~ sont isotropes p o u r Fr l'inegalite precddente s'ecrit encore iFc(z, ~) < 0 quel que soit z r 0 dans V~. I1 est clair que les V~satisfaisant it cette condition forment un ouvert f~ dans ~ ; ce qui precede implique que cet ouvert n'est pas vide (puisque tout point du riemannien symetrique G/K determine justement un point de f~). I1 est clair aussi que Vii ~ f~ implique Vii~ ~ ; sinon, en effet, il y aurait un z 4= 0 dans V~ r~ E, et on aurait z = ~, iF(z, z) < 0, ce qui est idiot. De ce qui precede resulte d o n c que G/K s'identifie it fL qui est une variete (analytique complexe) plongee dans (5. D'ofl la structure complexe de G/K. Le g r o u p e G des a u t o m o r p h i s m e s de F (i.e. des a u t o m o r p h i s m e s de E qui laissent F invariante) opere sur ~9 et sur f2 d'une maniere evidente, y laisse invariante la structure a n a l y t i q u e complexe, et sa maniere d ' o p e r e r sur fl est celle qui resulte, p a r t r a n s p o r t de structure, de son o p e r a t i o n sur G/K. Satisfaction gendrale. Profitant de l'absence de Dieudonne, on va traduire qa en matrices, p o u r faire le j o i n t avec Siegel et p o u r se mettre en etat de calculer q u a n d on ne peut pas faire

[1957c]

375 Groupes des formes quadratiques inddfinies et des formes bilin6aires altern6es

a u t r e m e n t (~a arrive encore quelquefois). O n prend une base (el . . . . R p o u r laquelle la matrice de F soit

,

e2n ) de E sur

soient E', E" les sous-espaces engendr6s sur R, respectivement, p a r (el . . . . . e,) et (e,+~ . . . . . e2,); ce sont des sous-espaces de E isotropes m a x i m a u x p o u r F. Si V~E f l , et que l, q), etc., aient le m6me sens que ci-dessus, @ sera positive non d6g~n6r6e sur E, d o n c sur U , ct on p o u r r a choisir dans E' n vecteurs o r t h o n o r m a u x p o u r q~; ils le seront alors aussi, dans E,, p o u r la forme hermitienne H = 9 + iF (puisque E' est isotrope p o u r F); ils formeront d o n c une base de E, sur C, ce qui entraine que E' et rE' sont suppldmentaires dans E. Alors Vii est transversal au complexifi6 E',, de E'; en effet, E'~ est l'ensemble des x' + iy', avec x' e E', y' c E'; si un tel p o i n t est dans Vii,on a y' = - zx' e E' r~ zE', donc x' = y' = 0. De marne Viiest t r a n s v e r s a l / t E',[. Choisissons darts Vi n vecteurs formant une base de Vi (sur C); 6crivons-les c o m m e " c o l o n n e s " (matrices ~ 2n lignes et 1 colonne) au moyen de (el . . . . , e z n ) pris c o m m e base de Ec sur C; cela d o n n e une matrice ~t 2n lignes et n colonnes (sur C), q u ' o n peut 6crire

V ' o/1 U, V sont deux matrices/t n lignes et n colonnes;

si on change les vecteurs de base choisis dans V~, cela revient/L multiplier U, V ~ droite p a r une m~me matrice carr6e inversible. Puisque V~est transversal/L E'~' la matrice U est de rang n, c'est-fi-dire inversible. Ecrivons que V~ est isotrope p o u r F ; cela s'exprime p a r la formule

:0, ou a u t r e m e n t dit ' U - V = tV. U. D e m~me, 6crivons que iFc(z, 2) < 0 p o u r tout z r 0 dans Vi; cela signifie (1/i)(~U 9V - ' V - U) > 0 (le premier m e m b r e est visiblement une matrice hermitienne). P o s o n s Z = V U - 1, matrice qui est ind6pendante de la base choisie dans Vii.La premi6re des relations ci-dessus s'6crit ' Z = Z ; Z est sym6trique. La seconde s'6crit (en m u l t i p l i a n t / t droite p a r U - ~, fi gauche p a r ' U - ~, ce qui ne modifie pas le fait que le premier m e m b r e est hermitien positif non d~g6n6r6) ( 1 / i ) ( Z - ~Z) > 0; a u t r e m e n t dit, si on 6crit Z = X + i Y avec X, Y sym6triques r6els, Y doit 6tre positive non ddg6ndr6e. O n a ainsi obtenu une bijection de fL donc en d6finitive de G/K, sur l'espace de Siegel ~ , form~ des matrices sym6triques Z = X + i Y sur C telles que Y > 0; ~ peut ~tre consid6r6 c o m m e un ouvert de C N, avec N = n(n + 1)/2, muni de la structure complexe induite p a r celle de C ~. L ' o p 6 r a t i o n de G sur ~ s'~crit imm6diatement. En effet, avec les notations ci-dessus, un 616ment de G s'6crira sous forme d ' u n e matrice carr6e

376

[1957c] Groupes des formes quadratiques inddfinies et des formes bilineaires altern6es

off A, B, C, D sont des matrices "5. n lignes et n colonnes sur R; cette matrice doit satisfaire fi la condition

qui exprime qu'elle laisse F invariante. Cette matrice op6re sur V~, d+finie par les matrices U, V, au moyen de la formule

ou autrement dit:

(U, V)--+ (AU + BV, CU + DV); elle opere donc sur Z par la formule

Z ~ (C + DZ). (A + B Z ) - I Enfin, on peut donner de l'op6ration sur ~ du groupe modulaire (sous-groupe de G form6 des matrices "a coeff• entiers) une interpr6tation int6ressante, et meme importante. En effet, F 6tant donnde dans E, les points de ~ sont, d'apr6s ce q u ' o n a vu, en correspondance biunivoque avec les structures complexes E, q u ' o n peut mettre sur E, pour lesquelles F est partie imaginaire d'une forme hermitienne H >> 0. Supposons donn6 en meme temps dans E un lattice A tel que F soit/t valeurs enti6res sur A x A ; E/A est alors un tore de dimension (r6elle) 2n, sur lequel F d6termine une classe de cohomologie enti6re de dimension (r6elle) 2. Toute structure complexe sur E d6termine sur E/A une structure de tore complexe (de dimension complexe n); pour que celui-ci soit une vari6t6 ab61ienne, il faut et il suffit qu'il existe dans E une forme hermitienne >>0 dont la partie imaginaire soit b. valeurs enti6res sur A x A; et, lorsqu'il e n e s t ainsi, il y a sur E/A un "diviseur positif" appartenant 'a la classe de cohomologie d6termin6e par cette partie imaginaire; muni de cette classe, E/A s'appelle alors une vari~t~ ab~lienne polaris~e. O n voit donc qu'/t tout point de ~ correspond sur E/A une structure de varidt6 ab61ienne polaris6e par F; pour qu'/t deux points corresponde la m6me structure, il faut et il suffit qu'ils se d6duisent l'un de l'autre par un automorphisme de E qui laisse invariants la forme F et le lattice A, donc un 616ment du groupe discret F des automorphismes de F qui sont fi coefficients entiers lorsqu'on prend pour base un syst6me de gdndrateurs de A. Autrement dit, les points de ~ / F (quotient de par la relation d'6quivalence d6finie dans | par le groupe discret F) sont en correspondance biunivoque avec les structures de vari6t6 ab61ienne polaris6e par F q u ' o n peut ddfinir sur E/A. Lorsque F est de d6terminant 1 sur A (c'est-fi-dire a tous ses diviseurs 616mentaires sur A 6gaux/~ 1), le groupe F qu'on obtient est le groupe modulaire de Siegel proprement dit; les vari6tds ab61iennes correspondantes sont dites "vari6t6s abdliennes polaris6es de la famille principale" (toute jacobienne est une telle variet6).

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377 Groupes des formes quadratiques inddfinies et des formes bilindaires altern6es

Bibliographie Minkowski, Hermann. Geometrie der Zahlen. Leipzig und Berlin, B. G. Teubner, 1910; New York, Chels,ea, 1953. Minkowski, Hermann. Diskontinuit~itsbereich ffir arithmetische Aquivalenz, J. ffir reine und angew. Math., t. 129, 1905, p. 220-274; Gesammelte Abhandlungen, Band 2, Berlin, B. G. Teubner, 1911, p. 53 100. Siegel, Carl Ludwig. Einffihrung in die Theorie der Modulfunktionen n-ten Grades, Math. Annalen, t. 116, 1939, p. 617-657. Siegel, Carl Ludwig. Einheiten quadratischer Formen, Abh. math. Sem. Hamburg Univ., t. 13, 1940, p. 209 239. Siegel, Carl Ludwig. Discontinuous groups, Annals of Math., t. 44, 1943, p. 674 689. Siegel, Carl Ludwig. Symplectic geometry, Amer. J. Math., t. 65, 1943, p. 1 86. Weyl, Hermann. Theory of reduction for arithmetical equivalence, 1., Trans. Amer. math. Soc., t. 48, 1940, p. 126 164; II., Trans. Amer. math. Soc., t. 51, 1942, p. 203-231. Weyl, Hermann. Fundamental domains for lattice groups in division algebras, Comment. Math Helvet., t. 17, 1944/45, p. 283-306.

[ 1958a] Introduction h l'6tude des vari6t6s k ihl6riennes (Pr6face) L a th6orie des vari6t6s k~ihl6riennes a pris un g r a n d essor depuis un q u a r t de sibele. Ces vari6t6s f u r e n t d6finies p o u r la p r e m i b r e fois, semble-t-il, dans une n o t e de K~ihler de 1933 (Hamb.Abh. 9, p. 173). Mais leur i m p o r t a n c e n ' a p p a r u t q u ' h la suite des p r e m i e r s t r a v a u x de H o d g e et s u r t o u t de l'expos6 d ' e n s e m b l e q u ' i l en d o n n a au c h a p i t r e V de son livre The theory and applications o[ harmonic integrals ( C a m b r i d g e 1941), off, i n d 6 p e n d a m m e n t de Kfihler, il e n t r e p r e n d l ' 6 t u d e s y s t d m a t i q u e de la m 6 t r i q u e (( kfihl6rienne )) q u ' o n p e u t d6finir sur t o u t e varidt6 alg~brique sans p o i n t m u l t i p l e plong6e dans un espace projeetif, et en tire des cons6quences tr+s i m p o r t a n t e s p o u r la g6omdtrie alg6brique. C'est p r i n c i p a l e m e n t dans la direction ainsi inaugurde p a r lui que les recherches se s o n t poursuivies depuis lots. On ne t r o u v e r a pas iei une m o n o g r a p h i e de ce sujet, mais, c o m m e l ' i n d i q u e le t i t r e , une simple i n t r o d u c t i o n , bas6e sur des cours professds h Chicago et h G S t t i n g e n d a n s les derni~res ann6es. Ce v o l u m e a u r a a t t e i n t son b u t s'il facilite au l e c t e u r l'~tude des t r a v a u x rdcents sur la question, et particuli~r e m e n t de ceux de K o d a i r a et ses dl~ves. J e n ' a i pu ndanmoins rdsister h la t e n t a t i o n d ' i n s d r e r un c h a p i t r e t r a i t a n t de la thdorie des fonctions t h ~ t a et des vari~t~s ab~liennes sur le corps des complexes. Cette th~orie p e u t ~tre considdrde c o m m e celle d ' u n t y p e p a r t i c u l i e r de s t r u c t u r e s kfihleriennes (les s t r u c t u r e s i n v a r i a n t e s sur un tQre), et c ' e s t m~me ce p o i n t de vue qui c o n d u i t h la d d m o n s t r a t i o n la plus n a t u r e l l e d ' u n des p r i n c i p a u x th~or+mes d'exist e n c e de la th~orie (le th~or~me d i t (( d ' A p p e l l - H u m b e r t )9. I1 n ' a p a s dt~ donnd de b i b l i o g r a p h i e ; on en t r o u v e de fort c o m p l e t e s dans plusieurs o u v r a g e s r~cents. J e me suis efforcd de rdduire au m i n i m u m la s o m m e de connaissances exigde du lecteur : quelques r~sultats dl~mentaires Reprinted from Introduction (~ l'~tude des variet~s kiihl~riennes, Hermann, Paris, by permission of Hermann, 6diteurs des sciences et des arts.

379

380

[1958a] l0

INTRODUCTION A L ' E T U D E DES VARIE,TES K,~HLERIENNES

d'analysc et de thdorie des fonctions ; quelques notions d'alg~bre et de topologie gdndrale (pour lesquelles il sera en gdndral renvoyd aux Eldmenls de N. Bourbaki) ; les d~finitions essentielles de la th~orie des vari~tds diff~rentiables (qu'on trouvera dans le volume de G. de R h a m , Varidlds di[]~rentiables, Paris, HerInann 1955, paru dans cette m~me collection) ; la ddfinition des op~rateurs . , 8 , A de la th~orie des formes harmoniques (expos~e dans le m~me volume) ; et, en quelques points, quelques notions sur la cohomologie enti~re. Encore ai-je, dans la mesure du possible, rappel~ les d~finitions et r~sultats dont il aura h ~tre fait usage. L'indication (( de Rham, w ~, renverra toujours au volume q u ' o n vient de citer ; les renvois Bourbaki seront faits sous la forme canonique. Des thdor~mes d'existence de la thdorie des formes harmoniques, qui jouent bien entendu un r61e essentiel dans le present volume, le lecteur n'a fi connaitre que l'existence des opdrateurs H e t G de de R h a m avec les propridtds formelles dnoncdes au n ~ 1 du Chapitre IV. Pour le cas particulier des tores, une ddmonstration directe de ces rdsultats, inddpendante de celle qui est donnde dans le volume de de R h a m pour le cas gdn~ral, est exposde au n ~ 2 du Chapitre IV, ce qui permettrait, si on le ddsirait, d'aborder la thdorie des fonctions th~ta sans s ' a p p u y e r sur la th~orie g~n~rale des formes harinoniques.

Paris, le 31 mai 1957.

[1958b] On the moduli of Riemann surfaces 1 TO E m i l Artin on his sixtieth birthday The p u r p o s e of the following pages is partly to clarify my own ideas on an interesting topic, at a stage when they are still unripe for publication, but chiefly to be present by p r o x y at Artin's b i r t h d a y celebration. In speaking of these ideas as " m y own", my intention is not to claim originality for them. They are little m o r e than a c o m b i n a t i o n of those of Teichmfiller with the ideas on the variation of complex structures, recently introduced by K o d a i r a , Spencer and others into the t h e o r y of moduli. The first concept to be elucidated is that of reinforcement of structure. Let S o be an oriented c o m p a c t surface of genus g > 1, given once for all. Let S be a Riem a n n surface of genus g, i.e. a surface of genus g, p r o v i d e d with a c o m p l e x - a n a l y t i c structure (or, what a m o u n t s to the same, an oriented surface of genus g with a conformal structure, or again an oriented surface of genus g with a class of conformally equivalent ds2). By a "class of m a p p i n g s " of S o into S, we shall u n d e r s t a n d a class of c o n t i n u o u s mappings, equivalent under h o m o t o p y ; a class will be called " a d m i s sible" if it contains at least one orientation-preserving h o m e o m o r p h i s m of S o onto S. A R i e m a n n surface S, p r o v i d e d with the a d d i t i o n a l structure defined on it by assigning one admissible class of m a p p i n g s of S o into S, will be called a Teichmtiller surface. P e r h a p s the most r e m a r k a b l e of Teichmfiller's results is the following: when p r o v i d e d with a rather obvious " n a t u r a l " topology, the set O of all classes of m u t u a l l y isomorphic Teichmtiller surfaces is h o m e o m o r p h i c to an open cell of real dimension 6g - 6. This global result will neither be used nor discussed in the following pages, the chief p u r p o s e of which is to consider the local p r o p e r t i e s of O and to define on it a " n a t u r a l " c o m p l e x - a n a l y t i c structure, of complex dimension 3g - 3, and a " n a t u r a l " H e r m i t i a n metric. The above "reinforcement of s t r u c t u r e " can be modified in various ways. Instead of admissible classes of mutually h o m o t o p i c mappings, one might wish to introduce classes of m u t u a l l y isotopic orientation-preserving h o m e o m o r p h i s m s of S o onto S; unless I am mistaken, k n o w n results in surface-theory imply that this w o u l d not actually differ from what we have done. O n the other hand, one can also consider a weaker type of reinforcement, in which two m a p p i n g s of S o into S are considered equivalent if they induce the same h o m o m o r p h i s m of the onedimensional h o m o l o g y group of S ~ H1(S~ into H i ( S ) ; as a t e m p o r a r y terminological prop, let us call a "Torelli surface" the surface S with the a d d i t i o n a l structure d e t e r m i n e d by an admissible class of m a p p i n g s for this wider concept of equivalence.

Part of this was done, I am somewhat ashamed to say, while on contract with the Air Force. The opinions of the Air Force do not necessarily coincide with mine.

381

382

[1958b1 On the moduli of Riemann surfaces

Call ~z~ the fundamental group of S ~ with an origin a ~ chosen once for all ; it is the group generated by 2g generators A ~ with the well-known relation, which we shall write as R(A ~. . . . . A~ = 1. Let f , f ' be two mappings ofS ~ into S; they induce h o m o m o r p h i s m s h, h' of ~o into ~(S,f(a~ and ~(S,f ,( a 0) ) , respectively. By considering the universal coverings of S o and S, one sees easily thatfandf' are homotopic if and only if h' can be derived from h by moving the origin of the fundamental group of S along a suitable path f r o m f ( a ~ t o f ' ( a ~ If, by an obvious "abuse of language", we agree to speak of " t h e " fundamental group ~(S) of S (which is then defined intrinsically only up to an inner automorphism), we may say that f a n d f ' are h o m o t o p i c if and only if they induce h o m o m o r p h i s m s of ~0 into ~(S) which are equivalent under inner automorphisms of ~z(S). In particular, an admissible class of mappings of S o into S will define a class of isomorphisms of ~o onto it(S), equivalent under inner automorphisms of ~(S); such a class will be called admissible; and the images of the A ~ under an isomorphism in such a class will be called an admissible set of generators of ~(S). Thus a Teichmfiller surface is defined by selecting on S an admissible set of generators of ~(S), provided two such choices are considered equivalent whenever they can be derived from each other by an inner automorphism. We shall denote by _S any Teichmtiller surface with the underlying Riemann surface S. Similarly, a Torelli surface _Swill be defined by selecting on S an admissible set of generators for the h o m o l o g y group Hi(S); here we have no equivalence relation between such sets. By the definition of an admissible set, the intersection-matrix for an admissible set of generators is

This implies in the well-known manner that we can define on the Torelli surface S_ a normalized set of differentials of the first kind, for which the period-matrix (for the given set of generators) has the form II1,Z(S)II, where Z(S) is a g x g matrix. More precisely, if ~ is the Siegel space of symmetric g x g matrices with positivedefinite imaginary part, Z(_S) is a point of ~ ; if S' is another Torelli surface with the same underlying Riemann surface S, Z(S_') will be a transform of Z(_S) by an element of Siegel's modular group. Thus we have a mapping _S ~ Z(_S) of the set Z of all classes of mutually isomorphic Torelli surfaces into the Siegel space ~. The set Z has an involutory a u t o m o r p h i s m S ~ h(S), viz. the one which leaves the underlying Riemann surface S of S unchanged and induces on Hi(S) the automorphism x --, - x (it is easily seen, e.g. by considering a hyperelliptic surface of genus g, that this automorphism changes one admissible set of generators of H1(S) into another). It is clear that Z(h(S_)) = Z(S), i.e. that _S and h(S) have the same image in the Siegel space. Moreover, Torelli's classical theorem asserts that two Torelli surfaces S, S' have the same image Z(S) = Z(S') in the Siegel space if and only if _S' = _S or S' = h(S), and that S = h(S) if and only if S is hyperelliptic. Call Z1 the image of Z by S_ --, Z(_S); we see that the inverse image of a point of Z 1 by that mapping consists of one or two points according as the corresponding

[1958b]

383 On the re|

of Riemann surl:aces

Riemann surface is hyperelliptic or not. Combining this mapping with the obvious " n a t u r a l " mapping of | onto E, we get a mapping of | onto El, which we write as =S-~ Z(S). Finally, if ~JJl is the set of all classes of mutually isomorphic Riemann surfaces of genus g, we have a natural mapping of 21 onto ~ . It will be seen that, when O is provided with its natural complex structure, S ~ Z(=S) is a holomorphic mapping of O into the Siegel space. Actually 21 is an analytic subvariety of ~, whose points are all simple except those corresponding to hyperelliptic Riemann surfaces (this, with an important additional statement concerning sets of local coordinates in the neighborhood of a simple point of Y~I, is Rauch's theorem). As to ~JJ~,there is virtually no doubt that it can be provided with a structure of algebraic variety (non-complete, of course, and with multiple points), the "variety of moduli", so that the natural mapping of | onto ~ is holomorphic. Let _S be a Teichmfiller surface; this consists of a Riemann surface S with a preferred choice of generators A1 . . . . . A2o for ~(S), these being defined only up to an inner automorphism. We can represent conformally the universal covering of S onto the upper half-plane H = { z l J ( z ) > 0}; it is well-defined up to a hyperbolic displacement. This determines a representation of ~r(S) as a discrete group F of hyperbolic motions, generated by 2g elements z--* aiz = (c~iz + fli)/(?iz + (~i), with :tg6i - fizT~ = 1 and R(al . . . . , a2o) = 1. An inner automorphism of ~(S), or a hyperbolic displacement in FI, will merely transform F by such a displacement; conversely, if two Teichmfiller surfaces determine two sets (al), (a'i) which differ merely by an inner automorphism of the hyperbolic group, they are isomorphic. As remarked by Siegel (Math. Ann. 133), one can select that automorphism in a unique manner so as to get f12o = 72g = 0, ~2g > l, fig = ~ O ; as the matrices for the a~ are determined only up to a factor + 1, one can further normalize them by taking fi~ > 0 for 1 < i _< 2g - 1. In that normalization, all the a~ are uniquely determined by the 69 - 6 numbers (ct~, fl~, 6i) for 1 _< i _< g - 1 and g < i _< 2g - 1 ; more precisely, the relations ~6~ - ~ i ~ i = 1, e(al, . . . , 0"20 ) = 1 together with those which express the normalization, determine the 7~ and the coefficients of o"g, cr2o by means of a set of algebraic equations with non-vanishing functional determinant in the neighborhood of any point of the coordinate space ~6r corresponding to a group such as F. Let O~ be the subset of N6~-6 consisting of the points which can be obtained in this manner; the above remarks show how to define a bijection of O onto 01. It will be seen that this bijection is a real-analytic isomorphism when | is provided with its natural complex-analytic structure; this implies that O1 is an open subset of ~ 6 g 6. Moreover, we shall give explicit formulas to determine the complex structure on O1 which can be derived from that of O by that bijection. In order to justify the statements that have been made so far, we shall make use of the Kodaira-Spencer technique of variation of complex structures. This can be introduced in an elementary manner in the case of complex dimension 1, which alone concerns us here; this, in fact, had already been done by Teichmfiller; but he had so mixed it up with his ideas concerning quasi-conformal mappings that much of its intrinsic simplicity got lost. Perhaps the worst feature of his treatment, in the eyes of the differential geometer, is that his extremal mappings are destructive

384

[1958b] On the moduli of Riemann surfaces of the differentiable structure; this corresponds to the fact that his metric on | is almost certainly not to be defined by a d s 2, even though it is presumably a Finsler metric. Let __So be a Teichmfiller surface. In order to avoid the use of coordinate neighborhoods on the underlying Riemann surface So, we represent So as l-I/F, where FI is the upper half-plane and F is a discrete subgroup of the hyperbolic group; _SO will then be defined by a preferred set of generators (~i) for F, which we may assume to be in normalized form. We consider a small "variation of structure", depending differentiably upon some real or complex parameters u; this can be defined as the conformal structure on H / F determined by a complex-valued differential form ~ = dz + # d~, where # is a complex-valued function of z and of the parameters u, such that any element a of the group F merely multiplies ~ by a scalar factor; this amounts to saying that # dS/dz is formally invariant under F, a property which we also express by saying that g is of type ( - 1, 1) for the usual complex structure of FI/F. When # is so given, the complex structure of the varied surface Su underlying the varied Teichmfiller surface S,, is the one for which ~ is a differential form of type (1, 0) at every point; for this to have a meaning, we must have ]#[ < 1 for all z e F I . We assume that # = 0 for u = 0, i.e. that, for u = 0, S u is no other than _So.The universal covering of S, is FI with the conformal structure determined by ~; for each u, this can be conformally mapped upon H with its natural conformal structure; call F , the differentiable h o m e o m o r p h i s m of H onto itself which realizes this conformal mapping, i.e. which is such that (for fixed u) dF,, differs from ~ only by a scalar factor, i.e. is of the form dF,, =j'~, w h e r e f i s a complex-valued function, everywhere r 0 in II. The conformal mapping F, can be normalized in various ways, e.g. by prescribing that a given point of I1 and a given direction through that point remain fixed under F,. A respectable firm of ellipticians, w h o m I consulted concerning the properties of F,, has assured me that it depends differentiably on the parameters u, and that it is real-analytic in the u's if this is assumed of #. N o w assume that u is a single real parameter; then (c~/~u),_ o is an "infinitesimal variation" in the sense of Kodaira-Spencer; this operator will also be denoted by a dot. It is determined by

which is again a complex-valued function of type ( - 1 , 1). The infinitesimal variation is trivial if and only if the varied structure can be obtained from the initial structure (that of So) by an infinitesimal deformation of the surface. The latter will be defined by a vector-field, i.e. by a complex-valued fuction ~ of type ( - 1, 0); and one finds at once that such a vector-field ~ determines the infinitesimal variation of structure given by v = ~/0~. Therefore we introduce the Kodaira-Spencer space for So, which we define as the quotient of the space of all functions v of type ( - 1, 1) by the space of functions v = 0~/~5 with ~ of type ( - i, 0), both being considered as vector-spaces over C. We shall denote by D = D(v) the element of that space determined by a given infinitesimal variation v = / i . Let ~o be a quadratic differential on So; this can be written as (~) = q. dz 2,

[1958b]

385

On the moduli of Riemann surfaces where q is holomorphic of type (2, 0). Notations being as above, consider the integral SS qv d5 dz, taken over So; this has a meaning, since the integrand is invariant under F. Stokes's theorem shows at once that this is 0 if v = ~?~/c~5with of type ( - 1, 0); therefore it depends only upon co and D = D(v), and m a y be written as ((~), D). For any v, one can solve the equation v = &p/(?~; the solution will be uniquely determined modulo a holomorphic function of z in Fl. If we again assume that v is of type ( - 1, 1), any solution ~0 of that equation will be such that, for every ~ e F, the function ~/J~ = (p -

dz

- ~o'~

(1)

d(~z)

(where (S stands for the function defined by (p~(z) = q)(az)) is holomorphic in H. These functions satisfy the relations dz

~'o, = 0 ; d ( ~

+ ~

(2)

for all a, z in F. Conversely, given a "cocycle" (~9,), i.e. a system of holomorphic functions ~ in 11, satisfying (2) for all a, z in F, let F operate on H x C by the formula 0 % t) =

o-z, d z

(t -

~(~))

;

then the quotient (11 • C)/F is a complex-analytic fibre-bundle with base So, the fibre being the plane C (with the group t ~ at + b). By a well-known theorem on fibre-bundles (which, in the present case, can also easily be verified directly), this has a differentiable cross-section t = ~0(z); this means that (1) has a solution ~0, which is a complex-valued, real-differentiable function. Another general theorem on fibre-bundles (Serre, GAGA), which could also, in this case, be verified directly without difficulty, tells us that (H x C)/F is an algebraic bundle over So; therefore it has a meromorphic cross-section, so that (1) has a meromorphic solution ~o'. We conclude that the Kodaira-Spencer space can be defined in any one of the following manners : (a) as above, as the quotient of the space of functions v of type ( - 1 , 1) by the space of functions v = c?~/c?z with ~ of type ( - 1, 0); (b) as the space of functions qo such that v = ~?q)/c~5is of type ( - 1, 1), divided by the sum of the space of all holomorphic functions and of the space of functions of type ( - 1, 0); (c) as the space of holomorphic cocycles ( ~ ) , i.e. of systems of holomorphic functions satisfying (2), divided by the space of trivial cocycles, i.e. of those for which (1) has a holomorphic solution (p; (d) as the space of meromorphic functions q/ such that all the functions ~o' - (dz/daz)tp '~ are holomorphic, divided by the sum of the space of holomorphic functions and of the space of meromorphic functions satisfying ~0' = (dz/daz)(p '~ for all a.

386

[1958b] On the moduli of Riemann surfaces

In (d), since we are taking the meromorphic functions modulo the holomorphic functions, it is only the principal parts (at the poles) that must be considered; the condition on q~' implies that these are determined once they are given in a fundamental domain. Now, notations being as above, we can write

(co, D)= f f q v d S d z =

~qqodz,

(3)

where ~ means the integral over S 0, or, what amounts to the same, over a fundamental domain for F in I1, and ~ means the integral over the positively oriented contour of such a fundamental domain. The latter may be taken as a polygon with 4(J sides, corresponding (in that order) to al, ao+ 1, ~ ~, a~+~l, a2 . . . . . ~2g~. It is clear that .~ q2 dz is 0 whenever 2 is a continuous function, defined only along the contour of integration, such that, formally, 2/dz takes the same values along corresponding sides of the contour; for then the same is true of q)~ dz, so that the integrals along corresponding sides cancel each other. In particular, we can take 2 = q) - q)', where ~0' is defined as above ; this shows that (co, D) can be written as qq/dz, i.e. that it is the sum of the residues of the meromorphic differential (~o'/dz)co over S 0. It is a well-known consequence of Riemann-Roch that this sum, for a given ~o', will be 0 for all quadratic differentials co if and only if there is on So a " m e r o m o r p h i c differential of degree - 1" q<'/dz having the same principal parts as (p'/dz at every point, and also that, given any quadratic differential co, one may choose the principal parts of ~o'/dz in the fundamental domain, and hence everywhere in H, so that ~ (q)'/dz)co # O. This proves that the space of quadratic differentials on So is the dual of the Kodaira-Spencer space for S 0, the duality between them being given by the bilinear form (co, D). One would have been led to the same conclusion by combining the general results of the Kodaira-Spencer theory with Serre's duality theorem, and specializing these to the case of complex dimension 1. N o w observe that the conformal mapping F , must transform 17 into another group F,, of hyperbolic motions, consisting of the elements or, = F~-1 o ~ o F,; the transforms a~(u) of the a~ are then a set of preferred generators for S,. This shows that the a~(u) depend differentiably upon u. The ai(u) m a y not be normalized ; in order to put them into normalized form, it may be necessary to modify F,,; it is easily seen that this will not affect the differentiability of F,, and therefore of the a,. Differentiation of the relation cr = F~ 1 o a o F, gives, for u = 0: = F a(az)

ko"

dz

dF,, for fixed u, differs from dz + I~ d5 only by a scalar factor, we have l~ = (c~F,/c~)/(c?Fu/~?z);differentiating this with respect to u for u = 0 gives v =

But, since

ti = (?P/c~5. Thus P is a solution of v = c~q~/(?~; therefore, by (1), we can take ~ = (dz/d~z)& This gives:

~,~ = (Tz + ~)(~z + [7) - (~z +/~)(fiz + 3).

[1958b ]

387 On the moduli of Riemann surfaces

N o w apply (3), and calculate ~ in the usual manner by combining together the integrals along corresponding sides of the fundamental polygon. We get:

,1~i

i=1

This is a bilinear expression in the di, fii, 9i, 3i on the one hand, and, on the other hand, in the periods of the vector-valued differential

dz qz dz q dz

qz 2

f'2 =

which has the property that f ~ = M~f~ for every a e F, M , being the constant matrix ~2

2~fl

fi2

72

276

32

In particular, if the d~i, fli, 9i, 75i are all 0, ((o, D) is 0 for all ~0, and therefore D = 0. All this applies equally well if the small variation depends upon real parameters in any number. In particular, take vl . . . . , v3o 3 such that the D(vi) are a basis of the Kodaira-Spencer space, and then take # = ~ i wivi, with complex wl . . . . , w30 3- The preceding calculations show that the ~ , fl~, 7~, 6~ will be dift'erentiable functions of the real and imaginary parts of the w~, with a non-vanishing jacobian. This proves our assertion that the image | of O in N6.o-6 is an open set, and that the real vector-space underlying the Kodaira-Spencer space may be identified with the tangent vector-space to ~60 6 at the point corresponding to =So. Loosely speaking, the "realization" | of | has the right differentiable structure. Our remarks even show (always on the basis of the ellipticians' assurances) that it has the right real-analytic structure. It is clear, too, that, if we select a basis col . . . . . , o39-3 for the quadratic differentials on So, the (COl, D) define 39 - 3 complex covectors which determine an almost complex structure on O, or on 01. It remains to be seen that this is a complex structure. This can be done in two ways. The simpler method has just been suggested to me by L. Bers, who observed that the 3g - 3 complex parameters w~ introduced above can in fact be used as complex local coordinates. As we have already shown that their real and imaginary parts can be used as real local coordinates, it only remains for us to show that, for every small value ofw = (wl . . . . . w3g 3), the complex vectors D~ in the Kodaira-Spencer space for Sw, defined by the differential operators O/~?w> are all 0. More generally, we shall show that this is the case whenever # is a real-differentiable function of z and of complex parameters wi in any n u m b e r and is holomorphic in all the w~. It is obviously enough to consider the case of one complex parameter w. Assume therefore that # = #(z, w) depends real-differentiably upon z and w and that it is holomorphic in w. For a given value w* of w, consider the vector D in the Kodaira-Spencer space for Sw,, defined by (~?/c?~)w-w*. Notations being the

388

[]958b]

On the moduli of Riemann surfaces same as before, the conformal mapping of the universal covering of Sw. onto gl is given by z* = Fw.(z), where Fw. is a differentiable h o m e o m o r p h i s m of II onto itself, such that dz ~ = dFw. is of the form f(z) 9(dz + tl(z, w*)d~), with a scalar factor f ( z ) . In terms of the variable z*, the conformal structure of Sw for any w may be defined by a differential form ~* = dz* + #*(z*, w)d~*

which has to differ from dz + I~(z, w)d~ only by a scalar factor. Then, for each quadratic differential co* = q*(dz*) 2 on Sw., (co*, D) is given by:

aJ

\~/w=w.

But a trivial calculation shows that /t* is again holomorphic in w; therefore ~?/~*/c?~ is identically 0, and D is 0. The second method depends upon the consideration of the integrals of the first kind. We again call in the elliptical engineer, who tells us the following: since the conformal structure of S. can obviously be derived from a d s 2 which depends differentiably upon the parameters u (viz. d s 2 = y 2 Idz + ~ d z I 2 with y = ..~(z), which is invariant under F), the space of real-valued harmonic differentials on S. has a basis consisting of differentials which depend differentiably upon the parameters. Take again the case of a single real parameter u, and denote (O/c?u). = 0 by a dot, as before. The differentials of the first kind on S. are those linear combinations of the harmonic differentials with constant complex coefficients which differ from dz + # d2 only by a scalar factor; clearly the space of such differentials is generated by those which depend differentiably upon u. Let q. = h.(dz + # dz) be one of these; it must satisfy dr/. = 0 and is invariant under F; for u = 0, it reduces to a differential of the first kind r/o = ho dz for So. As before, put v = / i and call (0 a solution of v = (?~0/t?2; differentiating r/. with respect to u for u = 0, we get /l = h " dz + h o v d~ = r . dz + d(ho ~o),

where we have put r = h - 0(h0 ~o)/c~z. This, too, must be invariant under F and must satisfy dO = O. Call pi(u) the periods of r/. along the cycles crl for 1 _< i _< 29; put p~ = p~(0); /~i is then the period of 0 along a~, Let r/~ be any differential of the first kind for So, with the periods p'~. We may write r/; = dJ; w h e r e f i s a holomorphic function in gl. Then r/or/; is a quadratic differential for So; and, if we put D = D(v) as before, we have:

r

: ff ho.

ff o

- f

Combining integrals along corresponding sides of the fundamental polygon, we get : 9

(r/or/'o, D) = ~ (P'o+iPi -- P'il}o+i). i=1

In particular, call r/J, for 1 < j < g, the normalized differentials of the first kind for =S.; by definition, they are such that I p / ( u ) l , for 1 _< j < g, 1 < i < g, is the unit-

[1958b]

389

On the moduli of Riemann surfaces matrix; and then, by definition, we have:

z(__x.) = IIp~+i(u)ll

(1 _< j _< g;1 _< i < g).

This gives II~ill = 0, II~+gll = Z. Substituting r/~, r/~ for r/0, rl• in the above formula, we get:

2 = -II(rt~rl~, D)ll. This proves that S --. Z ( S ) is a holomorphic mapping of O into 6. More precisely, it shows, not only that the coefficients po~+kof Z(=S) are holomorphic on O, but also that any 3g - 3 of them will be local coordinates in the neighborhood of a given point of _So if and only if the corresponding quadratic differentials ~/~tl~ are linearly independent. This is Rauch's theorem. It implies that Z(_S0) is a simple point of the image Y~I of O by Z if and only if the products ~/~r/~ generate the space of quadratic differentials, i.e. if and only if S o is not hyperelliptic. Onc can prove quite similarly that the subset of 521 consisting of the points which correspond to hyperelliptic Riemann surfaces is a non-singular analytic subvariety of ~ of complex dimension 29 - 1. It is clear now that the almost complex structure defined on O is a complex structure in the neighborhood of all the points which do not correspond to a hyperelliptic Riemann surface; by using well-known general theorems on analytic varieties, one can then extend this result even to the points which correspond to hyperelliptic surfaces. Finally, in order to define a " n a t u r a l " Hermitian metric on O, it is only necessary to define a Hermitian structure on the Kodaira-Spencer space for each So, or, what amounts to the same, on the dual of that space, i.e. on the space of quadratic differentials for So. This is done by putting, for any two quadratic differentials co = q d z 2, o)' = q' dz 2 for So: ((~), u/) = [ [ - q~fy2 dz d5 o ,/ with y = J ( z ) . In fact, the integrand can be formally written as e)~)/dS, where dS = y 2 dz dz is the hyperbolic area-element in H; it is therefore invariant under F, and the integral has an intrinsic meaning. This raises the most interesting problems of the whole theory: is this a Kfihler metric? has it an everywhere negative curvature? is the space O, provided with its complex structure and with this metric, a homogeneous space? It would seem premature even to hazard any guess about the answers to these questions.

[1958c] Final report on contract AF 18(603)-57 Within the framework of the project originally submitted to AFOSR, I eventually decided to concentrate on two lines of investigation: (I) The classical problem of the moduli of algebraic curves over complex numbers; (II) A study of the K;,ihler varieties topologically identical with the nonsingular quartics in projective 3-space (henceforward called K3 surfaces). In both directions, my results are still very fragmentary and incomplete; and I have had to postpone the arithmetical considerations which provided the original motivation for the whole project, in order to deal first with the function-theoretic aspects of the above questions. In both problems, the ideas of Kodaira and Spencer on the variation of complex structures have proved fundamental. I have much benefited from repeated consultation with them during my stays in Princeton, in January and February and again in June. It also turned out that Professor L. Bers had been engaged in a parallel investigation of problem (I); consultation with him on this topic has been very fruitful. On the other hand, various aspects of problem (II) have recently engaged the attention of Professor L. Nirenberg, Professor A. Andreotti, and Dr. Atiyah; I have learned a great deal from communications, written and oral, from all of them. In order to give, in what follows, a coherent account of these topics, it will be necessary to include much of the work of my colleagues, and it would be unpractical to try to unravel in detail what may belong to me and what belongs to each one of them. It should be understood that they deserve a large share of the credit for the work described in this report.

I. Moduli o f algebraic curves We consider curves of a given genus 9 -> 1. One basic concept is that of a Teichmtiller structure. If S, S' are two oriented surfaces of genus 9, we say that a class of mappings of S into S' in the sense of homotopy (or, briefly, a class C(S, S')) is admissible if it contains at least one orientation-preserving homeomorphism of S onto S'. Let So be an oriented surface of genus g, given once for all. By a Teichmfiller surface, we understand a Riemann surface of genus 9 (i.e. a surface of genus g, provided with a complex-analytic structure and oriented accordingly), together with an admissible class C(So, S). Isomorphism being defined for these in the obvious manner, we introduce, with Teichmfiller, a space T (the "Teichmfiller space"), whose points correspond in one-to-one manner to all the classes of mutually isomorphic Teichmfiller surfaces of genus 9. Teichmtiller's chief contribution was

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391 Final report on contract AF 18(603)-57

to define on T a certain topology, the " n a t u r a l " one in a sense described below, and then to prove that T, with this topology, is h o m e o m o r p h i c to an open cell of real dimension 69 - 6. So far, I have mainly been concerned with the local propperties of the Teichm/iller space and of its " n a t u r a l " complex-analytic structure. The definition of the latter depends upon ideas introduced by Teichmiiller himself, but which do not appear to have been fully understood until K o d a i r a and Spencer attacked similar problems for higher dimensions. In order to describe them, it is convenient to substitute for the above definition o f a Teichmiiller surface the following one. Let A1 . . . . , A20 be a set of generators, fixed once for all, of the fundamental group G o of So (with a given origin), satisfying the relation

R(A 1. . . . .

A2o ) = A1AzAllA2

1 ...

A2o~ = 1.

A set at . . . . . a20 of linear-fractional substitutions on z will be called admissible if it has the following properties: (a) it generates a discrete group G of hyperbolic substitutions acting on P; (b) S = PIG is a compact surface of genus 9; (c) there is an admissible class C(S o, S), mapping G o onto G considered as the fundamental group of S, which maps A i onto ai for 1 _< i _< 2 9. Each Teichmfiller surface S has a universal covering which can be mapped conformally onto P, and can therefore be represented as PIG ; the class C(So, S) which belongs to it defines an isomorphism of G o onto G; therefore, to each Teichmtiller surface, there corresponds at least one admissible set (o~ . . . . . 0029). Conversely, it is easy to see that two such sets will define isomorphic Teichmiiller surfaces if and only if they can be transformed into one another by a linear-fractional substitution. Using this fact, it is possible to normalize admissible sets in such a way that there is a one-to-one correspondence between classes of Teichmtiller surfaces (i.e. points of the space T) and normalized admissible sets (00i); this gives a one-to-one mapping of T onto a subset of the coordinate space R 6~ 6. It turns out that the latter subset is open, and that the mapping and its inverse are both indefinitely differentiable (and even, presumably, real-analytic) if T is provided with its " n a t u r a l " differentiable structure; the proof of the latter fact is due to L. Bers. N o w introduce a "variation of structure" as follows. Let/~ be any indefinitely differentiable complex-valued function in P such that I#l < 1; let Pu denote the upper half-plane with the modified complex structure for which dz + # dz is a differential form of type (1, 0); if the latter structure is invariant under G, i.e. if l~ dS/dz is formally invariant under G in an obvious sense, the complex structure of P~ can be projected onto a complex structure P~/G, making the latter into a Riemann surface S u, or rather a Teichmtiller surface if we keep the ai as the distinguished generators of G. Let F , , in that case, be the conformal mapping of P onto Pu; the Teichmfiller surface S~ is then the one defined by the admissible set 00i(/~) = Fj~-100IF~; S~ is isomorphic to S if and only if F , (which is defined only up to a fractional-linear substitution) can be chosen so that it commutes with all the 00~; in that case, the "variation of structure" defined by # will be called trivial. F r o m this, we get an "infinitesimal variation" if we take/~ to depend (differentiably) upon a real parameter t, so that/~ = 0 for t = 0; then v = (dlz/dt)t-o is called an infinitesimal variation of structure; it is clear that any function v in P,

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Final report on contract AF 18(603)-57 such that v dUdz is formally invariant under G, defines such a variation. The variation v will be called trivial if the finite variation # is tangent, for t = 0, to a trivial one, i.e. if there is a trivial variation y , depending upon t, such that d#/dt = d#'/dt for t = 0. It is easily seen that a necessary and sufficient condition for this is that there should exist a function { in P such that v = c3{/c3~and that {/dz is formally invariant. The infinitesimal variations v can be considered as the elements of a vectorspace V (of infinite dimension) over the complex numbers; the trivial variations make up a subspace V' of V. Standard procedures in c o h o m o l o g y theory and the theory of fibre-bundles (which, in a case like this one, depend merely upon elementary facts such as Stokes's theorem and the theorem of Riemann-Roch) show that V/V' is of finite dimension 3g - 3 and can be "canonically" identified with the dual space to the space of quadratic differentials of the first kind on S. As to the latter assertion, let co = q dz 2 be such a quadratic differential; in other words, we take for q a holomorphic function in P, such that q dz 2 is formally invariant under G. Then, for v in V, vq dz d5 is formally invariant under G, so that we may integrate it over P/G; Stokes's theorem shows that the integral is 0 for v in V', i.e. it depends only upon the class D of v modulo V'; denoting it by (co, D), one finds that this bilinear form defines a duality between V/V' and the space of quadratic differentials, as asserted above. At this point, one must make use of the fact (proved by Bers) that, if # depends differentiably upon some real parameters, the same will be true of the mapping function Fu and hence also of the coefficients in the substitutions ai(#)- it is now easy to calculate the effect of an infinitesimal variation on the coefficients of the cri, i.e. to calculate dai(#)/dt for t = 0, in terms of v = (d#/dt),= o. One can also calculate the effect of a given infinitesimal variation on the periods of the normalized integrals of the first kind on S. The conclusions one can derive from this are as follows. It is possible to provide T with a complex-analytic structure, of complex dimension 39 - 3, such that, whenever # depends holomorphically upon some complex parameters w~, the point of T which corresponds to S u depends holomorphically upon the w~; this observation is due to Bers, who also found, more precisely, that, if we take # = ~ Wiy2qi, with y = Im(z) and (ql . . . . , q3o 3) such that qi dz2 are a basis of the space of quadratic differentials of the first kind on S, then the wi can be taken as local complex coordinates in T in a neighborhood of the point corresponding to S. Furthermore, the quadratic differentials of the first kind on S can be identified with the covectors on T at that point ; and Petersson's hermitian metric, in the space of those differentials (which is no other than the space of automorphic forms of degree - 4 for the group G) defines an intrinsic Hermitian metric on T, which turns out to be a K~ihler metric. The facts concerning the mapping of T into R 6g 6 by means of the coefficients of the ~ri have already been stated. Finally, the periods of the normalized integrals of the first kind on S define a mapping of T into the Siegel space of symmetric g x g matrices with positive-definite imaginary part; the image of T under that mapping is a complexanalytic variety W, whose singular points are those corresponding to hyperelliptic Riemann surfaces; and one obtains a new proof for Rauch's theorem, stating which of the periods of the normalized integrals of the first kind can be used as local coordinates in the neighborhood of any given simple point of W.

[1958c]

393 Final report on contract AF 18(603)-57

H. The K3 surfaces We may start here from the observation (made independently, I believe, by Atiyah and myself) that, when a non-singular surface S in projective 3-space acquires a node, i.e. a conical double point, and the latter is desingularized by a standard dilatation, this process gives a surface S' which is homeomorphic to S. It was easy to surmise that the same is true when a surface acquires any number of distinct nodes; this, in fact, or rather a much more precise theorem, was proved by Atiyah. It shows, in particular, that the non-singular quartic in 3-space, the double plane with a non-singular sextic branch curve, and the desingularized K u m m e r surface, are all homeomorphic. Such surfaces will be called K3 ; they had already occurred in the work of the Italian geometers, and, more recently, in that of Kodaira. The Italians, in fact, had discovered an infinite sequence of families F, (n = 1, 2 . . . . ) of regular surfaces with P0 = 1 ; F 1 consists of double planes with a sextic branch curve; F2, of quartics in 3-space; F, consists of surfaces of degree 2n in projective (n + D-space, whose hyperplane sections are canonical curves of genus n + 1. There are very plausible arguments to indicate that all such surfaces are of type K3, although no complete proof for this seems to have been given yet. For K3 surfaces, the intersection matrix of two-dimensional cycles has the signature (19, 3), the determinant - 1, and is even (i.e. the self-intersection of every cycle is even); hence there is exactly one double differential of the first kind ; if we call it tl, then we must have dq = O, qz = 0 and t/f/>_ 0; if we assume (as seems very likely) that all K3 varieties (algebraic or not) constitute only one connected family, then the canonical class must be 0, so that ~/ 4= 0 and ~/f/> 0 everywhere; this implies that the complex structure is entirely determined by q. Conversely, let there be given, on a differentiable manifold of that nature, a complex-valued differential form t/ of degree 2, satisfying drl = 0, q2 = 0, and ~/f/> 0 everywhere; this determines a complex structure. It seems very plausible (but not at all easy to prove) that two such forms with the same periods must determine complex structures which can be transformed into one another by a difl'erentiable homeomorphism, homotopic to the identity; that all such structures are K~hlerian; and that the periods of rl do not have to satisfy any other condition than those which are implicit in the relations f]2 = 0, /7/7/ • 0. These conjectures (which have also been made independently by Andreotti) may also be expressed as follows. Let S be a class of such structures, two structures being put into the same class if and only if they can be transformed into each other by a differentiable homeomorphism, homotopic to the identity. Let (al . . . . . a22 ) be a minimal set of generators for the two-dimensional homology group with integral coefficients; iD/is as described above, let the Pl be its periods corresponding to the cycles ai; let P be the point with the homogeneous coordinates (Pa . . . . . P 2 2 ) in the complex projective space of dimension 21. Let F(x, y) be the symmetric bilinear forms in x = (xl ..... ")r Y = (Yl . . . . . Y22) whose matrix is the intersection-matrix of the cycles ag. Then the conditions q2 = 0, r/F/ > 0 imply that P is in the open subset H of the quadric F(x, x) = 0 which is determined by the inequality F(x, ~) > 0; H is a homogeneous space of complex dimension 20 for the orthogonal group determined by the real form F(x, x). Now, if we assign to each

394

[1958c] Final report on contract AF 18(603)-57

class S of structures of the given type the point P ~ H, we have a mapping of the set of all such classes into H; and Nirenberg, by an argument combining the Kodaira Spencer technique with techniques derived from the theory of elliptic equations, has proved that the image of that set in H must be open. The conjectures stated above would mean that the mapping is a one-to-one mapping of that set onto H. Furthermore, by a fundamental theorem of Kodaira, a given K3 structure will define an algebraic variety if and only if it has a K~hler metric whose fundamental form has integral periods, i.e. belongs to an integral cycle a; we must have F(a, a) < 0. For such a structure, the point P defined above must belong to the linear variety L~ defined by F(a, x) = 0, and therefore to the set Ha = H c~ L,. It is easy to see that for each integral a such that F(a, a) < O, Ha can be identified with the Riemannian symmetric space belonging to the orthogonal group of the quadratic form F o of signature (19, 2) induced by F on La. Again, one is led to conjecture that this establishes a one-to-one correspondence between such structures and Ha. Now it may happen that two classes S, S' may be distinct and still define isomorphic structures; this will be so when structures belonging to these classes can be transformed into one another by a differentiable homeomorphism, not homotopic to the identity. The latter will induce an automorphism of the homology group, and therefore a unit U of the quadratic form F(x, x), i.e. a matrix with integral coefficients belonging to the orthogonal group of F; let G be the group of all the units of F which can be obtained in this manner. Assuming the truth of the conjectures stated above, we see that two points P, P' of H will determine isomorphic structures if and only if they are equivalent under the group G. A similar statement will hold for Ha; here G has to be replaced by the subgroup Go of G consisting of the elements which leave a invariant. This shows that it is important to determine G, and in particular to determine whether G coincides with the group G of all the units of F. My results on this, too, are still incomplete. However, by applying the theory of theta functions to the Kummer surface considered as a model for K3 surfaces, it has been possible to reduce part of the problem to a purely arithmetical question which has been recently solved by M. Kneser. The result is that G is, at any rate, of finite index in G. Now, since H is not the Riemannian symmetric space for the orthogonal group of F, it follows that G does not act upon H in a properly discontinuous manner; hence there can be no theory of the moduli, in the ordinary sense, for the postulated connected family of K3 surfaces. This is as expected, and is analogous to the wellknown fact that there is no theory of moduli for complex toruses, but only for "polarized" abelian varieties. The situation is quite similar here. For, if one restricts oneself to the family of algebraic K3 surfaces polarized by assigning a K~ihler form belonging to a given integral cycle a, then it follows from what we have said that Go acts in a properly discontinuous manner on Ho, and is of finite index in the group of all units of the quadratic form Fo, so that HjGo is of finite measure. It is therefore to be expected that the automorphic functions in Ha, for the group G o, make up an algebraic function-field, the field of the moduli for the K3 surfaces of the given family. One interesting feature here is the occurrence, in a problem of moduli, of the automorphic functions belonging to the group of units of a quadratic form of signature (n, 2) (with n = 19 in the present case). This is believed to be

[1958 c ]

3 95

Final report on contract AF 18(603)-57 the first time that such a group has appeared in such context. Of course, before this can be more thoroughly investigated, it will be necessary to obtain full proofs for the conjectures stated above. After that is done, analogies with the theory of abelian varieties and of their fields of moduli (given by Siegel's modular functions) will undoubtedly suggest a number of further problems, of a function-theoretic and also of a number-theoretic nature; most fascinating, perhaps, are the possibilities suggested by the theory of complex multiplication. But this is still too remote to be discussed here.

[1958d] Discontinuous subgroups of classical groups (Notes by A. Wallace)

Introduction The object of this course is to prepare the way for a study of certain types of discrete subgroups of the real classical groups and the corresponding quotient spaces. The classical groups will be constructed in a rather special way which actually yields all these groups with only a small number of exceptions. The method consists in taking a semi-simple algebra A over the rationals with an involution or, extending A to an algebra A R over the real numbers, and considering the connected component G of the group of automorphisms of A R which commute with or. G is, in a natural way, an algebraic matric group, and a subgroup Gz of matrices in G whose elements are rational integers is a discrete subgroup. Discrete subgroups obtained in this way are to form the main object of study. An illustration of the kind of theorem to be studied is given in w where conditions for the compacity of G/Gz are worked out. The method of study of G/Gz involves introducing a second involution on AR which is positive (definitions in w and studying the set P(A~) of positive symmetric elements of A~ with respect to this involution. It turns out that G/Gz can be related to a subset of these positive symmetric elements, and that a special type of set W in P(AR) (an M-domain) covers the image of G/Gz, in a certain sense, only a finite number of times. Attention can then be transferred to the set W, which is arithmetically defined. The study of the set W depends on a study of P(A~) which generalizes classical results of Minkowski on the theory of positive definite quadratic forms and their equivalence under transformation by integral matrices. w167 of these notes are concerned with this theory. The next two sections give a list of the classical groups which can be obtained as indicated above. In w some results are obtained on algebras with involutions, and in w these are applied, along with the earlier results, to the construction and study of M-domains. It may appear that the results obtained in this way will depend on the particular way in which the group G is written as a matric group, since the definition of Gz certainly depends on this. However, the properties which are to be of interest eventually are only those which are invariant under commensurability; this can be defined as follows: Two discontinuous subgroups F and F' of a group G are said to be commensurable if F r~ F' is of finite index in each of them. Now if the group G is an algebraic matric group overQ, and is represented as a matric group in two different ways, then the two subgroups F' and F" of integral matrices in these two representations are commensurable. To prove this let G', G"

396

[1958d]

397 Discontinuous subgroups of classical groups

be the two representations of G as matric groups and write x' = 1, + (x'ij) for an element of G', x" = 1,, + (x~,) for an element of G". The isomorphism between G' and G" is expressed by equations xij = Pi~(x~,,), x~, = Q,u(xlj) where the Pij and Q , , are rational functions and in fact can be taken to be polynomials over Q with zero constant terms. If N is a c o m m o n denominator for all the coefficients in the P;i and Qa~,, then, for x;{, -= 0 (mod N), the corresponding x'ii will be integral. Thus F' contains F~, the subgroup of F" consisting of matrices -= 1,, (mod N). Similarly F" D F;e. N o w F~ is of finite index in F", and so the larger group F' c~ F" is of finite index in F"; and similarly in F'. Thus F', F" are commensurable. The result shows that, as far as properties invariant under commensurability are concerned, no generality is lost by the special method used here of constructing Gz. In particular it is easy to see that the compacity of G/Gz discussed in w is such a property.

