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-> v(t) yields a bijection [37,
•
(1) where Td is the greatest k-split torus of T (this can also be expressed by saying that
AUTOMORPHIC £-FUNCTIONS
the inclusion Td We have then
c
T induces
an isomorphism
Tik)f0 Tik) ..::::. T(k)fOT(k),
43 cf. [6]).
(2) The group rk operates on LTD via the cyclic group rk'/k which is generated by the image O" of a Frobenius element Fr. An unramified qJ is completely determined by qJ(Fr), which can be written qJ(Fr) = (t, Fr), where t E LTD is determined up to con jugacy by LTD. Thus qJ unr( T) = (Lye >
(1)
1
----.
D ----. G
----.
G
----.
1.
Since H1(Fk ; D) = 0, the map p. : G (k) � G(k) i s surjective. Let Gsc be the universal covering of the derived group f!§G of G. We have a commutative diagram 1
1
1�
1 ---> Gsc---> G ---> G / Gsc---> 1
�1 11
G
44
A. BOREL
Going 6ver to £-groups, we get
Since LG�c is of adjoint type (2.2), we see that ZL seen that
=
ker Lao. Moreover, it is easily
(2)
This yields a map This allows us to associate to a E Hl( W� ; ZL) a character aa of (G/G.c)(k) which is trivial on D(k), hence a character na of G(k) = G(k)/D(k). It can be shown to be independent of the choice of D. The map a >--+ na is compatible with restriction of scalars [37, 2. 12] and satisfies : (4) 10.3. Conditions on the sets n'P. (l) If n E n'P, then n(z) = x'P(z) . Id (z E C(G)) . If q/ = a . g:> (g:> , g:> ' E rt>(G) , a E H 1 ( W� ; ZL)) (see 6. 5), then n'P' = {na® n l n E U'P } . (3) The following conditions on a set U'P are equivalent : (i) One element of U'P is square-integrable modulo C(G). (ii) All elements of U'P are square-integrable modulo C(G). (iii) g:>( W�) is not contained in any proper Levi subgroup in LG. (4) Assume g:>(Ga) = {1 }. The following conditions on a set U'P are equivalent : (i) One element of U'P is tempered. (ii) All elements of U'P are tempered. (iii) g:>( Wk) is bounded. (5) Let H be a connected reductive k-group and r; : H -+ G a k-morphism with commutative kernel and cokernel. Let g:> E rt>(G) and g:>' = Lr; o g:>. Then any n E U'P, viewed as an H(k)-module, is the direct sum of finitely many irreducible admissible representations belonging to n'P' " 10.4. The unramified case. We say that g:> E rt>(G) is unramified if it is trivial, in the sense of 6.2, on Ga and on the inertia group I. If so, Im g:> may be assumed to be in LT. Therefore, if rt>(G) contains an unramified element, then G is quasi-split (see 8.2 (ii)). Assume now G to be quasi-split, to split over an unramified Galois extension (2)
AUTOMORPHIC L-FUNCTIONS
45
k' of k, and let rp E W(G) be unramified. There exists t E (LTotk such that rp(Fr) = (t, Fr), (1) (9. 5) and we have (w E W�), (2) rp (w) = (t, Fr) d wl where e : W� --. Z is the canonical homomorphism. The element t defines an un ramified character X of a maximal k-torus T of a Borel k-subgroup B of G (9. 5). It is then required that ll
, ( Vm, n) = (m, Fn). The notion of Dieudonne module can be extended to p-divisible groups. On applying D to N = (N1 � N2 � · · ·) we obtain a sequence of modules and maps (DN1 � DN2 � ·), and we define DN = proj lim DNn . This is a W[F, V]-module which is free of finite rank equal to height(N) as a W-module. In classifying p-divisible groups one begins by considering them up to isogeny : N and N' are isogenous if there is a surjective homomorphism N � N' with finite kernel or, equivalently, if there exists an injective homomorphism DN' � DN whose cokernel has finite length over W. If we write W' = W[Ifp] = W ®zp Qp , W'[F, F- 1 ] = W'[F, V] = W[F, V] ®zp Qp, and D'N for DN ®zp Q p regarded as a W'[F]-module, then we see that N and N' are isogenous if and only if D' N � D'N'. Let .A be the category of W'[F]-modules whose objects occur as D' N for some p-divisible group N. When k is algebraically closed one knows that .A has exactly one simple object Dl = W'[F]f(Fr - ps) for each rational number A , 0 � A � I , A = s/r, (r, s) = 1. Dl has dimension r over W', End(Dl) is the unique division algebra over Qp of degree r 2 , and any D E .A can be written uniquely as a finite direct sum D = (Dl1) m i E£> EB (Dl•) m • with distinct A;. Then A� o · · · , A1 are the slopes of D and m,.r,., where A; = s;/r,., is the multiplicity of A;. We sometimes write (Ds l r)m as flsm lrm. Thus fls lr may now be a multipl,e of a simple module ; it has slope sfr with multiplicity r and has dimension r over W'. When k is algebraically closed and N is a p-divisible group over k, the slopes of D' N are called the slopes of N. Clearly N is uniquely determined up to isogeny by its slopes and their multiplicities. For example, all p-divisible groups of height one are isogenous (in fact, isomorphic) to /lpoo or Qp/Zp because D'(ppoo) = D I and D'(Qp/Zp) = no are the only Dl of dimension one over W'. There is only one simple Dl of dimension two over W' ; it is fll / 2 = D'(A(p)) where A is a supersingu lar elliptic curve (cf. §5). Let k have algebraic closure k =f. k. Any p-divisible group N over k defines a p-divisible group N;; over k and it is known that DN;; � DN ® wk W;;. If k = Fq with q = p a then £a : DN � DN is W-linear and so its characteristic polynomial det(T - PIDN) = f Ht
46
A. BOREL
not fix pointwise any nontrivial torus in LT0, hence w = - Id ; if now f-l were sin gular, then the centralizer of p(C * ) would contain a semisimple subgroup H :1- { 1 } stable under Int n , the latter would leave pointwise fixed a torus S :1- { 1 } o f H, and Im rp' would be contained in the centralizer Z(S) of S, a contradiction. Since 1.1 = w · p = - f-l• we have A(rp( - 1)) = ( - 1)<2p, A>, for all A e X* (T). (3) Let o be half the sum of the roots a of G with respect to T such that (p, a) > 0. Then, Lemma 3.2 of [37] implies in particular (4) A(rp'( - 1)) = ( - 1)<26· A>, for all A E x.(T). It follows that (5)
f-l e o +
X*(T).
Therefore f-l is among the elements of X * (T) ® Q which parametrize the discrete series in Harish-Chandra's theorem. We then let Drp be the set of discrete series representations of G(R) with infinitesimal character X" · If G(R) is compact, then nrp consists of the irreducible finite dimensional representation with dominant weight f-l - o. In that case, no proper parabolic subgroup of LG is relevant ; hence (/)(G) consists of the rp considered here. 1 0.6. Let G = GLm k nonarchimedean. Let cjJ be an admissible representation of W�. If it is irreducible, then cp(Ga) = 1 . If it is indecomposable, then it is a tensor product p ® sp(m) , where m divides n, p is irreducible of degree n/m, and sp(m) is m-dimensional, trivial on 1 , maps a generator of the Lie algebra of Ga onto the nilpotent matrix with ones above the diagonal, zero elsewhere, and w e W� onto the diagonal matrix with entries a(w)i (0 � i < n) [9, 3. 1 .3]. If X is a character of Wk (hence of k *), and rp = x ® sp(n) , then Drp consists of the special representation with central character determined by X· In fact, the Weil-Deligne group came up for the first time precisely to fit the special representations of GL2 into the general scheme (see [9]). 11. Outline of the construction over R, C. We sketch here the various steps which yield the sets 0"' when k = R. For the proofs see [37]. We note first that we may always assume rp( Wk) c N(LT0), and we can write (9. 1)
rp(z) = zp zv
(z e C * ; fl.•
1.1 e
X*(T) ® C,
fl. -
1.1
e X * (T)).
1 1 . 1 . LEMMA . Let rp e (/)(G). Assume rp( W8) is not contained in any proper Levi subgroup in LG. Then (i) G has a Cartan k-subgroup C such that (!?)G n S)(R) is compact [28, 3. 1]. (ii) f-l is regular ; rp(C * ) contains regular elements [37, 3.3]. The group LC o may be viewed as a maximal torus of LGo ; hence there is an isomorphism LC -+- L T defined modulo an element .of W. Therefore rp defines an orbit of W in (/)(C), hence, by 9.2, an orbit X"' of W in X(C(R)). [Note that W, which is defined in G(C), operates on C(R), since C(R) n f')G is compact, hence on X(C(R)).] 1 1 .2. Let G0 = C(R)((!?)G)(R)t. Let A0 be the set of representations of G0 which
AUTOMORPHIC L-FUNCTIONS
47
are square-integrable modulo the center, and have infinitesimal character irreducible representa
X.< (A. E X!D). The induced representations 1r = Ifio?/Mn0) (no E A 0) are [37, p. 50]. By definition, DID is the set of equivalence classes of these tions [37, p. 54].
1 1 .3. Let p E tP(G). Let LM be a minimal relevant Levi subgroup containing Im p. It is essentially unique (8.6). We assume LM # LG ; we may view p as an ele ment of 1/>(M). By 1 1 .2, there is associated to it a finite set of ll!f>.M of discrete series representations of M. We may assume LM to be a Levi subgroup of a relevant parabolic subgroup Lp corresponding to P E fli(G/k). Then U = X * (T) ® R = X* (Lro) ® R. Let V be the subspace of elements of U which are orthogonal to roots of LM, and fixed under rk · It may be identified with the dual a; of the Lie algebra of a split compo nent A of P. Let � be the character of C(M) defined by the elements of ll!D,M· We may assume that 1 .; 1 E Cl(a;+). Let P1 be the smallest parabolic k-subgroup containing P such that 1 .; 1 , when restricted to ap1 , is an element of the Weyl chamber a g. Let M1 = z(ap) and P' = P n M1. Then P' is a parabolic subgroup of M1 . Moreover the restriction of le i to the split component M1 n Ap of P' is one ; therefore, for each p E n!f>.M • the induced representation Ind'}{l(p) is tempered. Let ll'!D be the set of all constituents of such representations. Then by definition, DID is the set of Langlands quotients J(Pl> lT) with lT E ll� (cf. [37, p. 82]). 1 1 .4. Complex groups. Assume now k = C. Then Wk = C * , and tP(G) may be identified to the set of homomorphisms of C * into LT0, modulo the Weyl group W, i.e., to ( 1) {(A, f.!. ) , where A., fl. E X * ( T) ® C, A. - fl. E X * (T)} modulo the (diagonal) action of W. In this case Im p is in the Levi subgroup LT of LB, which is the Lp of 1 1 .3. The set ll!f>.M consists of one character of T (cf. 9. 1). Choose P 1> M, as in 1 1 .3. Since the unitary principal series of a complex group are irreducible (N. Wallach), the set ll� consists of one element. Hence so does DID . Thus each DID is a singleton. The classification thus obtained is equivalent to that of Zelovenko. 1 1 . 5. Let G = GLn, k = R. In this case, it is also true that the tempered re presentations induced from discrete series are irreducible [22] ; therefore each set n� (cf. 9.3) consists of only one element, hence so does n!D and we get a bijection between IP(G) and UG(R). Let n = 2. If p is reducible, then Im p is commutative ; hence p factors through ( WR)•b = R * and is described by two characters f.!. • v of R * . Then DID consists of a principal series representation 7r(f.J., v) (including finite dimensional representations, as usual). In particular there are three p's with kernel C * , to which correspond respectively n(l , 1), n(sgn, sgn) and n( l , sgn), where sgn is the sign character. If p is irreducible, then p(z") may be assumed to be equal to (s0, ..-), where s0 is a fixed element of the normalizer of LTo inducing the inversion on it. p(R+) belongs to the center of LGo , and p(S) is sum of two characters, described by two integers. Then DID consists of a discrete series representation, twisted by a one-dimensional re presentation. 1 1 .6. As is clear from these two examples, the main point to get explicit knowl-
48
A. BOREL
edge of the sets nrp is the decomposition of representations induced from tempered representations of parabolic subgroups. This last problem has been solved by A. Knapp and G. Zuckerman [29], [30]. 1 1 .7. Remark on the nonarchimedean case. Langlands' classification [37] is also valid over p-adic fields [57]. In view of 8.6, it is then clear that the last step (1 1 . 3) of the previous construction can also be carried out in the nonarchimedean case. Thus, besides the decomposition of tempered representations, the main unsolved problem in the p-adic case is the construction and parametrization of the discrete series. 12. Local factors. 1 2. 1 . Let 7C E D( G(k)) and r be a representation of LG (2.6). Assume that 7C E Drp for some rp E f/J(G). For a nontrivial additive character cjJ of k, we let
e(s, 'JC, r) = e(s, 'JC , r, cp) = e(s, r o rp , cp), where on the right-hand sides we have the L- and e-factors assigned to the represen tation r o rp of Wk [60]. In the unramified situation of 1 0.4, this coincides with the definition given in 7.2. In view of what has been recalled so far, these local factors are defined if k is archimedean, or if k is nonarchimedean in the unramified case, or if G is a torus. 1 2.2. Let now G = GL,.. In this case there are associated to 7C E D ( G(k)) local factors L(s, '!C) and e(s, 'JC , cp) defined by a generalization of Tate's method, in [25] for n = 2, in [19] for any n, which play a considerable role in the parametrization problem and in the local lifting. A natural question is then whether these factors can be viewed as special cases of 1 2. 1 , where r = r,. is the standard representation of GL,., i.e., whether we have equalities (I) L(s, 7C) = L (s, 'JC , r,.), e(s, 'JC , ¢) = e(s, 'JC, r,., cjJ), with the right-hand side defined by the rule of 1 2. 1 . (a) Let n = 2 . It has been shown in [25] that the equivalence class of 7C i s charac terized by the functions L(s, 7C ® x), e(s, 7C ® X• cp), where X varies through the characters of k*. In this case, the parametrization problem and the proof of (I) are part of the following problem : (*) Given q E f/J(G), find 7C = 7C(fl) such that (I)
L(s,
'JC ,
(2)
L(s,
q
r) = L(s, r o rp),
® x) = L(s, 7C ® x),
for all x's, and prove that D(G(k)).
q
.....
e(s,
q
® X• ¢) = e(s, 7C ® X• cp)
7C(fl) establishes a bijection between f/J(G) and
This problem was stated and partially solved in [25]. The most recent and most complete results in preprint form are in [62] ; they still leave out some cases of even residual characteristic, although some arguments sketched by Deligne might take care of them (see [63] for a survey). As stated, the problem is local, but, except at infinity, progress was achieved first mostly by global methods : one uses a global field E whose completion at some place v is k, a reductive E-group H isomorphic to G over k, an element p E f/J(H/k) whose restriction to L(Hfkv) = LG is q, chosen so that there exists an automorphic representation 7C(p) with the L-series L(s, p) (see § 1 4 for the latter). This construe-
AUTOMORPHIC £-FUNCTIONS
49
tion relies, among other things, on Artin's conjecture in some cases, and [38] . In fact, it was already shown in [25] that (*) for odd residual characteristics follows from Artin's conjecture, leading to a proof in the equal characteristic case. At present, there are in principle purely local proofs in the odd residue characteristic case [6J] . Note also that the injectivity assertion is a statement on two-dimensional admissible representations of w;, namely, whether such a representation O" is determined, up to equivalence, by the factors L(s, O" o x) and e(s, O" o x , fjl). But, so far, the known proofs all use admissible representations of reductive groups [63] . (b) For arbitrary n, (I) has been proved in the unramified case, for special re presentations, and by H. Jacquet for k = R, C [24]. (c) Local L- and e-factors are also introduced for G = GL2 x GL2 in [21 ] , at any rate for products "" x 1e ' of infinite dimensional irreducible representations. Partial extensions of this to GLm x GLn for other values of m, n are known to experts. (d) For n = 3, 1e e D(G(k)) is again characterized uniquely by the factors L(s, "" ® x) and e(s, "" ® x. fjl) [27] , [46] . For n G 4 on, this is false [46] . How ever, it may be there are still such characterizations if X is allowed to run through suitable elements of ll(GLn_ 1 (k)) or maybe just ll(GLn_2(k)). 1 2. 3 . Local factors have also been defined directly for some other classical groups, in particular for GSp4 by F. Rodier [48], extending earlier work of M. E. Novodvorsky and I. Piatetskii-Shapiro, for split orthogonal groups, in an odd number 2n + 1 of variables by M. E. Novodvorsky [41 ] . In the latter case LGo = Sp2n, and in the unramified case, the local factors coincide (up to a translation in s) with those associated by 7.2 to the standard 2n-dimensional representation of the £-group. See also [42] . CHAPTER
IV. THE £-FUNCTION OF AN AUTOMORPHIC REPRESENTATION.
From now on, k is a globalfield, o = ok the ring of integers of k, A k or A the ring of adeles of k, V (resp. V.,, resp. V1) the set of places (resp. infinite places, resp. finite places) of V. For v e V, k., o. and Nv have the usual meaning. Unless otherwise stated, G is a connected reductive k-group. 13. The £-function of an irreducible admissible representation of GA. 1 3 . 1 . Let "" be an irreducible admissible representation of GA and r a representa
tion of LG. There exists a finite Galois extension k' of k over which G splits and such that r factors through LGo ><1 rk' tk · We want to associate to "" and r infinite Euler products L(s, ""• r) and e(s, ""• r), whose factors are defined (at least) for al most all places of k. Let v E v. By restriction, r defines a representation r. of L(Gfk.) = LGO )
L(s,
(2)
e(s,
1e,
'JC ,
r) r)
=
=
D.L(s , "" • • r.),
n.e(s, ""·· r., fjl.),
where f/J. is an additive character of k. associated to a given nontrivial additive character of k, and the factors on the right are given by I 2. I(l).
50
A.
BOREL
The local problem is solved for archimedean v's, and for almost all finite v's (see below) so that the factors on the right are defined except for at most finitely many v e V1. For questions of convergence or meromorphic analytic continuation this does not matter, and we shall also denote such partial products by L(s, 7C, r). By I 0.4, q;. is well defined if the following conditions are fulfilled : G is quasi-split over k., G(o.) is a very special maximal compact subgroup of G(k.), k' is unramified over k, and 7Cv is of class one with respect to G(o.). All but finitely many v e V1 satisfy those conditions [61]. 1 3.2. THEOREM [35]. Let 7C be an irreducible admissible unitarizable representation of GA and r be a representation of LG (2.6). Then L(s, 7C, r) converges absolutely for
Re s sufficiently large.
We may and do view r as a complex analytic representation of LG o � r,,,k> where k' is a finite Galois extension of k over which G splits (2.7). We let V1 be the set of v e V1 satisfying the conditions listed at the end of 1 3. 1 . We have to show that (1)
L'
=
v EITVt
L(s, 7Cv, r.),
converges in some right half-plane. Let Fr. be the Froberiius element of rk'v• lk.• where v' E v,, lies over v E vl . We have q;.(Fr.) = (t., Fr.), with t. e LTo (2) and (3) L(s, 7C0 , r.) = (det(I - r((t., Fr.))N;•)) - 1 . To prove the theorem, it suffices therefore to show the existence of a constant a > 0 such that (4) l.u l � (Nv) a for every v e V1 and eigenvalue ,u of r((t., Fr.)). Let n = [k' : k]. Since we may assume t. fixed under r,. (6 . 3), we have t� = (t., Fr.)n ; hence it is equivalent to show (4) for all eigenvalues ,u of r(t.). These are of the fonD. t;, where it runs through the set Pr of weights of r, restricted to LG0• Thus we have to show the existence of a > 0 such that j t. j Re A � (Nv) a for all v E vl and it E P,. (5) Let G' be a quasi-split inner k-form of G. Then LG = LG', and G is isomorphic to G' over k. for all v e V1 . We may therefore replace G by G' ; changing the nota tion slightly, we may (and do) assume G to be quasi-split over k. We then fix a Borel k-subgroup B of G and view LT as the £-group of a maximal k-torus T of G. For a cyclic subgroup D of r,,,,, let Vv be the set of v E vl for which r,. is equal to the inverse image of D in F,. The group U = X* ( T) 0 is then the group of one parameter subgroups of a subtorus S of T such that S/k. is a maximal k.-split torus of Gfk. for all v e V0• The group (6) Y = Hom(U, C * ) = Hom(X* (T)D, C * ) (v e V0), is independent of v, and is the Y of §6 for Gfk •. The root datum cp(Gfk.), which is determined by the action of D, is also independent of v e V0.
AUTOMORPHIC £-FUNCTIONS
51
Given y e Y, let y0 be a "logarithm" o f y, i.e., an element o f Hom(X* (T) D, C) such that (7) y(u) = NvYo (u) = Nv
(8)
= Nv
is well defined. If y has values in R+, then we choose y0 to be equal to its real part. The space a * is the dual of a = U ® R (the so-called real Lie algebra of Sfkv), and is acted upon canonically by ,. w as a reflection group. We let a * + be the positive Weyl chamber defined by B. Let Pv be the unramified character of T(kv), given by t ...... lo(t)lv, where I l v is the normalized valuation at v and o half the sum of the positive roots. Then its real logarithm p0 is independent of v e VD · In fact, it is a positive integral power of Nv whose exponent is determined by the kv-roots, their multiplicities, and the indices qa of the Bruhat-Tits theory [61]. But those are determined by the previous data and the action of rk. on the completed Dynkin diagram [61], which is also in dependent of v e VD. We write p0 instead of Pv. O · We have p0 e a * +. The representation 'lT:v is a constituent of an unramified principal series PS(xv), where Xv is an unramified character of T(kv), or, equivalently, of S(kv), determined up to a transformation by an element of k W. Thus we may assume Xv . o to be con tained in the closure �l'{a * +) of a * +. Since 'lT:v is unitary, the associated spherical function is bounded, and hence Re Xv . o is contained in the convex hull of k W(p0), i.e., we have (po - Xv . O • .A. ) � 0, for all .A. e a * +. (9) (See remark following the proof.) For .A. e X * (LT0), let .A.' be the restriction of .A. to X* (TD). In view of 10.4 and our conventions, we have then (10)
Let ;. (I I)
=
,. W(.A.') n �l'{a * +). Since Re Xv . O E �l'(a * +), we have Nv
Combined with (9), this implies ( 1 2)
If now .A. runs through P, there are only finitely many possibilities for A, whence (4), with a = sup(p0, J.) (.A. e Pr), for v e VD. Since V1 is a finite union of such sets, this proves (4). REMARK. The relation (9) is proved in [35, pp. 27-29] for the split case. For a general semisimple simply connected group, see I. Macdonald, Spherical functions on a group ofp-adic type, Publ. Ramanujan Institute 2, Madras, Theorem 4. 7 . I , or H. Matsumoto, Lecture Notes in Math., vol. 590, Springer-Verlag, Berlin and New York, Proposition 4.4. 1 1 . In fact, we have used it for a general connected reductive
52
A. BOREL
group but the reduction to the case of simply connected semisimple groups is easily carried out by going over to the universal covering of the derived group. 1 3.3. CoROLLARY. Let P be a parabolic k-subgroup of G, P = M · N a Levi decom position over k of P. Assume that n is a constituent of a representation Ind ��(11) inducedfrom a unitarizable irreducible admissible representation 11 of MA , viewed as a representation of PA trivial on NA- Then L(s, n, r) is absolutely convergent in some right half-plane.
We view LM as a subgroup of LG (3.3). Let r' be the restriction of r to LM. Let v E V1 be such that the conditions listed at the end of 1 3. 1 are satisfied by M, G, 11v and n• . Then, by the transitivity of induction, it follows that there exists Xv as in the above proof such that 11v (resp. n v) is the constituent of class 1 with respect to M(o.) (resp. G (o ,) ) of the principal series PS(x.) for M(kv) (resp. G (kv)) . Then L(s, nv , r ) = L(s, rJv, r') (7.2, 10.4). This being true for almost all v's, w e are reduced to 1 3.2. 14. The L-function of an automorphic representation.
14. 1 . A smooth representation of GA is automorphic if it is a subquotient of the regular representation of GA in Gk\GA- It is cuspidal if it consists of cusp forms. If so, it is unitary modulo the center. We let W.(G/k) denote the set of equivalence classes of irreducible admissible automorphic representations of G A- By Proposi tion 2 of [39], every n E W.(G/k) is a constituent of a representation induced from some cuspidal rJ E W.(Mjk), where M is a Levi k-subgroup of a parabolic k-subgroup of G. Combined with 1 3.3 this yields the n E W.(G/k) and r be a representation of LG. r) is absolutely convergent in some right half-plane.
14.2. THEOREM (LANGLANDS). Let
Then L(s,
n,
The L-function of an irreducible admissible automorphic representation will also be called an automorphic L-function. 14.3. There are several conjectures on the analytic character of L(s, n, r) for auto morphic n, all checked in some special cases, going back to the work of Heeke on L-series attached to Grossencharaktere and to modular forms. (a) If n E W.(G/k), then L(s, n , r) admits a meromorphic continuation to the whole complex plane. (b) Assume that n and G are such that the local solution to the local problem yields factors L and e at all places. It is then conjectured that there is a functional equation L(s, n , r) = e(s, n, r) · L(l - s, it, r), where it is the contragredient re presentation to n . (c) In a number of cases, it has been shown that : (* ) If n is cuspidal, r irreducible nontrivial, then L(s, n, r) is entire. Here and there, conjectures to the effect that this should be a general pheno menon have been stated. However, there are counterexamples. Heuristically, one sees this is likely to happen if n is lifted from a cuspidal representation of a reduc tive group H (in the sense of V below) and the restriction of r to LH contains the trivial representation. 14.4. (a) Let G = GLn and r = rn be the standard representation of GLn(C). Then 14.3(b), (c) are proved in [25] for n = 2, in [19] for n � 2, if L and e are de-
AUTOMORPHIC £-FUNCTIONS
53
fined to be the products of the L- and .s-factors mentioned in 12.4. As recalled in 12.4, these are the same as those considered here at almost all places, and for n = 2, at all places. (b) If G = GL2 x GL2 and r = r2 ® r2, similar results are established by Jac quet in [21]. (c) Let G = GL2 • If r: GL2(C) --+ GL3(C) is the adjoint representation, then 14.3(b), (c) are announced in [16] . This extends results of Shimura [54]. If r = Sym3 (r2), Sym4(r2), then 14.3(b) is stated in [15], in the context of the global lifting (see V) ; for Sym3(r2), it is also proved in [51], in the framework of 14.5 below. (d) Let k be a function field, G = GLm x GLn and r = rm ® rn· Let 1r: (resp. 1r:') be a cuspidal automorphic representation of the first (resp. second) factor. By the methods of [19], [26], [27], one can define L and e, and (Jacquet dixit) show 14.3(b), and also the holomorphy, except when m = n and 1r: is contragredient to ' 1r: . These methods also yield further examples for other groups and for other re presentations. It is expected that similar results hold over number fields. (e) 14.3(a) has also been checked when G = PSp(4) in some cases in [1], and, in general, in [42]. A functional equation is also established. 14.3(a), (b) are announced in [41] for orthogonal groups in an odd number of variables over functional fields, for the local factors mentioned in 12.3. For a survey and earlier references, see [43]. See also [44]. 14.5. We describe some cases in which 14.3(a) has been verified in [33] (see also [18] for a survey). Let C be a split k-group, of adjoint type, endowed with its canoni cal o-structure. Fix a Borel subgroup B of C and a maximal torus T of B defined over o. Let P be a maximal proper standard parabolic subgroup and P = M · N its standard Levi decomposition. Since C is adjoint, it is easily seen that C(M) is a torus. The group MfC(M) is semisimple, split over k, of adjoint type, of rank equal to rk(C) - 1 . We let G = M/C(M). The group LG o is simply connected (2.2(2)). We have a natural inclusion LG --+ LM, and L M is the Levi subgroup of a standard parabolic subgroup Lp = LM · U with unipotent radical U (3.3). Let A be the split component of P in T, and LA o the split component of Lpo in L T0• The group LA o acts on the Lie algebra u of U and its eigenspaces are irreducible LG0-modules. We let Fp denote the set of contragredient representations to these LG0-modules. The £-functions considered in [33] are of the form L(s, 1r:, r) with r e Fp and 1r: an ir reducible cuspidal automorphic representation of G. A number of examples are given in which L(s, 1r: , r) admits a meromorphic continuation. This is deduced from the results of [32] : let m be the length of a composition series of u with respect to M. Then, for suitable numbering of the elements of Fp and strictly positive integers a;, there is a relation M(s) = TI L(a;s, 1r: , r;) · L(sa; + 1 , 1r: , r;)- 1 , (I) l�i�m
where M(s) is the intertwining operator occurring in the theory of Eisenstein series with respect to P, and is known to have a merom orphic continuation to the complex plane [32]. If r = 1 , this and 1 3.2 yield the meromorphic continuation. In general, if we have the analytic continuation for all r;'s except one, (1) gives it for the remain ing one. 14.6. The converse problem is to what extent automorphic representations can be characterized by analytic properties of their £-functions, or to give analytic
54
A. BOREL
conditions on a given L-function which will insure that it is automorphic. The first main result was Heeke's characterization of the Mellin transform of a parabolic modular form. Then came Weil's extension of this theorem to congruence sub groups [64], [65], its generalization in the context of representations in [25], and the extension to GL3 [46], [27]. In those results, conditions are imposed on the L-func tions of 11: and of the twists 11: ® X of 11: by characters. However, the analogous statement is false from n = 4 on [46] . It may remain true if one imposes conditions on the twist 11: ® p of 11: by representations of GL,._ 1 or only of GL,._2 • For results in that direction, over function fields, see [45]. Note however that in the general problem outlined here, one wishes rather to turn things around and deduce the analytical properties of some given L-series by show ing directly that it is automorphic (see the seminars on base change and on zeta functions of Shimura varieties [17], [8], [40]). 1 4.7. Other problems. (I) One "representation the oretic" form of "Ramanujan's conjecture" is the following : if 11: = ®nv is an irreducible nontrivial admissible cuspidal automorphic representation (and G is simple), then each 'lr:v is tempered. It is now well known to be false for certain orthogonal or unitary groups, and even for one split group [20]. (2) Let 11: be a unitary irreducible representation of GA. If G = GL2, then its multiplicity in the space of cusp forms oL2 (G(k)\G(A)) is at most one, "multiplicity one theorem" [25]. In fact there is even a "strong multiplicity one theorem" [38] : given nv for almost all v's, there is at most one constituent 11: of the space of cusp forms with those local factors. The multiplicity one theorem has been proved for GL,. [52] and the strong form for GL3 [28]. It is unknown whether it is true for SL2 • On the other hand, there are counterexamples for some inner forms of SL2 [31]. CHAPTER V. LIFTING PROBLEMS.
Although the problems on automorphic L-functions discussed in § 1 4 are only partially solved, the solutions provide practically all cases in which an L-series (automorphic or not) has been proved to have meromorphic or holomorphic analy tic continuation with functional equation. This suggests trying, given an L-series and a reductive group G, to see whether G has an automorphic representation with the given L-series. Many instances of such questions can be viewed more precisely as special cases of the "lifting problem" or of the "problem of functoriality with respect to morphisms of L-groups." There is also a local version. For the sake of exposition, we shall start with the latter, but it should be borne in mind that the motivation and requirements stem from the global one, and that local and global are at present inextricably linked in many proofs. These questions were raised by Langlands in [35]. 15. L-homomorphisms of L-groups.
1 5. 1 . Let E be a field and H, G connected reductive £-groups. A homomorphism u : LH -+ LG over rk is said to be an L-homomorphism if it is continuous and if its restriction to L H0 is a complex analytic homomorphism of LH0 into LGO. Let E be local and G quasi-split. If rp e (/)(H), then u rp e (/)(G). In fact, condition 8.2(i) is clearly satisfied, by u rp, and so is 8.2(ii) because every parabolic subgroup of LG a
a
55
AUTOMORPHIC £-FUNCTIONS
is relevant, G being assumed to be quasi-split. Therefore qJ ...... u o qJ defines a map f/J(H) --+ f/J(G), to be denoted f/J(u). 1 5.2. Let E = k be a global field. For v E v, the Galois group rk. is a subgroup of rk ; hence the L-group of G viewed as a k0-group, to be denoted L(G/k0), is a subgroup of LG = L(G/k). Thus, in particular, the £-homomorphism u of 1 5. 1 defines by restriction an £-homomorphism u0 : L(Hfkv) --+ L(Gjk0), hence also a map f/J(uv) : f/J(Hfkv) --+ f/J(Gfkv) (v E V). The "lifting problem" is, roughly speaking, whether such maps are mirrored by maps of representations in the local case, or of automorphic representations in the global case. 1 5.3. EXAMPLE : BASE CHANGE. Let H be a split over E, F a finite Galois extension of E, and G = RFIEH. Then LGo is a product of copies of LH0, indexed and per muted by rFIE (5. 1). There is then a natural L-homomorphism u which is the iden tity on rE and the diagonal map on LH0• If E is a local field, then Wj.. is an open normal subgroup of W£, and the map f/J(u) may be viewed as given by the restric tion to Wj... 16. Local lifting.
1 6. 1 . Let k = E be a local field, G quasi-split over E, H a connected reductive E-group and u: LH --+ LG an £-homomorphism. The problem of local lifting is, roughly, to establish a correspondence ll(u) : ll(H(k)) --+ ll(G(k)) which preserves L- and e-factors. If the local parametrization problem of III is solved, then ll(u) is the map between indistinguishable classes which assigns llu•rp, G to llrp . H (({) E f/J(H)). The element ll E ll(G(k)) is said to be a lift of 1r E ll(H(k)) if ll E llu•rp, G• where (/) E f/J(H) is such that 7r E nrp.H · We have then (1)
L(s, ll, r)
=
L(s,
1r ,
r o qJ),
e(s, n, r, cp)
=
e(s, 'lr, r 0 u, cp).
for every representation r of LG. 1 6.2. The local lifting is thus viewed as a map between classes of £-indistinguish able representations rather than one between representations. However it is pos sible to single out one lifting under assumptions which, in the global case, are satisfied almost everywhere : assume H, G to be quasi-split, split over an unrami fied extension F of E, endowed with an oE-structure such that H(oE) and G(oE) are very special maximal compact subgroups, and 1r of class one with respect to H(oE). Then qJ such that 1r e Drp.H• and the set llu orp, G are well defined. Moreover, llu•rp, G contains exactly one element of class one (with respect to G(oE)), to be called the natural lift of 1r . 1 6. 3 . A full solution of the local parametrization problem does not seem to be in sight, and it is conceivable that it may require proving at the same time global results such as Artin's conjecture. Meanwhile, one wants to settle some approxima tions to it, notably to be able to prove some cases of Artin's conjecture. Note that if G = GLno then the sets Drp.G are either known or conjectured to consist of one element (1 2.2, 1 2.3). Such a lifting problem can then be stated as one of construct ing a map u * : ll(H(k)) = ll(G(k)) satisfying certain conditions. So far, there are two examples : (a) Base change (cf. 1 5.3) when H = GL2 and F is cyclic of prime degree over E [17], [38], [49], [56]. Besides some naturality conditions and 1 6.3, the main require-
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ments relate the characters of n and of the hypothetical u * (n). The results also describe the fibres and the image of u * . [Note that the results of [38] on this problem are used in [62], so that we cannot invoke the solution of the local parametrization problem (1 2.4) for GLdust to use the map U(u) of 16. 1 . If we could, then the local questions [38] would be mainly to relate the characters of n and ll(u)(n).] (b) H = GL2, G = GL3 , and (1) is given by the adjoint representation of LHo (see [16]). In this case, n = u * (n) must be trivial on the center of LG o and be such that the L- and e-factors of u * (n) ® X (X character of E *) are certain given functions. There is at most one such U ( 12.4(d)) . In [16], n is stated to exist, except possibly if E has even residual characteristic and n is "extraordinary. " 16.4. I n 16.3(a), the lifting problem was connected with the existence o f relations between characters. This is a direct connection between U(H) and ll(G), which is of great importance for the use of the trace formula in proving or using the local or global lifting. We now mention two other examples of such relations. Assume that G is a quasi-split inner form of H. There is then an isomorphism u : L H ...::. LG and an embedding C/J(u) : C/J(H) c: C/J(G). If f: H --+ G is a k,-isomorphism such that J- 1 · rJis an inner automorphism of G for every r e rk . then /establishes a bijection between conjugacy classes which are stable under rk . Using results of Steinberg [59], one then sees easily that maximal k-tori in H are isomorphic over k to maximal k-tori in G. This allows one in some cases to assign regular semisimple classes in G(k) to such classes in H(k), so that it makes sense to compare values of characters of H(k) and of G(k) on such classes. (a) Let k be either R or nonarchimedean with odd residual characteristic. Let G = GL2 and H be the group of invertible elements in the quaternion algebra over k. The sets Drp are singletons, C/J(u) assigns to a (finite dimensional) irreducible representation n of H(k) a discrete series representation n' of G(k). In this case, the semisimple classes of H(k) correspond to the elliptic classes in G(k). It is proved in [25] that the characters of n and n' differ only by a sign on those classes. (b) Let k = R. For rp e C/J(H), C/J(G), let X rp be the sum of the characters of the elements in nrp. Choose 'P E C/J(H) such that Drp consists of tempered representations. Then X rp and Xu•rp are equal on the regular semisimple classes of H(k), up to a sign depending only on H and G [53, 6.3]. 16.5. We could also take the Weil forms of the L-groups. In that case an L homomorphism, restricted to WE, is assumed to satisfy the obvious analogue of 8.2(i). Take in particular the case where H = { 1 }. Then u is just an element of C/J(G). The lifting problem in this case is part of the local problem of III. 17. Global lifting.
17 . 1 . Assume G to be quasi-split. Let H be a reductive k-group and u : LH --+ LG an L-homomorphism. Let uv : L(Hfkv) --+ L(Gfkv) and C/J(uv) : C/J(Hfk.) --+ C/J(Gfk.) be the associated maps ( v e V) (see 1 5. 1). Let n = ®. n. (resp. U = ®.D.) be an irreducible admissible representation of HA (resp. GA). Then Dis said to be a lift of n if n. is one of n. for every v e V (16.1). If that is the case, then, for every representation r of LG, we have (1)
L(s, ll, r)
=
L(s, n, r o u),
e(s, n, r)
=
e(s,
1C ,
r 0 u).
AUTOMORPHIC £-FUNCTIONS
57
It is also usually requested that n. be the natural lift (16.2) of 'R: v for almost all v's. The question is then whether every automorphic 1r: has a lift, which is automorphic, or, somewhat more ambitiously, whether there is a map u * : IJ!(H/k) --+ IJ!(G/k) with reasonable properties, which sends 1r: e W.(Hfk) onto a lift of 1r: . One also wants to describe the fibres and the image of u*. In that degree of generality, the problem appears to be inaccessible at present. However, there are many results, old and recent, which are very striking illustra tions of this principle, some of which will be extensively discussed in various semi nars. Here, for orientation, and to give an idea of the scope of the problem, I shall list briefly some special cases, referring to the literature or to other seminars for more details. REMARK. Let r be a representation of L H of degree n. Then it defines an £-homo morphism u : LH --+ LGLn = GLn{C) x rk in the obvious way. A positive answer to the lifting problem would imply in particular that if 1r: is an automorphic repre sentation of H, then L(s, 1r , r) = L(s, D, rn) where D is an automorphic representa tion of GLn and r n the standard representation. This would therefore to a large extent reduce the study of automorphic L-functions to those of GLn, with respect to the standard representation. 17.2. Let H = { 1 }, G = GLn. Then an L-homomorphism u is just a continuous complex n-dimensional representation of Fk . The question is then whether the Artin L-series L(s, u) is an automorphic L-series of GLn (with respect to the stand ard representation of GLn{C)), which should be cuspidal if u is irreducible. In view of known results on GLn (cf. 14.4) this would imply Artin's conjecture. For n = I, a positive answer is given by class-field theory. For n = 2, 3, a posi tive answer is equivalent to Artin's conjecture, since there are converses to Heeke theory [25], [65], [27], [46]. For n = 2, it has been proved for dihedral or tetrahedral representations of rk , and for some others over Q (see [38], [17], [15]). 17.3. Let k' be a Galois extension of k, n the degree of k' over k. Take H = Rk '/kGLh G = GLn. There is a natural homomorphism f: LHO XI rk 'lk into the normalizer of a maximal torus LTo of LG0• Since the former group is a quotient of LH, and LG = LGO X rk , we can define an L-homomorphism u : LH --+ LG by u(h , r) = (f(h), r) (h E LH0, r E rk). An automorphic representation of H is a Grossencharakter X of k'. The problem is then whether the Artin L-series L(s, x) is the L-series of an automorphic representation of G. If n = 2, k = Q, and k' is imaginary, this was proved by Heeke ; 1r: is associated to a cuspidal holomorphic automorphic form. If n = 2, k = Q, and k' is real quadratic, this was established by H. Maass. 1r: is then as.s ociated to a nonholomor phic automorphic form. For n = 3, this is proved in [26], [27] . 17.4. Base change. This is the global counterpart to 1 6.3(a). Let k' be a finite Galois extension of k. Assume H to be k-split and G = Rk'lkH. There is again an L-homomorphism u : LH --+ LG whose restriction to L H0 is a diagonal map. In this case G(A) and G(k) are canonically isomorphic to H(Ak,) and H(k') ; therefore the problem is to associate an automorphic representation of H(Ak' ) to an automorphic representation of H(Ak). Again, it should be a counterpart to the restriction to Wk' of homomorphisms wk --+ L H0 • If H = GL2 and k' is cyclic of prime degree, the lifting map u* for representa tions is constructed in [38], which also gives a description of its image and fibres.
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This extends work of Doi-Naganuma, Jacquet [21] (on the quadratic case) and of Saito [49], Shintani [55], [56] (cf. [17]). 1 7.5. Let G be quasi-split, and H an inner form of G. Then L H = LG and f/J(H/kv) c f/J(Gfkv) for all v's (8.3). Moreover, for almost all v's, H and G are isomorphic over k•. ; hence f/J(Hfkv) = f/J(G!kv) and D(H(kv)) = D(G(kv)). The question is then, given 1r = ®v1rv, is there an automorphic representation D = ®vllv of G such that Dv = 7rv for almost all v's ? If G = GL2 and H is the group o f invertible elements o f a quaternion algebra D over k, a positive answer is given by Jacquet-Langlands [25]. Note that, in that case, because of the "strong multiplicity one theorem," at most one D may be associated to a given 1r in this way. The possible D's are in fact the cuspidal automorphic representations for which Dv belongs to the discrete series for all v's over which D does not split (loc. cit.). 1 7.6. If G = GL2, G = GL3 and u is given by the adjoint representation, as in 13.4, the global lifting problem has been solved by Gelbart-Jacquet [16], the "local lifting" being the one of 1 6.2(b). 1 7. 7. Let M be a Levi k-subgroup of a parabolic k-subgroup P of G. Then L M imbeds naturally into LG (3.3), whence an L-homomorphism u : LM --. LG. If 1r is cuspidal, then the analytic continuation and residues of Eisenstein series [32] are known to yield a unitary u * (1r) in many cases, and, conjecturally, in general. 18. Relations with other types of L-functions.
1 8. 1 . In 1 7.2, the lifting problem amounts to identifying an Artin L-function with an automorphic L-function on GLn- One can also include in this problem more general representations of Weil groups if one passes to the Weil form of the L-groups. For simplicity, let us limit ourselves to relative Weil groups Wk'lk• where k' is a finite Galois extension of k over which H and G split. An L-homomorphism u : LHO � wk'/k --> LGO � wk'/k is then a continuous homomorphism compatible with the projections on Wk'lk• whose restriction to LHo is a complex analytic homo morphism into LGo , and such that, for w E Wk'lk• u(w) = (u'(w), w) with u(w) semi simple (cf. 8.2(i)). If H = { 1 }, an L-homomorphism is said to be an admissible homomorphism of Wk,lk into LG. In analogy with the definition of f/J(G) in the local case, we can con sider the set f/Jk,1iG) of equivalence classes of such homomorphisms, modulo inner automorphisms of LGo, and then pass to a suitable limit f/J(G) over k'. The lifting problem asks in this case to associate to any rp E f/J(G) an automorphic representation 1r, such that, for any representation r of LG, L(s, 1r, r) is equal to the Artin-Hecke L-series of r o u. In particular, is every Artin-Hecke L-series that of an automorphic representation of GLno with respect to the standard representation? If G is a torus, then [34] provides a positive answer. In fact, in this case the irredu cible admissible automorphic representations of G are the characters of G(k)\G(A), and [34] gives a homomorphism with finite kernel of f/Jk,1iG) onto the set of such characters. 1 8.2. In the same vein, it is natural to ask whether Hasse-Weil zeta-functions (or even L-functions of compatible systems of /-adic representations of Galois groups) can be expressed in terms of automorphic L-functions. For elliptic curves over function fields, it is a theorem. That it should be the case for ell iptic curves over
AUTOMORPHIC £-FUNCTIONS
59
Q is the Taniyama-Weil conjecture ; it has been checked in a number of special cases (see [2], [14] for surveys from the classical and representation theoretic points of view respectively). Apart from that, this problem has been pursued mostly for Shimura curves and certain Shimura varieties ; we refer to the corresponding semi nars for a description of the present state of affairs. Finally, one may ask whether it is possible to characterize a priori those auto morphic representations whose L-series have an arithmetic or algebraico-geome tric significance. A necessary condition if k is a number field is that for an infinite place v, ""• should be associated to a representation a. of Wk, whose restriction to C* is rational, C* being viewed as real algebraic group, i.e., be of type A0 in [3, 6.5]. If the L-series of 1r is to be an Artin L-series, then "" should even be of type A00 (loc. cit.), i.e., a. should be trivial on C* . Let k = Q. Then there are three pos sibilities for """' (1 1 .5). If """' = 1r(l , sgn), then "" corresponds to 2-dimensional representations of F0 with odd determinant by the theorem of Deligne-Serre [10], [50]. Modulo the Artin conjecture for such representations, the correspondence is bijective. However, I am not aware of any result for the other two possible values of """' · A positive answer would involve nonholomorphic automorphic forms. In [36], it is shown in many cases for GL2 over Q that the L-series of a representation of type A0 is that of a compatible system of /-adic representations of F0• Over a function field, there is no condition such as A0• In fact, for GL2, Drinfeld has shown that all irreducible admissible automorphic representations are associated to /-adic representations (see the lectures on his work by G. Harder and D. Kazhdan). REFERENCES I. A. N. Andrianov, Dirichlet series with Euler product in the theory of Siegel modular forms ofgenus two, Trudy Mat. Inst. Steklov 112 (1971), 73-94. 2. B. J. Birch and H. P. F. Swinnerton-Dyer, Elliptic curves and modular functions of one variable. IV, Lecture Notes in Math., vol. 476, Springer, New York, 1 975, pp. 2-32. 3. A. Borel, Formes automorphes et series de Dirichlet (d'apres R. P. Langlands), Sem. Bourbaki,
Expose 466, (1 974/75) ; Lecture Notes in Math., vol. 5 1 4, Springer, New York, pp. 1 89-222. 4. A. Borel and J.- P. Serre, Theoremes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1 964), 1 1 1 -1 64. 5. A. Borel et J. Tits, Groupes reductifs, Inst. Hautes Etudes Sci. Publ. Math. 27 (1 965), 55-1 5 1 . 6. P. Cartier, Representations of reductive P-adic groups, these PROCEEDINGS, part 1 , pp. 1 1 1-155. 7 . W. Casselman, GL., Proc. Durham Sympos. Algebraic Number Fields, London, Academic Press, New York, 1 977. 8. -- , The Hasse- Wei/ '-function of some moduli varieties of dimension greater than one, these PROCEEDINGS, part 2, pp. 141-163. 9. P. Deligne, Formes modulaires et representations de GL,, Modular Functions of One Variable, Lecture Notes in Math., vol. 349, Springer, New York, 1973, pp. 55-106. 10. P. Deligne and J.-P. Serre, Formes modulaires de poids 1, Ann. Sci. Ecole Norm. Sup. (4) 7(1 974), 507-530. 11. D. Flath, Decomposition of representations into tensor products, these PROCEEDINGS, part 1 , pp. 1 79-1 83. 12. F. Gantmacher, Canonical representations of automorphisms of a complex semi-simple group, Mat. Sb. 5 (1 939), 101-142. (in English) 13. S. Gelbart, Automorphic functions on adele groups, Ann. of Math. Studies, no. 83, Princeton Univ. Press, Princeton, N. J., 1 975. 14. --, Elliptic curves and automorphic representations, Advances in Math. 21 (1 976), 235292.
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15. S. Gelbar t , Automorphic forms and, Artin's conjecture, Lecture Notes in Math. , vol. 627, Springer-Verlag, Berlin and New York, 1977, pp. 241-276. 16. S. Gelbart and H. Jacquet, A relation between automorphic forms on GL, and GL, Proc. Nat. Acad. Sci . U.S.A. 73 (1 976), 3348-3350. 17. P. Gerardin and J.-P. Labesse, The solution of a base change problem for GL, (following Langlands, Saito, Shintani), these PROCEEDINGS, part 2, pp. 1 1 5-133. 18. R. Godement, Fonctions automorphes et produits eu/eriens, Sem. Bourbaki (1 968/69), Expose 349 ; Lecture Notes in Math . , vol. 1 79, New York, 1 970, pp. 37-53. 19. R. Godement and H. Jacquet, Zeta-functions of simple algebras, Lecture Notes in Math. , vol. 260, Springer, New York, 1972. 20. R. Howe and I. I. Piatetski-Shapiro, A counterexample to the 'generalized Ramanujan con jecture' for (quasi-) split groups, these PROCEEDINGS, part 1 , pp. 31 5-322. 21. H. Jacquet, Automorphic Forms on GL(2). II, Lecture Notes in Math . , vol . 278, Springer, New York, 1 972. 22. , Generic representations, Non-Commutative Harmonic Analysis, Lecture Notes in Math. , vol. 587, Springer, New York, 1 977, pp. 91-101 . 23. , From GL, to GL., U.S.-Japan Seminar on Number Theory (Ann Arbor, Mich., --
1975).
--
24. , Principal £-functions of the linear group, these PROCEEDINGS, part 2, pp. 63-86. 25. H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Math. , vol. 1 14, Springer, New York, 1 970. 26. H. Jacquet, I. Piatetskii-Shapiro and J. Shalika, Construction of cusp forms on GL3, Univ. of Maryland Lecture Notes, no. 1 6, 1 975. 27. , Heeke theory for GL3, Ann. of Math. (to appear). 28. H. Jacquet and J. Shalika, Comparaison des representations automorphes du groupe lineaire, C. R. Acad. Sci. Paris Ser. A 284 (1 977), 741-744. 29. A. Knapp and G. Zuckerman, Classification of irreducible tempered representations ofsemi simple Lie groups, Proc. Nat. Acad. Sci. U.S.A. 73 (1 976), 21 78-21 80. 30. , Normalizing factors, tempered representations and L-groups, these PROCEEDINGS, part 1 , pp. 93-105. 31. J. P. Labesse and R. P. Langlands, £-indistinguishability for SL, Institute for Advanced Study, Princeton, N. J. (preprint). 32. R. P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Math., vol. 544, Springer, New York. 33. --, Euler products, Yale Univ. Press, 1 967. 34. , Representations of abelian algebraic groups, Yale Univ., 1968 (preprint). 35. , Problems in the theory of automorphic forms, Lectures in Modern Analysis and Applications, Lecture Notes in Math. , vol. 170, Springer, New York, 1970, pp. 1 8-86. 36. , Modular forms and 1-adic representations, Modular Functions of One Variable. II, Lecture Notes in Math., vol. 349, Springer, New York, 1973, pp. 361-500. 37. , On the classification of irreducible representations of real algebraic groups (preprint). 38. --, Base change for GL, : The theory of Saito-Shintani with applications, Notes, lnst. Advanced Study, Princeton, N. J., 1 975. 39. , On the notion of an automorphic representation, these PROCEEDINGS, part 1, pp. 203--
--
--
--
--
--
--
207.
40.