[1959a] Ad61es et groupes alg6briques

On d6signera toujours par k, soit un corps de nombres algdbriques, soit un corps de fonctions alg6briques de dimension 1 sur un corps de constantes fini. On d6signera par k~ le compl6t6 de k par rapport 5. une valuation v; si une valuation est discr6te, on la ddsignera le plus souvent par un symbole tel que p, et on ddsignera par rp l'anneau des entiers p-adiques dans kp. O n ddsignera par S tout ensemble fini de valuations de k, contenant l'ensemble So des valuations non discr6tes (pour lesquelles le compldt6 est R ou C); bien entendu So est vide si k est un corps de fonctions. Si V e s t une varidt6 alg6brique, ddfinie sur un corps k, on identifiera, suivant l'usage, V avec l'ensemble des points de V sur le domaine universel, et on notera Vk l'ensemble des points de V rationnels sur k. Cette convention s'appliquera notamment si V e s t une varidt6 de groupe, une varidt6 d'alg6bre, une vari6t6 de corps, etc.; on dira par exemple que V e s t une varidt6 d'alg6bre de dimension n, ddfinie sur k, si, en tant que varidt6, c'est un espace a n n e de dimension n, sur lequel on s'est donn6, outre la structure additive usuelle, une structure multiplicative, c'est-'~-dire une application bilindaire (toujours suppos6e associative) de V x V dans V, ddfinie sur k; Vk est alors une alg6bre sur k au sens usuel; si Vk est un corps (donc, au sens usuel, une extension de k de degr6 n), on dit que Vest une vari6t6 de corps de dimension n sur k. Cette mani6re de parler est conforme fi l'usage ancien de Kronecker, Hilbert, etc. (qui ne se g~naient pas pour parler de l'616ment g6n6rique d'un corps). 1. Soit V une varidt6 ddfinie sur k; Vk, peut ~tre munie, d'une mani6re 6vidente, d'une topologie qui la rend localement compacte, et compacte si Vest compl6te (on commence par le cas oh V e s t une vari6t6 a n n e , Vkv 6tant alors considdrde comme partie fermde d'un espace vectoriel de dimension finie sur k~; on passe de lfi d'une mani6re 6vidente au cas d'une vari6t6 abstraite quelconque). Soit p une valuation discr6te de k. Si Vest une varidt6 a n n e , on notera V~ l'ensemble des points de Vkp dont les coordonndes sont dans r~; il est immddiat que c'est une pattie compacte de Vkp. Plus gdndralement, soit V une varidt6 abstraite, d6finie sur k; on sait que V admet toujours un recouvrement fini par des ouverts isomorphes/t des vari6tds affines V (i), ddfinies aussi sur k; autrement dit, on peut 6crire V = Ui Ji(V(~ oh less sont des isomorphismes (ddfinis sur k) des vari6t6s affines V (i) sur des ouverts de V (au sens de la topologie de Zariski, bien entendu). Pour toute valuation discr6te p de k, posons: = U i

c'est 15. une pattie compacte de Vk~. Posons maintenant, pour tout ensemble fini S

398

[1959a]

399 Adeles et groupes algebriques

de valuations de k contenant l'ensemble S O des valuations non discretes:

Vs = I1 vk, X [I veS

pr

off le second produit est 6tendu fi toutes les valuations de k (necessairement discretes) qui n'appartiennent pas fi S; c'est l• une pattie localement eompaete de l ~ Vk,~,et on a Vs = Vs, pour S ~ S'. On d~signera par VAk la limite inductive des Vs, c'est-~-dire la reunion des Vs munie de la topologie pour laquelle un systeme fondamental d'ouverts est form6 par la reunion de l'ensemble des ouverts dans t o u s l e s Vs. Cette notion est justifOe par le Jait que VA~ est dbfinie d'une manikre intrinsbque, c'est-'~-dire ne depend pas de la maniere dont on a 6crit V comme reunion finie d'images isomorphes de varietes affines; la verification de cette assertion est 616mentaire. On appelera VA~ l'espace adOlique associ6 fi V. Au lieu de A , on 6crira souvent A quand aucune confusion n'est possible. L'espace adelique associ6 fila droite affine n'est autre que l'ensemble des adeles (dits aussi "repartitions" ou "valuation-vectors") du corps k, avee sa topologie usuelle: cet ensemble (muni de sa structure topologique, et de sa structure d'anneau) sere note Ak, OU simplement A. La notion d'espace adelique poss6de des proprietes fonctorielles raisonnables. S i f e s t une application partout definie d'une variet6 V darts une variet6 W, definie sur k, on en deduit d'une maniere evidente une application continue de VA dens WA, Si par exemple on s'est donne sur V une loi de groupe algebrique, on en deduit une loi de groupe sur VA, et VA s'appellera le groupe adelique associe fi V. Par exemple, si Vest le groupe multiplicatifGm ?2 une variable sur k, le groupe adelique correspondant est le groupe des idbles de k au sens usuel. Si l'application f de V dens W e s t surjective, il n'en est pas de meme, en general, de l'application de VA dans WA qui lui est associee. Cependant, si Vest un fibre de base W, localement trivial sur k, au sens de la geometrie algebrique, et s i f e s t la projection de V sur sa base W, alors l'applieation correspondante de VA darts WA est surjective. En particulier, si G est un groupe et g u n sous-groupe de G, tous deux definis sur k, et q u ' o n pose H = G/g, l'application de GA darts HA, associee 5. l'application canonique de G sur H, n'est pas surjective en general; mais, si G est (sur le corps de base k) fibrO localement trivial sur H = G/g , alors on peut identifier canoniquement H A avec G A/gA . 2. Soit K une extension separable de k, de degr6 fini d; soit V une varidte affine ou projective de dimension n, definie sur K; on va indiquer comment on peut associer /tces donnees une varidte W de dimension nd, definie sur k, et qui, sur le domaine universel, est isomorphe au produit de V e t de ses conjuguees sur k. P o u r fixer les iddes, considerons le cas affine. Alors Vest definie par des equations: P,(X~ . . . . , XN) = 0

(1 _< /~ < m),

5. coefficients dans K. Soit (a~ . . . . . ad) une base de K sur k; posons: d

x~ = y~ a~ Yj~ j

1

(1 _< i _< N)

400

[1959a] Ad61es et groupes alg6briques

les Yji 6tant de nouvelles ind6termin6es; alors les P~, deviennent des polyn6mes fi coefficients dans K par rapport aux Yji, et peuvent donc s'6crire: d

Pu(X1 . . . . . X u ) = ~ ah Qnu(Yll . . . . . Ydu), h=l

off les Qhu sont des polyn6mes ~ coefficients dans k. Dans ces conditions, W est la vari6t6 d6finie par les 6quations Qhu(Yll . . . . . YdN) = 0

(1 < h _< d, 1 _< # < m)

dans l'espace affine de dimension dN. En effet, une v6rification facile montre qu'apr6s un changement de coordonn6es linOaire, ~ coefficients dans le corps compos6 de K et de tous ses conjugu6s sur k, ces 6quations sont pr6cis6ment celles qui d6finissent le produit de Vet de ses conjugu6es sur k. C'est d'ailleurs seulement cette derni6re v6rification qui exige que K soit s6parable sur k; s'il ne l'6tait pas, les d6finitions ci-dessus auraient encore un sens, mais d6finiraient une op6ration ayant des propri6t6s assez diff6rentes de celles qui nous int6ressent ici. On dira, dans les circonstances ci-dessus, que W e s t d6duite de V par restriction du corps de base de K h k, et on 6crira W = ~ ( V ) . I1 est imm6diat qu'il y a correspondance biunivoque (canonique !) entre VK et Wk. O n v6rifie aussi, sans difficult6, qu'il y a correspondance biunivoque (non moins canonique) entre VA,, et WAk. Cela permet, par exemple, dans t o u s l e s probl6mes relatifs aux espaces ad61iques sur les corps de nombres, de se ramener, si l'on veut, au cas off le corps de base est Q. 3. Soit G une vari6t6 de groupe de dimension n, d6finie sur k. I1 existe sur G une forme diff6rentielle ~o de degr6 n, invariante ~ gauche, d6finie sur k, et ~o est unique un facteur pr6s (facteur qui doit 6tre dans k); si xl . . . . . x, sont des fonctions sur G, d6finies sur k, qui soient des coordonn6es locales dans un voisinage de l'616ment neutre e (et, pour fixer les id6es, nulles en e), on pourra 6crire o~ = y dxl 9 .. dx,, off y est une fonction sur G, d6finie sur k, et finie en e. O n va montrer comment, si vest une valuation quelconque de k, on peut associer ~ Cn une mesure de Haar bien d6termin6e sur Gk,,, mesure qu'on notera ]~o I,,. P o u r cela on distinguera trois cas: (a) k~ = R; alors les xi peuvent servir de coordonn6es locales sur Gk,,au voisinage de e; au voisinage de e, y est fonction analytique r6elle de x ~. . . . . x,; dans ces conditions, y d x ~ . . , dx. peut s'interpr6ter c o m m e une forme diff6rentielle au sens usuel sur Gko au voisinage de e; c o m m e il est clair qu'elle est invariante ~ gauche, elle d6finit par translation une mesure de Haar sur Gk,,, q u ' o n note [~o [,. (b) k~ = C: les xi sont alors des coordonn6es locales complexes sur Gk~ au voisinage de e, et y est fonction analytique complexe des x~ au voisinage de e; on prendra [~o[~ = i"y~ dxl dx, ... dx, dx,. '(c) Soit p une valuation discr6te; Gkp est une vari6t6 analytique sur le corps valu6 complet kp; c o m m e pr6c6demment, x~ . . . . , x, peuvent fitre consid6r6es c o m m e coordonn6es locales au voisinage de e sur Gkp, c'est-~-dire qu'elles d6terminent un h o m 6 o m o r p h i s m e d'un voisinage de e dans Gkp sur un voisinage de 0

[1959a]

401 Ad61es et groupes alg6briques

dans l'espace k~,. Convenons, sur le groupe additif kp, de noter Idx Ip la mesure de Haar, norm6e par la condition que rp (anneau des entiers p-adiques) soit de mesure n 1; la mesure de Haar dans kp, produit des mesures Idxlp sur les n facteurs, sera notde Ddxl ... dx, Ip- On posera alors, au voisinage de e: I~o1~ = lyl~[dx~ ... dx, l~, off lYl~ est la valeur absolue p-adique de la valeur de y au point considdr6 (y est, c o m m e pr~c6demment, fonction analytique de xl . . . . , x, au voisinage de e, ce qui veut dire qu'on peut l'6crire comme s6rie convergente de puissances de x 1. . . . . x,). Naturellement, la valeur absolue p-adique est norm6e de la mani6re usuelle, c'est-fi-dire de fagon que l'automorphisme x ---, ax du groupe additif de k, multiplie la mesure de Haar par le facteur l a Iv. Les d6finitions ci-dessus se justifient du fait qu'elles sont ind6pendantes du choix des coordonn6es locales xl . . . . . x, au voisinage de e; c'est 6vident darts les cas (a) et (b), en vertu de la formule de changement de variables dans les int6grales multiples en analyse classique, et cela rdsulte immddiatement, dans le cas (c), de la formule correspondante en analyse p-adique (qui se d6montre encore plus facilement qu'en analyse classique). De ce qui pr6c6de, on peut, dans certain cas, d6duire la d6finition d'une mesure de Haar sur le groups addlique GA attach6 ~t G. En se reportant au paragraphe 1, il est 6vident que cela est possible chaque fois que, sur le produit

O < X ~0r %

vsS

il existe une mesure produit des mesures [colv; car il est 6vident que les mesures ainsi ddfinies sur Gs et sur Gs,, pour S ~ S', coincident sur Gs. Or, pour que la mesure en question soit ddfinie, il faut et il suffit que le produit infini

6tendu g toutes les valuations discr6tes de k, soit absolument convergent. Lorsqu'il en est ainsi, on notera ~],. Ice Iv la mesure ainsi obtenue. On observera que celle-ci ne d@end pas du choix de co; en effet, si on remplace co par cco, avec c e k • elle se multiplie par 1Iv Icl~, qui est 1 ("formule du p r o d u i t " d'Artin; rappelons que celle-ci se ddmontre comme suit: x ~ cx est un automorphisme du groupe additif Ak, qui multiplie la mesure de Haar par [Iv Iclv, et qui d'autre part induit un automorphisme sur k consid6r6 c o m m e sous-groupe de Ak, donc, par passage au quotient, d6termine un automorphisme du groupe compact Ak/k et par suite laisse invariante toute mesure de H a a r sur ce dernier; k 6tant discret dans A k, A k et Ak/k sent localement isomorphes, donc les mesures de Haar y prennent le m~me facteur). O n dira que G a la propri~t~ de convergence si 1 ~ [co I~ Y est ddfini; le cas off le produit infini 6crit ci-dessus, sans ~tre absolument convergent, est convergent lorsque les p sent ordonnds "naturellement" (c'est-fi-dire par ordre de grandeur croissante des normes) est int6ressant aussi; lorsqu'il en est ainsi, on dira que G a la propri~tO de convergence relative. Les exemples assez vari6s qu'on a trait6s

402

[1959a] Ad61es et groupes alg6briques

jusqu'ici rendent assez plausibles les conjectures suivantes: pour que G a i t la propri6t6 de convergence relative, il faut et il suffit que G n'admette pas d ' h o m o morphisme sur G,,, ddfini sur k; pour que G a i t la propri6t6 de convergence, il faut et il suffit que G n'admette pas d ' h o m o m o r p h i s m e sur Gin, ddfini sur le domaine universel. En particulier, t o u s l e s groupes unipotents, et t o u s l e s groupes semi-simples qu'on a pu traiter de ce point de vue, ont la propri6t6 de convergence. Le groupe Gm n'a pas la propridt6 de convergence relative. Le groupe addlique attach6 au groupe additif G, fi une variable n'est autre que le groupe additif de Ak; ce groupe a 6videmment la propri6t6 de convergence. Nous poserons : k/k

(on convient, une lois pour toutes, d'identifier d'une mani&e 6vidente une mesure de Haar sur un groupe localement compact F avec celle qu'elle d6termine par passage au quotient sur l'espace homog6ne F/7, lorsque y est un sous-groupe discret quelconque de F). On voit facilement que #k = IAI ~/2, o5 A est le discriminant de k, lorsque k est un corps de nombres alg6briques, et que #k = qO 1 si k est un corps de fonctions de genre .q sur un corps de constantes "~ q 616ments. Soit alors G u n groupe de dimension n sur k, ayant la propridt6 de convergence; nous poserons:

nG=

H Io 1 , L'

et nous dirons que c'est la mesure de Tamayawa sur G A. I1 r6sulte de ce qui pr6c6de que cette mesure est d6termin6e d'une mani6re unique (elle ne d~pend pas du choix de co). Soit K une extension s6parable de k, de degr6 rink Soit G u n groupe d6fini sur K ; soit G' le groupe ~K.k(G), d6duit de G par restriction du corps de base de K k; on a d6j/t observ6 que GAK s'identifie canoniquement/t G~k; on v6rifie facilement que Get G' ont simultan6ment la propri6t6 de convergence, et que les mesures de T a m a g a w a ~2G (sur GA,~) et ~2~, (sur G~,~) co'fncident lorsqu'on identifie ces groupes. C'est, entre autres, pour qu'il en soit ainsi qu'on a introduit le facteur ps dans la d6finition de ~G. 4. Toujours avec les m6mes notations, Gk s'identifie d'une mani&e ~vidente avec un sous-groupe discret de GA; on dit que c'est le groupe des ad61es principaux de G. O n s'est aper~;u, dans ces derniers temps, qu'une bonne partie des rdsultats les plus importants de l'arithm6tique classique pouvait s'exprimer en 6 n o n , a n t des propri6t6s de GA/Gk pour des groupes alg6briques G convenables; cette observation, faite d ' a b o r d par Chevalley, c o m m e chacun sait, pour le cas particulier du groupe des id~les (groupe Gin), est d'une importance capitale; le m6rite semble en revenir principalement "~ O n o et Tamagawa. Le cas off G est commutatif est celui de la th6orie classique des corps de nombres alg6briques; celui off G est le groupe orthogonal est celui de la th6orie des formes quadratiques; il semble donc qu'on touche au m o m e n t off ces deux th6ories, confondues/t leurs d6buts (la th6orie des formes quadratiques binaires ne diff6rant pas de celle des corps quadratiques),

[1959a]

403 Ad&les et groupes alg6briques

vont enfin se fondre de nouveau en une seule, 5. savoir la thdorie arithm6tique des groupes alg6briques. I1 n'est pas difficile de ddmontrer (cf. O n o [1]) que GA/Gk est compact lorsque G est unipotent. I1 en est de marne, d'autre part, pour certains groupes semisimples; en voici deux cas typiques: a. G est le groupe des 616ments de norme 1 sur le centre darts une varidt6 de corps non commutatif; b. G est le groupe orthogonal d'une forme quadratique ne repr6sentant pas 0. Marne lorsque GA/Gk n'est pas compact, il peut arriver que cet espace soit de mesure finie (pour une mesure de Haar quelconque sur GA); it semble plausible qu'il en soit ainsi pour les m~mes groupes dont on a conjectur6 plus haut qu'ils ont la propri6t6 de convergence relative. En tout cas, au moyen de la r6duction des formes quadratiques, on a pu ddmontrer cette propri6t6 pour un grand nombre de groupes semi-simples, et Ramanathan a annonc6 qu'il poss6dait une d6monstration s'appliquant fi tous les groupes semi-simples "classiques". Pla~ons-nous dans le cas off G a la propri6t6 de convergence; nous poserons: r(G) = ~

flc,,

d GA/Gk

et nous appellerons z(G), lorsqu'il est fini, le nombre de Tamagawa de G. Le m6rite essentiel de T a m a g a w a consiste 5- avoir ddfini T(G), d'abord dans le cas particulier ot~ G est le groupe orthogonal d'une forme quadratique (sur le corps des rationnels, puis sur un corps de nombres alg6briques), et 5- avoir reconnu les fairs suivants: a. P o u r ce groupe (plus pr6cisdment, pour la composante connexe de l'616ment neutre darts ce groupe), on a r(G) = 2; b. La formule T(G) = 2 est enti6rement 6quivalente 5. l'ensemble des r6sultats des trois c616bres m6moires de Siegel sur les formes quadratiques (C. L. Siegel, Uber die analytische Theorie der quadratischen F o r m e n [2], soit 194 pages, qui ne contiennent m~me pas une d6monstration compl6te pour le cas g6n6ral des formes de signature quelconque sur un corps quelconque). En fait, une lois qu'on a eu l'idde d'exprimer les choses dans ce langage, l'6quivalence de la formule z(G) = 2 avec les r6sultats de Siegel n'est pas trop difficile 5. vdrifier. Naturellement, T a m a g a w a ne s'est pas arr6t6 l~t. I1 a d ' a b o r d cherch6 5. r6diger, darts ce m~me langage, la d6monstration m~me du th6or6me de Siegel (l'id6e de traduire celle-ci dans le langage des id61es 6tait ddjs- venue 5- M. Kneser il y a quelques ann6es, mais celui-ci n'avait rien publi6 sur ce sujet). Un avantage essentiel de la nouvelle mdthode consiste en ce que le thdor6me 5-d6montrer est, du fair marne de son 6nonc6, birationnellement invariant, alors que chez Siegel (et, avant lui, chez Minkowski) il apparaissait comme lid 5. un "genre" de formes quadratiques. En particulier, en vertu des isomorphismes bien connus entre groupes classiques, les cas n = 3 et n = 4 se ram6nent ainsi ~t des questions analogues sur les alg6bres de quaternions, qu'on sait traiter directement; or c'6tait justement les cas qui donnaient le plus de difficult6 chez Siegel. Ces cas 6tant acquis, tout le reste de la d6monstration peut maintenant se pr6senter fort simplement, et presque sans calculs. D'autre part, on a commenc6 "aexaminer d'autres groupes semi-simples: groupe

404

[1959a ] Adeles et groupes algdbriques

spin (Tamagawa), groupes lin6aire sp6cial et projectif sur une alg6bre simple, etc. D a n s t o u s l e s cas q u ' o n a su traiter, on a trouv6 que ~(G) est un entier, et qu'il est dgal ~ 1 lorsque G est un groupe semi-simple "simplement connexe" (au sens alg6brique, c'est-g,-dire que tout g r o u p e isog6ne ~ G est une image h o m o m o r p h e de G). I1 en est bien ainsi, p a r exemple, si G est le g r o u p e des 616ments de n o r m e 1 sur le centre dans une vari6t6 d'alg6bre simple sur un corps de n o m b r e s ou bien sur un corps de fonctions.

Bibliographie 1. Ono, Takashi. Sur une propri6t6 arithm6tique des groupes alg+briques commutatifs, Bull. Soc. math. France, t. 85, 1957, p. 307 323. 2. Siegel, Carl Ludwig. Ober die analytische Theorie der quadratischen Formen, I : Annals of Math., t. 36, 1935, p. 527 606; II: t. 37, 1936, p. 230 263; III: t. 38, 1937, p. 212 291. 3. Tamagawa, Tsuneo, M6moire ',i para~tre aux Annals of Mathematics.

[1959b] Y. Taniyama (lettre d'Andr~ Weil) II est impossible, pour un math6maticien fran~ais de m o n ~ge, d'6crire sur T a n i y a m a sans songer aussit6t "~ Herbrand. Celui-ci aussi restera dans la m6moire de ceux qui l'ont connu c o m m e l'une des plus fortes personnalit6s parmi les math6maticiens de sa g6n6ration. Herbrand est mort h 23 ans dans un accident de montagne; au dire de ses compagnons, la prudence n'6tait pas en montagne sa qualit6 dominante; il semble que la vie n'importe plus beaucoup h ceux qui ont franchi certaines fronti6res de l'intelligence. Pour quiconque a connu Herbrand, il est difficile de croire que, s'il avait v6cu, notre math6matique, et particuli6rement notre th6orie des nombres, aurait tout h fait l'aspect qu'elle pr6sente aujourd'hui; il 6tait de ceux dont on attend, non seulement qu'ils r6solvent tel ou tel probl6me avant les autres, mais qu'ils enrichissent la science d'id6es que d'autres n'auraient point. T a n i y a m a aussi 6tait de ceux-l~. Sa personnalit6 rut pour moi l'une de celles qui domin6rent le colloque de T o k y o - N i k k o en 1955, et dont je conservai la plus forte impression a la suite du s6jour au Japon que je fis ~ cette 6poque; s'il 6tait clair qu'il s'y m61ait des 616ments dissonants, en conflit les uns avec les autres et avec le m o n d e ext6rieur, il n'y avait pas de raison de voir 1~ autre chose que le bouillonnement d'un temp6rament jeune qui n'a pas encore r6alis6 son harmonie interne; et j'attendais beaucoup, pour moi au moins autant que pour lui, du s6jour qu'il avait 6t6 invit6 ~ faire "~ Princeton. Je n'ai pas besoin de dire ici m o n 6motion et m o n chagrin quand j'ai appris queje ne devais plus le revoir. Du moins, plus heureux q u ' H e r b r a n d (dont le nom, en dehors de son oeuvre logique, reste attach6 seulement ~ deux ou trois r6sultats tr6s fins, tr6s en avance sur leur 6poque, et qui n'ont trouv6 leur explication que r6cemment), T a n i y a m a nous a laiss6 un travail de premier ordre, compl6tement achev6 dans le cadre qu'il s'6tait fix6, mais qui ouvre de vastes perspectives sur les plus importants probl6mes de l'arithm6tique moderne; c'est bien entendu de son m6moire de 1957 sur les fonctions L, paru darts le Journal of the Mathematical Society of Japan, que j'entends parler; trop modestement, il dit qu'il y suit "rues m6thodes," alors qu'en r6alit6, prenant pour point de d6part quelques observations que j'avais eu l'occasion de faire, et les joignant "~ ses propres r6sultats, il y d6veloppe des m6thodes enti6rement neuves et d'une grande port6e. Sans entrer ici dans des commentaires d6taill6s, j'observerai seulement que, pour bien apercevoir les id6es directrices du m6moire, il convient de commencer par le dernier chapitre. Lh il est montr6, au moyen de la r6duction modulo p, que, si A est une vari6t6 ab61ienne d6finie sur un corps de hombres alg6briques k, les repr6sentations l-adiques du groupe de Galois de k sur k (off/~ est la cl6ture alg6brique de k), d6termin6es par les groupes de points d'ordre / N sur A, ne sont pas ind6pendantes les unes des autres, mais sont li6es entre elles et avec la fonction z6ta de A par des relations tr6s pr6cises; et l'id6e peut-6tre la plus originale de T a n i y a m a a 6t6 de voir que ces relations, telles qu'il les formule, m6ritent d'fitre 6tudi6es en elles-m~mes, ind6pendamment de la vari6t6 A d'ofi

405

406

[1959b] Y. Taniyama on les a tirees. C'est cette etude qu'il a menee ~t bien, par une analyse des plus fines et ingenieuses, dans le cas "abdlien" off les representations/-adiques en question engendrent des algebres commutatives (cas qui se presente justement dans la multiplication complexe); ses resultats comprennent comme cas particulier ses thdoremes anterieurs sur les fonctions zeta des varietes abeliennes fi multiplication complexe; en meme temps, il 6claire par lfi d'un jour tout nouveau la theorie des caracteres "de type (A0)" dont j'avais seulement signal6 l'existence au cours du colloque de Tokyo-Nikko. Mais c'est bien entendu le cas non abelien dont l'etude l'attirait, sans qu'il air pu, semble-t-il, rien entreprendre ~ ce sujet; il est inutile d'ajouter que ce probleme ne semble abordable par aucune des methodes dont nous disposons actuellement. Pendant longtemps encore, sans doute, ce travail fournira des sujets de meditation aux arithmeticiens. J'ignore, au m o m e n t off j'dcris, s'il s'est retrouv6 parmi les papiers de T a n i y a m a des traces de recherches plus recentes. Mais, quand meme il n'en serait pas ainsi, son memoire de 1957 suffirait fi lui assurer dans l'histoire de notre science une place durable. A ceux qui l'ont connu personnellement, il laisse, avec un inoubliable souvenir, l'amer regret de n'avoir rien su ou rien pu faire pour le retenir parmi eux.

[ 1960a] De la m&aphysique aux math6matiques (h propos d' un colloque re'cent)

Les math6maticiens du XVlIIe si~cle avaient coutume de parler de la (( m6taphysique du calcul infinit6simal )), de la (( mdtaphysique de la thdorie des ~quations )). Ils entendaient par lh u n ensemble d'analogies vagues, difficilement saisissables et difficilement formulables, qui n~anmoins leur semblaient jouer u n r61e i m p o r t a n t hun m o m e n t donn6 dans la recherche et la d~couverte math~matiques. Calomniaient-ils la (( vraie )) mdtaphysique en e m p r u n t a n t son n o m pour d6signer ce qui, dans leur science, 6tait le moins clair? Je ne chercherai pas h 61ucider ce point. E n tout cas, le mot devra 6tre e n t e n d u ici en leur sens; h la (( vraie )) mdtaphysique, je me garderai bien de toucher. Rien n'est plus f6cond, tousles mathdmaticiens le savcnt, que ces obscures analogies, ces troubles reflets d ' u n e th6orie h une autre, ces furtives caresses, ces brouilleries inexplicables; rien aussi ne donne plus de plaisir au chercheur. U n jour v i e n t off l'illusion se dissipe; le pressentiment se change en certitude; les th6ories jumelles r6v~lent leur source commune a v a n t de disparaitre; comme l'enseigne la G~tgt on a t t e i n t h la connaissance et h l'indiffdrence en m~me temps. La m6taphysique est devenue math~matique, prate h former la mati~re d ' u n trait6 dont la beaut6 froidc nc saurait plus nous 6mouvoir. Ainsi nous savons, nous, ce que cherchait ~ deviner Lagrange, q u a n d il parlait de mdtaphysique h propos de ses t r a v a u x d'alg~bre; c'est la thdorie de Galois, qu'il touche presquc du doigt, h travers u n 6cran qu'il n'arrive pas k percer. Lh off Lagrange voyait des analogies, nous voyons des th6orbmes. Mais ceux-ci ne p e u v e n t s'dnoncer q u ' a u moyen de notions et de (c structures )) qui pour Lagrange n'dtaient pas encore des objets math6matiques : groupes, corps, isomorphismes, automorphismes, tout cela avait besoin d'&tre con~u et d6fini. T a n t que Lagrange ne fair que pressentir ces notions, r a n t qu'il s'efforce en vain d'atteindre h leur unit6 substantielle h travers la multiplicitd de leurs incarnations changeantes, 52

Reprinted by permissionof Hermann, 6diteurs des sciences et des arts

4O8

[1960a]

409 mat

h~matiques

il reste pris dans la m6taphysique. Du moins y trouve-t-il le fil eondueteur qui lui permet de passer d ' u n probl~me ~ l'autre, d'amener les mat6risux k pied d'oeuvre, de tout mettre en ordre en vue de la th6orie g6n6rale future. Grace h la notion d6eisive de groupe, tout eela devient math6matique ehez Galois. De m~me eneore, nous voyons les analogies entre le ealeul des diff6renees finies et le ealeul diff6rentiel servir de guide h Leibniz, h Taylor, h Euler, au eours de la p6riode h6roique durant laquelle Berkeley pouvait dire, avee autant d'humour que d'h-propos, que les ~ eroyants ~ du ealeul infinit6simal 6taient peu qualifi6s pour eritiquer l'obseurit6 des myst~res de la religion ehr6tierme, eelui-l~ 6t~nt pour le moins aussi plein de myst~res que eelle-ei. Un peu plus t~rd, d'Alembert, ennemi de toute m6taphysique en math6nmtique eomme ailleurs, soutint dans ses articles de l'Encyclop~die que la vraie m6taphysique du ealeul infinitesimal n'6tait pas autre chose que la notion de limite. S'il ne tira pas lui-m~me de eette id6e tout le patti dont elle 6tait susceptible, les d6veloppements du si~ele suivant devaient lui donner raison; et rien ne saurait 8tre plus elair aujourd'hui, ni, il faut bien le dire, plus ermuyeux, qu'un expos6 correct des 616ments du ealeul diff6rentiel et int6gral. Heureusement pour les ehereheurs, h mesure que les brouillards se dissipent sur un point, e'est pour se reformer sur un autre. Une grande pattie du eolloque de Tokyo s'est d6roul6e sous le signe des analogies entre la th6orie des nombres et la th6orie des fonetions alg6briques. Lh, nous sommes eneore en pleine m6taphysique. C'est de ees analogies, paree que j'en ai quelque exp6rienee personnelle, que je voudrais parler iei, avee l'espoir, vain peut-~tre, de donner aux leeteurs ~ honnStes gens ,~ de eette revue quelque id6e des m6thodes de travail en math6matique. D~s l'enseignement 616mentaire, on fait voir aux 61~ves que la division des polynSmes (~ une variable) ressemble beaueoup h la division des entiers et eonduit des lois toutes semblables. Pour les uns eomme pour les autres, il y a un plus grand eommun diviseur, dont la d6termination se fait par division sueeessive. A la d6eomposition des hombres entiers en faeteurs premiers correspond la d6eomposition des polynSmes en faeteurs irr6duetibles; aux nombres rationnels correspondent les fonetions rationnelles, qui, elles aussi, peuvent toujours se mettre sous forme de fractions irr6duetibles; eeUes-ei s'ajoutent par r6duetion au plus petit eommun d6nominateur, ete. I1 est done tout naturel de penser qu'il y a analogie entre les hombres alg~brlques (raeines d'6quations dont les eoeffieients sont des nombres entiers) et les ]onctions algdbriques d'une variable (raeines d'6quations dont les eoeffieients sont des polynSmes ~ une variable). Le fondateur de la doute 6t6 Galois s'fl qu'on trouve sur ee mort, d'ofl on peut

th6orie des fonetions alg6briques d'une variable aurait sans avait v6eu; e'est ee que permettent de penser les indications sujet dans sa e61~bre lettre-testament, 6erite k la veille de sa eonelure qu'il touehait d6jk h quelques-unes des prineipales 511

410

[1960a1 m

a

t

h

~

m

a

t

i

q

u

e

s

d~eouvertes de Riemann. Peut-6tre aurait-il donnd h cette thdorie une allure algdbrique, conforme ~ l'esprit des t r a v a u x contemporains d ' A b e l et de ses propres reeherches d'alg~bre pure. A u eontraire, Riemann, l'un des moins alg6bristes sans doute parmi les grands math6maticiens du xIxe si~ele, m i t la th6orie sous le signe du (( t r a n s c e n d a n t , (mot qui, pour le math6matieien, s'oppose t~ (calg6brique ,, et ddsigne t o u t ee qui a p p a r t i e n t en propre au continu). Les mdthodes tr~s puissantes mises en oeuvre p a r R i e m a n n amen~rent presque du premier coup la th6orie un degrd d'aeh~vement qui n ' a gu~re 6t6 d6pass6. Mais elles ne tiennent aucun eompte des analogies avee les hombres alg6briques, et ne p e u v e n t 6tre transposdes telles quelles en vue de l'6tude de eeux-ci, 6rude qui relive traditionnellement de l'arithmdtique ou th6orie des hombres, et qui, du v i v a n t ddjh de Riemann, 6tait en voie de d6veloppement rapide. C'est Dedekind, ami intime de Riemann, mais algdbriste eonsomm6, qui d e v a i t le premier tirer p a r t i des analogies en question et en faire un instrument de recherche. I1 appliqua avec succ~s, aux probl~mes trait6s p a r R i e m a n n p a r voie transeendante, les mdthodes qu'il a v a i t lui-m~me cr6~es et raises au point en vue de l'6tude arithm6tique des nombres alg6briques; et il fit voir qu'on peut retrouver ainsi la partie p r o p r e m e a t alg6brique de l'oeuvre de Riemann. A premiere vue, les analogies ainsi mises en 6videnee restaient superficielles, et ne paraissaient pas pouvoir porter sur les probl~mes les plus profonds de l'une ni de l ' a u t r e thdorie. H i l b e r t alia plus loin dans cette voie, ~ ce qu'il semble; mais, s'il est probable que ses 61~ves subirent l'influence de ses iddes sur ce sujet, il n'en est rest~ quelque trace que dans un eompte rendu obscur qui n'a m~me pas dt6 reproduit dans ses (Euvres complktes. Les lois non dcrites de la math6matique moderne interdisent en effet de publier des r u e s mdtaphysiques de eette esp~ce. Sans doute est-ce mieux ainsi; a u t r e m e n t on serait accabl6 d'articles encore plus stupides, sinon plus inutiles, que tous ceux qui encombrent h prdsent nos pdriodiques. Mais il est dommage que les iddes de H i l b e r t n'aient 6t6 ddveloppdes par lui nulle part. I1 y a v a i t loin encore, cependant, de l'arithm6tique, off r~gne le discontinu, h la th~orie des fonctions au sens classique. Or, en disant que les fonetions algdbriques sont raeines d'dquations dont les coefficients sont des polyn6mes, j ' a i volontairem e n t omis un point i m p o r t a n t : ces polynSmes eux-m6mes ont des coefficients; mais eeux-ci, quels sont-ils? Lorsqu'on traite de la division des polyn6mes darts l'enseignement 616mentaire, il v a sans dire que les coefficients sont des ((nombres ,~ : nombres (( rdels ,~ (rationnels ou non, mais donnds en tout eas, si l'on veut, p a r un d6veloppement d~eimal), ou, h u n niveau un peu plus 61ev6, nombres(( r6els ou imaginaires ,, ou, comme on dit, ((nombres complexes ~. C'est exclusivement de nombres complexes qu'il s'agit dans la th6orie riemannienne. Mais, du point de vue de l'alg6briste pur, t o u t ee qu'on demande a u x (( hombres ~ en question, c'est qu'ils se laissent combiner entre eux au moyen des quatre op6rations (ce que l'algdbriste exprime en disant qu'ils forment un ,~ corps ,~). Si on 54

[1960a]

411 m

a

t

h 6 m a t i

q

u

e

s

n ' e n suppose pas plus sur leur eompte, on obtient une thdorie des fonetions alg6briques, fort riche ddj~t (comme en t6moigne le volume r6cent et ddj~ classique q u ' a publi6 Chevalley sur ce sujet), mais qui ne l'est pas assez pour que les analogies avec les nombres algdbriques puissent btre poursuivies j u s q u ' a u bout. Heureusement il s'est trouv6 u n domaine intermddiaire entre l'arithmdtique et la thdorie riemannienne, et qui possbde, avec chacune de ces deux derni~res thdories, des ressemblances beaucoup plus 6troites qu'elles n ' e n ont entre elles; il s'agit des fonctions algdbriques (( sur u n corps fini )). Comme on le savait depuis Gauss, s'il ne s'agit que de pouvoir faire les quatre opdrations, il suffit d ' u n nombre fini d'dldments. I1 suffit par exemple d'en avoir deux, q u ' o n nommera 0 et 1, et pour lesquels on posera par convention la table d'addition et la table de multiplication que voici : 0+0=0, 0+1=1+0=1, 1+1=0; 0 • 0=0, 0 • 1~1 • O~ 1 X 1=1. Quelque paradoxale que puisse paraltre au profane la rSgle 1 -4- 1 = 0, quelque t e n t a n t qu'il soit de dire que c'est 1~ u n pur jeu de l'esprit qui ne rdpond ~ aueune (( rdalitd ,, u n tel syst$me est monnaie eourante pour le mathdmaticien; et Galois en ~tendit beaucoup l'usage en construisant les ~( imaginaires de Galois ,. P r e n a n t done les coefficients de nos polynSmes dans u n (~ corps de Galois ~, on construit des fonetions alg6briques dont la thdorie remonte ~ Dedekind mais s'est particuli~rement d~velopp6e depuis la th~se d'Artin. Pour dire en quoi elle consiste, il faudrait entrer darts des d~tails beaucoup trop techniques qui n ' a u r a i e n t pas leur place icio Mais on peut, je erois, en donner une idde imagde en disant que le mathdmatieien qui 6tudie ces probl~mes a l'impression de d~chiffrer une inscription trilingue. Darts la premiere colonne se trouve la thdorie riemannienne des fonctions alggbriques au sens classique. La troisi&me colonne, c'est la th$orie arithmdtique des nombres algdbriques. La eolonne du milieu est celle dont la ddeouverte est la plus rdeente; elle eontient la thdorie des fonetions alg6briques sur u n corps de Galois. Ces textes sont l'unique source de nos connaissances sur les langues dans lesquels ils sont dcrits; de ehaque colonne, nous n'avons bien entendu que des fragments; la plus complete et celle que nous lisons le mieux, encore ~ prgsent, c'est la premi&re. Nous savons qu'il y a de grandes diff6rences de sens d ' u n e colonne ~ l'autre, mais rien ne nous en avertit ~ l'avance. A l'usage, on se fait des bouts de dictionnaire, qui permettent de passer assez souvent d'une colonne ~ la colonne voisine. C'est ainsi q u ' o n avait ddchiffrd depuis longtemps, dans la derni~re colonne, le ddbut d ' u n paragraphe intituld (~ fonetion z6ta ~). Vers la fin de ee paragraphe, on croit lire une phrase tr~s mystdrieuse; elle dit que t o u s l e s zdros de la fonction se t r o u v e n t sur une certaine droite. Jamais on n ' a pu savoir s'il en est bien ainsi, ou s'll y a eu erreur de lecture. C'est le c~l~bre probl~me de 1' (( hypoth~se de R i e m a n n )), qui dans quelques mois sera tout juste centenaire. 55

412

[1960a1 ~

at

h ~ m a ~

q

ue~

La prineipale ddcouverte d'Artin, dans sa thtse, c'est qu'il y a, dans la seconde colonne, un paragraphe intittd6 aussi (( fonetion z6ta ~,, et qui est t~ peu de chose pros une traduction de celui qu'on connaissait d6jh; notre dietionnaire s'en est trouv6 beaueoup enrichi. Artin apergut aussi, dans eette colonne, la phrase sttr l'hypoth~se de Riemann; elle lui parut tout aussi mystArieuse que rautre. Ce nouveau probl~me, t~ premi$re rue, ne semblait pas plus faerie que le pr6e~dent. Ign rfiahtd, nous savons maintenant que la premitre colonne contenait ddjh tous les 616ments de sa solution. II n'dtait que de traduire, d'abord en th~orie (( abstraite ~ des fonctions algdbriques, puis dans le langage (( galoisien ~ de la seconde eolonne, des rfisultats obtenus depuis longtemps par Hurwitz en (( riemarmien ,,, et que les g~om~tres italiens avaient ensuite traduits dans leur propre langage. Mais les meilleurs sp6eiahstes des th6ories arithm~tique et (( galoisienne ~ ne savaient plus lire le riemannien, ni h plus forte raison l'italien; et il fallut vingt arts de recherches avant que la traduction f6t rnise au point et que la d6monstration de l'hypoth~se de Riemann dans la seeonde eolonne ffit complttement ddehiffr6e. Si notre dietionnaire 6tait suftisamment c0mplet, nous passerions aussitx3t de I~ la troisi~me colonne, et l'hypoth~se de Riemann, la vraie, se trouverait d~montrde, ere aussi. Mais nos cormaissanees n'atteignent pas jusque lh; bien des d~ehiffrements patients seront encore ndcessaires avant que la traduction puisse 6tre faite. Au tours du colloque auquel il a $t6 fair allusion plus haut, il a 6t6 beaucoup diseut6 de (( m~taphysique ~ tr propos de ces probltmes; un jour eelle-ci fera place h une thdorie mathdmatique dans le cadre de laquelle ils trouveront leur solution. Peutgtre, comme c'6tait le cas pour Lagrange, ne nous manque-t-il, pour franehir ee pas d6cisif, qu'une notion, un concept, une (( structure ~. D'ing6nieux philologues ont bien trouv~ le secret des archives de Nestor et de celles de Minos. Combien de temps faudra-t-il encore pour que notre pierre de Rosette, h nous autres arithm~tieiens, rencontre son Champollion?

56

[ 1960b] Algebras with involutions and the classical groups

IT has been knowr~ for a long time t h a t there is a close connection between semisimple algebras with involutions and the classical semisimple Lie groups and Lie algebras. B u t the precise degree of generality of this relationship does not seem to have been ascertained anywhere, at least explicitly, in the printed literature on this subject. I n the first part of the present paper, it will be shown how this can be done, at a n y rate over a groundfield of characteristic 0, b y borrowing some elementary techniques from m o d e r n algebraic geometry. Then, taking for our groundfield the field R of real numbers, we shall give, for the l~iemannian s y m m e t r i c spaces a t t a c h e d to the classical groups, an i n t e r p r e t a t i o n which rests u p o n the use of algebras with involution. This was already implicit in Siegel's f u n d a m e n t a l work on discontinuous groups, and it is hoped t h a t our results will help to achieve a b e t t e r u n d e r s t a n d i n g of t h a t work and to clarify also some of its arithmetical aspects.

PART

I

~EMISlMPLE GROUPS OVER A FIELD OF CHARACTERISTIC 0. 1.

I n P a r t I, all spaces, varieties, groups are to ba understood

in the sense of algebraic geometry; b y this we mean t h a t t h e y are all allowed to have points in the universal domain, and not only in their field of definition. I f V is a v a r i e t y (for instance a group), t Work.supported (in part) by the O.U.I%.P.A.F. The author is greatly indebted to his colleague Mr. M. for a counterexample and other helpful remarks, to Mr. P. (the famous winner of many cocycle races) for the main idea of Part I, to others for conversations totally unrelated to the problems considered here, and to the Indian iV~atheraatieal Society for kindly allowing this paper to rest for over two years in their editorial officesbefore letting it take its flight into the world. Reprinted by permission of the editors of J.

Ind. Math. Soc.

413

414

[1960b] 5'90

ANDRE WEIL

d e f n e d over a field k, we shall denote b y Vk the set of points of V with coordinates in k (in group-theory, this differs from the usage of Chevalley, according to which a group G, defined over an infinite field/~, is always identified with the set which we call G~). We assume the universal domain to be of characteristic 0 ; without restricting the generality, one could take it to be the field of complex numbers. Within the universal domain, we select a groundfield k and a normal algebraic extension K of k, of finite degree d; we call g the Galois group of K over /~, and we write ~ for the imago of an element ~ of K u n d e r an automorphism a ~ g ; we have (~a), = ~a~ for all a, ~ in g. I f V is a vector-space of dimension n (over the universal domain), defined over k, Vk and VK are vector-spaces of dimension n over k and over K, respectively. We have Vk a VK, and we m a y identify VK with the tensor-product V~ | K t a k e n over k; g operates in an obvious m a n n e r on VK, and Vk consists of the elements of VK which are invariant under g. I f V' is the dual space of V (over the universal domain), and if T is a n y tensor-product (V | V | ...) | | ...), over the universal domain, of f~etors identical either with V or with V', t h e n T~ is the tensor-product similarly built up from V~ and its dual V~ over/~; and a similar s t a t e m e n t holds for T K. B y a c o c y d e , we shall u n d e r s t a n d a mapping a - + $'~ of g into the group of automorphisms of the vector-space V, such t h a t : (a) for each a E g, Fo is defined over K ; (b) for all a, ~ in ~, we have Fa~ = ( F o y o F~. Let (Fo) be such a cocycle; for e v e r y x ~ VK, p u t :

O n e s e e s at once t h a t x [~] = (xt~])[~], which means t h a t the group g can also be made to operate on VK b y (x, a ) - + x E~ These operations are k-linear, b u t not K-linear; we have (~x) t~] = ~~ t~] for x e VK and ~ e K. F r o m this, it follows t h a t the set W of those elements of VK which are invariant under all operations x - + x ["], i.e. which satisfy x ~' = F o ( x ) for all a, is a vector-space over k. I f ql . . . . . a~ are linearly independent over k in W, it is easy to see, b y induction on m, t h a t t h e y are linearly independent over K in

[1960b]

415 ALGEBRAS W I T H

INVOLUTIONS

59l

VK; for otherwise, because of the induction assumption, there would be a relation E ~ i a ~ = 0 , with the ~ in K and not all in k, a n d ~m = 1 ; a p p l y i n g to this the operation x - + x [~], we get Z i ~ a~ = 0, hence Z i ( ~ - - ~) a~ = 0, and therefore, because of the induction assumption, ~ = a s s u m p t i o n on the a~. Now k; t a k e a basis (~1 . . . . . ~ ) elements Z~ ~ x [~l are in W ~

~ x [~]

c:~B

~i for all i a n d ~, which contradicts the t a k e for al, ..., a m a basis for W o v e r of K over }; for e v e r y x e V~, the for i = 1 . . . . . d, so t h a t we can write : ~

(1~i

d)

~=1

with % ~ k. I t is w e l l - k n o w n t h a t the d e t e r m i n a n t [ ~ [ is not O, and therefore these equations can be solved for the x ["1, yielding expressions for t h e m as linear combinations of the a with coefficients in K ; in particular, x itself is such a linear combination. We h a v e t h u s shown t h a t a~ . . . . , am is a basis for IrK o v e r K , so t h a t in particular we m u s t h a v e m = n. T a k e now a basis b1.... , b, of Vk over k ; let (I) be the a u t o m o r p h i s m of V which m a p s bi onto a i for 1 ~< i ~< n ; it is defined over K. Combining the relations a i = O(b~), a~ : F~(ai), b~ = bi, we get $'~(O(bi) ) = O~(b~) for all i, hence F~ o 9 : 9 ~ or F~ = (I)~ r (which can be expressed b y saying t h a t "all eocycles are trivial " ). N o w lot t, t', ... be elements of " t e n s o r - s p a c e s " T, T ' , .... i.e. of tensor-products built u p f r o m factors identical with V or V'. More precisely, a s s u m e that t e T k, t' ~ T ' k, . . . , and that all the teasers t, t', ... are i n v a r i a n t u n d e r every one o f the a u t o m o r p h i s m s $',; b y this we m e a n of course t h a t t is i n v a r i a n t under the canonical extension of F , to T, etc. Similarly, write (I) for the canonical extension of ~) to T, T ' , ... and p u t tl = (I)-l(t), etc. We have: t~ = ( r

= r

= tl

for all a, hence t 1 e T~, and similarly t' 1 e T'~, etc. The m a i n application which we have in view concerns the case of algebras a n d of algebras with involution. B y an algebra A , we u n d e r s t a n d a vector-space V with the additional structure deter-

416

[1960b] 592

ANDRE W E I L

mined on it b y a bilinear mapping of V • V into V, or, what amounts to the same, b y an element t of the tensor-space T = V' | V' | V; if, in addition to this, we prescribe an endomorphism ~ of V, or, what amounts to the same, an element t' of T ' ~- V' | V, such t h a t is an i n v o l u t o r y antiautomorphism (or, as we shall say more briefly, an involution) of the algebra A, t h e n we speak of V, with the structure determined on it b y t and t', as the algebra with involution (A, ,). All algebras will be assumed to be associative and to have a unit-element, usually denoted b y 1. We say t h a t the algebra A, with the underlying vector-space V and the multiplicative structure determined b y the element t of V' | V' | V, is defined over k if V and t are defined over k; t h e n A k is an algebra over k in the usual sense, and A K is the algebra derived from A k b y extending the groundfield from k to K. The same holds for algebras with involution. As our universal domain is assumed to be of characteristic 0, it is known t h a t A, is somisimple if and only if A (as an algebra over the universal domain) is so. The center Z of a semisimple algebra A, defined over k, is a commutative semisimple algebra, also defined over k; Z k is then the center of A k. We say t h a t A is absolutely simple if it is simple as an algebra over the universal domain ; then it is isomorphic to a m a t r i x algebra Mn over the universal domain. On the other hand, we say t h a t a semisimple algebra A, defined over k, is simple over k if it is not a direct sum of subalgebras of A, all defined over k; this will be so ff and only ff A k is simple as art algebra over k, or also if and only if the center Z of A is simple over k, or again ff and only if Z k is a field. I t is clear t h a t the groups of automorphisms of algebras and of algebras with involution are algebraic groups. As a special case of the results p r o v e d above, we have now the following theorem : THEOREM 1. Let A be an algebra (resp. an algebra with involution) defined over a field k; let K be a Galois extension of k with the Galois group g. Let (2'0) be a cocycle of fi, consisting of automorphisms of A. Then there is an algebra (rosp. an algebra with involution) A 1, defined

[]960b]

417

ALGEBRAS WITH INVOLUTIONS

593

over k, and an isomorphism r of A 1 onto A , defined over K , such that F a ---- r o Cp-:for all (~ Gg.

In,fact, let V be the underlying vector-space of A; and let t be the tensor (resp. let t, t' be the pair of tensors) on V which defines the structure of A. Define t: and (I) (resp. t:, t~ and (I)) as above by means of the cocycle (F~); and define A 1 as the algebra (resp. the algebra with involution) defined on V by t: (resp. by tl, t~). These will satisfy all the conditions in our theorem. 2. It is our purpose to show, by means of Theorem 1, that, with few exceptions, the classical semisimple groups over any field of characteristic 0 can be represented as groups of automorphisms of semisimple algebras with involution ; in fact, it will turn out that there is almost a one-to-one correspondence between these two classes of objects. We begin by discussing the classical simple groups over the universal domain. Consider first the projective linear group PL(n) in n variables, i.e. the factor-group of GL(n) by its center (to be consistent with our notation, we omit any mention of the underlying field when the latter is the universal domain). Let A be the direct sum of two algebras, both isomorphic to the matrix algebra M~ of order n (i.e. consisting of all n • n matrices); on A, consider the involution defined by (x, Y) - + (t y, tx) ' where X , Y are two matrices of order n, and tX, t y are their transposes. Using the classical theorem of Skolem-2qoethor, one sees immediately that the automorphisms of the algebra with involution A (i.e., the automorphisms of the algebra A which commute with the given involution) make up an algebraic group G, consisting of two connected components G0, G:; GO consists of all the automorphisms of A which leave each component M , of A invariant, and more precisely of the automorphisms: (X, Y ) ---+ (X, y),(M) = ( M - 1 X M ,

tM. Y. t M - 1 )

where M is an arbitrary invertible matrix ; the mapping M - + r is then a homomorphism of GL(n) onto Go whose kernel is

418

[1960b] 594

ANDRE WEIL

the center of GL(n), so that GO may be identified with P L ( n ) . As to (71, it consists of those automorphisms of A which exchange its two components, and may also be defined as the coset of (7o in (7 which contains the automorphism (X, Y ) - + (Y, X). The inner automorphisms of G induce automorphisms on Go ~hieh are either inner automorphisms of Go or products of such automorphisms with the automorphism induced on G o by (X, Y ) - + ( Y , X); the latter may be written as r r and it is easy to see that, for n > 3, this is not an inner automorphism It is well known that these are all the automorphisms of G0 = PL(n). As our calculation also shows that no automorphism of A, other than the identity, induces the identical isomorphism on Go for n > 3, it follows that, for n > 3, every automorphism ofG 0 can be derived, in one and only one way, from an automorphism of A. This implies that, for n > 3, if A ' is an algebra with involution, isomorphic to A, and G' o is the connected component of the identity in the group of automorphisms of A', every isomorphism of G0 onto G'0 can be derived, in one and only one way, from an isomorphism of A onto A'. In order to obtain in a similar manner the orthogonal and sympleotio groups, or rather their quotients by their centers, take A = Mn; sine0 X - + t X is an involution on A, the most general antiautomorphism of A is of the form X - - + . F - I . t X . . F , which is involutory if and only if t F = h F , with ~ in the center; this implies ~9.= 1, so that our involution is given by X - + F -1. t X . . F with F invertible and t.F - 4- .F. An automorphism X - + M - 1 X M of the algebra A commutes with that involution if and only if .F = t M . . F . M ; let G be the group consisting of such automorphisms. I f t $ , = _ F, n must be even, and the matrices M satisfying .F -= t . M . F . M are of determinant 1 (as follows from the consideration of the pfaffian) and make up a connected algebraic group, the symplectic group Sp(n) ; G is the quotient PSp(n) of that group by its center. It is known that in that case (7 has only inner automorphisms ; and we see, just as above, that every automorphism of G

[1960b]

419 ALGEBRAS W I T H I N V O L U T I O N S

595

can be derived in one a n d only one w a y f r o m an a u t o m o r p h i s m of the given algebra with involution. T a k e now the case tF ~ F ; as our underlying field is the universal domain, it would be no restriction to t a k e for F the u n i t - m a t r i x I n. The matrices M for which F = t M . F . M m a k e u p the o r t h o g o n a l group O(n), with two connected c o m p o n e n t s O + ( n ) , O - ( n ) consisting of the matrices M in the group with the d e t e r m i n a n t + I a n d - - 1 , respectively; a n d the connected c o m p o n e n t GO of the identity in 0 is the quotient PO+(n) of O+(n) b y its center. [f n is odd and ~ 3, the center of O(n), consisting of :t: 1~, h a s one element in each one of those components; therefore G is connected v~nd m a y be identified with O+(n). I t is k n o w n that, also in this c~se, G has only inner a u t o m o r p h i s m s , a n d our f u r t h e r conclusions are the same as before. Finally, t a k e the case in which n is even a n d ~ 4. T h e n the center of O(n), consisting again of 4- ln, is contained in O + ( n ) ; therefore G has two connected components. Hero it is k n o w n t h a t the g r o u p of inner a u t o m o r p h i s m s of G o is of index 2 in the g r o u p of all a u t o m o r p h i s m s of Go, e x c e p t for n =: 8, in which case it is of index 6. On the o t h e r hand, it is easily seen t h a t the inner a u t o m o r p h i s m s of (7 induced by elements of G~ determine on Go a u t o m o r p h i s m s which are not inner ones of G0. Therefore, if we leave aside the exceptional ease n = 8, we can again conclude t h a t e v e r y a u t o m o r p h i s m of G O can be derived in one a n d only one w a y f r o m an a u t o m o r p h i s m of the algebra w i t h involution A. N o w observe t h a t e v e r y semisimple algebra A o v e r the universal d o m a i n is a direct sum of m a t r i x algebras, a n d t h a t e v e r y involution of A m u s t either leave a c o m p o n e n t of A i n v a r i a n t or interchange it with a n o t h e r one. Thus, o v e r the universal d o m a i n , e v e r y semisimple algebra with involution is, in an obvious senso~ the direct s u m of algebras with involution of one o f the t y p e s discussed above. On the o t h e r hand, ff Go is the connected c o m p o n e n t O f the i d e n t i t y in the g r o u p of a u t o m o r p h i s m s o f the algebra with involution A, it is clear t h a t the a u t o m o r p h i s m s in Go m u s t t r a n s f o r m each c o m p o n e n t of A into itself. F r o m this

420

[1960b] 596

ANDRE WEIL

it follows immediately t h a t GO must be a direct product of groups of the various types considered above, and t h a t conversely e v e r y such p r o d u c t can be obtained in this manner. Some groups, however, are obtained in this m a n n e r more t h a n once, because of the well-known isomorphisms between groups of the various families ; those are as follows : (a)

M2 admits an inv~176

with J = ( --10 ~ )

which is invariant under all automorphisms and antiautomorphisms of M 2 ; this follows for instance from the fact that, for any invertible m a t r i x M in M2, we have

J-l.tM.J

----dot(M). M -1.

Therefore SL(2) is identical with Sp(2), hence PL(2) with PSp(2). (b)

P O + ( 3 ) is isomorphic with PSp(2).

(c)

PO +(4) is isomorphic with the product of itself.

(d)

PO +(5)

(e) PO +(6)

PO +(3)

with

is isomorphic with PSp(4). is isomorphic with PL(4).

In view of these circumstances, let us restrict our list of groups and algebras to the following: (I) (~roups: all semisimple groups, with center reduced to the neutral element, which, when decomposed into a direct product of simple groups, contain no factor isomorphic either to one of the exceptional groups or to P0+(8). (II) Algebras with involution: all semisimple algebras with involution which, when decomposed into a direct sum, consist of summands isomorphic to one of the following : (a) M~, $ M~ for n ~ 3, with an involution exchanging the two summands ; (b) M2n for n >~ 1, with the involution X - + j - a . eX. j determined b y an invertible alternating m a t r i x J ; (c) M s with the involution X - + tX, f e r n :=- 7 or n ~ 9.

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421 ALGEBRAS WITH INVOLUTIONS

597

Then it follows from what we have p r o v e d t h a t each elm of the groups in ore. list is isomorphic to the connected component of the identity in the group of automorphisms of one of our algebras with involution, and that, if A and A' are two such algebras and G and G' are the corresponding groups, a n y isomorphism between G and G' is induced b y a uniquely determined isomorphism between A and A'. The latter s t a t e m e n t holds in particular for A == A', (7 -- (7'. 3. L e t (7 be a connected algebraic group, defined over the groundfield 1~. Lot us assume t h a t (7 is semisimplo, has a center reduced to the neutral element, and that, when G is decomposed into a p r o d u c t of simple factors over the universal domain, none of these factors is isomorphic to one of the five exceptional groups or to PO+(8). F r o m the results ill w2 it follows t h a t there is an algebra with involution A0, defined over the prime field, such t h a t the connected component of the identity Go in the group of automorphisms o f A 0 is isomorphic to G over the universal domain. Let f be an isomorphism of (7 onto Go; take for K a normal algebraic extension of/c, of finite degree, over which f is defined. Notations being now the same as in w1, f~ o f - 1 is an automorphism o f Go, defined over K. B y the results of w2, there is a uniquely determined automorphism $'~ of A o which induces the automorphism f " o f - 1 on Go; it is therefore invariant u n d e r all automorphisms of the universal domain over K ; the characteristic being 0, this implies t h a t it is defined over K. I t is obvious t h a t (F~) is a cocycle. Therefore, b y T h e o r e m 1, there is an algebra with involution A d e f n e d over b, and an isomorphism O, defined over K, of A onto A 0, such t h a t F , = (Pr (I)-1 for all a. Let G' be the connected component of tlm identity in the group of automorphisms of A ; r determines an isomorphism, which we again call (I), of G' onto G o ; t h e n ~ ----(I)-1 o / i s an isomorphism, defined over K, of (7 onto G'. Moreover, we have (I)~ o r ___f~ o f - 1 for all a. This can also be written as ~ = ~ . Therefore ~ is defined over b, and we m a y use it to identify G with G'. We have thus proved t h a t any group G

of the given type can be represented as the connected component of the

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identity i~ the group of automorphisms of a 8emisimple algebra with involution defined over k. 4. We shall now determine when the connected component Go of the identity in the group of automorphisms of a semisimple algebra with involution (A, ~) is not a semisimple group. Over the universal domain, this is immediately a p p a r e n t from the the results of w3. I n fact, P L ( n ) is simple except for n = 1, in which ease it is reduced to 1; PSp(n) is defined only if n is even, and is always simple; and P O + (n) is semisimple except for n - - 1 and n = 2. Therefore Go is semisimple, and has a center reduced to the identity, provided no component of (A, ~) is isomorphic to M~ with the involution X - + tX. We shall say t h a t a semisimple algebra with involution is non-degenerate if it has no such component and if at the same time it has no c o m m u t a t i v e component. Then we have the following t h e o r e m : THEOREM 2. Let (A, ~) be a non-degenerate semisimple algebra with involution; let Go be the connected component of the identity in its group of automorphisms, and let Uo be the connected component of 1 in the multiplicative group of the elements u of A such that u'u----1. Then Go is a semisimple group, isomorphic to the quotient of Uo by its center; and the center of Go is reduced to the identity. We have already shown t h a t Go is semisimple a n d has a center reduced to the identity. As the group of automorphisms of the center Z of A is finite, e v e r y element o f Go must induce the i d e n t i t y o n Z and is therefore (by the classical theorem of Skolom-Noothor) a n inner automorphism x--+ v - 1 xv, with an invertible v in A. C_,allj(v) this automorphism ; ff we write t h a t it commutes with ~, we find t h a t z = v'v m u s t be in Z; it must then be in the multiplicative group H consisting of those invertible elements o f Z which are even for ~. Call V the group of those elements v o f A for which v'v is in H, and U the group of those elements u of A for which u'u = 1. Consider the homomorphism (h, u) -+ hu o f / / • U into V; one sees at once t h a t it maps H • U onto the group of those elements v of A for which v~v is i n / / 3 ; but, over the universal domain,

[1960b]

423 ALGEBRAS W I T H

INVOLUTIONS

599

we h a v e H ~ --- H (since H is c o m m u t a t i v e a n d the characteristic is not 2), a n d therefore t h a t h o m o m o r p h i s m m a p s H • U onto V. F r o m this, it follows t h a t U a n d Y h a v e the same image j(U) = j ( V ) u n d e r j. As we h a v e seen t h a t G o is contained in j ( V ) , our conclusion follows. Incidentally, we observe t h a t the classical Cayley t r a n s f o r m a t i o n m a y be used to s t u d y the g r o u p U o of T h e o r e m 2. L e t us say t h a t an e l e m e n t x of an algebra with involution (A, ~) is even (:for ~) if x' = x, and odd (for t)if x ' = x; as the characteristic is not 2, A is the direct s u m of the spaces A +, A - of even a n d odd elements for e. N o w let u be a generic element of U 0 over a field of definition k for (A, ~) ; write w = (1 - - u). (1 + u ) - l ; t h e n w is an odd element of A for ,. Conversely, if w is a generic element of A - over ]r the formula u = (1 - - w).(1 ~- w) -1 defines a generic element o f U 0 o v e r k. Thus these formulas define a birational correspondence between U 0 and A S. E v e r y algebra with involution o v e r a field ]c can be w r i t t e n as (A~, ~), where (A, ~) is, in the sense explained above, a n algebra with involution, defined o v e r It; we h a v e already observed t h a t , if the latter is semisimple, the f o r m e r is semisimple, a n d conversely. We shall say t h a t the f o r m e r is non-degenerate if the latter is so. T h e relation between the structures of these two algebras will now be briefly discussed, particularly in order to find when (A k, ~) is non-degenerate. To decompose the semisimple algebra A~ over ]c into simple c o m p o n e n t s is the s a m e as decomposing A o v e r k, i.e. writing it a s a direct s u m o f sub-algebras, defined o v e r k, in such a w a y t h a t none o f the s u m m a n d s can be split a n y f u r t h e r in the same manner. E a c h s u m m a n d is t h e n t r a n s f o r m e d into itseff or interchanged with a n o t h e r one b y the involution L. T h u s it is enough to consider the case in which (Ak, ~) is simple, which m e a n s t h a t A~ is either simple or the direct sum of two simple algebras Bk, Uk interchanged

424

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by e. I n the latter case, let K be the center of Bk; call d the degree of K over/c; if we consider B k as a vector-space over K, its dimension must be of the form n 2. Then A splits into the direct sum of two algebras B, C, which m a y be regarded as the tensor-products o f B~ and C~ with the universal domain over ]c, and which are interchanged b y ~. Over the universal domain, B is the direct sum of d algebras isomorphic to M s. Degeneracy can only occur for n -- 1, i.e. when A~ is commutative. Next, assume t h a t A k is simple; let K be its center, and K + the set of all even elements of K for ~; K + is a field; let d be its degree over ]c. I f K is not the same as K +, it is an extension of K + of degree 2; call n 2 the dimension of A~ as a vector-space over K. Then A is, over the universal domain, the direct sum of 2d algebras isomorphic to Ms; b y considering the center of A, one sees at once t h a t none of these 2d components is invariant u n d e r ~. Therefore degeneracy occurs only for n = 1, i.e. again when A~ is commutative. Finally, assume t h a t A k is simple and t h a t ~ induces the identity on the center K of A k, so that, with the above notation, we have K = K +. Let d be the degree of K over ]c, and n ~" the dimension of A~ as a vector-space over K ; t h e n A is the direct sum of d algebras isomorphie to M~, each of which is invariant under ~; it is nondegenerate whenever n ~ 3. I f n = 1, A k is c o m m u t a t i v e and degenerate. I f n = 2, A k is a quaternion algebra over K (which m a y be isomorphic to M s (K)); let 8 be the dimension of A / a s a vectorspace over K. Then one sees at once that, in each one of the d simple components of A over the universal domain, the odd elements for e make up a vector-space of dimension 3; comparing this with the results of w2, we find t h a t we have 3 = 3 or 3 = 1 according as c, in each one of these components, is of the t y p e X ' ~ j - a . tX. j with t j = _ j or of the t y p e X ' = tX. Thus degeneracy occurs if and only if ~ : 1. One verifies easily t h a t this is so if and only if A~, as an algebra over K, can be generated b y an e v e n element u and an odd

element v such t h a t

uv =

-- vu, u ~ ~ K,

v 2 ~ K.