--
--
, Automorphic representations, Shimura varieties, and motives. Ein Miirchen, these part 2, pp. 205-246. 41. M. E. Novodvorsky, Theorie de Heeke pour les groupes orthogonaux, C. R. Acad. Sci. Paris Ser. A, 285 (1 975), 93-94. 42. , Automorphic £-functions for the symplectic group GSp(4), these PROCEEDINGS, part 2, pp. 87-95. 43. M. E. Novodvorsky and I. I. Piatetskii-Shapiro, Rankin-Selberg method in the theory of automorphic forms, Proc. Sympos. Pure Math., vol . 30, no. 2, Amer. Math. Soc. , Providence, R. I., 1 977, pp. 297-301 . 44. I. Piatetski-Shapiro, Euler subgroups, Lie Groups and Their Representations, Proc. Summer School, Budapest, 1 97 1 . PROCEEDINGS,
--
AUTOMORPHIC L-FUNCTIONS
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45. --, Zeta-functions of GL., Notes, Univ. o f Maryland, College Park, Md. 46. --, Converse theorem for GL3, Notes, Univ. of Maryland, College Park, Md. 47. , Multiplicity one theorems, these PROCEEDINGS, part 1, pp. 209-212. 48. F. Rodier, Les representations de GSp(4, k), oil k est un corps local, C. R. Acad. Sci. Paris 283 (1976), 429-431 . 49. H . Saito, A utomorphic forms and algebraic extensions of number fields, Lecture Notes in --
Math, vol. 8, Kinokuniya Book Store Co., Ltd., Tokyo, Japan, 1975. 50. J.-P. Serre, Modular forms of weight one and Galois representations (prepared in collabora tion with C. J. Bushnell), Proc. Durham Sympos. Algebraic Number Fields (London), Academic Press, New York, 1977, pp. 193-268. 51. F. Shahidi, Functional equations satisfied by certain L-functions, Compositio Math. (to appear). 52. J. A. Shalika, The multiplicity one theorem for GLm Ann. of Math. (2) 100 (1974), 171-193. 53. D. Shelstad, Characters and inner forms of a quasi-split group over F (to appear). 54. G. Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. (3) 31 (1975), 79-98. 55. T. Shintani, On li/tings of holomorphic automorphic forms, U.S.-Japan Seminar on Number Theory, (Ann Arbor, Mich., 1975). 56. --, On liftings of holomorphic cusp forms, these PROCEEDINGS, part 2, pp. 97-1 10. 57. A Silberger, The Langlands classification /or reductive p-adic groups (to appear). 58. T. A. Springer, Reductive groups, these PROCEEDINGS, part 1 , pp. 3-27. 59. R. Steinberg, Regular elements of semi-simple algebraic groups, Inst. Hautes Etudes Sci. Pub!. Math. 25 (1965), 49-80. 60. J. Tate, Number theoretic background, these PROCEEDINGS, part 2, pp. 3-26. 61. J. Tits, Reductive groups over local fields, these PROCEEDINGS, part 1, pp. 29-69. 62. J. B. Tunnell, On the local Langlands conjecture for GL(2), Thesis, Harvard Univ. 63. , Report on the local Langlands conjecture for GL,, these PROCEEDINGS, part 2, pp. 1 35-1 38. 64. A. Wei!, Ueber die Bestimmung Dirichletscher Reihen durch Funktionalg/eichungen, Math. Ann. 168 (1967), 149-167. 65. , Dirichlet series and automorphic forms, Lecture Notes in Math., vol. 1 89, Springer, New York, 197 1 . --
--
THE INSTITUTE FOR
AovANCED STUDY
Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 63-86
PRINCIPAL L-FUNCTIONS OF THE LINEAR GROUP
HERVE JACQUET Regard the group Gn =; GL(n) as an algebraic group over some local or global field F. Then LG� = GL(n, C). Let r n denote the natural representation of this last group on Cn ; the £-functions attached to rn play a central role in "Langlands philosophy". We review certain aspects of their theory, taking into account recent results on classification of representations. 1. Local nonarcbimedean theory. We let F be a local nonarchimedean field. When this does not create confusion, we write Gn for Gn(F) and use a similar notation for any F-group. (1 . 1) Let 1C be an admissible representation of Gn on a complex vector space V; we denote by 1t the representation contragredient to 1C, V the space on which it operates, and < · , · ) the canonical invariant bilinear form on V x V. The represen tation 1t is admissible, and irreducible if 1C is. Moreover ('it)- � 7C. The functions (v E V, ii E V) (1 . 1 . 1) g f-------> (1C(g)v, ii) and their linear combinations are the coefficients of 1C. Clearly if/is a coefficient of
1C, then the functionfV defined by ( 1 . 1 .2)
is a coefficient of 'it. Let M(p x q, F) be the space of matrices with p rows, q columns and entries in F; denote by .?(p x q, F) the space of Schwartz-Bruhat functions on M (p x q, F). If/ is a coefficient of 1C, f/J is in .?(p x q, f), and s in C set ( 1 . 1 . 3)
Z (f/J, s,J) =
J f/J(g) j det gj • f(g) dxg.
The integral is extended to the group Gn and dx g is a Haar measure on this group. Below (Propositions ( 1 .2), ( 1 .4)) we state again the results of [R. G.-H.J.]. (1 .2) PROPOSITION. Suppose 1C is irreducible. (1) There is s0 such that the integrals (1 . 1 . 3) converge absolutely in the half-plane Re(s) > s0• (2) If the residual field of F has q elements, then the integrals represent rational AMS (MOS) subject classifications
(1 970). Primary 12A85, 22E50, 22E55, 1 2A80, 1 2B35. © 1 979, American Mathematical Society
63
64
HERVE JACQUET
functions of q-s ; as such, they admit a common denominator which does not depend on f or fP. (3) Let ¢ =F 1 be an additive character of F. There is a rational function r(s, 11:, ¢) such thatfor all coefficientsf of 11: and all fP Z(fPA , 1 - s + t (n - 1),JV) = r (s, 11: , ¢) Z(fP, s,j),
where fP/\ denotes the Fourier transform offP with respect to ¢ .
In (3) rpA is defined by (1 .2.4)
fPA (x) =
J fP(y) ¢ [tr(yx)] dy,
where dy is the self-dual Haar measure on M(n x n, F). On the other hand the left-hand side in (3) has a meaning by (1) and {2) applied to it. Besides we could formulate Proposition (1 .2) for the pair (n, it) rather than for 11: , the situation being symmetric in 11:, it ; this symmetry will be apparent at each step of the proof anyway. ExAMPLE (1 .2.5). Suppose 11: is cuspidal. This condition is empty and therefore always satisfied if n = I ; in this case 11: is just a quasi-character of Fx and we know Proposition (1 .2) from [J.T. 1] or [A.W. 3] . If n > 1 , the coefficients of 11: are compactly supported modulo the center of Gn ; we can then exploit this fact to obtain Proposition {1.2), the proof being essentially the same as in the case n = 1
([R.G.-H.J.], [H.J. 1])
(1 .3) The proof of Proposition (1 .2) will be given in §2. For the time being, we derive some simple consequences of Proposition (1 .2). If/and fP are as above and h is in G"' then the functions/1 and fP 1 defined by fi(g) = f(gh), fP 1 (x) = fP(xh) (1.3.1) are functions of the same type. Moreover if we assume (1 .2. 1), ( 1 .2.2), then :
( 1 . 3.2) It follows that the subvector space J(n) of C(q-•) spanned by integrals (1 .3.3)
Z(fP, s + f(n - 1), f)
is in fact a fractional ideal of the ring C [q-s, q•]. Furthermore if we take/such that
f(e) =F 0 and rp with support in a small enough neighborhood of e, we find that (1 .3.3) is actually independent of s and =F 0. Thus l(n) contains the constants ; in other words it admits a generator of the form P(q-• )- 1 with P E C [X]. We will
normalize P by demanding that P(O)
=
1 and will set
(1 .3.4) ExAMPLE ( 1 . 3.5). If 11: is supercuspidal then L(s, n) lxl1• Then (loc. cit.) L(s, 11:) = (1 - q-s-t)- 1 .
=
Assume now Proposition (1.2) for (n, it). Because J(n) (which justifies the notation). Set (1.3.6) e(s, 11:, ¢) Then (1 .2.3) reads
= r(s, 11:,
1 unless n =F
= 1
and n(x)
=
0, the factor r is unique
f/J)L (s, n)/L(l - s, it).
PRINCIPAL L-FUNCTIONS
(1 .3.7)
Z (l/J/\, 1 - s +
f(n -
65
1),JV)fL(1 - s, ft) = e(s, 11: , cjJ) Z {l/J , s +
(
!n
- 1),/)/L(s, 11:).
In view of the definition of the £-factors, this implies that both e(s, 11: , ¢) and its inverse are in C[q-•, q•]. Thus e(s, 11: , ¢) is a monomial in q-•. Let also w be the central quasi-character of 11: , that is to say the homomorphism (I) : px --+ ex such that 11:(a) = 11:(a · 1 ,.) = w(a)1 v for a e P .
(1 .3.8)
If the assertions of(l .2.3) are true for one choice of ¢, they are true for any other choice ¢'. In particular (1 .3.9)
e(s, 11: , ¢') = w(b) l b l <•-l l2l " e(s, 11: , ¢) if cjJ '(x) = cjJ(xb).
By exchanging 11: and 1r we find also {if Proposition (1 .2) is true for (11:, 1r)) : e( I - s, ft, cjJ)e(s, 11: , ¢) = w( - 1).
(1 .3. 10)
Moreover let us denote, for any quasi-character X• by 11: ® X the representation g ...... 71:(g)x(det g) of G,. on V. Let also aF or a be the module of F. Then : (1 .3. l l) (1 .3. 1 2)
L(s, 11: ® a t) = L(s + t, 11:),
e(s, 11: ® at , ¢) = e(s + t, 11: , ¢).
The r-factor satisfies similar identities. Again F is a local nonarchimedean field. (2. 1) Let P be an F-parabolic subgroup of G,.. To be specific we take P to be standard of type (nh n2, · · · , n,) , so that the matrices p in P have the form 2. Induced representations.
ml
(2. 1 . 1 )
mz
U;j
m; e
p= 0
G,;
.
m,
We also set U = Up (unipotent radical of P), M = Mp = PfUp. Most of the time we identify M to the subgroup of p e P such that u;i = 0. Then, in a natural way, (1 � i � r ) . (2. 1 .3) If a,. , 1 � i � r, is an admissible representation of G,;, we can form the representa tion a = x a,. of M = P/U, regard it as a representation of P trivial on U, and induce it to G,. The representation (2. 1 .2)
(2. 1 .4)
e
= l(G, P; a) = l(G, P; a h az, · · · , a,)
66
HERVE JACQUET
that we obtain is admissible. Its contragredient � is equivalent to f
(2. 1 . 5)
= /(G, P; 0') = J(G, P; 0' 1 > O'z,
···,
O' r) ·
More precisely let W be the space of q . Then the space V of e consists of all func tions F from G, to Wwhich are smooth and satisfy F(pg) = op(p)l i ZF(g),
(2. 1 .6)
where op is the module of P. The group G,. operates by right-shifts on V. The space V' of f is defined similarly in terms of the space W of 0' . For F e V, F e V' the function tp(g) = ( F(g), F(g)) satisfies tp(pg) = op(p)tp(g).
(2. 1 .7)
Denoting by tp 1------>
(2. 1 .8)
J
P\G
tp(g) dg
a positive right-invariant form on the space of continuous functions satisfying (2. 1 .7), we may set (F, F) =
(2. 1 .9)
J
P\G
( F(g), F'(g)) dg
and this is a pairing which allows us to identify V' to Let RF be the ring of integers of F and let
V and e '
to �.
(2. 1 . 10)
Since G,.
=
P Km we may take (2. 1 .8) to be ·
(2. 1 . 1 1)
1------>
tp
JKn tp (k) dk.
(2.2) It will be convenient to have a description of the coefficients of e in terms of those of (J. So let/be the coefficient of e determined by Fe V, F E v = V' ((1 . 1 . 1)) ; then the function H: G,. x G, -+ C defined by H(gi> g2) = (F(g 1 ), F(g2)) satisfies the following conditions : (2.2. 1) (2.2.2)
H(ulmgh Uzmgz) = op(m)H(gh gz) for g; E G,., U; E Up, m E Mp;
for any gh g2 the function m is a coefficient of (J ® o}(Z ; H is K,.
(2.2.3)
Moreover/is given by (2.2.4)
f(g) =
J
P \G
x
1------>
H(mgh g2)
K, finite on the right.
H(hg, h) dh =
JK H(kg, k) dk.
Conversely, if H is any function satisfying (2.2. 1)-(2.2.3), then the function f defined by (2.2 . 4) is a coefficient of '/C. The coefficientfV of � is then given by
PRINCIPAL £-FUNCTIONS
67
(2.2.5) H satisfies (2.2. 1 )-(2.2.3) with a instead of q. (2.3) Before formulating the main theorem of this section we remark that if 11: is
and
an admissible representation which is perhaps not irreducible but admits a central quasi-character ((1 .3.8)), then the assertions of(1 .2) make sense for (11: , :it'), although they may fail to be true. Below we assume that each (J; admits a central quasi character ; thus � admits also a central quasi-character. (2.3) PROPOSITION. With the notations o/(2. 1 ) suppose the assertions of (l.2) are true for each pair (q; , a,.). Then they are true for (�. � ) . Moreover :
I (�) = IJ J(q;) , r(s, �. cp) = IT r(s, (J;, cp). i
i
The proof will occupy (2.4)-(2.6). (2.4) Letfbe given by (2.2.4). Then, exchanging the order of integrations, we get
JK dk JGn f/J(g) I det g j•+Cn-1) 12H(kg, k) dx g = J dk J f/J(k- 1 g) j det g j•+en-1 ) 12 H(g, k) dxg K Gn = J dk' J f/J(k- 1 pk')j det pj •+Cn-1) 12 H(pk', k) d1p.
Z(f/J, s + f(n - 1 ) , f) = (2.4. 1)
dk
P
If moreover we define (p being as in (2. 1 . 1)) (2.4.2) (2.4.3)
1/f(m�o m2, . . . , m, ; k, k') =
J f/J(k-lpk') ® du,.i,
h(m�o m2, . . . , m, ; k, k') = H(pk ' , k)o pl12(p),
then, after integrating in the variables u,.i, integral (2.4. 1) can be written as (2.4.4)
J dk dk' J 1/f(m�o m2, . . . , m, ; k, k')h(m� o . . . , m, ; k, k') . n j det m,. j•+ Cn;-1)/2 ® dxm,. . '
Because of the K-finiteness of the functions involved, for f/J and H given, (2.4.4) can be written as a sum over a finite set of K x K of the inner integrals. But for given k and k', 1/f(m�o mz, . . . , m, ; k, k') is a finite sum of products n ..w,.(m;) with If!,. E !/(n,. x n,., F) ; similarly h(m�o m2, • · · , m, ; k, k') is a finite sum of products n ,. .fi (m,.) where .fi is a coefficient of (J; ((2.2.2)) . Thus (2.4. 1) is a finite sum of products (2.4.5)
IJZ(f/J,., s + !(n,. j
-
1),/;),
where f/J,. is in !/(n,. x n,., F) and .fi is a coefficient of q ,. . So we have proved (2. 1 . 1 ), (2. 1 .2) for �. and even the inclusion /(�) c n .. /(q;). (2.5) Now we prove the reverse inclusion. Starting with an expression (2.4. 5) we can certainly find f/J E !/(n x n, F) so that with the notations of (2. 1 . 1) :
68
HERVE JACQUET
J f/J(p) ® duii = I,J f/J,{m;).
(2.5.1)
Moreover there are two K-finite functions r; and r; ' on K such that
JJ f/J(k-lxk')r;(k)r;'(k') dk dk' = f/J(x).
(2. 5.2)
Then (2.4.5) is equal, for large Re s, to (2.5.3)
J f/J(k-lpk')i det p i •+Cn-1) 12 I,J .fi(m;)o}t2(p)r;(k)r;'(k') d1p dk
dk' .
Let dh be the normalized Haar measure on the compact group K n P. Changing k to hk, k' to h' k' with h and h' in K n p and then integrating over (K n P) X (K n P), we find that (2.5.3) is equal to (2. 5.4)
J dh dh' J f/J(k-lh-lph' k')i det P i s+Cn-1) 12 ·
IT .fi(m;)o lf,2(p)r;(hk)r;'(h'k') d1p dk dk'. j
Now change p to hph'- 1 and write h, h' in the form (2. 1 . 1) with h; E Kn; > h; E Kn; instead of m; . We see that (2. 5.4) is equal to
J f/J[k-1pk']Hl (p, k, k')i det p i •+Cn-1) 12 d1p dk dk'
(2.5.5)
where H1(p, k, k')
=
JJ r;(hk)r;'(h'k') l} f.{h;
·
m; · h;-1 ) o),12(p) dh dh' .
It is easily verified that there is a function H satisfying (2.2. 1)-(2.2.3) such that (p E P, k E K, k' E K'). H(pk, k') =: H1(p, k, k') If I is the corresponding coefficient of � {(2.2.4)), then, comparing (2.5.4) with (2.4. 1 ), we find that IT Z{f/J;, s + !(n; 1), f.-) = Z{f/J, s + t(n 1), !) . j
-
-
So we have proved that /(�) = 0 ;I(u;) . (2.6) Now we pass to the functional equation. In (2.4. 1), let us replace 1 by 1v and f/J by f/J". Then H is replaced by ii {(2.2. 5)) and the identity (2.4. 3) gives : (2 . 6 . 1 ) h(m!l, mz-1, · · · m;-1 ; k', k) = ii(pk, k')o-p112(p). ,
Similarly (2.4.2) gives (2.6.2)
lf!"(mh m2,
···,
m, ; k', k)
=
J f/J1''(k-1pk') ® du;i,
where lf!" denotes the Fourier transform of 1/! with respect to each one of the vari ables m; . Instead of (2.4. 1) and (2.4.4) we have now for Re s large enough
69
PRINCIPAL £-FUNCTIONS
Z(r[JA, s + -!-(n - l),JV) =
(2.6. 3) =
J dk dk' S r[JA(k-1pk') J detpJ s+(n-1l 12H(pk', k) d1p J dk dk' J lf!A(mt. m2, · · · m, ; k', k) ,
" h(mll , mz1, · m;1 ; k', k) n j det m; J s+(n;-1)/Z ® d X m; . " "
i
,
The functional equations for the representations O"; and the remarks we made on h and If! imply now Z(rtJA, I - s + -t (n - l), JV) = fl r(s, a; , sf!)Z(f/J, s + -t (n - 1), /) . i
This concludes the proof of (2.3). (2.7) Again let the notations be as in (2.3). From (2.3) applied to � and � we get (2.7. 1) L(s, �) = fiL(s, a,.) , L(s, �) £
=
fiL(s, 0',). £
Then the last assertion of (2.3) reads e(s, �. ¢)
(2.7.2)
=
fl e(s, 0"; , ¢). i
Let 1C be an irreducible component of � ; then any coefficient of 1C is a coefficient of � and it is an irreducible component of �- It follows that Proposition ( 1 .2) is true for (1C, it). Moreover (2.7.3) I(1e) c /(�), !( it) c /(�), r (s, 1e, ¢) = r(s, �. ¢). Thus there are two polynomials P, P in C [X] such that (2.7.4) L(s, 1e) = P(q-s)L(s, n L(s, it) = P(q-s)L(s, �), P(O) = P(O) = 1 . Note that P(q-s) divides L(s, �) -1 in C[q-s]. Moreover 1C) L(s, �)P(q-s) (s, �. ¢) e(s, 1C , ¢) - r (s, 1C , ¢) L(1L(s, - s, it) - r L(l - s' �) P(q- l+s) (2.7.5) - e(s, �. ¢) P(q-s)s . P(q-l+ ) _
_
Since the e-factors are units of the ring C[q-s , qs], we find that
P(X) = Il (1 a;1qX) . (2.8) In general, if 1C is an arbitrary irreducible admissible representation of G" ' it is a component of an induced representation of the form (2. 1 .4) where the 0"; are cuspidal. Since (1 .2) is true for each O"; ((1 .2.5)), it is true for (�, �) ((2.3)) and thus for (1e, it). So (1 .2) is completely proved. 3. Computation of the £-factor. We have established ( 1 .2) for any irreducible admissible representation 1C of Gn ; we also know r(s, 1e , ¢) ((2.7.2)) . It remains to compute the £-factor. (2.7.6)
-
70
HERVE JACQUET
(3. 1) Square-integrable representations. We have seen that if 71: is cuspidal then L(s, 71:) = 1 unless n = 1 and 71: = at, in which case (3. 1 . 1)
We now compute L(s, 71:) when 71: is essentially square-integrable. Recall that 71: is square-integrable if it admits a central character w and its coefficients (which transform under w) are square-integrable modulo the center. A representation 71: is essentially square-integrable if it has the form 71: = 7l:o ® at where 7l:o is square integrable and t real. We now review the work of Bernstein and Zevelenski on the construction of such representations. Let r be a divisor of n so that n = jr. Let P be the standard parabolic subgroup of Gn of type (j, j, . . · , j). Let also -r be a cuspidal representation of G, . Set for 1 ;;;; i ;;;; r, a; = -r ® ai- 1 , Then the induced representation (3. 1 .2)
admits a unique essentially square-integrable component 'Jl:, All essentially square integrable representations 71: are bbtained in this way and r, -r are uniquely deter mined by 71:. PROPOSITION (3. 1 .3). With the above notations L(s, 71:) = L(s, -r) . Indeed suppose L(s, -r) = 1 . Then L(s, a;) = 1 and by (2.7.1) L(s, �) = 1 . By (2.7.4) we have then L(s, 71: ) = 1 . Suppose now L(s, -r) # 1 . Then j = 1 , r = n, 1: = a.t and our assertion is nothing but Proposition (7. 1 1) of [R.G.-H.J.]. REMARK (3. 1 .4) . If 71: is square-integrable then the poles of L(s, 71:) are in the half-plane Re(s) ;;;; 0. This follows from (3. 1 .3) or from Proposition ( 1 . 3) of [R.G.-H.J.].
(3.2) Tempered representations. Let again P be a parabolic subgroup of type (nb n2, . . . , n,). Let now a,, be a square-integrable representation of Gn,· Then it is
well known that the induced representation (3.2. 1)
is irreducible (cf. for instance [H.J. 2]). The irreducible representations of this type are precisely the tempered ones. Note that if 71: is equivalent to another representa tion ' (3.2.2) 71: = I(G, P ' ; a � , a � , . . · , a � , ) where the ai are square-integrable, then P' and P are associate. More precisely
r = r ', P = MUp, P = M' UP' , and there is an inner automorphism of Gn taking M to M' and a = x a; to a' = x a;. ; in other words one passes from (M, (a;)) to ( M', (ai)) by a permutation of the diagonal blocks of M and M'. Conversely if
P' and the a;. are related to P and the a; in this way, then 71:' is equivalent to 'Jl:. By (2.3), L(s, 71:) = IT L(s, a;), L(s, 7t) = IT L(s, ii;), (3.2.3)
i
t"
.s(s, 71:, ¢) = Il .s (s, a, , ¢). i
71
PRINCIPAL £-FUNCTIONS
REMARK (3.2.4). It follows from (3.2.3) and (3. 1 .4) that the poles o f L(s, n ) for n tempered are contained in the half-plane Re(s) � 0. A representation n is said to be essentially tempered if it has the form n = no ® at where no is tempered and t is real. Then (3.2.5) L(s, n) = L(s + t, n0). (3.3) Langlands construction. We now review the work of Silberger and Wallach which extends the results of Langlands to the p-adic case. Let Q = MQ UQ be a parabolic subgroup of type (PI> p2 , . . . , p,) and for each i, I � i � r, -.,. an irreducible essentially tempered representation of Gm;· Set -. = x -.,. . Then -.,. = -.,., 0 ® at; where -.,., 0 is tempered and t,. real. We assume that (3.3.1) t 1 > t2 > . . . > t,.
Then the induced representation (3.3.2) 'lJ = l(G, Q ; -. b -.z, . . . , -. ,) has a largest proper subrepresentation 'l}' (possibly {0}) ; the irreducible representa tion n = 'l}/'1)' is noted (3.3.3) Every irreducible representation n has the form (3.3.3) where P (standard) and the -.,. are uniquely determined. The space V' of '1)1 may be described explicitly. Let W be the space of -. and W the space off. Then V' is the space of F in the space V of t; such that (3.3.4)
J (F(ug),
v)
du
=
o,
The integral is absolutely convergent and extended to ()
=
()Q
=
UQ, where
Q = t Q is the parabolic subgroup opposed to Q. It easily follows that the coef ficients of n can be obtained by integrals similar to (2.2.4). Namely let H: Gn x --->
C be a function satisfying the following properties : H(u 1 mgb u2mg2) = H(gb g2), u 1 E UQ, u2 E DQ, m E MQ ; (3.3.5) Gn
(3.3.6)
for any gb g2 the function m ,_. H(mgb g2) is a coefficient of ,. ® o}j2 ;
H is Kn x Kn finite on the right. (3.3.7) Then the function f defined by the convergent integral
(3.3.8)
f(g) =
JMIG H(hg, h) dh = JU- xK H (ukg, k) dk du
is a coefficient of n and all coefficients of n can be obtained in this way for suitable
H ( see (3.6.6) and (3.6.7) for convergence questions) . Instead of Q we may consider Q. Then if condition (3.3. 1) is replaced by the
similar condition with the inequalities reversed, the representation analogous to (3.3.3) is defined. For instance the quotient
72
HERVE JACQUET
(3.3.9) of the induced representation (3.3.10) is defined. Its coefficients are given by integrals (3.3.8) where H satisfies (3. 1 .5) (3. 1 .7) with (Q, i') instead of (Q, -r) . In particular suppose f is the coefficient of 7C defined by (3.3.8) where H satisfies (3.3.5)-(3.3.7) for (Q, -r). Then (3.3. 1 1) with H(gb g2)
JV(g) = = H(g2, g 1 ) .
jV is a coefficient of 7C' and
JMIG H(hg, h) dh,
But H indeed satisfies (3.3.5)-(3.3.7) for (Q,
i').
Thus
7C' = if:. (3.3. 1 2) Of course Q is also conjugate to the standard parabolic subgroup Q' of type (n" nr-b . . · , nl ) so that (3.3. 1 3) (3.4) THEOREM. Let the notations be as in (3.3). Then L(s, 7C)
=
IT L(s, ..-,.) , L(s, if:)
c(s, 7C, ¢)
=
=
IT L(s, i';),
IT c(s, -r; , ¢).
It is enough to prove the assertion relative to L(s, 7C) ; indeed the one for L(s, if:) can be obtained by exchanging 7C and if: and the last assertion follows then from (2.7) and (cf. (2.3)) r(s, r;, ¢) = IT ; r(s, ..-,., ¢). The proof will occupy the rest of this section and (3.5). By (2.3) and (2.7) there are polynomials P and P such that L(s, 7C) = P(q-s) IT L(s, ..-,.) , L(s, if:) = P(q-s) IT L(s, i';) .
It follows from (3.2.4) and (3.3.1) that L(s, ..- 1 )- 1 and IT £(1 - s, i';) prime. Thus P(q-s) is prime to L(s, ..-1 )- 1 ((2.7.5)). So if P # 1 then (3.4. 1)
-
1
are relatively
IT L(s, -r;)
2�i�r
is not in 1(7C). Therefore it will suffice to show that (3.4. 1) is indeed in 1(7C). (3.5) The identity L(s, 7C) = IT L(s, -r;) (3.5. 1) l�i�r
is trivial for r = 1 . Assuming it is true for r - 1, let us prove it for r. As we have just seen, it suffices to show (3.4. 1 ) is in 1(7C). Set n 1 = P h n2 = p2 + · · · + p,. Let Q' be the parabolic subgroup of type (p2, · · · , p ,) in G"2. Then the representation (3.5 . 2) is defined. Set also 0" 1 = -rb O" = 0" 1 x O"z and let P be the parabolic subgroup of type (nb n2) in Gn. Then 7C is actually a quotient of
PRINCIPAL £-FUNCTIONS
73
(3.5.3) More precisely it is easy to see that the coefficients of 77: are given by the integrals (3.3.8) where H is any function satisfying (3.3.5)-(3.3.7) with (P, a) instead of (Q, -.). If/is a coefficient of 77: defined in this way, then Z(�, s + }(n - 1),/) (3.5.4) = dk du l det g i•Hn-l)/2 �(k-l u-lg)H(g, k) dxg, G. where u is integrated over (] = Up. This is also
JJ
Js
J
JJ l det , 1 det �; �J-1 12 [(; �J k', J . JJ [ (: � ) ]
dk dk'
. a
(3. 5.5)
m l i •Hn - ) /2 1
l
n
� k-
1
1 m2
l
m2 1 •+
xy
k' dx dy .