Thus a necessary and sufficient condition for the non-degeneracy of a simple algebra with involution (Ak, e) over /c is t h a t it should

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425 ALGEBRAS WITH INVOLUTIONS

601

not belong to the t y p e just described and t h a t A~ should not be commutative. A necessary and sufficient condition for the nondegeneracy of a semisimple algebra with involution (Ak, ,) over/c is then t h a t none of its simple components should be of either one of these types. II.

ALGEBRAS WITH INVOLUTION OVER THE REAL FIELD

6. F r o m now on, we shall not need a universal domain, as we shall be operating with algebras over a fixed groundfield, mostly the field R of real numbers. I f A is an algebra over any ground field/r we denote b y ' t r ' the trace of the regular representation of A. In other words, if L u is the ondomorphism x - + ux of the underlying vector-space to A, tr(u) is the trace of L u. Then tr(xy) is a symmetric bilinoar form on A • A ; according to a well-known criterion, it is non-degenerate if and only if A is absolutely semisimplo (i.e. if it is semisimple and remains so under a n y extension of the groundfiold). The trace tr(x) is invariant u n d e r all automorphisms of A ; if A is semisimplo, the right-hand and left-hand regular representations are equivalent, and t h e n the trace is also invariant under all antiautomorphisms o f A, or, as we shall say more briefly, all antimorphisms of A. I f u is a n y element of A, its " minimal polynomial " P is the polynomial o f smallest degree, with coefficients in the groundfield k, such t h a t P ( u ) = 0; P is also the minimal polynomial for the endomorphism L~ of the underlying vector-space to A. The " s p e c t r u m " Su of u is the set of all the distinct roots of P in the algebraic closure of k. L e t us say t h a t u is semisimple if L is so (or, what a m o u n t s to the same, if u generates an absolutely semisimple subalgebra of A) ; this will be the ease if and only if all roots of P are simple. Assume t h a t u is semisimple, and t h a t its spectrum S, is contained in k; let f be a n y k-valued function, defined on k or on a subset o f k containing S u. T h e n there is a polynomial Q, with coefficients in k, coinciding with f on S~; moreover, Q is uniquely determined modulo P ; therefore the element Q(u) o f A does not depend upon the choice of Q ; this will be denoted by f(u) ; f --+ f(u)

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is clearly an isomorphism of the ring of k-valued functions on Su onto the subalgebra of A g e n e r a t e d b y u. I f f is a k-valued function, defined on Su, the s p e c t r u m of f(u) is f(Su); and, ff g is a k-valued function, defined on f(Su), we h a v e g(f(u)) = h(u), with h == g o f. The t r a n s f o r m u ~ of art element u of A b y an a u t o m o r p h i s m or a n t i m o r p h i s m A of A has the same s p e c t r u m S u a n d the same m i n i m a l polynomial as u, and is somisimplo if u is so. Therefore, if S u is contained in k a n d f is a k-valued function on S u, we h a v e f ( u ~) = f(u) ~. L o t u be a n invertible element of an algebra A. We shall denote b y j(u) the inner a u t o m o r p h i s m determined b y u, i.e. defined b y the formula X ~

X j(u) "-- % - - l x u .

I f u, v are b o t h iuvertible, we h a v e j(uv) = j(u)j(v). I f A is a n y a u t o m o r p h i s m or a a t i m o r p h i s m of A, A-lj(u)A is the inner autom o r p h i s m j(u') with u ' = u ~ if ~ is an a u t o m o r p h i s m a n d u'---(u~) -1 if it is an a n t i m o r p h i s m . We shall write e(2) = 1 w h e n e v e r is an a u t o m o r p h i s m , e()t) = - - 1 whenever it is an a n t i m o r p h i s m , a n d define a s y m b o l u [~] b y the formula

so t h a t we have, in all cases

A-1 j(U) ~ =j(u[;~]). L e t A be a semisimple algebra, a n d let ~ be an a n t i m o r p h i s m of A. T h e n tr(x ~ y) is a non-degenerate bilinear f o r m on A x A. We h a v e (x ~ y)~ = y~ x ~, and hence tr(x ~ y) = tr(y ~ x~), which shows t h a t the bilinoar f o r m tr(x ~ y) is s y m m e t r i c if a n d only if ;~2 __ 1, i.e. if a n d only ff ~ is an involution. More generally, let A be a n a n t i m o r p h i s m of A; t a k e a e A, b e A, a n d consider the bilinear f o r m tr(ax ~ by). Obviously, it is non-degenerate if a n d only ff a, b are invortible, i.e. if t h e y are not zero-divisors. Assume now t h a t ~ is an involution, and t h a t a a n d b are invortiblo ; the formula

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427 ALGEBRAS WITH

I/~/VOLUTIONS

603

tr(ax ~ by) ~ tr(x "~bya) = tr(y ~ b~ xa ~) shows t h a t the f o r m tr(ax ~ by) is s y m m e t r i c if a n d only if, for all y, we h a v e bya = b ~ ya ~, i.e. y = b -1 b ~ ya ~ a -1 ; this is so if a n d only if we h a v e b ~ ~- bz - i , a ~ = az, where z is an invertible element in the center Z of A; as a~-----az implies a = z~ a ~, z m u s t satisfy z z ~ - 1 . L e t , be an involution of the semisimple algebra A. As tr(x' y) is a. non-degenerate s y m m e t r i c bilinoar f o r m on A • A, we can use it to a t t a c h an " a d j o i n t " L ' to e v e r y e n d o m o r p h i s m L of the underlying v e c t o r - s p a c e ; this is defined, as usual, b y the f o r m u l a : tr((Lx)' y) : t r ( x ' ( L ' y)). I n particular, the formula tr((ux) ~y) = tr(x~(u ~y)) shows t h e n t h a t the adjoint o f L~ is L ~ , a n d in particular t h a t L~ is self-adjoint if and only if u = u '. LEMMA 1. Let (A, ~) be a semisim21e algebra with involution over a field k of characteristic O. Let A +, A - be the subspaces of even and of odd elements of A f o r ~. T h e u A + and A - are the orthogonal complements of each other for each one of the symmetric bilinear f o r m s tr(xy) and tr(x' y). The formula t r ( x y ) = tr(y ~x ~) : tr(x' y~) shows at once t h a t , if one of the two elements x, y is in A + a n d the o t h e r in A - , 2tr(xy) a n d therefore tr(xy) m u s t be 0. Conversely, assume for instance t h a t x is orthogonal, with respect to tr(xy), to all vectors y such t h a t y~-=- 9 where e is + 1 or - - 1. T h e n it is orthogonal to y + e y' for all y ~ A, so t h a t we h a v e 0 = tr(x(y + 9 y~)) ----tr((x ~- e x~)y) for all y c A , a n d therefore x + 9 0 since tr(xy) is non-degenerate. The p r o o f for tr(x ~y) is quite similar. Of course L e m m a 1 remains valid for every groundfield of characteristic other t h a n 2, provided tr(xy) is non-degenerate, i.e. provided A is assumed to be absolutely somisimple.

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7. F r o m now on, we shall deal exclusively with semisimple algebras o v e r R; such a n algebra is isomorphic to the direct s u m of m a t r i x algebras o v e r R, o v e r C (the field of complex numbers) and over K (the division-algebra of quaternions). I f A is a n y algebra o v e r R, we say t h a t a n involution ~ on A is positive if tr(x~x) > 0 for all x other t h a n 0 in A ; tho existence of such an involution implies t h a t tr(x~y) is non-degenerate and therefore t h a t A is semisimple. PI~O1,OSITION 1.

Let A be a semisimple algebra over R; then there exists at least one positive involution on A; and all positive involutions on A coincide on the center Z of A. W r i t e A as a direct s u m of simple c o m p o n e n t s A i. A positive involution :r on A m u s t t r a n s f o r m each A i into itseff; for, if it t r a n s f o r m e d Ai, say, into Aj with j ~=i, x~x would be 0 for all x in A s. F r o m this one concludes a t once t h a t it is enough to p r o v e our proposition in the case o f a simple algebra A, which we can write as a m a t r i x algebra Mn(D ) over a division algebra D which can b e R , C o r K . I f D i s R o r K , the center Z i s R , a n d e v e r y a u t o m o r p h i s m or a n t i m o r p h i s m of A induces the identity on Z. I f D = C, we h a v e Z = D, a n d e v e r y a u t o m o r p h i s m or a n t i m o p r h i s m of A m u s t induce on Z either the identity or the a u t o m o r p h i s m z - + ~ , where ~ is the i m a g i n a r y conjugate of z; as we h a v e tr(z) = n 2 (z-]-z) for e v e r y z e Z, a n y positive involution :r on A m u s t induce z - + z on Z, fur, if it induced the identity, one would h a v e tr(z~z) = n~(z ~ + ~ ) , a n d this is < 0 for z = i. This proves the second assertion in our proposition. As to the first one, observe t h a t x~ > 0 for all x :/: 0 in D i f w e write x for x if D ----R, for the i m a g i n a r y conjugate of x if D = C, a n d for the q u a t e r n i o n conjugate of x if D = K ; also, in all three cases, x - + ~ is an involution on D, and X - + t 2 ~ is as usual, denotes the transpose p u t r(X)----Z~xi~; if, as always, regular r e p r e s e n t a t i o n of M~(D) seen t h a t we h a v e

an of we as

involution on M~(D) ff t~, X. F o r X----(x~j) in M~(D), denote b y t r the trace of the a n algebra over R, it is easily

[1960b]

429 ALGEBRAS ~VITH I N V O L U T I O N S

605

tr(X) = ~, [~-(x) + ~(x)]

with ~ equal to n/2 i f D = R , to n i f D = C , Therefore, if X -- (a0), we have

and t o 2 n i f D = K .

tr(t~. X) = 2y . ~ xox~, which shows that X--+tX is a positive involution on Mn(D ). This completes the proof. COROLLARY. I f ~, fl are two positive involutions on an al9ebra A over R, or is an inner automorphism of A. The assumption implies that A is semisimplo and that a-1fl is an automorphism of A; by prop. 1, it induces the identity on th0 center; b y the Skolem-Noethor theorem, it must therefore be an inner automorphism. If A is any semisimplo algebra over R, the automorphism of the center Z of A induced on it by all positive involutions of A will be denoted by z - + ~ ; if Z~ is any simpl0 component of Z, z --~ ~ induces the identity on it if it is isomorphic to R, and the imaginary conjugate if it is isomorphic to C. Let ~ be a positive involution on the algebra A over R. We have seen above that, ff u----u =, L~ is seff-adjoint for tr(x~y), i.e. for the quadratic form tr(x~x); as the latter is positive, it follows from well-known theorems that u is then semisimplo and has a real spectrum. Now assume that an element a of A is such that the bilinear form tr(x=ay) is symmetric, non-degenerate and positive; as it is symmetric and non-degenerate, a must be invertible and even for ~; that being assumed, the formula

tr(x=ay) = tr((L~x)~y) shows that the positivity of the bilinear form tr(x~ay), i.e. that of the quadratic form tr(x=ax), is the same thing as the positivity of the seff-adjoint operator L~ for the form tr(x~x); and this is positive if and only if all the characteristic roots of La(i.o., all the elements of its spectrum S~) are > 0. When that is so, we say that a is positive for ~; and we denote by P(~) the set of all such

430

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ANDRE WE.IL

elements. F r o m the results of no. 6, it follows t h a t , if f is a n y real-valued function, defined on the set of real n u m b e r s > 0, f(a) is defined for all aEP(~), a n d satisfies f(a) ~ = f(a); this will be so, in particular, for f(t) = t p, for a n y p c R, a n d for f(t).= log t. I f f(t) > 0 for all t > 0, ~hen f(a) is in P(~) for all a eP(~), since the s p e c t r u m of f(a) is the imago u n d e r f of the s p e c t r u m of a. Thus, for e v e r y p ~ R , we h a v e a m a p p i n g a - + a p of P(~) into itself. F o r all p , p ' i n R , wohaveaP+P'=aPa r a n d (aP) r = a pC. I n particular, we note that, for a eP(~), the only b ~P(a) such t h a t b 9" = a i s

b

=

a 1/2.

PROPOSITION 2. Let o~ be a positive involution on an algebra A over R. Then the set P(a) of positive elements for r162is a convex open subset of the vector-space of even elements for r162 ; and the group A* of invertible elements of A operates on P(~) by (x,a) ~ x~ax. I t is clear t h a t P(~) is convex. T a k e a eP(~) ; if p is the smallest element o f the s p e c t r u m S~ of a, i.e. the smallest charactreistie value of La(with respect to the quadratic form tr(x~x)), we h a v e p > 0 and

tr(x~ax) ~ P tr(x~x) for all x E A. Now, in the space of even elements o f ~ in A, i.e. of elements u of A such t h a t u ~ = u, t a k e a neighborhood U of 0 such t h a t , for e v e r y u E U, all characteristic values of L,~ are > - - p a n d < p ; then, for e v e r y u ~ U, a q- u is in P(~); this shows t h a t P(~) is open in the space of even elements. The last assertion in our proposition is obvious. P~OPOSITION 3. Let ~ be a positive involution on an algebra A over R ; let ~ be any automorphism or antimorphism of A. Then ~-1 ~ is a positive involution on A; and ~ maps P(cr onto P(h-I~2). The first assertion is obvious. F u r t h e r m o r e , we h a v e a e P(a) if a n d only if tr(x ~ ay) is s y m m e t r i c and positive. As the trace is i n v a r i a n t u n d e r 2, this is the same as to say, if ~ is an a u t o m o r p h i s m , t h a t tr(x ~ a ~ y~) is s y m m e t r i c a n d positive ; ~his p r o p e r t y will be unaltered if we replace x, y b y x ~-1, y~-l, a n d is therefore equivalent

[1960b]

431 ALGEBRAS V~ITH INVOLUTIONS

607

to the s y m m e t r y and positivity of tr(x ~-1~ a ~ y), which proves our proposition in this ease. Similarly, if ;~ is an antimorphism, a 9 P(r162 is equivMent to the s y m m e t r y and positivity o f tr(y a a a x~); replacing x, y b y y~-l~, x~-l~, we get our conclusion as before. 8. We can now determine as follo~s the set of all positive involutions on a semisimple algebra A over R. PROPOSITION 4. Let cr be a positive involution on an algebra A over R. Then, for every a e P(r162 the formula fl

:

~j(a)

:

j(a -1/~) (zj(a 1/2)

determines a positive involution fl on A . Conversely, i f fl is any positive involution on A , it can be expressed by that formula with an a 9 P(~). For every a e P(:r the element b = a 11~is also in P(r162 and the formulas of no. 6 give j(a) = j ( b ~) and r162j(b-1) :r = j ( b ) ; this gives at once ~j(a) = j ( b -1) r162 ; as this is the transform of ~ by the automorphism j(b), it is a positive invohttion. The p r o o f of the converse will be derived from the following lemma : Lv,~MA 2. On every semisimple algebra A over R, there is a positive involution ~ such that, i f a and b are any two elements of A , the following properties are equivalent: (i) the bilinear form tr(ax ~ by) in x, y is no~-dege~erate, symmetric, and positive ; (ii) there is an element z of the center Z of A such that z~z = l, z a e P ( ~ ) , z - t b 9 P(r162

One can see at once t h a t (ii) implies (i) whenever ~ is a positive involution on A. I n fact, as tr(ax: by) does not change if we replace a, b b y za, z -1 b with z E Z, we m a y replace the assumption (ii) by the assumption t h a t a, b are in P(~). We have already seen

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in no. 6 t h a t tr(ax ~ by) m u s t t h e n be n o n - d e g e n e r a t e a n d s y m m e t r i c ; p u t t i n g n o w u ---- a 1/9, v = b 1/2, we h a v e t r ( a x ~ bx) = t r ( u 2 x ~ v 2 x) = t r ( ( v x u ) ~ v x u ) • 0 since u ~ : u, v ~ = v, w h i c h p r o v e s t h e positivity. Now, in order t o c o n s t r u c t a n i n v o l u t i o n for w h i c h t h e c o n v e r s e is true, it is clearly e n o u g h to consider the case in w h i c h A is simple, i.e. of the

f o r m M n ( D ) with D = R ,

the involution X-+tX

C or K ; we shall s h o w t h a t

has t h e n t h e required p r o p e r t y .

I n fact,

a s s u m e t h a t A, B are t w o m a t r i c e s in Mn(D), such t h a t t r ( A t X JBY) is n o n - d e g e n e r a t e , s y m m e t r i c a n d positive; as we h a v e seen in no. 6, t h e first t w o a s s u m p t i o n s i m p l y t h a t A, B are invertible a n d t h a t t ~ = z - 1 1~, A = z A , where z is an invertible e l e m e n t o f the center, i.e. w h e r e z is a n o n - z e r o scalar, a n d a real one if D is R or K. Moreover, z m u s t satisfy z.~ -- 1, w h i c h implies z = 4- 1 if D = R or K. I f D ~ C, t a k e ~ e C such t h a t ~2 = z, a n d p u t A 1 = CA, B1 = ~ - I B ; t h e n A 1, B 1 also satisfy (i), a n d we h a v e t ~ 1 = A I , tJB1 = B1; therefore, after so m o d i f y i n g A, B in t h e case D----C if necessary, we m a y a s s u m e t h a t t ~ __ 9 A, t/~ = 9 B, with 9 = 4- 1 for D = R or K a n d 9 = 1 for D ----C. N o w t h e p o s i t i v i t y a s s u m p t i o n in (i) m e a n s t h a t t r ( A t X B X ) :> 0 for all X :/= 0; p u t t i n g t = r ( A t X B X ) , w h e r e v, as above, d e n o t e s t h e usual t r a c e o f a m a t r i x , this can be w r i t t e n as t + t > 0. B u t , p u t t i n g W = A t X B X , t = ~(tW) = T(tX. 9 B . X .

we h a v e

9 A ) = t,

so t h a t o u r p o s i t i v i t y a s s u m p t i o n a m o u n t s t o v ( A t X B X ) > 0 fo X ~ 0. T a k i n g t w o c o l u m n - v e c t o r u = (u~), v = (vi), p u t :

f ( u ) = q~Au, g(v) = ~vBv ; these are m a t r i c e s o f o r d e r 1, i.e. scalars (in D). T h e p o s i t i v i t y a s s u m p t i o n , applied to t h e m a t r i x X ----v. tu ---- (vi ~j), gives

9 (A ~ X B X ) = ~(f(u) g(v)) =Au) g(v)> 0 for all u ~: 0, v r 0. I f D = R a n d 9 -- - - 1, we h a v e f ( u ) = O, g(v) = 0 for all u, v; therefore this case c a n n o t occur. I f 9 ---- 1, f ( u ) a n d g(v) are real ; t h e y m u s t h a v e t h e s a m e sign for all u r 0, v ~ 0, so t h a t ,

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after replacing A, B b y -- A , - - B if necessary, we can write our assumption as f ( u ) > 0 for all u ~ O, g(v) > 0 for all v ~ 0; as one sees immediately, this implies t h a t A, B are positive for the involution X --+ *X. Finally, if D = K and r ------ 1, p u t q = f ( u ) , r = g(v); t h e n we have q = -- q va 0, 7 ---- -- r ~ 0; applying the positivity assumption to u, 0v with 0 e K, we see t h a t we must have q 0r 0 > 0 for a l l 0 va 0; as one can always find 0 such t h a t 0 r this is impossible.

0------q,

We can now complete the proof of Prop. 4 for the ease of an involution 0r having the p r o p e r t y stated in L o m m a 2. L e t fl be a n y positive involution on A. B y the corollary of Prop. 1, r162 fl is an inner a u t o m o r p h i s m j ( a ) of A, so t h a t we m a y write fl ~ ~j(a) with an invertible a. As we have seen in no. 6, the antimorphism fl = ~j(a) is an involution if and only if tr(x~y) is symmetric ; therefore it is a positive involution if and only if the bilinear form tr(x~y) = t r ( a - a x~,ay) is non-degenerate, symmetric and positive, i.e. if (a -a, a) satisfies condition (i) of the lemma. B u t then, b y our assumption on a, there is z e Z such t h a t z - a a is in P(:r Replacing a b y z - a a does not change j(a) ; we have therefore proved t h a t fl is of the form ~j(a) with a e P(~), hence also of the form j(b) - 1 ccj(b) w i t h b = a 11~. This proves Prop. 4 for the particular :r which we have been considering. I t also proves t h a t all positive involutions on A are transforms of this involution ~ b y inner automorphisms. Since the p r o p e r t y of expressed b y Prop. 4 is obviously invariant under automorphisms, our proof is thus complete. One m a y observe t h a t the p r o p e r t y of ~ expressed in L e m m a 2 is also invariant u n d e r automorphisms ; therefore (i) and ( i i ) a r e equivalent whenever ~ is a positive involution. 9. B y Prop. 4, if ~ is a positive involution on A, the mapping a - + ~j(a) maps P(~) onto the set of all positive involutions on A ; this shows in particular t h a t the latter set is connected if it is provided with its natural topology as a d o s e d subset of the space of all endomorphisms of the underlying vector-space to A.

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One can also define as follows a one-to-one mapping, and more precisely a h o m e o m o r p h i s m , of a closed subset of P(cr onto the set of positive involutions on A. We observe first t h a t , if ~, fl are two positive involutions, a n d if we write fi as the t r a n s f o r m of ~ b y a n inner a u t o m o r p h i s m j ( b ) of A, then, b y Prop. 3, P(fl) is the imago of P(~) u n d e r j(b). I n particular, if Z is the center of A, and if we write P ( Z ) = Z c~ P(~), P ( Z ) is independent of the choice o f ~. I f A is the direct sum of the simple algebras Ai, and if, for each i, ei is the unit-element of A i, it is easily seen t h a t P ( Z ) consists of the elements Z~ ti ei where all the ti are real a n d > 0.

Let ~r be a positive involution on the algebra A over R; let a, a' be two elements of P(~r Then we have j(a) -~j(a') if and only if a - la' is in P(Z). LEMMA 3.

P u t b = a 1/2 a n d z---- a - 1 a'. I f z is in Z, we h a v e az-= b~zb ; b y Prop. 2, if this is in P(a), z m u s t be in P(~), hence in P(Z). The converse is obvious. Now, calling again A i the simple components of A, a n d ei the unit-element of A i for each i, write N~ for the n o r m of the regular r e p r e s e n t a t i o n of A i o v e r R ; if d~ is the dimension of A i o v e r R, we have, for t e R , Ni(tei) -- t% F o r e v e r y x i e A i , write vi(xi)-([Ni(xi) ll/di)ei; and, for e v e r y x = Y , i x i i n A, with x i e A i for all i, write v(x) : Z i vi(xi). Then v is a m a p p i n g of A into P(Z), inducing the identity on P ( Z ) a n d such t h a t v(xy) = v(x)v(y) for all x, y in A. Therefore, for e v e r y invertible x in A, there is one a n d only one element z of P ( Z ) such t h a t v(zx) = 1, viz. z = v(x) -1. F u r t h e r m o r e , if a is a semisimple elemenr of A with a positive s p e c t r u m (e.g. if a is in P(~) for some positive involution cr we h a v e v(a~) : v(a) p for all p e R ; for this ,is t r u e if p is a n integer, hence if it is rational, a n d therefore b y continuity in the general case. I f ~ is a n y a u t o m o r p h i s m or a n t i m o r p h i s m of A, we h a v e v(x ~) = v(x) ~ for all x e A. We shall denote b y PI(~.) the sot of all elements a of P(~) such t h a t v(a) ---- 1. Combining Prop. 4 with L e m m a 3 a n d some trivial topological considerations, we get t h e following 9

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I~OPOSITIO~ 5.

Let :r [J be two positive involutions on the algebra A over R. Then there is one and only one element a of PI(~) such that fl = r162 ; and an element a' of P(~) is such that fl = :r i f and only i f it is of the form az with z e P ( Z ) . Moreover, the mapping a----~:cj(a) induces on P1(:r a homeomorphism of PI(U) onto the set of all positive involutions on A .

10. W e shall d e n o t e b y ~ t h e g r o u p o f all a u t o m o r p h i s m s a n d a n t i m o r p h i s m s o f the semisimple a l g e b r a A o v e r R. F o r )~ e ~ , t h e n o t a t i o n e(•), u [~] will be used in t h e sense explained in no. 6. L~MA 4. Let r162 be a positive involution on A ; let ~ be an element of f~ commuting with ~. Then, for a ~ P(~), the involution ~j(a) commutes with ~ i f and only i f a TM ----za with z ~ P ( Z ) ; for a E Pi(~), ~ j(a) commutes with ~ i f and only if a TM = a. Clearly, aj(a) c o m m u t e s w i t h ~ if a n d o n l y i f j ( a ) does so, i.e., b y t h e f o r m u l a s o f no. 6, ff a n d o n l y if j ( a TM) is t h e same as j(a) ; this a m o u n t s t o s a y i n g t h a t a TM m u s t be o f t h e f o r m za w i t h z E Z. B u t , as ~ c o m m u t e s w i t h ~, it t r a n s f o r m s P(~) into itseff (by P r o p . 3), so t h a t a TMis in P(~) w h e n e v e r a is in P(m). A p p l y i n g L e m m a 3, we get t h e first assertion in o u r l e m m a . I f v(a) = 1, we h a v e v(a TM) = 1, hence u(z) = 1 for a TM -~ za ; for z ~ P(Z), this implies z = 1. LEMMA 5.

I f tWO positive involutions ~, fl commute with each other,

they coincide. I n L e m m a 4, t a k e )~ = ~, a n d w r i t e fl as ~j(a) with a E P I ( ~ ) ; w e get a -1 --~ a, i.e. a ~ -~ 1, hence a ---- 1. I t is clear t h a t e v e r y element )~ o f fr t r a n s f o r m s a positive i n v o l u tion ~ into a positive i n v o l u t i o n 2 ~ - ~ , to w h i c h we can a p p l y t h e results p r o v e d a b o v e ; in particular, we can write it as ~j(a(h)) w i t h a (~) e PI(~). W e shall n e e d t h e following p r o p e r t y o f t h e m a p p i n g - + a(%) :

Let ~ be a positive involution on A . For every ~ ~ ~ , let a(h) be the element of P~(a) such that %:r = =j(a(%)). LEMMA 6.

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Then, for all ~, ix in ~, we have a(A~) = aC~). a(~)t~-11. Call a ' t h e r i g h t - h a n d side o f t h e f o r m u l a to be p r o v e d . B y the definition o f a(~), a(~), we h a v e

~(galx-1) A-1 = }t~}t-~. A j(a(~)) Z -1 = ~j(a(A)) j(a(/x) t~-ll) = aj(a') so t h a t we m u s t h a v e a(Atx) = za', w i t h z in t h e c e n t e r Z. particular, if we replace t L b y A- I , we see t h a t we m u s t h a v e

In

a(~ -~) = z~ a(A)-E~l with z, ~ Z. P u t a = a(A), b = al/2; mj(a) is t h e same as j(b) -~ o:j(b), a n d therefore t h e definition o f a = a(~) can be expressed b y s a y i n g t h a t j(b)A c o m m u t e s w i t h ~ ; therefore it t r a n s f o r m s P(~) into itself. I n p a r t i c u l a r , a j(b)~ must" be in P(~) ; this is no o t h e r t h a n a *. F r o m this it follows a t once t h a t a -[~l is in P~(~) ; in t h e a b o v e f o r m u l a , z, m u s t therefore be equal to 1, a n d we h a v e : a(A -1) = a()l)-[~], which is n o t h i n g else t h a n t h e special case /x = A-1 o f t h e f o r m u l a to be p r o v e d . N o w we go b a c k to t h e g e n e r a l case. W e h a v e s h o w n t h a t a(A/x) : za', w i t h z ~ Z. L e t us w r i t e t h a t za' is in P(~) ; this a m o u n t s to s a y i n g t h a t t h e bilinear f o r m F(x, y) = tr(x ~ za()t) a(/x) [~-11 y) = ~r(a(2)x ~ - 1

za(/x) [~-11 y)

is s y m m e t r i c , n o n - d e g e n e r a t e a n d positive. As t h e t r a c e is i n v a r i a n t b y 2, F(x, y) can be w r i t t e n , if )t is a n a u t o m o r p h i s m , as

F(x, y) = tr(a(A) ~ x ~ z ~ a(/x) y~). B u t in t h a t case a(A) ~ is t h e s a m e as a(~l-1) -1 a n d is therefore in P(~) ; w r i t i n g it as c * w i t h c e P(~), we get

F(x, y) -= tr((x * c) ~ z ~ a(/x) (y~ c)) ; t o s a y t h a t this is s y m m e t r i c , n o n - d e g e n e r a t e a n d positive is to s a y t h a t z ~ a(/~) is in P(~) ; as a(/x) is in P ( a ) , z ~ m u s t therefore be in

P(Z). Similarly, if )t is a n a n t i m o r p h i s m , we use t h e fact t h a t a(2) ~

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is the same as a()t -1) in order to write it as c 2, with c e P(~), and t h e n write F as F ( x , y) = t r ((y~ e) ~ a(/i) -1 z ~ (x ~ c)), again with the same conclusion as before, viz. z ~ E P ( Z ) , i.e. z ~ P ( Z ) . Since obviously v(z) is 1, this implies z = 1, which completes our proof. 11. T~EORE~ 3. Let A be a semisimple algebra over R ; let K be a compact subgroup of the group of all automorphisms and antimorphisms of A . T h e n there is at least one positive involution on A which is invariant under K , i.e. which commutes with every element

elK. Assume first t h a t K is a group of automorphisms of A ; and choose on A a positive involution ~. As in L e m m a 6, we call a()t), for )~ ~ K, the elemen~ of Pi(a) such t h a t Ag~t- i -- ~ j(a(A)).

B y Prop. 5, this is uniquely defined, and A - + .a(2) induces a continuous mapping of K into P(~) ; b y L e m m a 6, this satisfies the relation a()l/i) = a (2). a (/i) ~-i for all A,/~ in K. Now p u t a : [ a(/~) d/z, ,1

K

where d/~ denotes the H a a r measure on K. As P(~) is open and convex in the vector-space of symmetric elements for a, a is in P(a). If, in the integral which defines a, we replace /z b y A/~ with a fixed A e K, and apply the above formula for a ()t/~), we get a -~ a (~). a ~'-1. On the other hand, we have ,~:r

A- i -~ r162

) )~j(a) )~-i : ~j(a(h)a a-i) ;

these two formulas, taken together, show t h a t the positive involution ~j(a) commutes with ~. As A is a r b i t r a r y in K, this proves our conclusion in the present case.

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N o w assume t h a t K contains a t least one a n t i m o r p h i s m p; then, if K o is the group of all a u t o m o r p h i s m s of A contained in K , we h a v e K = K 0 u p K 0, p 2 ~ K o a n d K o p - - p K o. B y w h a t we h a v e p r o v e d above, we can choose a positive involution ~ w h i c h ' c o m m u t e s with e v e r y element of K-o. I f we write, as before, ~ - 1 in the form aj(a(2) ), with a()t) e Pl(a), for e v e r y A c K , we h a v e a(2) = 1 for A e K 0. P u t a = a ( p ) ; b y L e m m a 6, we h a v e a ( p A ) = a for all A E K 0 , i . e . a ( / ~ ) = a for all / z r 0. As we h a v e p K 0 = K op, this gives, for ~ ~ K0 :

a "= a(~ p) = a(,~), a(p) ~-1 = a ~-1, a n d therefore a = a ~. Similarly, again b y L e m m a 6, we have 1 = a(p 2) = a.(aP) - 1,

a n d therefore a -- a p. Now p u t b -- a 1/2 and fl = aj(b). B~ L e m m a 4, f~ o o m m u t e s with all elements of K 0. Moreover, we h a v e

p fl p - 1 = ~j(a). p j(b) p - ~ = aj(a) j(b[p-ll). As a ==-a p, we h a v e b = bp ; the r i g h t - h a n d side of the last formula is therefore equal to ~j(b), i.e. to ft. This shows t h a t fl c o m m u t e s with all elements of K . CogencY. Let (A, t ) b e a 8emisimple algebra with involution over R. Let K be a compact subgroup of the group of automorphisms of (A, t). Then there is a positive involution on A which commutes with t and with all elements of K. Our a s s u m p t i o n s i m p l y t h a t K U t K is a c o m p a c t subgroup o f the g r o u p of all a u t o m o r p h i s m s and a n t i m o r p h i s m s of A ; our assertion is therefore an i m m e d i a t e consequence of T h e o r e m 3. 12. F r o m now on, we shall deal with a somisimplo algebra with involution (A, ~) o v e r R ; a n d we shall denote b y G its g r o u p of a u t o m o r p h i s m s . B y T h e o r e m 2 of P a r t I, the connected c o m p o n e n t o f the i d e n t i t y in G is semisimple if (A, t) is non-degenerate; and it h a s been explained in P a r t I hi w h a t sense one m a y say t h a t " a l m o s t all " somisimple real groups can be obtained in this manner.

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The corollary of Theorem 3, applied to the case in which K is reduced to the identity, shows t h a t there is at least one positive involution on A which commutes with ~. F r o m the results proved above, we can also deduce at once the following : PROPOSI~ON 6. Let (A, ~) be a semisimple algebra with involution over R. Let ~, fl be two positive involutions on A, both commuting with ~. Then there is one and only one element a of PI(~) such that fl = ~j(a), and it is such that a ' ~ - a - l ; conversely, for every such element a, aj(a) is a positive involution commuting with ~. Furthermore, we have fl-~j(a-1/~):cj(al/~); and the mapping p-+j(aP), for p E R~ is an isomorphism of the additive group R onto a one-parameter group of automorphisms of (A, ~). This is in fact an immediate consequence of Prop. 5 and L e m m a 5. The set of all positive involutions of A commuting with ~, provided with its " n a t u r a l " topology (as explained in no. 9) will be denoted b y R ; the group (7 operates on it b y the law (~,~)-+~-~.~

()~e(7, ~ e R ) ,

and Prop. 6 shows t h a t it operates on it transitively. We shall denote b y K(~) the subgroup of G consisting of the elements of G which commute with ~; R is therefore isomorphic to G/K(~). I t will be shown t h a t K(~) is a maximal compact subgroup of G, so t h a t R is essentially the Riemannian symmetric space attached to G. We first show t h a t R is homcomorphie to an open cell. This will be done b y defining a " t r a c e operator ", corresponding to the " n o r m o p e r a t o r " v defined in no. 9. Write once more A as the direct sum of the simple algebras A i ; call Z~ the center o f A i and ni 2 the dimension of A i as a vector-space over Z i ; denote b y S i the trace in A t over Z~, i.e. the trace of the regular representation of A i considered as an algebra over Z i. Lot x be a n y element of A ; write it as x --- Zixi, with x, e A i for all i; we p u t :

a ( x ) : a ( ~ . , x i ) = ~ n~-~Si(xi) ;

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a is then a linear mapping of A into Z, which induces the identity on Z ; and, if A is a n y a u t o m o r p h i s m or antimorphism of A, we have a(x ~) ~- a(x) ~ for all x in A. Furthermore, we write

~0(x) = ~[~(x) + ~(x)] where z - + ~ is as defined in no. 7. I f u is a n y element of A, and if e~ is defined as Z u~/n !, (ro(u) is nothing t h a n log v(e u) ; we need only a special case of this, which we formulate as a lemma :

LEMMA 7. Let u be a semisimple element of A with real spectrum ; then we have a o (u) ----log v(e~). I t is clearly enough to consider the case in which A is simple ; its center Z can be identified with R or (2. Let d be the dimension of A over R ; d. a 0 is then the trace of the regular representation of A over R, while, for a n y x e A, d. log ,(x) is the same as log I N (x)[, where N is the n o r m o f the regular representation of A over R. I f u is a semisimple element of A with real spectrum, we can, b y choosing a suitable basis for A, write L~ as a diagonal m a t r i x ; let r 1, ..., r a be its diagonal coefficients. Then, if v = e~, L~ is the diagonal m a t r i x with the diagonal elements e% I n the regular representation of A over R, the trace of u is Zr~ and the n o r m of v is IIe% This proves the lemma. I n particular, let ~ be a positive involution on A ; e v e r y even element u for ~ is semisimple with real spectrum, and, if we put, for such an element, v = e~, v is in P(~) ; conversely, u is given in terms of v by u = log v. Then L e m m a 7 shows that, for such a pair u, v, the relations % ( u ) = 0, v(v) : 1 are equivalent. On the other hand, if u, v is such a pair, and if ~ is a n y involution on A, the relations u ' = - - u , v ~ - - v -1 are equivalent. Combining this with Prop. 6, wo got the following "decomposition t h e o r e m " : THEOREM 4. .Let G be the group of automorphisms of a semisimple algebra with involution (A, ~) over R . Let ~ be a positive involution on A , commuting with e ; let K(r162 the subgroup of G consisting of the elements of G which commute with r162; let W be the vector-space of the elements w of A such that w ~ ~ w, %(w) -~ O, w' -=- -- w. T h e n

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the mapping (w, ~) -+j(ew)~,for w E W, ,~ ~ K(~), is a homeomorphism of W • K(a) onto G. F o r w e W, p u t c -- ew ; f r o m w h a t we h a v e seen above, it follows t h a t t h e m a p p i n g w - + c = ew is a h o m e o m o r p h i s m o f W o n t o t h e set o f all elements c o f Pa(~) such t h a t c r = e -1. Then, b y L e m m a 4, j(e w) is in G, so t h a t j(ew)2 is in G for e v e r y )~ in K(~). L e t n o w /Z be a n y e l e m e n t o f G ; t h e n /Z g/z- 1 is a positive involution c o m m u t i n g w i t h ~, a n d can t h e r e f o r e be w r i t t e n as ~j(a) with a e PI(~), a ' = a - ~. P u t t i n g c = a - ~/2, we g e t

/Z ~ /Z-1 : o~j(a) =: j(e):cj(c) - ~, w h i c h s h o w s t h a t ~ = j(e)-I/Z c o m m u t e s with g ; as /Z a n d j(c) b o t h c o m m u t e w i t h ,, 2 is t h u s in K(~). P u t t i n g w = log c, we get /Z =j(ew)2. This d e c o m p o s i t i o n o f /Z is u n i q u e ; f o r , if we h a v e / Z ----j(c') ?t' with c' e PI(~), c ' ' : - c ' - 1 , ~' e K(~), then, w r i t i n g t h a t )~' c o m m u t e s w i t h ~, we get

/z ~ / z - ~ = j ( ~ ' ) ~j(c')-~ =

~j(c '-~)

which, c o m p a r e d with t h e f o r m u l a w r i t t e n above, gives c ' - * = a a n d t h e r e f o r e c' = a-1/2, b y P r o p . 5. W e h a v e t h u s p r o v e d t h a t t h e m a p p i n g in o u r p r o p o s i t i o n is bijective. Trivial topological considerations will t h e n show t h a t it is bicontinuous. COrOLLArY. Let (A, e) be a semisimple algebra with involution over R. Let R be the set of all the positive involutions on A which commute with ~. Then R, provided with its natural topology, is homeomo~Thic to an open cell. As we h a v e o b s e r v e d above, P r o p . 6 implies t h a t identified w i t h G/K(:r

R can be

if ~. is a n y positive i n v o l u t i o n c o m m u t i n g

w i t h ~ on A. L e t f be t h e canonical m a p p i n g o f G o n t o G/K(a). T h e o r e m 4 s h o w s t h a t t h e m a p p i n g w-+f(j(ew)) is a h o m e o m o r p h i s m o f W o n t o G/K(a). I f the i n v o l u t i o n , induces the i d e n t i t y on the center Z o f A, it is e a s y t o see t h a t t h e relation a ' - - - - a - a , for a n e l e m e n t a o f P(~), implies v ( a ) = 1; a n d similarly t h e relation w ' - - - - - - w, for

442

[]960b] 618

ANDRE WEIL

an even element w for ~, implies ao(W) -~ O. I n t h a t case, it is frequently advantageous, instead of the transcendental mapping w = log c, to use the Cayley transformation t -- (1 -

~). (1 + c) - 1 ,

c -

(1 -

t ) . (1 + 0 - 1 ,

where t satisfies t = t ~ = - - t ' and the inequalities t tr(x ~ tx) i < tr(x~x) for all x r 0 in A ; the set T of these elements t of A is a convex open subset of the space determined b y t = t ~ = - t', and one sees, just as in the proof of Theorem 4 and its corollary, t h a t the formulas written above determine a homeomorphism between T a n d R. 13. W h e n dealing with algebraic groups over R, one must be careful to distinguish between connected components in the algebraic and in the topological sense ; for the example of GL(n, R) shows t h a t the group of points with real coordinates in an algebraic group, defined over R, which is irreducible and therefore connected in the algebraic sense, need not be connected in the topological sense. F o r a similar example where the group is the group of automorphisms G o f a semisimple algebra with involution over R, one m a y take the algebra M ~ + z(R) with the involution X ' = S - ~. tX.S, S being the m a t r i x S----

0

--

?)

'

the algebraic connected component of the i d e n t i t y in G is PO + (S), and it is easy to see t h a t PO+(S, R) is not topologically connected. On the other hand, it is well-known t h a t the set of real points on a n y algebraic variety, defined over R, consists at most of a finite n u m b e r of connected components in the topological sense; this must t h e n be the case, in particular, for all the groups which we have considered so far. As above, let G be the group of automorphisms of a semisimple algebra with involution (A, ~) over R ; for e v e r y positive involution commuting with ~ on A, denote again b y K(a) the group of the elements of G which commute with ~. We shall denote b y G' and b y

[1960b]

443 ALGEBRAS W I T H

INVOLUTIONS

619

K'(a), respectively, the connected components of the identity in G and in K(~) in the topological sense.

Notations being as above, G'/K'(~) is isomorphic to G/K(~), and we have K'(~) -= G' n K(~). LEMMA 8.

P u t K" = G' n K(~) ; call f the canonical mapping of G onto G/K(:r Theorem 4 shows t h a t f(G') is G/K(~); therefore it is simply connccted. But trivial topological considerations show t h a t f(G') is the same as G'/K". On the other hand, K'(~) is obviously the topological connected component of the identity in K". Therefore G'/K'(~) is a covering space of G'/K" ; as the latter is simply connected, this implios t h a t K'(~) = K". T~EOREM 5. Let (A, L) be a semisimple algebra with involution over R ; let G be its group of automorphisms, and G' the topological connected component of the identity in G; let R be the space of all positive involutions commuting with ~ on A ; and let G act on R by (~,~)-+A-l~fori~G,~ ~ R. For each ~ R , let K(~) be the group of those elements of G which commute with ~, and let K'(:r be the connected component of the identity in K(~). Then, for each ~ R, K(g) (resp. K'(~)) is a maximal compact subgroup of G (resp. of G') ; conversely, all maximal compact subgroups of G (resp. of G') are of that form, and are transforms of one another by inner automorphisms of G (resp. of G'). Moreover, R is isomorphic both to G/K(~) and to G'/K'(~) for each ~ ~ R. We have Mready seen that R is isomorphic to G/K(:r ; therefore, by L e m m a 8, it is also isomorphic to G'/K'(~.). I t is clear t h a t K(~) and K'(~) are closed subgroups of the group of all automorphisms of the vector-space underlying A ; as the elements of K(~) and K'(~) commute with ~, t h e y leave invariant the positive quadratic form tr(x ~ x ) ; therefore these groups are compact. Assume t h a t K'(~) is not a maximal compact subgroup of G' ; then it is properly contained in a compact subgroup K of G'. B y the corollary of Theorem 3, K must be contained in K(fi) for some fi ~ R, and therefore also in K'(fl), so that K'(~) is properly contained in K'(fi). By Prop. 6, fi is the transform of ~ b y some inner automorphisra

444

[1960b] 620

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j(b) of A, belonging to G' ; this implies at once t h a t K'(fl) is the image of K'(~) under the inner automorphism of G' determined by j(b), and therefore that these two groups have the same dimension ; as t h e y are topologically connected Lie groups, this shows t h a t K'(~) cannot be properly contained in K'(fi). Similarly, assume t h a t K(~) is not a maximal compact subgroup of G ; then, just as before, we see t h a t K(g) must be properly contained in some K(fi), which implies t h a t K'(~) is contained in K'(fl) ; therefore, as shown above, K'(~) must be the same as K'(fi). As before, we see t h a t K(fl) is the image of K(~) under an inner automorphism of G ; therefore these groups have the same n u m b e r of connected components, this n u m b e r being finite as we have seen before. Since the connected component of the identity in K(~.) and in K(fl) is the same, viz. K'(~), this shows that K(~) cannot be properly contained in K(fi). On the other hand, the corollary of Theorem 3 shows t h a t e v e r y compact subgroup of G is contained in some group K(~) ; therefore, if it is maximal, it must be of the form K(~) ; and a similar proof holds for G'. 14. In order to prove our last result, we have to consider the Lie algebras of the groups discussed above. With the notations of Theorem 5, the Lie algebras of G and of K(~) will be denoted b y g and b y ~(~), respectively. I f Z is the center of the semisimple algebra A, the group of automorphisms of Z is finite ; this implies t h a t any automorphism of A, sufficiently close to the identity, induces the identity on Z and is therefore an inner automorphism. F r o m this, one easily deduces the well-known fact t h a t the Lie algebra of the group of automorphisms of A consists of the inner derivations

for all u e A. I f u and v are in A, we have D~ = D~ if and only if u--v is in Z ; from this, it follows t h a t e v e r y inner derivation can be written in one and only one way as D u with a(u) = 0, a being as defined in no. 12 ; one m a y therefore identify the Lie algebra of the

[1960b]

445 ALGEBRAS ~VITH I N V O L U T I O N S

621

g r o u p of a u t o m o r p h i s m s of A with the s u b s p a c e of A d e t e r m i n e d b y a(u) = O. I f • is a n y a u t o m o r p h i s m or a n t i m o r p h i s m of A, a n d if e()~) h a s the s a m e m e a n i n g as in no. 6, one sees a t once t h a t 2 - 1 Du 2 is the inner d e r i v a t i o n Du, with u' = e()t) u ~. I n p a r t i c u l a r , .Du c o m m u t e s with ~ if a n d only if z - - u - - e ( ) t ) u z i s in Z ; w h e n t h a t is so, we h a v e z ~ ~(z) = ~ ( u ) -

e(~) ~ ( u ) ~,

which shows t h a t z m u s t t h e n be 0 if a(u) -- 0, or also if ~ is a n inner a u t o m o r p h i s m o f A ; in b o t h t h e s e eases, therefore, D , c o m m u t e s w i t h u if a n d o n l y if u = e(2) u z.

L~MMA 9. A s s u m e that (A, ~) is non-degenerate, and let ~, fl be two elements of R . T h e n ~(~) = t(fl) implies ~ = ft. F r o m the results p r o v e d a b o v e , it follows t h a t t h e Lie a l g e b r a g of G consists of t h e inner d e r i v a t i o n s D~ for a(u) = 0, u' = - - u; let U be t h e v e c t o r - s p a c e d e t e r m i n e d b y t h e l a t t e r conditions. L e t V(~), W(~) be t h e s u b s p a c e s of U consisting of the e l e m e n t s o f U which a r e o d d for ~ a n d e v e n for ~, r e s p e c t i v e l y , a n d let V(fl), W(fi) be defined similarly. T h e n ~(cr ~(fl) consist o f t h e inner d e r i v a t i o n s D v for v e V(~) a n d for v ~ V(fl), r e s p e c t i v e l y . B y o u r a s s u m p t i o n , D~ c o m m u t e s w i t h fl for e v e r y v E V(~.) ; as a(v) -~ 0 for v e V(~), this implies t h a t v ~ = v a n d t h e r e f o r e v e V(fl). T h e r e f o r e our a s s u m p t i o n can be e x p r e s s e d as V ( ~ ) = V(fl). N o w t a k e w e W(~) ; as w is e v e n for ~r L e m m a 1 of no. 6 shows t h a t , w i t h r e s p e c t to t h e bilinear f o r m tr(xy), w is o r t h o g o n a l to all t h e o d d e l e m e n t s for ~, a n d in p a r t i c u l a r to all the e l e m e n t s o f V(~), i.e. of V(fl). On t h e o t h e r h a n d , as w is o d d for ~, it is o r t h o g o n a l to all e v e n e l e m e n t s for ~. N o w let B be t h e space of o d d e l e m e n t s for fl ; for a n y b e B, w r i t e : b0 = 89 -~- b~), b 1 = 89 - - b'), b~. = ~(bl), b~ = b 1 - - b 2. T h e n we h a v e b = b o -}- b~ + b 8 ; b 0 is e v e n for ~. As fl c o m m u t e s with ~, b 1 is o d d for fl ; t h e r e f o r e b~ is in V(fl). Moreover, b~ is o d d for fl a n d is in t h e center, so t h a t it is o d d for ~ (by P r o p . 1 of no 7).

446

[1960b] 622

ANDRE WEIL

Thus bo, b~ a n d b a are all orthogonal to w. This proves t h a t w is orthogonal to e v e r y b ~ B ; b y L e m m a 1 of no. 6, w m u s t therefore be even for ft. Thus W(~) is contained in W(fl); exchanging with fl, we see t h a t W(cr = W(fl). As the odd a n d the even elements for u a n d for fi in U are the same, it follows t h a t ~ and fl coincide on U, or in other words t h a t ~ - 1 fi induces the identity on the Lie algebra g of G. Now, b y Prop. 6, ~ - 1 fl is an inner a u t o m o r p h i s m of G' ; since it induces the identity on the Lie algebra of G, it m u s t therefore be in the center of the algebraic connected component of the identity G Oin G. B u t we h a v e a s s u m e d t h a t (A, c) is non-degenerate; and it follows f r o m T h e o r e m 2 of P a r t I that, when t h a t is so, G o is semisimple with a center r~duced to the identity. This completes our proof.

Notations and assumptions being as in Theorem 5, assume also that (A, ~) is non-degenerate. Then, for ~ e R, K'(~) is its own normalizer in G' ; K(cr is the normalizer of K'(~) and is its own normalizer in G; and the mappings ~ - + K'(cr ~-+ K(~) are bijeetions of .R onto the sets of maximal compact subgroups of G' and of G, respectively. T~EOREM 6.

The latter s t a t e m e n t follows at once f r o m L e m m a 9. Now, for e R a n d A ~ G, p u t fi --- A-1 :cA ; A t r a n s f o r m s K ' ( a ) into K'(fl) ; therefore, if A t r a n s f o r m s K'(~) into itself, L e m m a 9 shows t h a t we m u s t have ~ ~ fi, i.e. t h a t A m u s t be in K(~). This proves the theorem. 15. I f (A, ~), when decomposed into simple components, has no s u m m a n d on which ~ induces a positive involution, the group G' has no c o m p a c t factor, so t h a t the space R -- G'/K'(~) is the R i e m a n n i a n s y m m e t r i c space a t t a c h e d to the g r o u p G'. I n fact, it follows f r o m the results of P a r t I t h a t one can obtain in this m a n n e r all the R i e m a n n i a n s y m m e t r i c spaces a t t a c h e d to the semisimple groups which h a v e no c o m p a c t factor and no factor isomorphic to a n exceptional Lie group. Unfortunately, the latter are (for the t i m e being, a t least) still beyond the scope of the m e t h o d discussed in this paper.

[1960b]

447 ALGEBRAS W I T H I N V O L U T I O N S

623

Finally, we o b s e r v e that, if a positive i n v o l u t i o n ~ is chosen in t h e space R, a n d all o t h e r p o i n t s o f R are w r i t t e n as fl = ~j(a) w i t h a e PI(~), a' = a -1, the i n v a r i a n t m e t r i c in R is e x p r e s s e d b y

ds ~ = t r ( a - 1 da. a - 1 da), a n d t h e geodesic joining ~ to fl = ~j(a) ~ j ( a p) for p e R. The Institute for Advanced Study Princeton, :N'ow Jersey

consists o f t h e p o i n t s

[ 1960c] On discrete subgroups of Lie groups

1. L e t G be a topological g r o u p and F an a r b i t r a r y group; one m a y t h i n k o f F as being provided w i t h the discrete topology. Consider the space G (r) of all m a p p i n g s of F into G; this is the same as the product lIve~Gv, w h e r e Gv is the same as G for e v e r y 7 e F, and will be provided w i t h t h e usual p r o d u c t topology. The set .~ = ~ ( F , G) of all r e p r e s e n t a tions of F into G m a y be described as the subset of G (r~, consisting of all t h e m a p p i n g s r of F into G which s a t i s f y r(77') ~ r(~/)r(7 ') for e v e r y pair 7, 7' of e l e m e n t s of F; this is a closed subset of G (F) and will be provided w i t h the topology induced on it by t h a t of G(~); w i t h t h a t topology (the so-called " t o p o l o g y of pointwise c o n v e r g e n c e " ) , ~ will be called the space of r e p r e s e n t a t i o n s of F into G. If F is g e n e r a t e d by a family of e l e m e n t s (%)~ea, indexed by a set A, a r e p r e s e n t a t i o n r of F into G is uniquely det e r m i n e d by the e l e m e n t s r(%), so t h a t t h e r e is a one-to-one correspondence b e t w e e n .9t and a c e r t a i n subset of the set G (A) of all mappings of A into G. More precisely, F is t h e n a homomorphic image of the f r e e group F' w i t h the g e n e r a t o r s (7")~ea; let ~ be the homomorphism of F' onto F which maps 7" onto % for e v e r y a e A; let A' be the kernel of q~; the elem e n t s of A', being e l e m e n t s of F', are " w o r d s " w(7') in the 7", and we have then, for e v e r y such " w o r d " , w(7) = e, w h e r e z is the n e u t r a l elem e n t of F; these are the " r e l a t i o n s b e t w e e n the g e n e r a t o r s % of F " . The space 9~' = ~ ( F ' , G) is t h e n in an obvious one-to-one correspondence with G (A), and it is a trivial m a t t e r to v e r i f y t h a t this is a homeomorphism w h e n G (~) is provided w i t h t h e p r o d u c t topology in a m a n n e r similar to t h a t described above. T h e n ~ is in an obvious one-to-one correspondence w i t h t h e closed subset ~, ef ~', consisting of all the r e p r e s e n t a t i o n s of F' into G which m a p A' into the n e u t r a l e l e m e n t e of G; and it is again a trivial m a t t e r to v e r i f y t h a t this is a homeomorphism. We will i d e n t i f y .~' w i t h G <~), and ~ w i t h ~ , by m e a n s of t h e s e correspondences. L e t us a s s u m e f u r t h e r t h a t (wo)ae~ is a family of e l e m e n t s of A', such t h a t A' is g e n e r a t e d by the w~ and by t h e i r images u n d e r all i n n e r a u t o m o r p h i s m s of F'; if we w r i t e each such e l e m e n t as a " w o r d " w~(7), we say t h e n t h a t the relations b e t w e e n t h e % are " g e n e r a t e d " by the " f u n d a m e n t a l relat i o n s " w~(7) = ~; and ~ , as a subset of ~ ' = G (~), is t h e n the set of the e l e m e n t s (g~) of G c~) which s a t i s f y t h e relations w ~ ( g ) = e. P a r t i c u l a r i n t e r e s t a t t a c h e s to those r e p r e s e n t a t i o n s r of F into G which 369 Reprinted from Ann. of Math. 72, 1960, by permission of Princeton University Press.

449

450 370

[1960c] ANDRE WEIL

are injective (or, as one also says, faithful) and such t h a t r(F) is a discrete subgroup of G with compact quotient space G/r(F); of course G has then to be locally compact. Clearly a representation r has these properties if and only if there are a neighborhood U of e in G such t h a t r - l ( U ) = {z} and a compact s u b s e t / ~ of G such t h a t G = K . r ( F ) . Let us denote by ~0=~0(F, G) the set of all such representations, considered as a subset of :~. It has been conjectured by A. Selberg that, if G is a semi-simple Lie group, .~0 is an open subset of ~J~'. This will now be proved for all Lie groups. More precisely, the following theorem will be proved:

Let G be a connected Lie group; let F be a discrete group, and ro an injective representation of I' into G, such that r0(F) is discrete in G with compact quotient-space G/ro(F). Then there are a neighborhood U of e in G, a compact subset K of G, and a neighborhood 1I of ro in the space ~ of all representations o f f in G, such that r - l ( U ) = {~} and G = K . r ( F ) for all r c ~. Moreover, L' has then a finite set of generators with a finite set or f u n d a m e n t a l relations. The last s t a t e m e n t is included only for the sake of completeness, since it is an immediate consequence of the fact t h a t the fundamental group of any compact manifold has the property in question (incidentally, a proof for this is included in what follows) and t h a t the same is t rue of the fundamental group of any connected Lie group. In view of these facts, !~ can be described as has been done above, using a finite set of generators ( % ) a n d a finite set of fundamental relations (we); then the space denoted above by ~ ' is the product of finitely many Lie groups, isomorphic to G, and is therefore a connected real-analytic manifold; ~{ is the subset of .~' defined by the finitely many real-analytic relations w~(g) = e, and is t h e r ef o r e a real-analytic subset of ~ '. The theorem stated above implies, as we have said, t h a t the subset of ~ which has been denoted by ~o is open on ~.)~;if G is the hyperbolic group (the quotient of SL(2, R) by its center), it is known t ha t ~J~0is actually a manifold; the question naturally arises w h e t h e r this is true for ~0, or r a t h e r for each connected component of ~0, whenever G is a Lie group. 2. As will be seen in no. 11, the general case of our theorem can be reduced to the case when G is simply connected. When G is such, G/ro(F) is a compact manifold V whose f u n d a m e n t a l group is isomorphic to F. We first deal with some purely topological aspects of this situation. In nos. 2-5, we will denote by V any connected manifold (it would be enough for our purposes to assume t h a t V is connected and locally simply connected). L e t V be the universal covering of V, with the projection

[1960c]

451 ON DISCRETE SUBGROUPS

371

p onto V; let P be the f u n d a m e n t a l group of V, considered as a g r o u p of a u t o m o r p h i s m s of l?, so t h a t V can be identified with V/17. We w r i t e x7 for the image of a point x of l? u n d e r an e l e m e n t 7 of I', and e for the n e u t r a l e l e m e n t of P. Assume t h a t (U~)~e~ is a family of connected open subsets of 17, indexed by a finite set I, with the following properties: (a) the sets U~7, f o r i c I, 7 ~ [', are a covering of V (i.e., t h e i r union is I?); (b) f o r every p a i r (i, j) there is at most one element 7 o f [" such that

U~7 meets

~7~.

F o r each i, p u t U~ = p(U~); applying (b) to 4 = j, we see t h a t U~Teannot m e e t U~ unless 7 = s; this means t h a t p induces on U~ a bijeetive mapping, and t h e r e f o r e a homeomorphism, of U~ onto U~. B y (a), t h e sets U~ are a covering of V; call N t h e n e r v e of t h a t covering; this is the set of all subsets J of I, such t h a t Flje~U~ is not e m p t y . In particular, we have {i, j} e N if and only if p(U~) m e e t s p ( U 0 , i.e., if and only if t h e r e is 7 in P such t h a t U~7 meets U~ by (b), w h e n t h a t is so, 7 is uniquely d e t e r m i n e d by t h a t condition; this e l e m e n t 7 will be d e n o t e d by%~. I t i s e l e a r t h a t T ~ = e for all i i n I , and t h a t % 7 ~ = z for all {i, j} e N. A subset J of I is in N if and only if, for some j in J, t h e r e are a point x in Uj and e l e m e n t s 7~ of P such t h a t x is in [7,~%,, for all h in J, and t h e n the same is t r u e for all choices of j in J . B u t t h e n we m u s t have %, = 7,,~j for all h in J . T h e r e f o r e J is in N if and only if, for some j in J, all pairs {j, h} w i t h h in J are in N, and the sets UhT,~, for h e J, have a n o n - e m p t y intersection; and t h e n the same is t r u e for all j in J . In particular, t a k e J = {i, j, k} ; this is in N if and only if {i, j} and {i, k} are in N and t h e r e are points u~ in U~, u, in ~ , u~, in U~, such t h a t u~ = ujT~ = u~%~; b u t t h e n u is in /J.j and i n g~Tk~7~.~, SO t h a t we have As V is connected, the n e r v e N i s connected; this means t h a t I c a n n o t be w r i t t e n as t h e union of two disjoint n o n - e m p t y subsets I', I", such t h a t no pair {i', i"}, with i' in I ' and i " in I", belongs to N. In f a c t , if this w e r e not so, the unions of the U , and of the U < would be disjoint n o n - e m p t y open subsets of V. We select some e l e m e n t i0 of I as " o r i g i n " , and, for each i in I, we select a " c h a i n " C~, i.e., a sequence 4o, i1,... , i ~ = i , w i t h the origin i0 and the end-point i~ = i, such t h a t , for 1 < ~ < m, {4~_, i~} is in N. We p u t S~ = 7~0qTq% . . . 7%~_1~ for the chain C~. We t a k e for C~0 the chain io, io, so t h a t S~ = s. Now, for e v e r y {i, j} in N, p u t a~:j = S~%~S2h These elements s a t i s f y the relations ( ~ = G ~ j ~ when-

452

[1960c] 372

ANDRE

WEIL

ever {i, j, k} is in N (which implies, for i = j = k, t h a t a . = e for all i, and, for i = k, t h a t a ~ a ~ = e for all {i, j} in N), and, for every chain C~ = (i0, . . . , i~), the relation a~0~ " ' " a ~ _ ~ = e. It will now be shown t h a t the a u generate F, and that the relations we have j u s t w r i t t e n generate all the relations between them. 3. As to the first assertion, we use the f a c t t h a t the nerve N-of the covering of V by the U,7 m u s t be connected. Now a pair {(i, 7), (J, 7')} is in N if and only if U~7 meets U~7', i.e., if and only if {i, j} is in N and 7 ---- 7 ~ 7 ' . L e t 7 be a n y element of F; there m u s t be a chain of N , with the origin (io, 7) and the end-point (i0, e); if this chain consists of the elements (i~, 7~), with 0 -_< l~ _-< m, we m u s t therefore have i0 -- i~, 70 = 7, 7~ = e, and, for 1 __
which, t o g e t h e r w i t h the relations i.~,= io, 70 = 7, 7.~ = e, a~0 = e, and w i t h the definition of the a~, implies t h a t we have 7

~

O'10~lO'll/2 " ~ "

0"i,1~1'~ 0

9

Now we prove our assertion about the relations between the a , , L e t F* be a group g e n e r a t e d by elements a n, w i t h the f u n d a m e n t a l relations e n u m e r a t e d at the end of no. 2; the relations F ( a ~ ) = a,j determine a homomorphism of F* onto F, and we have to show t h a t this is an isomorphism. L e t W be the union of the disjoint open sets U, • {(i, 7*)} in the product 12 • I x F*, where I a n d F* are provided w i t h the discrete topology. In W, we introduce an equivalence relation R as follows. Two points (u~, i, 7"), (uj, j, 7'*), w i t h u~ c U~, u~ e Uj, will be called equivalent under R if and only if p(uO = p(uj) and 7* = ~ 7* '* 9 This is reflexive, because a~ = e* for all i (where e* is the n e u t r a l element of F*), and symmetric, because a~jaj~* * = e* for all {i, j} in N; in order to prove transitivity, let ( u . i, 7*) be equivalent to (uj, j, 7'*), and the l a t t e r point to (u~, k, 7"*); t h e n p(u~) = p(uj) = p(u~), so t h a t {i, j, k} is in N ; also, we have 7* = a~7'*, 7'* = q*J ~ ~'"* l , which, in view of the relations between the a* ~j~ implies 7"=a*~7"*; this proves t h a t (u~, i, 7*) and (u~, k, 7"*) are equivalent. One sees at once t h a t the equivalence relation R is open; therefore V* = W / R is a manifold. We can define a mapping of W into I~, by p u t t i n g f ( u , i, 7*) -u ~ F ( 7 * ) ; if (u~, i, 7*) and (uj, j, 7'*) are equivalent under R, we have p(u,) = p(u~), which implies uj = u~%j and t h e r e f o r e u j ~ ~ = u~3;'a~, and 7* = a~7'*; therefore those two points have the same image in I2 under

[1960c]

453 ON DISCRETE

SUBGROUPS

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the mapping f , so t h a t f determines a mapping ~ of V* =: W/R into I?. The inverse image of U~7 by f consists of the points (u j, j, 7*) of W such t h a t u ~ j l F ( 7 *) = u~7 for some u~ c U~; when t h a t is so, we have p(u~) = ~j *~ ]. p(uj) and {i, j} e N, so t h a t (uj,j, 7*) is equivalent under R to (u~, i, a*7 Therefore the inverse image of U~7 by qJ consists of the union of the images in V* of the subsets U~ x {(i, 7*)} of W for which F(7*) = ~7; as those images are disjoint open subsets of V*, and as q~ induces on each one of t h e m a homeomorphism onto U~7, this proves t h a t V*, with the mapping ~ onto I?, is a covering of ~?; moreover, we see t h a t ~ is bijective if and only if F is so. As I? is simply connected, this implies t h a t F is bijective if and only if V* is connected. 4. In order to prove t h a t V* is connected, it will be enough, since the U~ are connected, to show t h a t the nerve of the covering of V* by the images of the sets U~ • {(i, 7*)} is connected. This will be done by constructing a " c h a i n " of sets U~ x {(iv, 7~)}, with i _< ~ _< n, beginning w i t h a given set /J~ x {(i, 7*)} and ending up with U~0 • {(i0, z*)}, such t h a t the images in V* of a n y two consecutive sets of the chain have a common point. The latter condition will be fulfilled if and only if, for every ~, {i~_. i.,} e N and ~/~-1 = G~,,_?..* %,*" when t h a t is so, we have

w i t h i~ = i, and i~ = i0. Conversely, if we can write 7" in t h a t form, we only have to denote by 7* the product of the last n - ~ factors in the right-hand side in order to have a chain fulfilling the required conditions. Now, since F* is g e n e r a t e d by the a,~, we can write for 7* an expression consisting of factors a~'~ and (a~*~)-~; using the defining relations for F*, we can replace each factor (aj*~)-~ by a~*j, so t h a t 7* now appears as a product of factors a~,~*. For each factor a~, in t h a t expression, we use the relations corresponding to the chains Cj = (J0, J , " " , J~), C~ = (h0, h , . . . ,h~), w i t h j0 = h0 = i0, j~ = j, h~ = h, in order to rewrite it as G j9A ~

G *iOj 1G *JlJ2 9 " ,

G*J p _ l j G j *h G h *h q _

1 ' ' ' G *tt2A 1G *hll 0 ",

s u b s t i t u t i n g for the aj*,~ the expressions in the right-hand sides, we get for 7* an expression which is of the desired form except t h a t it begins w i t h i0 instead of i; multiplying this to the left w i t h the inverse of the relation corresponding to the chain C~, we get w h a t we w a n t . 5. We add some remarks to the facts proved in nos. 2-4. Firstly, a covering such as (U0, used above, can be characterized as follows, in

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t e r m s of V a l o n e . L e t us say t h a t a connected open subset U of V is h o m o t o p i c a l l y f i a t on V if e v e r y closed p a t h contained in U is homotopic to 0 on V; this will be so if and only if the inverse image p - ' ( U ) of U i n for p consists of disjoint c o n n e c t e d components, each of which is mapped bijectively onto U by p. Now let (U~)~e~ be a finite covering of V by connected open sets U,, such t h a t , w h e n e v e r U, N Uj is not e m p t y , U~ U U~ is h o m o t o p i c a l l y f i a t on V; and take for /J~ any connected component of p-l(U~); these will satisfy conditions (a) and (b). The description given above of the f u n d a m e n t a l g r o u p I' of V by means of the g e n e r a t o r s a~ and of the relations w r i t t e n in no. 2 depends only upon the n e r v e N of the covering (U0; moreover, a f t e r the group F has been so constructed, no. 4 proves t h a t l? itself is isomorphic to the q u o t i e n t of the union of the open subsets U~ x {(i, 7)} of V x I • F by the equivalence relation b e t w e e n points (u~, i, 7), (u~, j , 7') of t h a t union which is given by u~ = u j, 7 = a~7'. It would make no difference in this construction if we substit u t e d the %~ for the a~j, since this m e r e l y a m o u n t s to t r a n s f o r m i n g the equivalence relation by the mapping (u, i, 7) -~ (u, i, ~7) of the union of the sets U, x {(i, 7)} onto itself. One m a y also note the following consequence of these results. By a eochain o f I i n a g r o u p G, let us u n d e r s t a n d a m a p p i n g (i, j)--~ x u into G of the set of those pairs (i, j) for which {i, j} is in N ; this will be called a cocycle if x~ = x~jxj~ w h e n e v e r {i, j, k} is in N ; two cocycles (x~j), (y~j) will be called e q u i v a l e n t if t h e r e is a mapping i - ~ z~ of I into G such t h a t y~ = z~x~jzj ~ for e v e r y {i, j} c N. Then the classes of equivalent cocycles of I in G are in a one-to-one correspondence with the classes of representations of F into G; as usual, two such r e p r e s e n t a t i o n s are called equivalent if t h e y can be derived f r o m one a n o t h e r by an inner a u t o m o r p h i s m of G. 6. Now we go back to the Lie group G. A n y q u o t i e n t of G by a disc r e t e subgroup m a y be considered as a Clifford-Klein f o r m of G. In order to give a precise c o n t e n t to this concept, we introduce the notion of Gs t r u c t u r e on a manifold. L e t n be the dimension of G; let w , . . . , w~ be a basis for the r i g h t - i n v a r i a n t differential forms of d e g r e e 1 on G. These s a t i s f y the M a u r e r - C a r t a n equations: Re(w)

=

dw~ -

]~j~c~j,~(ojw~

=

0 ,

w h e r e the c,j~ are the constants of s t r u c t u r e for G. By a G-manifold, we shall u n d e r s t a n d an analytic v a r i e t y V of dimension n, provided with n differential forms ~,, s a t i s f y i n g the equations R~(rj) = 0 and linearly ind e p e n d e n t at e v e r y point of V; the 7i, are called the s t r u c t u r a l forms of

[1960c]

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V. If V and V' are two G-manifolds, a mapping cp of V into V' will be called a G-mapping if it is analytic and if the inverse images by ~o of the structural forms for V' are those for V; if cp is bijective, then its inverse is also a G-mapping, and cp is called a G-isomorphism. Every G-mapping of V into V' is a "local isomorphism" in the sense t h a t each point of V has a neighborhood which is mapped G-isomorphically by cp onto its image in V'. It follows from Frobenius's theorem on completely integrable systems that, if V and V' are G-manifolds, a a point on V and a' a point on V', there is a G-isomorphism of a neighborhood of a on V onto a neighborhood of a' on V' which maps a onto a'; by the same theorem, any two such isomorphisms must coincide in some neighborhood of a on V; from this it follows that, if V is connected, two G-mappings of V into V' which coincide at one point must coincide e ve r yw he re. The group G itself has a natural G-structure, determined by the forms w,; the automorphisms for t h a t structure are the right-translations. Therefore the G-structure of G can be transported to the quotient G/P of G by any discrete subgroup F (this being understood as the space of right cosets xF). Ev er y point of a G-manifold has a neighborhood which is isomorphic to a neighborhood of e (the neutral element of G) on G. A Gmanifold V is called complete if there is an open neighborhood U of e in G with the following property: to every point a of V, there is a neighborhood W of a and a G-isomorphism of U onto W, mapping e onto a. The group G itself, and every group locally isomorphic to G, are complete Gmanifolds; every compact G-manifold is complete. If V i s a connected and simply connected G-manifold, and V ' a complete G-manifold, there is one and only one G-mapping of V into V' which maps a given point a of V onto a given point a' of V'; if at the same time V is complete, then, for such a mapping, V becomes a covering manifold of V' and therefore its universal covering. If we take for G the simply connected Lie group with the given structure, then this shows t h a t every complete, connected and simply connected G-manifold is isomorphic to G. If now V is any complete connected G-manifold, its universal covering may be identified with G; if F is the fundamental group of V, it operates on G by G-automorphisms, i.e., by right-translations; this means t h a t F can be identified with a discrete subgroup of G, and V with G/F. This applies in particular to every compact and connected G-manifold. Let U be a connected open subset of the simply connected group G; let V be the quotient G/F of G by a discrete subgroup F; let U' be an open subset of V, and assume t ha t there is a G-isomorphism ~ of U onto U'. L e t p be the canonical mapping of G onto V = G/F; take a point a in U;

456 376

[1960c] ANDRE WEIL

as p is surjective, t h e r e is a point b of G such t h a t p(b) = ~o(a). T h e n ~, and t h e m a p p i n g of U into Vdefined by u--~ p(ua
~/(8, s") = ~/(s, s')v(s', s"),

[1960c]

457 ON D I S C R E T E S U B G R O U P S

377

and a similar relation with v instead of 7. 8. As K is compact, we can choose a finite subset S of K such t h a t K is contained in the union of t h e sets Us for s e S; t h e n the sets Us, for s e S, have the properties (a) and (b) s t a t e d for the sets U~ in no. 2. The n e r v e of the covering of V by the sets p(Us) is the intersection N(S) of N ( K ) with the set of all subsets of S. F o r each T in N(S), we can choose, once and for all, elements u~(t) of U such t h a t all the elements ur(t)t, for t e T, have the same image in V; this means t h a t we have ur(t) =u~(t')v(t', t) for all t, t' in T. Call q~ the m a p p i n g of U x S into V defined by q~(u, s) = p(us); as this is surjective, and as V is compact, t h e r e is a compact subset X of U x S such t h a t ~p(X) = V; for each s e S, call X~ the compact subset of U such t h a t Xs x { s } = X ~ ( U x {s}). L e t U ' b e t h e ball ~ x ~ < p ' 2 , w h e r e p ' is t a k e n < p and so close to p t h a t U' contains all the sets Xs and all the points u~(t) for T e N(S), t ~ T. Then U' • S contains X, so t h a t e ( U ' x S) = V, which means t h a t the sets p(U's), for s e S, are still a covering of V; t h e r e f o r e the sets U's, for s e S, still have the properties (a), (b) of no. 2. For T e N(S) and t e T, the point u~(t)t is in U't; this shows t h a t T is in the n e r v e of the covering of V by the sets q~(U's); as this n e r v e is obviously contained in N(S), it is still N(S). We can now define G and V by means of the non-connected manifolds U' x S x F, U' x S, with suitable identifications. Call f the mapping of U' x S x F into G defined by f(u, s, 7) = usT; one sees a t once t h a t two points (u, s, 7), (u', s', 7') have the same image by f in G if and only if we have {s, s'} e N(S), 7 = 7(s, s')7', u' = uv(s, s'); if we write R for this relation, it follows f r o m this t h a t R is an equivalence relation and t h a t G can be identified with (U' x S x F)/R; as R is compatible w i t h the Gs t r u c t u r e of U' x S x F, G can be identified with ( U ' x S x F)/R, not only as a topological space, b u t even as a G-manifold. Similarly, V, as a G-manifold, can be identified w i t h ( U ' x S)/R', w h e r e R ' is the equivalence relation b e t w e e n pairs (u,s), (u',s') of U ' x S defined by

{s, s'} e N(S), u' = ur(s, s'). F u r t h e r m o r e , we can use the sets Us to define a set of g e n e r a t o r s for F in the m a n n e r explained in no. 2. In order to do this, we have to choose an e l e m e n t so of S, and, for each s e S, a chain C~ as described in no. 2; t h a t being done, we define elements ~(s), a(s, s') of F, in the m a n n e r explained there; the a(s, s') are t h e n g e n e r a t o r s for F, and the relations are those s t a t e d in no. 2. 9. We now modify as follows the equivalence relations R, R ' defined

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above. Once and for all, we choose a neighborhood gt of e in G, such t h a t U ' g t c U, u~.(t)~-lc U' for every T e N(S) and every t e T, and t h a t the closure of Xz~ -1, for every s c S, is a compact subset Ys of U'. Now take a representation 7 -~ 7 of F into G; this will be given by assigning the images ~(s, s') of the generators a(s, s') of F in t h a t representation, which can be taken arbitrarily, subject to the relations b e t w e e n the a(s, s'). We put ~(S, S t) = S~(S)--I~(S, 8')~(S?)S '-1

~(s, s') = ~(s, s')r(s, s') -~ . As a consequence of the relations b e t w e e n the ~(s, s'), we have ~(s, s ) = e for all s e S, ~(s, s')e(s', s ) = e for all {s, s'} e N(S), ~(s, s " ) = ~ ( s , s')~'(s', s") for all {s, s', s"} e N(S). Moreover, if the representation 7 -~ 7 is close enough to the identity mapping of F onto itself, the elements w(s, s') will be in ~2 for all {s, s'} e N(S). It will now be shown that, when this is so, the ~(s, s') g e n e r a t e a discrete subgroup F of G, isomorphic to F, such t h a t G/F is compact. In fact, call /~ the relation b e t w e e n elements (u, s, 7), (u', s', 7') of U ' x S x F , given by {s, s'} e N ( S ) , 7 = 7 ( s , s')7', u'=ue(s, s'); call/~' the relation b e t w e e n elements (u, s), (u', s') of U' x S, given by {s, s'} e N(S), u' = u~(s, s'). We first prove t h a t these are equivalence relations. It is clear t h a t t h e y are reflexive and symmetric. Assume that /~' holds for (u, s), (u", s") and also for (u', s'), (u", s"); this can be w r i t t e n as {s, s"} e N(S), {s', s"} e N(S), and u " = u~-(s, s") = u'~-(s', s " ) ,

with ~ = uw(s, s'), ~' = u'w(s, s'), so t h a t ~, ~' are in U ' ~ , hence in U. Therefore those relations imply t h a t {s, s', s"} is in N ( S ) ; in view of the relations b e t w e e n the ~(s, s'), it follows at once from this t h a t /~' holds for (u', s'), (u", s"). The same proof shows t h a t / ~ is also an equivalence relation. As the equivalence relations R, R ' are open and are compatible with the G-structures of U' x S x F and of U' x S, the quotients = (U' x S• F)/R, V = ( U ' x S)/[~' are G-manifolds. We denote by f a n d by ~ the canonical mappings of U' x S x F onto G and of U' x S onto IF, respectively. We now prove t h a t IF is compact, by showing t h a t it is the union of the images under ~ of the compact subsets Ys x {s} of U' x S. In fact, let (u, s) be any point of U' x S; as V is the union of the sets p(X,s), there is an element s' of S and a point x of Xs, such t h a t (x, s') is equiva-

[1960c]

459 ON DISCRETE SUBGROUPS

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lent to (u, s) for R', which means t h a t {s, s'} is in N(S) and t h a t u = xv(s', s). Then, if we put y = xw(s', s) -~, y is in X~,~2-1, hence in Ys,, and we have u = yv(s', s), which shows t h a t (u, s) is equivalent to (y, s') for /~'. This proves our assertion. The sets ~ ( U ' x {s}), for s e S, are an open covering of 17. We now prove t h a t the n e r v e of this covering is still N(S). In fact, assume first t h a t T is in t h a t n e r v e ; t h e n e v e r y pair {t, t'} of elements of T m u s t be in N(S), and t h e r e are elements u~ of U' for e v e r y t c T, such t h a t u , = u~(t, t') for all t, t' in T. T a k i n g to in T, and p u t t i n g ~ = u~w(t, to) for all t e T, we g e t u~0 = ~ r ( t , to) for all t e T; as the ~ are in U'~2, hence in U, this implies t h a t T is in N(S). Conversely, let T be any set in N ( S ) ; t a k i n g to in T, p u t ~ = uT(t)w(t, t0)-'; t h e n we have, by the definition of the e l e m e n t s u~(t), U~o = ~P(t, to) for all t c T; as the ~ are in U', this shows t h a t T is in the n e r v e of the covering of IF by t h e sets

~ ( U ' x {s}). 10. As IF is compact, it is a complete G-manifold. It has the covering by the sets ~ ( U ' x {s}), w i t h the n e r v e N(S). I f {s, s'} is in N(S), the union of the sets ~p(U' x {s}), cp(U' • {s'})is t h e q u o t i e n t of the union of t h e two sets U' x {s}, U' x {s'} by t h e equivalence relation induced on t h a t union b y / ~ ' ; it is a t once seen t h a t this q u o t i e n t is isomorphic to U' U U'~(s, s'); as explained in no. 6, this implies t h a t it is homotopically flat on IP. By the results in no. 5, this shows t h a t the f u n d a m e n t a l g r o u p of lP is isomorphic to P, and f u r t h e r m o r e t h a t the universal covering of IF is the manifold c o n s t r u c t e d as explained in no. 5; but it is at once seen t h a t this is n o t h i n g else t h a n G. We have t h u s proved t h a t is connected and simply connected, as well as complete. As such, it is isomorphic to G and m a y be identified with it; as the isomorphism b e t w e e n G and G is d e t e r m i n e d only up to right-translation, we m a y assume the identification to be made in such a w a y t h a t the point f ( e , So, e) of G is identified with the point So of G. As two G-isomorphisms of U' into G can differ only by a right-translation, the sets f ( U ' x {(s, 7)}), in the identification we have just made of w i t h G, m u s t a p p e a r as r i g h t - t r a n s l a t e s of U', i.e., as sets of the f o r m U'p(s, 7). We will now d e t e r m i n e the factors p(s, 7), or a t a n y r a t e the f a c t o r s p(so, 7) and p(s, e), since t h a t will be e n o u g h for our purpose. Since we have f ( e , So, e) = so, we have p(So, e) = so. L e t {s, s'} be in N ( S ) ; t h e n ~p(U' x {s}), ~ ( U ' x {s'}) have a point in common, which is the image by ~ of a point (u, s) of U' x {s} and of a point (u', s') of U' x {s'}, so t h a t we have u = u'f(s', s). Take a n y 7 in F, and p u t 7' = 7(s', s)7; t h e n the points (u, s, 7), (u', s', 7') have the same

460

[1960c] 380

ANDRI~ WEIL

image by f . On the other hand, by the definition of p(s, 7), we have f ( u , s, 7) = up(s, 7),

f ( u ' , s', 7') = u'p(s', 7') 9

This gives p(s', 7(s', s)7) = ~(s', s)p(s, 7) 9 If we call 8(s), ~(s', s) the images of 8(s), 7(s', s) in the given representation 7 --~ 9 of F, the definition of ~(s', s) can be w r i t t e n ~(s', s) = ~(s')q(s', s)~(s) -~

w i t h )~(s) defined by ~(s) = s~(s)-~(s) ;

we note that, by the definition of the elements 8(s), we have 3(So) = e, hence "8(s0) = e and ~(so) = so. Now, p u t t i n g p'(s, 7) = )~(s)-ap(s, 7), we can write as follows the relations obtained above: p'(s', 7(s', s)7) = ~(s', s)p'(s, 7) 9 L e t so, sl, . . . , s~ be such t h a t {s~-l, s~} ~ N ( S ) for 1 =
By induction on m, we deduce from the above formula t h a t we have p'(s0, 7) = @ ' ( s ~ , e) .

It was proved in no. 3 t h a t one can find a sequence so, s , . . . , s~ w i t h the property stated above, and such t h a t s~ = so and t h a t 7 is a n y given element of F. As we have p'(so, e ) = e, this shows t h a t we have, for every 7 in F, p'(so, 7) = ~, and therefore p(So, 7) = s0~. On the other hand, I? is the quotient of G, i.e., of G, by the f u n d a m e n t a l group F of I?, considered as operating on G; and the construction of the universal covering in no. 5 shows t h a t an element 7 of t h a t f u n d a m e n t a l group operates on G by mapping the class of (u, s, 7'), for the relation/~, onto the class of (u, s, 7'7); in particular, it maps f ( e , so, e) = So onto f ( e , so, 7) = p(so, 7) = s0~, i.e., it operates on it by right-multiplication by 7. But the operation on G of a n y element of the f u n d a m e n t a l group of is an automorphism of the G-structure and is therefore a right-translation; this shows t h a t it is the right-translation ~, and t h a t 17 is the

[1960c]

461 ON DISCRETE SUBGROUPS

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same as G/F, w h e r e P is the image of F by the r e p r e s e n t a t i o n 7 -~ 7. We now observe t h a t f ( U ' x {(so, e)})=- U'so is m a p p e d isomorphically onto its image ~p(U' • {so}) by the projection of G onto 1?. In view of w h a t we have just proved, this means t h a t we c a n n o t have u'so = u"So~ unless 7 = e; t h e r e f o r e we c a n n o t have ~ e so ~U'so unless 7 = e. F u r t h e r m o r e , applying the relations found above to the chain Cs -(so, sl, - - . , s~) w i t h s~ = s, we g e t

p'(So, ~(s)) = ~(s)p'(s, e) which, in view of w h a t we have a l r e a d y proved, gives p'(s, e ) = e and p(s, e) = )~(s). Now I? is the image of the union in G of the sets f(U'x

{(s, e ) } ) =

U'p(s, e)-= U's~(s)-l~(s),

which are respectively contained in the sets Us provided the ~(s) are close e n o u g h to the ~(s). I f we call K ' the closure of the union of the sets Us for s e S, this means t h a t we have G = K ' P , provided the r e p r e s e n t a t i o n 7 - ~ 7 is t a k e n close enough to the i d e n t i t y m a p p i n g of F onto itself. This completes the proof of t h e t h e o r e m s t a t e d in no. 1, for the case w h e n G is simply connected. 11. L e t us still denote by G the c o n n e c t e d and simply connected Lie g r o u p w i t h the given s t r u c t u r e , and let G' be a connected Lie group, locally isomorphic to G. We can t h e n i d e n t i f y G' w i t h a q u o t i e n t G/Z, w h e r e Z is a discrete subgroup of the c e n t e r of G, and the f u n d a m e n t a l g r o u p of G' is isomorphic to Z. As it is k n o w n t h a t G' is homeomorphic to the p r o d u c t of a maximal compact s u b g r o u p G" of G' w i t h an open ball, Z is also the f u n d a m e n t a l g r o u p of a compact g r o u p G" and is t h e r e fore finitely g e n e r a t e d . Call p the canonical m a p p i n g of G onto G'=G[Z. L e t 11' be a discrete subgroup of G' w i t h compact quotient-space G'/F'; p u t F = p-'(F'). T h e n F is a discrete subgroup of G, containing Z; we can i d e n t i f y F' w i t h F/Z and G/P w i t h G'[P', so t h a t G/P is compact; we m a y t h e r e f o r e apply to G and F t h e t h e o r e m in no. 1. L e t (70 be a set of g e n e r a t o r s for P, such t h a t t h e r e is a finite set R~(7) = e of fundam e n t a l relations b e t w e e n them; these m a y be chosen for instance as described above. L e t (~j) be a finite set of g e n e r a t o r s for Z; each of these can be expressed in t e r m s of the 7~; for each, t a k e one such expression, say ~j = Fj(7). T h e n F' is g e n e r a t e d by the 7[ = p(%) and the relations Rx(7') = e', Fj(7') = e' are clearly a f u n d a m e n t a l set of relations for the 7~, if e' denotes the n e u t r a l e l e m e n t of G'. L e t r0, r~ be the i d e n t i t y mappings of F and of F', respectively, onto themselves. L e t U be an open neighborhood of e in G, such t h a t p in-

462

[1960c]

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ANDRI~ WEIL

duces on U a h o m e o m o r p h i s m of U onto U' = p(U); let q~ be the homeomorphism of U' onto U, inverse to p. L e t r ' be a r e p r e s e n t a t i o n of [~' into G'; assume, first of all, t h a t , for e v e r y i, r'(7[) is in U'7',, and p u t P

=

t

t-I

9

T h e n we have p ( ~ ) = r'(7~). As the r'(7~) s a t i s f y the f u n d a m e n t a l relations w r i t t e n above for F', the e l e m e n t s R~(~), FJ(7) are all in Z; t h e y depend continuously upon the 7~, hence upon r', and coincide respectively w i t h e and w i t h the ~ if r ' = r3 if we t a k e r ' so close to r ' t h a t R~(~) is in U for e v e r y ~ and N~(7) in U~'j for e v e r y j , then, since U A Z = {e}, we m u s t have R~(~) = e, F~(7) = ~J for all ~ and j . This means t h a t t h e r e is a r e p r e s e n t a t i o n r of F into G, such t h a t r(%) = % for e v e r y i, and t h a t this induces on Z the i d e n t i t y mapping of Z onto itself. Moreover, it is clear t h a t r depends continuously upon r'. We have t h u s defined a continuous mapping r ' --~ r into ~ ( F , G) of a neighborhood of r ' in .~(F', G'), such t h a t , for e v e r y r', r induces t h e i d e n t i t y m a p p i n g on Z and t h a t p(r(7)) -- r'(p(7)) for e v e r y 7 e F. I t is now a trivial m a t t e r to v e r i f y t h a t , since the t h e o r e m in no. 1 holds for G, F and r0, it holds also for G', F' and r~. 12. Now, going back to the notations of no. 1, we consider again a connected Lie group G (which we do not assume to be simply connected) and a discrete group F with a finite set (71, " " , 7~) of g e n e r a t o r s and a finite set R~(7) = z of f u n d a m e n t a l relations b e t w e e n t h e m . As explained t h e r e , we can i d e n t i f y the space ~ = ~(F, G) of r e p r e s e n t a t i o n s of F into G w i t h the real-analytic subset of G (~) defined by the relations R~(g, 9 9 g~) = e; and the set ~t0 of those injective r e p r e s e n t a t i o n s r e for which r(F) is discrete w i t h compact quotient-space is an open subset of ~ . L e t X be a topological space or a differentiable or real-analytic manifold; let f be a continuous resp. differentiable resp. real-analytic m a p p i n g of X into G r such t h a t f ( X ) c ~ 0 ; for each x e X, write r~ instead of f i x ) ; r~ is then, for e v e r y x e X, a r e p r e s e n t a t i o n of F into G, w i t h the p r o p e r t i e s specified above. We m a k e F operate on the space X • G by p u t t i n g , for e v e r y x e X, g e G: (x,

=

(x,

g.

9

I t is clear t h a t F operates on X • G continuously resp. differentiably resp. real-analytically. Take any x e X, g e G; since in a n y case the mapping x--~r~ is continuous, the t h e o r e m in no. 1 shows t h a t t h e r e is a neighborhood Y of x in X, and a neighborhood U of e in G, such t h a t , for

[1960c]

463 ON D I S C R E T E SUBGROUPS

383

e v e r y y e Y, r~(7) e U implies 7 = e; if now U' is a neighborhood of e in G, such t h a t U ' - ' U ' c U, one sees a t once t h a t the neighborhood Y x g U ' of (x, g) in X x G has no point in common w i t h its image u n d e r a n y elem e n t 7 of G, o t h e r t h a n e. This means t h a t we can define the q u o t i e n t S = ( X x G)/F as a topological space, or as a differentiable or real-analytic manifold, as the case m a y be; X x G is t h e n a covering of S (its universal covering, if both X and G are simply connected). As the operations of F on X x G are compatible w i t h t h e projection f r o m X x G to X, t h a t projection d e t e r m i n e s a m a p p i n g p of S onto X, which is continuous, resp. differentiable, resp. real-analytic; for e v e r y x e X, p-'(x) is no o t h e r t h a n the compact manifold G/r~(F). By t h e t h e o r e m in no. 1, for e v e r y x c X, t h e r e is a neighborhood Y of x and a compact subset K of G such t h a t p - ' ( Y ) is the image of Y x / s in the canonical m a p p i n g of X x G onto S; this implies at once t h a t , to e v e r y compact subset K, of X, t h e r e is a compact subset Ks of G such t h a t p-l(K1) is the image of K, z Ks, hence compact; in o t h e r words, p is a p r o p e r mapping of S onto X. All this applies to the case X = ~0, f being t h e i d e n t i t y m a p p i n g of ~0 onto itself. Also, p has locally, in a sufficiently small neighborhood of each point of S, the same local properties as the projection f r o m X x G to X ; in particular, if X is a differentiable manifold, p is a differentiable m a p p i n g which has e v e r y w h e r e maximal r a n k (i.e., the Jacobian m a t r i x of p has e v e r y w h e r e a r a n k equal to the dimension of X ) . We now apply this to the case w h e n X is an open interval, say --1 < x < 1, w i t h its n a t u r a l analytic s t r u c t u r e . By means of a locally finite covering of S, we can p u t on S a differentiable ds~; using G r a u e r t ' s t h e o r e m , 1 we can even do this real-analytically. This ds ~ can be mapped back into X x G; this defines on X x G an analytic ds ~, i n v a r i a n t u n d e r F. The e l e m e n t a r y t h e o r y of differential equations shows now t h a t the orthogonal t r a j e c t o r i e s of the fibres {x} x G in X x G, i.e., the curves which (for the given ds ~) are a t e v e r y point o r t h o g o n a l to the fibre t h r o u g h t h a t point, m a k e up a f a m i l y of c u r v e s which can be w r i t t e n as g = F ( x , go), this being the equation of the orthogonal t r a j e c t o r y t h r o u g h the point (0, go); and F is an analytic f u n c t i o n of (x, go). If, for each x, we p u t F~(go) = F ( x , go), F~ is t h e n an analytic homeomorphism of G onto itself. As the differential equations of the orthogonal t r a j e c t o r i e s are i n v a r i a n t u n d e r F, we have, for e v e r y ~, e 1~: F

(g0 9

=

F

(g0) 9

i H. Grauert, On Levi's problem and the imbedding o.t real-analytic manifolds, Ann. of Math., 68 (1958), 460-471. It will be shown elsewhere that the same could also be done w i t h o u t m a k i n g use of G r a u e r t ' s theorem.

464

[1960c] 384

ANDR~ WEIL

Dividing by the equivalence relations determined in G by the operations of r0(F) and of r~(F), respectively, we see t h a t F~ determines an analytic homeomorphism of G/to(F) onto G/r~(F). Now it is known ~ that, on any connected real-analytic set, any two points can be joined ]~y a succession of analytic arcs. Therefore: With the notations of no. 1, let r, r' be two points in the same connected component o f ~o. Then there is an analytic homeomorphism F o f G onto itself, such that

F(g.r(7)) = F(g). r'(7) f o r every g ~ G and every 7 e F; and this determines an analytic homeom o r p h i s m o f G/r(F) onto G/r'(F). INSTITUTE FOR ADVANCED STUDY

2 A. H. Wallace, Sheets of real analytic varieties, Canad. J. Math., 12 (1960), 51-67 (see lemma 5.2(4), p. 66). The same result is already proved in H. Whitney et F. Bruhat, Quelques propri~t5s fondamentales des ensembles analytiques-rdels, Comment. Math. Helv., 33 (1959), 132-160 (see Prop. 2, p. 141), but the formulation, as given there, is slightly weaker than what we need.

[ 196 lb] Organisation et d6sorganisation en math6matique Le sujet que je voudrais aborder ce soir n'est pas proprement math6matique. La mathdmatique poss6de cette particularit6 de n'6tre pas comprise par les nonmath6maticiens. Mes coll6gues historiens de l'Institute for Advanced Study Princeton se plaignent de temps en temps qu'ils ne comprennent rien fi ce que j'6cris, tandis que je regarde parfois ce qu'ils 6crivent, en me figurant le comprendre, et que j'aie m6me l'audace d'avoir une opinion lb,-dessus. Mais aujourd' hui je sors des math6matiques. Toutes les critiques sur ce que je vais vous dire ne m'6tonneront pas du tout. Pourtant je dois vous pr6venir d'avance que je vais parler seulement pour les mathdmatiques et non pas pour les autres sciences; ce que je vais dire ne pourra 6tre appliqu6 ft, l'astronomie ni/t la physique. Un caract6re particulier des math6matiques est qu'il y a toujours eu et il y a encore aujourd'hui un certain degr6 d'unit6. Dans d'autres domaines, il y a tendance ~i des spdcialisations vari6es. I1 n'y avait pas autrefois de s6paration claire entre l'astronomie, la m6canique, la physique et m~me la chimie. Mais ces branches se sont s6par6es par la suite, et ~t pr6sent, m~me dans l'int6rieur de la physique, il se forme des sp6cialit6s qui sont compl6tement distinctes et entre lesquelles il y a peu de communication. En mathdmatique aussi, il y a bien entendu des sp6cialitds qu'on nomme analyse, g6omdtrie ou alg6bre. Mais ces sp6cialitds n'ont pas d'existence permanente et les lignes de ddmarcation se d6placent et changent continuellement. Parler de sp6cialit6s rigidement 6tablies en math6matique est certainement un non-sens. Les math6maticiens qui se sp6cialisent 6troitement se condamnent ~ une vue incompl6te de la math6matique contemporaine; l'essentiel est que chacun, m6me dans sa sp6cialit6, ait une connaissance suffisante de l'ensemble des math6matiques. I1 y a / t chaque 6poque un certain hombre de tels mathdmaticiens: aux alentours de 1900-1910, citons seulement Hilbert et Poincar6; dans la gdndration suivante il y avait Hermann Weyl. II me parait extr~mement important qu'il existe de tels mathdmaticiens et qu'il n'y ait rien qui encourage des chercheurs fi se compartimenter 6troitement. C'est 1~ le premier point de vue. Pour aborder le second point de vue, je commencerai par exlbliquer la diff6rence entre ce qu'on appelle "organisation" et ce qu'on appelle "organisme." Un organisme est quelque chose de vivant et une organisation est quelque chose de bureaucratique et m6canique. I1 est d'ailleurs essentiel que des organismes soient sains. Par exemple un cancer peut ~tre un organisme vivant ~ sa mani6re, mais qui n'est pas particuli6rement d6sirable pour qui lui donne l'hospitalit6. Quand je dis organisme, j'entends un organisme sain et c'est cela que je veux opposer au mot organisation. Reprinted fromBull. Soc. Franco-Jap. des Sc. 3, 1961, pp. 25-35, by permissionof the editors. 465

466

[1961b] Organisation et d6sorganisation en math6matique

Quand je parle de d6sorganisation, je tiens ",ice qu'il soit clairement entendu qu'elle n'est nullement incompatible avec la naissance et le d6veloppement d'organismes sainement constitu6s. J'ai connu quelques-uns des organismes de ce genre. E'un fi la naissance duquel j'ai pris une certaine part, c'est le groupe qu'on d6signe sous le nom de Bourbaki; l'autre fi la cr6ation duquel je n'ai pris aucune part, mais/t la vie duquel je participe actuellement, c'est l'Institute for Advanced Study fi Princeton. Je parlerai d'abord de ce dernier. Comme il y a 1/t un excellent exemple de ce qu'on peut appeler un organisme par opposition/t une organisation, il ne sera pas inutile que j'en dise quelques mots. I1 est caract6ristique d'un organisme du genre de ceux dont je parle en ce moment, qu'~, leur naissance ils sont cr66s sur une petite 6chelle sans aucune id6e pr6con~ue du but qu'ils recherchent. L'Institute a 6t6 cr66 aux envivons de 1930 par Flexner, qui a eu l'id6e de choisir les math6matiques comme sujet de recherches pour deux raisons. D'abord c'est leur universalit6: on peut faire appel ~ des math6maticiens de toutes les parties du monde et les diff6rences de langues sont si peu importantes entre eux qu'au bout de tr6s peu de temps ils s'entendent toujours. Le second point est que les math6matiques ne cofitent pas cher; qu'elles ont un rendement maximum pour une somme d'argent donn6e. La quantit6 d'argent qu'il faut pour les recherches mathdmatiques est infime en comparaison de ce que cofitent les sciences expdrimentales. Je ne parle pas de la physique nucldaire qui n'6tait qu'~ ses ddbuts/t cette 6poque-lfi. Mais d6jfi le radium cofitait tr6s cher. II n'y avait pas de cyclotron, mais il y avait un tas de gadgets qui cofitaient des sommes astronomiques. En math6matique, heureusement, on peut encore se d6brouiller avec un crayon et du papier. MSme une bibliothbque math6matique reprdsente une d6pense insignifiante aupr6s d'un laboratoire pour un physicien ou pour un chimiste, un h6pital pour un mddecin, etc. Flexner a donn6 de l'importance aux mathdmatiques pour ces raisons, et il a fait venir Hermann Weyl, cit6 tout ~ l'heure comme un exemple de math6maticien qui a compris et p6n6tr6 l'ensemble des math6matiques de son 6poque. C'est lui qui a d6velopp6 ~ ses d6buts la partie mathdmatique de l'Institute for Advanced Study, et fond6 la position importante qu'elle occupe aujourd'hui, celle qui consiste ~ servir de "clearing house" pour les id6es math6matiques. I1 est absolument essentiel, pour l'existence mSme des mathdmatiques, qu'il y ait dans le monde au moins un endroit qui serve de clearing house--un endroit off les id6es s'dchangent constamment non seulement darts l'int6rieur d'une mSme sp6cialit6 mais d'une spdcialit6 fi l'autre. Je ne sais pas ou 6tait le clearing house au temps d'Archim6de. C'6tait peut-Stre ",i Alexandrie. Mais comme plusieurs des m6moires d'Archim6de sont 6crits sous forme de lettres adress6es 5_ d'autres mathdmaticiens contemporains, il est manifeste que l'6change d'iddes entre math6maticiens de diverses rdgions du monde hell6nique constituait un facteur essentiel du progr6s math6matique ~ cette 6poque. Au d6but du 19e si6cle, le principal clearing house s'est trouv6/t Paris. I1 y est rest6 jusque vers 1880. A ce moment-l/t, les activitds de ce genre se partageaient entre Paris et G0ttingen, et pendant une dizaine d'ann6es apr6s la premiare guerre mondiale, c'6tait essentielle-

[1961b1

467 Organisation et d&organisation en mathamatique

ment G6ttingen qui a servi de clearing house. Vers 1930, Hitler a tout d6moli en Allemagne; l'activit6 de G6ttingen est tomb6e 5- z6ro tr6s rapidement. Paris, pour toutes sortes de raisons, a 6t6 en tr6s mauvaise condition pour reprendre ce r61e 5- ce moment-15-, ll y avait donc une place 5- prendre, et cette place pouvait 6tre prise 5- relativement peu de frais par l'Institute 5- Princeton, car il ne faut pas beaucoup d'argent pour faire venir un math6maticien d'une pattie du m o n d e 5l'autre. H e r m a n n Weyl a dirig6 les activit6s de l'Institute sans aucune id6e d'organisation et sans m~me consciemment avoir cette id6e de constituer 5- Princeton le clearing house. I1 partageait avec ses coll6gues yon Neumann, Veblen et Alexander l'id6e d'inviter les math6maticiens sans distinction de sp6cialit6s, de nationalit6s, seulement /l titre individuel, quand on pensait qu'ils pouvaient avoir quelque chose d'int6ressant/l dire. Q u a n d ces math6maticiens se sont trouv6s ensemble, ils ont commenc6 5. 6changer des id6es les uns avec les autres, et peu 5- peu le clearing house s'est form6. I1 a fallu ~ peu pr6s cinq ou six ans pour devenir ce q u ' o n peut appcler un organisme bien vivant et solide. La guerre l'a mis un peu en sommeil, mais apr6s la guerre, l'Institute a repris sa vie et s'est d6velopp6 un peu plus quantitativement, toujours sans aucune tentative de r6gle, de bureaucratie et de tout ce qu'on n o m m e organisation. J'ajoute que depuis une dizaine d'ann6es, Paris a recommenc6/~ jouer aussi, tr6s heureusement, ce r61e de clearing house grfice/~ des circonstances favorables: d ' a b o r d le n o m b r e de math6maticiens fran~ais de premier plan qui se sont trouv6s 5- Paris a beaucoup augment6 vers 1950, un grand h o m b r e de ceux qui se trouvaient en province sont venus /t Paris. Je n'ai pas besoin de dire que la centralisation excessive peut avoir des inconv6nients, mais en tout cas, la concentration de math6maticiens de premier plan 5- Paris pr6sente des avantages consid6rables; c'est un facteur de nature 5- attirer des math6maticiens 6trangers 5- Paris. Le C N R S a beaucoup aid6 5. cet 6gard. Mais j'ai constat6 avec regret qu'au cours des derni6res ann6es le C N R S s'est organis6 de plus en plus. L'invitation d'un math6maticien ne se fait plus qu'5- travers toutes sortes de formalit6s. Heureusement il est apparu dans ces derni6res ann6es encore un autre institut 5- Paris : l'Institut des Hautes Etudes Scientifiques. II est n6 en se modelant explicitement sur la ligne de l'Institute 5- Princeton. I1 n'est jamais tr6s bon de se modeler sur un autre institution au d6part, mais comme il est encore tout petit, il est compl6tement libre de bureaucratie autant queje sache, et il y a espoir qu'il se d6veloppe en un organisme vivant. Je vous ai promis de parler aussi d'un autre organisme: Bourbaki. Q u a n d ce groupe commen~a 5. se r6unir aux environs de 1934, il 6tait tr6s innocent et tr6s ignorant. I1 avait l'id6e naive que les math6matiques 6taient bien organis6es telles qu'elles 6taient dans les trait6s classiques, et qu'il 6tait temps de les refaire un peu mieux dans un esprit plus moderne. Q u a n d ce groupe commenga 5- discuter de cette id6e, il n'a pas fallu trois mois pour d6couvrir que la premi6re chose 5- faire 6tait de tout d6sorganiser. Nous avons donc mis toutes les math6matiques dans un chapeau et les avons secou6es 6nergiquement. Alors un tas de sujets ont commenc6 5- se grouper par eux-m6mes pour notre propre stup6faction, d'une mani6re naturelle mais 5- laquelle nous n'6tions pas habitu6s. Nous avons

468

[1961b] Organisation et d6sorganisation en mathdmatique

laiss6 faire et nous avons poursuivi dans cette voie en conservant dans nos discussions le caract6re soigneusement d6sorganis6. Dans une r6union de ce groupe, il n'y a jamais eu de pr6sident de s6ance. Chacun parle qui a envie de parler et tout le m o n d e a l e droit de l'interrompre. Si l'on s'apergoit qu'il n'a rien d'intdressant 5dire, on s'arrange pour le faire taire; et s'il veut continuer/t parler, il est oblig6 de crier fort. Ceux qui ont entendu le po6me symphonique "Till Eulenspiegel" de Richard Strauss peuvent avoir une idde de ce qui se passe; il y a 1~ un th6me du hdros Till Eulenspiegel qui repara~t de temps en temps, et/~ ce Till Eulenspiegel il arrive tout le temps des catastrophes 6pouvantables: les cuivres se mettent 5- tonner, son th6me dispara~t compl6ment, et puis, quand les cuivres sont essouttt6s, le th6me Till Eulenspiegel revient et finalement c'est lui qui gagne. Ce genre de chose est aussi arriv6 dans nos discussions, mais ne croyez pas que l'6tat actuel de Bourbaki refl6te ndcessairement l'opinion de celui de nous qui criait le plus fort. J'ai seulement voulu mettre en valeur le caract6re anarchique de ces discussions qui s'est conserv6 dans toute l'existence de ce groupe. Vous avez le droit de demander c o m m e n t un groupement aussi d6sorganis6 et aussi anarchique a pu arriver 5- des r6alisations concr6tes d'une vingtaine de volumes parus sous le n o m de Bourbaki. Remarquez bien qu'aucun de ces volumes n'est le produit d'un travail individuel. La bonne organisation aurait sans doute voulu qu'on assigne 5- chacun un sujet ou un chapitre, c o m m e il se fait dans les volumes bien organis6s. Mais l'id6e ne nous est jamais venue de faire cela. En math6matique qui a une structure logique, la structure vivante et organique nous parait beaucoup plus importante qu'une structure organisationnelle, si j'ose risquer ce barbarisme. C'est indispensable que le travail soit le travail collectif du groupe. I1 nous a paru 6tonnant 5-nous-m6mes qu'en travaillant dans ces conditions anarchiques, on soit arriv6 5- des r6sultats aussi concrets. C'est url ph6nom6ne assez 6tonnant que nous avons l'habitude d'attribuer 5- la coop6ration mystdrieuse de notre maitre Nicolas Bourbaki, mais il serait tr6s difficile de vous expliquer par quelle voie se produit cette coop6ration. Ce qu'il y a 5- retenir de cette expdrience sur un plan concret, c'est que tout effort d'organisation aurait abouti 5- un trait~ c o m m e tous les autres; la naissance d'un organisme vivant est un ph6nom6ne dont les conditions sont difficiles 5- d6crire et qui ne peut p a s s e r6p6ter 5- volont6. Je pense qu'il est temps d'arriver 5- une esp6ce de conclusion. Je tiens/t souligner encore une fois que j'ai parl6 seulement pour les math6matiques. Dans un autre domaine, en astronomie par exemple, il est parfaitement 6vident que pour explorer syst6matiquement toutes les rdgions du ciel, il faut une collaboration de t o u s l e s observatoires; et par cons6quent il est indispensable d'avoir une organisation qui assigne ses tfiches 5- chaque observatoire. Mais je consid6re que cela n'est pas n6cessaire en mathdmatique. Ce n'est pas par hasard, je pense, que dans les deux pays que je connais le mieux, la France et les Etats-Unis, les mathdmaticiens les plus s6rieux et les plus distingu6s ont fini par se d6sint6resser compl6tenent du fonctionnement de la soci6t6 math6matique de leur pays respectif. Ma conclusion est que, ce qu'il faut d6sirer, ce sont des circonstances favorables 5. la naissance d'un organisme sain, mais toute tentative d'organisation, autant que je puisse voir, en mathdmatique, non seulement n'est pas de nature/t favoriser la naissanse de tels

[1961b]

469 Organisation et d6sorganisation en math6matique

organismes, mais serait plut6t de nature fi emp~cher cette naissance. C'est pourquoi, en math6matique, je suis compl6tement en faveur de la d6sorganisation.

(R~sumO par la ROdaction de la confOrence.faite par M. A. Weil le 10 mai 1961 c) la Maison fi'anco-japonaise de Tokio)

[ 1962a] Sur la th6orie des formes quadratiques

Dans le premier de ses c616bres m6moires ([4]) sur les formes quadratiques, Siegel a 6tabli un th6or6me fondamental qui peut s'6noncer sous la forme suivante. Convenons de d6signer par N(S, 7") le nombre de solutions de l'6quation matricielle SIX] = T lorsque & T sont deux matrices sym6triques donn6es, /t coefficients entiers rationnels, d'ordre m et d'ordre n respectivement (avec n ~< m), et que la matrice X, / t m lignes et n colonnes, est assujettie ~ ~tre /t coefficients entiers rationnels; suivant l'usage, S[X] d6signe la matrice tX. S.X, off tX est la transpos6e de X. Si S et T sont les matrices de formes d6finies positives, il est clair que N(S, T) est fini; on posera dans ce cas :

t~(S, T) -----N(S, T)/N(S, S). Supposons que S e t T soient d6finies positives (c'est-/t-dire que les formes quadratiques de matrices S, T soient telles); et choisissons, pour chaque classe de formes du ((genre)) de S, un reprdsentant Si; le th~orSme de Siegel s'6crit alors comme suit :

(1)

y.

-F

=

1-[ av(s, r); V

dans cette formule, le coefficient ),(S) ne d6pend que du genre de S; v parcourt l'ensemble des (> du corps Q des nombres rationnels, ensemble qu'on peut considdrer comme ayant pour 616ments les nombres premiers rationnels et le symbole Go; enfin, pour chaque v, av(S, T) d6signe, en un certain sens, la <<mesure>) de 1'ensemble des solutions de l'6quation S[X] = T dans le comp16t6 Qv de Q pour la place v (compl6t6 qui est le corps des r6els s i v = 0% et le corps Qv des nombres p-adiques siv est le nombre premier p), et ne d6pend que du genre de S et de celui de T.

9

471

472

[1962a] Dans les m6moires suivants de la m~me s6rie, Siegel 6tendit la formule (I) aux formes quadratiques, d6finies ou non, sur un corps de nombres alg6briques; dans tous les cas, la structure de la formule qui exprime son th6or6me fondamental reste la m~me, mais les nombres fz(Si, T) qui y figurent ne peuvent se d6finir en g6n6ral au moyen des nombres de solutions enti~res de certaines 6quations; on les d6finit au moyen des volumes de ~<domaines fondamentaux>> pour certains groupes discontinus. Ce qui nous int6resse pour l'instant, c'est qu'on aboutit dans tous les cas une formule du type (1), et en particulier que le premier membre comporte toujours une sommation sur routes les classes du genre de S. Dans un travail ult6rieur ([5]), Siegel, en appliquant aux formes inddfinies une m6thode nouvelle, obtint pour celles-ci une formule plus pr6cise, qui donne la valeur de #(S, T); compte tenu de (1), ce nouveau r6sultat peut s'exprimer en disant que t o u s l e s termes du premier membre de (1) ont m~me valeur. Cependant, ce r6sultat n'est d6montr6 que moyennant une restriction suppl~mentaire sur le nombre n de variables de la forme T; supposant que S e t T sont ~t coefficients entiers rationnels et que S est 6quivalente, sur R, h une forme h p carr6s positifs et m -- p carr6s n6gatifs, Siegel est oblig6, pour appliquer sa m6thode, de supposer que

(2)

_~__3) n ~ inf (p, m -- p,

I m>2n+2 "

r >1 n

I1 6tait naturel de se demander si ces derniers r6sultats de Siegel sont vraiment ind6pendants des pr6c6dents, et si l'in6galit6 (2) est n6cessaire pour assurer leur validit6. Le but du pr6sent travail est de r6pondre h ces questions par la n6gative. Nous prendrons pour point de d6part l'id6e f6conde de Tamagawa, qui consiste ~t introduire dans la th6orie des groupes alg6briques le langage des <~ad61es>>. I1 6tait d6jh connu, d'apr6s Tamagawa, que le (<nombre de Tamagawa>> du groupe orthogonal (d6fini au moyen d'une forme quadratique non d6g6n6r6e sur un corps de nombres alg6briques) est toujours 6gal ~t 2, et que ce r6sultat est 6quivalent h la formule (1) pour le cas T = S. Le calcul du nombre de Tamagawa du groupe orthogonal est d'ailleurs effectu6 compl6tement dans mon cours de Princeton ([6]), par une m6thode qui, si elle est en substance assez proche (au langage pr6s) de la d6monstration de la formule (1) par Siegel, en est logiquement 10

[1962a]

473 ind6pendante. Ce r6sultat 6tant suppos6 acquis, nous nous proposons ici de d6montrer ce qui suit : (A) La formule (1), p o u r S et T non-d6g6n6r6es et n ~< m -- 3 (la m~me m6thode, convenablement modifi6e, permettrait de traiter aussi le c a s n = m -- 2, et celui off T e s t de d6terminant 0); (B) Le fait que si, dans le premier membre de (1), on groupe ensemble t o u s l e s termes correspondant ~ des formes Si appartenant ~ un m~me ((genre spinoriel)~, la somme obtenue a la m~me valeur p o u r t o u s l e s ((genres spinoriels)) contenus dans le genre de S. Lorsque S est une forme inddfinie, il r6sulte d ' u n th6or6me de Eichler et Kneser (cf. [2]), qu'il y a identit6 entre les ((genres spinoriels~) et les classes du genre de S; en ce cas, (B) permet donc de conclure que tousles termes du premier membre de (1) ont m~me valeur, ce qui est le r6sultat m~me de Siegel, relatif aux formes ind6finies, affranchi de la condition (2) et soumis seulement aux conditions 6nonc6es ci-dessus dans (A). Pour le c a s n = 1, ces r6sultats se trouvent d6j~ dans un travail r6cent ([3]) de M. Kneser; comme celui-ci a bien voulu me le communiquer, il les avait en fait obtenus p o u r n quelconque (y compris certains cas off l'on a n = m -- 2 ou det (T) -- 0); sa m6thode ne diffSre pas substantiellement de celle qui sera suivie ici. Si je me suis permis n6anmoins de revenir sur la question, c'est principalement afin de mettre en 6vidence le fait que la condition (2) ne tient pas h la nature du probl6me, ce qui ne peut apparaitre clairement tant q u ' o n se b o r n e au cas n = 1. 1. C o m m e corps de base, nous adoptons un corps de nombres alg6briques k, dont les compl&ions distinctes seront d6sign6es par kv, v p a r c o u r a n t l'ensemble des ((places)) de k; on notera p u n id6al premier de k, ainsi que la place correspondante, de sorte que les k~ d6signent toutes les compl&ions p-adiques de k; et on notera A l'anneau des ad61es de k. Une lois p o u r toutes, nous nous plagons dans un espace vectoriel de dimension m sur k, que nous identifions h k m par le choix d ' u n e base; nous nous y donnons une forme quadratique non d6g6n6r6e, d6termin6e, p o u r la base choisie, par la matrice sym6trique S ~t m lignes et m colonnes, h coefficients dans k. On d6signera par G le groupe o r t h o g o n a l O+(S), form6 des matrices X de d6terminant 1 satisfaisant ~ S[X] = S; suivant l'usage, on note Gg, Gk v 11

474

[1962a] ou plus bri6vement Gv, et GA, les groupes des solutions de det X = 1, S[X] ~- S, ~t coefficients dans k, dans kv, et dans A, respectivement; ces groupes sont munis de leurs topologies naturelles. Supposant d6sormais m t> 3, nous d6signerons par G;, le groupe des commutateurs de Gv (c'est-/t-dire, bien entendu, le sous-groupe de Gv engendr6 par les commutateurs d'616ments de Gv) et par G~ celui de GA. Nous noterons r la ((norme spinorielle)) de G; celle-ci d6finit, pour tout v, une repr6sentation continue de Gv dans le groupe fini kv/(kv) 2, et, sur GA, une repr6sentation continue de GA dans Ik/(lx) 2, Ix d6signant le groupe des idbles de k. Si S est d'indice/> 1 dans kv (c'est-/i-dire si la forme quadratique d6finie par S ((repr6sente z6ro>) dans kv), on sait que G~ n'est autre que le noyau de v dans G~; il en est ainsi en toute place h l'exception au plus des places/~ l'infini si m i> 5, et en presque toute place p o u r m =- 3 ou 4. Si S est d'indice 0 (c'est-/t-dire si la forme S ((ne repr6sente pas z6ro)~) dans kv, G~ est encore le noyau de ~ saul si m = 4 et si v e s t un id6al premier de k, ne divisant pas 2; dans ce dernier cas, G~ est un sous-groupe ouvert de Gv, d'indice 2 dans le noyau de v, et Gv/G~ est un groupe de type (2,2,2). On conclut aussit6t de l~t que G.~ est un sous-groupe ferm6 de GA, d'indice fini (6gal h une puissance de 2) dans le noyau G~i de ~; p o u r m ~ 4, on a G,~ = G j ; dans tous les cas, t o u s l e s 616ments de GA/G~ sont d ' o r d r e 2. De plus, au moyen d ' u n th6or~me &((approximation faible)) t o u t / t fait 616mentaire, on v6rifie imm6diatement que, m~me p o u r m = 4, on a G,[ G~c = GJi Gx; comme c'est seulement le groupe G,[ Gx qui nous importe dans ce qui suit, on pourrait donc, sans rien changer, y substituer GJi /i G,i. N o t o n s enfin (bien que cette remarque ne soit pas utile p o u r notre objet) que la norme spinorielle d6terlnine un isomorphisme de GA/G,~ Gx sur le groupe Ix/k* (ID z, dont la structure est ddtermin6e par la th6orie du corps de classes (c'est le groupe de Galois du compos6 de toutes les extensions quadratiques de k). 2. Pour tout sous-groupe ouvert Ga de GA, o n peut consid6rer les classes bilat6res dans GA suivant les sous-groupes Go et Gk; l'ensembte de ces classes s'appellera par d6finition le genre orthogonal d6termin6 par G e t Go. Si les Ui sont des repr6sentants des classes bilat6res en question, on aura d o n c :

(3) 12

GA = U Go U~ Gk.