Here m; ranges over Gn;- We are going to see that given �2 e 9'(n2 x n2, F) and a coefficient /2 of o-2 there are H and � so that Z( �2, s + t(n2 - I), /2) is equal to (3. 5.5). Since, by the induction hypothesis, L(s, o-2) is equal to (3.4. 1), it will follow that (3.4. 1) belongs to 1(77:) which will conclude the proof. There is a coefficient/1 of 77: 1 and a �1 in 9'(n1 x n1, F) with support in a compact neighborhood of e such that (3.5.6) There are also �12 e 9'(n 1 x n2, F), �2 1 e 9'(n2 x nh F), with supports in a neigh borhood of 0 so that the function � e 9'(n x n, F) defined by m (3.5.7) � 1 y = � � (m i )�I 2(Y)�21(x) �2(m2)
(
m2
x
)
satisfies (3.5.8) There are also two K-finite functions � and e on K such that
JJ
(3.5.9) Then : Z(�2,
s+ =
(3.5.10)
-!(n
�(k-lzk') � (k)f(k') dk dk' = �(z).
- 1 ) , /2 )
JJ l det
l
m l •+ (n ,-l ) /2
· dx m1dx m2 ·
JJ
JJ { (; k
-l
l det m2 1•+
t!zl m2
� xy) k' ] dx dy.
HERVE JACQUET
74
The proof is then finished as in (2.5). Namely let dh and dh' be the normalized Haar measures on K n P and K n P respectively. For h E K n P, h' E K n P set h
=
hi =
(�1 �J
* \. (0h;_ h;) '
then (3.5.10) is also equal to
JJ dk J J l det m1 l•+
(3.5. 1 1) ·
JJ ,.,[''lk-l(mxm1 1 m2 Y+ xy) k'] dx dy
where
[(
) ] (� �J 1 2JJ �(hk)�'(h'k')f1(h1m1h;_-1)fz(hzmzh�-l) dh dh' .
l 0 H1 m 0 mz , k ' k' (3.5. 1 2) = op 1 -
There is H satisfying (3.3.5)-(3.3.7) for (P, u) so that 0 m 0 (3.5. 1 3) H Om 1 m2 k' ' k = H1 0 1 mz ' k' k' · Comparing (3.5. 1 1) to (3.5.5) we obtain our assertion. (3.6) Unramified representations. Suppose 7C is unramified, that is contains the unit representation of K. Then Bn denoting the parabolic subgroup of type (1 , 1 , 1 , . . . , 1), there are u; e C so that 7C is the unique unramified component of
[(
(3.6. 1) We may even assume u 1 � n-tuple of the f.J.; in the form
) ]
u2 �
• • •
)
[(
� un .
]
Changing notations we may write the
where n = n 1 + n2 + . . . + n , 1J{ = a1i).{, A.{ is a character, ti is real and t1 > t2 > · · · > t,. Then 7: i = l(Gni• Bn i ; lJ { , lJ� , , lJ�) is an essentially tempered unramified representation of Gni and in fact 7C = J(G, Q ; 't:I o 7: z, . . . , 7: ,), (3.6.2) if Q is the parabolic subgroup of type (n 1o n2, . . , n ,) . This follows from an explicit computation : if H satisfies the conditions (3.3.5)-(3.3.7) for (Q, X 7:i) and H(k, k' ) = 1 then • • ·
•
(3.6.3)
J H(u, e) du
=f.
o.
Thus the "J-component" ofrJ = l(G, Q; 1: 1o 7:z, . . . , 1:,) is unramified ; since � � 1J we arrive at (3.6.2).
PRINCIPAL £-FUNCTIONS
Thus (3.6.4)
75
L(s, 11:) = ITL(s, p;), L(s, it) = ITL(s, pi1), j
e(s, 11: , ¢) = Ile(s, p;, ¢) ( = I ifthe exponent of¢ is zero). j
For more precise results see [R.G.-H.J., §6]. REMARK (3.6.5). Let the notations be as in (2.3). Suppose U ; is irreducible un ramified. Then � admits a unique unramified component 11: . It easily follows from (3.6.4) that L(s, 11:) = TI L(s, u;) with similar relations for L(s, it) and e. REMARK (3.6.6). Let 11: be a tempered representation of Gn. Let Pn = I(Gm Bn ; 1 , 1 , · · · , 1) and Bn be the spherical function attached to p. Then any co efficient of 11: is majorized by a multiple of Bn. Let the notations be now as in (3.3). Then the coefficients of 7:; are majorized by a multiple of Bn; ® at;. Moreover if H satisfies (3.3.5)-(3.3.7) for (Q, 1:) then IHI � cH0 where H0 satisfies (3.3.5) and H0 (mk, k') = o}{Z(m) ij Bn;(m;) . I
Thus as far as convergence is concerned we may replace 7:; by Pn; ® at; and H by H0 > 0. Together with the next lemma, this takes care of all problems of conver gence. LEMMA (3.6. 7). If {) is a bounded set of Y(n x n, F), then there is (/)0 � 0 in Y(n x n, F) so that (/)0(k 1 xk2) = (/)0(x) for k; E K and 1 (/) 1 � (/)0 • We may assume that {) contains also all functions x �---+ (/)(k1 xk2) with (/) E {). Then there is (/) 1 � 0 so that 1(/) 1 � (/) 1 for all (/) in {) [A.W. 1 , Lemma 5]. It suffices to take (/)0(x) = JJrp1(k1 xk2) dk1 dk2 where dk; is the normalized Haar measure on K. (3.7) According to "Langlands' philosophy" there should be a "natural bijec tion" u �--+ 11: (u) between the n-dimensional semisimple representations of the Weil Deligne group w;.. and the irreducible admissible representations of Gn(F). More over, one should have for 11: = 11:(u) L(s, 11:) = L(s, u) , 1t" (u) = 11:(0'),
e(s, 11:, ¢) = e(s, u , ¢), 11: (u ® x) = 11: (u) ® x ·
It is clear that if the map u �---+ 11: (u) could be defined for the irreducible representa tions of the Weil group, then the conjecture would be proved (cf. §5 below). In §§4, 5 the ground field F is local archimedean. (4. 1) We let Kn be O(n, R) if F = R and U(n) if F = C. We donote by @n the Lie
4. Local archimedean theory.
algebra of the real Lie group Gn(F). We consider only admissible representations of the pair (@" ' Kn) {as in [N.W.]) although we will often allow ourselves to speak of a representation of the group Gn(F) ; in any case, we assume the reader to be thor oughly familiar with the relation between representations of the group Gn(F) and of the pair (@m Kn). Let 11: be an admissible representation of (@"' Kn) ; then one can define the con tragredient representation it, the coefficients of 11: , and its central quasi-character, even though 11: is not a representation of the group (cf. [H.J.-R.L., §§5, 6]). Again if /is a coefficient of 11: thenjV ((1 . 1 .2)) is a coefficient of it.
76
HERVE JACQUET
As in the nonarchimedean case let Y(p x q, F) be the space of Schwartz func tions on M(p x q, F). We also introduce the subspace Yo = Y0(p x q, F) of functions (/) of the form (P being a polynomial), if F = R, 1J(x) = P(x,.i)exp( - n- L: xri)
1J(z) = P(z,.i, zii)exp( - 2n- L: z;Ai), if F = C. We can consider, for a given n-, the integrals ( l . 1 .3) where (/) is in Y(p x q, F) and f is a coefficient of n- . (4.2) PROPOSITION. Suppose n- is an irreducible admissible representation of (@m Kn). (1) There is s0 so that the integrals ( 1 . 1 .3) converge absolutely in the half-plane Re(s) > s0• (2) For (/J in Y0(n x n, F), the integrals are meromorphicfunctions ofs in the whole complex plane . More precisely they can be written as polynomials in s times a fixed meromorphic function of s . (3) Let ¢ # 1 be an additive character of F; there is a meromorphic function r(s, n-, ¢) so that,for all coefficients! of n- and all (/) E Y0(n x n, F), Z(1JA, 1 - s + l (n - l), JV) = r(s, n-, ¢)Z(1J, s, f). Again (/)!\ is defined by (1 .2.4). If (/) is in Y0(n
x
n, F) and ¢ is defined by
¢(x) = exp( ± 2in-x), if F = R, ¢(z) = exp[ ± 2in-(z + z)], if F = C, then (/)!\ is still in Y0 ; the left-hand side of (4.2.3) has then a meaning by (4.2.2) applied to i't. If ¢ is not given by (4.2.4) then some adjustment has to be made, that we leave to the reader (cf. (1 .3. 1 1), (1 .3. 12)). From now on, we assume ¢ is given by (4.2.4). EXAMPLE (4.2.5). If n = 1, then n- is just a quasi-character of px and (4.2) is proved in [J.T.] or [A.W. 2]. (4.3) The proof of Proposition (4.2) will be given below in (4.4). For the time being, we derive some simple consequences of (4.2). By Lemma (3.6.7), the convergence of (1 . 1 .3) for Re s > s0 is actually uniform for (/) in a bounded set ; thus, for Re s > s0, (4.2.4)
(4.3.1)
1J �--+ Z((!J, s, f)
is a distribution, depending holomorphically on s . Since Y0(n x Y(n x n, F), it follows that iff # 0 then there is at least a (/) E Yo � 0.
n , F) is dense in so that Z((/J, s, f)
Suppose (4.2.2) has been proved. Let L(s) be a meromorphic function of s so that, for all (/) in Y0, Z(1J, s + t(n - l ), f)/L(s ) is a polynomial ; let also /be the subvector space ofC[s] spanned by those ratios. By what we have just seen I # {0}. Moreover if (/) is in Y0, so is the function (/J' defined by (4.3.2)
77
PRINCIPAL £-FUNCTIONS
Let w be the central quasi-character of 70 ; if we differentiate with respect to t the identity (4.3.3)
Z((/), s + -! (n - 1),/) =
and then set t
J (/)(xe-1)f(x) idet xi •+
x
w
0, we arrive at a relation of the form a i= 0. (4.3.4) (as + b) Z((/), s + !(n - 1), f) + Z((/)', s + -! (n - 1), f) = 0, This shows that I is an ideal. Let P0 be a generator of I and L(s, 70) = P0(s) L(s). We see that the ratios Z((/J, s + -!(n - 1),/)/L{s, 70) are again polynomials, but this time they span C[s] as a vector space. Up to multiplication by a constant, these properties characterize L{s, 70). Assume now (4.2) proved for the pairs (70, :ir). Then r is uniquely determined and one can define a factor e as in (1 .3.5) ; then the functional equation (4.2.3) reads as in (1 .3.6). In view of the definition of the £-factors, this implies that both e(s, 70, ¢) and its inverse are in C[s]. Thus e(s, 70, ¢) is just a constant if cjJ is given by(4.2.4) ; otherwise it is an exponential factor ((1 .3.9)). REMARK (4.3.5). For the time being the L- and e-factors are defined up to multi plication by constants ; of course these constants are related since r is intrinsically defined. For n = 1 , one may take the factors to be those given in [J.T.l , 2]. REMARK (4.3.6). Relations (1 .3.9)- (1 .3. 12) apply to the archimedean case. (4.4). Let the notations be as in (2. 1), the ground field F being now R or C; it goes without saying that a; is now an admissible representation of (®n; • Kn,.). The induced representation .;((2. 1 .4)) can still be defined [N.W.] and its coefficients are given by the rule of (2.2), the only difference being that in (2.2.2) g; must be in Kn and in (2.2.3) H must be C"" on Gn x Gn . The remarks made before (2.3) apply and, as in (2.3), we have : PROPOSITION (4.4. 1). With the notations of (2. 1) suppose that each a; admits a central quasi-character and the assertions of (4.2) are true for each pair (a;, fi;). Then they are true for (�, �). Moreover, r(s, �. ¢) = TI ;r(s, a;, ¢) and one can take =
L(s, �) = TI L(s, f7;),
L(s, �) = TIL {s, a;), i
i
e(s, �. ¢) = Tie(s, a;, ¢). i
The proof is similar to the proof of (2.3). It suffices to observe that if (/) is in !/0(n x n ; F) the functions x �--+ (/)(k1 xk2), k; E Kno span a finite dimensional vector space of !/0 and for each k, k' in Kn the functions (2.4.2) belong to the space ® !/0(n; x n;, F). Notations being as in (4.4. 1 ), let 70 be an irreducible component of �. Then any coefficient of 70 is a coefficient of � and :ir is a component of �. Thus (4.2) is true for (70, :ir). More precisely there are polynomials R and R such that L(s, 70) = R(s) TI L(s, a;) , L(s, :ir) = R(s) TI L{s, fi;) (4.4.2) i
i
78
HERVE JACQUET
while (4.4.3) Since e(s, (4.4.4)
r(s, 11:, cp) = Tir(s, a,. , cp). i
11:,
cp) and e(s, a, cp) are constants R and R have the form R(s) = TI (s - s,), R(s) = cTI{l - s - s,). i
i
Finally any given irreducible admissible representation 11: of Gn is a component of some induced representation � as in (4.4. 1) with n,. = I . Since (4.2) is then true for each (a;, a,.) , it is true for (11:, :it') and (4.2) is proved. (4.5) PROPOSITION. For all s, L(s, 11:) #- 0. PROOF. Let 11: be an irreducible admissible representation of Gn so that 11: is an irreducible representation of some induced representation � as in (4.4. 1 ) with n,. = I . We first show that for any (/) in Y(n x n, F) the ratio (4.5.1) Z(f/>, s + f(n - 1), f)/L(s, 11:) continues to an entire function of s. Indeed let s0 be such that for Re(s) > s0 (resp. Re(s) < I - s0) the integrals Z(f/>, s + f(n - 1), f) (resp. Z(f/>, I - s + t(n - 1 ), /V)) converge absolutely. Select s 1 o s2 such that s0 < s 1 < s2 • Let also Q be a polynomial such that Q(s )L(s, �) has no pole in the strip 1 - s2 � Re(s) � s2• Let f/>; be a sequence of Yo converg ing to some element (/) of Y. Then, for a given f and a given polynomial R, the following limits are uniform in the domain s 1 � Re(s) � s2 : lim R(s)Z( f/>,., s + -!(n - 1 ), f) = R(s )Z( f/>, s + t(n - 1 ), f ), ( 4. 5.2) i-+oo
(4.5.3)
i, j-+oo
lim R(s)Z(f/>; -
f/)b
s+
t(n
- 1), f) = 0.
Similarly r/Jt;' - (/)If � 0 so that� uniformly in the domain I - s2 � Re(s) � 1 lim R(s)Z((/)If' - (/)If, 1 - s + t(n - I) , JV) = 0. (4.5.4)
St .
i, j-oo
Indeed by using repeatedly (4.3.4) one can reduce these assertions to the case R 1 ; they are then obvious. On the other hand, Q(s)Z(f/>,. - f/>i> s + !(n - 1), f)
;� �� �) Q(s)Z((/)If' - f/>J, 1 - s +
= e(s, �. cp) - 1 L( The classical formula
F(x + iy) r(x' - iy)
�
I Y I "'-x'
�
!(n
=
- 1), fV).
+ oo )
shows that Q(s)L(s, �)/L(l - s, �) is bounded by a polynomial in the previous strip. Thus we also find, uniformly for 1 - s2 � Re(s) � I - sb lim Q(s)Z(f/>,. - (/)b s + t(n - 1) , / ) = 0. (4.5.5) i, j-+oo
79
PRINCIPAL L-FUNCTIONS
Now fixj and let c for s 1 � Re(s)
�
>
0 be given ; choose N so that for i, j
�
J Q(s)Z(qJ; - qli, s + -!-(n - 1), / J �
c,
N
s2 or 1 - s2 � Re(s) � 1 - s 1 . But now
Q(s)Z(qJ; - qli, s + -!-(n - 1), / ) = Q(s)R(s)L(s, .;) where R is another polynomial. Since L(s, .;) decreases rapidly, on any vertical strips, we find that given (i, j) there is M so that, for j l m(s) l � M, (4.5.6)
J Q(s)Z(qJ; - qli, s + -!-(n - 1) , / )J �
s.
Thus by the maximum principle (4.5.6) is satisfied for i, j � N and l - s2 � Re(s) � s2• It follows that Q(s)Z( qJ;, s + -!-(n - 1 ),f) is a Cauchy sequence for the topo logy of uniform convergence on the strip 1 - s2 � Re(s) � s2• Its limit Z(s) is holomorphic on 1 - s2 < Re(s) < s2 and coincides with Q(s)Z(qJ, s + -!-(n - l ),f) on s 1 � Re(s) < s2• Since s2 is arbitrarily large the holomorphy of (4 . 5.1) follows. Suppose now L(s0, n") = 0 and / :f. 0. Then for each qJ, the meromorphic func tion Z( qJ, s + -!-(n - I), f) vanishes at s0 ; but if qJ has compact support contained in Gn then Z(qJ, s + -!-(n - 1), f) is defined for all s by the convergent integral (1 . 1 .3). So we find that (1 . 1 .3) vanishes for s = s0 and qJ E C�(Gn) ; therefore/ = 0, a contradiction. Thus L(s0, n) :f. 0. Q.E.D. COROLLARY (4.5.8). Suppose .; is as in (4.4. 1) and n is an irreducible component of .; . Let R and R be as in (4.4.2). Then if s; is a zero of order m of R (resp. R) it is a pole of order � m of L(s, .;) (resp. L(s, �) ). 5. Computation of the £-factor ; arcbimedean case. Recall there is a "natural bijection" A �--+ n{A) between the set of classes of semisimple representations of the Weil group Wp of degree n and the admissible irreducible representations of Gn(F) ([R.L.], [A.K.-G. Z] , [N.W.]). In particular n(A)- = n-0). (5. 1) THEOREM. For any A of degree n, let n be n(A). Then L(s, n)
L(s, A), s(s, n, ¢)
=
=
L(s, it) = L(s, h s (s, A, ¢).
The proof of this theorem will occupy all of §5. Since the £-factor has not been normalized, it would be more correct to say that one can take L(s, n) to be L(s, A) and that r (s, n, ¢) = s(s, A, cj;)L(l - s, � )/L(s, A). (5.2) Square-integrable representations. Suppose A is irreducible. Then n{A) is essentially square-integrable and conversely. More precisely, if n = 1 then A is a quasi-character of px , A = n, and our assertion follows from the definition of the factors L and c for A. Suppose n :f. 1 ; then n = 2 and our assertion has been proved in [H.J.-R.L., Theorem (1 3. 1), Lemmas ( 1 3.24) and (5. 1 7)] ; actually in this case, the proof is a refinement of (4.4. 1). (5.3) Tempered representations. Suppose A is unitary ; then n is tempered and conversely. More precisely A = EB A; where A; is irreducible of degree n; . Set rr; • '=
·
80
HERVE JACQUET
n(A;). Then n = l(G, P ; a1o a2, (nb nz , · · · , n,). By (4.4. 1)
(5.3 . 1 )
· · · , a ,)
where P is the parabolic subgroup of type
L(s, n) = IJ L(s, a;) = IJ L(s, A;) = L(s, A) .
The factors L(s, i) and e(s, n, ¢) are computed similarly so our assertion follows again. The case of the essentially tempered representations follows from (1 .3. 1 1), ( 1 .3 . 1 2), and the relation n(A ® x) = n(A) ® X · (5.3.2) REMARK (5.3.3). It follows from (5.3.1) that, when n is tempered, the poles of L(s, n) are in the half-plane Re(s) � 0. (5.4) General case. In general, we may write A = E9 p.;, p.; = p.;, o ® at; where p.;, o is unitary of degree p;, t; real. We set then -r;, o = n(p.;,0), -r; = n(p.;) so that by (5.3.2) -r; = -r;, o ® a1 i . We may assume (3.3 . 1 ) is satisfied. Then if Q is the para bolic subgroup of type (Pb p2, • • • , p,) the induced representation r; of (3.3.2) admits a unique irreducible quotient noted as in (3.3.3). That quotient is n = n(A) . By (4.4; 1) and (5.3) we already know that r(s, n, ¢) = IJ r(s, -r;, ¢) = t(s, A , ¢)L(1 - s, �)/L(s, A) . i
So it will suffice to compute L(s, n). For exchanging then A and � we will get L (s, i) (since 1r = n(�) ) and the t-factor from the r-factor. So it will suffice to show that L(s, n) = TI 1:;;; :;; , L(s, -r;). This is trivial if r = 1 . So we may assume r > 1 and our assertion true for r - 1 . A priori, we have L (s, n) = P(s) IJ L(s, -r;), L(s, i) = P(s) IJ L(s, f;)
where P and P are polynomials related by (4.4.4). Let s0 be a zero of order u of P. By (4.5.8) it is a pole of order � u of TI ;L(s, -r;) and by (4.4.4) a pole of order � u of TI ;L(l - s, f;). But L(s, -r1) and TI L( l - s, f;) cannot have a common pole ((5.3.2), (5.3.3), (3.3. 1)). Thus it will suffice to show that any pole of order u of TI ;;;:;2 L(s, -r;) = TI ;;;:; 2 L(s, p.,-) which is not a pole of L(s, -.1) is a pole of order � u of L(s, n). This will be proved in (5.5) and (5.6). (5.5) Let now the notations be as in (3. 5). So set n1 = PI > n2 = n - n1o a1 = n(p.1) = -r1o O"z = n(p.z EB • . . $ p.,), a = a1 x a2. By the induction hypothesis (5.5. 1 )
L(s, O"z)
=
2 �i� r
IT L(s, p.;).
Again n is a quotient of (3.5.3) where P = MU is the parabolic subgroup of type (nh n2). The coefficients of n are given by the absolutely convergent integrals (5.5.2)
f(g) =
JAnG H(hg, h) dh = JKx U H(ukg, k) dk
where D = Up, P = tp and H: Gn ing properties : (5.5.3)
_
x
Gn --+ C is any function satisfying the follow u1 e U, u2 e D, m e M;
PRINCIPAL £-FUNCTIONS
o}(2,
81
(5.5.4) for k; E Km the function m f-> H(mkb k2) is a coefficient of 0' (8) H is Coo and Kn x Kn finite on the right. (5.5.5) This being so iff is given by (5.5.2) then Z(
=
J l det md s+ Cn1-l l 1 det m21 s+ Cnz-ll/2Ji (m1) y 1 f2Cm2)
2
This multiple integral converges absolutely if Re(s) is large enough. We are going to show that (5.5.7) For any
0,
(5.5.8) Then there is a function H satisfying (5.5.3) -(5.5.5) such that (5.5.9)
H [(�l �2) k', kJ o¥2(�1 �J J Jfl (hlmlh�-l)f2 (h2m2h�-l)�'(h'k')�(hk) dh dh' . =
If/is the coefficient of n corresponding to H by (5.5.2) we get, as in (3. 5), (5. 5. 10)
A(
=
Z(
This establishes (5.5.7) for
(; l �J :r
0
and is in Y'0 if
0
(5. 5. 1 2)
B(
=
r(s, n , cf;)A(
where B is obtained by replacing in the definition of A the pair (P, 0') by the pair (P, 6'). It is then clear that (5.5.7) can be proved as the holomorphy of (4.5. 1) (cf. proof of(4.5)). To begin with it is clear that it can be obtained from (P, 6') as n is obtained from
82
HERVE JACQUET
(P, a). (Cf. (3. 3). ) More precisely iff is a coefficient of 77: given by (5.5.2) then JV is given by (3.3. 1 1) and H (loc. cit.) satisfies (5. 5.3)-(5.5.5) for (P, a). This being so set B(f/J , s, /1 , /2) =
(5. 5. 1 3)
J j det m1 j •+Cn1-1l 12 j det m2 j •+C"2-1l 12 f1 (m 1) /2(m2) (/J[(m 1 ; yx �: 2)J dxm 1 dxm2 dx dy,
each time f.· is a coefficient of a,.. Then for (/) E Yo (5.5.7) applies to B with L(s, ft) instead of L(s, 77:). Now let (/) E Yo and a coefficient /; of 0"; be given ; choose � . e and H as above so that (5.5.9) is satisfied as well as (5.5. 10) where f is given by (5.5.2). Then
JJ (JJA[k-lzk']e(k)�(k') dk dk' = (/JA(z). Moreover n - being given as in (3.3. 1 1) by H(gh g2) = H(g2, g1), we find
(� �2) JJf�[hlmlh�-1]f'/[h2m2h2-1] �(h'k')�'(hk) dh dh'.
= o}(2 1
SincefV is given by (3.3. 1 1) we get B (f/Ji\, 1 - s,f �, f '{)
=
Z(f/JA, 1 - s + t(n - 1), JV).
Comparing with (5.5.10) this concludes the proof of (5.5. 1 2) and (5.5.7). (5.6) Let d be the space of functions on X = Gn 1 x {M(n2 x n2 , F)} which, in an obvious sense, are of compact support with respect to the first variable and of Schwartz type with respect to the second variable. We give to d the obvious topolo gy. In particular C;'(X) is a dense subset of d. On the other _hand we may regard d as a subspace of the space of Schwartz functions on {M (n 1 x nh F)} x {M (n2 x n2, F)}. For f/J1 in C;'(Gn 1), f/J12 in C';'(M (n 1 x n2, F)), f/J21 m C;'(M (n2 x nh F)), and f/J2 in Y(n2 x n2, F) we may set (5.6. 1) We let do be the subspace of d spanned by these functions ; it is easily checked that do contains a dense subspace of C;'(X) so is dense in d. For 7/f in d and coefficientsfh /2 of 0"1> a 2 we set (5.6.2)
U (7/f, s, fl , /2) =
JJ 7/f(mh m2) j det md s+ (nJ-1)/2 j det m2 j s+ Cnz-1)/2
· Ji (m 1)/2(m2) dxm 1 d x m2.
The integral converges absolutely for Re(s) large enough. By (5.5.7) we know that (5.6.3) For 7/f in d0, the ratio U(7/f, s, /1 , /2) /L (s, 77:) continues to an entire function ofs. On the other hand, we are going to prove that
PRINCIPAL £-FUNCTIONS
E-3
(5.6.4) For 1/f in d, the ratio U(l/f, s, fi> f2)/L(s, O'z) continues to a holomorphic function of s. As such, it is a continuous function of(l/f, s) on d x C. Let us observe that the first assertion of (5.6.4) is trivial for 1/f in the (algebraic) tensor product (5.6.5) For if 1/f = 1/f 1 ® 1/f2 the ratio is the product of an absolutely convergent integral by Z(l/f2, s + -}(n2 - 1 ), /2 ) /L(s, O'z) which is a polynomial. Moreover assume s0 is a pole of order u of L(s, 0'2) ; then there aref1 , f2 and 1/f E d1 so that the ratio of (5.6.4) has a pole of order u at s0. Taking at the moment (5.6.4) for granted, assume s0 is not a pole of L(s, 0' 1 ) and is a pole of order < u of L(s, n). Since do is dense in d, it follows from (5.6.3) that the distribution U( , s,fb f2), which depends meromorphically on s, has a pole of order < u at s0 • The same is true of the (scalar) meromorphic functions U(l/f, s, fi> f2) . By taking 1/f in d1 we get a contradiction. This proves (5. 1). It remains therefore to prove (5.6.4). As noted, the first assertion is trivial for 1/f in d1, the ratio being then the product of a polynomial by a function bounded in vertical strips. Now there is a relation (5.6. 6) a #- 0, (as + b) U(f/J, s, /1, /2) + U(f/J', s, /1 , /2) = 0, where (/J' is in d1 if (/J is and depends continuously on f/J. Next introduce for Re s sufficiently small the integral V(f/J, s, /1 , /2) ·
=
J 1/f(mb m2) i det mds+ (n!-1) /2 i det m211-s+ (nz-l) /2f1(m1)f2V(m2) dxm1 dxm2.