[1962a]

475 I1 r6sulte d'ailleurs des travaux de Siegel que les classes d'un genre orthogonal sont toujours en nombre fini; mais nous n'aurons pas faire usage de ce r6sultat. Si de plus G,~ est comme ci-dessus le groupe des commutateurs de GA, l'ensemble Ga G.~ Gk est un sous-groupe ouvert de GA; les classes suivant ce sous-groupe s'appelleront les sous-genres du genre orthogonal d6termin6 par G e t Go; chacun d'eux est 6videmmerit r6union de classes du genre en question (qu'on appellera, par abus de langage, les classes de ce sous-genre). Comme Go GA Gk contient le groupe des commutateurs de GA, les sous-genres forment le groupe commutatif GA/GoGAG~; d'aprbs ce qui prdcbde, c'est un groupe discret dont tousles 616ments sont d'ordre 2. De plus, comme GA/Gg est de mesure finie (6gale au nombre de Tamagawa de G si la mesure est normalis6e convenablement), des r6sultats g6n6raux connus sur la mesure de Haar montrent que GA/G~G.~Gk est de mesure totale finie, donc que c'est un groupe fini puisqu'il est discret; bien entendu, cela r6sulte aussi, si l'on veut, du fait que les classes d'un genre sont en nombre fini. On en conclut que les sous-genres forment un groupe de type (2,2, ..., 2). Supposons en particulier que la forme quadratique d6finie par S soit ((ind6finie )), c'est-~-dire qu'il y air au moins une place /~ l'infini v de k telle que la forme S repr6sente z6ro dans kv. I1 s'ensuit alors du th6orbme d'approximation de M. Kneser (cf. [2]) que le noyau de la norme spinorielle, donc ~ plus forte raison le groupe GA, est contenu dans G~ Gk, donc aussi dans U-1 Go UGk quel que soit U ~ GA; cela donne UGA s Go UGg, et par suite

Go UG,~ G~ C G~ UGk. Comme G~ est le groupe des commutateurs de Ga, le premier membre n'est pas autre chose que le sous-genre UG9 GA Gg. On a donc montr6 que, dans le cas d'une forme ind6finie, il n'y a pas de distinction ~t faire entre classes et sous-genres d'un genre orthogonal. 3. Pour retomber sur les notions classiques de la th6orie des formes quadratiques, on supposera que k est le corps Q des nombres rationnels, que S est ~ coefficients entiers rationnels, et qu'on a pris Go = H Gary), off Gut v) ---- Gv quand v est la place ~ l'infini de k, et off Gate) d6signe, pour tout nombre premier p, l'ensemble des 616ments de Gp ~ coefficients entiers p-adiques. Soit alors R 13

476

[1962a]

le r6seau Z m des points fl coordonn6es enti6res dans Qm; pour tout p, soit de m~me Rp = Z~ (c'est l'adh6rence de R dans Q~ ). Pour chaque i, soit R(~) le transform6 de R~ par l'automorphisme de Q~ d6fini par la coordonn6e de Ui -1 relative ~t p (coordonn6e qui, par d6finition de GA, est un 616ment de G~). I1 y a alors, comme on sait, un r6seau R(~) et un seul dans Qm dont l'adh6rence dans Q'~ est R(~) quel que soit p. Soit S, la matrice qui exprime la forme quadratique d6finie par S lorsqu'on prend pour base de Qm, au lieu de la base canonique, une base (arbitrairement choisie) du r6seau R(t). On v6rifie facilement, dans ces conditions, que les S, sont des repr6sentants des classes du genre de la forme quadratique S, au sens classique de ces roots. Quant aux (( sous-genres )), ils ne sont pas autre chose dans ce cas que les ((genres spinoriels)) introduits par Eichler dans la th6orie des formes quadratiques (cf. [11). Un peu plus g6n6ralement, prenons pour k un corps de nombres : alg6briques; soit R u n r6seau de k m, c'est-~t-dire un module de type fini sur l'anneau des entiers de k, contenant m vecteurs lin6airement ind6pendants sur k; et, p o u r tout id6al premier p de k, soit R~ l'adh6rence de R darts k~. Posons G(~) = Gv pour toute place ~t l'infini de k; et, pour tout id6al premier p de k, prenons pour G ~) le groupe des 616ments de G~ qui transforment R~ en lui-m~me. Consid6rons le genre orthogonal d6fini par G e t par Go =/-/Gr ~. Les Ul 6tant comme plus haut des repr6sentants des classes de ce genre, d6signons de nouveau par R~ ) le transforms de R~ par la coordonn6e de U, -~ relative ~t p, et par Rr *) le r6seau dont l'adh6rence dans k'~ est R~ ~ quel que soit p. Les r6seaux R(i), dans l'espace k m muni de la forme quadratique S, constituent ~ nouveau des repr6sentants des classes du genre de S, au sens classique. On notera d'ailleurs qu'on peut, sans changer Go, remplacer les Rp par des r6seaux c~R~ respectivement homothdtiques aux R~o, condition que l'on ait c~ =- 1 pour presque tout p; en proc6dant ainsi, on pourra par exemple faire en sorte que la forme quadratique d6finie par S soit ~t valeurs enti~res sur R~ (ou bien que la forme bilin6aire d6finie par S soit h valeurs enti~res sur R~ x Rp, ces conditions 6tant 6quivalentes s i p ne divise pas 2), et qu'il n'y ait pas de r6seau homoth6tique/t R~, contenant R~ et distinct de R~, qui satisfasse ~t la m~me condition. I1 est clair qu'alors les r6seaux R~ ) ont la m~me proprietY. Ici encore nos ((sous-genres)~ ne sont autres que les ((genres spinoriels)). 14

[1962a]

477

4. D6sormais, nous conviendrons une fois pour toutes de prendre pour Go un sous-groupe ouvert de GA de la f o r m e / / G ( ~ ), off les G(~) sont soumis aux conditions suivantes : (a) G(~) - - Gv pour toute place ~t l'infini v de k; (b) pour tout id6al premier p de k , G(~) est un sous-groupe ouvert de G p ; (c) pour presque tout p, G(~~est le groupe des 616ments de G~ ~ coefficients entiers dans kp. On sait d'ailleurs que, si 0 est une repr6sentation lin6aire fid61e de G, ayant k pour corps de rationalit6, la condition (c) est 6quivalente h la suivante : (c') pour presque tout p, G(~) est le groupe des 616ments X de G:o tels que 0 (X) soit h coefficients entiers dans k~. On d6signera par Go~ le produit 17 Gv, 6tendu aux places h l'infini de k, et par Ge le groupe compact 1I G(~), off le produit est &endu aux id6aux premiers de k; on a Go = Go~ • Go. Sur le groupe alg6brique G, on choisira, une lois pour toutes, une ((jauge)), c'est-~t-dire une forme diff6rentielle (au sens alg6brique) invariante h droite et h gauche, de degr6 6gal h la dimension de G, ayant k pour corps de rationalit6. Cette forme d6termine, comme on sait (cf. [6]) une mesure de Haar mr sur chacun des groupes Gv; de plus, si on pose, pour tout id6al premier p ae k, tz(p) = mp(G(~)), le produit i n f i n i / / # ( p ) est absolument convergent. On peut donc d6finir sur Go la mesure produit des my; la mesure de Haar m sur GA qui coincide sur Go avec ID I-dlm(O)12Hmv (Off D est le discriminant de k, et dim ( G ) = m ( m - 1)/2) est ind6pendante du choix de Go, et du choix d'une jauge sur G, et s'appelle la mesure de Tamagawa sur GA. On en d6duit, par passage au quotient, une mesure (qui sera encore not6e m) sur le quotient de G.4 par n'importe quel sous-groupe discret de GA, et par exemple sur GA/Gk. Par d6finition, le ((nombre de Tamagawa)) de G est m (GA/Ge); on sait qu'il est 6gal ~t 2. i1 s'ensuit que m(GoGe/Ge) a une valeur finie, n6cessairement > 0 puisque Go est ouvert; cette valeur est 6gale h m(F) si F est un <(domaine fondamental)) pour Ge dans Go Ge, c'est-~t-dire une partie bor61ienne de Go Ge qui rencontre en un point et un seul route classe h droite suivant Ge contenue dans Go Gk; on satisfait ~t cette condition en prenant pour F un domaine fondamental pour le groupe Go ----- Go c, Gk dans Go; par exemple, on pourra prendre F = Fo • Gc, off F0 est un domaine fondamental dans Go~ pour la projection de Go sur le premier facteur Go~ du produit Go = Goo • Go. Si on note m~ la mesure de Haar dans Go~,produit de I D I-dim(a)/2 par le produit des mesures 15

478

[1962a] mv pour toutes les places ~t l'infini v de k, on a alors m ( F ) = mo~(Fo)Hff(p). Ce r6sultat peut aussi s'dcrire

m(ao GzjGk) = mo~(Goo/Go)Hff(,p), off, pour simplifier, on a 6crit Go au lieu de la projection de Go sur Go~. Si, dans cette formule, nous substituons U -1 Ga U ~t Gm off U est un 616ment quelconque de GA, elle reste 6videmment valable (les ,u (p) gardant les m~mes valeurs) h condition de substituer ~ Go le groupe

Go(U) = U-1GaU ~ Glc; ici encore, par abus de notation, nous ne distinguons pas entre Go(U) et sa projection sur G~. Comme m est invariante h gauche, cela donne : (4)

m(Ga UGklG,~)

-~

moo(GoolGo(U) )H ff(p).

Si dans cette formule on substitue '~ U les 616ments Ul fignrant dans (3), c'est-~t-dire les repr6sentants des classes du genre consid6r6, et q u ' o n fasse la somme des formules obtenues, il vient

(5)

m(GA/Gx) = H ff(p)~ m~o(Go~/Go(UO). i

C'est 1~ le nombre de Tamagawa de G; comrne nous le rappelions tout ~t l'heure, il est 6gal h 2. Dans le cas (( classique )) q u ' o n a consid6r6 au n ~ 3, ce r6sultat n'est pas autre chose que le th6orSme fondamental de Siegel sur la ((mesure d ' u n genre)), c'est-&-dire la formule (1) pour T = S; dans ce cas, les groupes Go(UO sont les ((groupes d'unit6s)) des classes du genre de S. I1 est 6vident d'ailleurs q u ' o n a u r a r obtenu une formule analogue en sommant seulement sur les classes d ' u n sous-genre : la formule serait la mSme, h un facteur prSs 6gal au nombre des sous-genres (done ~t une puissance de 2). En particulier, s'il s'agit d'une forme ind6finie, chacun des termes du second membre de (5) a la valeur 21-v si 2~ est le nombre des sous-genres (ou, ce qui revient au m~me dans ce cas, le nombre des classes). 5. Soit maintenant T u n e matrice sym6trique ~t n lignes et n colonnes, de d6terminant non nul, ~t coefficients dans k; une fois pour routes, nous supposerons que l'on a 1 ~< n ~< m -- 3 (done m >1 4). Nous d6signerons par H la vari6t6 affine d6finie, dans 16

[1962a]

479 l'espace ~t m n dimensions des matrices X ~t m lignes et n colonnes, par l'6quation S[X] = T; nous supposerons que H a au moins un point rationnel sur k, ou, ce qui revient au m~me en vertu d ' u n th6or6me classique de Hasse, que H a des points rationnels sur chacun des corps kv. On notera He, Hv, HA les ensembles de points de H ~ coordonn6es dans k, dans kv et dons A, respectivement. Choisissons une lois p o u r toutes un 616ment M de H~c, et d6signons par g le sous-groupe de G form6 des 616ments X de G tels que X . M = M . Si ml . . . . . mn sont les n colonnes de la matrice M, et si Vest le sous-espace de k m engendr6 par les vecteurs mi, T e s t la matrice, par rapport '2 la base (ml, ..., mn) de V, de la forme quadratique induite sur V par la forme d6termin6e par S dons kin; comme on a suppos6 T de d&erminant non nul, les m/ sont lin6airement ind6pendants, et le sous-espace W de k m orthogonal ~t V par rapport ~t S est suppl6mentaire de V dans k m, donc de dimension m -- n; de plus, la forme quadratique induite par S sur W est non d6g6n6r6e. Le groupe g est alors le sous-groupe de G qui laisse invariants tous les points de V, et il s'identifie d ' u n e mani~re 6vidente au groupe orthogonal de la forme induite par S sur W. La vari6t6 H n'est pas autre chose, dons ces conditions, que l'espace homog6ne G/g; plus pr&is6ment, le th6orSme de Witt montre, comme on sait, que l'application X - + X . M d6termine, par passage au quotient, des isomorphismes de Gk/g~ sur Hk, de Gv/gv sur Hv et de GA/gA s u r HA : en particulier, X - + X . M est une application ouverte de GA s u r HA, et, p o u r tout v, c'est une application ouverte de Gv sur Hv. Soit Go un sous-groupe ouvert de GA, et soit go = ga c~ Go; go est un sous-groupe ouvert de gA, et il est clair que l'intersection ga ,', Go UG~ de gA avec une classe du genre d6termin6 par G et Go est une r6union (qui peut &re vide) de classes du genre d6termin6 par g et go. Plus pr6cis6ment, consid6rons, dans HA, l'ensemble ouvert U - 1 G o M = (U -1 Go U) 9 U - 1 M , qui est une orbite p o u r le sous-groupe ouvert U -1 Go U de GA; et posons : Ho(U) = Hg ~ U - 1 G o M . L'ensemble Ho(U), qui peut ~tre vide, est 6videmment stable pour le sous-groupe Go(U) ---- Gk ~ U -1 GoU de Gg, et peut donc s'6crire 17

480

[1962a]

comme r6union d'orbites par rapport h ce groupe; si les MQ sont des repr6sentants de ces orbites, on pourra 6crire chacun d'eux sous la forme Me = U - 1 X e M avec Xe E Go, et aussi sous la forme Me = Y o M avec Ye e G~ puisque Gg op~re transitivement sur Hg. Si, dans ces conditions, on pose Vo = X~-~UYe, on v6rifie facilemerit que l'on a (6)

gA ,', G~ UG~---- IA g~ V~gk

et que les ensembles qni figurent au second membre sont disjoints. 6. Si de plus Ga satisfait aux conditions (a), (b), (c) du d6but d u n ~ 4, il est clair que go y satisfera aussi; on pourra alors appliquer la formule (4) d u n ~ 4 aux termes du second membre de (6). Pour cela, on choisira une jauge dans g, au moyen de laquelle on d6finira la mesure de Tamagawa m' dans ga, des mesures de H a a r mb dans les groupes gv, et, comme au n ~ 4, une mesure de Haar m~o dans le groupe go~. On peut alors, d'une mani6re et d'une seule, choisir une jauge dans H (c'est-&-dire une forme diff6rentielle au sens alg6brique, invariante par G, de degr6 6gal h la dimension de H, et ayant k pour corps de rationalit6) de telle sorte que la jauge dans G soit formellement le produit des jauges dans g e t dans H. Au moyen de cette jauge, on d6finira des mesures invariantes m~, m.~ dans les espaces homog6nes Hv, HA, et on aura, au sens de la th6orie des espaces homog~nes, m~ -- mv/m~, m ~ = mA/rn~4. Cela pos6, soit W un 616ment quelconque de GA; dans la formule (6), remplagons U par WU et Go par WGo W - l ; il vient : gA r~ W G t l U G k = [.J (gA ~ W G t l q

W-1) Vogk,

off les V0 sont d6termin6s comme suit: on 6crit l'ensemble H• ,~ U -1 Ga W - 1 M comme r6union d'orbites Go(U)" Mo distinctes par rapport au groupe Go(U) = Gx ,', U -1 Go U,

et, pour chaque ~, on 6crit Me -

18

U-1XoW-1M---- Y o M

[1962a]

481 avec Xq ~ Go, Yo ~ G~, puis Vo = W X -1 UYQ. Appliquons (4) au r6sultat ainsi obtenu; il vient

m'((gA ta WGa UGk)/g~) = ~ m~(go~/(gk ~ y~IGo(U) Ye))

]--I m'~(gp ~ Wp G~~) w~l),

off Wp d6signe la coordonn6e de W relative ~t p. Mais, pour chaque p, W~ G(~) W~1 M est une partie ouverte compacte de Hp, orbite de M par rapport au groupe W:oG (~) W~ 1" comme le stabilisateur de M dans ce groupe est l'intersection de celui-ci avec gp, on a donc :

m'~(gp c~ Wp G(~~) W~1) = mp(W~ G (p) W~I) 9m~(Wp G(o~) W ; 1 i ; 1. Au second membre, le premier facteur est 6gal ~t m~(G(~)), c'est-hdire h #(p), et le second ~t m~(H~P)) -1 si l'on a pos6, pour tout v 9

H(av) = G (v) Wv 1M. De plus, nous ~crirons aussi :

.o = [-[.(:'= Go w- M. C'est l~t une orbite de Go dans HA, et r6ciproquement toute orbite de Go dans HA s'6crit ainsi pour un choix convenable de W. On notera qu'en vertu de r6sultats g6n6raux 616mentaires sur les vari6t6s ad61iques (cf. [6]), H(o~) est, pour presque tout p, l'ensemble des points de H ~t coordonn6es dans l'anneau des entiers de k~; il est facile d'ailleurs de v6rifier cette assertion directement. La formule obtenue plus haut peut s'6crire maintenant : (7)

m'((ga ~ WG~ UGk)/gD 1--[ m~ (H~~))

= ]--[ #(.p) ~, m~(g~/(g~ c~ Y~IGo(U) Yo)), et les YQ qui figurent au second membre s'obtiennent en 6crivant (8)

He ~ U-1Ho = I,.J Go(U)Me,

Q

Me = YQM,

YQ e G~, 19

482

[1962a] off la r6union qui apparait au second membre de la premiere de ces formules est une partition du premier membre en ensembles disjoints. 7. Comme dans la formule (3) d u n ~ 2, d6signons par Ui des repr6sentants des classes du genre d6termin6 par Go dans GA. Si, pour chaque i, on substitue. Ui /1 U dans (8), on pourra 6crire (9)

H~ ~ UT,1Ho = (J Go(U~) "Mio, Mio = Y~oM, Yto c G~; 0

bien entendu, le nombre de termes qui figurent au second membre de la premibre de ces formules d6pend de i (pour certaines valeurs de i, il peut s'annuler). Si maintenant on fait la somme des formules obtenues en substituant Ui h U dans (7), on obtiendra (10)

m'(gA/gg) ]-~ m~ (H(ff )) 20

-= ]--[ tt(p) ~_, m~(go~/(gk ,~ Y ~ Go(U 0 YIQ)). On peut obtenir une formule plus pr6cise en faisant porter la sommation, non pas sur tousles U,, mais seulement sur les repr6sentants Uj des classes d ' u n sous-genre UGo G:4 G~; on 6crira 27' au lieu de X, pour indiquer que la sommation porte seulement sur ces Uj, pour un sous-genre U d o n n 6 . Au lieu du facteur m'(gA/gD du premier membre de (10), il faudra alors 6crire (i 1)

m'((gA ~ WUGo G'A Gk)lgg).

Mais on a GA = gA Go Gd G~; cela r6sulte par exemple du fait que, puisque m -- n >/ 3, g contient un sous-groupe g' isomorphe h u n groupe orthogonal ~ 3 variables, donc isomorphe au quotient d ' u n groupe de quaternions de norme ~ 0 par son centre; pour ce dernier, la (<norme spinorielle>> n'est autre que la norme quaternionique, d'ofi r6sulte aussit6t que tout id61e dont les coordonn6es l'infini sont > 0 est norme spinorielle d ' u n 616ment de gA, donc ~t plus forte raison d ' u n 616ment de gA; cela donne GA =gA Go~G'fi, off G~i est le noyau de la norme spinorielle dans GA, &Off (puisque G~i C GA GD le r6sultat annonc6. I1 s'ensuit qu'il y a un 618ment Z de ga tel que WU ~ Z G o G d G~; on a alors

gA r~ WUGo G~ G~ ~ Z " (gA r~ Go GA Gg), 20

[1962a]

483 ce qui prouve que (11) a une valeur ind6pendante de WU; comme la somme des valeurs de (11), lorsqu'on y substitue ~ U des reprdsentants de t o u s l e s sous-genres, est m'(gA/gk), on voit que (11) a la valeur 2-Vm'(gA/g~), Off 2~ est le nombre des sous-genres dans G.4. Cela donne : (12)

2-:'m'(gA/g~) l---[ mp(H(~)) p t

= [ I .(p) P

,, r # Go(V,)Y.)), J,~

off, comme il a 6t6 expliqu~, la sommation du second membre porte sur les repr6sentants Uj des classes d'un sous-genre UGaGAGk. En particulier, s'il s'agit d'une forme ind6finie S, chaque sous-genre ne contient qu'une classe, et il n'y a pas ~ sommer sur j au second membre. Dans les formules (10) et(12), les premiers membres contiennent les <(composantes ~)H{~) d"-une orbite arbitraire Ho = Go W-~M de Go dans HA, et les seconds membres contiennent les 616ments Yie qui sont d6finis ~ partir de cette orbite au moyen de (9). Consid6rons alors plus g6n6ralement, au lieu d'une orbite, toute partie de HA de la forme K = 11 Kv, o~ les Kv sont des parties des Hv satisfaisant aux conditions suivantes : (a) pour tout v, Kv est stable par rapport ~ Gr ) (ce qui implique que K est stable par rapport Go, et que, lorsque vest une place A l'infini, K~ = Hv); (b) pour tout id6al premier p de k, Kp est compacte; (c) pour presque tout p, Kp est l'ensemble des points de Hp dont les coordonn6es sont des entiers de k~. Ces conditions impliquent que K est r6union finie d'orbites de Go dans Ha. Cela pos6, si on substitue Kp ~ H (~) dans le premier membre de (10), ou dans celui de (12), ce premier membre devient une fonction additive de K. De m5me, si on substitue K h Ho dans (9), l'ensemble des Yio d6fini par cette relation d6pend additivement (en un sens 6vident) de K, de sorte que les seconds membres de (10) et de (12) en d6pendent additivement aussi. I1 S'ensuit que, si on substitue K ~t Ha dans (9), et Kp ~t H ~ ) dans (10) et (12), les formules obtenues restent toujours valables. En d'autres termes, (10) et (12) restent valables pourvu qu'on suppose que les H ~ ) (au lieu d'etre des orbites pour les groupes G(g)) satisfont aux conditions (a), (b), (c). Par exemple, si S e t T sont h coefficients entiers dans k, et si on a pris pour G(g), pour tout p, 21

484

[1962a]

l'ensemble des 616ments de G~ ~t coefficients entiers dans ku, on satisfera aux conditions impos&s en prenant aussi p o u r H(~ ), p o u r tout p, l'ensemble des 616ments de Hp ~t coefficients entiers dans kv. On notera que le calcul ci-dessus ne d6pend nullement de la connaissance du n o m b r e de T a m a g a w a m'(gA/gk); nous avons eu besoin seulement de savoir que ce n o m b r e a une valeur finie. Pour retrouver le r6sultat final de Siegel, il suffit de remplacer ce n o m b r e par sa valeur (6gale h 2, c o m m e on sait) dans les formules obtenues; (10) et (5) donnent les th6or6mes de Siegel dans le cas de la th6orie g6n6rale des formes quadratiques, tandis que (12) contient le r6sultat relatif aux formes ind6finies. Si T e s t de d6terminant nul, ou bien si n = m -- 2, le groupe g cesse d'&re semisimple. Presque rien n'est cependant /t changer aux calculs ci-dessus tant que c'est l'extension d ' u n groupe orthogonal par le groupe additif d ' u n espace vectoriel (ce qui est le cas si T est de rang r, si m -- 2 n -k r i> 3, et si on prend p o u r H la vari&6 form6e par les matrices X de rang n qui satisfont S[X] = T). De toute fa9on, les consid6rations expos6es ici se pr& tent ~t des g6n6ralisations assez vari6es, qu'il n'entrait pas dans notre p r o p o s d'examiner p o u r l'instant.

BIBLIOGRAPHIE

[1] M. EICHLER, Quadratische Formen und orthogonale Gruppen, J. Springer, Berlin 1952. [2] M. Kr,msm~, Klassenzahlen indefiniter quadratischer Formen in drei oder mehr Veranderlichen, Arch. d. Math. 7 (1956), 323-332. [a] M. KNESER, Darstellungsmasse indefiniter quadratischer Formen, Math. Zeitschr. 77 (1961), 188-194. [4] C.L. SIEGEL, l~ber die analytische Theorie der quadratischen Formen : (I) Ann. of Math. 36 (1935), 527-606; (II) Ann. of Math. 37 (1936), 230-263; (III) Ann. of Math. 38 (1937), 212-291. [5] C.L. SmGEL, On the theory of indefinite quadratic forms, Ann. o f Math. 45 (1944), 577-622. [6] A. WF.IL,Adeles and algebraic groups, Lecture-notes, Institute for Advanced Study, Princeton 1961.

22

[ 1962b] On discrete subgroups of Lie groups (II) 1. This is a continuation of m y p a p e r [7] with the same title, which will be r e f e r r e d to as D ~, and w h i c h h a s to be s u p p l e m e n t e d in t h e m a n n e r described below in Appendix I. Indeed, t h e p r e s e n t paper is n o t h i n g else t h a n a combination of t h e ideas of D E with the m e t h o d of variation of s t r u c t u r e , as applied to special t y p e s of semisimple groups by Calabi and by Calabi-Vesentini in t h e i r r e c e n t work 1. We shall mainly be concerned w i t h semisimple groups w i t h o u t compact components, i.e., with connected semisimple Lie groups h a v i n g no conn e c t e d compact normal subgroup. H o w e v e r , we begin by considering a n y connected Lie g r o u p G h a v i n g a t least one discrete subgroup w i t h compact quotient; this implies t h a t G is unimodular (it would even be e n o u g h for this to assume t h a t G has a discrete subgroup H s u c h t h a t G/H has a finite m e a s u r e , for the m e a s u r e d e t e r m i n e d on G / H b y a r i g h t - i n w x i a n t m e a s u r e on G). L e t n be the dimension of G; choose a basis )(1, . . . , X~ for the space of r i g h t - i n v a r i a n t vector-fields on G; we have

(1)

[ x . x~] = ~ 1

c~X~

(1 <=~, ~ <=n),

w h e r e the c~, are the c o n s t a n t s of s t r u c t u r e of G. The X~ will be considered, in t h e usual m a n n e r , as differential operators a c t i n g on functions on G. To say t h a t t h e y are r i g h t - i n v a r i a n t means t h a t t h e y c o m m u t e w i t h r i g h t - t r a n s l a t i o n s , so t h a t t h e y are t h e infinitesimal operators belonging to the group of l e f t - t r a n s l a t i o n s on G. The dual basis to the X~, in t h e space of differential f o r m s on G, consists of the f o r m s w x given by i(X~)w~ = g~, w h e r e i is t h e interior product and ( ~ ) is t h e u n i t - m a t r i x . T h e n we have, for e v e r y f u n c t i o n f on G: (2)

df = ~=~ Xff. w~

o

The w ~ are a basis for t h e r i g h t - i n v a r i a n t differential f o r m s on G. On G, we have t h e r i g h t - i n v a r i a n t differential f o r m w~A - . - A w~; G being o r i e n t e d so as to m a k e this positive, we can use it as volume-element on G and denote it, as such, by d~2. As G is unimodular, this m u s t be invaria n t u n d e r left-translations, or, w h a t a m o u n t s to the same, u n d e r t h e Cf. [3], [4] and [1]. I a m also g r e a t l y indebted to Calabi for p e r m i s s i o n to m a k e u s e of n o t e s on s e m i n a r lectures g i v e n by h i m on this s u b j e c t in P r i n c e t o n in 1958-59. 578 Reprinted from Ann. of Math. 75, 1962, by permission of Princeton University Press.

486

[1962b]

487 579

DISCRETE SUBGROUPS

infinitesimal operators X~; in the usual notation, this can be w r i t t e n

O(X~).co 1 A . . .

Aw ~=0

(1-<~-
Using H. C a r t a n ' s well known i d e n t i t y O(X) = i(X)d + di(X), we get: d(to 1 A " " A w ~-~A w ~+~A " " A w ~ ) = 0 . T h e r e f o r e , if H is a n y discrete subgroup of G such t h a t G/H is compact, we have, by Stokes's formula:

(3)

fa/nX~'f'dt2----

f~/~'+-d(f'w'A

...

A o/'-' A o)~'+' A - . .

A w") = 0

for all ~ and all f u n c t i o n s f on G/H. 2. Now consider a group F and a o n e - p a r a m e t e r f a m i l y t--+ rt of r e p r e s e n t a t i o n s of F into G; we assume t h a t ro is injective, r0(F) is discrete in G, G/ro(F) is compact, and also t h a t t -+ r~(7) is of class C~ f o r every 7 e F. We can t h e n apply the results of D ~, no. 12, s u p p l e m e n t e d by Appendix I of this paper; this says t h a t t h e r e is an open i n t e r v a l I cont a i n i n g 0, such t h a t , if F is made to act on I x G b y

(t,

= (t, x .

(t G I,

c C,

r),

it o p e r a t e s f r e e l y and properly on I x G. T h e n the quotient-space S - (I x G)/F is a s e p a r a t e d manifold of class C ~, and t h e m a p p i n g (t, x) -~ t of I x G onto I d e t e r m i n e s on S a p r o p e r m a p p i n g 7v of S onto I; f o r each t e / , t h e "fibre" 7v-~(t) can be identified in an obvious m a n n e r w i t h the compact manifold G/r~(F). The vector-fields Xx d e t e r m i n e in an obvious m a n n e r n vector-fields on I x G; as these are i n v a r i a n t u n d e r F, t h e y d e t e r m i n e n vector-fields on S; these vector-fields, on I x G and on S, will also be denoted by Xx; t h e y are, a t e v e r y point (t, x) of I x G (resp. a t e v e r y point M o f S), t a n g e n t to the fibre t x G (resp. 7v-I(~(M))) t h r o u g h t h a t point, and t h e y s a t i s f y (1). We can also m a k e G act on I x G by s - (t, x) = (t, sx); as these operations c o m m u t e w i t h those of F, t h e y d e t e r m i n e operations of G on S; t h e corresponding infinitesimal t r a n s f o r m a t i o n s are t h e linear combinations of t h e Xx 'with c o n s t a n t coefficients. 3. Now let ~ be t h e class of all vector-fields Y of class C~ on S such t h a t Y~r = 1. By an obvious identification, this m a y also be considered as t h e class of t h e vector-fields Y of class C ~ on I x G, i n v a r i a n t u n d e r F, such t h a t Yt = 1 (where t m e a n s t h e f u n c t i o n (t, x) -+ t on I x G). On I x G, a vector-field Y satisfies Yt = 1 if and only if it can be w r i t t e n as Y = 8/8t + }-]~xq~X~, w h e r e t h e ~o~ are functions on I x G; b u t this does not show how to choose the cp~ so t h a t Y m a y be i n v a r i a n t u n d e r F. How-

488

[1962b]

580

ANDRI~ WEIL

ever, it is easily seen t h a t ~ is not e m p t y ; in fact, by means of a locally finite covering, put on S a d s ~ of class C~, and t a k e for Y, a t e v e r y point M of S, the v e c t o r which is orthogonal to the fibre 7c-t(z(M)) t h r o u g h M and satisfies Y~ = 1; this belongs to ~. If Y e ~, the vector-fields in are those of the f o r m Y § Y':.x~ X ~ , w h e r e t h e ,px are functions of class C~onS.

T a k i n g Y e ~, w r i t e now [Xx, Y] as a linear combination of Y and the Xx. Since we have YTc = 1 and XxTc = 0, we have [Xx, Y]7~ = 0, so t h a t Y c a n n o t occur in [Xx, Y]; this gives: (4)

[Xx, Y] = ~ , . n: l f x .t~ X ~ .

The Jacobi i d e n t i t y gives: (5)

Xxf~ -- X,f~

= G p ( c [ , f ; + c ; p f f -- c~pfg) .

Also, if we s u b s t i t u t e Y' = Y + ~ x PxXx for Y, t h e f ~ are replaced by (6)

f ' ~ = f ~ § Xxq~~ § ~ p c~p~p .

Suppose now t h a t , for two vector-fields Y, Y' in ~, we have [X~, Y ' - - Y] = 0 for all ~; since Y' -- Y is a linear combination of t h e X~, this means t h a t Y ' -- Y induces, on each fibre t x G of I x G, a vector-field which comm u t e s with all the Xx, i.e., a l e f t - i n v a r i a n t vector-field. This m u s t a t the same time be i n v a r i a n t u n d e r P, i.e., u n d e r t h e r i g h t - t r a n s l a t i o n s r~(7) for all 7 ~ F. As r i g h t - t r a n s l a t i o n s act on t h e l e f t - i n v a r i a n t vector-fields by the adjoint group of G, this means t h a t , for each t, Y' -- Y is a left-inv a r i a n t vector-field belonging to t h e subalgebra n~ = ~ ( r j F ) ) of t h e Lie algebra ~q of G which consists of the vectors i n v a r i a n t u n d e r Ad(r~(7)) for e v e r y 7 e F. 4. W e n o w m a k e the a d d i t i o n a l a s s u m p t i o n t h a t n, = {0} f o r a l l t c I (for our immediate purposes, it would be enough to assume t h a t t h e dimension of ~t~ remains c o n s t a n t for all t in some neighborhood of 0). As shown in Appendix II, this is c e r t a i n l y the case w h e n e v e r G is semisimple w i t h o u t compact components. Then, if Y, Y', f ; , , f ' ~ are as above, f ' ~ = f ~ for all k., ~ implies Y' = Y. Now we seek to d e t e r m i n e Y' e ~ so t h a t , for each t e L t h e i n t e g r a l

I.. ~ ~ x , ~ ( f'~)~df~ is a minimum. Choosing Y c ~ arbitrarily, and p u t t i n g Y ' = Y § ~p~X~, we can express this by saying t h a t the ~x are to be d e t e r m i n e d for each t in such a w a y t h a t the integral

f~--l(t) ~ ~(f~ + x ~ + ~

[1962b]

489 581

DISCRETE SUBGROUPS

should be a minimum. F o r a given t, this is a variational problem of classical type, and standard t e c h n i q u e s show (in view of t h e f a c t t h a t f ' [ = f ~ implies Y ' = Y) t h a t this has for each t a unique solution. I owe to H h r m a n d e r t h e proof of the f a c t t h a t this solution is of class C~, not only on each fibre ~ l(t), b u t even on t h e manifold S; his proof will be found below in Appendix III. Replacing now Y by Y', we m a y assume t h a t Y itself is t h e solution of our variational problem; t h e n we m u s t have, for all choices of t h e ~x, t h e E u l e r equations +

~--l(t )

=

0

9

if we t r a n s f o r m this by m e a n s of (3), we get: (7)

~

X~f~ + ~ . ~ c~f~ = 0 .

8. Now let us assume t h a t t h e r e is a subgroup G' of G, connected or not, such t h a t t h e l e f t - t r a n s l a t i o n s by elements of G' leave the quadratic f o r m s ~-~ (wa) = invariant; this is to say t h a t , for e v e r y s e G', Ad(s) induces on the Xa an orthogonal s u b s t i t u t i o n Xa -~ ~ a~X,. Then, if Ys is the t r a n s f o r m of the vector-field Y u n d e r t h e operation (t, x) --* (t, sx) of s, (4) gives

[ ~ ~X~, r~] = E ~ , ~ f ~ x ~ , which, in view of t h e o r t h o g o n a l i t y of (a[), m a y also be w r i t t e n

[Xx, Y~] = ~,,~.pfi~o~v~X,. This shows a t once t h a t ~--~.x,,(f~)~ is u n c h a n g e d if Y is replaced by Y , so t h a t Y~ is also a solution of t h e variational problem by which we determined Y. As t h a t solution is unique, this proves t h a t Yis i n v a r i a n t u n d e r G'. In particular, if G' is a Lie subgroup of G, Y m u s t be i n v a r i a n t u n d e r the infinitesimal t r a n s f o r m a t i o n s of G'; this is the same as to say t h a t [X, Y] = 0 for all t h e vectors X in t h e Lie algebra of G'. Also, this shows t h a t Y is i n v a r i a n t u n d e r t h e c e n t e r of G. 6. The above results will now be applied to the case w h e n G is a conn e c t e d semisimple group w i t h o u t compact components, w i t h t h e Lie a l g e b r a ~; as proved in A p p e n d i x II, our assumption n: = {0} is verified in t h a t case. L e t f be t h e Lie a l g e b r a of a maximal compact subgroup of Ad(G); we can consider it in an obvious m a n n e r as a subalgebra of ~; let K be t h e connected subgroup of G w i t h the Lie algebra ~; let n - r be its dimension. L a t i n indices i, j , . . . , will r a n g e f r o m 1 to r , while the Greek indices a,/~, . . . will r a n g e f r o m r + i to n (and t h e indices ~,/~, . - . f r o m 1 to n as before). We choose X , . . - , X , so t h a t the X~, for

490

[1962b] 582

ANDRI~ WEIL

r + 1 <= (~ <= n , are a basis of f and t h a t t h e Killing f o r m of G is r

2

_

2

It is k n o w n t h a t X~ -* --X~, X~ -* X~ is t h e n an involutory a u t o m o r p h i s m of g; t h e r e f o r e , a m o n g t h e c ~ ~ cr can be :~0. We ~, only t h e c~,c~-, , c~j, shall w r i t e c ~ , c~. instead of c~, c~. W r i t i n g t h a t t h e Killing f o r m is i n v a r i a n t u n d e r t h e adjoint r e p r e s e n t a t i o n , one sees t h a t the c~,~ are altern a t i n g in all t h r e e indices a , / 9 , 7, and also t h a t we have 6t3

This implies t h a t the infinitesimal operators X~ annul the f o r m ~V" : . . d t vt(o~ \ / ~ so t h a t K leaves t h a t f o r m invariant; as it leaves t h e Killing f o r m inv a r i a n t , t h e f o r m ~ x (~ ~ is also i n v a r i a n t u n d e r K. The above f a c t s imply t h a t [X~, [X~, X~]] is a linear combination of t h e X~; we write, as usual [X,, [Xj, X~]] = E : : ~ R ~ , X ~ , w h e r e t h e R~jk are given by

t h e y can be i n t e r p r e t e d geometrically as t h e Riemann c u r v a t u r e t e n s o r for the r i e m a n n i a n s y m m e t r i c space G / K . T h e y have t h e obvious symm e t r y properties (9)

R,~jk = R~k~ = --R~.~0 = - - R ~ j

;

moreover, the Jacobi i d e n t i t y b e t w e e n X~, Xj, Xk gives the well-known relations (10)

R~

+ R~

+ R,~j = 0 .

Finally, w r i t i n g t h a t the coefficient of (o*(o~ in the Killing f o r m is 2~j, we get

which can also be w r i t t e n as =

7. Now we come back to t h e solution Y of t h e variational problem of no. 4. Applying to K the r e s u l t of no. 5, we see t h a t [X~, Y] = 0 for all a, i.e., f ~ = 0 for all ~, c~. W r i t e f~j, f ~ instead of f~, f~; t h e equations (5), (7) become: (12)

X~f~

-- X~f,~ = E ~ ( c ~ f ~

-- c ~ i ~ f ~ )

[1962b]

491

583

DISCRETE SUBGROUPS

(13)

x~fo~ - X~f~ = ~

(14) (15) (16)

~

(c~A~

x,A~ + ~

- co~A~)

~c~fo~ = o

(we omit one additional relation which is not needed for our purposes). Now put:

1~ ~.j,~(X~f~i

09 = ~

-- X i f { k ) ~

We want ~ to evaluate the integral I(r + ~F)d~2, taken on the fibre

7c-1(t).

J

Using (12), we see that we have ,~ = ~ . ~ . ~ X ~ A ~ ( X ~ f .

-

Xjf~)

= ~ , ~ , j , ~ (c~ikf~j" X~f,~ -- c ~ f ~ "

X~f~) ,

and therefore, using (3):

with 9 ' = ~,,,,j.~ (coJ~.

X ~ f , ~ - c,,~f,~. X k f , ~ ) .

Similarly, using (14), we can write

and therefore, as above,

with ,t"= ~.~.~

c ~ , k f ~ , xif~,o 9

Now, using (13) and (15), one finds: (P' § ~F' = - - r ( f ) ,

where F is the quadratic form in the f~j given by F(f)

= - - ~ . j . ~ . ~ (R~j,~kf~f~k + R, kj~f, jf~k + R,k~kf, j A ~ ) ,

e T h i s decisive step in our proof is modelled a f t e r t h e w o r k of Calabi on discrete g r o u p s of d i s p l a c e m e n t s in s p a c e s of c o n s t a n t n e g a t i v e c u r v a t u r e ; cf. footnote 1.

492

[1962b]

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ANDR]~ WEIL

which, in view of (9), (10) and (11), can also be w r i t t e n F ( f ) = }_~,,j (Aj) ~ + ~ , , j ~,~ R,~kjf~jf~k 9

8. It will now be proved t h a t F ( f ) >= 0; more precisely, the quadratic f o r m F is positive and n o n - d e g e n e r a t e unless a t least one of t h e simple components of the Lie algebra of G is of dimension 3. In fact, we can w r i t e f = f ' + f " , w h e r e f ' = (f:j) is s y m m e t r i c and f " = (f[S) is altern a t i n g in the indices i, j. If we define the bilinear f o r m F ( f ' , f " ) by F ( f ' , f " ) = F ( f ' + f " ) -- F ( f ' ) -- F ( f " ) ,

we have, in view of (9): F(f',f")

P " ~ ~er ,err . ~f~f~5 + Z.~,j.~, ~R ~k~J~jJhk ,

= ~

as t h e first sum changes sign if we e x c h a n g e i, j , and the second sum does so if we e x c h a n g e i with j and h w i t h k, this is 0. T h e r e f o r e F ( f ) = F ( f ' ) + F ( f " ) . Also we have, for a similar reason

and t h e r e f o r e , using (8), (9) and (10): ~

.

~ h ~ .R i l t k.JrJ,i j, Jrh,. k, -- .

1 X-~

2. Z--'l J ~ ~

R ~ j n krJ ~, ,j 2r~, k,

~

1

-~o~

Ig-" c c ~ i J r,,l~ ~..g~,j J i 3"]

"

This proves t h a t F ( f " ) >= O, and also t h a t F ( f " ) can be 0 only if f " = 0. Now we w a n t to show t h a t F ( f ' ) > O. In fact, if we make on the X~ a n y orthogonal substitution X~-* ~ j ~ i X ~ , this will c h a n g e f~j into ~ ~f~, and t h e r e f o r e the quadratic f o r m Y~,~,J~T~T~ into

we m a y t h e r e f o r e choose t h a t orthogonal substitution in such a way t h a t a t a given point, the quadratic f o r m ~ , ~ f ~ T~ T~ becomes a diagonal f o r m Y~ a~ T:. T h e n we have, a t t h a t point, fl[~ = a ~ j . B y considering t h e effect of the same substitution on the R ~ , one sees immediately t h a t it does not c h a n g e the value of the f o r m F ( f ' ) . T h e r e f o r e , in order to show t h a t F ( f ' ) >= O, it is enough to show t h a t this is so a t a point w h e r e fi'~ = a ~ . Now, a t such a point, we have F(f') = ~

a~ + ~ , ~ r~a~a~ ,

w h e r e we have put r~j :

--R~H : ~-~'~( c ~ ) ~ ,

so t h a t we have r~j = r ~ >= 0 and also, in view of (11):

[1962b]

493 585

DISCRETE SUBGROUPS

(17)

~jr,j----1

L e t ,o be an eigenvalue of the quadratic f o r m ~ , j e i g e n v e c t o r belonging to p, we h a v e

(l
r~ja~aj; if (v~) is an

pv~ = )-~j r~jvi

(1 _< i _< r) ,

and t h e r e f o r e , in view of (17),

I pv~l <=supj I vjl

(1 ~ i ~ r) ,

which obviously implies I P l =< 1. T h e r e f o r e F ( f ' ) >=0 for all f', as we had asserted. 9. In no. 7, we h a v e proved t h a t

f~_,(:) (ap § W)dt2 = -- i~-~(t)F(f)dt2 ; ap and W are obviously =>0, and we have j u s t proved t h a t F ( f ) ~ O. T h e r e f o r e ap, W and F ( f ) are 0. In view of the results of no. 8, this implies t h a t f " = 0, f = f'. As (P = 0, (12) gives g~Jk = g~kj for all i, j, k if we p u t g~

= ~

c~,~f.~ .

B u t we have g~jk = --gj~k; as the substitutions (ijk)~(jik), (ijk)-*(ikj) g e n e r a t e t h e s y m m e t r i c group in t h e t h r e e l e t t e r s i, j, k, one sees now at once t h a t g~j~ m u s t be both s y m m e t r i c and a l t e r n a t i n g in i, j , k, so t h a t it is 0. P u t W ~ = ~ f ~ k X ~ ; as we h a v e g ~ J k = 0 , we have [ Wk, X~] = 0 for all i, so t h a t Wk is an e l e m e n t of f which c o m m u t e s w i t h all t h e X~. Now one verifies i m m e d i a t e l y t h a t t h e vectors W in f w i t h t h a t p r o p e r t y m a k e up a normal subalgebra m of g; as this is contained in f, t h e corresponding subgroup of Ad(G) is both compact and normal, which contradicts the assumption t h a t G is w i t h o u t compact components unless m = {0}; t h e r e f o r e we have Wk = 0 for all k, hence f~k = 0 for all a , k. 10. Now we derive some f u r t h e r consequences f r o m F ( f ) ---- 0 w h e n f = (f~j) is s y m m e t r i c in i, j . If we w r i t e g as the direct sum of simple algebras, and divide up the r a n g e s of t h e indices )~, c~, i accordingly, the c ~ are 0 unless all t h r e e indices belong to one and t h e same simple f a c t o r of ~, so t h a t R~j~k is 0 unless all four indices belong to t h e same simple factor. T h e r e f o r e F ( f ) splits up into the similar f o r m s belonging to the simple f a c t o r s of g, plus the sum ~ (f~j)~ e x t e n d e d to all the pairs (i, j) in which i and j belong to distinct simple factors of g. As F ( f ) = O, this shows t h a t f~j = 0 unless i, j belong to one and the same simple f a c t o r of

494

586

[1962b]

A N D R ~ WEIL

fl; and it r e m a i n s for us to d e t e r m i n e w h e n F ( f ) is 0 in t h e case of a simple Lie a l g e b r a g; this c a n n o t occur unless - 1 is an e i g e n v a l u e of t h e f o r m ~ , r j r ~ j a ~ a j for some choice of t h e basis (Xx) of g. L e t (v~) be a n e i g e n v e c t o r b e l o n g i n g to t h e e i g e n v a l u e --1; we h a v e (1 <- i _< r) ;

--v~ = ]~,j r , v j

in v i e w of (17), this can be w r i t t e n Y:.j r~j(v~ + v j) = 0

(1 ~ i =< r ) .

P u t v = sup~lv~[; t h e n we m u s t h a v e r~j = 0 w h e n e v e r [v~l = v and v~ + v~. :~ 0. T h e r e f o r e , if I v~l -- v, t h e r e is a v a l u e of j for w h i c h vj ---v~; in o t h e r words, t h e sets of values of i f o r which v~ = v and f o r which v~ --- - - v a r e not e m p t y . A f t e r r e - o r d e r i n g t h e X~ if n e c e s s a r y , we m a y now a s s u m e t h a t v~----v for l _ < i _ < s , v ~ = - - v for s+l_0, t>0. Then the matrix (r~j) is of t h e f o r m

(o 00) ~

0

0

N

w h e r e M, N a r e r e s p e c t i v e l y an (s, t ) - m a t r i x and a s q u a r e m a t r i x of order r -- s -- t, and ~M is t h e t r a n s p o s e of M. In v i e w of t h e values o b t a i n e d f o r t h e r~j, r ~ = 0 implies c~; = 0, h e n c e also e a~3 ~ 9= 0 f o r all c~. T h e r e f o r e , in t h e r e p r e s e n t a t i o n of t~ d e t e r m i n e d b y X~ ~ (c~j), ~ e v e r y e l e m e n t X of ~ is m a p p e d onto a m a t r i x of t h e f o r m

p(X) =

0

.

0

I t is k n o w n t h a t p m u s t be irreducible (in f a c t , if a p r o p e r s u b s p a c e m of t h e space g e n e r a t e d b y t h e X~ is stable u n d e r p, one varifies a t once 3 t h a t m + [m, m] is a n o r m a l s u b a l g e b r a of g; t h e r e f o r e , as g is simple, m m u s t be {0}). In p a r t i c u l a r , we m u s t h a v e s + t = r, so t h a t p is of t h e form

(o

I f X , X ' a r e in ~, p ( [ X , X ' ] ) m u s t be of t h a t f o r m ; as we h a v e 8 T h i s a r g u m e n t is taken from E. Cartan's classical Mdmorial volume, "La Thgorie des Groupes Finis et Continus et l'Analysis Situs," Paris, 1930. T h e discerning reader will already have noticed that our whole approach to the subject of semisimple Lie groups, including the notation, is derived from Ch. IV of that volume.

[1962b]

495 DISCRETE SUBGROUPS

p([X, X']) = [p(X), p(X')] =

587

A B ' -- A ' B 0 ) 0 BA' -- B ' A '

this implies t h a t p([X, X']) = 0. B u t t h e kernel of p is t h e same normal s u b a l g e b r a ro of g which was considered in no. 9; as this is {0}, we see t h a t f is abelian; as it has t h e f a i t h f u l irreducible r e p r e s e n t a t i o n p, this implies t h a t r = 2, n -- r = 1, and t h a t g is the Lie a l g e b r a of dimension 3 belonging to t h e hyperbolic group. T h e r e f o r e , if this is not so (g being still assumed to be simple), F ( f ) = 0 implies f = 0. Going back now to t h e general case of a semisimple group, we see t h a t , unless g has a simple f a c t o r of d i m e n s i o n 3, all the f ~ are O, so that the vector-field Y on I x G satisfies [X~, Y] = 0 f o r all ~. In o t h e r words, t h e vector-field Y is t h e n i n v a r i a n t u n d e r all l e f t - t r a n s l a t i o n (t, x ) - * (t,sx) inlx G. 11. Consider now t h e differential systems, in S and in I x G, whose solutions are t h e c u r v e s which, at e v e r y point, are t a n g e n t to Y. As t h e vector-field Y on S is n o w h e r e t a n g e n t to t h e fibre ~ l(t), and as t h e s e fibres are compact, one sees a t once t h a t , in S, e v e r y such curve cuts each fibre once and only once; t h e r e f o r e t h e same is t r u e in I x G, so t h a t , in I x G, e v e r y solution of our differential s y s t e m is a cross-section of I x G, i.e., t h e image o f / b y a m a p p i n g t -* (t, ~:(t)) of class C ~. We shall d e n o t e by t -* (t, ~(t, s)) t h e uniquely d e t e r m i n e d solution of t h a t s y s t e m in I x G which goes t h r o u g h t h e point (0, s). As Y is i n v a r i a n t u n d e r P o p e r a t i n g on I x G to t h e r i g h t in t h e m a n n e r explained in no. 2, we have, f o r all t~LseGand TeP: (18) ~(t, sr0(7)) = ~(t, s)r~(7) . On t h e o t h e r hand, if G' is a group of l e f t - t r a n s l a t i o n s leaving Y i n v a r i a n t , we have, for all t e L s e G and s' e G': (19)

~(t, s's) = s'~(t, s) .

L e t us assume first, as at t h e end of no. 10, t h a t ~ has no simple f a c t o r of dimension 3. In t h a t case, as we have seen t h e r e , Y is i n v a r i a n t u n d e r all l e f t - t r a n s l a t i o n s by G, so t h a t (19) gives ~(t, s) = s~(t, e); t h e n (18) gives, for all 7 and t:

r~(7) = ~(t, e)-lro(7)~(t, e) . This can be expressed by saying t h a t r~ differs f r o m r0 only by an inner a u t o m o r p h i s m of G. In D I, we a g r e e d to denote by ~0(P, G) the space of t h e injective repr e s e n t a t i o n s r of P into G such t h a t r(F) is discrete and G/r(P) is compact. In particular, let F be a discrete subgroup of G such t h a t G[P is compact,

496

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588

ANDRI~ WElL

so t h a t the identity mapping r0 of P belongs to 9~0(F, G); f r o m now on, the connected component of ro in .~0(P, G) will then be denoted by ~1(1", G). With this notation, let r, r ' be a n y two points in ~l(P, G); as shown in D ~, t h e y can be joined t o g e t h e r by a succession of analytic arcs contained in ~I(F, G). We have just proved t h a t , in a sufficiently small neighborhood of any point on such an arc, all points belong to r e p r e s e n t a t i o n s which differ only by inner automorphisms. An obvious compactness a r g u m e n t gives now: THEOREM 1. Let G be a connected semisimple Lie group without compact components, whose Lie algebra has no simple factor of dimension 3. Let F be a discrete subgroup of G such that G/F is compact. Then ~tt1(P, G) consists of the representations of F into G induced on F by the inner automorphisms of G. 12. In order to get a more complete result, we need additional information on the n a t u r e of the solution (f~j) of our variational problem in the case of a simple group G of dimension 3; as shown in no. 5, this m u s t be i n v a r i a n t u n d e r t h e c e n t e r of G, so t h a t it is enough to consider the case w h e n G is its own adjoint group, i.e., w h e n it is t h e quotient of SL(2, R) by its c e n t e r {___12}. H e r e we have r = 2, n = 3; it is easily seen t h a t the c~, are uniquely d e t e r m i n e d by t h e conventions in no. 6, e x c e p t for the sign, and t h a t this m a y be chosen so t h a t we have [ X . X d = X~,

[ X . X d = X~,

[ X . Xd = - X ~ .

Then K is the compact subgroup of dimension 1 with the infinitesimal t r a n s f o r m a t i o n X3. I t is well-known t h a t t h e space H of left-cosets of K in G can be identified w i t h the half-plane Ira(z) > 0 in the plane of a complex variable z; we do this so t h a t K is t h e coset corresponding to the point i = ~ / - - 1 in H (we shall no longer need i as an index, so t h a t this notation will cause no confusion). Then, if s is the image in G of t h e e l e m e n t (ca

~ ) o f SL(2, R ) , s acts on H by Z

---->Z8

--

-

az+c bz § -

and t h e coset Ks in G corresponds to t h e point r = (ai + c)/(bi + d) of H. Now, if we combine the results of no. 9 w i t h (13), we see t h a t , for a n y solution of our variational problem, we m u s t have f ~ = f ~ and fl~ + f ~ = 0. P u t F = f ~ + i 9 f~; t h e n (14), (16) give X~F = - i . X y , X y -- 2 i . F. An easy calculation, which we omit, shows t h a t t h e most general solution for t h e s e differential equations is

[1962b]

497

589

DISCRETE SUBGROUPS

F ( s ) = (bi + d)-4~p(v) ,

where q) is any holomorphic function in H; F is invariant under a righttranslation s ~ SSo if and only if the "quadratic differential" cP(v)dv ~ is invariant under so acting on H as we have said. For a given r r 0, it is clear t h a t the subgroup G' of G which leaves ~P(v)dv ~ invariant is closed. Assume t h a t G' is not discrete; then there is a one-parameter subgroup G~ of G leaving q~(~)dv~ invariant; therefore, in a suitable neighborhood of any point of H where r is not 0, the holomorphic differential (Pll:dv is invariant under G1. It is now easily seen that, in any neighborhood of a point of H, every holomorphic differential which is invariant under a oneparameter subgroup of G must be either 0 or of the form d v / ( A v ~ + B v + C), where A, B, C are constants. Therefore, when we assume that q~ r 0 and t h a t G' is not discrete, ~Pdv ~ must be of the form q~dv ~ = ( A v ~ + B y + C)-~dv ~ .

This is invariant under an element s of G, corresponding to (a

b)in

SL(2, R), if and only if the binary form A u ~ + B u y + Cv ~ is invariant under (u, v) ~ ( a u + cv, bu + dr). One finds at once that G' has then at most two connected components, and t h a t GIG' cannot be compact. Therefore, i f r 4= 0 a n d G/G' is c o m p a c t , G' m u s t be d i s c r e t e i n G. 13. Now we go back to the problem discussed in nos. 3-10, and we begin by assuming t h a t G is a product of simple groups G ("), none of which is compact. As the f ; satisfy (5) and (7), they also satisfy the equations

obtained by applying Xx to (5), using summation with respect applying (1) and (7). For each value of t e L this is an elliptic the fibre 7c-1(t). Therefore, if, for a sequence (t~) of values of to 0, we have, on z-l(t~), a solution f~(t~) of the system (5), we assume for instance t h a t we have, for each t~: (20)

f ~-~(~,~)~ ^ , ~ f --t~x(t~) 2d a

to ~,, and system on t, tending (7), and if

= 1,

the known a p r i o r i estimates for solutions of elliptic systems (el. [5], [6]) show t h a t a suitable subsequence of the sequence f ~ ( t ~ ) converges towards a solution f'~, of the same system on rc-~(0), and t h a t the convergence is uniform in the f~(t,,) and their derivatives up to any order. Now we choose a basis of the Lie algebra of G consisting of bases X2~ for the Lie algebras of the simple factors G ~ of G; Y being as before, we

498

[1962b]

590

ANDRI~IWEIL

know t h a t [X/n~, Y] is 0 unless G on)has the dimension 3, and also t h a t , for each p, [X~ ~, Y] is a linear combination of the X~n) corresponding to the same value of p; therefore we m a y write [X(~n), Y] = ~ , f ~ ) X ~ n) .

Applying (12) and (16), we see t h a t X~P)f(,~') = 0 for p ~ p'; therefore, for each p, t h e f ~ ) , considered as functions on I • I L G on), depend upon t h e / - c o o r d i n a t e and the GCnLcoordinate alone. For each p, put: ap(t) :

~.~(f(~) d , f l(t) ~ 2 . we have an(t) = 0 for all t e I unless G (n) is of dimension 3. L e t p be such t h a t an(t) is not identically 0 in a n y neighborhood of t = 0, and let (t~) be a sequence of values of t, t e n d i n g to 0, such t h a t a,(t~) 4= O. P u t now tk~,\~m]

~

n\

Y~in")(t,) = 0

m!

dg, h\

m]

9

(p' 4= p

or

p" 4: p) ;

this defines, for each t , , a solution of the s y s t e m (5), (7) which satisfies (20); as we have seen, it m u s t (after the sequence (t,) hus been replaced by a suitable subsequence) converge towards a solution ( f ) of the systern (5), (7) on ~-~(0), other t h a n 0, such t h a t f ~ U ' ) = 0 unless p ' = p" = p; this determines a solution of the same system on G, invariant under right-translations by elements of r0(F). But t h e n the functions j~cn~) define a solution of the corresponding s y s t e m on G (n~, other t h a n 0, which is invariant under t h e projection of r0(P) on G (n>. L e t G' be the group of all the right-translations in G ('> which leave t h a t solution invariant. Then r0(F) is contained in the group of the elements of G whose G(nLcoordinate is in G'; as G/ro(F) is compact, and as we have seen in no. 12 t h a t G' is closed in G on),this implies t h a t G(~>iG' is compact, and therefore, by t h e final result in no. 12, t h a t G' is discrete. This proves t h a t , unless the projection o f r0(P) on G (n) is discrete i n G Cn),all the fJ~) m u s t be 0 i n some neighborhood o f t = 0 i n L 14. For convenience, we shall identify F w i t h r0(F) from now on. Collect into a partial product G" of G all the simple factors G (n) of G of dimension 3 such t h a t the projection of F on G (n~ is discrete; let G' be the product of all the other simple factors of G. We know t h a t , if G ~*~is a n y one of the factors in G', all the fJ?,) are 0 for t in a suitable neighborhood of 0, which we m a y assume to be L Then we have IX, Y] = 0 for every X in t h e Lie algebra of G', so t h a t Y is invariant under left-translations by elements of G'; therefore, if ~(t, s) is as in no. 11, (19) is valid for all t e L s e G , s' ~G'.

[1962b]

499 DISCRETE SUBGROUPS

591

Call F', F " the projections of F on G', G"; and call A', A" its intersections w i t h G', G" considered as subgroups of G. The definition of G" implies t h a t F " is discrete in G"; t h e r e f o r e we can apply corollary 3 of A p p e n d i x II. This shows in p a r t i c u l a r t h a t F' is discrete in G', t h a t A' has finite index in F', and t h a t G'/A' is compact. Now, if we apply (18) and (19) to a n y ~' e A', we g e t (21)

~(t, s6') = ~(t, s)rt(6') ,

~(t, $'s) = F~(t, s ) .

F o r s = e, t h e s e relations show t h a t , if we modify r, by t h e inner autom o r p h i s m of G d e t e r m i n e d b y ~(t, e), i.e., if we replace it b y ~(t, e). f t . ~(t, e) -1, r, induces t h e i d e n t i t y on A'. Then, if 7 = (7', 7") e F, t h e projection of rt(7) on G" depends only upon t and 7" and m a y be w r i t t e n as r['(7"), w h e r e r 7 is a o n e - p a r a m e t e r f a m i l y of r e p r e s e n t a t i o n s of F " into G" such t h a t r~' is t h e identity. Now, applying (21) to a n y s = s" e G", we see, in view of t h e f a c t t h a t rt(F) = 6' and #'6' = 6's", t h a t ~(t, s") c o m m u t e s w i t h e v e r y 6' e A'; t h e r e f o r e , for e v e r y s" e G", t h e projection of ~(t, s") on G' is in the normalizer N(A') of A' in G'; as t h a t projection is e for t = 0, and N(A') is discrete b y corollary 1 of A p p e n d i x II, this shows t h a t ~(t, s") is in G" for all t e I and s" e G". T h a t being so, if we t a k e s = e and 7 e A" in (18), we see t h a t r~(7) e G" f o r all t e I and 7 e A"; t h e r e fore, if 7 = (7', 7") e F, t h e projection of r,(7) on G' depends only upon t and 7' and m a y be w r i t t e n as r;(7'), w h e r e r; is a o n e - p a r a m e t e r f a m i l y of r e p r e s e n t a t i o n s of F' into G' such t h a t r ' is t h e identity. As r; is t h e i d e n t i t y on A', corollary 2 of A p p e n d i x II shows t h a t it is the i d e n t i t y on F'. This gives, for all 7 = (7', 7") e F: =

If now we apply t h e results of D ~j u s t as in no. 11, and if we use t h e noration ~I(F, G) in t h e m a n n e r explained t h e r e , we see t h a t we h a v e proved t h e following t h e o r e m : THEOREM 2. Let G be a product of connected non-compact simple Lie groups. Let F be a discrete subgroup of G such that G/F is compact. Let G" be the product of those simple factors G (") of G which are o f dimension 3 and such that the projection o f F on G (p) is discrete in G (p), and let G' be the product o f all the other simple factors of G. Then the projections F', F" o f f on G', G" are discrete in G' and in G", respectively; G'/F' and G"/F" are compact; and F is o f finite index in F' • F". Moreover, i f j is the injection m a p p i n g of F into F' x F", and i f ~ ' is the set o f the representations of F' into G' induced on F' by the inner automorphisms o f G', then (r', r") --) (r' • r") o j is a homeomorphism o f

500

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ANDR]~ WEIL

~ ' • ~I(F", G") onto ~RI(F, G). Here we have denoted by r ' • r " the representation of F' • F" into G' • G" which maps (7', 7") onto (r'(7'), r"(7")). 15. In order to obtain a complete result for a product of simple noncompact groups, we still have to deal w i t h the case G = G". The result is as follows: THEOREM 3. Let G be a produ.ct of connected non-compact simple Lie groups Gp of dimension 3. Let F be a discrete subgroup of G such that G/F is compact and that, for every p, the projection Fp of F on Gp is discrete in G~. Then, for each p, G~/Fp is compact, and F is of finite index in I L F~. Moreover, i f j is the injection mapping of F into I L Fp, (rp) --~ ( I L r~) o j is a homeomorphism of I L ~l(Fp, Gp) onto ~l(F, G). Here we have denoted by II~ r~ the representation of rip F~ into G which maps (7,) onto (r,(7~)). The first assertion in our theorem follows at once from corollary 3 of Appendix II by induction on the n u m b e r of the factors in G. The second p a r t will be proved in our usual m a n n e r by dealing first w i t h a one-param e t e r family r, of representations of F into G. Notations being the same as before, we shall write, on I • G: Y:

0 +~ 0t

~/~/v~l. ~ ~ '

t h e n the ~pk'~satisfy the equations (6) w i t h f , f ' replaced by 0, f ; this can be w r i t t e n X ~k~'1~1 = 0 "Y/z

(p' :~ p)

9

The l a t t e r equations show t h a t ~ ~'~ , , as a function on I • H , G~, depends only upon t h e / - c o o r d i n a t e and the G ; c o o r d i n a t e , so t h a t we can write

y_

a + ~ ot

.

'~

(t,x~)

. X ~

at the point (t, (x,)) of I • 1-in G,. This implies t h a t the solutions (t, ~(t, s)) of the differential system determined by Y on I • G are of the form (22)

}(t, (sp)) = ((},(t, s~))

where, for each p, (t, }p(t. s~)) is the solution t h r o u g h (0, s~) of the differential system determined in I • G~ be the vector-field ~ ~p~tt Y~ = - ~ y + 2_.,~,'e~ , , x0). X~p)

[1962b]

501 DISCRETE SUBGROUPS

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I f we a g a i n i d e n t i f y F w i t h r0(F), we see now a t once, by combining (18) w i t h (22), t h a t t h e Gp-coordinate of r~(7), for 7 = (%) c F, depends only upon t and % a n d m a y be w r i t t e n as r~PS(%), w h e r e r~') is a o n e - p a r a m e t e r f a m i l y of r e p r e s e n t a t i o n s of F~ into G,. The conclusion of T h e o r e m 3 follows n o w b y m a k i n g use of the r e s u l t s of D I in t h e s a m e w a y as in no. 11. 16. N o w t a k e f o r G a n y connected semisimple g r o u p w i t h o u t c o m p a c t c o m p o n e n t s , and let Z be its c e n t e r . N o t a t i o n s b e i n g as before, it has b e e n s h o w n in no. 5 t h a t Y is i n v a r i a n t u n d e r Z, so t h a t (19) gives ~(t, zs) = z~(t, s) f o r z ~ Z; if now z ---- r0(7) f o r some 7 e F, a comparison w i t h (18) s h o w s t h a t z -- rt(7) for all t e L L e t n o w F be a discrete s u b g r o u p of G such t h a t G/F is c o m p a c t ; b y m a k i n g use of t h e r e s u l t s of D ~ in t h e s a m e w a y as in no. 11, we see now t h a t , w h e n e v e r r, r ' a r e in ~I(F, G) and an e l e m e n t 7 of F is such t h a t r(7) = z e Z, t h e n r'(7) = z. T h e r e f o r e e v e r y r e ~ ( F , G) c a n be e x t e n d e d to an injective r e p r e s e n t a t i o n r* of t h e g r o u p F* = Z . F into G b y p u t t i n g r*(zT) -- z . r(7) for z e Z, 7 e F. Moreover, as Z - F is c o n t a i n e d in the n o r m a l i z e r N ( F ) of F in G, corollary 1 of A p p e n d i x I I shows t h a t it is discrete in G; as G/F is c o m p a c t , G/F* is so; for similar reasons, t h e g r o u p r*(F*) ---- Z - r(F) is discrete in G, and G/r*(F*) is c o m p a c t , f o r e v e r y r e ~I(F, G). T h u s we h a v e proved t h a t e v e r y e l e m e n t of ~ ( F , G) induces t h e i d e n t i t y on F R Z, and t h a t r -* r* is a h o m e o m o r p h i s m of ~ ( F , G) onto ~ ( F * , G). F r o m now on, a s s u m e t h a t F contains Z. L e t Z1 be a s u b g r o u p of Z; p u t G' = G/Z1; call p t h e canonical h o m o m o r p h i s m of G onto G'; and p u t F' ----2(F), so t h a t F = p-~(F'). As e v e r y r e ~ ( F , G) induces the i d e n t i t y on Z, t h e relation p o r = r ' o p d e t e r m i n e s a m a p p i n g r --* r ' of ~ ( F , G) into ~li(V', G'), which is obviously continuous. Choose n o w (as in D ~, no. 11) a finite set (%) of g e n e r a t o r s of F, such t h a t P is defined b y a finite set of relations Rx(%) = e b e t w e e n t h e %; choose a finite set (~j) of g e n e r a t o r s f o r Z1; and, f o r e a c h ~'j, choose an e x p r e s s i o n ~'j = Fj(%) of ~'j in t e r m s of t h e %; t h e n , if we p u t 7'~ = p(7~), F' h a s t h e g e n e r a t o r s 7~ and is defined b y t h e r e l a t i o n s Ra(73 = e', F j ( 7 ~ ) = e' b e t w e e n t h e m . Take any r'0 e N~(F', G'); p u t s~ = r~(7'~); and, f o r each i, choose s~ e G such t h a t p(s~) = s~. L e t V be a neighborhood of e in G, such t h a t V R Z1 = {e}; choose an open neighborhood U of e in G such t h a t R,~(u~s~) ~ V . Rx(s~) and Fj(u~s~) e V . F~(s~) for all ?~, j w h e n e v e r all t h e u~ a r e in U, and also such t h a t p induces on U a h o m e o m o r p h i s m of U onto its i m a g e U ' in G'; call q> t h e inverse of t h a t h o m e o m o r p h i s m . L e t 1I' be t h e open neighborhood of r~ in ~tt~(F', G'), consisting of t h e r e p r e s e n t a t i o n s r ' such t h a t r'(7~) e U's~ for all i. T h e n it is e a s y to see t h a t if, for a n y r ' e t t ' , t h e r e is an r ~ N~(F, G) such t h a t p o r = r ' o p, this m u s t be g i v e n by f o r m u l a s

502

[1962b]

594

ANDRl~ W E I L

r(7~) = z; ~. q~(rt ( Tt i ) t--I s i ) . s~, where the zi are elements of Z1 satisfying the relations Rx(z~) = R~(si) ,

F~(z~) = ~ j . Fi(s~) .

As these relations are independent of the choice of r' in W, one concludes immediately from this, and from obvious continuity considerations, t h a t the inverse image of 11' under the mapping r -+ r ' is the union of neighborhoods of the points in the inverse image of r~, and t h a t this mapping determines, on each one of these neighborhoods, a homeomorphism onto 11'. In other words, for this " na t ur al " mapping, !~(P, G) is a covering space of ~ ( F ' , G')% In particular, we can apply this to the case Z~ = Z; G' is t hen t h e adjoint group of G and is a product of simple groups, so t h a t the structure: of ~I(F', G') is fully determined by Theorems 2 and 3. In view of t h e known facts on fuchsian groups, this shows for instance t h a t ~ ( F , G) is a manifold. Alternatively, we may write G - G J Z , where G~ is the simply connected group with the same infinitesimal st ruct ure as G, and Z~ is a subgroup of the center of G1; then, if F~ is the inverse image of F in G , we see t h a t ~l~(F~, G~) is a covering space of ~R~(F, G); here again G1 is a product of simple groups, so t h a t the s t r u c t u r e of ~(P~, G~) is again giver~ by Theorems 2 and 3.

APPENDIX I Through an oversight, the main theorem of D e, as formulated there in no. i, is not quite strong enough for the application which is made of it in D r, no. 12; this is to be corrected now. We wish to show that the theorem in question is valid with the following addition: L e t G, F, r0, ~, 11 be as i n that theorem; then, J b r e v e r y g ~ G, there is a neighborhood W o f g i n G, a n d a neighborhood 11' o f ro i n 1I, s u c h that the u n i o n o f the sets r - l ( W ) , f o r all r e 1I', is a f i n i t e set. Assume first t h a t G is simply connected. L et all assumptions and notations be as in nos. 7-10 of De; put W --- soIU'-~sog. A s , m a p s U' x S • F onto G, we can write sog = us7, with u e U', s c S, 7 e F. If the representation 7 -* ~ is close enough to the identity, the point ~ determined by us7 = ~s~(s)-I~(s)~ 4 If G h a s no c o m p o n e n t of d i m e n s i o n 3, t h e s e two s p a c e s a r e a c t u a l l y isomorphic. T h i s follows f r o m T h e o r e m I and f r o m a r e s u l t of Borel [2, Corollary 4.4]. If the s a m e could be proved for g r o u p s of d i m e n s i o n 3, t h e n T h e o r e m s 2 a n d 3 would s h o w t h a t it r e m a i n s t r u e for all s e m i s i m p l e g r o u p s .

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503 DISCRETE SUBGROUPS

595

will be in U'. Now assume t h a t , for such a r e p r e s e n t a t i o n , and for some 7' e F, U is in W; this means t h a t we have ~'so~' -= sog = ~s3(s)-1~(s)7 w i t h some ~' e U'. This can also be w r i t t e n as

so,

-1)

s, e ) ,

w h i c h a m o u n t s to saying t h a t (~', so, 7'7 -1) and (~, s, e) are equivalent for /~; this implies t h a t {So, s} e N ( S ) and 7'7 -1 = 7(s0, s). So 7' m u s t have one of t h e finitely m a n y values 7(s0, s)7. This proves our assertion w h e n G is simply connected. Reasoning as in D ~, no. 11, one e x t e n d s it a t once to the g e n e r a l case. Now let assumptions and notations be as in D ~, no. 12. F r o m w h a t we h a v e proved above, it follows t h a t , w h e n P is made to act on X • G by

(x,

= (x,

it operates p r o p e r l y on X x G. This m e a n s t h a t , given a n y two compact subsets K, K ' of X • G, t h e r e are at most finitely m a n y elements 7 of F such t h a t K 7 m e e t s K ' . T h e r e f o r e t h e space S -- ( X x G)/F is s e p a r a t e d (i.e., " H a u s d o r f f " ) . While this f a c t had not expressly been s t a t e d in no. 12 of D ~, some of t h e assertions made t h e r e would not m a k e sense unless this w e r e so. Once t h e above addition to the main t h e o r e m of D ~ has been obtained, it is a trivial m a t t e r to s t r e n g t h e n it as follows: one can choose W a n d p, so that the sets r-~( W ) , f o r r e 1I', are all e m p t y i f g r ro(r) a n d all equal to {g} i f g e roW).

APPENDIX I I

L e t G be a connected Lie group of dimension n; in D ~, no. 6, we defined a G - s t r u c t u r e on a manifold V of dimension n as being given by n everyw h e r e linearly i n d e p e n d e n t differential f o r m s w ~ on V s a t i s f y i n g t h e M a u r e r - C a r t a n equations for G. I f t h e X~, a t e v e r y point of V, are t h e vectors defined by i(Xx)w ~ = 3~, t h e Xx m a k e up n e v e r y w h e r e linearly i n d e p e n d e n t vector-fields on V, s a t i s f y i n g the s t r u c t u r a l equations (1) for G; a G - s t r u c t u r e m a y be considered as given b y n such vector-fields, just as well as by s t r u c t u r a l f o r m s cox. The g r o u p G itself is always to be considered as c a r r y i n g t h e G - s t r u c t u r e d e t e r m i n e d by t h e r i g h t - i n v a r i a n t vector-fields Xx (cf. no. 1); t h e n , if F is a discrete subgroup of G, G/F (the space of r i g h t cosets sF in G) carries a G - s t r u c t u r e , d e t e r m i n e d in an obvious m a n n e r b y t h a t of G.

504 596

[1962b] ANDR~ WEIL

As observed in D ~, no. 6, the automorphisms of the G-structure of G are the right-translations; if F is a discrete subgroup of G, a right-translation x ~ xs determines an automorphism of G/F if and only if sPs -1 = F, i.e., if and only if s belongs to the normalizer N(F) of F; and it is easily seen (using the elementary facts noted in D ~, no. 6, i.e., essentially nothing more than Frobenius's theorem) t h a t all automorphisms of G/F can be obtained in this manner. The group of automorphisms of G/F may t herefore be identified with N(F)/F. L et N0(F) be the component of e in the closed subgroup N(F) of G; it is a connected Lie group. For each 7 e F, the image of N0(F) by x ~ XTX-1 must be a connected subset of F, containing 7, and is t h e r e f o r e {7}. Thus N0(F) is also the component of e in the centralizer Z(F) of F (consisting of the elements of G which commute with e ve ry 7 c F). In particular, the Lie algebra ~(F) of N0(F), which may be identified with those of Z(F) and of N(F)/F, consists of the vectors X i n the Lie algebra ~ of G which are invariant under Ad(7) for every 7 c F (as usual, we denote by Ad(s) the automorphism of g induced by the inner automorphism x ~ sxs -~ of G). Now assume t h a t G/F is compact, i.e., t h a t t here is a compact subset K of G such t h a t G = K F . Then, if p is any representation of G in a finitedimensional vector-space A over R, any vector a c A which is invariant under p(7) for eve r y 7 e F has a compact orbit under p(G). In this situation, we can apply the following lemma: LEMMA. Let p be a r e p r e s e n t a t i o n o f a topological group G i n a finited i m e n s i o n a l vector-space A over R. Let A' be the set o f all the vectors i n A whose orbit u n d e r p(G) is r e l a t i v e l y compact. Then A' is a subspace o f A , i n v a r i a n t u n d e r p(G); a n d p induces on A ' a r e p r e s e n t a t i o n p' o f G such that p'(G) is contained i n a compact group o f a u t o m o r p h i s m s o f A r,

The first assertion is obvious. Now let al, . . . , a,, be a basis for A'; as the vectors p(x)al belong to a bounded subset of A' for all x e G, G induces on A' a relatively compact set of endomorphisms of A', hence also a relatively compact subset of the group of automorphisms of A'. Now let again G be a Lie group with the Lie algebra g; call c the set of the vectors in g whose orbits under the adjoint group are relatively compact. It is clear t h a t this is not only a vector-subspace of .q but a Lie subalgebra of g, invariant under Ad(G) and even under all automorphisms of g. The adjoint representation x -~ Ad(x) of G induces on c a representation whose kernel is the centralizer C of c in G; in view of the lemma, this implies t h a t G/C has an injective representation into a compact group. In the case with which we are mainly concerned in this paper, G is con-

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n e c t e d and semisimple w i t h o u t compact components, so t h a t it has no non-trivial r e p r e s e n t a t i o n into a compact group; t h e r e f o r e , in t h a t case, we have C = G, hence c = {0}. I f now F is a discrete subgroup of G such t h a t G/F is compact, it follows f r o m w h a t has been said above t h a t e v e r y v e c t o r in n(F) has a compact orbit u n d e r the adjoint group, so t h a t n(P) c c; t h e r e f o r e : THEOREM. Let G be a connected semi-simple Lie group without compact components; let F be a discrete subgroup of G such that G/F is compact; let ~(F) be the set of all the vectors in the Lie algebra of G which are i n v a r i a n t under Ad(7) for every 7 e r. Then ~(F) = {0}. This is of course contained in a d e e p e r r e s u l t proved by Borel for t h e case w h e n G/F is m e r e l y assumed to have finite m e a s u r e [2, Corollary 4.4]. COROLLARY 1. Let G and F be as in the theorem; then the normalizer N ( F ) of F in G is discrete, G/N(P) is compact, and F is of finite index in N ( F ) . In fact, the first assertion a m o u n t s to saying t h a t N0(F) = {e}, and this is e q u i v a l e n t to our t h e o r e m . The o t h e r assertions follow f r o m this a t once.

COROLLARY 2. Let G and F be as in the theorem, and let A be a subgroup of finite index of F. Then the space of the representations of F into G which induce the identity on A is discrete. The subgroup A' of F which induces the i d e n t i t y m a p p i n g on the homogeneous space F/A is of finite index in F (at most equal to d! if d is the index of A in F) and is a normal subgroup of F; replacing A by A', we see t h a t it is e n o u g h to consider t h e case w h e n A itself is normal in F. As A is of finite index in F, G/A is compact, so t h a t N(A) is discrete by corollary 1. Now, if a r e p r e s e n t a t i o n r of F into G induces t h e i d e n t i t y on F, we have r(F) c N ( A ) . T h e r e f o r e , if r, r ' are t w o such r e p r e s e n t a tions, and if (%) is a finite set of g e n e r a t o r s for F, we m u s t have r(7~) = r'(%) for all i, and t h e r e f o r e r = r', as soon as r ' is close enough to r. COROLLARY 3. Let G', G" be two connected semisimple Lie groups without compact components; let F be a discrete subgroup of G = G' • G" w i t h compact quotient. Let A', A" be the intersections of F with G' and w i t h G" considered as subgroups of G, and let P', F" be its projections on G' and on G". Assume that F" is discrete in G". Then P' is discrete in G'; G'/P' and G"/F" are compact; F has a finite index in F' • P", and A' • A" has a finite index in F. L e t K be a compact subset of G such t h a t G = K F ; call K ' , K " its projections on G' and on G". F o r a n y x' in G', we can w r i t e (x', e") = k7

506

[1962b]

598

ANDR]~ WEIL

w i t h k e K and 7 = (7', 7") e F; t h e n 7" belongs to K ''-~ f3 F", which is a finite set since F " is discrete and K " is compact. Choose a finite n u m b e r of elements % = (7~, 77) of F so t h a t K ''-1 g / F " = {7;', " " , 7"}. Then, if x', k, 7 are as above, t h e r e is an i such t h a t 7 = %$' with 5' e A'. This gives

x' e (U~ K'73. A', which shows t h a t G'/A' is compact. T h e r e f o r e , by corollary 1, N(A') is discrete in G'. On t h e o t h e r hand, one sees a t once t h a t F' is contained in N(A'), so t h a t it is discrete. E x c h a n g i n g now G' and G" in the above proof, we see t h a t G"/A" is compact. Our o t h e r assertions are now obvious.

APPENDIX III

(This appendix reproduces, w i t h minor verbal changes, a communication o f L. Hgrmander to the author, and is published here with his permission). L e t notations be as in nos. 2 and 3; t a k e a n y vector-field Y0 e ~ on S, and consider, as in no. 11, t h e solutions of t h e differential s y s t e m d e t e r m i n e d on S by t h a t vector-field. F o r e v e r y point M o f S, call O ( M ) the point w h e r e t h e solution of t h a t s y s t e m which goes t h r o u g h M cuts t h e fibre 7c-1(0). Then, for each t, 9 induces on t h e fibre ~-l(t) a h o m e o m o r p h i s m of class C ~ of t h a t fibre onto 7~-~(0), and the m a p p i n g M - ~ (re(M), ~ ( M ) ) is a h o m e o m o r p h i s m of class C ~ of S onto I • 7~-~(0). P u t V = ~-~(0); t h e n t h e m a p p i n g of 7c-~(t) onto V induced by (P will map t h e vector-fields Xx onto n vector-fields Xx(t) on V; similarly, if Y and the f ~ are as in no. 3, it will map t h e functions induced by t h e f ~ on z-l(t) onto functions f~(t) on V; it maps t h e volume e l e m e n t d 3 on ~-l(t) onto a volume e l e m e n t d~2, = ~(t)~d~2 on V, w i t h a density ~(t) ~which is n o w h e r e 0; and t h e X~(t), t h e f~(t) and ~(t) are all of class C ~ on I • V. A f t e r replacing t h e Xx(t) by $(t)Xx(t), w r i t i n g f,~(t) instead of ~(t)f~(t), c,~p(t) instead of $(t)cL and q~ instead of 9 ~, we can now s t a t e t h e variational problem of no. 3 as follows: f o r each value of t, the functions q~ are to be chosen so as to

m i n i m i z e the integral (23)

fv~

~ (f~x(t) § Xx(t)q% § ~

c~xp(t)q%)2d~2 .

H e r e all t h e d a t a are assumed to be of class C ~ on I • V; for each t, t h e Xx(t) are e v e r y w h e r e linearly i n d e p e n d e n t vector-fields; and one wishes to show t h a t t h e problem has a unique solution, of class C ~ on I • V, u n d e r t h e assumption t h a t , for each value of t, the s y s t e m

[1962b]

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SUBGROUPS

X~(t)9~ + ~ p c,xp(t)9, = 0

(24)

has no solution of class C ~ on V. o t h e r t h a n O. On t h e space L s of functions of i n t e g r a b l e square on V, we use the / r

~

\V2

n o r m II g II = (W d ~ )

; similarly, for a s y s t e m g = (g~x) of such funetions,

N~

we use t h e n o r m given by

II g IIs =

~ x . , II g,x ]is = f ~'-x,~ (g~x) sdt2 ;

w i t h this norm, t h e space of such s y s t e m s will also be denoted by L ~ (this will cause no confusion). On t h e o t h e r hand, for q~ = (q%), we introduce t h e norm given by Ill e Ills = ~ , ~

II X~(O)e~ 11s + ~

1l 9~ IIs ,

and we call H t h e space of the q) = (q)~) for which this is finite, i.e., for which all the q~. are in L s and t h e i r first derivatives, in t h e distribution sense, are also in LL W e shall now prove t h a t t h e inequality (25)

Ill 9 llts G C s ~ x ~ . 1IXx(0)q)~ + ~;~'~pGxp(0)~Pp Iis

holds, for a suitable choice of t h e c o n s t a n t C, for all q~ ~ H. In f a c t , if this w e r e not so, t h e r e would be a sequence q~"} w i t h Ill 9 c*~Ill = 1 such t h a t t h e r i g h t - h a n d side of (25) tends to 0. By a well-known principle based on Poinear6's i n e q u a l i t y (el. e.g., Courant-Hilbert, Vol. 2, pp. 488490), e v e r y sequence for which III q~(*)Ill is bounded has a subsequence which c o n v e r g e s in LS; t h e r e f o r e we m a y assume t h a t q~(~) c o n v e r g e s in L s tow a r d s a limit ~p. As t h e r i g h t - h a n d side of (25) t e n d s to 0 for this sequence, this implies t h a t t h e Xx(0)q~ converge in LL T h e r e f o r e ~p is in H, we h a v e ]l] q~ ]]I = 1, and ~p is a solution of t h e s y s t e m (24) w i t h t = 0. As (24) for t ---- 0 implies ~

X~(0)(X~(0)~ + ~

c~(0)~) = 0,

and as this is an elliptic system, t h e t h e o r y of elliptic systems (cf. e.g., [5] or [6]) shows t h a t q~ can be modified on a null-set so as to become C ~. We have t h u s obtained a non-zero solution of (24) for t = 0, of class C% a g a i n s t our assumption. This proves (25). F r o m the c o n t i n u i t y of t h e d a t a in t, it follows now at once t h a t we have the inequality (26)

Ill ~ Ill~ < 4 c s ~ x ~ I1xx(t)~p, + ~ , G~,(t)q~p lIs

for all q~ e H and all t in some neighborhood I ' of 0 in L F r o m now on. we assume t h a t t is in I'.

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ANDRI~ WEIL

F r o m (26), it follows t h a t the mapping G, of H into L 2 given by

9 ~ o~(9) = (x~(t)9~ + Ep c~(t)~p) has a closed range. L e t P(t) be the projection on this range in LL Then the problem of minimizing (23) has the unique solution

~p -_ _G(t)-ip(t)f in H, and it follows from (26) t h a t Ill q~JII =< 2C l[f]r. The solution q~ of our variational problem m u s t also s a t i s f y the Euler equations of t h a t problem

E . c ~ f ~)d~'~

E.

for all ~ = ( ~ ) of class C~ on V. This is a w e a k form of a system (27)

L~q) = F ~ ,

where the leading t e r m in L~ is - - ~ x X:9~, so t h a t this is an elliptic system; F~ is the effect of an operator of the first order acting on f , so t h a t F~ is of class C ~ on I ' • V. Therefore, for each t e I', ~ can be modified on a null-set so t h a t it will be of class C ~ on V (cf. [5], [6]). The s y s t e m (27) has no other solution t h a n 9 in H. In fact, assume now t h a t ~p is a n y solution of (27) in H. In view of the definition of L~, we have

f r~.~ (X~qg~ + ~_,p c~p~)~dFt = I r ~ ( L ~ 9 ) ~ d F t = f ~

F~q>~dFt .

Combining this w i t h (26), we get

Ill 9 Ill 2 < 4 c ' II F l l . I1 ~ iI < 4C ~ II E l l . Ill ~ Ill, and t h e r e f o r e (28)

lll~l]l =< 4C"I[F]I 9

In particular, F = 0 implies q~ -- 0, as we asserted. If a = ( a , . . . , c~.), where the c~ are integers ~ 0 , we shall write, as usual, I a ] = ~ a~; and we write X~ for the differential operator of order [a I given by x g = x~(o) o ~ . . . x ~ ( o y ~ .

W i t h this notation, the proofs for the r e g u l a r i t y of solutions of elliptic systems (cf. again [5], [6]) give estimates supv I Xg~P~ I -<- C~(Ill 9 [[1 +

E,~l_<_,~l+.+~ supv~ IXgF~ 1) ,

w i t h C~ independent of t. In view of (28), we get now

[1962b]

509 DISCRETE SUBGROUPS

(29)

601

supv I Xgg~ I ~ C~' ~B~ml~l+~+l supv,~ I X~F~ l ,

which is valid w h e n e v e r (27) has the solution 9 e H. Until now we have only used t h e f a c t t h a t the data, and in p a r t i c u l a r t h e F , , are of class C ~ on V for each t. Now, in order to discuss the differentiability in t, we exhibit t h e dependence upon t in (27) b y w r i t i n g it as L~(t)~(t) = F~(t). P u t ~ ( t , h) = h - l [ 9 ( t + h) -- 9(t)] ;

this is a solution of t h e s y s t e m (30)

L~(t)~(t, h) = G.(t, h ) ,

w h e r e we have p u t G~(t, h) = h-1[F~(t § h) -- F~(t)] -- h-l[L~(t § h) -- L , ( t ) ] ~ ( t + h ) . As ~(t, h) is in H, we can apply (29), s u b s t i t u t i n g ~(t, h), G~(t, h) f o r 9, F , . In view of (29) and of our assumptions on the differentiability of t h e d a t a on I • V, this gives a t once a u n i f o r m bound f o r I X g ~ ( t , h) I f o r each ~ (and for all t, t + h in I'); this shows t h a t t h e Xgq)~(t) are Lipschitz-continuous in t. F o r h -* 0, G,(t, h) t e n d s to a limit G~(t); for each t, this is of class C ~ on V, moreover, f r o m t h e differentiability of the data on I • V, it follows t h a t , for e v e r y ~, X : G ~ ( t , h) t e n d s u n i f o r m l y to XgG~(t) for h -~ 0. Consider t h e s y s t e m L~(t)~ = G~(t), which is just (27) differentiated f o r m a l l y w i t h r e s p e c t to t; as (30) has t h e solution ~(t, h), and as t h e r a n g e of an elliptic s y s t e m is closed, this has a solution @ in H. T h e n we h a v e L~(t)[~(t, h) -- ~] --- G~(t, h) -- G~(t) . Applying t h e e s t i m a t e s (29) to this system, we see now t h a t Xg@~(t, h) t e n d s u n i f o r m l y to X g ~ , for e v e r y ~, for h ~ 0. T h e r e f o r e we have ~ -Oq~/Ot, and this is of class C ~ on V for each t. R e p e a t i n g the same a r g u m e n t for ~ and G~, we find t h a t 0:9/0t ~ is of class C ~ on V, etc. This proves t h a t 9 is of class C ~ on I ' • V. Since t h e same conclusion m u s t hold t r u e for some neighborhood I " of a n y point t of L we see t h a t q~ is of class C~ on I x V, as was to be proved. REMARK. The assumption t h a t (24) has no solution can be replaced by t h e assumption t h a t t h e dimension of t h e space of solutions of (24) is t h e same for all t e I; this dimension is necessarily finite by Ascoli's t h e o r e m . One m e r e l y has to replace H by a s u p p l e m e n t in H of t h a t space of solutions for t = 0. If no such assumption is made, t h e r e s u l t is false, as shown by t h e following example. Take V -- R/Z (the real line modulo 1);

510

[1962b]

602

ANDRE WEIL

~fdx ~

let f be a n y function of class C ~ on V such t h a t

O, and consider

the problem of minimizing the integral

dq9

tg) ~dx .

This has a unique solution 9(t) for t 4= 0, but

fg(t)dx

tends to co for

J

t~0. This concludes I-Ibrmander's communication and completes the proof of the facts which were needed in no. 4: of this paper. It is natural to conjecture that, if the data of the variational problem in no. 4 are real-analytic, the solution is real-analytic; but Hbrmander informs me t h a t a proof for this would presumably require more delicate arguments; a n y w a y , it would not be true unless one assumes t h a t n~ = {0}, or at least t h a t the dimension of nt is constant. As the a r g u m e n t I had in mind in writing the f o o t n o t e on page 383 of D ~ depended upon this, I m u s t now withdraw t h a t footnote; I see no other w a y at present of proving the real-analytic equivalence of the fibres considered there t h a n by using Grauert's theorem. INSTITUTE FOR ADVANCED STUDY BIBLIOGRAPHY A. BOREL, On the curvature tensor of the hermitian symmetric manifolds, Ann. of Math., 71 (1960), 508-521. 2. - - , Density properties for certain subgroups of semisimple groups without compact components, Ann. of Math., 72 (1960), 179-188. 3. E. CALABI, On compact riemannian manifolds with constant curvature, I, Proc. Symp. Pure Math. III (Differential Geometry), Providence, 1961, 155-180. 4. - - , and E. VESENTINI, On compact locally symmetric K~hler manifolds, Ann. of Math., 71 (1960), 472-507. 5. K.O. FRIEDRICHS, On the differentiability of the solutions of linear elliptic differential equations, Comm. Pure and Appl. Math., 6 (1953), 299-325. 6. L. NIRENBERG, Remarks on strongly elliptic partial differential equations, Comm. Pure and Appl. Math., 8 (1955), 649-6?5. 7. A. WEIL, On discrete subgroups of Lie groups, Ann. of Math., 72 (1960), 369-384. 1.

[1962c] Algebraic Geometry Algebraic geometry is concerned with such geometric loci as can be defined, in terms of some suitable coordinate system, by algebraic relations between coordinates, that is, by relations involving only the four species of ordinary arithmetic (as opposed to relations involving trigonometric, exponential, or other transcendental functions, infinite series, or any other limiting process). For instance, in plane geometry, an algebraic curve is one which can be defined by an equation f ( x , y) = 0, where f is a polynomial in the coordinates x, y. Straight lines, circles, conic sections are algebraic curves. Curves which cannot be so defined (for example, the sinusoid and the cycloid) are called transcendental and fall outside the scope of algebraic geometry. This basic distinction between algebraic and transcendental curves was first made by Ren6 Descartes (1596-1650), who called them, respectively, geometric and mechanical curves.

Degree and Dimension A first principle of classification of algebraic curves is according to their degree, that is, according to the degree of the polynomial f ( x , y) in the equation of the curve f ( x , y) = 0. As every change of coordinates is expressed by linear formulas (expressing the new coordinates x', y' linearly in terms of the old ones x, y, and conversely), the degree of a curve does not depend upon the choice of coordinates; one expresses this fact by saying that the degree is a geometric character attached to the curve. The curves of degree 1 are no other than the straight lines in the plane; those of degree 2 are the conic sections (including circles and the degenerate conic sections consisting of pairs of straight lines). These facts were already known to Pierre de Fermat (1601-1665), who may thus, together with Descartes, be considered as the founder of algebraic geometry. Similarly, in space geometry, an algebraic surface will be a locus defined by one equation f ( x , y, z) = 0 between the three space coordinates x, y, z, where the lefthand side is a polynomial in these coordinates. The degree of f is again a geometric character of the surface, called its degree; it is equal to 1 in the case of a plane, to 2 in the case of a quadric. On the other hand, in order to define a space curve, one must have at least two equations f (x, y, z) = O, g(x, y, z) = O. More generally, if any kind of element can be determined by coordinates Xx. . . . . x, in such a way that every choice of the coordinates xl . . . . . x, determines exactly one element of that kind, then any number of equations fl(xl . . . . . x.) = 0. . . . . fh(xl, - . . , x,) = 0

(1)

(where the left-hand sides are polynomials) is said to determine an algebraic locus of such elements (also called a variety, or a bunch of varieties, or a closed Reprinted withpermissionof the EncyclopediaAmericana,copyrightO 1962,The AmericanaCorporation.

511

512

[1962c] Algebraic Geometry

algebraic set). Clearly, if (xl, . . . , x,) is any solution of the system (1), it is also a solution of every equation of the type F~fl + -.. + Fhfh = 0, where F 1. . . . . Fh are arbitrary polynomials in the coordinates. If there is such an equation involving only some of the coordinates but not all of them, say x l , . . . , xr with r < n, one says that the latter are not independent (for the given locus). The maximal number of independent coordinates, for a given locus, is called the dimension of the locus. An algebraic curve is a locus of dimension 1 ; an algebraic surface is of dimension 2; a locus of dimension 0 must consist of a finite number of points. This fundamental concept also goes back to Fermat. Universal Domain In using, as above, the language of analytic geometry, it is ordinarily understood that the coefficients and variables are taken to be numbers (real or imaginary). Actually, in order to write polynomial equations, no more is required than a domain whose elements can be combined together by the operations of addition and multiplication, with the properties which are ordinarily assumed for them; such a domain is called a ring. However, in order to deal with its specific problems, algebraic geometry allows itself a domain where all algebraic equations in one unknown have a solution (in the language of modern algebra, an algebraically closed field) and which contains algebraically independent elements (elements which do not satisfy any algebraic relation) in arbitrarily large number; this is called a universal domain. Such is the domain consisting of all complex numbers (real or imaginary), whereas the field of real numbers fails to satisfy the requirements of algebraic geometry because it is not algebraically closed (the equation x 2 + 1 = 0 has no solution in it). The necessity of introducing points with imaginary coordinates, in order to give a coherent formulation of the basic principles and results of elementary algebraic geometry, was first realized during the first decades of the 19th century; this is the true meaning of Poncelet's principle of continuity. One frequently distinguishes between classical algebraic geometry, which takes as its universal domain the domain of complex numbers, and modern or abstract algebraic geometry, which makes no restriction on the universal domain other than those implied in the definition. A peculiar feature of modern algebraic geometry, and a paradoxical one from the point of view of the classical theory, is the special attention given to universal domains of characteristic p, where p may be any prime (if 1 + 1 + 9-. + 1 = 0, 1 being repeated p times in the left-hand side, the domain is said to be of characteristic p, otherwise of characteristic 0). This is justified by the occurrence of such domains in all applications of algebraic geometry to problems concerning congruences in number theory. An algebraic plane curve, given by an equation f(x, y) = 0, is called irreducible if the polynomial f is irreducible, that is, if it is not the product of two polynomials of lower degree. If it were such a product, say g(x, y)h(x, y), the curve would be the union of the two curves defined respectively by g(x, y) = 0 and by h(x, y) = 0. In general, an algebraic locus is called irreducible if it is not the union of two other algebraic loci; an irreducible locus is also called a variety.

[1962c]

513 Algebraic Geometry

Rationality; Birational Correspondences An irreducible curve f ( x , y) = 0 is called rational if x, y can be expressed as rational functions x = R(t), y = S(t)

(2)

of a suitable parameter t. A straight line, being given by an equation ax + by + c = 0, is rational, since it can be parametrized by x = t, y = - ( a / b ) t - (c/b). Every conic is a rational curve. So is, for instance, the curve y2 _ x 3 = 0, of degree 3, since it can be parametrized by putting x = t 2, y = t 3. On the other hand, it can be shown that the curve yZ - x 3 - 1 = 0 is not rational. This concept of rationality had occurred already in the early stages of integral calculus, since it is fundamental for the classification of integrals of the type f F(x, y)dx

(3)

where F is a rational function of x, y, and y is the algebraic function of x determined by the equation f ( x , y) = 0. In fact, if the curve f ( x , y) = 0 is rational, (3) can be written, by means of (2), as the integral of a rational function of t, and therefore can be integrated by rational and logarithmic functions. This is the case, for instance, whenever y is given as y = x / a x 2 + bx + c, since this can be written a s y2 = a x 2 + bx 3- c, which is the equation of a conic. On the o t h e r hand, the integral ~ ( d x / ~ 1) cannot be expressed by rational and logarithmic functions (it is an elliptic integral). If the curve f ( x , y) = 0 is rational, it can be shown that one can choose, as parameter t in (2), a rational function of x, y; as t m a y be taken to be one coordinate of a point on a straight line, this is expressed by saying that the formulas (2), together with the formula expressing t in terms of x, y, define a birational correspondence between the curve f ( x , y) = 0 and a straight line. More generally, consider two curves C, D, respectively given by two equations f ( x , y) = O, 9(u, v) = 0. If it is possible to express the coordinates of a variable point (x, y) of C in terms of the coordinates of a variable point (u, v) of D by rational functions x = R(u, v), y = S(u, v)

(4)

and at the same time those of (u, v) in terms of those of (x, y), also by rational functions u = R ' ( x , y), v = S ' ( x , y),

(5)

then one says that the formulas (4), (5) determine a birational correspondence between C and D, and that C and D are birationally equivalent. A rational curve is thus one which is birationally equivalent to a straight line.

The Genus of a Curve By means of the formulas (4), (5), every integral of type (3) can be transformed into a similar one in terms of u, v, and conversely. Thus, from the point of view of the

514

[1962c] Algebraic Geometry

classification of integrals of type (3), birationally equivalent curves are interchangeable. An integral of type (3) is called of the first kind if it remains finite for every choice of the limits of integration and if the same is true of its transforms under any change of variables of type (4). While this definition seems to depend upon the concepts of integral calculus, it is easily seen that, in substance, it is purely algebraic and remains valid even in abstract algebraic geometry. Integrals of type (3) which are not of the first kind are said to be of the second or of the third kind, according to the way in which they can become infinite. As shown by the examples given above, the degree of a curve is not a birational invariant; that is, it need not be the same for two birationally equivalent curves. F r o m the point of view of birational equivalence, the main element of the classification of algebraic curves is the genus, which is defined as the maximal number of independent integrals of the first kind belonging to the curve. The rational curves are those of genus 0. Curves of genus 1 are called elliptic curves, because they can be parametrized by means of elliptic functions (when the universal domain is the field of complex numbers); they can also be characterized as those nonrational curves which are birationally equivalent to curves of degree 3. The concept of genus, and the essential results concerning integrals of algebraic differentials (also called abelian integrals), including their classification into three kinds and a study of their periods, were given by B. Riemann (1826 1866) in a famous paper (1857). Riemann used transcendental methods, involving Dirichlet's principle (an existence theorem for the solutions of certain types of partial differential equations, a full justification for which was given only much later, in 1900, by David Hilbert). Algebraic proofs for the same results were given later by R. Dedekind and H. Weber (1882) and many other authors. The concepts of rationality and birational equivalence, and also those concerning the integrals of algebraic differentials, can be extended to algebraic surfaces and varieties of higher dimension. However, while the work of Riemann and of his successors may be said to have given a fairly complete picture of the theory of algebraic curves, the same is far from being the case for higher-dimensional varieties. Some of the reasons for this will now be expl-ained.

Simple and Multiple Points As shown in the elements of differential calculus, if (a, b) is a point of the curve given by f(x, y) = 0, the tangent to the curve at that point has the equation fro" (x - a) + f~,. (y - b) = 0

(6)

where the coefficients f'a, f ; are the values taken at the point (a, b) by the partial derivatives o f f with respect to x and to y. The usual proof for this depends upon a limiting process. From the algebraic point of view, however, (6) may be taken as the definition of the tangent, the derivatives of a polynomial being taken to be defined by the rules given in differential calculus for their calculation. Such a definition of the tangent, however, becomes meaningless at a point (a, b) for which f'a and .f~ both vanish; when that is so, the point is said to be multiple; otherwise it is called simple. The simplest types of multiple points are called

[1962c]

515

AlgebraicGeometry nodes (Fig. 1) and cusps (Fig. 2). If an irreducible curve of degree n has no other multiple points (including its points at infinity in the sense of projective geometry) than r nodes and s cusps, its genus is (nt) =

1)(n-2) 2

--r--s.

Figure 1

Figure 2

Similarly, on an algebraic surface, a point is called simple if the formula for the tangent plane at that point, as given by the rules of differential calculus, remains formally meaningful; otherwise it is called multiple. These definitions can be extended without difficulty to curves, surfaces, and higher-dimensional varieties in spaces of arbitrary dimension. At a simple point, the method of power-series expansions can be applied (provided such expansions are interpreted in a purely formal manner, leaving aside all questions of convergence which are foreign to algebraic geometry). This is the main reason why algebraic geometry has many results concerning the properties of algebraic varieties at a simple point, while it is comparatively helpless when it has to deal with multiple points. Further Problems and Methods

The question arises whether a variety V can be transformed, by a birational correspondence, into another one, V', without multiple points. In order to be of any use, such a transformation must take into account, not only the points at finite distance, but also, in some suitable sense, the points at infinity. The classical way of doing this, and, even now, the most satisfactory in many respects, makes use of the concepts of projective geometry; thus, in the problem stated above, V and V' have to be interpreted as varieties in projective spaces. In the case of a curve, the problem turns out to have one and essentially only one solution (in the sense that, if V' and V" are two solutions, the birational correspondence which must exist between V' and V" determines a one-to-one correspondence between the points on V' and V'). Whether the problem always has a solution in the case of higher-dimensional varieties is still in doubt, but the solution can certainly not be unique in the sense explained above. For this reason, algebraic geometry, when it has been concerned with such varieties, has concentrated largely on those problems which do not depend too much on the choice of a model. Methods for dealing with these problems can be classified roughly into algebraic and transcendental methods. In the latter, the field of complex numbers is taken as

516

[1962c] Algebraic Geometry

universal domain, and its properties are freely used (including all available results of function theory, topology, and the theory of partial differential equations ; these methods generalize those which were so successfully applied by Riemann to the theory of curves; among their best-known exponents, one must mention Henri Poincar6, l~mile Picard, S. Lefschetz, W. V. D. Hodge and K. Kodaira. Algebraic methods, while they sometimes appear to be less powerful, are of greater scope, since they involve no restriction on the universal domain; they include the socalled geometric methods of the Italian school of algebraic geometry, which dominated the field between 1890 and 1925 (its most prominent representatives being Corrado Segre, Federigo Enriques, Guido Castelnuovo, Francesco Severi). While the work of this school has been criticized for its frequent lack of rigor and its weakness in the elucidation of fundamental concepts, later work has largely justified it by showing most of its methods to be essentially sound and by applying them to further problems, which are of too technical a nature to be discussed here.

[ 1964a] Remarks on the cohomology of groups 1. This is a sequel to two recent papers [3a, b] where I proved some theorems on the deformation of certain types of discrete subgroups of Lie groups; the method consisted in proving (in [3a]) a general result on the small deformations of such groups, and then combining this with the determination of their infinitesimal deformations, as was (in substance) done in [3b]. I have now noticed that, by using an e l e m e n t a r y lemma of a general nature, one m a y deduce the results of [3b] more directly f r o m the knowledge of the infinitesimal deformations, or, w h a t amounts to the same, of the relevant cohomology groups. The first purpose of the present note is to indicate briefly how this can be done. For simplicity, the lemma in question will be formulated for analytic manifolds, since the real-analytic case is alone relevant to the above-mentioned problems; the lemma is actually valid, not only for analytic manifolds over a n y complete valued field, but also for manifolds of class C" (with any r => 1) over the reals, and for non-singular algebraic varieties in the sense of a b s t r a c t algebraic g e o m e t r y . By a morphism, we understand a mapping of a manifold into a manifold, of the type indicated by the case under consideration (analytic, or of class C ~, or, in the case of algebraic varieties, a n y everywhere defined mapping in the sense of algebraic geometry). LEMMA 1. Let U and V be analytic m a n i f o l d s and (W,)~e~ a f a m i l y of analytic manifolds; let f be a m o r p h i s m of U into V, and, f o r each ~, let F~ be a m o r p h i s m of V into W~ such that FLof is a constant m a p p i n g w i t h the constant value cj and put X = A~ F~-~(c,). Let a be a point of U, and b = f ( a ) its image in V by f ; let A, B be the tangent vector-spaces to U at a, and to V at b, respectively; call M the image of A in B under the tangent linear m a p p i n g to f at a. Also, f o r each e, call LL the kernel of the tangent linear m a p p i n g to F, at b, and put L = ~ L~. Assume now that L = M; then there is an open neighborhood 1/'oof b in V such that, i f V' is any open neighborhood of b i n V0, X ~ V' is a submanifold of V' and coincides w i t h the image f ( U ) N V' of f-~(V') u n d e r f . In some open neighborhood of a in U, take a submanifold U~ of U with the same dimension as M, containing a, and transversal at a to the kernel of the t a n g e n t linear mapping to f at a; then, if A~ is the t a n g e n t linear v a r i e t y to U~ at a and if f~ is the mapping of U~ into V induced by f , the t a n g e n t linear mapping to f l a t a is an isomorphism of A1 onto M. On the other hand, for each 149 Reprinted from Ann. of Math. 80, 1964, by permission of Princeton University Press.

517

518

[1964a]

150

ANDR]~ WEIL

~, take local coordinates x: ~ in a neighborhood of c. with the value 0 at c,; then each function x : % F , is defined in a neighborhood of b on 17; and, among these functions, we can choose finitely m a n y functions 9~ whose differentials at b are linearly independent, and such t h a t L is defined by the equations dg~ = 0. We can now choose an open neighborhood V0' of b in V where all the 9~ are everywhere defined, and where their common zeros make up a manifold Y; L is then the t a n g e n t linear v a r i e t y to Y a t b, and X ~ 1/-' is contained in Y. As f maps U into X, this implies t h a t f maps f-l(V0') into Y, and t h a t f l maps f~-l(V0') into Y. In the analytic case now under consideration, the assumption L = M implies then t h a t there is a neighborhood of a in U~ which f~ maps isomorphically onto a neighborhood of b in Y; the same would be true in the case of C~-manifolds. In the case of algebraic varieties, the assumption L = M implies t h a t a is an isolated point of fi-~(b), and hence, by a theorem of Chevalley (re-stated and proved as L e m m a i in M. Rosenlicht, Trans. A.M.S. 101 (1961), p. 212), t h a t the set-theoretic image under f l of any neighborhood of a in U1 is a neighborhood of b in Y. Thus, in any case, the image of U~ under f l contains an open neighborhood of b in Y, which we m a y write as Y n Vo, where 170 is an open neighborhood of b in V0'. Let now V' be any open neighborhood of b in Vo; then the image of f - ' ( V ' ) under f is contained in X ~ V', which is contained in YV/V'; on the other hand, it contains the image of f ; ~ ( V ' ) under fl, which contains Yf~ V'; therefore it is the same as YN V' and the same as X V / V ' . This completes the proof. 2. Now let p be a representation of a group F, i.e., a homomorphism of P into the automorphism group of a finite-dimensional vector-space V over a field K. Let z be a mapping of F into V; for each v e P, call p'(v) the automorphism of the affine space underlying V which is given by x --~ p(v)x + z(v). Then p' is a homomorphism of F into the group of automorphisms of t h a t affine space if and only if z satisfies the relation (1 )

z(~/7 ') = z(7) § p(7)z(~/) ,

for all v, ~" in F; one expresses this by saying t h a t z is a l-cocycle (we shall say more briefly a cocycle) of P in V, when F operates on V by (% x) --* p(v)x. Let t~ be the translation x -~ x + a in V; then p'(v) = t o p ( 7 ) t : ~ is equivalent to (2)

z(v) = a - p(~/)a ;

if this is so for all v, z is a coboundary; the quotient of the space of coeycles by the space of coboundaries is the cohomology space HI(F, V). Let (~%)~e~ be a family of generators for P. Let F' be the free group generated by a family of generators (v:)~e~ indexed by the same set A; let A' be the kernel of the homomorphism of F' onto F which maps v" onto v~ for every

[1964a]

519

151

COHOMOLOGY OF GROUPS

~ . The e l e m e n t s of A ' c a n be considered, in t h e u s u a l m a n n e r , as " w o r d s " w(~/') in t h e 7"; f o r e v e r y such w o r d , w e h a v e w(7) -- ~, w h e r e e is t h e n e u t r a l e l e m e n t of F . I f (wx) is a f a m i l y of e l e m e n t s of A', such t h a t A ' is g e n e r a t e d b y t h e w~ a n d b y t h e i r t r a n s f o r m s u n d e r all i n n e r a u t o m o r p h i s m s of r ' , t h e n one s a y s t h a t F is defined b y t h e g e n e r a t o r s % w i t h t h e r e l a t i o n s wx(7) -- ~; w h e n t h a t is so, a h o m o m o r p h i s m r ' of F ' into a g r o u p G w i t h t h e n e u t r a l e l e m e n t e h a s t h e c o n s t a n t v a l u e e on 2~', a n d t h e r e f o r e d e t e r m i n e s a h o m o m o r p h i s m r of F into G, if a n d o n l y i f r ' ( w x ) = e f o r all )~. Since such a h o m o m o r p h i s m r ' is u n i q u e l y d e t e r m i n e d b y t h e r~ = r'(7:), a n d t h e s e can be chosen a r b i t r a r i l y , w e m a y also s a y t h a t a h o m o m o r p h i s m r of F into G is u n i q u e l y d e t e r m i n e d b y t h e e l e m e n t s r~ -- r ( % ) , a n d t h a t t h e s e can be chosen a r b i t r a r i l y p r o v i d e d t h e y s a t i s f y t h e r e l a t i o n s wx(r) = e. I n p a r t i c u l a r , l e t a g a i n p b e a r e p r e s e n t a t i o n of F, a n d let z a n d p ' be as a b o v e . T h e n w e see t h a t z is a cocycle if a n d o n l y if w x ( p ' ( 7 ) ) = e f o r all ~. I n o t h e r w o r d s , t h e cocycle z is u n i q u e l y d e t e r m i n e d b y t h e v e c t o r s z~ -- z(%), a n d t h e s e can be chosen a r b i t r a r i l y p r o v i d e d t h e y s a t i s f y t h e r e l a t i o n s wx(p'(~/)) = e. Moreover, z is t h e n a c o b o u n d a r y i f a n d o n l y if t h e r e is a v e c t o r a such t h a t (2) is satisfied w h e n e v e r one s u b s t i t u t e s one of t h e % f o r 7. I n o r d e r to w r i t e in a c o n v e n i e n t f o r m t h e c o n d i t i o n s we h a v e j u s t f o u n d f o r t h e z~, w r i t e e a c h w o r d wx(7) as t h e p r o d u c t 81 . . - 6~r

of a s e q u e n c e of

f a c t o r s , each one of w h i c h is e i t h e r of t h e f o r m 7~ or of t h e f o r m 7~1; a n d p u t , f o r 1 - i -< n(~)

(3)

u ~ = p(~i - . . ~-1) z~

i f ~ = 7~ ,

u ~ = - - p ( ~ - . . 6~) z~

if ~ = 7; ~ .

T h e n i t is e a s i l y seen t h a t t h e r e l a t i o n s w x ( p ' ( 7 ) ) = e f o r the.z~ a r e e q u i v a l e n t to t h e f o l l o w i n g ones: (4)

,_,~=1 u ~ = 0 .

T h e s e a r e t h e r e f o r e t h e r e l a t i o n s w h i c h d e t e r m i n e t h e cocycles of F in V; H ' ( F , V) is 0 if a n d o n l y if e v e r y s o l u t i o n of t h e s e e q u a t i o n s is a c o b o u n d a r y , i.e., of t h e f o r m z~ = a -- p ( % ) a f o r some choice of a . 3. N o w a s s u m e t h a t F is f i n i t e l y g e n e r a t e d a n d t h a t (7~)~e~ is a finite s e t of g e n e r a t o r s f o r F, i n d e x e d b y a finite s e t A . L e t R b e t h e set of all h o m o m o r p h i s m s of P into a g r o u p G; if w e i d e n t i f y each r e R w i t h t h e p o i n t (r~) of G ~), R is t h e set of t h e e l e m e n t s of G (~) w h i c h s a t i s f y t h e r e l a t i o n s w x ( r ) = e. W e shall c o n s i d e r o n l y t h e f o l l o w i n g t w o cases: (a) G is a Lie g r o u p ; t h e n G TM is also a Lie g r o u p , a n d R is a r e a l - a n a l y t i c s u b s e t of GC~); in t h i s case, w e t a k e f o r K t h e field R of r e a l n u m b e r s ;

520

[1964a l

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ANDRI~, WEIL

(b) G is an algebraic group; then G ~ is also an algebraic group, and R is a closed subset of G r in the Zariski topology; in this case, we take for K the universal domain. In both cases, each w~ may be considered as a morphism of G ~A~into G, and we have R = Fix w;~(e). On the other hand, for a given r e R, consider all the transforms r ' of r by the inner automorphisms of G; these are given by r : = x r ~ x 1, with x ~ G, and the mapping x --~ (xr~x -~) is a morphism of G into G ~ , which maps G into R. We shall now apply Lemma 1 to these morphisms. This requires the determination, for the present situation, of the spaces denoted by L and by M in Lemma 1. We identify the t a n g e n t vector-space to G at any point of G with the tangent vector-space to G at e, i.e., with the Lie algebra ~ of G, by means of a righttranslation. Then G operates on g by means of the adjoint group; more precisely, the inner automorphism x -~ s x s -~ of G induces on the t a n g e n t vector-space of G at e the automorphism X--~ Ad(s)X. It is now easily seen that the image M of the t a n g e n t vector-space ~ to G at e, under the tangent mapping to the morphism x -~ (xr~x -~) at e, consists of the vectors (Z~) c g(~ of the form Z~ = X -- Ad(r~)X, where X is a n y vector in g. On the other hand, write again w~(~) as 5~ . . . 5~x~, where the 5~ are as above; then the image of a t a n g e n t vector (Z~) to G (~) at r, under the t a n g e n t mapping to the morphism wx of G (A~ into G, is ~ ux~, where the u~ are given by (3), provided one substitutes (Z~) for (z~) and Ador for p in (3). Therefore, a f t e r this substitution is made, the kernel Lx of the t a n g e n t mapping to wx at r is defined by (4); and those equations, taken for all ~, define the linear variety L = Ax Lx. In other words, L is the space of the cocycles of F in g, when F operates on ~ by (% X) -~ Ad(r(~'))X, while M is the space of coboundaries for the same group; L - M is equivalent to H~(F, g) -- 0. In view of Lemma 1, this gives the following theorem: I f H~(F, g) = 0, t h e r e is a neighborhood o f r i n w h i c h e v e r y e l e m e n t o f R i s the t r a n s f o r m o f r by a n i n n e r a u t o m o r p h i s m o f G.

As a consequence, in order to prove Theorem 1 of [3b], one has only to verify that, under the assumptions of that theorem, H~(F, g) = 0; in substance, this is precisely what is done in nos. 6-10 of t h a t paper (cf. also [4]). 4. In the cases covered by Theorems 2 and 3 of [3b], H~(F, .q) is not 0; nevertheless, these theorems can be proved by a similar method, viz., by applying Lemma 1 to the situation described above in nos. 2-3, and combining this with the information about H~(F, g) contained in [3b] and with the direct determination of this group for the case in which F is a discrete subgroup, with compact

[1964a]

521 COHOMOLOGY OF GROUFS

153

quotient-space, of the 3-dimensional hyperbolic group. It seems hardly worthwhile to give a detailed proof along these lines, which would not lead to any new result; but I shall take this opportunity for giving some results about the cohomology of the groups F of the last-mentioned type, since this also plays a role in other investigations (cf. e.g., [1] and [2]). As in [1], we consider more generally the discrete subgroups F of the hyperbolic group for which the quotient-space has a finite measure. It is known that these groups can be obtained as follows. To begin with, take a free group A with 2g + n generators A1, . . . , A~, BI, . . . , B~, C, - . . , C~ (where g ~ 0, n >= 0); call E t h e neutral element of that group. Put R0 = E, and define elements (or "words") R , . . . , R~+~ of that group inductively by the following formulas

(5) (6)

R~ - R~(A, B, C) = R~ 1Ac~Bc~A~IB~1 R~+~ = R~+~(A, B, C) = R~+,,_~C.~

(1 ~< c~ _< g ) , (1 -< v -< n) ;

actually, for 1 ~ c~ _< g, the word R~ contains only the A~ and B~ for 1 =_
(7)

R~+~(a, b, c) = e ;

c~=e

(1-v-
where e is the neutral element of F; we shall denote by k the homomorphism of A onto F which maps A~, B~, C~ onto a~, b~, c~, respectively, for 1 _< c~ _< g, l_<~_
(8)

A'~ = F~(A, B)

= ~.~-~17 B~ ~1~-~.~

(1 _< c~ _< g ) ,

(9)

B" = G~(A, B)

= R~A~Rs

(1 _< ~ _< g ) ,

(10)

C'~ = H~(A, B, C) = R~+~_~R~-.~

(1 _< ~ _< n ) .

Then a trivial induction shows that we have the relation (11)

R~ = R~(A', B', C') = R : ~

(1 __
which gives at once

F~(A', B') = A~ ,

G~(A', B') = B~

H~(A', B', C') = C~

(1 -< ~ --< g ) , (1 = v =< n) ;

this shows that there is an involutory automorphism 9 of A which maps A~, B~, C~ onto A:, B:, C', respectively. From (6), (10) and (11), it follows that we have

522

[1964a]

154

ANDRI~I WEIL s --I ,I~C~ = R.+~_~C~- - s ~ R,+~-I,

--1 ~(R,+~) = R,+,~,

which implies that ~P maps the kernel of k onto itself; therefore the relation q~ok -- ~o~ determines an involutory automorphism 9 of F. Put a'~ = 9(a~), b: -- 9(b~), c'.~-- ~p(c~). Assume now t h a t A operates on a vector-space V by ( X , x) --+ X . x ; then, for any cocyle z of A in V, we have (12)

z(R~) -- ~ = ~ (A'~ -- E ) R ~ B ~ . z ( A ~ ) - ~ = ~ (B'~ -- E)R~_~A~.z(B~)

(1 _< ~ _< g ) ,

(13)

z(Rg+~) ---~z(Rg) -]- Et*=l Rg+t~-l"Z(C~)

(1 --< . --< n) ,

and also, for every integer s >__0: (14)

z(C~) = ( E + C.~ + . . . + C~-1).z(C~) .

In particular, if p is a representation of F in a vector-space V over a field K, we can apply (12), (13) and (14) to the representation po~ of A in V. On the other hand, the remarks in no. 2 show that a coeycle z of F in V is uniquely determined by the vectors z(a~), z(b~), z(c,~), and that these can be chosen arbitrarily provided they satisfy the relations obtained by writing 5(R~+~) = 0 ,

~(C~) = 0

for the cocycle 5 of A in V defined by 5(A~) -- z(a~) ,

5(B~) = z(b~) ,

5(C,) = z(c,) .

6. For each v, call F, the subgroup of P generated by c,; it is isomorphic to Z if s~ = 0, and cyclic of order s~ if s, > 0. In view of (14), and with the notations which have just been explained, the relation 5(C~,) = 0 can be written

(15)

~o

~ p(c'~)z(c~) = o ,

which expresses that z(c~) determines a cocycle of F~ in V; this is a coboundary if and only if z(c~) is of the form (16)

z(c~) = w~ -- p(c~)w~

with w,~ e V. We shall say that a cocycle z of F in V is p a r a b o l i c if, for every ~, it induces a coboundary on F~, i.e., if z(c~) is of the form (16) for every ~. E v e r y coboundary of F in V is of course a parabolic cocycle; we shall write P~(F, V) for the quotient of the space of parabolic cocycles by the space of coboundaries of F in V. It is easily seen (and, of course, well-known) that every solution of (15) is of the form (16) if s~ is not a multiple of the characteristic of K; if that is so for every ~, every cocycle of F in Vis parabolic, and P~(F, V) = H~(F, V); in particular, this is so if K is of characteristic 0 and none of the s~ is 0.

[1964a]

523 COHOMOLOGY OF GROUPS

155

From the above remarks, it follows t h a t a parabolic cocycle z is uniquely determined by the vectors z(a~), z(b~), w~, and t h a t these can be chosen arbitrarily, provided they satisfy the relation which was written above as 5(Rg+~) = 0; this can at once be written in t e r m s of the z(a,~), z(b~), w~ by using (12), (13) and (16). In order to write it more conveniently, put

(17)

u~ = p(R~(a, b, c)b~) z(a~) v~ = p(R~_i(a, b, c)a~) z(b~)

(1 -< ~ <= g ) , (1 s ~ ~< g) ,

t~ = p(Rg+~(a, b, c)) w~

(1 -< ~ -< n) .

The relation in question can then be written as follows (18)

E o [p(a:) -- l~]uo - E o [p(b:) -- l~]v~ + E ~ [p(c') - l~]t~ = 0 ,

where 1. = p(e) is the identity automorphism of V. Call F ( u , v, t) the left-hand side of (18); F i s a linear mapping of V 2~+~ into V. Let d be the dimension of V, and let i' be the codimension in V of the image of V :~+~ under F; then the kernel of F, i.e., the space of the solutions (u, v, t) of (18), has the dimension D - (2g § n)d -- d § i'. On the other hand, for each ~, call e,~ the rank of the endomorphism 1~ -- p(c~) of V. Then the kernel of the mapping w~--~ z(cO defined by (16) has the dimension d - e~; therefore, in the mapping of the space of solutions (u, v, t) of (18) into the space of parabolic cocycles which is defined by (16) and (17), the dimension of the kernel is ~--]~(d - e~). This gives, for the space of parabolic cocycles, the dimension (19)

P=(2g--1)d

§247

In order to give a more convenient interpretation for i' than the one given above, let V' be the dual space to V; write <x, x'> for the bilinear form on V • V' which defines the duality between V a n d V'; as usual, denote by ~p(s), for s e F, the transpose of p(s), i.e., the automorphism of V' defined by < p ( s ) x , x ' > = <x, ' p ( s ) x ' > .