Then, for 1/f E d 1o V (1/f�'� , s, fl , fz) = r(s, 0'2, ¢) U(f/J, s, /1 , /2). Again V satisfies a relation like (5.6.6). Finally, d 1 is dense in d. These remarks being made the proof of (5.6.4) is similar to the proof of the holomorphy of (4.5. 1). This concludes the proof of (5. 1). 6. Global theory. The field F is now an A-field. (6. 1) Let n = ®v nv be an irreducible admissible representation of Gn(A) ; again n may be a representation of some other object than the group [I.P.S.]. Then ir = ®irv · Set (6. 1 . 1) L(s, ir) = IJL(s, ir0). L (s, n) = IT v L(s, n0), v Suppose that the central quasi-character w of n is trivial on px . Then set (6. 1 .2) nv, ¢ v). c(s, n) = Ilc(s, v Here ¢ = TI ¢v is a nontrivial character of A/F; almost all factors in (6. 1 .2) are one and the product does not depend on ¢. (6.2) THEOREM. Suppose n is automorphic. Then the infinite products (6. 1 . 1) con verge absolutely in some right half-space. They continue to meromorphic functions of
84
HERVE JACQUET
s in the whole complex plane . As such they satisfy L(s, n) = s(s, n)L(l - s, it). If F is a number field, they have finitely many poles and are bounded at infinity in vertical strips . If F is a function field whose field of constants has 0 elements, they are rationalfunctions of o-s. PROOF. If n is cuspidal this is Theorems 3, 4, 5, VII, §6 of [A.W. 2] for n = 1 , and Theorem (1 3.8) of [R.G.-H.J.] for n > 1 . If n > 1 and n i s not cuspidal, there is a standard parabolic subgroup of type (nb n2, · · · , n,) , and, for every i, an irredu cible automorphic cuspidal representation a; of Gn; such that the following condi tions are satisfied : for any place v, n. is an irreducible component of the induced representation e. = /(G., P. ; a1., a2., · · · , a,.) . (Cf. [R.L.].) Thus, for any place v, L (s, n.)
=
P.(s) IT L(s, a .- . •) ,
L (s, it.)
=
P.(s) IT L(s, a.- . •) .
r(s, n., ¢.)
=
IT r(s, a.- . •• ¢.),
j
j
j
where P. is a polynomial in s if F. = R or C, and a polynomial in q;;• if F. is non archimedean with residual field of cardinality q • . Moreover, for almost all v, n . and the a.- . • are unramified so that ((3.6.5)) P. = P. = 1 , s(s, n., ¢.) = 1 , almost all v. Thus L (s, n)
=
L(s, it)
=
P.(s) IT L(s, a,-) , IT v j
ITv P.(s) ITj L(s, a,.),
Pv(1 - s) s(s, n) - IT IT. s(s, a,-) . v pv(s) ' _
Our assertions are then obvious, except the one on the boundedness of L(s, n) in the number field case, which follows from the Phragmen-Lindelof principle. BIBLIOGRAPHY [A.A.] A. Andrianov, Zeta functions of simple algebras with non abelian characters, Uspehi Mat. Nauk 23 (1968), 1-66 = Russian Math. Surveys 23 (4) (1968), 1-65. [1.8.-A.Z. 1] I. N. Bernstein and A. V. Zelevinsky, Representations of the group GL(n, F) where F is a non archimedean local field, Uspehi Mat. Nauk 31 (3) (1976), 5-70 = Russian Math. Surveys 31 (3) (1976), 1-68. [1.8.-A.Z. 2] -- , Induced representations of the group GL(n) over a p-adic field, Funkcional Anal. i Prilozen 10 (3) (1 976), 74-75 = Functional Anal. Appl. 10 (3) (1976), 225-227. [1.8.-A.Z. 3] -- , Induced representations of reductive p-adic groups. II, preprint. [M.E.] M. Eichler, Allgemeine Kongruenzklasseneinteilungen der /deale einfacher A lgebren uber algebraischen Zahlkiirpern und ihrer L-Reihen, J. Reine Angew. Math. 179 (1 938), 227-25 1 . [G.F. 1 ] G . Fujisaki, On the zeta-functions of a simple algebra over the field of rational numbers. Fac. Sci. Univ. Tokyo, Sect. 1 7 (1 958), 567-604. [G.F. 2] --, On the £-functions of a simple algebra over the field of rational numbers, Fac. Sci. Univ. Tokyo, Sect. 1 9 (1 962), 293-3 1 1 .
PRINCIPAL £-FUNCTIONS
85
[S.G.) S . Gelbart, Fourier analysis o n matrix spaces, Mem. Amer. Math. Soc., vol. 108, Provi dence, R.I., 1 971 . [R.G.) R. Godement, Les fonction C des algebres simples. I, II, Sem. Bourbaki, 1 957/1 958, Exp. 1 7 1 , 1 76. [R.G.-H.J.) R . Godement and H. Jacquet, Zeta functions of simple algebras, Lecture Notes in Math., vol. 260, Springer-Verlag, 1972. [E.H.) E. Heeke, Uber Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produkten twicklung. I, II, Math. Ann. 114 (1937), 1-28 ; 3 1 6-35 1 . [K.H.) K. Hey, Analytische Zahlentheorie in Systemen hyperkomplexer Zahlen, Dissertation, Hamburg, 1929. [K.I.) K. Iwasawa, A note on £-functions, Proc. Intemat. Congr. Math., Cambridge, Mass., 1 950, p. 322. [H.J. 1) H. Jacquet, Zeta functions of simple algebras (local theory), Harmonic Analysis on Homogeneous Spaces, Mem. Amer. Math. Soc., Providence, R.I., 1973, pp. 381-386. [H.J. 2] , Generic representations, Non-Commutative Harmonic Analysis, Lecture Notes in Math., vol. 587, Springer-Verlag, Berlin, 1 977, pp. 91-101 . [H.J.-R.L.) H. Jacquet and R. Langlands, Automorphic forms on GL{2), Lecture Notes in Math., vol. 1 14, Springer-Verlag, Berlin, 1 970. [M.K.) M. Kinoshita, On the �-functions of a total matrix algebra over the field of rational num bers, J. Math. Soc. Japan 17 (1 965), 374-408. [A.K.-G.Z.] A. W. Knapp and G. Zuckerman, Classification theorems for representations of semi-simple Lie groups, Non-Commutative Harmonic Analysis, Lecture Notes in Math., vol. 587, Springer-Verlag, Berlin, 1977, pp. 1 38-1 59. [R.L.) R. P. Langlands, On the classification of irreducible representations of real algebraic groups, mimeographed notes, Institute for Advanced Study, Princeton (1 973). [H.M. 1) H. Maass, Uber eine neue Art von nicht-analytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Functionalgleichungen, Math. Ann. 121 (1 949), 1411 83. [H.M. 2) , Zetafunktionen mit Grossencharakteren und Kugelfunktionen, Math. Ann. 134 (1957), 1-32. [G.M. 1) G. N. Maloletkin, Zeta functions of a semi-simple algebra over the field of rational numbers, Dokl. Akad. Nauk SSSR 201 (3) (1971), 535-537 Soviet Math. Dokl. 12 (6) (1971), 1 678-1681 . [G.M. 2) , Zeta /unctions of parabolic forms, Mat. Sb. 86 (1 28) (1971), 622-643 = Math. USSR-Sb. IS (4) (1971), 61 9-641. [A.S.) A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. 20 (1956), 47-87. [H.S.) H. Shimizu, On zeta functions of quaternion algebras, Ann. of Math. (2) 81 (1965), 1 66193. [G.S.] G. Shimura, On Dirichlet series and abelian varieties attached to automorphic forms, Ann. of Math. (2) 76 (1962). [E.S.] E. Stein, Analysis in matrix spaces and some new representations of SL(N, C), Ann. of Math. (2) 86 (1 967), 461-490. [T.T.] T. Tamagawa, On the zeta functions of a division algebra, Ann. of Math. (2) 77 (1963), 387-405. [J.T. l) J. Tate, Fourier analysis in number fields and Heeke's zeta-functions, Algebraic Number Theory, edited by J.W.S. Cassels and A. Frohlich, Thompson, 1967, pp. 305-347. [J.T. 2) --, Number theoretic background, these PROCEEDINGS, part 2, pp. 3-26. [N.W.) N. Wallach, Representations of reductive Lie groups, these PROCEEDINGS, part 1, pp. 7186. [G.W.) G. Warner, Zeta functions o n the real genera/ linear group, Pacific J. Math. 4S (1973), 681-691 . [A.W. 1) A. Weil, Fonctions zeta et distributions, Sem. Bourbaki, no. 3 1 2, (1965-66), 9 pp. [A.W. 2) --, Sur certains groupes d'operateurs unitaires, Acta Math. 111 (1 964), 144-21 1 . [A.W. 3) , Basic number theory, Springer-Verlag, Berlin, 1967. --
--
=
--
--
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[A.W. 4] A. Weil, Dirichlet series and automorphic forms, Lecture Notes in Math., vol. 1 89, Springer-Verlag, Berlin, 1 97 1 . [E.W.] E . Witt, Riemann-Rochscher Satz und �-Funktion im Hyperkomplexen, Math. Ann . 110 (1 934), 1 2-28. [A.Z. 1] A. V. Zelevinskii, Classification of irreducible non-cuspidal representations of the group GL(n) over p-adic fields, Funkcional Anal. i Prilozen 11 (1) (1 977), 67 Functional Anal. Appl. 1 1 (1) (1 977). [A.Z. 2] Classification of representations of the group GL(n) over a p-adic field, preprint. (Russian) =
-- ,
COLUMBIA UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 2, pp. 87-95
AUTOMORPHIC L-FUNCTIONS FOR SYMPLECTIC GROUP
G Sp(4)
MARK E. NOVODVORSKY* 0. Introduction. The paper represents a talk delivered at the Summer Institute of the American Mathematical Society at Corvallis in 1 977. Let k be a global field, P the set of its normalized valuations, A = 11 pE P kp the adele ring of k, G a reductive algebraic group over k, GA = Il pEP Gp the group of ideles of G, 1t: = ®pEP 7t:p a class of isomorphisms of irreducible admissible repre sentations of GA• p a finite dimensional representation of Langlands' dual group LG. R. P. Langlands [1] defined £-function L(1r:, p, s) as a certain Euler product, L(1r:, p, s) = pITP L(7t:p, p, s), (0. 1) E convergent for all complex numbers s with sufficiently large real part if 1r: is uni tarizable. He conjectured that if 1t: is the class of an automorphic representation, then L(1r:, p) can be continued to a meromorphic function on the whole complex plane which satisfies the functional equation (0.2) L(1r:, p, s) = c(7t:, p, s) L(1r: * , p, 1 - s), 1r:* being the contragradient representation to 1t: and c(7t:, p, s) = pEIT c(7t: P• p, s) P another Euler product whose existence is a part of the conjecture. However, the definition of the local factors L(7t:p , p, s) depends on a parametrization problem (cf. Borel's paper, Automorphic L-function, these PROCEEDINGS, part 2, pp. 27- 61) and, therefore, does not work at present until all the irreducible representations 1t: P of the groups G P corresponding to nonarchimedean valuations p are of class 1 . Therefore, in order to prove Langlands' conjecture one must first construct the Euler factors L(1r:p) and c(7t:p). A general approach to this problem was suggested by I. I. Piatetski-Shapiro [2]. Here we develop the ideas of [2] for the symplectic group G = G Sp(4) and prove Langlands' conjecture for the standard 4-dimensional representation p of its Lang lands dual group LG = G Sp(4, C). We consider also the group G = G Sp(4) x GL(2) and prove Langlands' conjecture for generic cuspidal automorphic repre sentations 1r: (cf. Piatetski-Shapiro's paper Multiplicity one theorems, these PRoAMS (MOS) subject classifications (1 970). Primary lOD lO, 22E55 .
* Supported by NSF Grant MCS76-021 60Al . © 1979, American Mathematical Society
87
88
MARK E. NOVODVORSKY
CEEDINGS, part 1 , pp. 209-21 2) and the standard 8-dimensional representation p of its Langlands dual group G Sp(4, C) x GL(2, C) (i.e. p is the tensor product of the standard 4- and 2-dimensional representations of the factors). Our results are complete only for functional global fields k . For number fields k we obtain meromorphic continuations of these automorphic L-functions but, because of certain difficulties with archimedean valuations, not the functional equations (0.2). Since the methods of [2] are based on generalized Whittaker models, we consider generic and hypercuspidal automorphic representations separately (cf. the quoted paper of Piatetski-Shapiro). The results for generic representations of G Sp(4) (§ 1) can be extended to all split orthogonal groups of Dynkin type Bn (Sp(4) covers S0(5)) ; in this form they were announced by the author (for char k > 0) in [10] and [11]. The results for hyper cuspidal representations of G Sp(4) (§2) were obtained by I. I. Piatetski-Shapiro and the author. The results for G Sp(4) x GL(2) (§3) were announced by the author (for char k > 0) in [12]. In this paper we assume that char k "# 2. 1. G Sp(4), !generic representations. Let G � G Sp(4) be the group of similitudes of the bilinear form
of four variables over k, Z the subgroup of all upper triangular unipotent matrices from G, cjJ a nondegenerate l character of the group ZA of ideles of Z which is trivial on principal ideles Zk, p a nonarchimedean valuation of the field k. The space of all locally constant complex functions W on the group Gp which satisfy the equation (1 . 1 ) W(zg) = cp(z) W(g) 'V z E Zp, g E Gp, is denoted "'f/'q,(Gp) ; Gp acts in this space by right translations. A class 11:p of isomor phisms of irreducible admissible representations of the group Gp is called non degenerate if it contains the subrepresentation of GP in an irreducible Gp-invariant subspace of "'f/'q,( Gp) ; this subspace then is called a Whittaker model of 11:p and de noted by "'f/'q,(71:p) · THEOREM (1. M. GELFAND, D. A. KAZDAN (3], F. RODIER (4]). A Whittaker model of a nondegenerate class 71:p is unique . Let 11: p be a nondegenerate class of isomorphisms of irreducible admissible re presentations of the group Gp. Its restriction to the center Cp of Gp is proportional to a character, denote it wp . In view of the canonical isomorphism of the center C of the group G Sp(4) and the multiplicative group, (1 .2)
C3
c J a
a
- a e k',
'That is, rp is not trivial on the ideles of any horocycle subgroup of Z.
AUTOMORPHIC £-FUNCTIONS FOR G SP (4)
89
OJp can be considered as a character of k0 . The contragradient class �; coincides with �P ® OJp(up) where u is the unique homomorphism of G Sp(4) into k* whose square is determinant (u is the factor of similitude). Therefore, 'if'"¢C�;) coincides with the set of functions W*(g) = W(g) · OJj;1 (up(g) ), (1 .3) We introduce the subgroups
and consider the integral ( 1 .4)
,fp( W, s) =
J (U·Hl p W(uh) ii a il •-1 1 2d(uh),
1. The integral (1 .4) is absolutely convergent in a vertical half-plane and defines ,Ip( W) as a rational function in qp•, qp being the number of ele ments in the residue field of the valuation p. All the functions ,/p( W), W e 'if'"¢(�p), admit a common denominator. There exists a rational in qp• function r(�p) such that ,fp( W, s) = r(�p. s),fp{(.8( W))*, 1 - s), THEOREM
Re s
( 1 . 5)
�
0
{3 =
1 0 0 1
1 0 0 1 0
We denote by Qp(� p) the polynomial in q"js with constant term 1 which is the com mon denominator of smallest degree for functions ,1p( W), W e 'if'"¢(�p), and define (1 .6)
L(�p. s) = [Q(�p. q-•)]- 1 , e(�p. s) = r (�p. s) · L(�p. s)/L(�;. 1 - s).
Let � = (?!)p = P �p be the class of isomorphisms of an automorphic generic cuspidal representation of the group G A- Then all its nonarchimedean components �p are nondegenerate and the Euler product (1 .7) L(�. s) = IT L(�p. s), pEP\S
S being the subset of all archimedean valuations from P, is absolutely convergent in a vertical half-plane Re s � ; it coincides with the Euler product of Langlands
90
MARK E. NOVODVORSKY
corresponding to the standard 4-dimensional representation of Langlands' dual group LG � G Sp(4, C) if k is a functional global field and differs from it by a meromorphic (in the whole complex plane in s) factor if k is a number field. The factors e(7rp) are of the form aqj/" and for almost all p E P\S are identically equal to I ; therefore, the Euler product (1 .8)
e(7r, s) = n e(7rp, s) p EP\S
is, in fact, finite and defines e(n') as an entire function without zeros. THEOREM 2. The function L(n) admits meromorphic continuation to the whole complex plane. If char k = p > 0, then this continuation is a polynomial in p• and
p-• satisfying the functional equation
(1.8)
L(n, s)
= e(7r, s)L(n *, I
-
s).
2. G Sp(4), hypercuspidal representations. This case has been investigated in co operation with I. I. Piatetski-Shapiro. For hypercuspidal representations, Whittaker models associated to nondegener ate characters of maximal unipotent subgroups do not work, and one has to con sider other subgroups ai:J.d generalized Whittaker models. It is convenient for our purposes to consider a different realization of the group G = G Sp(4). We choose a 2-dimensional semisimple k-algebra K and a 2-dimensional bilinear skew-sym metric form B in 2 variables over K; G can be realized now as the group of simili tudes of the form trK 1 kB. Now G contains the subgroup G1 � {g e GL(2, K) : det g e k } (2. I) of all K-linear transformations from G ; the subgroup of all upper triangular uni potent matrices from G1 is denoted by U1, the unipotent radical of the normalizer of U1 in G is denoted by U; U is a 3-dimensional commutative horocycle subgroup in G. The algebra K has a unique nontrivial k-automorphism
(2.2) it induces a k-automorphism of the form trKi k B which is denoted T' =
{(� �), 1C E K *}
c
-r, -r E
G. We put
G1 ;
obviously, T' is isomorphic to the multiplicative group K* of the algebra K. The groups G1, U1, and T1 are normalized by -r. Let p be a nonarchimedean valuation of the field k. We take a nontrivial charac ter X of the factor group up/ u; and a quasi-character �I of the group r; whose re striction on the subgroup
is unitary. If f is -.-invariant, we continue it to a character � of the group Tp , T = Tl (2.3) If �� is not -.-invariant, we put
U
-.T�
AUTOMORPHIC £-FUNCTIONS FOR G SP(4)
T = T', � = e ' (2. 4) In both cases we define Z = T· U, cj;(tu) = �(t) · x(u) (2.5)
V
91
t E Tp, u E Up ;
Z is an algebraic k-subgroup of G, cf; is a quasi-character of the group Zp. As in § 1 , we denote by "If"q,(Gp) the set of all complex locally constant functions W on the group Gp satisfying the equation ( 1 . 1), and an irreducible admissible subrepresent ation of Gp in "If" q, by right translations is called the Whittaker model of its class of isomorphisms 1rP and denoted if/'q,('Kp). THEOREM 3 . Every class 1rp of isomorphisms of irreducible admissible representa tions of the group Gp is either nondegenerate or has Whittaker model "If"q,(1rp)for some algebra K, subgroup T, and character cf; of the described type; for every fixed cf; this model is unique.
REMARK 1 . One class 1rp can have Whittaker models corresponding to several of the characters cf; described in §§ l and 2. REMARK 2. The group Tp = r; U -. r; is the stabilizer of X in the normalizer of the group Up in Gp; so, X can be lifted to either a character or a 2-dimensional ir reducible representation of the group fP Up · Theorem 3 can be reformulated as the theorem of the existence and uniqueness of Whittaker models associated to these 1 - and 2-dimensional representations ; such reformulation might be preferable logically but leads to more bulky constructions for Euler factors in case of 2-dimen sional representations. In the case when Z and cf; are defined by the formula (2.3), the uniqueness part of Theorem 3 was proven by I. I. Piatetski-Shapiro and the author [8] and by the au thor [9]. For supercuspidal representations 'Kp the uniqueness part of Theorem 3 follows from F. Rodier's Theorem 1 [5]. Now we put •
(2.6)
H=
{(" 1 ),
"E
k*
}
c
G'
c
GL(2, k)
and consider the integral (2.7)
fp( W, s)
=
s Hp W(h) jj h jj • dh,
THEOREM 4. The integral (2. 7) is absolutely convergent in a vert1cal half-plane Re s � and defines fp( W) as a rational function in q-p•. All the functions fp(W}, W E "If"q,('Kp) , admit common denominator.
As in § 1 , we denote by Qp('Kp) the normalized common denominator of the smallest degree and put
(2.8) THEOREM 5. Euler factor Lp('Kp) does not depend on the choice of the subgroup Z and the character cf; of Zp (particularly, if the class 1rp is nondegenerate, the factors L(1Cp) defined in §§1 and2 coincide) .
92
MARK E. NOVODVORSKY
Let 11: = @pEP 11:p be the class of isomorphisms of an automorphic cuspidal representation of the group GA. Choosing for every nonarchimedean p a Whittaker model "'f"q,(11:p). we obtain Euler factors L(11:p) and define L(11: , s)
(2.9)
pEP\S L(11:p, s),
= IT
S being the set of archimedean valuations of k ; this product is absolutely conver gent in a vertical half-plane Re s � , it coincides with Langlands' Euler product corresponding to the standard 4-dimensional representation of Langlands' dual group L G � G Sp(4, C) if k is a functional field, and differs from it by a meromor phic (in the whole complex plane in s) factor if k is a number field.
THEOREM 6. The function L(11:) admits meromorphic continuation to the whole complex plane . .lf char k = p > 0, then this continuation is a rational in p-• function satisfying the functional equation
L(11: , s) = .>(11: , s)L(11:* , 1
(2. 10)
where .>(11:) is an entire function in s without zeros.
-
s)
In fact, .>(11:) is an Euler product of some factors .>(11:p) appearing in local func tional equations ; these equations are rather bulky and will appear in a joint work with I. I. Piatetski-Shapiro. ·
3. G Sp(4) x GL(2), generic representations. In this section we use a modification of H. Jacquet's treatment of induced representations applied in [7] to automorphic £-functions on GL(2) x GL(2). The subgroups C and Z of G Sp(4), the character ¢ of ZA, the space "'f" q,( G Sp( 4, kp)) , p nonarchimedean, Whittaker model "'f" q,(11:p) for a nondegenerate class 11:p of irreducible admissible representations of G Sp(4, kp), the homomorphism a, and the character wp in this section are the same as in § 1 . We define the subgroup
(3. 1 )
H=
{(: g ;)
e G Sp(4) , a , b, c, d e k, g e GL(2)
}
and also its homomorphisms onto the group GL(2, k) : ¢1 :
(3 2) .
¢z :
(: g ;) - g, (: ;) - (: ;} g
We denote by Z the subgroup of all upper triangular unipotent matrices in GL(2,kp). It is the image of H n Z under the homomorphism ¢ �o and the kernel
AUTOMORPHIC £-FUNCTIONS FOR G SP(4)
93
of this homomorphism considered on the group (H n Z) A belongs to its com mutator subgroup ; therefore, every character of the group ZA defines under ¢ 1 a character of ZA which we denote with the same symbol. As in § 1 , we denote by "'f/¢(GL(2, kp)) the space of all locally constant complex functions W satisfying the equation W(zg)
(3 .3)
rp(z) W(g) ,
=
z E Zp, g E GL(2, kp)
and by "'f/ �(fc p), fcP a class of an infinite-dimensional irreducible admissible repre sentations of GL(2, kp), a subrepresentation of this group in "'f/ sb(GL(2, kp)) which belongs to fcp ; such a subrepresentation exists and is unique (cf. H. Jacquet, R. P. Langlands [6]). We denote by tiJp the restriction of fcp on the center of the group GL(2, kp) ; we consider tiJp as a character of the multiplicative group kt in view of the canonical isomorphism The space "ffl¢{fc;), 1c; � fcp ® w-p1 (det) being the contragradient representation to fcP • consists of the functions W*(g)
(3 .4)
W(g) · w-p1(det(g) ) ,
=
g E GL(2, kp).
For every locally constant complex function � on the plane kp E9 kp we define its Fourier transform iJ(x, y)
(3.5)
=
Jkpffikp �(u, v) a(yu - xv) d(u, v)
where a is a fixed nontrivial character of the group Afk of classes of adeles ; for every complex quasi-character fl. of the multiplicative group kt we put h E Hp.
(3 .6)
If the integral (3 .6) is absolutely convergent, then the function/satisfies, obviously, the equation (3.7)
1((: g ;) h, � ) .
fl.
=
p(d- 1 )/(h , � . r) ,
Therefore, we can consider the integral (3 . S)
.,! ( W, W, � . fi. l (s) , s) = fi.t (K, s)
=
J W(h) W(iflt(h)) f(h, �. fi. l) II ifJz(h) jj s dh , II " 11 25 w(K) tiJ(K),
w E "'f/q,(77:p), w E "'f/vifcp).
and a similar one for .,t ( W* , W *, iJ, p2(s) , s ), pz(K , s) = II " jj Zs w- l (K) w- l (!C). THEOREM 7.
The integrals (3.6) and (3 . 8) are absolutely convergent in a vertical
94
MARK E. NOVODVORSKY
half-plane Re s � and define .f( W, W, �) and .f( W *, W*, fi>) as rational/unctions in q-p•. There exists a rational in q;• function 7(11:P • itp) such that .f ( W, W, �. ,u 1 (s), s) = r('ll:p , itp, s) .f ( W, W*, fP, ,u 2(1 - s), 1 - s) (3 . 9) "f/ W E "'f/¢(11: p), JV E "'f/ ;p(it p), �. All the rational functions .f( W, W, �) (for different W e "'f/¢(11:p). W e "'f/;p(itp), �) admit a common denominator. As in § 1 , we denote by Q(11:P • itp) the normalized common denominator of the smallest degree in q-s and define L('ll:p ® itp, s) = [Q('ll:p , itp, q-•)]- 1 , ( 3.10) e('ll:p ® itp, s) = r('ll:p , itp, s) · L('ll:p ® itp. s)jL(11:; ® it;, 1 - s). If 11: ® it is the class of isomorphisms of a cuspidal generic irreducible representa tion of the group G Sp(4, kp) x GL(2, kp), 11: = @pEP 'll:p , it = @pEP itp, then we put (3. 1 1)
L(11: ® it, s)
=
e('ll: ® it, s)
=
IT L('ll:p ® itp, s),
p EP\S
IT e('ll:p ® itp, s),
p EP\S
S being the set of archimedean valuations of k. As before, the first product is ab
solutely convergent in a vertical half-plane Re s � . For a functional field k it coincides with Langlands' £-function L(11:, p, s) if p is the tensor product of the standard 4- and 2-dimensional representations of the complex groups G Sp(4, C) and GL(2, C) ; for a number field k these £-functions differ by a factor merom or phic in the whole complex plane. The second product (3. 1 1) is, actually, finite and defines e('ll: ® it) as an entire function without zeros. THEOREM 8 . The function L(11: ® it) admits meromorphic continuation on the whole complex plane. If char k = p > 0, L(11: ® it) is a rational function in p-• satisfying the equation L(11: ® it, s) = e('ll: ® it, s) L(11:* ® it*, 1 - s). (3. 12) REFERENCES 1. R. P. Langlands, Problems in the theory of automorphic forms, Lecture Notes in Math., vol. 1 70, Springer-Verlag, Berlin and New York, 1 970. 2. I. I. Piatetski-Shapiro, Euler subgroups, Proc. Summer School Bolya-Janos Math. Soc. (Buda pest, 1970), Halsted, New York, 1975. 3. I. M. Gelfand and D. A. Kazdan, Representations of the group GL(n, K) where K is a local field, Inst. Appl. Math., no. 242, Moscow, 1 971 . 4. F. Rodier, Modele de Whittaker des representations admissibles des groupes reductifs p-adiques deployes, C. R. Acad. Sci. Paris Ser. A 275 (1 972), pp. 1045-1048. 5. -- , Le representations de G Sp(4, k) on k est un corps local, C. R. Acad. Sci. Paris Ser. A 283 (1 976), pp. 429-43 1 . 6. H . Jacquet and R . P . Langlands, Automorphic forms on GL(2) , Lecture Notes i n Math., vol . 1 1 4, Springer-Verlag, Berlin and New York, 1970.
AUTOMORPHIC £-FUNCTIONS FOR G SP(4)
95
7 . H. Jacquet, Automorphic forms on GL(2). Part 2, Lecture Notes in Math., vol. 278, Springer Verlag, Berlin and New York, 1 972. 8. M. E. Novodvorsky and I. I. Piatetskii-Shapiro, Generalized Bessel models for the symplectic group ofrank 2, Mat. Sb. 90 (2) (1 973), 246-256. 9. M. E. Novodvorsky, On the theorems of uniqueness of generalized Bessel models, Mat. Sb. 90 (2) (1973), 275-287. 10. , Theorie de Heeke pour les groupes orthogonaux, C. R. Acad. Sci. Paris Ser. A 285 (1977), 93-94. 11. , Fonctions / pour des groupes orthogonaux, C. R. Acad. Sci. Paris Ser. A 280 (1975), 1421-1422. 12. , Fonction / pour G Sp(4), C. R. Acad. Sci, Paris Ser. A 280 (1 975), 1 91-192. --
--
--
PURDUE
UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 2, pp. 97-1 10
ON LIFTINGS OF HOLOMORPIDC CUSP FORMS
TAKURO SHINTANI Introduction. In [10], H. Saito calculated the trace of "twisted" Heeke operators acting on the space of holomorphic cusp forms with respect to the Hilbert modular group over a cyclic totally real abelian field of prime degree. He discovered a strik ing identity between his "twisted trace formula" and the ordinary trace formula for the Heeke operators acting on the spaces of elliptic modular forms. Applying his identity he showed that elliptic modular forms are "lifted" to Hilbert modular forms (for the origin of "lifting" type results, see [2]). No less significantly, he char acterized the space spanned by lifted forms. In the U.S.-Japan symposium "Ap plications of automorphic forms to number theory" which was held at Ann Arbor in June 1 975, the author reported a representation theoretic interpretation and generalization of Saito's work (see [13]). Results presented in the author's talk were immediately generalized by Langlands in [8]. Moreover, Langlands discovered an unexpected application of "lifting theory" to the theory of Artin L-functions. The present paper consists of two sections. In § I , we reproduce (with slight modifications) what the author presented at Ann Arbor. The second section is devoted to a few supplementary remarks. The author wishes to express his hearty thanks to Professor H. Saito, who gave him detailed expositions of [10] before its publication. Two of the author's previous papers [14] and [15] are also motivated by [10]. Notation. For a ring R, R x is the group of units of R. We write G(R) = GR = GL(2, R). For a p-adic field k, o(k) = ok is the ring of integers of k. We denote by qk the cardinality of the residue class field of k. For each t e k set d(tx) = ak(t) dx = i t i k dx, where dx is an invariant measure of the additive group of k. For each locally compact totally disconnected group G, C0(G) is the space of compactly supported, locally constant functions on G. Let k be a field and F be a field extension of k. We denote by N[ the norm mapping from F to k. 1.
1 . 1 . Let k be either a finite algebraic number field or a p-adic field. Let F be a commutative semisimple algebra over k of prime degree /. We assume that the group of automorphisms of F over k contains a cyclic subgroup g of order I gen erated by q . Then F is isomorphic either to a cyclic field extension of k or to the direct sum of /-copies of k. Set GF = GL(2, F) and Gk = GL{2, k). We may regard AMS (MOS) subject classifications (1 970). Primary lOOxx ; Secondary 100 1 5 , 10020. © 1979, American Mathematical Society
97
98
TAKURO SHINTANI
g as a group of automorphisms of GF (with the fixed point set Gk) in a natural man ner. Two elements x and y of GF are said to be a-twistedly conjugate in GF if there exists a g E GF such that y = g"xg- I . We denote by xGF· " the set consisting of all elements of GF which are a-twistedly conjugate to x in GF. The set of a-twisted conjugacy classes in GF is denoted by '&',(GF). The usual conjugate class in GF containing x is denoted by xG F . For each x E GF, set N(x) = x"1- 1 . x"1-z · · · x" · x.
It is easy to see that N(g" · xg- I) = gN(x)g- I . We denote by '&'(Gk) the set of conjugacy classes in Gk. For the proof of the next lemma see Lemma 3.4 and Lemma 3.6 of [10].
LEMMA 1 (SAITO) . The notation and assumptions being as above, the mapping xGF, " ...... N(x)GF n Gk is an injection from '&' ,( GF) into '&'( Gk). If F is not a field, the mapping is a bijection. In the following, for each c E '&',(GF), we denote by N(c) E '&'(Gk) the image of c under the mapping given in Lemma 1 . An x E Gk is said to be regular if the group of centralizers of x in Gk is a two-dimensional k-torus. We call a conjugate class in Gk regular if it consists of regular elements. A a-twisted conjugacy class c in GF is said to be regular if N(c) is regular. We denote by '&';(GF) (resp. '&''(Gk)) the set of regular a-twisted conjugate classes (resp. regular conjugate classes) of GF (resp. Gk).
1 .2. We keep the notation in 1 . 1 and assume that k is a p-adic field. Let G'F be the semidirect product of GF with g. More precisely, G'F is the group with the underly ing set g x GF with the composition rule given by (r, g)(z-', g') = (z-z-', g• 'g') (z-, z- ' E g, g, g' E GF). It is immediate to see (via Lemma I) that (a, g 1 ) and (a, g2) are conjugate in G'F if and only if g 1 and g2 are a-twistedly conjugate in GF· Denote by Gk (resp. GF) the set of equivalence classes of irreducible admissible unitariz able representations of Gk (resp. GF). For each R E GF and z- E g, let R• be the re presentation of GF given by R•(x) = R(x•). Then R< E GF. Thus, the group g op erates on GF. Denote by G} the subset of g-fixed elements of GF. A representation of Gp. is said to be admissible if its restriction to GF is admissible. Let R- be an irre ducible admissible unitarizable representation of the group Gp.. As is well known, the restriction R of R- to GF is either irreducible or a direct sum of /-mutually inequivalent irreducible representations of GF. In the former case R- is said to be of the first kind. Assume R- to be of the first kind and set J.,. = R-((a, I)) . Then (V g E GF).
(1 . 1)
Thus R E Gj!.. The relation (1. 1) characterizes the operator J, up to a multiplication by an lth root of unity. Conversely let R E GJ.. Then there exists a unitary operator of order I which satisfies ( 1 . 1). Hence, R is extended to an admissible irreducible unitarizable representation R- of Gp. of the first kind by setting R-((am, g)) = J�R(g). For rp E C0(GF), denote by R-(rp) the operator given by
-
R (rp)
=
LFR-((a, g)) rp(g dg, )
ON LIFTINGS OF HOLOMORPHIC CUSP FORMS
99
where dg is the invariant measure of GF normalized so that the total volume of GocF) is equal to l . It is shown that there exists a locally summable function x(R-) on GF such that trace R-(rp)
=
JGFrp(g)x(R-)(g) dg.
It is proved that x(R-)(g) depends only upon the a-twisted conjugate class of g in GF and that x(R-) is defined only on 'G'�(GF). For each r E Gk, denote by X r the char acter of r. Recall that x,(x) depends only upon the conjugate class of x and that Xr is defined only on 'G''(Gk), assume further that the characteristic of the residue class field of k is not equal to 2. THEOREM I . Let the notation and assumptions be as above. (I) For each R E GJ, there exist an extension R- of R to a representation of G"j and an r E Gk such that (1.2) where N is the norm mapping from 'G';(GF) to 'G''(Gk) given in Lemma I . (2) For each r E Gk , there uniquely exists R E GJ whose suitable extension R- to a representation of G"j satisfies ( 1 .2). For each r E Gk, we call R E G�, which is related to r by ( 1 .2), the lifting of r from Gk to G �- (Jacquet introduced in [4] the notion of lifting from a different viewpoint.) REMARK I . An analogue of Theorem 1 for finite general linear groups was given in [14]. Furthermore, an analogue of Theorem I for the case of (F, k) = (C, R) is given in [15] (resp. [8]) by a local (resp. global) method. Let us describe the lifting from Gk to GJ in a concrete manner. If F is isomorphic (as a k-algebra) to the direct sum of I copies of k, GF is isomorphic to the direct product of /-copies of Gk. It is immediate to see that the lifting of r E Gk is given by r@ · · · @ r (the same is true even when the characteristic of the residue class field of k is equal to 2). Next, we consider the case when F is a cyclic field extension of prime degree /. First, let us recall a description of Gk. For details, see §§3 and 4 of Chapter 1 of [5]. For quasi-characters p1 and p2 of kx such that p1p21 i= ai l , let p(f.J.I> p2) be the cor responding irreducible representation of Gk in the principal series. If p1p21 = ak , let a(pb p2) be the corresponding special representation of Gk. For a quadratic extension K of k and a quasi-character w of Kx such that w' i= w (' denotes the conjugation of K with respect to k), let n-(w, K) be the corresponding absolutely cuspidal representation of Gk. If the characteristic of the residue class field of k is not equal to 2 (as we are now assuming), it is known that each infinite dimensional irreducible admissible representation of Gk is equivalent to some of p(f.J.I> p2) , o-(f.J. I> p2) and n-(w, K). For each quasi-character f.J. of kx, we denote by p the quasi character of Fx given by p = f.J. o N�. For a quadratic extension K i= F of k and a quasi-character w of K x, denote by w the qu isi-character of L x (L = K · F) given by w =
w · Nf.
PROPOSITION I . The notation and assumptions being as above, the lifting R of r E Gk to G� is given as follows :
1 00
TAKURO SHINTANI
(1) Ifr = p(f.i.t> f.1.2) (resp . o{f.i.h f.1.2)) , R = p(!Jh !-'2) (resp. R = o{!Jh !-'2)) . (2) Ifr = n-(w, K) and K =1- F, R = n-(cu, L), where L = F K. (3) Ifr = n-(w, K) and K = F, R = p(w, w'). REMARK 2. The proof of the first part of Proposition 1 is straightforward (even when the characteristic of the residue class field of k is equal to 2). 1 .3. We keep the notation in 1 .2. In particular, k is a JJ-adic field (the residue class field of k may be of characteristic 2). For an x e GF, let Z11(x) be the subgroup of GF consisting of all elements of g of GF satisfying g "xg - 1 = x. Normalize invariant measures on GF and Z11(x) so that total volumes of their maximal compact open subgroups are all equal to 1 . Denote by dg the invariant measure on GF f Z11(x) given as the quotient of the normalized invariant measure of GF by that of Z11(x). Forf e C0(GF), set ·
( 1 . 3) It is shown that the integral is absolutely convergent. Moreover, for c e ((;f;(GF),
A11(f, x) = A11(f, y) for any x, y e c. For each c e ((;!�(G), we put
(1 .4) A11(/, c) = A11(f, x), where x is an arbitrary element of c. For each x e Gk, let Z(x) be the subgroup of centralizers of x in Gk. We normalize invariant measures on Gk and Z(x) so that the total volumes of their maximal compact open subgroups are all equal to 1 . Denote by dg the invariant measure on Gk/Z(x) given as the quotient of the normalized invariant measure of Gk by that of Z(x). For fe C0(Gk), set ( 1 . 5)
A(f, x) =
J Gk!Z (x) f(gxg-1) dg.
It is known that the integral is absolutely convergent. For each c e ((;f'(Gk), we put (1 .6)
A(f, c) = A(f, x),
where x is an arbitrary element of c. It is easy to see that the right side of (1.6) is independent of the choice of x e c. PROPOSITION 2. The notation being as above, for each le C0(GF), there exists an f E C0(Gk) which satisfies the following conditions (I) and (2) : (1) For each regular c E C11(GF), A11(f, c) = A(f, N(c)) . (2) For each c E ((;f ' (Gk) - N (((;f;(GF)), A(f, c) = 0. REMARK 3. In Proposition 2, assume I is the characteristic function of Go (Fl · Assume further that F is either the unramified cyclic extension of degree I of k or is isomorphic to the direct sum of I copies of F. Then one may putfto be the char acteristic function of Go (kl . REMARK 4. The correspondence I f--0 f in Proposition 2 is, in a sense, dual to the lifting of irreducible characters of Gk given in Theorem 1 . 1 .4. Let k be a totally real algebraic number field of degree n and let kA (resp. k;l)
ON LIFTINGS OF HOLOMORPHIC CUSP FORMS
101
be the ring of adeles (resp. the group of ideles) of k. Denote by koo (resp. kA. o) the infinite (resp. finite) component of kA- Then kA = koo EB kA, O and k;,j_ = k� x k;4, 0• Moreover, both groups Gkoo and GkAo O are embedded into GkA in a natural manner and GkA. = Gkoo X GkA, O · Let { oo h · · ·, oo n } be the set of infinite places of k. For each g E GkA' set g = googO (goo E Goo, go E GkAo o) and goo = (goo, h . . · , goo,n), where goo, ; E GR is the component of g"" corresponding to the infinite place oo ,. For goo ,; we introduce the following standard parametrization sin 0; goo, ,. = t; + t · 1 X1 ; VY; 1y-:-1 -sC?mS 0; 0'· cos 0t· - ' 'V "
( )( )(
(t;
·
t:
)(
)
> 0, X ; E R, Y ; > 0, 0; E R). Let 0; be the differential operator on GkA given by
For 0 = (O h · · ·, O n) E Rn , we denote by tc(O) the element of Gk"" whose ith component is 0; sin 0, ( -cos sin 0; cos 0;) ·
Let X be a unitary character of the group k;,j_fkX and let r be a function on the set of infinite places of k with values in the set of positive integers. Set r; = r( oo ,) . Denote by S(x, r, kA) the space of C-valued functions f on G kA which satisfy the following conditions ( 1 .7), ( 1 . 8) and ( 1 .9) . ( 1 . 1) f(g) is bounded, smooth with respect to goo and locally constant with respect to g0• Furthermore, OJ = fr,{r; 2)/4 ( 1 � i � n) . -
( 1 . 8) ( 1 .9)
(( ))
! r z z g = x (z)f(g) (V r E Gk,
(
f (gtc(O) ) = exp v' - 1
vz
jj r,o,) J(g)
E k�). (V 0 E Rn).
Denote by p(k) the set of finite primes of k. For each v E p(k), let k. be the comple tion of k with respect to v and let o. be the ring of integers of k •. Then the group GkA. o is isomorphic to the restricted direct product n .ak . Normalize the Haar measure of GkAo O so that the volume of IT .Go. is equal to 1 . For each q> E C0(Gk11, 0), we denote by T(q>) the linear operator on S(x, r, kA) given by ( T(q>)f) (g) = fGkA• 0 f(gx)q>(x) dx. It is shown that Trp is of finite rank. Let F be a totally real cyclic extension of k of prime degree /. Set l = x NfJ. Then x is acharacter of the group F},/Fx . Extend r to a function r on the set of infinite places of F by setting r(w) = r( w), where w is any infinite place of F and w is its restriction to k. For each -r E g = Gal{F/k) (the Galois group of F with respect to k) and any function f on GFA' denote by Jrf the function on GFA given by (Jrf) (g) = f(gr) (g E GF) · It is easy to see that Jr leaves the space S{x , r, FA) invariant. For each v E p(k), set F. = k. ®k F. Let o (F.) be the maximal compact subring of F•. The group g operates on F. in a natural manner. If v remains to be a prime in F, F. is a cyclic field extension of k• . If v splits in F, F. is isomorphic to the direct sum of .
a
1 02
TAKURO SHINTANI
/-copies of k •. In both cases, F. is embedded in FA, o in a natural manner. The group GFA, o is the restricted direct product n .aF.. For each g E GFA, O• we denote by Kv E GF. the v-component of g. In the following we choose and fix a generator o of g once and for all. Let fP be an element of Cif'(GFA, ol ) of the form]ffJ(g) = n .lp.(g.), where fPvE Cif'(GF) and fPv is the characteristic function of G•
The notation and assumptions being as above (in particular, fPv and · · · , rn > 2,
'Pv are related by ( 1 ) and (2) of Proposition 1 ) , if rh r2 , ( 1 . 1 0)
REMARK 5. In [10, Theorem 5.6], k is the rational number field and F is a tamely ramified totally real abelian field of degree /. Furthermore, fP is assumed to be unramified. However, the correspondence fPv -+ rp. is explicitly described for each v. In that point, Theorem 5.6 of [10] is more precise than Theorem 2. REMARK 6. If I # 2, S(XX•• r, kA) is isomorphic to S(x. r, kA) as GkA, 0-module. 1 . 5. Let us consider the representation theoretic meaning of both sides of the equality ( 1 . 1 0) of Theorem 2. Via the right regular representation : g �--+ Tg, the group GFA, o acts on S('l,. r, FA)· By a theorem of Jacquet-Langlands (see Proposi tion 1 0.9 and Proposition 1 1 . 1 . 1 of [5]), the space S(z, r, FA) decomposes into an algebraic direct sum of irreducible mutually inequivalent GFA, 0-submodules. Denote by GFA, 0(r, 'l,) the set of equivalence classes of irreducible representations of GFA, o realized on irreducible GF A, 0-submodules of S(l, r, FA). For each tr: e GFA, 0(r, 'l,). denote by V(tr:, r, 'l,) the irreducible GFA, 0-submodule of S(l, r, FA) on which tr: is realized. We have S(l, r, FA) = .E " V(tr:, r, 'l,) (an algebraic direct sum), where the summation with respect to tr: is over GFA, 0(r, 'l,). Denote by tr:" the re presentation of GFA, o given by tr:"(g) = tr:(g "). The obvious relation J" TK. T�" implies that tr:" e GFA, 0(r, l) for each tr: e GFA, o (r, 'l,) and that V(tr:", r, tr:) = J;;-1 V(tr:, r, 'l,). Take a tr: e GFA, 0(r, 'l,). It is known that, for each v e p(k), there exists tr:. e GF. such that tr: is equivalent to the restricted tensor product ®vEJl (kJ 'Irv (except for a finite number of v, tr:v is an unramified representation of GF.). Denote by GFA, 0(r, l• g) (g = Gal(F/k)) the subset of GFA, 0(r, z) consisting of all tr: such that tr:" � tr:. If tr:" # tr:, J" induces a cyclic permutation among V(tr:", r, 'l,) (-r e g). Thus the trace of the restriction of JaT(fP) to the subspace of S(l, r, FA) spanned by { V(tr:, r, 'l,) ; tr: e GFA. 0(r, 'l,) - GFA, 0(r, t• g)} vanishes. If tr:" � tr:, tr:� � tr:v for each v e p(k). There exists a linear operator Ju{tr: .) of order I on the representation space of tr:v such that =
1rv(g") = Ju(tr: .}- 1tr:v (g)Ju (tr:v) {V g E GF.).
ON LIFfiNGS OF HOLOMORPHIC CUSP FORMS
103
Such an operator J"('rr0) is unique up to a multiplication by an /th root of unity. For unramified 1r0, normalize J" (7r0) so that it fixes 1rv{G.) -invariant vectors in the representation space of 1r0• We may assume that the system of linear operators {Jq(1rv) ; v E p(k)} is normalized so that J(f j v(1r, r, x) � ®vE p (k) Jq(1rv). For q> = Ti v'Pv E C()(GFA, o), set 1rv(f/>v) = fGF. rpv{x)1rv(x) dx. Then 1rv(f/>v) is a linear oper ator of finite rank acting on the representation space of 'lrv · Moreover Trace J" T(rp) j v( 1r, r, x) = IJ v trace J" (1rv) 7r:v(f/>v). Hence, (1 . 1 1)
where the summation with respect to 1r is over all GFA. 0(r, l• g). In a similar man ner, we have (1 . 1 2) for every (/) = n v(/)v E C()(GkA' o). where the summation with respect to 'If: = ®v'lf:v is over all GkA. 0(r, x) and
J
'lf:v((/)v) = Gk• q; v{X) 7r: 0(X) dx. Recall that in equalities ( 1 . 10), ( 1 . 1 1) and (1 . 12), 'Pv and (/)v are related by the equalities (1) and (2) of Proposition 2 (V v e p(k)) . Thus, it is now natural to infer that a local implication of these equalities is Theorem 1 . Furthermore, a global consequence is the following representation theoretic version of Theorem 3 of [10]. THEOREM 3 . Assume rh . . . , rn > 2. (1) For each 'If: = ®v'lf:v E G�A. 0(r, x) (7r:v E Gk.), there uniquely exists 1r = ®1rv e Gt;A, 0(r, l• g) (1rv e GF) such that for each odd place v of k, 'lrv is the lifting of 'lf:v from Gk. to GJ We call 1r the lifting of 'If: from G�A. o(r, X) to Gt;A, o(r, l• g). (2) If I i= 2, for each 1r e Gt;A, 0(r, l• g), there uniquely exists 'If: e G�A. 0(r, x) such that 1r is the lifting of 'If: . (3) If I= 2, for each 1r e Gt;A· 0(r, l• g), there exists a 'If: e Gt. 0(r, x) or GkA. 0(r, XX I ) ( X I is the character of order 2 of kJ./P which corresponds to F in class field theory) such that 1r is the lifting of 'If: . Moreover, 'lf: I and 'lf:2 E GkA, o(r, x) have the same lifting to GFA. o(r, l· g) if and only if 'lf: I = 1C2 or 1C 1 = 1C2 ® X I (det). 2. In this section we expose the proof of Proposition 2. Then we indicate how Theorem 1 and Theorem 3 are made plausible by Theorem 2 (the proof of Theorem 2 is a (more or less obvious) modification of proofs of Theorem 1 and Theorem 5.6 of [10]). 2. 1 . In the following three subsections we use the notation in 1 . 1 and 1 .3 without further comment. In particular, k is a p-adic field. Assume that F is isomorphic to the direct sum of /-copies of k. Then we may assume that GF is the direct product •.
104
TAKURO SHINTANI
of l copies of Gk and that, for each x = (xh · · · , x1) E GF, :xa is given by xa (x1 , xh · · · , x1_ 1 ). For any f E GF, set, for any x E Gk, (2. 1) f(x) = f((xz , . . . , x1)- 1 x, Xz, . . . , x1 ) dxz . . . dx1 •
=
J (Gk) l - I
I t i s easy to see that jE C0(Gk) and that A a(f, c) = A (J, N(c)) for arbitrary c E c;(GF) · Thus the proof of Proposition 2 is quite straightforward for this case. 2.2. We summarize known results on orbital integrals on Gk. We denote by qk the cardinality of the residue class field of k and by 7T: a generator of the maximal ideal of o(k). Denote by Qk the set of isomorphism classes of two dimensional semisimple algebras over k. For each K E Qk, choose an embedding of K into M(2, k) as a k-algebra. Via the embedding, we identify K with a subalgebra of M(2, k). For each x E K, denote by x' the image of x under the unique nontrivial k-algebra automorphism of K with respect to k. It is well known that � ' (Gk) = U U (t) Gk (disjoint union), K eQk t
where the union with respect to t is over a complete set of representatives of (Kx - kx) with respect to the action of automorphism groups of K with respect to k. Furthermore,
V
�
{ e z ) zC � Yk} u
(disjoint union).
Let o(K) be the maximal o(k)-order of K. For each nonnegative integer m, we denote by o(K) m the unique o(k)-order of K such that [o(K), o(K) m1 = qf. Let o(K); be the group of units of o(K) m . For each t E o(KY - o(kY, there uniquely exists a nonnegative integer i such that t E o(K)f - o(K)ft-1 • Set i = i(t) for t E o(KY - o(kY . For each/ E C 'Q(Gk), set a(z) = f(z) and {J(z) = A (J, z(1 D) (z E kx) (cf. (1. 5)). Then both a and {3 are locally constant, compactly supported functions on kx . Furthermore, for each K E Qk, we denote by IPK a function on (Kx - kx) given by IPK(t) = A(J, t) (note that IP K does not depend upon a choice of an embed ding of K into M(2, k)). Then IPK is a locally constant function on Kx - kx . The next lemma is a version of Lemma 6.2 of [7] (see also Theorem 2. 1 . 1 and Theorem 2.2.2 of [12]). We include a proof. LEMMA 2. Let the notation and assumptions be as above. Then a triple { rp K ; (K E Qk), a, {3} satisfies the following conditions : (I) The support of IP K is relatively compact in Kx . (2) (/)K(t) = (/)K(t') ("f/ t E Kx - kx). (3) For each z E kx there exists a neighborhood U(z) of z (s.t. z-1 U(z) c o(KY ) such that (2.2)
(/)K(t)
=
{1 -
}
C(K) a(z) + C(K)q�
for any t E U(z) - U(z) n kx, where we put C(K) = [o(K)X, o(K)f']. PROOF. Choose an w E o(K) so that { 1 , w } is an o(k)-basis of o(K). An o(k) basis for o(K)m is given by { 1 , '!T:mw } (m = 0, 1 , 2, . . . ). For each t E K, there uni quely exists
105
ON LIFTINGS OF HOLOMORPHIC CUSP FORMS
c (t) =
(��� :��)
such that
Gw) = t(t) (!}
Then the mapping t ,_. c (t) is an embedding of K into M(2, k) as a k-algebra. It is known that every proper o(K)m-ideal in K is principal (see Proposition 1 of [3]). Thus, Gk =
U Go (k) (1
m=O
71:
)
m t(Kx) (disjoint union).