By definition, i' is the dimension of the space of vectors x' e V' such t h a t
v , t), x ' > = 0

for all (u, v, t). This is clearly equivalent to the relations p ( a'~) x' =

x',

~p(b~)x ' '

=

x' ,

~p(cOx ' '

=

x' .

As the a'~, b'~, c'~ make up a set of g e n e r a t o r s for F, this shows t h a t i' is the d i m e n s i o n of the space of the vectors i n V ' w h i c h are i n v a r i a n t u n d e r ~p(s) f o r all s ~ F. On the other hand, call i the d i m e n s i o n of the space of the vectors i n V w h i c h are i n v a r i a n t u n d e r p(s) f o r all s e F. This is the dimension of the

524

[1964a] 156

ANDRI~ WEIL

kernel of the mapping a --~ z of V onto the space of coboundaries of F which is defined by (2); therefore that space has the dimension d - i, and we get, for the dimension p of the space P1(F, V), the formula p=(2g-2)d§247247 This gives the dimension of the cohomology space Hi(F, V) in the cases in which all cocycles are parabolic (e.g., if no s~ is (i and K is of characteristic 0). 7. As in no. 3, these results can be applied to the study of the space of the representations of F into a group G of one of the types introduced there, i.e., a Lie group in case (a) and an algebraic group in case (b). We assume that none of the s~ is a multiple of the characteristic of the field K (i.e., in case (a), that none of them is O); then every cocycle of F is parabolic; more precisely, as we have already observed in no. 6, the cohomology space HI(F~, V), for any representation of P~ in a vector-space over K, is 0. For each ~, call W~ the set of the elements w of G which satisfy w~ = e; the theorem at the end of no. 3 shows that W~ is an analytic submanifold of G in case (a), and that it is the union of finitely many mutually disjoint non-singular subvarieties of G in case (b); moreover, it shows that the connected component in W~ of any point w of W~ consists of the points x w x -1 for x e G, and that the tangent linear variety to that component at w is the image My of g under the tangent linear mapping to x ---* x w x -1 at e. Now let R be the set of the homomorphisms of F into G; for each r e R, put r~ r(a~), r: = r(b~), r~' = r(c~), and identify r with the point (r~, r:, r") of G 2~ • W, where W = II~ W~. We can consider R,+~ as defining, in an obvious manner, a mapping of G:~ • W i n t o G, and the set R is then nothing else than R j ~ ( e ) . By the same kind of calculations as those in nos. 3, 5, 6, it is easy to determine the tangent linear mapping to R~+~ at any point, or, what amounts to the same, the t a n g e n t linear variety S to the graph of R~+~ at any point (r, R~+~(r)) of that graph, and in particular, at the point (r, e) of this graph for any r ~ R; if we take again for p, as in no. 3, the homomorphism Ador of P into the group of automorphisms of g, and if we write again F ( u , v, t) for the left-hand side of (18), we find that S is the space of all vectors of the form

(~(a~), z(bo), ~(~), F(u, v, t)) when one takes for (z(a,), z(b~)) all the vectors in fi2~, for (w~) all the vectors in g~, and for (u, v, t) and (z(c~)) the vectors defined in terms of these by (16) and (17). With the notations of no. 6, S has then the dimension 2gd + ~ e ~ , d being the dimension of G. The points of S for w h i c h F ( u , v, t) = 0 are no other than the cocycles of F in g, and make up a subspace of S whose dimension P is

[1964a]

525 COHOMOLOGY OF GROUPS

157

given by (19); its codimension in S is d if and only if i' = 0. This shows t h a t the g r a p h of R,+. is t r a n s v e r s a l to (G 2~ • W) • e at (r, e), for a given r e R, if and only if i' = 0. Therefore, w h e n i' = O, t h e r e is a neighborhood o f r i n w h i c h R is a n a n a l y t i c s u b m a n i f o l d o f G 2~ • W i n case (a), a n d a n o n - s i n g u l a r s u b v a r i e t y o f G ~ • W i n case (b), w i t h the d i m e n s i o n P = (2g - 1)d § ~ e ~ . INSTITUTE FOR ADVANCED

STUDY BIBLIOGRAPHY

1. G. SHIMURA,S u r les int@raIes attach~es aux formes automorphes, J. Math. Soc. Japan, 11 (1959), 291 311. 2. A. WEIL, Ggn~ralisation des fonctions ab~liennes, J. Math. Pures et Appl. (IX), 17 (1938), 47 87. 3. - - , On discrete subgroups of Lie groups, (a) Ann. of Math., 72 (1960), 369-384; (b) ibid. 75 (1962), 578-602. 4. Y. MATSUSHIMA and S. MURAKAMI,On vectorbundle-valued harmonic forms and automorphic f o r m s on symmetric riemannian manifolds, Ann. of Math., 78 (1963), 365-416. (Received June 5, 1963)

Commentaire N.B. Dans ce commentaire, une r6f6rence telle que [1938a] renvoie/l l'article du texte portant cet indicatif (cf. Table des Mati6res) ; [1938a]* renvoie au commentaire de l'article en question.

[1951c] Review of "Introduction to the theory of algebraic functions of one variable" (by C. Chevalley) Le livre de Chevalley sur les fonctions algabriques parut en 1951, et on m'en d e m a n d a le comptc-rendu pour le Bulletin of the A.M.S. La qualita de ce livre et de la contribution qu'il apportait fi son sujet atait hors de doute; mais il tamoignait d'une attitude qui me paraissait dapassae, et je pris cette occasion pour marquer la difference entre le point de vue de Chevalley, qui avait 6t6 celui de l'acole algabrique allemande, et le point de vue gaomatrique auquel je m'atais rallie depuis longtemps. Quant ~ l'usage qu'on fit un peu plus tard de citations tronquaes, extraites de m o n compte-rendu, pour faire achec ~t une candidature de Chevalley ~t la Sorbonne (cf. [1955e]), est-il besoin de dire que je ne m'en sentis nullement responsable ? On notera, pp. 390 392, une presentation de certains rasultats de Chevalley, diffarente de la sienne. II s'agit lh d'une extension aux integrales de 3 e espace de rues anciennes observations sur les intagrales ~ (logf)d(log g) (cf. [1939a]*). On obtient ainsi une ganaralisation des relations bilinaaires de Riemann, ct en marne temps unc expression purement algabrique de la dualita entre les groupes d'homologie H l ( S - P, Q) et HI(S - Q, P), o e j e continuais 5. rechercher l'analogue, en thaorie classique, de la loi de raciprocit6 pour les corps de fonctions en caracteristique p. En efl'et, arrive h ce point, on n'est pas loin des "jacobiennes genaralisaes" de Rosenlicht et de la thaorie du corps de classes (cf. [1950a]*).

[1952a] Sur les theoremes de de Rham J'ai dejfi dit ([ 1940b]*, [1947b]*) combien, de longue date, j'attachais d'importance aux theorames de de Rham, et combien [cur pratique m'atait devenue familiare. J'avais fini par voir lfi une pierre de touche de toute thaorie homologique; et lorsque par exemple Eilenberg me fit part en 1944 de la theorie axiomatique qu'il venait de b~ttir avec Steenrod, ma premiere question fut pour demander si celle-ci rendait compte des thaorames de de R h a m ; ma deception fut grande d'apprendre qu'il n'en etait rien. Une fois au Bresil, rues raflexions sur la thaorie de Hodge d'une part (v. [1947b]), et de l'autre mes calculs sur les " W - g r o u p e s " (v. [1951b]) m'avaient amena fi me constituer une technique d'algebre homologique, peu raffinee mais qui a suffi ~t rues besoins. Sans doute y avais-je ate grandement aide par les notions que j'avais acquises aux Etats-Unis, au contact d'Eilenberg, sur les groupes d'EilenbergMacLane. D'autre part, lors d'un rapide voyage '5. Paris en juin et juillet 1945, je m'atais rencontre briavement avec Leray, tout juste revenu de captivita, et il m'avait parle de sa "cohomologie "~ coefficients variables"; c'est ainsi du moins 526

Commentaire

527

que le souvenir m'en 6taft rest6, ct cette idde, bien que vague dans mon esprit, m'avait frapp6. Enfin, je ne pouvais ignorer que Bourbaki avait commenc6 envisager sdrieusement la composition d'un livre sur la "topologie combinatoire" comme on disait alors, c'cst-',i-dire principalement sur l'homologie; il devait y renoncer plus tard devant l'afftux des ouvrages, dont quelques-uns de bonne qualit6, qui allaient parMtre dans la d6cade suivante; mais en attendant ce sujet 6tait/~ l'ordre du jour. C'est ainsi que j'obtins au ddbut de 1947 l'essentiel des r6sultats qui forment le contenu de [1952a]; j'en fis part aussit6t /i Dieudonn6, qui m'avait rejoint au Brdsil l'ann6e pr6cddente, et fi Cartan qui 6tait alors/~ Strasbourg. Celui-ci, de son c6t6, venait d'dtudier de pr6s les travaux de Leray, de sorte que ma lettre trouva chez lui un terrain tout pr6par6; il y r6pondit sur le champ par une s6rie de lettres et de papiers qui constitu6rent bient6t une thdorie compl6te de la cohomologie; c'6tait la thdorie dite des "carapaces", qui allait former le sujet de son tours de Harvard de 1948, puis de son sdminaire de 1948 49 fi I'Ecole Normale. Bien entendu les thdorbmes de de Rham y rentraient comme cas particulier. Pendant quelques anndes je crus que ces d6veloppements, et ceux qui suivirent (surtout le Sdminaire Cartan de 1950 51) avaient rendu superflue ma ddmonstration. Je finis par penser qu'un rdsultat de cette importance m6ritait d'Stre trait6 s6par6ment ; de plus, mon thdor6me au sujet du type d'homotopie (w s'y rattachait naturellement et pouvait pr6senter quelque intdrSt. D'ailleurs, ~ l'occasion d'une s6ance de s6minaire/l Chicago, N. Hamilton avait apport6/t ma ddmonstratiqn une amdlioration substantielle (cf. [1949d]*); cela m'encouragea ~ la r6diger pour les Commentarii, qui avaient d6j/t accueilli [1947b].

[1952b] Sur les "formules explicites" de la th~orie des nombres premiers Ayant d6montr6 l'"hypoth6se de Riemann" pour les corps de fonctions de dimension 1, je ne pouvais pas ne pas chercher '~ en tirer des lumi6res sur l'hypoth~se de Riemann classique. Du reste je m'6tais persuad6, d'abord qu'on ne d6montrerait jamais celle-ci pour le corps des rationnels sans la d6montrer en m~me temps pour tousles corps de nombres, puis que la clef en serait un th6or6me de positivit6, analogue en quelque mani6re au th6or~me de Castelnuovo, et qui du m~me coup entrainerait la conjecture d'Artin. Je me placais ainsi, je crois, dans la ligne de Hilbert; Hellinger m'a racont6 en effet que celui-ci, apr6s avoir d6montr6 la rdalit6 des valeurs caract6ristiques d'un noyau symdtrique en enseignant pour la premi6re lois sa th6orie des 6quations int6grales, avait ajout6: "Et avec ce th6or6me, Messieurs, nous ddmontrerons l'hypoth6se de Riemann." C'est dans cet esprit que j'avais repris une lois de plus l'examen des analogies entre corps de nombres et corps de fonctions. Ce sujet se prSte fi un grand nombre d'exercices, et les "formules explicites" de la th6orie classique des nombres premiers en fournissent un qui est assez simple (si on en laisse de c6t6 les subtilit6s) pour qu'on puisse en d6gager le formalisme alg6brique et l'6tendre aux fonctions L. C'est ce qui est fait dans [1952b], o/l il est montr6 que l'hypoth6se de Riemann pour l'ensemble des fonctions L sur un corps de nombres 6quivaut fi la positivit6 d'une certaine distribution sur le groupe des classes d'idSles du corps. I1

528

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e n e s t de marne pour les corps de fonctions, de sorte que pour ceux-ci cette positivit6 est un fair acquis. I1 n'est pas plus difficile, en principe, de procdder de m6me fi l'6gard des fonctions L " d ' A r t i n - H e c k e " (cf. [1951b]*), en ajoutant la conjecture d'Artin/~ l'hypoth6se de Riemann. C'est ce qui est fait dans [1972] au moyen d'un formalisme quelque pcu clarifi6 et simplifi6; on obtient ainsi sur les " W - g r o u p e s " des distributions dont la positivit6 6quivaut ~ l'ensemble des conjectures en question. A la lumi6re de travaux rdcents ol) des cas particuliers de la conjecture d'Artin ont pu ~tre traitds avec succ6s, on peut se demander ~ prdsent si elle n'est pas moins rebelle que sa compagne. N6anmoins, c o m m e je l'ai indiqu6 dans [1972], l'interpr6tation qui est donn6e de l'une et de l'autre dans ce dernier travail, jointe l'analogie entre les corps de nombres et les corps de fonctions, me semble fournir actuellement l'argument le plus convaincant en faveur de leur validit6 ~ toutes deux dans le cas classique. P o u r ceux qui, en attendant une solution d6finitive de ces probl~mes, s'int6resseraient 5. des rdsultats partiels, fussent-ils num6riques, voici deux points qu'il n'est peut-~tre pas inutile de signaler. D'une part, si le scepticisme 5- l'6gard de l'hypoth6se de Riemann classique a beaucoup diminu6, j'ai constat6 qu'il subsiste encore vis-/l-vis des fonctions L de Hecke. Pourquoi un sp6cialiste des ordinateurs, en ce temps o/l on les fait fonctionner un peu 5. tort et 5- travers, ne s'amuseraitil pas 5- calculer des z6ros des plus simples de ces fonctions, fi savoir celles qui appartiennent au corps de Gauss Q(i)? D'autre part, on s'est acharn6 5. jalonner dans la bande critique des domaines o/1 les zdros de ~(s) ne peuvent pdndtrer. Mais, c o m m e il apparait dans [1952b], l'hypoth6se de Riemann 6quivaut au fait que, pour une classe suffisamment large de fonctions @(s) positives sur la droite R e ( s ) = 89 la somme ~ qb(p) est positive. N'y aurait-il pas un exercice instructif pour un analyste dans la recherche de fonctions qb pour lesquelles il en est ainsi? Q u a n t fi la notion de distribution qui intervient dans ce travail, et dont je pris pr6texte pour le dddier 5- un analyste, elle 6tait encore relativement nouvelle, marne dans le cas des variables r6elles, puisque Laurent Schwartz ne les avait introduites pour celles-ci qu'en 1945. A la fin de [1952b], p. 265, je me suis m6me exprim6 c o m m e si cette notion avait ddjs- 6t6 d6finie sur tousles groupes localement compacts; elle ne le fut que par Bruhat en 1961 (Bull. Soc. Math. Fr. 89, pp. 43 75). I1 est vrai quej'en avais ddjfi fait l'essai dans un cas particulier, pendant m o n sdjour au Brdsil, en ddfinissant pour mon usage ce que j'appelai plus tard la "distribution d ' H e r b r a n d " ; mais l/t il s'agit seulement d'un groupe totalement discontinu, oO la notion de distribution est 616mentaire et se confond avec celle de fonction simplement additive des parties ouvertes compactes du groupe 6tudi6.

[1952c] Fibre-spaces in algebraic geometry II n'a pas 6t6 jug6 utile de reproduire ces notes, dues ~ A. Wallace, d'un cours profess6 en 1952 fi Chicago. I1 suffira d'en r6sumer ici le contenu: w R6vision de la notion de "vari6t6 abstraite" (cf. la 2 c 6dition de [1946a]). w Ddfinition des varidtds fibrdes avec groupe structural opdrant sur les fibres, au

Commentaire

529

moyen de "fonctions de transition" ",i valeurs dans le groupe.-w Classification des vari6tds fibr6es fi groupe G,, = GL(1) op6rant "naturellement" sur la droite projective D. w Surfaces rdgl6es: (a) classification des vari6tds fibr6es sur une courbe (i.e. fi base de dimension 1), fibr6es par D, appartenant au groupe affine, x ---, a x + b, op6rant sur D; (b) existence d'une section sur toute surface r~gl6e (vari6t6 fibr6e sur une courbe, fi groupe P G L ( 2 ) opdrant sur D). w Vari6t6s fibr6es fi groupe C • sur une vari6t6 analytique complexe (cf. [1958a], Chap. V). [ 1952 d] Jacobi sums as "Gr6ssencharaktere"

Peu avant la guerre, si mes souvenirs sont exacts, G. de Rham me raconta qu'un de ses 6tudiants de Gen6ve, Pierre Humbert, 6tait all6 fi G6ttingen avec l'intention d'y travailler sous la direction de Hasse, et que celui-ci lui avait propos6 un probl6me sur lequel de R h a m d6sirait mon avis. Une courbe elliptique C 6tant donn6e sur le corps des rationnels, il s'agissait principalement, il me semble, d'dtudier le produit infini des fonctions zSta des courbes Cp obtenues en r6duisant C modulo p pour tout nombre premier p pour lequel C~, est de genre 1 ; plus pr6cis6ment, il fallait rechercher si ce produit poss6de un prolongement analytique et une 6quation fonctionnelle. J'ignore si Pierre Humbert, ou bien Hasse, avaient examin6 aucun cas particulier. En tout cas, d'apr6s de Rham, Pierre Humbert se sentait d6courag6 et craignait de perdre son temps et sa peine. J'avoue avoir pens6 que Hasse avait 6t6 par trop optimiste. Je dis fi de Rham, non seulement que le probl6me me semblait bien trop difficile pour un d6butant, mais mSme que je ne voyais aucune raison pour que le produit en question efit les propri6t6s que Hasse lui supposait. Par la suite, Pierre Humbert se tourna vers Siegel et la th6orie des formes quadratiques; je ne sais si ma conversation avec de R h a m contribua fi ce changement d'orientation. II mourut jeune, peu apr6s la guerre. Q u a n d je revis Hasse H a m b o u r g , / t l'occasion d'une conf6rence sur le sujet mSme de [1952d],je n'eus pas l'impression qu'il efit gard6 aucun souvenir de cet 6pisode ni du probl6me propos6 ,i9 Pierre Humbert. Au Brdsil, cherchant des applications de l'hypoth6se de Riemann, je fis des calculs sur les fonctions zSta des courbes hyperelliptiques sur un corps fini (cf. [1948c]), et en particulier celle de y2 = X4 + 1 sur Fp. C'est l',i un exercice facile; la fonction zSta, ou pour mieux dire sa partie non triviale, est 1 + p U 2 s i p - 3 (mod 4); s i p - 1 (rood 4), c'est (1 - c~U)(1 - ~ U ) , o i l c~ = a + bi, ~ = a - b i sont les facteurs premiers de p dans Z(i), norm6s par a - 1 (mod 4). Q u a n t cette derni6re condition, mes notes de cette 6poque renvoient au m6moire de Gauss sur les r6sidus biquadratiques; c'est sans doute ce qui me d o n n a l'id6e, un peu plus tard, d'6tudier ce m6moire de plus pros (cf. [1974a], p. 106). C o m m e l'indiquent ces notes, il s'ensuit imm6diatement que le produit infini de Hasse pour la courbe y2 = X 4 + 1 est une fonction L de Hecke relative au corps Q(i). La conjecture de Hasse commen~ait fi prendre forme/t mes yeux. Les choses en rest6rent l/t, il me semble, jusqu'fi une discussion que j'eus avec Chevalley au cours des ann6es suivantes, ll ne croyait pas que les fonctions L de Hecke eussent un r61e fi jouer en th6orie des nombres. Je soutenais le contraire (cf.

530

Commentaire

[1936i]* et [1951b]*) et citai en exemple le produit de Hasse pour la courbe y2 = X 4 + 1, peut-Stre aussi pour y2 = X s _ 1, en ajoutant qu'il devait s'agir lfi d'un ph6nom6ne bien plus g6n6ral. L/lje m'avan~ais beaucoup; ce qui finalement me d o n n a confiance dans cette id6e, ce rut une lois de plus l'analogie entre corps de nombres et corps de fonctions; j'observai (v. [1950b], p. 99) que la conjecture de Hasse pour les courbes sur un corps de nombres correspond exactement fi rues conjectures de 1949 pour les surfaces sur un corps fini, au sujet desquelles il ne me restait plus aucun doute. Parvenu ",ice point, je me devais d'examiner les courbes pour lesquelles je savais calculer explicitement la fonction zSta, c'est-Mdire avant tout les courbes Y" = aX" + b. En ce cas, tout se ramenait ais6ment 5. un probl~me concernant les " s o m m c s de Jacobi", et plus pr6cisdment ~ la ddtermination d'une racine de l'unit6 qui entre dans l'expression de ces sommes; j'ignorais qu'fi la terminologie pr6s une bonne partie de la question avait 6t6 trait6e un si6cle plus t6t par Eisenstein (v. [1974d], p. 2). En tout cas, je trouvai que les fonctions z6ta des courbes ym = aXn + b sur un corps de nombres s'expriment bien, c o m m e j e l'esp6rais, par dcs fonctions L de Hecke. Le cas des courbes y 2 = a X 4 + b e t y 2 = a ? ( 3 + b permettait de supposer qu'il pouvait en 8tre de mSme pour toute courbe elliptique /~ multiplication complexe, et un calcul sommaire montrait que ce cas se ramenait, en premi6re approximation, ~ la d6termination d'un signe _+ 1. Cette observation, communiqu6e ~ Deuring, d o n n a lieu par la suite 5, sa belle s6rie de recherches sur les fonctions zSta de ces courbes (GOtt. Nachr. 1953, no. 6; 1955, no. 2; 1956, no. 4; 1957, no. 3). Pour des rdsultats sur les sommes de Jacobi qui contiennent ceux de [1952d] comme cas particuliers, v. [1974d].

[1952e] On Picard varieties Depuis le colloque de Paris en 1949 (v. [1950a]), je m'effowais d'approfondir et de d6velopper les id6es q u e j'y avais esquiss6es, ainsi que les r6sultats du travail d'Igusa (Am. J. of Math. 74 (1952), pp. 1 22) dont Kodaira, arriv6/t Princeton en 1949, m'avait transmis le manuscrit fi la fin de la mSme annde. Kodaira aussi s'int6ressait fi ces questions; en 1950, fi la suite d'un s6minaire de G. de R h a m sur les formes harmoniques et les courants dans le cas r6el, il exposa d'importants r6sultats sur les vari6t6s kfihl6riennes (v. G. de R h a m et K. Kodaira, Harmonic Integrals, I.A.S. 1950). Au congr6s de Cambridge, Hodge exposa ses id6es du moment. J'y renouai connaissance avec lui; ce fut le point de ddpart d'un 6change de lettres, des plus instructifs pour moi, dans les mois qui suivirent. J'eus aussi vers la meme 6poque une abondante correspondance avec Kodaira, de Rham, Laurent Schwartz et Chow sur les sujets en question. En ce qui concernait la "vari6t6 d'Albanese" (nomm6e "dual Picard variety" dans [1952e]), il n'6tait besoin que d'une mise au point assez facile. Je crus 61argir un peu le cadre des vari6t6s projectives, o/1 s'6tait plac6 Igusa, en introduisant la notion de "vari6t6 de Hodge", formellement moins restrictive. Je ne pr6voyais pas que bient6t Kodaira d6montrerait l'identit6 substantielle des deux notions, dans un travail qu'on peut qualifier de tour de force (Ann. of Math. 60 (1954), pp. 28 48).

Commentaire

531

En tout cas, du point de vue de la m6thode, je n'eus pas tort sans doute d'6tablir cette distinction. P o u r la "vari6t6 de Picard" et ses liens avec les classes de diviseurs, Igusa, Kodaira et moi pouvions nous inspirer de Poincar6, des Italiens, de Lefschetz. Mes vieilles connaissances les fonctions multiplicatives y jouaient leur r61e (cf. [1934b,c]*); ce n'est pas 6tonnant, puisque ces fonctions, sur une vari6t6 V, ne sont pas autre chose que les sections des vari6t6s fibr~es sur V, de groupe G,, = GL(1, C) = C • (la fibre 6tant C ou bien la droite projective), et puisque la vari6t6 de Picard de V est la vari6t6 des modules pour ces fibr6s (cf. [1938a]*, [1949c], [1950a], [1952c]). Mais il ne ressortait pas clairement du travail d'lgusa si l'application canonique induit bien une application holomorphe sur toute famille alg6brique ou alg6broide de diviseurs. J'eus quelque peine/l l'6tablir ([1952e], p. 883) et n'y parvins pas m6me compl6tement; fi cette 6poque off on ne disposait pas du th6or6me d'Hironaka, ni de la th6orie des courants 61abor6e par la suite par Federer, ni m6me du th6or+me de Lelong sur les courants d6finis par les variet6s analytiques, ma technique n'6tait pas ad6quate fi ce que j'avais en vue. M6me pour pouvoir appliquer la th6orie de l'homologie aux sous-vari6t6s d'une vari6t6 V, je ne pouvais m'appuyer que sur un travail in6dit, et qui malheureusement l'est rest6; c'6tait la th6se de m o n 6tudiant N o r m a n Hamilton, alors en voie d'ach6vement, mais qui, je ne sais plus pourquoi, ne fut soutenue qu'en 1955. Il a fallu, c o m m e on sait, un travail de Borel et Haefliger (Bull. Soc. Math. Fr. 89 (1961), pp. 461 513) pour combler cette lacune. A plus forte raison les ressources de la technique 6taient-elles insuffisantes pour ce qui, d6s 1950, me paraissait ouvrir le plus de perspectives nouvelles,je veux dire la th6orie des "jacobiennes interm6diaires". C'est l~-dessus que porte presque toute la correspondance dont il a 6t6 fait mention plus haut, et oO je ne cessais de rbclamer en vain fi mes amis, et particuli6rement 5. Schwartz et ~ de Rham, les r6sultats sur les courants dont je voyais quej'avais besoin. Ici quelques explications vont &re n6cessaires. Soit A l'espace des formes harmoniques r6elles de degr6 2p + 1 sur la vari6t6 V de dimension complexe n; on peut l'identifier au groupe de cohomologie H 2p+ I(V, R). Dans A, soit Ale r6seau image de H 2p+ ~(V, Z), ensemble des formes p6riodes enti6res. Le complexifi6 A c de A est somme directe des sous-espaces Xa,b formes respectivement par les formes de type de Hodge (a, b) avec a + b = 2p + 1; soit Pa.b le projecteur de Ac sur Xa, b. Soient 0 < v < p, a = 2p + I - v, b = v; alors Pa, b + Pb.a est l'extension fi A c d'un projecteur de A sur un sousespace A~ dont X,,.b + Xb, ~ est le complexifi6, et les projecteurs Pa, bet Pb.a d6terminent sur A v des isomorphismes (r6els) sur X,. b et sur Xb. a, respectivement. Transportant b. A~, au moyen de ceux-ci, la structure complexe de Xa.b ou bien celle de Xb.a, on obtient sur Av deux structures complexes, imaginaires conjugu6es l'une de l'autre. C o m m e A est somme directe des A v, on peut ainsi y d6finir 2 p§ structures complexes distinctes, qu'on peut consid6rer c o m m e d6termin6es par les op6rateurs de carr6 - 1: ,u

J~ = i~/sv(P2p+ v=D

l_v,

v --

Pv,2p+l-v)

532

Commentaire

avec e~ = + 1 p o u r 0 < v < p. P o u r chacune de ces structures, A/A devient un tore complexe. Hodge, dans son livre (Harmonic Integrals, pp. 192 196), avait 6tabli un th6or6me (reproduit, aux n o t a t i o n s pr6s, dans m o n livre [1958a], pp. 77-78, c o m m e corollaire du th. 7) d o n t voici l'essentiel. O n peut d6finir dans A une forme altern6e B(x, y), fi valeurs enti~res sur A x A, s ' a n n u l a n t sur A u x A~ p o u r # ~ v, et telle que, p o u r 0 < v < p, la forme induite p a r B(x, J~y) sur A~ x A~ soit sym6trique et d6finie (positive ou ndgative suivant le choix de e.~.); le calcul m o n t r e aussi que ces formes sont toutes positives si on a pris

J~ = - C = - ~

i"-bp~,b.

N a t u r e l l e m e n t ce rhsultat subsiste p o u r d~ = C, p o u r v u q u ' o n remplace B p a r - B ; le choix d~ = - C est dhtermin6 p a r le fair q u ' o n obtient ainsi la variht6 d ' A l b a n e s e de V p o u r p = n - 1 et sa variht6 de P i c a r d p o u r p = 0. En vertu des thhor6mes connus sur les varihths abhliennes (cf. p. ex. [1949d] ou [1958a]), il s'ensuit que, p o u r 0 _< p < n, le tore complexe ainsi dhfini p a r J~ = - C sur A/A est une variet6 abhlienne; p o u r tout choix de d~, autre que + C , cette assertion ne serait plus vraie en ghnhral. Ce sont ces variht6s qui sont dhfinies dans [ 1952e] sous le nora de " j a c o b i e n n e s " et que, p o u r 0 < p < n - 1, on n o m m e souvent " j a c o b i e n n e s i n t e r m h d i a i r e s ' . C o m m e je m'en aper~us plus tard et c o m m e le m o n t r e n t des exemples trhs simples (en particulier le cas off V est un p r o d u i t de courbes elliptiques), elles ont un grave dhfaut: leurs m o d u l e s ne sont pas fonctions h o l o m o r p h e s de ceux de V, ce qui semble ruiner tout espoir d'une interprhtation alghbrique. C o m m e Griffiths l'a montr6, on r e m h d i e / i cet inconvhnient p a r un autre choix de d~, mais cela d o n n e des totes complexes qui en ghnhral ne sont pas des varihths alghbriques. Q u o i qu'il en soit, m a construction, j u s q u ' a ce point, ne faisait usage que de thhorhmes de H o d g e qui ne soulhvent pas de discussion. M a i s ressentiel p o u r moi dans rues rhflexions sur ce sujet 6tait l'application, heuristiquement dhfinie, du g r o u p e des cycles algebriques h o m o l o g u e s ~ zhro, de c o d i m e n s i o n p + l, dans la j a c o b i e n n e dhfinie p a r les formes de degr6 2p + 1. L'idhe de cette a p p l i c a t i o n a dfi me venir, consciemment ou non, de Volterra et de ses "fonctions de ligne". Encore tout jeune, il avait observ6 qu'une forme diffhrentielle fermhe de degr6 2 dans un d o m a i n e de l'espace ordinaire, inthgrhe sur une surface S d o n t le c o n t o u r est la difference L - L 0 de deux lignes fermhes, dhtermine une fonction additive de la ligne fermhe L, et qu'il y avait l/l un i m p o r t a n t sujet d'htude. II avait vu aussi que cette notion s'htend d'elle-m6me aux d i m e n s i o n s quelconques, et qu'elle fait intervenir des questions d ' " a n a l y s i s s i t u s " / t cause des phriodes de la forme en question; 1~ il dut se souvenir des lemons de son maitre Betti. Il avait dhvelopp6 longuement (loc. cit., [1947b]*) le lien entre cette idhe et sa thhorie des " f o n c t i o n s conjuguhes" (c'est-h-dire en s o m m e des formes h a r m o n i ques) qu'il prhsentait c o m m e ghnhralisation.de la thhorie de R i e m a n n des fonctions analytiques, et p a r laquelle il prhludait aux travaux de H o d g e (cf. [1947b]*). Ce que je cherchais ~ faire en 1950 rentrait dans ce p r o g r a m m e ; un cycle alghbrique X de c o d i m e n s i o n p + l, h o m o l o g u e a zhro sur V, brant donnh, j'inthgrais sur un "cycle singulier" Q de b o r d X les formes h a r m o n i q u e s de degr6 2n - 2p - l; cela

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donnait dans l'espace A dual de l'espace de ces formes une "fonction de ligne" f ( X ) , bien d6finie modulo p6riodes, donc une application canonique f du groupe des cycles X dans la jacobienne intermediaire relative aux formes de degr6 2p + 1. C'est pourquoi, aux noms varies qu'on a proposes pour cette application, je substituerais volontiers le nom d ' " application de Volterra" s'il y fallait absolument un nom propre. Par la suite D. Liebermann, gr~.ce aux moyens techniques qu'ont fourni les travaux plus r6cents (Lelong, Borel-Haefliger, Hironaka, Federer), a pu mettre au point ce qui chez moi 6tait rest6 ~ l'6tat d'esquisse provisoire et r6ussir l',i off j'avais 6chou6, je veux dire dans la d6monstration de l'analyticit6 (au sens complexe) de l'application en question; en voici l'essentiel. Soit X~ un cycle alg6brique positif qui varie analytiquement en fonction de param6tres z; V 6tant suppos6e projective, on se ram6ne, au moyen des coordonn6es de C h o w de Xz, au cas off z varie dans une vari6t6 alg6brique, puis on voit ais6ment qu'il suffit de traiter le cas off celle-ci est une courbe Z, et ou X~ est une vari6te irr6ductible. Soit p + 1 la codimension de X~, et soit Y la vari6t6 de codimension p d6crite par X~ quand z varie sur Z. D'apr6s Hironaka, on peut consid6rer Y c o m m e image ~p(Y') d'une vari6t6 Y' sans point singulier, et Xz c o m m e image (p(X'~) d'un diviseur X'z sur Y'. Soient a un point fixe et z un point variable sur Z;joignons-les par un chemin 2. Q u a n d t varie sur 2, X~ d6crit un cycle singulier Q' de dimension topologique 2n - 2p - 1 sur Y', de bord X'z - X'a, dont l'image Q = q~(Q') dans V a pour bord X z - X a. L'image canonique de X, dans la jacobienne interm6diaire est donn6e par l'int6grale sur Q des formes diff6rentielles co de degr6 2n - 2p - 1 sur V qui satisfont dco= O, d(Cco) = 0. Mais il revient alors au m6me d'integrer sur Q' les formes co' = ~o-1(~o), et celles-ci satisfont 6videmment h d~o' = 0, d(Cco') = 0. II s'ensuit q u ' o n est ramene au cas des diviseurs sur la vari6t6 Y', pour lesquels le r6sultat est connu en vertu de la th6orie de la vari6t6 de Picard et d'un th6or6me de Kodaira (loc. cir. [1952e], p. 893), et ft. des propri6t6s "fonctorielles" 6videntes des jacobiennes interm6diaires. Telle est dans ses grandes lignes la demonstration de Liebermann (Am. J. of Math. 90 (1968), pp. 1165-1199; v. surtout pp. 1190-1193). Elle fait voir en particulier qu'avec les notations ci-dessus on a co' = 0 chaque lois que la d6composition de ~o suivant les types de Hodge ne fait appara~tre que des types (a, b) autres que (n-p1, n - p) et que ( n - p, n - p 1). Avec les notations introduites pr6c6demment, cela revient ~t dire que l'image canonique de X~ - X , , donc de tout cycle alg6briquement 6quivalent ~ z6ro, est dans l'image de l'espace Ap dans le tore A/A. Mais toutes les structures complexes d6finies sur A par les op6rateurs J~ coincident sur A r au signe pr6s; ce signe est d6termin6 par le fait que l'application canonique doit 6tre holomorphe. Du point de vue de cette application, le choix de J~ est d o n c sans importance. II n'est pas difficile de voir qu'il y a dans le tore complexe A/A un tore complexe maximal Bp contenu dans l'image de Ap, ind6pendant du choix de J~, et que celui-ci contient l'image de tout cycle alg6briquement 6quivalent ~. z6ro sur V. Ce tore est une vari6t6 ab61ienne, puisqu'il en est ainsi de A/A pour J~ = - C ; c o m m e d'autre part, d'apr6s Griffiths, il y a un choix de J~ pour lequel les modules de A/A sont fonctions holomorphes de ceux de V, il est pr6sumer qu'il en est de m6me de ceux de Bp.

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Je reviens maintenant /t ma correspondance avec Hodge; celui-ci disait avoir, lui aussi, cherch6 sans succ6s une interprdtation des vari6t6s ab61iennes d6termin6es par les matrices des p6riodes des formes harmoniques. En revanche, c'est lui qui me fit connaitre l'exemple si frappant des hypersurfaces cubiques darts p4: "I am sure", m'dcrivait-il en novembre 1950, "that the period matrix of the Picard integrals on this surface" [la surface d6finie par les droites sur la vari6t6] "is equivalent to the "3-dimensional" Riemannian matrix of the variety." Je ne sais o/l j'en 6tais de mes r6flexions ",i ce moment, mais son exemple, qui sans doute lui venait de T o d d (v. J. A. Todd, Proc. Ed. Math. Soc. (II) 4 (1935), pp. 175 184; cf. G. Fano, Atti R. Acc. Torino 39 (1904), pp. 597-615 et 778-792), m'encouragea grandement; c'est fort peu apr6s cette lettre qu'apparaissent dans les miennes les "jacobiennes interm6diaires" et l'application canonique. J'allai m6me jusqu'fi penser que celle-ci pouvait 6tre toujours surjectivc, et fis part fi Hodge de cette "conjecture". I1 ne tarda pas/t la d6molir; voici ce qu'il m'6crivait en f6vrier 1951 : "... I now feel that I was a little optimistic in suggesting that this" [l'hypersurface cubique] " m i g h t be a guide to the general s i t u a t i o n . . . Let X = E - F ' . . . F and F' will belong to the same algebraic system on V . . . . Hence there exists a one paramctcr family of analytic cycles including F and F' which lie in an analytic cycle Z ofp + 1 complex dimensions. We can then take Q c Z. But every harmonic form of type (r, s), r + s = 2p + 1, on V vanishes on Z, and hence on Q, unless r = p, s=p+ 1 or r = p + 1, s = p . Hence the image of X lies in {U~~ = 0 , ( r , s ) r (p, p + 1) or (p + 1, p)} where U~~ is the coordinate in Jr(V) defined by the form C'cst seulement en relisant cette correspondance en 1978 que je me suis aper~u que le paragraphe ci-dessus contenait, clairement exprimde, l'idde de Liebermann, qui bien entendu n'en a rien su. Sans doute il n'aurait gu6re 6t6 possible, en 1951, de l'exploiter fi fond; il n'en est pas moins remarquable que Hodge me l'ait communiqu6e et qu'il n'en ait rien tir6, ni moi non plus; je me contentai/l cet 6gard de la remarque qui forme la conclusion de [1952e], p. 893. Le reste de cette correspondance concernait d'une part mes tentatives pour 6tablir les propridtds "fonctorielles" des jacobiennes intermddiaires et de l'application canonique; pour la plus grande partie cela a 6t6 fait par giebermann et ne pr6sente pas de difficult6 s6rieuse, v u l e s moyens dont on dispose actuellement. D'autre part je m'effor~ai de trouver des conditions purement algdbriques, n6cessaires et suffisantes pour qu'un cycle algdbriquement 6quivalent/t z6ro ait pour image 0 dans la jacobienne interm6diaire correspondant a sa dimension. Ce probl6me cst toujours ouvert, que je sache, et jc doute que les conjectures que je formulai sur ce sujet en 1951 aient eu quelque valeur. 11 me reste fi signaler une inadvertance fi la fin de la page 82 de mes VariOtOs kiihlbriennes ([1958a]), ol) je reproduis d'apr6s [1952e] la d6finition des jacobiennes; c'est bien - C, comme dans [1952e], et non pas C, qui ddfinit la structure complexe de celles-ci. [1953] Th6orie des points proches sur les vari6t6s diff6rentiables Je me suis trop avanc6 au ddbut de cette conf6rence (p. 112) en attribuant/t Nicolas

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Bourbaki les iddes qui y sont exposdes. En rdalit6, j'avais, sur ce sujet, soumis ',i Bourbaki un avant-projet assez d6taill6, avec un chapitre d'application fi la th6orie locale des groupes de transformations et des groupes de Lie. Bien que Bourbaki efit re~u ensuite d'un autre de ses collaborateurs un manuscrit qui traitait de ces questions tout au long par la marne mdthode, il ne crut pas devoir en adopter le principe. I1 n'cst rien r6sult6 de 1~ que I'expos6 ci-dessus, dont l'occasion fut un colloque organis6 fi Strasbourg par Ehresmann. Depuis les premiers tfitonnements du calcul infinit6simal et de la gdomdtrie diff6rentielle, les g6om6tres ont pens6 "points infiniment voisins" (ou "points proches"; le terme latin est "proximus"), quitte/~ r6diger tout autrement lorsque cette notion rut tombde en discr6dit ; il y a 1~ un chapitre d'histoire qui mdriterait d'etre 6crit. Une tangente a toujours 6t6 pens6e c o m m e droite qui coupe la courbe en deux points infiniment voisins; si Leibniz a pu 6prouver quelque difficult6 adapter cette idde intuitive au voisinage du second ordre (cf. N. Bourbaki, ElOments d'histoire des mathOmatiques, p. 212, note (**)), ce n'6tait 1/t qu'une 16gdre erreur de parcours. De m6me, d6s les ddbuts de la th6orie des groupes de transformations, il a 6t6 question de "transformations infinitdsimales", et il n'est pas de gdom6tre qui n'ait senti que le crochet, non seulement s'obtient 'a partir du c o m m u t a t e u r par passage ",ila limite (cf. [1939b]), mais qu'en un certain sens "c'est" un commutateur. La marne idde est apparue en g6om6trie algdbrique dans la th6orie des points singuliers des courbes planes; lfi elle a 6t6 mise sur une base solide par l'emploi de transformations quadratiques, mais elle pr6existait/~ celui-ci. On peut en voir d'autres exe.mples dans T h o m s o n et Tait (loc. cit. [1948a]*), et aussi dans les potentiels dits de double couche de la physique mathdmatique classique, puisque ccux-ci provienncnt d'aimants infinitdsimaux transversaux ~ une surface le long de laquelle ils sont r6partis avec une densit6 superficielle continue; il est bien connu qu'il y a lfi une des origines de la notion moderne de distribution. II y a eu fi notre 6poque diverses tentatives pour donner de la prdcision +,ices iddes vagues. Je ne veux pas parler ici de la " n o n - s t a n d a r d analysis", qui ne parvient ~ son but que par une modification substantielle des axiomes de l'analyse classique et dont l'utilit6, que je sache, n'a pas 6t6 d6montr6e d'une mani6re convaincante. Mais, de m6me qu'on pcut voir dans l'6tude des alg6bres de Lie une thdorie du voisinage du premier ordre de l'616ment neutre dans les groupes de Lie (avec cette r6serve cependant que d6j/L le crochet se trouve dans le voisinage du second ordre), les groupes formels et les hyperalg6bres sont fi considdrer c o m m e des techniques pour 6tudier, et cela en route caractdristique, les voisinages infinitdsimaux de tout ordre de l'616ment neutre. La thdorie des sch6mas elle-m6me, d6s qu'elle traite de sch6mas non "r6duits" (c'est-/t-dire d6finis localement par des iddaux qui ne sont ni premiers ni intersections d'iddaux premiers), est/t envisager sous le m6me aspect. Vers 1952, Bourbaki avait commenc6 /t solliciter de ses collaborateurs des rapports sur la thdorie 61ementaire des groupes de transformations, des groupes de Lie et des espaces homog6nes. Je me proposai, sinon de renouveler ce sujet, du moins de le traiter dans un langage nouveau o6 la notion intuitive d'61dments infiniment voisins prendrait un sens mathdmatique pr6cis. J'esp6rais d'ailleurs pouvoir ensuite exploiter mort id6e dans la gdomdtrie alg6brique de caract6ristique

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p, off l'ins6parabilit6 cr6ait des difficult6s que les moyens classiques n'arrivaient pas 5- surmonter (cf. [1955a,b]*) et semblait devoir se prSter 5- un traitement analogue. Ce sont ces m6mes questions relatives fi la caract6ristique p qui motiv6rent dans le m~mc temps les recherches de Dieudonn6 sur les hyperalg6bres; il y avait 15- plus qu'une simple coincidence, car Dieudonn6 et moi, aprSs avoir 6t6 fort proches l'un de l'autre 5- $5o Paulo en 1946 et 1947, l'6tions de nouveau 5Chicago depuis qu'il avait accept6 une chaire ~ Northwestern University. Je ne sais s'il y a u r a r int6rSt 5- rechercher les liens qui doivent exister entre les diverses notions dont je viens de parler. Peut-6tre celle de "point proche" qui fair le sujet de [1953] avait-elle I'avantage d'6tre plus directement bas6e sur l'intuition g6om6trique; ce n'6tait sans doute pas suffisant pour justifier l'introduction d'une notation nouvelle quelque peu biscornue. Ainsi dut en juger Bourbaki. [1954a] Remarques sur un m6moire d'Hermite

J'avais connu Ostrowski ~t G6ttingen en 1927; alors d6jfi son 6rudition 6tait 16gendaire, et j'avais eu l'occasion d'y avoir recours. Sollicit6 d'envoyer une contribution fi un volume qui allait lui ~tre d~di6, je crus ne pouvoir mieux faire que d'offrir une note qui n'6tait qu'un simple excrcice, mais tir6 d'un m6moire d'Hermite c618bre autrefois et 5- peu prSs inconnu de nos jours. Depuis que j'avais r6dig6 [1948a] au Br6sil, j'6tais rest6 m6content de n'avoir pu d6finir la jacobienne d'une courbe que sur une extension non pr6cis6e du corps de base (v. [1948b], n ~ 37; cf. aussi [1955a,b]). En 1949, Chow avait annonc6 qu'il savait rem6dier 5- cet inconv6nient par l'emploi des "coordonn6es de Chow"; Matsusaka avait m~me ensuite appliqu6 cette m6thode avec succ~s 5_la vari6t6 de Picard; mais il n'avait pas, 5-ce qu'il me semblait alors, dissip6 toutes les obscurit6s du sujet (cf. [1954d]*). D'ailleurs, s'il 6tait plausible que la jacobienne J d'une courbe C ffit d6finie sur le corps de base, maints exemples montraient qu'il ne peut en ~tre de mSme, en g6n6ral, de l'application canonique de C dans J. Ce n'6tait pas choquant, puisque cette application n'est canonique qu'5- une translation pr6s. Mais il s'imposait de clarifier mes id6es par l'examen des courbes de genre I 5- la lumiSre de l'exp6rience que j'en avais acquise autrefois; sur ce sujet F. Ch'~telet m'avait pr6c6d6, mais son travail (Ann. Univ. Lyon (ILIA) 89 (1946), pp. 40-49) avait 6chapp6 5- mon attention. Je savais bien que, du point de vue arithm6tique (c'est-5--dire sur le corps des rationnels), ces courbes se classifient suivant leur degr6, la r6duction de l'une d'elles 5- un degr6 plus bas n'6tant en g6n6ral pas possible ou demandant du moins la solution d'un difficile problSme diophantien. AprSs les courbes qui ont un point rationnel, et se ramSnent donc 5. la forme dite de Weierstrass, viennent les courbes y2 = f ( x ) 0~1f est de degr6 4, puis les cubiques planes, puis les intersections de quadriques dans p3 (ces trois types 6tant ceux qui apparaissent chez Fermat), etc. A m e s debuts j'avais reconnu le lien entre ces types de courbes et la methode de r6solution des 6quations de genre 1 par n-section, la bisection conduisant en particulier aux courbes y2 = f(x), OU encore, sous forme homog6ne, y2 = f(X, Z) (cf. [1929]). Je me trouvais ainsi renvoy6 ~t Hermite. J'y trouvai en effet ce que j'y cherchais, c'est-5--dire les formules qui permettent, au moyen des invariants et

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covariants de la forme biquadratique binaire f ( X , Z), d'6crire la jacobienne de y2 = f(x, 1) et les relations entre les deux courbes. On obtient ainsi un exemple typique d'espace homog6ne principal, sur lequel j'allais pouvoir modeler la th6orie g6n6rale de ces espaces. J'examinai aussi du m~me point de vue le cas des cubiques planes, qui n'offre pas plus de difficult6 ',i condition d'avoir recours aux Higher Plane Curves de Salmon (Chap. V, n ~ 217-232); avec les notations de Salmon pour les invariants et covariants de la forme cubique ternaire f ( X , Y, Z), la jacobienne de f = 0 est j2 ___403 + 108S~H 4 _ 2 7 T H 6. Par chance, depuis l'ann6e off je pr6parais le concours de l'Ecole Normale, je poss6dais, en traduction frangaise, ce livre, celui qui le pr6c6de (Conic Sections) et celui qui le suit (Analytic Geometry of Three Dimensions), trois ouvrages qui encore pr6sent sont une mine d'informations utiles. J'avais d6j/l tir6 de ce chapitre, au trimestre d'6t6 1948, un cours d'introduction a la gbom6trie algebrique dont j'ai parfois regrett6 qu'il n'ait pas 6t6 r6dig6. L'essentiel en 6tait la th6orie des groupes alg6briques de dimension 1, pr6sent6s comme cubiques planes (non singuli6res,/t point double ou bien/l rebroussement suivant qu'il s'agit du genre 1, de G,, ou de Q), au moyen de la "r6siduation" de Sylvester (loc. cit. Chap. V, n ~ 158-161) qui y tient lieu du th6or6me de Riemann-Roch.

[1954b,c] Mathematical Teaching in Universities; The mathematical curriculum L'article [1954b] est/t peu de chose pr6s identique au r6sum6 d'une conf6rence faite en 1931 ~ la r6union annuelle de la Indian Mathematical Society (cf. photographie, page 131 du tome I). J'y ai seulement substitue ~ l'Inde la Pold6vie; ce pays, comme on salt, est 6troitement associ6 ~ la 16gende de Nicolas Bourbaki. Le r6union de 1931 eut lieu /~ Trivandrum dans l'extr~me Sud de l'Inde; y assistaient en majorit6 des brahmanes du Sud, dont certains fort orthodoxes. J'6tais seul Europ6en pr6sent; cela posa un petit probl6me fi l'occasion des repas, qui 6taient pris en commun. La jeune g6n6ration le r6solut avec bonne humeur en faisant discr6tement s6cession et m'accueillant en sa compagnie. Quant ~ [1954c] qui a 6t6 ins6r6 ici pour ne pas le s6parer de [1954b], il dut 6tre r6dig6 pendant les premi6res ann6es de mon s6jour/t Chicago, lorsque Marshall Stone y r6organisait l'enseignement. C'6tait un projet pour une brochure /t distribuer aux 6tudiants; il ne fut pas adopt6 et est rest6 in6dit jusqu'/~ pr6sent.

[1954d] Sur les crit~res d'~quivalence en g~om~trie alg~brique Depuis 1949, je regardais la construction d'une th6orie alg6brique de la vari6t6 de Picard comme la tS.che la plus urgente en g6om6trie alg6brique abstraite. Les travaux des Italiens et ceux de Lefschetz m'avaient donn6/l penser que la clef s'en trouverait dans une 6tude attentive des relations d'6quivalence (lin6aire, alg6brique, num6rique) entre diviseurs (cf. [-1950a], p. 440, tome I). En ce domaine, je ne voulais me tier qu'/t ce que je saurais d6montrer par moi-m6me; parmi mes confr6res g6om~tres, seul Zariski me paraissait m~'rit~r H n ~ eonfiance totale, mais ses

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travaux 6taient orient6s dans une direction trop diff6rente de la mienne pour que j'eusse l'occasion d'en faire usage. De leur c6t6, Chow 5, Johns Hopkins, Ndron et Samuel en France, Matsusaka au Japon s'int6ressaient activement au m~me sujet. De temps 5- autre nous nous tenions mutuellement au courant de nos progr6s. I1 n'dtait pas question de priorit6 entre nous; il me semble que cette maladie, qui a parfois 6t6 le fldau de la vie scientifique, est devenue plus b6nigne de nos jours, parmi les math6maticiens tout au moins; en tout cas nous en 6tions heureusement indemnes. Comme il 6tait naturel, Chow avait recours avant tout 5- la mdthode des "coordonndes de Chow" qu'il avait introduite en 1937, en collaboration avec v. d. Waerden. C'est aussi par cette m6thode que Matsusaka rdussit fi construire effectivement la vari6t6 de Picard (Jap. J. of Math. 21 (1951), pp. 217-235 et 22 (1952), pp. 51-62). Pour moi, afin de re'assurer d'une base solide, je crus devoir reprendre avant tout les criteres d'equivalence, dus principalement 5- Severi, qui sont expos6s dans les classiques Algebraic Surfaces de Zariski, et les munir de d6monstrations inattaquables, valables en route caractdristique. C'est cette laborieuse mise au point qui fait l'objet de [1954d] ; pour ma commodit6 et celle de rues amis j'avais commenc6 par en formuler les rdsultats dans une note ([1952f]) qu'il n'a pas paru utile de reproduire dans ce volume. Toute raise au point court le risque d'etre ennuyeuse, et celle-ci n'y 6chappe pas. Une bonne pattie de ses r6sultats aurait 6t6 la consdquence de la th6orie des vari6t6s de Picard et d'Albanese et de leurs propri6t6s "fonctorielles" si l'on avait su 6tablir celle-ci directement. Mais je n'en voyais pas encore le moyen (cf. [1955a,b]*). [1954e] Footnote to a recent paper

Depuis [1949b], l'un de mes passe-temps favoris 6tait de calculer des exemples de " m e s " conjectures. Non que j'eusse longtemps conserv6 des doutes ~ leur sujet; j'aurais presque 6t6 tent6 de dire, comme Euler (Opera (1), vol. If, p. 384) et comme Severi (Hamb. Abh. 9 (1933), p. 357) que " p o u r n'atre pas ddmontrde, la chose n'en 6tait pas moins certaine." Mais il y avait quelque plaisir 5- obtenir chaque fois le r6sultat attendu et 5- tirer de calculs arithm6tiques la valeur de nombres de Betti dont ensuite les topologues ne manquaient pas de me confirmer l'exactitude. Je n'ai jamais cru utile de publier ces exercices; si [1954e] fait exception, c'est que l'auteur de l'article auquel je r6pondais avait cru trouver un contreexemple 5, rues conjectures, qu'il avait mal comprises. De plus, mon calcul faisait apparaitre un lien entre l'intersection, suppos6e lisse, de deux quadriques darts P" pour n impair, et une certaine courbe hyperelliptique, et ce point pouvait m6riter d'etre examin6 de plus pr6s; en effet L. Gauthier en donna peu apr6s l'explication (Rend. Sere. Mat. Torino, 14 (1954-55), pp. 325-328); cf. aussi U. V. Desale et S. Ramanan, Inv. math. 38 (1976), pp. 161 185. [1954f] Number of points of varieties in finite fields

Mes conjectures de 1949 impliquaient que, si Vest une vari6t6 lisse de dimension n, projective ou tout au moins complete, sur un corps/l q 616ments, le nombre de

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ses points rationnels est qn,/t une erreur prSs qui est de l'ordre de q" 1/2 (et mSme de q"- 1 si Vest "rdguli6re'). Serge Lang 6tait/t Chicago, arriv6 de Princeton off il avait et6 l'616ve d'Artin; il desirait s'initier /~ la g6ometrie algebrique. Je lui proposai d'examiner ensemble ce qu'on pouvait dire du terme d'erreur, dans l'6valuation ci-dessus, "a la lumiSre des r6sultats ddj~, acquis. En effet nous ffimes bient6t en 6tat de donner dans [1954f] une estimation grossi6re, mais d6jb, utile, valable mSme pour des varidtds non lisses et non compl6tes.

[1954g] On the projective embedding of abelian varieties Dans son c616bre m6moire du prix Bordin, Lefschetz avait obtenu au moyen des fonctions thSta le plongement projectif des vari6t6s ab61iennes. Pour le cas des jacobiennes, j'avais, sans le savoir, retrouv6 son r6sultat et sa mdthode dans ma th6se ([1928], Chap. II); il me le fit gentiment observer quelques ann6es plus tard, tout en m'encourageant vivement fi continuer dans cette vole. J'ai fini par suivre son conseil. Pour un volume qui devait lui 8tre dddi6, il semblait tout indiqu6 de reprendre sa mdthode et de la transposer fi la g6om6trie algdbrique abstraite ; il n'y avait plus grande difficult6 fi cela.

[1954h] Abstract versus classical algebraic geometry Au congres d'Amsterdam, Fun des nouveaux laur~ats de la m6daille Fields, qu'on n'avait jamais vu muni d'une cravate, crut cet ornement n6cessaire pour sa pr6sentation ~ la reine de Hollande; il est fait allusion ~ce "happening" dans ma conf6rence (p. 550, 1.6 5 du bas). Celle-ci cut un caractSre quelque peu improvis6, du fait que je n'avais pas compt6 assister au congrSs; la d6fection, au dernier moment, de Fun des conf~renciers fit que je re~us des organisateurs un appel pressant auquel il efit 6t6 discourtois de me d6rober. En passant, je me laissai aller fi un coup d'oeil rdtrospectif (p. 555, 1.22 24) sur mes anciennes recherches au sujet du th6or6me de Castelnuovo; le peu que j'en dis serait fi rectifier d'apr6s [1940b]*. En la circonstance, la recherche d'une plus minutieuse exactitude m'aurait entrain6 trop loin. Quant ",i rues commentaires sur l'hypothSse de Riemann en dimension > 1, cf. [1941]*. Mon espoir qu'on la tirerait d'un th6orSme de positivit6 valable en gdom6trie alg6brique abstraite ne s'est pas trouv6 vdrifi6, malgr6 une ingdnieuse suggestion de Serre (Ann. of Math. 71 (1960), pp. 392 394); la d6monstration de Deligne est toute arithmdtique. De mort c6t6, j'ai observ6 depuis lors qu'en vertu de la thdorie de Hodge un th6or6me de positivit6 est valable en caract6ristique 0, dans la dimension topologique 2, pour une surface fibr6e par des courbes elliptiques et une correspondance compatible avec la fibration (cf. [1967a]*); mais je n'ai p u l e d6montrer par voie algebrique, et de toute faqon il est trop sp6cial pour qu'on puisse en tirer ne serait-ce qu'une conjecture plausible pour des cas plus gdndraux.

[1954 i] Poincar6 et l'arithm6tique A l'occasion du eongres d'Amsterdam, il se tint ~, la Haye une petite r~union en commemoration du centenaire de la naissance de Henri Poincar6, et on me

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demanda d'y parler de Poincar6 arithmdticien. Lit aussi je dus improviser, mais je savais de quoi je parlais, et j'eus tout loisir, rentr6 chez moi, de r6diger un texte moins inad6quat. Ii se trouva que, dans cette rdunion, on avait tout simplement omis la gdomdtrie alg6brique ! Je me gardai bien d'attirer l'attention sur cet oubli, dont je fis mention ensuite dans mon texte 6crit (p. 4, 1.3-8). Plus tard, l'un des organisateurs m'a dit son regret de n'y avoir pas pens6/t temps et fait appel/~ moi. C'6tait bien ce que j'avais craint; une telle tfiche efat exig6 une bien plus longue pr6paration.

[1955a,b] On algebraic groups of transformations; On algebraic groups and homogeneous spaces Grfice entre autre fi l'impulsion donn6e par Marshall Stone (cf. [1949c]* et [1954b,c]*), il s'6tait cr66 au d6partement de math6matique de Chicago une atmosph6re scientifique des plus stimulantes; coll6gues, visiteurs, 6tudiants y contribuaient, chacun /t sa mani6re. Malgr6 des d6buts peu prometteurs (cf. [1948c]* et [1949b]*), et malgr6 un cliinat et un environnement ddprimants, je n'avais pas tard6 fi m'y trouver 5~l'aise. Du reste, l'attention se concentrait sur un sujet plut6t que sur un autre suivant la pr6sence de tel ou tel visiteur. L'ann6e 1954 1955 vit assemblde it Chicago une constellation hors du commun; Borel, Hertzig, Lang, Matsusaka, Murre, Swinnerton-Dyer, Zeeman s'y trouv6rent r6unis; Dieudonn6 et Rosenlicht 6taient nos voisins g Northwestern. Ce fut l'ann6e des groupes alg6briques et des vari6t6s ab61iennes. Pour un auditoire de cette qualit6,je voulus d'abord, dans mon cours, compl6ter ce qui 6tait rest6 inachev6 dans [1948b] au sujet de la construction des groupes alg6briques, c'est-/t-dire principalement ce qui a trait aux groupes de transformations, aux espaces homog6nes et ~ la d6termination pr6cise du corps de d6finition. C'est ce qui fait l'objet de [1955a]. Le contenu de [1955b] en forme la suite naturelle; 1/i je pus m'appuyer, non seulement sur rues propres observations au sujet des courbes elliptiques (cf. [1954a]), mais surtout sur des rdsultats rdcents de Chow concernant les jacobiennes (Am. J. of Math. 76 (1954), pp. 453 476). J'obtins ainsi un ddbut de th6orie des espaces homog6nes principaux attachds ~i un groupe alg6brique, dont le lien avec la cohomologie galoisienne, anticip6 d6j/t en partie par F. Chittelet (loc. cir. [1954a]*), devait apparaitre clairement quelques anndes plus tard (cf. J. Tate, Son. Bourb. n ~ 156, D6c. 1957). La suite de mon cours de 1954 1955 fut consacrde aux varidtds ab61iennes, et plus pr6cis6ment fi la th6orie des varidt6s de Picard et d'Albanese, fondde sur ce que je nommai le "th6or6me de la balangoire" ou "seesaw principle," directement emprunt6 it Severi, et le "th6or6me du cube". I1 n'y manquait plus gu6re que la bidualit6 (toute varidt6 abOienne est la duale de sa duale, c'est-it-dire de sa varid6 de Picard) et les pr6cisions suppldmentaires sur les isog6nies insdparables et leurs noyaux que j'attendais de ma notion de point proche (cf. [1953]) et qui en fait sont venues de la thdorie des schdmas. Heureusement je pus me dispenser de r6diger ces recherches; Serge Lang m'offrit fort opportun6ment de les incorporer dans l'ouvrage qu'il se proposait d'6crire sur ce sujet, ce it quoi je l'autorisai volontiers. Je lui sugg6rai seulement

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d'attendre au moins de pouvoir y ins6rer une d6monstration de la bidualit6, qui /~ mon avis ne pouvait tarder. I1 ne me crut pas, et son livre (Abelian Varieties, Interscience Publ., N.Y./London) parut en 1959. L'attente n'aurait pas 6t6 longue, car deux d6monstrations de la bidualit6 furent publi6es aussit6t apr6s (M. Nishi, Mere. Coll. ofSc., Kyoto (A) 32 (1959), pp. 333-350; P. Cartier, Ann. of Math. 71 (1960), 315-361; cf. P. Cartier, S~m. Bourb. n ~ 164, Mai 1958). On notera que j'etais rest6 fidele au principe 6nonce dans l'introduction de [1948b]: "L'6tude des jacobiennes et la th6orie g6n6rale des vari6t6s ab61iennes se prStent un mutuel secours, de sorte qu'il convient de les mener de front." Depuis lors, Mumford, dans ses Abelian Varieties (Oxford/Bombay 1970), a fait voir qu'fi pr6sent on peut construire toute la th6orie sans jamais faire mention des jacobiennes. I1 n'y a d'ailleurs pas lieu de s'en 6tonner. Toute la th6orie de la jacobienne est d~j',i contenue en puissance dans le th6or~me de Riemann-Roch pour les courbes; il 6tait donc /~ pr6voir qu'on pourrait se dispenser d'y avoir recours d6s qu'on disposerait de moyens techniques impliquant ce th6or6me comme cas particulier; c'est ce que montrerait, j'imagine, une analyse attentive du livre de Mumford.

[1955c,d] On a certain type of characters of the id~le-elass group of an algebraic number-field; On the theory of complex multiplication Depuis que j'6tais/t Chicago, j'avais nou6 avec les math6maticiens japonais, par correspondance ou personnellement, des relations de plus en plus cordiales, et je suivais les travaux de plusieurs d'entre eux avec beaucoup d'attention. Kodaira et Iwasawa, en route pour le congr6s de Cambridge, 6taient venus me visiter pendant 1'6t6 de 1950. A ce mSme congr6s j'avais retrouv6 Iyanaga, perdu de vue depuis 1933, et nou6 avec lui les liens d'une amiti6 qui m'a 6t6 pr6cieuse en mainte circonstance depuis lors. Pour Igusa, on a vu (cf. [1950a]*) que ses premi6res recherches s'6taient rencontr6es avec les miennes et que j'avais eu fi m'occuper de leur publication. J'avais souvent vu N a k a y a m a pendant les ann6es qu'il passa Urbana, et j'ai ddj/t dit (v. [1951b]*) comment en 1951 il me sauva, sinon la vie, du moins l'honneur. Enfin, apr6s un 6change de lettres qui avait dur6 plusieurs anndes, Matsusaka 6tait venu me rejoindre fi Chicago en 1954. En cons6quence, quand les math6maticiens japonais entreprirent pour la premi6re fois d'organiser chez eux une r6union internationale qu'ils d6di6rent ~ Takagi et/t la th6orie des nombres, je fus particuli6rement heureux d'Stre invit6 fi y participer. Ce fut la plus belle, la plus joyeuse et la plus f6conde rdunion mathdmatique laquelle il m'ait jamais 6t6 donn6 d'assister; seul le congr6s international de Zurich en 1932 m'a laiss6 un souvenir quelque peu comparable. Est-ce seulement un effet de l'fige? I1 me semble qu'fi force de se multiplier, ce genre de r6union, depuis une quinzaine d'ann6es, n'a pu manquer de s'affadir. Au pays andhra, dit-on, les vieillards se plaignent que les piments n'ont plus le gofit si fort qu'ils l'avaient autrefois... Je m'embarquai fi Seattle, pour Yokohama, sur le Hikawa-Maru, ",i pr6sent d6funt. C o m m e contribution au colloque, j'apportais quelques id6es que je croyais neuves sur l'extension aux varidtds abdliennes de la th6orie classique de la multiplication complexe. Comme chacun sait, Hecke avait eu l'audace,

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stupdfiante pour l'6poque, de s'attaquer fi ce probl6me d6s 1912; il en avait tir6 sa th6se, puis avait pouss6 son travail assez loin pour d6couvrir des ph6nom6nes qui lui avaient paru inexplicables, apr6s quoi il avait abandonnd ce terrain de recherche dont assur6ment l'exploration 6tait prdmatur6e. En 1955,/t la lumi6re des progr6s effectuds en g6om6trie algdbrique, on pouvait espdrer que la question 6tait mfire. Elle l'6tait en effet;/t peine arriv6 5. Tokyo, j'appris que deux jeunes japonais venaient d'accomplir sur ce marne sujet des progr6s d6cisifs. Mort plaisir 5. cette nouvelle ne fut un peu tempdr6 que par ma crainte de n'avoir plus rien 5. dire au colloque. Mais il apparut bient6t, d'abord que Shimura et Taniyama avaient travaill6 inddpendamment de moi et m6me ind6pendamment l'un de l'autre, et surtout que nos rdsultats 5. tous trois, tout en ayant de larges parties communes, se completaient mutuellement. Shimura avait rendu possible la reduction modulo p au moyen de sa th6orie des intersections dans les varidt6s d6finies sur un anneau local (Am. J. of Math. 77 (1955), pp. 134 176); il s'en 6tait servi pour l'6tude des vari6t6s abdliennes/t multiplication complexe, bien qu'initialement, 5. ce qu'il me dit, il efit plut6t eu en rue d'autres applications. Taniyama, de son c6t6, avait concentr6 son attention sur les fonctions z&a des vari6tds en question et principalement des jacobiennes, et avait g6n6ralis6 5. celles-ci une bonne partie des rdsultats de Deuring sur le cas elliptique. Quant 5. ma contribution, elle tenait surtout 5. l'emploi de la notion de "vari6t6 polarisde"; j'avais choisi ce terme, par analogie avec les "varietds orientdes" des topologues, pour designer une structure suppldmentaire qu'on peut mettre sur une varidt6 compl6te et normale quand elle admet un plongement projectif. Faute de cette structure, la notion de modules perd son sens.

I1 fut convenu entre nous trois que je ferais au colloque un expos6 g6n6ral ([1955d]) esquissant ~i grands traits l'ensemble des r6sultats obtenus, expos6 qui servirait en m6me temps d'introduction aux communications de Shimura et de Taniyama; il fut entendu aussi que par la suite ceux-ci r6digeraient le tout avec des d6monstrations d6taill6es. Leur livre a paru en 1961 sous le titre Complex multiplication of abelian varieties and its application to number theory (Math. Soc. of Japan, Tokyo); mais Taniyama 6tait mort tragiquement en 1958, et Shimura avait dfi l'achever seul. D'autre part, tout en restant loin des r6sultats de Taniyama sur les fonctions z6ta des varidtds "de type C M " (comme on dit 5. prdsent), j'avais aper~u le r61e que devaient jouer dans cette thdorie certains caract6res de Hecke privildgids, baptises (faute de mieux) "caract6res de type (Ao)", ainsi que les caract6res 5. valeurs ~3-adiques qu'ils permettent de d6finir (cf. [1955c], p. 6). Je trouvai 1/t une premi6re explication du ph6nom6ne qui avait le plus 6tonnd Hecke; il consiste en ce que, d6s la dimension 2, les modules et les points de division des vari6t6s de type CM d6finissent en g6n6ral des extensions abdliennes, non sur le corps de la multiplication complexe, mais sur un autre qui lui est associ6. Ce sujet a 6t6 repris et plus amplement ddvelopp6 par Taniyama (J. Math. Soc. Jap. 9 (1957), pp. 330-366); cf. aussi [1959b].

[1955e] Science Fran~;aise? A chacun de mes s6jours 5. Paris, annuels depuis 1948 (saufen 1955 quand je visitai le Japon), mes amis et moi ne manquions pas de faire d'ambres constatations sur le

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triste 6tat o~ nous semblait tomb6e la recherche scientifique en France; n'y faisaient exception que les math6matiques, qui 6taient florissantes, et quelques groupes de travail qui se distinguaient ~/t et l/t. Ces r6flexions n'6taient pas nouvelles; fi mon retour d'Am6rique en 1937 j'avais d6j& compos6 un article rest6 in6dit; c'est le texte [1938c] (v. tome I, pp. 232 235), et Delsarte, & la veille de la guerre, avait publi6 dans la Revue Rose une s6rie d'articles sur le m6me sujet (loc. cir. [1939a]*; cf. [1971b]). Vers 1954, il fur question de savoir qui parmi nous attacherait le grelot. C o m m e Delsarte me le fit observer, l'op6ration, pour certains de nos amis, aurait pu pr6senter quelque risque; ce risque 6tait presque nul pour lui, r6solu qu'il 6tait ~ n'avoir jamais d'ambitions "sorbonnardes" ni acad6miques, et tout ~ fait nul pour moi dans la mesure ou ma situation, & cheval sur deux continents, m'assurait une enti6re ind6pendance: "je suis oiseau, voyez mes ailes; je suis souris, vivent les rats." Gr'~ce ~ Henri Cartan, en effet, j'avais 6t6 r6int6gr6 en 1945 dans le syst6me universitaire frangais, mais en qualit6 de "d6tach6"."Voyez-vous, Weil," me disait /t S~o Paulo un coll6gue franqais un jour que nous sortions de chez notre attach6 culturel, "il y a dans le monde les attach6s, et il y a les d6tach6s." A l'occasion de la publication des 6crits de ma s~eur, j'6tais entr6 en relations avec les gens de la N.R.F., Albert Camus d'abord, puis surtout Brice Parain, qui efit 6t6 le Socrate de notre 6poque si notre 6poque avait m6rit6 un Socrate. Celui-ci fit publier mon article apr6s m'avoir donn6 d'utiles conseils en vue de sa r6daction d6finitive. A pr6sent je peux bien r6v61er un secret de Polichinelle; les professeurs A et B (pages 8-10) s'appelaient Siegel et Chevalley. L'homme d'6tat dont il est question au d6but 6tait Mend6s-France. I1 r6pondit & l'envoi de mon article par une lettre attrist6e: " . . . Mais, h61as, que de choses ~ faire, que de probl6mes/l r6soudre, qui doivent tous 6tre abord6s en m~me temps en raison de leur importance, et en raison de tousles ajournements dont ils ont souffert jusqu'ici." Un autre homme politique r6pondit sur un ton bien diff6rent, mais pour dire au fond la m~me chose: "Votre article est tout &fair juste, et ses conclusions exactement celles auxquelles j'ai abouti et que dans ce r6gime affreusement d6cadent je ne sais comment faire triompher." Le premier venait d'etre chef du gouvernement. Le second fut premier ministre quelques ann6es plus tard. Bien entendu il n'en fut ni plus ni moins, jusqu'& l'explosion de 1968, dont l'universit6 frangaise n'a pas fini de mesurer les cons6quences. [1956] The field of definition of a variety Ce travail, consacr6 & la "descente du corps de base", prend la suite de [1955a,b], qu'il permet dans une large mesure d'am6liorer et de simplifier. Dans [1961a] (Chap. I, w pp. 4 10), la question est plus amplement 61ucid6e au moyen du "foncteur" RK/k, qui associe & une vari6t6 V de dimension d, d6finie sur une extension s6parable K d'un corps k, une varibt6 W = RKjk(V ) de dimension d. [K:k], d6finie sur k, telle que Wk = V~. Du m6me coup apparait la possibilit6

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de " t o r d r e " une varidta (ou un groupe, ou une algabre) au moyen d'un groupe d'automorphismes; cette idde fut utilisae par exemple par D. Hertzig dans sa thase (Chicago 1957; cf. Proc. Am. Math. Soc. 12 (1961), pp. 657-660; cf. aussi [1960b], Part I, pp. 589-601).

[1957a] Zum Beweis des Torellischen Satzes Andreotti, lors d'une visite 5. Chicago, attira mon attention sur le thdorame de Torelli, dont il avait donna une damonstration dans le style italien, quelques annaes auparavant. Sur ce sujet, on consultera utilement l'excellent petit livre de Mumford, Curves and their Jacobians (U. of Mich. Press 1975), pp. 68 94 et 103-104.

[1957b] Hermann Weyl (1885-1955) Quand de Rham me demanda au nom de l'Enseignement Math~matique une notice nacrologique sur Hermann Weyl, je ne crus pas pouvoir m'en charger seul;j'atais rest6 trop ignorant de tout ce qui concerne les groupes semisimples et leurs reprasentations. Heureusement Chevalley accepta de collaborer avec moi, se chargeant notamment de toute la portion (pp. 178-184) consacrae/t ce sujet.

[1957c] (1) R~duction des formes quadratiques; (2) Groupes des formes quadratiques indafinies et des formes bilinaaires altern~es En 1957 58 je passai toute une annae fi Paris; Satake et Shimura y arrivarent en automne; Cartan en profita pour faire porter son saminaire sur les fonctions automorphes, et principalement celles qui appartiennent aux sous-groupes arithmatiques des groupes simples. Evidemment c'6tait avant tout 1' oeuvre de Siegel qu'il s'agissait d'etudier; je me chargeai de deux exposas praliminaires sur la thaorie de la raduction. Commenter Siegel m'a toujours paru l'une des tfiches qu'un mathamaticien de notre temps pouvait le plus utilement entreprendre. Tel avait dfi atre aussi l'avis de Hermann Weyl, qui y avait consacre plusieurs travaux (v. ses Ges. Abh., n ~ 120, vol. III, pp. 719 757; n ~ 126, vol. IV, pp. 46 74; n ~ 136, vol. IV, pp. 232-264). En ce qui me concerne, [1946b] avait 6ta mon premier essai dans ce genre; [1957c], puis [1958d], [1960b], [1961a], [1964b], [1965] ont suivi. A lire Siegel, j'avais 6t6 frapp6 entre autre par le fait que la compacit6 des domaines fondamentaux pour les groupes classiques est lice fi l'absence de zaros pour les formes qui les daterminent; c'est en partie pour me convaincre du degra de ganaralit6 de cette observation que j'avais entrepris l'examen datailla de la thaorie de la raduction telle que je la trouvais exposae dans le mamoire de Siegel de 1940 (Ges. Abh. n ~ 33, vol. II, pp. 138 168) et chez H. Weyl (loc. cir.). Traditionnellement, depuis Gauss, cette thaorie s'est proposa de dacrire l'espace off opare un groupe discret comme une mosaique de domaines juxtaposas, transformas les uns des autres par le groupe. A cette idee s'apparentait la methode de la

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triangulation des vari6t6s, que Poincar6 avait voulu mettre 5. la base de la topologie combinatoire, et qui avait fini par se heurter de front 5' la "Hauptvermutung". La topologie aurait pu y faire naufrage si la m6thode "cartographique" (cf. [1946a]*), dite parfois "m6thode de (~ech ", n'avait pris le relais au moment opportun ; cela n'a pas emp6ch6 ensuite la triangulation et la Hauptvermutung de reprendre leurs droits, mais dans le r61e plus modeste qui leur revenait. C'est la m6me r6action salutaire qui avait amend H. Weyl 5' ddfinir les surfaces de Riemann par voisinages empi6tant les uns sur les autres, plut6t que par les barbares coupures des analystes de jadis. "Quiconque coupera une surface de Riemann la tuera", avait coutume de dire Bourbaki. Selon Bourbaki, une notion bien comprise de structure quotient d6finie par des 616ments locaux devait rendre d6suets la plupart des pavages et d6coupages traditionnels. N6anmoins, 15' aussi le fanatisme n'est pas de mise; si la th6orie de la r6duction avait accord6 5' ces pavages une importance excessive, ils peuvent avoir du bon, par exemple lorsqu'il s'agit de compactification, et je les traitai un peu trop dddaigneusement dans [1957c] (v. pp. 366 et 371). Mais il m'importait surtout de mettre en valeur ce que je jugeais essentiel parmi les r6sultats de Siegel, c'est-5'-dire le crit6re de compacit6, le volume fini de l'espace quotient, et l'existence de "domaines de Siegel" (domaines poss6dant le propridt6 (M) de [1957c], p. 367). Quant aux d6monstrations qui en ont 6t6 donndes ensuite, cf. [1958d] et [1958d]*. [1958a] Introduction d l'~tude des vari~t~s k/ihl~riennes Cet ouvrage, qui ~tait en germe dans [1947b] et [1949d], a 6t~ compos~ 5' Chicago 5' partir de 1955; une bonne partie du contenu des chapitres I, II, IV avait 6t6 expos6e 5' diverses reprises dans mes cours de Chicago, puis 5' G6ttingen dans une s6rie de conf6rences qui avait 6t6 r6digde par l'un de mes auditeurs et multigraphi6e. Je puisai largement aussi dans les 6crits de G. de Rham et de Kodaira, en particulier leur sdminaire de Princeton (Harmonic Integrals, I.A.S. 1950) et les Varibtbs DifJ?rentiables de de Rham (Hermann 1955). A celui-ci j'empruntai notamment les op6rateurs G e t H, dont les propridt6s formelles, rappel6es dans mon chapitre IV, p. 66, impliquent les th6or6mes d'existence de Hodge et peuvent ainsi en tenir lieu. C'est d'autre part la contribution de Kodaira au s6minaire de Princeton qui est 5' la base de mon chapitre V; j'aurais pu l'alleger beaucoup si j'avais voulu me contenter de l'application qui en est faite aux tores complexes au chapitre VI. [1958b,c] On the moduli of Riemann surfaces; Final Report on contract AF18(603)-57 I1 y aurait presque un livre, ou du moins un bel article, fi 6crire sur rhistoire des modules et des vari6tds de modules; il faudrait pour cela remonter j usqu'aux d6buts de la th6orie des fonctions elliptiques. Ici, comme partout ailleurs dans ces commentaires, je bornerai mes remarques aux aspects auxquels j'ai touch6 ou bien qui m'ont touch6 personnellement. La th6orie des modules des courbes, inaugur6e par Riemann, a fait fi notre 6poque deux pas en avant d6cisifs, d'abord en 1935 du fait de Siegel (Ges. Abh.

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n ~ 20, w vol. I, pp. 394-405), puis par les remarquables travaux de Teichmtiller; sur ceux-ci, il est vrai, il continua quelque temps 5- planer des doutes qui ne furent lev6s d6finitivement que par Ahlfors en 1953 (J. d'An. Math. 3, pp. 1 58). D'ailleurs on finit par se rendre compte que la ddcouverte par Siegel (loc. cir.) des fonctions a u t o m o r p h e s appartenant au groupe symplectique atteignait en premier lieu les modules des vari6t6s ab61iennes (cf. p. ex. [1957c], pp. 376) et par ricochet seulement ceux des courbes, par l'intermddiaire de leurs jacobiennes et du th6or6me de Torelli. Grfice 5- Siegel, on disposait ainsi d'un premier exemple de thdorie des modules pour des vari6t6s de dimension > 1. On doit 5- K o d a i r a et Spencer d'avoir d6couvert (Ann. of Math. 67 (1958), pp. 328-466) que les progr6s de la cohomologie permettaient, non seulement d'aborder un nouvel aspect du marne probl6me, mais encore, du point de vue local tout au moins, de s'attaquer au cas g6n6ral des vari6t6s 5structure complexe. Toutes ces questions m'int6ressaient fort; il 6tait naturel pour moi de m ' y essayer aussi. En 1957-58, j'dtais/t Paris, en cong6 de l'Universit6 de Chicago; il en avait 6t6 convenu ainsi depuis longtemps, et l'Universit6, tout en s'y pr&ant de bonne grfice, avait insist6 pour financer ce cong6 au moyen d'un " c o n t r a t " avec l'aviation militaire amdricaine (U.S. Air Force), c o m m e il se faisait c o u r a m m e n t / t cette 6poque. N o n sans un peu de r6pugnance, je consentis /L cet arrangement, auquelje gagnai un transport gratuit par avion militaire pendant l'hiver de 1958. A cette occasion j'eus 5- me prdsenter, au milieu d'une foule de colonels et de majors, au sergent qui remettait les billets aux passagers, et, pri6 d'indiquer m o n grade, je r6pondis " O n l y a professor", sur quoi il me dit s6rieusement: " I think a professor is just as good as a colonel." Colonel ou professeur, j'6tais cens6 faire m o n rapport 5- la fin de m o n " c o n t r a t " et ne vis pas de raison de me d6rober 5- cette " o b l i g a t i o n " ; ce rut [1958c]. Auparavant, on m'avait demand6 une contribution au recueil d'articles qu'on voulait offrir 5- Artin en mars 1958 pour son soixanti6me anniversaire; cela me ddcida 5- r6diger mes observations, tout incompl6tes qu'elles fussent, sur les modules des courbes et ce que je n o m m a i "l'espace de Teichmiitler"; sur le m6me sujet, je fis en mai un expos6 au s6minaire Bourbaki (non reproduit ici, car il ferait double emploi avec [1958b]), et j'en r6sumai les r&ultats dans la premi6re partie de m o n rapport. Mais je m'aper~us bient6t qu'en plus d'un point je m'6tais rencontr6 avec Ahlfors et avec Bers; ceux-ci poursuivirent ces recherches, et ne tard6rent pas 5- me d& passer. Q u a n t aux questions pos6es 5- la fin de [1958b], ils ont fait voir que la rdponse est positive pour les deux premi6res, et n6gative pour la troisi6me. Dans la seconde partie de m o n rapport, il s'agit des vari6tds kfihldriennes dites K3, ainsi nomm6es en l'honneur de K u m m e r , Kfihler, K o d a i r a et de la belle m o n t a g n e K2 au Cachemire. De bonne heure j'avais 6t6 intrigu6 par l'exemple donn6 par Severi, au moyen d'une quartique dans p3, d'une surface poss6dant un groupe infini d ' a u t o m o r p h i s m e s lid au groupe des unit& d'un corps quadratique rdel (v. ses Op. Mat. n ~ XLVIII, vol. II, pp. 259-285); un m o m e n t j'avais m~me esp6r6 trouver 15, un moyen d'engendrer des extensions abdliennes de ce corps, donc une g6ndralisation de la multiplication complexe; c'6tait sans doute trop beau pour 6tre vrai. M o n attention fut de nouveau attir6e sur les K3 par une observation de

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Milnor, qui, lors d'une visite fi Chicago, demanda pourquoi les surfaces de Kummer, apr6s d6singularisation, ont m6mes nombres de Betti que les quartiques non singuli6res dans p3. Ce genre de consid6rations amenait tout naturellement aux id6es d6crites dans [1958c]; mais encore fi pr6sent il reste beaucoup b~ faire, je crois, avant que ce sujet ne soit compl6tement 6lucid6.

[1958d] Discontinuous subgroups of classical groups En 1957, pendant mon absence de Chicago, j'avais accept6 l'offre d'une chaire l'Institute de Princeton "Adater de l'automne 1958; il rut convenu qu'avant de prendre cong6 d6finitivement de Chicago j'y passerais le trimestre d'6t6. Andrew Wallace, qui y avait ~t6 mon ~l~ve (cf. [1952c]) et qui 6tait alors professeur Toronto, vint m'y rejoindre:je lui proposal de r6diger mon cours. II h6sita d'abord; reals, quand il m'entendit parler de la "m6thode babylonienne" de r6duction (cf. [1957c], p. 365), ce terme lui plut tellement qu'il se d6cida aussit6t. Ce cours ne fut pas autre chose que le d6veloppement de ce qui 6tait esquiss6 dans [1957c]. Suivant toujours Siegel et Weyl, je donnai d'abord, d'apr6s Siegel, la d6finition unifi6e des groupes classiques au moyen des alg~bres semisimples fi involution (cf. [1960b] et [1960b]*); puis je proc6dai fi la constructfon, pour ces groupes, de "domaines de Siegel", c'est-fi-dire de domaines satisfaisant fi la condition (M) de [1957c]. Pour le cas g6n6ral des groupes semisimples, cf. Borel et Harishchandra (loc. cit. [1965]); pour une version "ad~lique" des m~mes r~sultats, v. Godement, Sdm. Bourbaki, n ~ 257, mai-juin 1963.

[1959a] Ad61es ct groupes alg6briques Depuis quelques ann~es, la notion de vari6t6 ad61ique, ou en tout cas celle de groupe ad61ique, 6tait "dans Fair"; un cas particulier en 6tait apparu par exemple dans un probl6me de Kuga (n ~ 16 d'une collection de probl6mes qui fut distribu6e aux participants du colloque de Tokyo-Nikko en 1955). Pour les groupes alg6briques lin6aires, la d6finition g6n6rale du groupe ad61ique fut donn6e en 1957 par Ono (loc. cir. [1959a]) qui l'appliqua utilement aux groupes commutatifs, g6n6ralisant ainsi les ~nciens r6sultats de Chevalley sur le groupe des id61es. Mais la port6e de cette m6thode m'apparut seulement a la lecture d'une communication de Tamagawa fi Mautner que celui-ci me fit voir, en 1958 je crois, lots d'un de ses passages fi Paris. En quelques pages, Tamagawa y d6finissait, pour le groupe orthogonal, la mesure et le nombre auxquels son nom est rest6 attach6 et montrait que le calcul du "nombre de Tamagawa" pour ce groupe 6tait fi lui seul l'6quivalent du principal r6sultat de Siegel dans la th6orie des formes quadratiques. De mon c6t6 j'6tais persuad6 depuis longtemps que la th6orie des nombres classique avait fait fausse route en s6parant trop nettement des autres les places l'infini (cf. [1951b], [1952b] et la pr6face de [-1967c]). En ce qui concerne les formes quadratiques, cela ressortait d6jfi des conclusions de Hasse et de celles de Siegel; que la notion d'ad61e y trouvfit sa place n'avait rien pour 6tonner; Tamagawa en donnait l'6clatante confirmation. Je crus opportun de reprendre la question fi ses d6buts, tout en demandant fi Tamagawa de me renseigner plus

548

Commentaire

amplement sur sa mdthode et ses r6sultats; [1959a] est bas6 sur la correspondance que j'eus avec lui et sur rues propres r6flexions sur le m6me sujet. Pour de plus amples d6veloppements, v. [1961a]. [1959b] Y. Taniyama J'appris la mort de Taniyama 'fi Princeton en novembre 1958, alors que tout semblait pr6vu pour qu'il v~nt y passer l'ann6e 1959-1960. La lettre reproduite ici est extraite d'un fascicule d'une revue japonaise que ses amis d6di6rent b, sa m6moire.

[1960a] De la m~taphysique aux math~matiques Apr6s la mort d'Enrique Freymann, la maison d'6ditions Hermann, qu'il avait si longtemps et si brillamment pilot6e fi travers vents et mar~es, rut reprise par Pierre Ber6s, personnalit6 bien diff6rente, mais, elle aussi, non d6pourvue de relief. Parmi bien d'autres entreprises, il lan~a la revue Sciences (aujourd'hui d6funte), destin6e en principe fi publier des expos6s de vulgarisation d'un niveau 61ev6. La math6matique ne s'y prate gu6re, mais il insista aupr6s de moi d'une mani6re si pressante que je me laissai convaincre. Je me souvins opportun6ment de la lettre que j'avais adress6e fi ma soeur en 1940, de ma retraite de Bonne-b~ouvelle, et qui s'dtait retrouvde parmi ses papiers; c'est celle qui a 6t6 reproduite ici ([1940a]). La partie de cette lettre qui porte sur le r61e des analogies en math6matique me parut convenir /t l'occasion. En vingt ans, l'hypoth6se de Riemann n'avait rien perdu de son attrait ni de son myst6re; rien ne m'empachait de mettre mon texte sous le signe du colloque de Tokyo-Nikko (cf. [1955c,d]); il n'y fallut que de ldg6res retouches. On notera, p. 54, que je continuai/t me rdfdrer seulement/t un compte-rendu de Hilbert quand j'aurais dfi citer le XII e probl6me de sa c616bre conf6rence de 1900; fi 9 cela il n'dtait plus d'excuse (cf. [1940a]*). Quant au compte-rendu en question, j'ai le souvenir prdcis qu'il se cache dans le Jahrbuch; mais, apr6s quelque temps perdu rdcemment ",i une recherche infructueuse, j'ai renonc6 fi le retrouver.

[1960b] Algebras with involutions and the classical groups De longue date l'ceuvre d'Elie Caftan m'avait persuad6 de l'importance des espaces riemanniens sym6triques dans la th6orie des groupes semisimples et surtout dans celle des groupes simples non compacts. J'avais aperqu de bonne heure, et fait observer/t Siegel, le r61e jou6 implicitement par cette notion dans ses travaux sur les formes quadratiques ind6finies (v. ses Ges. Abh., n ~ 33, vol. II, pp. 138 168); si on la d6gage des notations matricielles auxquelles Siegel est toujours rest6 attach6, on volt qu'elle intervient comme espace des involutions positives permutables avec une involution donn6e dans une alg6bre de matrices. Par la suite Siegel avait 6tendu sa m6thode aux autres groupes classiques (Ges. Abh. n ~ 60, w vol. IIi, pp. 167-178; cf. [1958d] et [1965]). Ces questions 6taient &l'ordre d u j o u r quand j'arrivai f Princeton et y retrouvai Borel en 1958. Depuis quelques ann6es, lui et Chevalley avaient fait faire fi la th6orie des groupes alg6briques des progr6s surprenants, et Chevalley avait montr6

Commentaire

549

(/t ma grande d6ception, je l'avoue) que, sur un corps alg6briquement clos de caract6ristique p > 1, il n'y a pas plus de groupes simples que sur le corps complexe. Cependant ils n'avaient pas tir6 au clair la classification de ces groupes sur un corps de base quelconque. La vraie difficult6 de ce probl6me tient/l la prdsence des groupes exceptionnels et au d6sir bien naturel de ne pas traiter ceux-ci comme des monstres mais de les englober darts une th6orie g6n6rale off ils n'apparaissent plus comme tels. Pour cela il faut, comme on dit, "soulever des poids et arracher des racines"; c'est un exercice qui a toujours ddpass6 mes forces et que j'ai dfi,/~ regret, abandonner/t des col16gues mieux douds pour cela. Du moins, comme me le fit observer un amateur de cocycles tr6s connu, mon travail de 1956 sur la descente du corps de base, en donnant le moyen de "tordre" une structure par un groupe d'automorphismes, permettait de retrouver 616gamment l'id6e de Siegel, suivant laquelle tous les groupes classiques peuvent atre ddcrits comme groupes d'automorphismes d'alg6bres fi involution. Lorsque le corps de base est le corps des r6els, on retrouve 6galement ainsi, sous forme g6om6trique, la construction donn6e par Siegel pour les espaces riemanniens symdtriques correspondants. En r6digeant ce travail en 1958-59, mon secret espoir 6tait de pouvoir inclure dans mon expos6 au moins quelques-uns des groupes exceptionnels; pour cela je voulais substituer, aux alg6bres associatives classiques, l'alg6bre de Cayley ou "alg6bre des octaves", ~ laquelle justement Borel avait consacr6 l'un de ses premiers travaux. Pour bien faire j'aurais souhait6 aussi pouvoir traiter les formes exceptionnelles que peut prendre le groupe orthogonal/t 8 variables sur les corps admettant des extensions de degr6 3. Apr6s quelques essais infructueux, je dus y renoncer. J'avais destin6 ce travail au volume "jubilaire" du Journal of the Indian Mathematical Society; il rut insdr6 darts la seconde partie de ce volume, qui parut seulement en 1961 ; c'est 15, l'explication de la remarque finale de la note, p. 589, que les 6diteurs voulurent bien ne pas prendre en mauvaise part. Quant aux initiales de cette note, elles se rdf6rent 5, l'Office of Useless Research of the Poldavian Air Force (cf. [1954b]).

[1960 c] On discrete subgroups of Lie groups A Princeton mon coll6gue Atle Selberg s'6tait pos6 depuis quelque temps la question de l'arithm6ticit6 des sous-groupes discrets, /t covolume fini, des groupes semisimples; il avait formul6/l ce propos diverses conjectures, dont certaines portaient plus particuli6rement sur les groupes /t quotient compact, et avait obtenu des rdsultats partiels mais encourageants (cf. sa communication au Bombay Colloquium on Function Theory, T.I.F.R. 1960, pp. 147 164). I1 y a lfi une s6rie de probl6mes qui n'ont pas encore 6t6 tous rdsolus, mais sur lesquels de grands progr6s ont 6t6 faits dans les derni6res ann6es, principalement gr/tce fi G. A. Margulis (cf. J. Tits, SOm. Bourbaki, n ~ 482, f6v. 1976, qui contient une abondante bibliographie). Lorsqu'on se borne au cas des groupes "co-compacts" c'est-5~-dire/~ quotient compact, il s'agit, gdomdtriquement parlant, du probl6me des formes de CliffordKlein d'une structure gdom6trique homog6ne donn6e; si la structure en question

550

Commentaire

est celle d'espace riemannien /t courbure constante n6gative, on retrouve bien le probl6me classique des formes de Clifford-Klein de la g6om6trie hyperbolique. S'agissant d'un groupe semisimple quelconque, on peut, comme le faisait Selberg, appliquer cette id6e/l l'espace riemannien sym6trique associ6, pourvu du moins que le groupe discret op6re sans point fixe; d'apr6s un th6or6me de Selberg (loc. cit.), on peut toujours se ramener/t ce cas en rempla~ant au besoin le groupe donn6 par un sous-groupe d'indice fini. Pour moi, en abordant cette question, je trouvai plus avantageux d'op6rer dans le groupe de Lie lui-m~me, consid6r6 comme muni de la structure de G-vari6t6 d6finie dans [1960c], p. 374. J'y 6tais d'autant plus enclin que, dans l'avant-projet destin6/l Bourbaki dont il a 6t6 fait mention dans [1953]*, j'avais 6tudi6 ces structures avec quelque d6tail, sous le nom d'"espaces de Lie"; ce sont en somme les espaces qui sont "localement homog6nes principaux". Le passage au quotient par un groupe discret F donne alors sans autre hypoth6se une G-vari6t6, compacte si F est co-compact. Dans ses recherches, Selberg avait fait usage du pavage d'un espace riemannien sym6trique par des domaines fondamentaux poly6draux. I16tait plus naturel pour moi d'avoir recours ~ la m6thode des recouvrements ouverts, qui en mainte circonstance, combin6e ou non avec un minimum de cohomologie, m'avait rendu de si utiles services (v. [1941]*, [1946a], [-1949d], [1952a], [1952c], [-1955a]; cf. [1957c]*). Encore une fois elle r6pondit bien ~ ce que j'en attendais. Les ouverts dont j'avais besoin 6taient fi peu pr6s les "group-chunks" de [1955a], et je n'eus qu'~ imiter la proc6dure suivie dans [1952a], w pour v6rifier en toute g6n6ralit6 l'une des conjectures de Selberg; c'est le th6or6me 6nonc6 dans [1960c], p. 370.

[1961a] Ad61es and algebraic groups J'ai dit plus haut (v. [1959a]*) l'impression que m'avait faite en 1958 la d6couverte de Tamagawa; j'y consacrai mon cours fi l'Institute en 1959-60; c'est ce cours, excellemment r6dig~ par Demazure et Ono, et augment~ d'un appendice sur le groupe G2 par Demazure, qui forme le contenu du volume [1961a]. Les deux premiers chapitres traitent de ce qu'on pourrait appeler la g6om6trie et l'analyse ad61iques; les deux autres traitent des groupes classiques, et principalement du calcul du nombre de Tamagawa. Siegel, lorsqu'il avait d~couvert ses th6or6mes sur les formes quadratiques, avait donn~ des d6monstrations compl6tes pour les formes fi coefficients rationnels; pour les formes fi coefficients dans un corps de nombres alg~briques, il avait trait6 des cas particuliers typiques, puis avait laiss6 au lecteur le soin de suivj:e ses indications dans le cas g6n~ral. La m6thode ad61ique, elle, permet de traiter d'un seul coup tous ces cas et m6me celui off le corps de base est un corps de fonctions de caract6ristique p > 1. En ce sens, la premi6re d6monstration compl6te des th6or6mes de Siegel qui ait 6t6 publi6e est celle qui est contenue dans [1961 a], avec le compl6ment que devait lui apporter [1962a]; cela suffirait, s'il en 6tait besoin, &justifier l'emploi de la m6thode ad61ique dans ces questions. Quant fi l'intervention de fonctions zfita au Chap. IV de [1961a] (sugg6r6e elle aussi par des travaux de Siegel; v. ses Ges. Abh., n ~ 30-31, vol. II, pp. 41-96), elle

Commentaire

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prend la place des 6valuations d'int6grales qui forment chez Siegel la "partie analytique" des d6monstrations. C'est ainsi que, dans la d6termination du nombre de classes par Dirichlet, les fonctions L avaient pris la place des 6valuations de volumes dont s'6tait servi Gauss dans sa solution (rest6e in6dite de son vivant) du mfime probl6me.

[1961b] Organisation et d6sorganisation en math6matique Au cours d'un nouveau s~jour au Japon, on me demanda de faire fi la Maison Franco-Japonaise de Tokyo (celle m~me dont Delsarte devint le directeur un peu plus tard) une causerie en fran~ais pour un public plus litt6raire que scientifique. J'en profitai pour m'attaquer au monstre de l'organisation et de la planification scientifiques, qui de nos jours pr6tend tout r6genter. "Le veau se cogne au chine", comme dit Soljenitsyn . . . .

[1962a] Sur la th6orie des formes quadratiques J'ai dit plus haut que la d6termination du nombre de Tamagawa pour lc groupe orthogonal 6quivaut au th6or~me principal de Siegel sur les formes quadratiques; mais ce n'est pas tout fi fait exact. Siegel avait donn6 une formule pour le hombre de repr6sentations d'une forme cp fiv variables par une f o r m e f ~ n variables, sur un corps de hombres alg6briques, pour tout v <_ n; si on d6signe par S(n, v) ce r6sultat, on peut le transformer en un 6nonc6 6quivalent relatif ~ l'ad61isation de l'espace R(.s ~p) des repr6sentations de ~p p a r f s u r un "domaine universel"; celui-ci est un espace homog6ne O(f)/OOp) par rapport au groupe orthogonal O(f) d6fini par f Tamagawa avait fait voir que son 6nonc6, qui donnait la valeur 2 pour le "nombre de T a m a g a w a " de O(f), 6quivalait fi l'6nonc6 S(n, n) de Siegel. Une lois ce point acquis, la question se posait de savoir si les 6nonc6s S(n, v), pour 1 _< v _< n - 1, peuvent s'en d6duire par des raisonnements formels sur des espaces homog~nes. C o m m e il est montr6 dans [1962a], la r6ponse est affirmative.

[1962b] On discrete subgroups of Lie groups (II) Ici encore il s'agit d'id6es qui 6taient "dans Fair" et ne demandaient qu'fi s'incarner. Selberg avait pos6 la question de la rigidit6 des sous-groupes discrets des groupes semisimples; j'avais conclu moi-m~me, dans [1960c], que la question de la rigidit6 des sous-groupes discrets et co-compacts d'un groupe de Lie G simplement connexe se famine fi l'6tude des petites d6formations d'une G-vari6t6 compacte G/F de groupe fondamental F, celui-ci pouvant ~tre consid6r6 comme donn6 par des g6n6rateurs et des relations en hombre fini. C'est 1~ en somme un probl~me de "modules", susceptible d'etre abord6 par l'une ou l'autre des m6thodes dont j'avais d~jfi appris fi faire usage (cf. [-1958b,c]), et par exemple par celle de Kodaira et Spencer. C o m m e le savaient bien les g6om6tres classiques, la premi6re chose fi faire dans un probl6me de d6formation est de le lin6ariser en examinant d'abord les d6formations infinit6simales (of. [1953]*). Si le probl6me lin6aris6 sc trouve 6tre

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Commentaire

localement trivial, il convient de l'aborder globalement au moyen d'un recouvrement ouvert; il s'6nonce alors en termes de cohomologie; c'est lfi en somme le principe de la m6thode de Kodaira et Spencer. Pour aller plus loin, il devient tout indiqu6 de chercher ~ faire usage de la m6thode des cocycles harmoniques, qu'on pourrait appeler m6thode de Hodge; elle consiste ~ normaliser chaque cocycle dans sa classe de cohomologie par une condition de minimum. Lorsque celle-ci d6termine le cocycle d'une mani6re unique au moyen d'une 6quation elliptique, on dira qu'un tel cocycle est harmonique. I1 reste ensuite, d'une part fi 6tudier directement l'espace des cocycles harmoniques, et d'autre part fi revenir en arri6re et examiner quelles conclusions on peut tirer de 1~ pour la d6termination des d6formations finies de la structure dont on 6tait parti. Tel est le sch6ma de d6monstration que je suivis dans [-1962b] ; il 6tait directement inspir6 des travaux de Calabi et Vesentini cit6s dans [-1962b], p. 578, note 1. Un premier pas consiste fi faire usage de [-1960c] pour se ramener au cas d'une d6formation d6pendant diff6rentiablement d'un seul param6tre; diff6rentiant par rapport a celui-ci on obtient (p. 579) une d6formation infinitesimale repr6sent6e par un champ de vecteurs Y sur chaque fibre ~ - l ( t ) ; ~ son tour celui-ci est remplac6 (pp. 580-581) par un champ harmonique. Enfin, quand le groupe G est semisimple sans composante compacte, la d6termination des champs harmoniques peut 8tre effectu6e compl6tement au moyen de la th6orie des alg6bres de Lie. Conform6ment aux conjectures de Selberg, on trouve ainsi que le sous-groupe F de G, discret et co-compact, est rigide chaque fois que G n'a pas de composante de dimension 3; s'il en a, le "manque de rigidit6" de F provient exclusivement de ce qui 6tait d6jfi connu dans le cas des groupes fuchsiens.

[1962c] Algebraic Geometry I1 m'a parfois 6t6 propos6 de collaborer fi des encyclop6dies; le plus souvent cette offre s'accompagnait de conditions aux limites peu attrayantes. Exceptionnellement l'Encyclopedia Americana mc laissa toute libert6 pour r6diger un article sur la g6om6trie alg6brique; celui-ci parut dans l'6dition de 1962 e t a 6t6 r6imprim6, 16g6rement 6court6 je crois, dans les 6ditions suivantes. I1 est reproduit ici dans sa premi6re version.

[1964a] Remarks on the cohomology of groups C o m m e il a 6t6 indiqu6 plus haut, la d6monstration, dans [1962b], se divisait en deux parties: l'une qui ramenait le probl6me des petites d6formations fi celui des d~formations infinit6simales, et l'autre ot~ ce dernier, apr~s avoir 6t~ formul6 en termcs cohomologiques, 6tait trait6 au moyen de la th6orie des alg6bres de Lie par la m6thode des cocycles harmoniques. Je m'aper~us par la suite que la premi6re partie pouvait ~tre remplac6e par un raisonnement tr6s simple et g6n6ral, reposant sur un lemme 616mentaire de calcul diff~rentiel (le lemme 1 de [1964a]). En effet, une lois acquis les r6sultats de [1960c], il ne s'agit plus que d'6tudier les repr6sentations, dans un groupe de Lie connexe G, d'un groupe F donn6 par des g6n6rateurs et des relations en nombre fini. Plus

Commentaire

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pr6cisdment, tout se ram6ne fi l'6tude locale de ces repr6sentations au voisinage de l'une d'elles. Elles forment un ensemble analytique rdel; leurs d6formations infinit6simales peuvent atre regarddes comme des cocycles de F dans l'alg6bre de Lie de G; les cobords correspondent aux ddformations "triviales" ddtermindes par les automorphismes intdrieurs de G. Or c'6tait lfi justement une question que j'avais examin6e de pr6s lorsque je travaillais/~ mon m6moire de 1938 au Journal de Liouville. II est vrai qu'alors je m'6tais born6 au cas off le groupe G est, soit GL(n, C), soit SL(n, C), soit le groupe unitaire U(n, C), et off le groupe discret F est un groupe fuchsien sans 616ment parabolique; en ce temps-lfi, bien entendu, je ne disposais ni du langage ni de l'appareil de la cohomologie. Dans l'intervalle, Eichler, puis Shimura avaient repris certaines de ces questions, en se pla~ant fi un point de vue quelque peu diff6rent du mien. Le m o m e n t me parut o p p o r t u n pour exhumer rues vieux papiers et compldter/~ mon tour quelques-uns des r6sultats de Shimura; c'est l'objet des n ~ 4 7 de [t964a]. Q u a n t / l la justification qu'on peut en tirer de certaines des affirmations de [1938a], v. [1938a]*. On notera que, dans [1964a], p. 153, j'aurais dfi exclure le cas "trivial" 9 = 0, n < 2, off tout ce qui suit est en d6faut. D'autre part, pour comparer les r6sultats de [-1964a] avec ceux de [1938a], il faut tenir compte du fait que, pour G = GL(n, C), ou G = U(n, C), l'entier not6 i' d a n s [ 1964a], p. 155, est > 0; plus prdcis6ment, il est "en g6ndral" 6gal/l la dimension du centre de G, ce qui entraine une correction dans la formule finale, p. 157.

Appendix I: Correspondence, by XXX A correspondent, who wishes to remain anonymous, writes as follows: 9 . . U n a notissima congettura di F. Severi (Rend. P~,l. 28 (1909), p. 45) asserisee ehe "ogni variet~ dotata di p~nti multipli si pu() considerate come

lira ire di una senza singolarit~, apparlenente allo stes.~o .~'pazio." Secondo una notizia orora diffusa da Naneago dall'agenzia " U n i t e d Press," l'ipotesi d e l F i l h s t r e autore sarebbe staia eonfutata dall'egregio geom e t r a franecse l~enato Thorn, basandosi sull'esempio dei toni del 3 ~ or(line nello spazio S~ e mediante assai delicate eonsiderazioni topolog'iehe. Forse non dispiaccr'~ ai lettori del Suo pregiato periodieo trovare qui una trattazione geometriea elementare dell'esempio di Thorn. Di fatti, si d e t e r m i n e r a n n o tutte le variet~ del 3 ~ ordine in uno spazio S,~ q u a h n q u e . Questo trarr~ con s~, come conseguenza immediata, la falsit~ dell'ipotesi suddetta. Sia Vr una variet/l del 3~ ordine nello spazio S~, di dimensione r < ~ - - 1, non eontenuta in un iperpiano. Sia C il eono, proiettante la Vr da un punto sempliee qualunque M di V,.; sia M' un altro punto di V~, sempliee sul eono C. I1 cono C ~ del seeondo ordine; quindi la sua proiezione da M ' una variet~ lineare di dimensione r - I - 1 , sieehg la V~ ~ contenuta in una varieth lineare di dimensione r - l - 2 . J~ dunque r ~ n - - 2 , e C h~ ]a dimensione n - - 1 . Lo stesso vale per il eono C', proiettante V~ da 3iq I eoni C, 6" sono distinti, giaeehg M" ~ sempliee sul eono C ; la loro intersezione dunque una varieth ridueibile del 4=.~ ordine, spezzata nella V~ e una varieth lineare. Siano A ~ 0, B ~ 0 le equazioni di quest'ultima. Allora si possono serivere le equazioni dei eoni C, C' nella f o r m a :

A P = BQ,

(1)

A P ' = B(2',

denotando con P, Q, P', (2' quattro forme lineari helle coordinate omogenee hello spazio. Sia s il numero di forme indipendenti fra ]e sei forme A, B, P, (2, P', (2'; questo ~ almeno 4, giaceM, se fosse 2, i eoni C, C' non sarebbero irriducibili, e, se fosse 3, la loro intersezione si spezzerebbe in quattro variet~ lineari distinte o coincidenti. I valori possibili per s sono quindi 4, 5 e 6. Received June 3, 1957. 951 Reprinted from the Am. Y. of Math. 79, 1957, pp. 951-952, by permission of the editors, 9 1957 The Johns Hopkins University Press.

555

556

Appendix I 952

x.x.x.

Sia L la varietlt lineare, di dimensione . n - - s , definita dalle equazioni A~B=0, P=Q=P'=Q'~o. Ogni varlet5 lineare, proiettante da L un punto dell'intersezione di (7 e C', 6 eontenuta in questa, come si vede subito sulle equazioni (1). P r o i e t t a n d o V,- da L, si ottiene quindi una variet5 W del 3 ~ ordine di dimensione s - - 3 in uno spazio S,-1; e V,. non ~ altro the il cono proiettante g da L. Nello spazio S, 1, si pu6 prendere per coordinate omogenee s forme indipendenti fra le forme A, B, P, Q, P', Q'; allora le equazioni (1) vi definiscono una varietg del 4 ~ ordine, spezzata nella W e una variet~ lineare. Adesso distinguiamo ire casi: (a) s = 6 : IF ~ la varietg di Segre Wa immersa in S~, immagine birazionale senza eceezione del prodotto di un piano e una t e t r a ; come 6 nolo, priva di punti multipli. (b) s ~ 5: IF ~ sezione iperpiana della sopradetta IF;. 1~3 faerie vedere the tutte le sezioni iperpiane irridueibili della IFa sono proiettivamente equivalenti fra di loro; denotando (:on IF., una di esse, 5 anch'essa una variet~ razionale senza punti multipli. (c)

s = 4: IF ~'~ la ben nota c~;biea razionale IF~ immersa in Sa.

Cosl ~ dimostrato ehe ogni variet~'~ del 3 ~ ordine appartiene a uno dei seguenti tipi : 1~ le variet~k di dimen,~ione r, eontenute in una varieta lineare di dimensione r q- 1 ; 2~

le tre variet'5 razionali W;~,, W.,, IV1 enumerate disopra;

3 ~ i eoni proiettanti una di queste tre da una variet~ lineare di dimensione qualunque. Evidentemente, una variet& del secondo o terzo tipo non pu6 essere limite di una varietg del primo tipo, e perci6 una variet~ del terzo tipo, di dimensione maggiore di 3, non pub essere limite di varietfi prive di punti multipli. X. X. X.

Appendix II: Correspondence, by R. Lipschitz We have received the following letter, purporting to come from an ultramundane correspondent :

SIR, It iS sometimes a matter of wonder, to us in Hades, that what we had believed to be our best work remains buried under thick layers of dust in your libraries, while the very talented young men in the mathematical world of the present day strive manfully against problems which are by no means as novel as they think. For instance, it is not so long ago that the very remarkable algebraic systems discovered by my friend Professor Clifford shortly before leaving your world have again attracted the attention of your algebraists after many years of oblivion. When, during my lifetime, I first became interested in them, I, too, fancied that they were new; I soon found out my mistake, and hastened to acknowledge Professor Clifford's prior discovery. It is now a matter of great satisfaction to me to hear that his name has been given to them, as a fitting tribute to his memory among the living. On the other hand, as Professor Clifford has told me himself, it had not occurred to him to apply these algebraic systems to the study of the substitutions which transform a sum of squares into a sum of squares (or, as my young friend and colleague Hermann Weyl would say, of the orthogonal group); he kindly insists that this idea was wholly mine. As you may well believe, we have often discussed this topic since I had the honour of joining the distinguished company of the mathematicians in the Elysian Fields; incidentally, without the many delightful conversations which I have had with him, I should hardly be able now to write to you in English (a feat which ! could have accomplished only with g r e a t difficulty d u r i n g m y lifetime). I t is not, h o w e v e r , in o r d e r to a s s e r t m y claims to f a m e in this m a t t e r t h a t I a m n o w a s k i n g f o r the h o s p i t a l i t y of y o u r j o u r n a l . In w h a t y o u a r e pleased to call t h e n e t h e r world, we a r e h a p p i l y f r e e f r o m v a i n g l o r i ous feelings. B u t it m a y be u s e f u l to a f e w of y o u r c o n t e m p o r a r i e s to h a v e t h e i r a t t e n t i o n d r a w n upon s o m e f o r m u l a s c o n t a i n e d in m y m e m o i r U n t e r s u c h u n g e n iiber die S u m m e n von Q u a d r a t e n (a b r i e f a c c o u n t of which m a y be f o u n d in t h e B u l l e t i n des Sciences M a t h ~ m a t i q u e s f o r 1886), since t h e y would be s o u g h t in vain, unless I a m m u c h m i s t a k e n , in v a r i o u s 247 Reprinted fromAnn. of Math. 69, 1959, by permissionof Princeton UniversityPress. 557

558

Appendix H

248

CORRESPONDENCE

learned volumes r e c e n t l y published on this v e r y subject. U n f o r t u n a t e l y , it appears t h a t t h e r e is now in y o u r world a race of vampires, called r e f e r e e s , who clamp down mercilessly upon m a t h e maticians unless t h e y k n o w the r i g h t passwords. I shall do m y best to modernize m y l a n g u a g e and notations, b u t I am well a w a r e of m y shortcomings in t h a t r e s p e c t ; I can assure you, a t a n y r a t e , t h a t m y intentions are honourable and m y results i n v a r i a n t , probably canonical, p e r h a p s even functorial. But please allow me to a s s u m e t h a t the characteristic is not 2. Call e , - 9 ",G the g e n e r a t o r s of P r o f e s s o r Clifford's algebraic system; this m e a n s t h a t e l - - 1 for all i, and eje~= --v,ej for i < j. For each set [ = {i~, - - - , i,,,} of indices, w r i t t e n in t h e i r n a t u r a l o r d e r < i~< . . - < i , ~ = n ,

1~i, put e(f)

=

% % . . - %,~,

with e(I) = 1 if m = 0; the set I and the unit e(I) will be called even if m is even, odd if m is odd. L i n e a r combinations of e v e n (resp. odd) units will be called e v e n (resp. odd) quantities. Now t a k e an a l t e r n a t i n g m a t r i x X = ( x ~ ) , and assume at first t h a t the d e t e r m i n a n t of E + X ( w h e r e E is the unit m a t r i x ) is not 0. My learned and illustrious colleague P r o f e s s o r C a y l e y was, I believe, the first one to o b s e r v e t h a t , if X is such a m a t r i x , the f o r m u l a (1)

U=

(E--X)-(E+X)-'

defines an o r t h o g o n a l m a t r i x U, and t h a t c o n v e r s e l y X can be expressed in t e r m s of U by the f o r m u l a

(2)

X = (E--

U).(E

+ U ) -~ .

For each e v e n set J----{j~, - . . , j~,} of indices ( w r i t t e n , as always, in t h e i r n a t u r a l order), p u t

x(J) =

1 ~ 2Pp!

s(J, H)xI,~1, x~:/, ""x1%~_,,~

w h e r e the s u m m a t i o n is e x t e n d e d to all p e r m u t a t i o n s H of d, and ~(J, H) is q 1 or - 1 according as the p e r m u t a t i o n is e v e n or odd; p u t x ( J ) = l for p -- O. Consider the e v e n q u a n t i t y

(3)

~ = ~S(J)e(J)

w h e r e the s u m m a t i o n is e x t e n d e d to all e v e n sets of indices J. On the o t h e r hand, t a k e two vectors ~ = ( ~ , , - . - , G ) , ~-(r~,, . . . , ~5,) such t h a t 5 = U~; by the definition of U, this can be w r i t t e n as

Appendix H

559 CORRESPONDENCE

249

or a g a i n as

S~e~ § ~ j xLje~ej.$ jej = ~e~ § ~

7ijej.x~je~ej .

Multiply this to t h e l e f t w i t h 1

2p ~ - Xl~lT~2XhS~z4

"

9

9

Xlb21)_l&21ehi~lt2

*

9

9

el~2~~

a n d t a k e t h e s u m o v e r all s e q u e n c e s ( h , . . . , h,~, i) of distinct indices in odd n u m b e r . I t is easily verified t h a t t h e r e s u l t can be w r i t t e n as

c o n v e r s e l y , if (5) holds, a c o m p a r i s o n of the coefficients of the units et on b o t h sides will s h o w t h a t (4) is satisfied, so t h a t ~ = US. Thus, to e v e r y o r t h o g o n a l m a t r i x U which can be e x p r e s s e d as in (1), we h a v e associated an e v e n q u a n t i t y ~ such t h a t (5) is e q u i v a l e n t to rj = US; as (2) shows, this will be so w h e n e v e r d e t ( E + U ) is not 0. N o w , for e v e r y set I of distinct indices i , . . . , i~,, d e n o t e b y D(I) t h e d i a g o n a l m a t r i x w h o s e diagonal coefficients 3~ a r e such t h a t a~ is - I or + 1 a c c o r d i n g as i is in I or not. I f A is an a r b i t r a r y m a t r i x , one can easily v e r i f y t h e i d e n t i t y ~flet(E

+ D(J)A) = 2~-~(1 + d e t ( A ) ) ,

w h e r e t h e s u m is t a k e n o v e r all e~zen sets J ; in p a r t i c u l a r , if det(A) is not -- 1, a t least one t e r m on t h e l e f t - h a n d side m u s t be o t h e r t h a n 0. T h e r e f o r e , if U is an o r t h o g o n a l m a t r i x of d e t e r m i n a n t + 1, t h e r e is a t l e a s t one of the o r t h o g o n a l m a t r i c e s U' = D ( J ) U which can be e x p r e s s e d b y (1), so t h a t w e can associate w i t h it an e v e n q u a n t i t y t2' such t h a t t h e relation is e q u i v a l e n t to w i t h ~ e q u a l to -- l or + l a c c o r d i n g a s i i s in J or not. B u t t h e n (5) will be e q u i v a l e n t to ~ = US p r o v i d e d we p u t t 2 = e ( J ) t 2 ' . I f U had b e e n a n o r t h o g o n a l m a t r i x of d e t e r m i n a n t --1, one could h a v e r e a c h e d a similar conclusion, p r o v i d e d n is e v e n , b y considering odd instead of e v e n sets. N o t only h a v e we t h u s p r o v e d t h a t , to e v e r y o r t h o g o n a l s u b s t i t u t i o n U of d e t e r m i n a n t + 1 , t h e r e is an e v e n q u a n t i t y ~q such t h a t (5) is e q u i v a l e n t to ~.= US, b u t we h a v e also g i v e n an explicit m e t h o d f o r the c o n s t r u c t i o n of t2 (which, as m a y readily be seen, is uniquely d e t e r m i n e d

560

Appendix H

250

CORRESPONDENCE

by this condition up to a scalar factor). Therein, I believe, lies t h e a d v a n t a g e of the m e t h o d followed in m y m e m o i r . To e v e r y q u a n t i t y ~ , let us associate a n o t h e r one ~ * b y p u t t i n g (ehei ~ - ' - ei~)* = e~ 9

e~2e~I ,

and e x t e n d i n g this to all q u a n t i t i e s by l i n e a r i t y . If, in f o r m u l a (1), we s u b s t i t u t e - X for X, this c h a n g e s U into U -~ and ~2 into t2*. F r o m this, one concludes at once t h a t , if St is t h e q u a n t i t y associated w i t h a n y o r t h o g o n a l s u b s t i t u t i o n , ~st* m u s t be a scalar; in m y memoir, this scalar was called the n o r m of t~ and d e n o t e d by N(t2); now, as I a m told, it is k n o w n as the spinor norm. In particular, if t2 is g i v e n by (3), its n o r m is g i v e n b y N(a)

= aSt*

jx(J) .

=

Since x(J) is the pfaffian of the m a t r i x consisting of t h e x,j for i e J , j e J , its s q u a r e x(J) ~ is the d e t e r m i n a n t of t h a t m a t r i x . F r o m this it follows at once t h a t N ( ~ ) is n o t h i n g else t h a n t h e d e t e r m i n a n t of E + X; e x p r e s s i n g X in t e r m s of U by (2), we g e t N(t~) = 2 ~ . d e t ( E + U) -1 Similarly, if J is an e v e n set such t h a t t h e d e t e r m i n a n t of E + D ( J ) U is not 0, p u t U' = D(J)U, and call X ' and ~2' t h e m a t r i x and t h e q u a n t i t y derived f r o m U' as X and ~2 w e r e d e r i v e d f r o m U. T h e n we h a v e N(t2') = 2 ~. d e t ( E + D(J)U) -1 . On the o t h e r hand, we see as above t h a t ~ and e(J)~' can differ only b y a scalar factor; c o m p a r i n g t h e coefficients of e(J) in these two quantities, we g e t

t2 = x( J)e( J)t2' , so t h a t we h a v e d e t ( E + D(J)U) = 2~N(t2)-lx(J)'~. As I m e n t i o n e d in m y memoir, t h e s e f o r m u l a s can be combined into a single one as follows. L e t t = ( t l , . . . , t~) be a vector; w r i t e diag(t) for t h e diagonal m a t r i x with the diagonal coefficients $i, . . . , t~; and put, for each e v e n set of indices J : A(J) = d e t ( E + D(J)diag($)). T h e n we h a v e N(t2) d e t ( E + diag(t)U) -- ~ j

A(J)x(J)~;

m o r e o v e r , as this f o r m u l a is h o m o g e n e o u s in t h e coefficients of t h e e v e n

Appendix H

561

251

CORRESPONDENCE

quantity 12, it remains valid if t2 is multiplied by a scalar factor. Therefore it holds whenever U is an orthogonal matrix of determinant + 1, provided is an even quantity, given by (3), such t h a t (5) is equivalent to ~]= US. I can still vividly recall my pleasure when I first came across this result in bygone days. But I fear t h a t I am becoming garrulous, and t h a t your patience with me may be exhausted by now. I have the honour to be, etc. R. LIPSCHITZ