Hence, for anyfE C0(Gk) and any t E Kx - kx , (2. 3) where we set f-(x) = fc co.> f(uxu- 1) du, cm = [o(K) X , o(K),';;] ,
(cf. the proof of Lemma 7.3.2 of [5]). It is sufficient to prove the lemma for z = I . Since f is locally constant and compactly supported, there exists a neigh borhood U of I in o(KY such that (m = 0, 1, 2, · · · )
and that (n = 0, 1 ,
for all t E
u - u
· · ·
)
n k x . We note that a{1) = f(lz),
� {1)
and
ncoZ
= I; q-,;n f�
(0I 71:1n) (m = 1 , 2,
Hence we have
{
· · ·
)
.
}
rpK (t ) = I - C(K) a(l) + C(K)q �Ct) -1 f3(l ) qk - 1
for any t E U - U n kx . A triple {a, f3, rpK ; K E Qk} of locally constant compactly supported functions a, f3 on kx and a system {rpK ; K E Qk} of locally constant functions (/JK on Kx - kx is said to be an admissible triple if it satisfies the conditions (1), (2) and (3) of Lem ma 2. The next lemma, which is also a version of Lemma 6.2 of [7], follows from Corol lary 1 . 1 .4 of [12] and the previous lemma. LEMMA 3. Let { a, f3, rpK ; K E Qk } be an admissible triple. Then there exists an fE C0(Gk) such that rpK(t) = A(f, t) for any t E Kx - kx. Moreover, for such an f, a(z) = f(z 1 2) and �(z) = A (f, z(1 D). ·
106
TAKURO SHINTANI
2.3. Let F be a cyclic field extension of prime degree I of k. For each K E Qk, set L = K ®k F. Then L is a two-dimensional semisimple algebra over F. A pre scribed embedding of K into M(2, k) is naturally extended to an embedding of L into M(2, F). Furthermore, the norm mapping N f from F to k is naturally ex tended to the norm mapping Nk from L to K. Set L 1 = { t E LX, Nkt = 1 } and F1 = L 1 n F and L' = {t E U ; Nkt E Kx - kx } . The following description of � (GF) is due to H. Saito (see Lemma 3.5 and Remark 3.8 of [10]). LEMMA 4.
..
(1) where the union with respect to t is taken over a complete set of representatives for
L' I L1 with respect to the action of the automorphism groups of L with respect to F. (2) If I "# 2,
��( GF) -
�
..(GF) = zEFUXfF1 (z · 12)GF• " U zE FUXfF1 z ( l � )GF• 11 •
If I = 2,
For each / E C0(GF), we define functions a and (3 on kx as follows : For z E N[P , set z = N[z (z E P) and a (z) = A (f, z · 1 2) and (3(z) = A (f, z(I D) 1 / l k (cf. ( 1 . 3)). For z E P - Nf FX, set (3(z) = 0. If 1 "# 2, set a(z) = 0. If / = 2, set
..
..
It follows from Lemma 1 and Lemma 4 that a and (3 are well-defined functions on kx . Moreover, they are both locally constant and compactly supported. For each K E Qk, we define a function rpK on K x - kx as follows : For t E (Kx - k x ) n Nk L, set t = Nk t (t E U) and rpK(t) = A ..(f, t). For t ¢ (Kx - kx) n NkL x , set rp K(t) = 0. Lemma 1 and Lemma 4 again guarantee that rpK is a well defined function on Kx - kx . The proof of Proposition 2 is now reduced to the following Lemma 5. LEMMA 5. The notation being as above, {a, (3; rp K ; K E Qk} is an admissible triple. For a subgroup V of Kx (K E Qk), set (2.4)
The next lemma will be applied for the proof of Lemma 5. LEMMA 6. For a given K E Qk and a given compact subset C1 of GF, there exist an open subgroup V of Kx and a compact subset C2 ofGF such that g " wg- 1 E C 1 for some w E W( V) implies g"g- 1 E C2 • PROOF. There exists an open compact subgroup V1 of Kx which satisfies the fol lowing conditions :
I07
ON LIFfiNGS OF HOLOMORPHIC CUSP FORMS
The mapping : x ,__. x1 is an isomorphism from V1 to Vf whose inverse mapping .
���
(2. 5) ( V1 is so small that the series is absolutely and uniformly convergent on V1 ). Denote by c3 the image of cl under the norm mapping : X N(x) (cf. 1 . 1). Since N(g"wg- 1 ) = gN(w)g- 1 ; g "wg- 1 E ch for some w E W(VI) (cf. (2.4)) implies gw1g- 1 E c3. Put c4 = c3 n {gw1g-I ; w E W( VI), g E GF} · On C4 , the binomial series (2. 5) is absolutely and uniformly convergent. Hence the closure C5 of the
......
image of C4 under the mapping :
x ,__. x l i i
is compact. It is easy to see that
gwg- 1 = (gwlg- 1 ) 1 1 1 for w E W( Vl). Hence g"wg- 1 E cl for some w E W( VI) implies g"g- 1 E C1 C51. Thus, we may put C2 = C1 C51 and V = V1 • PROOF OF LEMMA 5. It is easy to see that f/JK satisfies the conditions (I ) and (2). For a t0 E Nf ·FX, take z E px such that t0 = Nfz. Let C0 be the support off and
choose an open compact subgroup V of Kx which has the property described in Lemma 6 for C1 = z- 1 C0• Then there exists a compact subset C2 of GF such that zg"wg- 1 E C0 for w E W(V) implies g"g-I E C2• ( W( V) is given by (2.4).) Let {y,- ; i E I} be a complete set of representatives for the double coset Go C Fl\GFfGk. It follows from Lemma I that the mapping : g ,__. g "g- 1 is a homeomor phism from GFfGk onto the closed subset {g E GF, N(g) = I } of GF. Hence, there exists a finite subset /0 of I such that (2.6) zg" wg- 1 E C0 for a w E W( V) implies g E U G(oF)Y;Gk. iEJo
Then, for t E v - v n kx' Zu(t) = K X if v is small enough and I: {-t ;A (f.· , t), f{J K(tot 1) = A 11(/, zt) = iEJ o where tJ.i1 is the invariant volume of Gk n y;:-1 G(oF)Y; in Gk, and f.· E C0(Gk) is
given by
Since /0 is finite, it follows from Lemma 2 that there exists a neighborhood V1 of I in Kx such that, for any t E VI - VI n k and any i E Io,
c
V
{ q��)Jt :(I ) + C(K)qi(t)-l A(.t:. ( I D).
A(.{;, t) = I -
However, (2.6) implies that I: /;( I ) ,u; = Au(!. z) = a(to), iEJo
�o,u.-A( f.· , e D) = Au( !. ze D) = l l l ;l �(to).
There exists a smaller neighborhood Vz c VI such that, for t E Vz - Vz n p ' i(ti) = ord l + i(t). Set U = {t0t 1 ; t E V2}. Then U is a neighborhood of t0 in Kx . We have proved that, for any t E U - U n kx,
108 (2.7)
TAKURO SHINTANI
{ q��\ } a(to) + C(K)q iCtt( / H �(to).
rpx(t) = 1 -
Now take a to E kx - NfP . Set L = F ®k K. If ! -:f. 2, Nf,_P n kx = NfP . Hence there exists a neighborhood U of t0 in K such that U n Nt,_Lx = 0 . Hence, rpx(t) = 0 for any t E u - u n k x . Since a( to) = �(to) = 0, the condition (3) is satisfied. Next assume l = 2. If K is not a field, there exists a neighborhood U of to in KX such that u n NfP = 0. Hence rpx(t) = 0 for any t E u - u n k x . Since � (to) = 0 and C(K) = qk - 1 ' the equality (2. 7) is valid for any t E u - u n
kx.
Now assume that K is a quadratic extension of k . Then there exists s0 E Lx such that t0 = Nf,_s0• Furthermore, Z..(s0) is isomorphic to the multiplicative group of the division quaternion algebra over k. The proof of the following Sublemma is quite similar to that of Lemma 6. SUBLEMMA. The notation and assumptions being as above, for a given compact subset C1 of GF, there exist an open subgroup V of Kx and a compact subset C2 of GF such that g 11s0vg- 1 E C1 for some v E V implies g11s0g- 1 E C2. Normalize invariant measures of Z.,.(s0) and Kx so that volumes of their maximal compact subgroups are all equal to 1 . It follows easily from the Sublemma that there exists a neighborhood U1 of s0 in Lx such that the following integral is absolutely convergent for s E U1 and give� rise to a locally constant function on U 1 : f cF1xx f(g11sg- 1 ) dg. If t = Nf,_s if= k x , the above integral is equal to rpx(t) = A.,.(f, s). If s = s0, the above integral is equal to A.,.(f, s0) x f z.csol !K x die, where die is the quotient measure of the invariant measure of Z.,.(s0) by that of Kx . The volume of Z.,.(s0)/Kx is equal to 1 or 2 according as K is ramified or not. On the other hand, C(K) is equal to qk or qk + I according as K is ramified or not. Since a(t0) = - (qk - l)A.,.(f, s0) and �(t0) = 0, we have shown that there exists a neighborhood U of t0 such that (2. 7) is valid for any t E u - u n k X . The proof of Lemma 5 is now complete. REMARK. The proof of Lemma 5 shows the following further relations between f and f of Proposition 2 when F is a field extension of k. If z = Nfz (z E P), f(z · l z) = A.,.(f, z · 12) and If kx 3 z if= NfP , A(f, z( 1 D) = 0 and f(z · l z) = 0 = - (qk -
( (z 1))
l) A.,. !,
if I
-:f.
2,
if I = 2.
2.4. When F is isomorphic to the direct sum of I copies of k, the correspondence f -+ f of Proposition 2 is made explicit by (2. 1). When F is the unramified cyclic extension of k of degree I, one can make the correspondence explicit iff is bi-G. (F)
invariant. In more detail, denote by L0(Gk, G. (k) ) the set of all functions f E C0(Gk) which are right and left G. (k) -invariant. Then L0(Gk, G. (k) ) becomes a commutative
ON LIFTINGS OF HOLOMORPHIC CUSP FORMS
109
algebra with respect to the convolution product. For indeterminates X and Y, let C[X, Y, x-r, y- I ]0 be the subalgebra consisting of all elements of C[X, Y, x-r , y-1 ] which are symmetric with respect to X and Y. For each /E L0(Gk, Go Ckl), put F(f, k)[X, Y] = 1: m, nE Z Cmnxm yn, where
Then it is known that the mapping : f ,.....
F(J, k)
is an algebra isomorphism from
L0(Gk, Go Ckl ) onto C[X, Y, x- I , Y- 1 ]0 (cf. Theorem 3 of [11]).
Let F be the unramified cyclic extension of
L0(GF, Go CFl ), there uniquely exists a 'A(/)
=
F('A(f),k)(X,
Y). The mapping : f ,.....
Lo(GF, GoCFl ) into Lo(Gk, Go Ckl ).
E
k
of degree l. For each f E
L0(Gk, GoCkl ) such that F(f, F) (X1, Yl)
'A(f)
is a C-algebra homomorphism from
The homomorphism 'A is discussed in [6] in a more general context. The following proposition is implicit in Saito [10]. In fact, considerable parts of 3.4-3. 1 3 and 5. 1-5.4 of [10] are devoted to the proof of Proposition 3.
PROPOSITION 3. The notation and assumptions being as above, f E L0(GF, Go CFl) and 'A(f) E L0(Gk, G0) are related by (1) and (2) of Proposition 2.
2.5. Let M be a finite subset of �(k) containing all primes of k which ramify in F. Denote by C0(GFA. 0,M) the subspace of C0(GFA, 0) spanned by q> fl v
(2. 8)
Denote by GFA. 0(r, X· M, g) (resp. GkA. 0(r, X· M)) the subset of GFA. 0(r, l• g) (resp. GkA. 0(r, x)) consisting of all n = Q9 nv (resp. n = Q9 nv) such that nv (resp. nv) is an unramified irreducible representation of GF, (resp. Gk,) for any v r/= M. For q> = fl q;v E C0(GFA. o' M), we have, by ( 1 . 10), (1 . 1 1) and (1 . 1 2) that (2.9)
/-1
ll: TIv trace J,lnv) 1rv((/> v) = l:0 l: TI trace nv( l/)v), n
j= 1t V
where the summation with respect to n is over all GFA. 0(r, l• M, g) and the sum mation with respect to n is over all GkA, 0(r, XXi • M). For each n = ®nv e GFA. 0 (r, l• M, g) denote by X(n) the subset of U}-;;;1 GkA. 0(r, XXi • M) consisting of all n = ®nv which satisfy the following condition : For each v ¢ M, the lifting of n. from Gk, to G!J.., is n • .
1 10
TAKURO SHINTANI
Then equalities (2.8) and (2.9) together with "the strong multiplicity one theo rem" (see Theorem B of [9] and Theorem 2 of [1]) show the following : (2. 10) Here I must confess that I was too optimistic at Ann Arbor. I was erroneously con vinced that both Theorem 1 and Theorem 3 are immediate consequences of the equality (2. 10). Actually, highly nontrivial considerations are necessary to derive these theorems from (2. 10). Anyway, far-reaching generalizations of Theorem 1 and Theorem 3 are established in [8] . REFERENCES 1. W. Casselmann, On some results of Atkin and Lehner, Math. Ann. 201 (1 973), 301-3 14. 2. K. Doi and H. Naganuma, On the functional equation of certain Dirichlet series, Invent. Math. 9 (1 969), 1-14. 3. Y. Ihara, Heeke polynomial as congruence �-functions in elliptic modular case, Ann. of Math. (2) 85 (1 967), 267-295. 4. H. Jacquet, Automorphic forms on GL(2). II, Lecture Notes in Math., vol. 278, Springer
Verlag, 1 972. 5. H. Jacquet and R. Langlands, Automorphic forms on GL(2), Lecture Notes in Math., vol. 1 14, Springer-Verlag, 1 970. 6. R. Langlands, Problems in the theory of automorphic forms, Lecture Notes in Math., vol . 1 70, Springer-Verlag, 1 970, pp. 1 8-61 . 7. , Modular forms and 1-adic representations, Lecture Notes in Math., vol. 349, Springer Verlag, 1 973, pp. 361 -500. 8. , Base change for GL(2), the theory of Saito-Shintani with applications, Institute for Advanced Study, Princeton, N.J., 1 975. 9. T. Miyake, On automorphic forms on GL, and Heeke operators, Ann. of Math. (2) 94 (1971), 1 74-1 89. 10. H. Saito, Automorphic forms and algebraic extensions of number fields, Lectures in Mathema tics, vol. 8, Kyoto Univ., Kyoto, Japan, 1 975. 11. I. Satake, Theory of spherical functions on reductive algebraic groups over IJ-adic fields, Pub!. Math. No. 18, Inst. Hautes Etudes Sci . , 1 963. 12. J. Shalika, A theorem on semi-simple IJ-adic groups, Ann. of Math. (2) 95 (1 972), 226-242. 13. T. Shintani, On liftings of holomorphic automorphic forms (a representation theoretic in terpretation of the recent work of H. Saito), Informal proceedings (available at Univ. Tokyo) for U.S.-Japan Seminar on Number Theory (Ann Arbor, 1 975). 14. , Two remarks on irreducible characters offinite genera/ linear groups, J. Math. Soc. Japan 28 (1 976), 396-414. 15. , On irreducible unitary characters of a certain group extension of GL(2, C), J. Math . Soc. Japan 29 (1 977), 1 65-1 88. --
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UNIVERSITY O F
TOKYO
Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 2, pp. 1 1 1-1 1 3
ORBITAL INTEGRALS AND BASE CHANGE
R. KOTTWITZ Let F be a local field of characteristic 0 and let E be either F x x F (I times) or a cyclic extension of F of degree l, where l is a prime. Embed F in F x x F diagonally. Let (r}F be the valuation ring of F, and let (r}E be (r}F x x (r}F if E = Fx x F, and let it be the valuation ring of E if E is a field. If E is a field, let r = Gal(E/F), and let (J be a generator of r. If E = F X X F, let (J be the auto morphism (xi> · , Xr) 1-+ (Xz , Xt , xl ) of E, and let r be the group of automor phisms of E generated by O" ; again F is cyclic of order l. Let G = GL2 • The action of F on E induces an action of F on G(E). We use the embedding of F in E to identify G(F) with G(E)F. Define a norm map N: G(E) -> G(E) by putting Ng = ga r - ! · · · gag. This map was introduced by Saito [2]. It de pends on the choice of O". · · ·
· · ·
· · ·
· · ·
• •
0 0
0 0
·
0
,
LEMMA I. Let g, x E G(E). Then
(i) (ii) (iii) (iv)
N(g-axg) = g-l(Nx)g ; (Nx) " = x(Nx)x- 1 ; det(Nx) = NE;F(det x) ; Nx is conjugate in G(E) to an element of G(F).
The first three statements are easy calculations, and it is not hard to get (iv) from (ii). The equality (i) suggests the following definition : x, y in G(E) are (}"-conjugate if there exists g E G(E) such that y = g-"xg. Statements (i) and (iv) together say that N induces a map from ()"-conjugacy classes in G(E) to conjugacy classes in G(F). This map is always injective, and it is surjective if E = F x · · · x F. It should also be noted that x is ()"-conjugate to y if and only if (O" , x) is conjugate to (0", y) in the semidirect product r IX G(E). Choose Haar measures dg and dgE on G(F) and G(E) respectively. If F is non archimedean, normalize dg, dgE so that meas(KF) = meas(KE) = I , where KF = G((r}F), KE = G((r}E). For any element r of G(F), let Gr denote the centralizer of r in G. We now give a definition which is due to Shintani [4]. if
DEFINITION. Let qJ E C;;' (G(E)) andjE C:;'(G(F)). We say thatf is associated to qJ
(A) AMS (MOS) subject classifications (1 970). Primary 22E50. © 19 79, American Mathematical Society
Ill
1 12
R. KOTTWITZ
whenever No
=
T and T is a regular element of G(F) ; f(g-l rg) dg J dt G7(F)\G (F)
(B)
=
o
for every regular element T of G(F) which is not a norm from G(E). In (A) and (B) dt is a Haar measure on GrCF). REMARKS. ( I ) For each regular T e G(F) which is a norm, it is enough to check condition (A) of the definition for only one o. (2) For any element 0 E G(E), let GnE) = {g E G(E) : g-17og = o}. It is not hard to see that if T = No is a regular element of G(F), then Ci'a(E) = G7(F). For T e G(F), define e(r) to be - 1 if T is central and is the norm of an element of G(E) which is not a-conjugate to a central element of G(E), and define e(r) to be 1 otherwise. It can be shown that iff is associated to rp, then
J
(A')
�m�m
f(g- 1 Tg) dg dt
= e(r)
f
�� ��
rp(g-u og) dgE dt'
whenever No = T belongs to G(F), where dt' is a Haar measure on G�(E) which depends only on the Haar measure dt on G7(F) ;
J
(B')
G7(F)\G (F)
f(r1 rg)
'1t
=
o
for every element T of G(F) which is not the norm of some element of G(E). LEMMA 2. (i) Let rp E C�(G(E)). There is at least one fe C�(G(F)) which is
associated to rp. (ii) Let f e C�(G(F)). Then there exists some rp E C�(G(E)) to which f is asso ciated (f and only if fc7(F)\G CF) f(g-I rg) dgfdt = 0 for every element T of G(F) which is not a norm from G(E) (or equivalently, for every regular element T of G(F) which is not a norm from G(E)).
Assume that F is nonarchimedean, and let YfF = Yf( G(F), Kp) be the Heeke algebra of complex-valued compactly supported functions on G(F) which are hi invariant under Kp. Let YfE = Yf(G(E), KE)· Saito [2] introduced a C-algebra homomorphism b : YfE -+ YfF which we will now describe. A functionfin .Yt'p gives rise to a functionfv on the set Dp of isomorphism classes of unramified irreducible admissible representations of G(F) by putting fV(11:) = Tr 7e(f) for 1C E Dp. The set Dp is an algebraic variety over C (it is isomorphic to C * x C* divided by the action of the symmetric group S2, which acts on the product by permuting the two factors). The map f ...... f V is a C-algebra isomorphism, called the Satake isomorphism, of YfF with the algebra of regular functions on the variety Dp. This discussion applies to E as well ; if E is a field, then DE is again isomorphic to C* X C* divided by s2, and if E is F X X F, then DE is Dp X . • •
X
• . .
Dp.
There is a map of algebraic varieties from DF to DE ; if E = F x · · · x F the map is 1C ...... 1C ® • · • ® 1C, and if E is a field it is 7C(-zo) ...... 7C(Res ��-.) where 7C('Z") is the unramified representation of G(F) corresponding to the unramified representation -. of the Weil group Wp of F, and 7C(Res ��-.) is the unramified representation of
113
ORBITAL INTEGRALS AND BASE CHANGE
G(E) corresponding to the restriction of 1: to the Weil group WE of E. So we get a C-algebra homomorphism from the algebra of regular functions on DE to the algebra of regular functions on DF, and hence also a C-algebra homomorphism b : yt'E -+ yt'F • REMARK. If E = F x · · · x F, then .Yt'E � .Yt'F® · · · ®.Yt'F, and b(fi ® · · · ®fz) = II* . . . *h ·
LEMMA 3. IfF is nonarchimedean and E is unrami.fied over F (E = F x allowed), then b( rp) is associated to rp for all rp E .Yt'E ·
···
x F is
This lemma is easy to prove for E = F x · · · x F. It was first proved by Saito [2] when E is a field ; it was subsequently proved by Langlands [1] using the build ings of SL2(F) and SL2{E). Let A be the group of diagonal matrices contained in GL2• To regular elements of A there are associated weighted orbital integrals which appear in the trace formula for GL2 over a global field. We need to introduce a function ).F on G(F) whose logarithm is the weight factor in these integrals. Let g E G(F) and write g = a(� f)k with a E A(F) and k E KF. Then .A.F(g) = I if x E {f)F and .A.F(g) = l x l -2 otherwise. The function .A.E is defined in the same way on G(E) in case E is a field. LEMMA 4. Suppose that F is nonarchimedean and that E is an unrami.fied extension
field of F. Then for any o E A(E) such that No is regular, andfor any rp E :Yt'E, I
J
Amwm
b(rp) (g- 1 (No)g) Iog AF(g)
�!
=
J
'!
rp(g-" og)l og AE(g) E
Amw �
where da is any Haar measure on A(F).
This is proved in §3 of [1]. REFERENCES I. R. P. Langlands, Base changefor GL(2), Notes, Institute for Advanced Study, 1 975. 2. H. Saito, Automorphic forms and algebraic extensions of number fields, Lectures in Math
--
ematics, Kyoto University, 1975.
3. , Automorphic forms and algebraic extensions of number fields, U.S.-Japan Seminar on Number Theory (Ann Arbor, 1 975). 4. T. Shintani, On /iftings of holomorphic automorphic forms, U.S.-Japan Seminar on Number Theory (Ann Arbor, 1 975) (see these PROCEEDINGS, part 2, pp. 97-1 10). RICHLAND, WASHINGTON
Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 2, pp. 1 1 5-133
THE SOLUTION OF A BASE CHANGE PROBLEM FOR
GL(2)
(FOLLOWING LANGLANDS, SAITO, SIDNTANI)
P. GERARDIN AND J. P. LABESSE These Notes present a survey of the results on the lifting of automorphic repre sentations of GL(2) with respect to a cyclic extension of prime degree ofthe ground field, and of some of its applications to the Artin conjecture, with some sketches of proofs. §§ 1-5 are devoted to the definitions and results on the lifting, §6 to the proof of the Artin conjecture in the tetrahedral case. The first part ends up with three appendices describing respectively two-dimensional representations of the Weil group (Appendix A), representations of GL(2) over a local field (Appendix B) and a global field (Appendix C). Part II gives some indications on the proof of the results on lifting. The main tools are the orbital and twisted orbital integrals, and a twisted trace formula [Sa]. The main references for the lifting are [Sa], [S-1], [S-2], [L]. As a side remark, we would like to point out that the study of the example of the general linear group over a finite field [S-2] is illuminating. I. DEFINITIONS, THEOREMS, APPLICATIONS 1. Notation. Let F be a local or global field. Then WF is the Weil group of F and, for F a p-field, W� is the Weil-Deligne group of F [T]. We recall that there exists a canonical surjective homomorphism WF -+ CF which identifies W�b with CF.
In all these notes, E is a Galois extension of F, cyclic of prime degree /, and F its Galois group (the "split case", where E = F X F X . . . X F /-times and r is generated by CT : (xh x2, . . . , x1) f-+ (x2, . . . , xr. x1 ) is handled easily and is left to the reader). For F local, choose a nontrivial character ¢ of the additive group of F, and de fine ¢E1 F = ¢ o TrE1 F · In the following we shall use systematically the notation given by Borel [B) about £-groups : r/J(G), . . . , and by Tate [T] for Weil groups, the L- and e-factors. 2. Base change for GL( l ) . From abelian class-field theory, there is a commuta
tive diagram
AMS (MOS) subject classifications (1 970). Primary 100 1 5 , 1 2A70 ; Secondary 20E50, 20E55 . © 1 9 79, American Mathematical Society \
1 15
1 16
P.
GERARDIN AND J. P. LABESSE
where (}" is a generator of r. The one-dimensional representations of WF are given by the quasi-characters of Wfl-b = CF ; by composing a quasi-character of CF with the norm, a map is defined called the lifting (or more precisely, the base change lift) : x >--> XEI F = X o NEI F which sends the set d(F) of quasi-characters of CF in the set d(E) of quasi characters of CE · The group r acts on the set of quasi-characters of cE by : (rO)(z) = O(zr),
the group f of characters of r can be identified with the set of characters of CF which are trivial on NE 1 FCE ; this group f acts on the set of quasi-characters of CF by multiplication : X >--> x(, ( E f . Then the exactness of the second line of the above diagram implies the following result : PROPOSITION I . The lifting X >--> X EI F defines a bijection from the orbits of f in d(F) onto the invariant elements by F in d(E) ; moreover, the following relations hold: L(X EI F) = fl L(x (), E C F
e(X E 1 F) = IT. e(x () for F global, CE F
e(X EI F• ¢ El F) = J...E I F(¢) - 1 IT. e(x(, ¢) for F local. r;, E r
3. Base change for GL(2) on the L-groups. 3 . 1 . Let G be the group GL(2) over F, and GE 1 F be the group over F defined by restriction of scalars of the group GL(2) over E, so that GE 1 F(F) = GL(2, E) ; the group GE 1 F is quasi-split, and there is a natural map : LG = GL(2, C) X Fr----- LGE I F = GL(2, cy � FF ;
here GL(2, C)F is the set of applications from F in GL(2, C), and FF = Gal(F/F) acts by permutations of the coordinates [B, §5]. Given any p E W(G), its restriction to W£ defines an element PE l F E W(GE 1 F) ; the set W(GE 1 F) can be identified with the set W(G/E) [B], and the above map defines the application (/)(G) ---> (/)(GEl F) = (/)(GIE), p
f------->
PE l F
called the base change. For F local, let FrF be a Frobenius element in FF ; when E is unramified, then FrE = Fr� E rE is a Frobenius element for E. Moreover, if p is unramified and defined by FrF >--> s, where s is a semisimple element in GL(2, C), then PEl F is unramified and is given by FrE >--> s 1 • The properties of L- and e-factors with respect to induction [T] show that : L(pEI F) = fl. L(p ® (), C EF
e(PEI F) = IT e(p ® (), r.Er
for F global,
e(pEI F• ¢EI F) = J...E I F(¢) - 2 IT e(p ® ( , ¢ ), for F local. CEF
BASE CHANGE PROBLEM FOR
GL(2)
1 17
3.2. From the classification of the two-dimensional admissible representations of
w;.. (see Appendix A), one has the following result :
PROPOSITION 2. (a) The lifting p E l/J(G) t-> PE I F E (/J(GE I F) has for image the set of F-invariants in l/J(GE 1 F). (b) In the following cases, the lifting is given by
(f-l EB )))E l F = /-lEI F EB ))E l F• {lnd �k O)EI F = Ind ��E OKEI K for E =f. K, (Ind �k O)EI F = 0 EB (JO for E = K, 1 =1- a E r, (x ® sp(2))E I F = XEIF ® sp(2) (for F nonarchimedean). (c) Given a nondecomposable p E l/J(G), the representations which have the same lifting are the p ® ' for all ' E f ; for p = il EB f-l• the representations which have the same lifting are the il' EB 1-l'' for all ,, ,, E f.
4. Base change over a local field. In this section, we denote by a a generator of F = Gal(E/F). 4. 1 . Let ll(G) be the set of classes of admissible irreducible representations of G(F). There is a conjectural bijection l/J(G) � ll(G) [B]. The base change map l/J(G) -+ l/J(GE1F) must reftect a map ll(G) -+ ll(GE1F) · The definition of base change for representations of G(F) will be given in 4.3 ; since the image of l/J(G) is the set of F-invariant elements in l/J(GE1 F), one studies first the admissible irreducible repre sentations of G(E) equivalent to their conjugates by r. 4.2. Let 7t be an admissible irreducible representation of G(E) such that t1 7t � 7t ; then there exists an operator C on the space o f 7t such that c- 1 7t(z)C = 7t(zt�) , z E G(E), and CZ = Id. This operator is determined up to an lth root of unity. The mapping 7t' : (am , z) �--> Cm 7t(z) defines an extension 7t' of 7t to the semidirect pro duct F � G(E). PROPOSITION 3. This representation has a character given by a locally integrable function Tr 7t' on F � G(E).
On a x G(E), the character Tr 7t' defines a a-invariant distribution on G(E), i.e., invariant under a-conjugation : z �--> y-t�zy, y, z E G(E). Let us state some properties of the a-conjugation. For z E G(E), put or simply N(z) if no confusion can arise. PROPOSITION 4. (a) NE1 F . tJz is conjugate in G(E) to an element of G(F) ; (b) z �--> NE I F , t1z defines an injection of the set of a-conjugacy classes of G(E) into
the set of conjugacy classes of G(F) ; (c) the elliptic classes of G(F) obtained by NE / F , tJ are those with determinant in NE l FEx ; the hyperbolic classes of G(F) obtained are those whose eigenvalues are norms of Ex ; any unipotent class of G(F) is in the image of N. 4.3. Definition of the base change for GL(2) over a local field. Let 7r: E ll(G) and 7t
1 18
P. GERARDIN AND J. P. LABESSE
be an irreducible admissible representation of G(E) which is equivalent to its con jugate by a. Then it is called a base change lift of 11:, or a lifting of 11:, if either (a) 11: = 7r:(f.t , v) and it = 7r: (f.tEIF • vE l F), or (b) there exists an extension it' of it to F 1>< G(E) such that Tr it'(a x z) = Tr 11:(x) for z e G(E) whenever NE I F.a z is conjugate in G(E) to a regular semi simple element x e G(F). Some of the notation in the following theorem is explained in Appendix B. THEOREM 1 (BASE CHANGE FOR GL(2) OVER A LOCAL FIELD). (a) Any 1C E fl(G)
has a unique lifting 11:E I F e fl(GE1F), and any 11: e D(G)fixed by F is a lifting ; (b) the lifting is independent of the choice of the generator a of F; (c) 1CE I F = 1T:eJF � 11:' = 11: ® C for a C e f, or 11: = 7r:(f.t , v), 11:' = 7r:(f.t ', v') with f.l-lll' and v-I v' in F; (d) w"EIF = (w")E I F • (11: ® X)E I F = 11:E I F ® XEI F for any one-dimensional repre sentation X of FX , (11:E I F)v = (11:Y)E I F (contragredient representations) ; (e) for E => F => k with E and F Galois over k, 7(11:EIF) = (r11:)E I F for any r E Gal(E/k), with image r in Gal(F/k) ; (f) at least for p e C/>(G) not exceptional, 11:(p)E I F = 11:(pE I F) · S. Global base change. 5. 1 . Let D(G) be the set of classes of irreducible admissible automorphic repre sentations of G(AF) = GL(2, AF), where AF is the ring of adeles of the number field ·
F. From the principle of functoriality [B], the base change on £-groups should re flect a map from D(G) to D(GEl F), the set of irreducible admissible automorphic re presentations of G(AE) = GL(2, AE) = GE1 F(AF) ; such a map must be compatible with the local data. For any place v of F, put Ev = E ® F Fv ; it is a cyclic Galois extension of F or a product of 1 copies of F; in this latter case, define the lifting 7r:Er!F. of 11: e ll(G) by 1CEr!Fv = 1C ® · · · ® 11: (I times). 5.2. Definition of the global base change for G L(2) Let 11: e D(G), it e D(GE1F) ; then it is called a lifting of 11: (or more precisely a base change lift of 11:) if, for every place v of F, itv is the lifting of 11: 0 • 5.3. The notations used in the following theorem are those of Appendix C. THEOREM 2 {GLOBAL BASE CHANGE FOR GL(2)). (a) Every 1C E ll(G) has a unique lifting 1CE I F E ll( GEIF) ; (b) a cuspidal it e D(GE 1 F) is a lifting if and only if it is fixed by r, and then, it is a .
lifting of cuspida/ representations ; a cuspidal 11: e D(G) has a lifting which is cuspidal except for I = 2 and 11: = 11:(Ind �k 8) and then 11:E I F = 11:(8, 118) ; (c) for a cuspida/ 11: e ll(G), the representations 11:', which have 11:E1 F for lifting are the 11:' = 11: ® C with C e F; (d)
(J)"EIF = (w1r)E I F (11: ® X)EIF = 1CEJF ® X E!F (1CEIF) V = (1i")EI F
(central quasi-characters), (twisting by a quasi-character), (contragredient representations) ;
(e) for E => F => k with E and F Galois over k, 7 (11:E I F) Ga!(F/k) image of f e Gai(E/k) ;
= (711:)EI F for
any r e
BASE CHANGE PROBLEM FOR
GL(2)
1 19
(f) i/ 11: = 7r:(p) for some p e cp(G), then 71:E IF = 71:(PE I F). REMARK. There are examples of noncuspidal n e /l(GE1F), fixed by F which are not liftings (cf. [L, § 10]). 6. Artin conjecture for tetrahedral type. Let p be a two-dimensional admissible representation of the Weil group of the number field F; we assume that its image modulo the center : Wp ---+ GL(2, C) ---+ PGL(2, C) is the tetrahedral group �4. This group is solvable : 1 ---+ D4 ---+ 2f4 ---+ C3 ---+ I . The action of the cyclic group C3 on the dihedral group D4-the so-called mattress group-is given by the cyclic permutations of its nontrivial elements. The inverse image of D4 in WF is a normal subgroup of index 3, hence is the Weil group WE of a cubic Galois extension E of F:
1
--->
WE ---> WF --->
1
___,
n
14
___,
1
2f4
___,
Gal(E/F) ---> 1
11
c3 -----. 1
The restriction PE l F of p to WE has for image the dihedral group D4 ; it is induced from a one-dimensional representation of a subgroup of index 2 in WE• so that there is a corresponding cuspidal automorphic representation 11:(pE 1 F) of GL(2, AE). The inner automorphisms of 2f4 give an action of C3 on D4 , and the action of r = Gal(E/F) fixes the class of PE l F• hence also the class of 11:(pE 1 F). From Theorem 2 (5.3), this representation is the base change lift of exactly three classes of irreducible cuspidal automorphic representations of GL(2, AF), and their central character has for base change lift the central character of 11:(pE 1 F), which is equal to det PE l F = (det p)E1 F ; there is only one of them, say 11:, with, the central character det p. The Artin conjecture for the representation p is the holomorphy of the corres ponding L-function : s ..... L(s, p). According to Jacquet-Langlands [J-L, Theorem 1 1 . 1, p. 3 50], the L-function s ..... L(s, 11:) corresponding to cuspidal 11: is holo morphic ; hence, the Artin conjecture will be proved for p if we show the equality of these two L-functions. From [J-L, pp. 404--407], this will be done if the following assertion is shown to be true : (c0) 71:v = 11:(p.) for each archimedean place v of F, and for almost all v. As E is cubic over F, each infinite place v of F splits in E, so for wl v, we have the equations : which are more generally true for any place v of F which splits in E. Now if v does not split and is unramified in E, and if moreover Pv is unramified, then so is the restriction PEviFv of Pv to wE. • and the representation (71:E I F)v = 7r: (PE.rF) is unramified ; since Ev/F. is unramified this representation is the base change lift of unramified representations. This shows that 71:v is unramified ; call p"• a two-dimensional representation of WE. • such that 71:v = 11:(p".). We have shown that our assertion (c0) is equivalent to : (c 1 ) if v does not split and is unramified for p and E, Pv and Prr. are equivalent. The adjoint representation of PGL(2) defines an injection of PGL(2) in GL(3),
1 20
P.
GERARDIN AND J. P. LABESSE
hence a morphism A : GL(2) --+ GL(3). We observe now that the condition (c1 ) is equivalent to the apparently weaker condition : (cz) if v is unramified for p and E, the three-dimensional representations Ap. and Ap"• are equivalent. In fact, call a (resp. b) the image of a Frobenius in WF. through p. (resp. p") ; if (c2) is satisfied, a E Cxb ; but det a = det b, hence a = ± b. If a = - b then, since p. and p"• have the same restriction to WE.• as is conjugate to - as, that is Tr{a3) = 0. Hence A(a3) is of order two, and this means that A(a) is of order 6 ; but the image of Ap. is in the tetrahedral group which has no element of order 6 ; so we have a = b, hence p. = p"• ' The introduction of Ap is motivated by the crucial observation, due to Serre, that this three-dimensional representation is induced by a one-dimensional representation of WE ; in fact, the tetrahedral group leaves invariant the set of the three lines joining the middles of the opposite edges of the tetrahedron. This means that Ap is induced by the one-dimensional represen tation () of the stabilizer of one of these lines (obtained by restriction of Ap) ; but this stabilizer is the pull-back of the dihedral group D4 c �4 in WF, which is the subgroup WE : Ap = Ind �� (}, From [J-PS-S), to such a three-dimensional irreducible monomial representation
Ap of WF is associated an irreducible cuspidal automorphic representation 11:(Ap) of GL(3, AF). On the other hand, the morphism A reflects a lifting from irreducible
cuspidal automorphic representations of GL(2, AF) to automorphic representations of GL(3, AF) ; and, here, the representation A11: corresponding to 11: is cuspidal [G-J-2]. To prove (c2) it suffices to show the condition : (c3) the lifting A11: is equivalent to 11:(Ap). There is a practical criterion given by [J-S] to prove the equivalence of such repre sentations : 11: 1 and 11:2 are equivalent if and only if L(s, 11: 1 x 7r2) has a pole at s = 1 , where L is the L-function attached to the representation of GL3(C) x GL3(C) in GL9(C) given by the tensor product. We shall prove that almost all local factors of L(s, A11: x 11:(Ap) Y) and of L{s, 11:(Ap) x 11:(Ap)v) are equal ; by nonvanishing properties of local factors [J-S], this is enough to prove that L(s, A11: x 11:(Ap)V) has a pole at s = I, and hence (c3) will be proved. If v is split, then 11:. = 11:(p.) and, at least when 11: v and P v are unramified, (A11:). = 11:(Ap.) : the local L-factors are then equal. If v does not split in E and is unramified for E and p, the two local L-functions are those associated to the nine-dimensional representation of WF given by Ap"• ® ((Ap) ';') and Ap. ® ((Ap) ';') ; but we know that Ap. = Ind ��= () . . Now if U (resp. V) are representations of a group G (resp. a subgroup H) one has U ® Ind]} V � Ind ]}( V ® Res ]}U) ; also recall that the two representations p"• and Pv have equivalent restrictions to WE. ; hence Ap"• ® (Ap)'j is equivalent to (Ap.) ® (Ap)'j so that they have the same L-factor. This concludes the proof of the Artin conjecture for the tetrahedral case. The above proof is taken from a letter of Langlands to Serre (December 1 9 75) ; it also contains some indications on a method to handle the octahedral case ; however the latter requires some results on group representations which are not ; yet available. Still, a partial result is obtained [L, § 1 ] :
BASE CHANGE PROBLEM FOR
GL(2)
121
Assume that p is of octahedral type ; we use the fact that @;4 has a normal sub group W4 of index 2 ; hence there is a quadratic extension E of F for which the res triction PE l F is of type w4 . By the above theorem, 7r (PEI F) exists and, by Theorem 2, it is the base change lift of two cuspidal admissible irreducible automorphic representations 1r' and 1r" of G(AF). Assume now that F = Q, that E is totally real and that the complex conjugations in Gal(Q/Q) are sent by p into the class of (A -�) ; in such a case the components of 1r' and 1r" at the real place verify 1r' = 1r::O = 7r (p�) with P� = 1 EEl sign ; then 1r' and 1r" correspond to holomorphic automorphic forms of weight one. This situation has been studied by Deligne-Serre [D-S], and they show that 1r' = 1r(p') and 1r" = 1r(p") for some representations p', p" of WF in GL(2, C) ; now, by Theorem 2, 1r(p')EIF = 1r(p'EIF), and the same is true for 1r" ; this shows that p' and p" are the two representations which lift to PEl F ; hence either p = p' or p". Thus one concludes that either 1r' = 1r(p) or 1r" = 1r(p), and this gives oo
THEOREM 4. For a two-dimensional representation of W0 which is of octahedral type and which sends the complex conjugation on (A -�), and such that the above quadratic field E is real, the Artin conjecture is satisfied.
Appendix A. List of the two-dimensional admissible representations of W� [D).
Notations. F is a local (resp. global) field, CF is Fx (resp. the group of ideles
classes of F) ; W� is the Weil-Deligne group of F [T]. For F global, v a place of F, there is an injection WF, --> WF which defines an application p ,_. Pv from the two-dimensional admissible representations of WF into those of WF, ; if another representation p' of WF satisfies p� Pv for all but a finite number of places, then p' is equivalent to p, and p� Pv for all v. The two dimensional admissible representations of WF are classified by the image of the inertia group in PGL(2, C), called the type of the representation. (1) Cyclic type : f.1. E9 v is the sum of the two one-dimensional representations of WF defined by f.1. and v ; f.1. (£) v � v (£) p ; det(p (£) v) = pv ; (f-t (£) v) ® X (f-!.X) EEl (vx) ; (p E9 v)v = f..l.- 1 E9 v - 1 The L- and e-functions verify : �
�
.
e(p (£) v) = e(p)e(v) L(p EEl v) = L(p)L(v), e(p (£) v , ¢) = e(p , ¢)e(v, ¢) (if F is local), f.1. (£\ v = (8) . ( f-t v EEl v.) (if F is global).
(if Fis global),
(2) Dihedral type : r = Ind�f (), where () is a quasi-character of CK , and K a separable quadratic extension of F; r is irreducible if and only if () =F 11() (I =F u E Gal(K/F)) . For () = 110, let X be a quasi-character of CF such that () = X o NK 1 F and let t1 be the character of CF with Kernel NKl FeK· Then Ind�f 0 = x <£l xt1 ; det 1: = t1 . O lcF ; 1: ® X = Ind�f (O x 0 NK I F) ; f = Ind�:;:o - 1 ; L(r) = L( ()) , e(r) = e(O) (F global), e(r, ¢) = AK I F (¢ ) e( () , ¢ o TrK 1 F) (F local) ; for F global, r = @rv with rv = Ind�k!:' Ov for K./Fv quadratic, and rv = O v EEl Ov for K. = Fv x F• . The equivalences are : Ind�l () 1 Ind�f 8 2 ¢> either K1 = K2 and () I > ()2 conjugate by Gal(K/F), or K1 =F K2, 0{1011 and 02"2021 are of order 2 and 01 o NK 1 K2tK 1 = 82 o NK 1 K2tK2• (3) Exceptional type : the image of the inertia group in PGL(2, C) is W4 (tetra hedral type), @;4 (octahedral type), or W5 (icosahedral type) ; they occur only for F �
1 22
P. GERARDIN AND J. P. LABESSE
global or F nonarchimedean local of even residual characteristic; in this latter case, the icosahedral type does not occur. (4) Special type (occurs only for F nonarchimedean local) : X ® sp(2) for a quasi-character X of Fx and sp(2) the representation of W.F defined by z E C >->
(� �)
and w E WF >->
( � �) . ' I
Appendix B. List of admissible irreducible representations of GL(2, F), F local field [J-L].
Notations. G = GL(2, F), I I is the absolute value defined by the dilatation of the Haar measure on F: d(ax) = lal dx, and ¢ is a nontrivial character of the additive group of F. Representations. (1) Principal series p(p , ].1), where p , ].1 are quasi-characters of Fx . (a) Definition. Let p(p , ].1) be the representation of G by right translations in the space of smooth functions for G such that f(ang)
=
p(u) ].l(v) luv-1 1 1 /2 f(g)
for any g E G, a = (8 � E G, n E (ij �). When this representation is irreducible, then 1C(p , ].1) is p(p , ].1). When p(p , ].1) is reducible, there are exactly two irreducible subquotients ; one is finite dimensional and 1C(p , ].1) is this one. The other one is denoted a(p , ].1). (b) Equivalences. 1C(p , ].1) 1C(].I, p). For F = C, any irreducible admissible re presentation of G is equivalent to a 7C(p , ].1). (c) Finite dimensional representations. F nonarchimedean : they are one-dimen sional, and are the 7C(p , ].1) for w-1 = 1 1 ±1 : 7C(p , ].1) (x) = l det x l ± 112].1 (det x), x E G : the corresponding representations a(p , ].1) are called the special representations ; F = R : They are the 7C(p, ].1) with p ].l- 1 (a) = lal±(n+I) · sign(a) n for integers n � 0 ; F = C : they are the 7C(p , ].1) with /-l].l-1 (a) = [an+l (a) m+1 ] ± 1 for integers n � 0, �
m � 0. (d) Other properties.
Restriction to the center : w" c p . v l = p v ; twisting by a quasi-character of P : 7C(p , v) ® x = 1C(f-tX• vx) ; contragredient representation : 1C(p , ].l)v 7C(p - 1 , v-1) ; local factors : L(1C(p , v)) = L(p) L(v), e(7C(p , v) , ¢) = e(p , if;) e(v, ¢) . (2) Wei/ representations. 1C(t"), 1: = IndU::J;' (}, with (} a quasi-character of K x , and K a separable quadratic extension o f F. (a) Definition. Let GK be the subgroup of index two in G defined by those elements which have a norm of Kx for determinant. Fix a nontrivial character ¢ of the additive group of F. Then 7C(7:) is the class of the representation of G induced by the following representation r((}, ¢) of GK, in the space of smooth functions f on Kx such thatJ( t - 1 x) = (} (t)f(x) for t, x E Kx , NKl F t = I : �
(r((}, if;) (� �) f )(x) (r((} , ¢) (� �)! )(x) 1
=
if; (uNK 1 FX) f(x),
= AK t F(¢)
U
E F,
JK J(y)if;K ! F(xya)d�y,
BASE CHANGE PROBLEM FOR
(r(O , ¢) (� �) t )Cx) = O(b)f(bx)
GL(2)
1 23
for a = NK1 pb , b e Kx ,
where ¢ K 1 F = ¢ o TrK 1 F • y" is the conjugate of y by the nontrivial element u of Gal(K/F), dr/Jy is the self-dual Haar measure on K with respect to the character ¢ K 1 F and A.K 1 p(¢) is the unitary part of the local factor e( &, ¢), where (1 is the nontrivial character of px which is trivial on NK 1 pKx . (b) Equivalences. (I ) n(Ind!tk 0) = n(Ind!tk "0) ; ( 2) n(Ind!tk 0) - n(x. xa) if 0 = "0 and x is a quasi-character of px such that O (a) = x(NKI F a) ;
(3) Other equivalences : n(Ind ;;1 fh) = n(Indltk2 02) for K1 =I= K2 , if and only if 01"10}1 and 0 2"2 021 are of order 2 and satisfy 0 1 o NK1 KztK1 = 0 2 o NK 1K?JK2• (c) Characterization. If a nontrivial character X of px fixes a class 1C of irreducible admissible representations of G : n ® X = n, then X is of order 2, attached to a separable quadratic extension K of F and 1C = 1C(z') where 1: = Ind!tk 0 for a suitable 0, and conversely (cf. [L, §5]). (d) Other properties. For F = R, or nonarchimedean with odd residual characteristic, any irreducible admissible representation is a 7C(A., p) of a 7C(Ind�k 0) ; restriction to the center : m" < • > = (1 · O lpx ; twisting by a quasi-character of px : 1C(7:) ® x = 7C(Indltk 0 . x a NKl p), contragredient representation : 7C(f) = n(Indltk o- 1 ), local factors : L (1C(7:)) = L(O) , e (n(7:), ¢) = AK I F(¢)e(O , sVKIF)· (3) Exceptional representations. They occur only for F nonarchimedean of residual characteristic 2, and, up to twisting by quasi-characters of px, their number is 4( 1 21 - 2 - 1)/3 for F of characteristic 0, infinite for F of characteristic 2. They are supercuspidal (see complements below) (cf. [Tu]). (4) Special representations. They occur only for F nonarchimedean, and are the infinite dimensional subquotient u(p , ))) of the reducible p(p, ))), that is for p))-1 = I 1 ± 1 ; one has u(p, ))) ""' u()), p), u(p, ))) = xu( l l 1 1 2 , 1 1-1 1 2) for x = p i 1 - 1 1 2 . Complements.
(1) Representations u(p , ))). When the induced representation p(p, ))) is not irreducible, u(p, ))) denotes any representation equivalent to the unique infinite dimensional subquotient of p(p, ))) : for F = R, the representations u(p, ))) are the representations 7C(Ind �� 0), 0 =1: "0. (2) For a two-dimensional admissible representation p of Wp, there is at most one irreducible admissible representation 1C of G often denoted 1C(p) when it exists such that m,. = det p , L(1C ® x) = L(p ® x) and e (n ® X• ¢) = e(p ® X• ¢) for any quasi-character X of px ; for p reducible, p = p EB )) , then 1C = 1C(p, ))) ; for p = X ® sp(2) then n = u(p, ))) with f.t = x l p n, )) = X 1 1-1 1 2 ; for p dihedral p = Indltk 0, then 1C = 7C(Indltk 0) ; in the remaining cases, that is when p is ex ceptional, the existence of n is still not completly settled, but base change techni ques were used to prove it in many instances. Conversely, any 1C should be a n(p). Appendix C. Irreducible admissible automorphic representations of GL(2) [J-L]. Notations. F is a global field, Ap its ring of adeles, Cp = A'j./Fx the group of ideles classes, I I the absolute value on Ap.
1 24
P. GERARDIN AND J. P. LABESSE
(I) Noncuspidal representations. (1 . 1) :n:(A, fl.) for A, fl. Grossencharakters of F: they are the following represen tations : :n:(A, fl.) = ®.:n:(A., f.J. v) ; the one-dimensional representations are the :n:(A, fl.) for Af.J.- 1 = 1 1± 1 . (1.2) Any noncuspidal irreducible admissible automorphic representation :n: of GL(2, AF) has the following form : there are two Grossencharakters A and fl. of F,
and a finite set S of places of F, such that the components :n:. of :n: are given by V ¢ S, V E S, :7t' v = O' (A., fl.v ), :7t' 0 = :n: (A v, f.J.v) ,
where u(A. , f.J. v) denotes the infinite dimensional subquotient of the reducible p(A., fl. v) (Appendix B). (2) Cuspidal representations (examples). (2. 1) :n:{'L') with .... = IndJH () for a separable quadratic extension K of F and a
Grossencharakter () of K, not fixed under Gal(K/F), is the representation :n:{'L') = ® :n:(Ind�k'• o.) •
= () . Ei?> () . for K. = F. x F• . Properties. (1) Let :n: e D(G) ; in order that there exist a nontrivial GrosseD
with Ind�k' () .
charakter X of F such that :n: ® X = :n: , it is necessary and sufficient that there exist a separable quadratic extension E ofF and a Grossencharakter () of E such that (a) X is the character of CF with Kernel NErFCE ; (b) :n: = :n:(Ind�� ()) (in particular :n: = :n:('L', 'L'X) if () = a(), where .... is a GrosseD charakter of Fwhich has () for lifting to E) ; (2) :n:(Ind��� 0 1 ) = :n:(Ind�f2 02) <=> either K1 = K2 and ()2 = 0 1 or a()h or K1 =1: K2 then () 1 , a1 0 1 1 and 02 · a()21 are of order 2, and (0 1 ) o NK1K2rK1 = (02) o NK1 Kz/Kz· {3. 1) More generally let p be a two-dimensional admissible representation of WF ; we say that :n: = ®:n: . is :n:(p) if :n:. � :n:(p.) for all v. The existence of such :n: when p is irreducible is related to the Artin conjecture for the p ® X• where X is any quasi-character of CF [J-L, § 1 2]. (3.2) Of course there are many other, more complicated, types of cuspidal representations : think of the classical L1 for example. II. BASE CHANGE FOR G4, A SKETCH OF THE PROOF. 1. The trace formula. In all the following we shall use notations close to those of (G-J-1] in these PROCEEDINGS. Let F be a number field, E a cyclic extension of prime degree, put Z1 = NElF Z(AE), where AE denotes the ring of adeles of E, and Z1 {F) = z1 n Z(F) ; as usual Z is the center of G4 = G and is identified with the multiplicative group. Since E/F is cyclic one has Z1 (F) = NErFZ(E) . Choose a character co of Z1 /Z1 (F) and consider the space £2 (Z1 · G(F)\G(A), co) = £2 of functions on G(F)\G(A), which transform on Z 1 according to co : rp(zrg)
=
co(z)rp(g),
z e zh r e G(F),
and square-integrable on Z1 · G(F)\G(A). In such a situation, which is slightly more general than the one studied in [G-J-1]
BASE CHANGE PROBLEM FOR
GL(2)
1 25
(where E = F), one defines in an obvious way the spaces L5 and L�p · The re striction of the natural representation of G(A) in L2 to L5 EB L�P will be denoted by r. Iff E 'tf':,"'(Z1 \G(A), w- 1 ) , the space of smooth functions on G(A) compactly sup ported modulo Z1 which transform according to w- 1 on Zh the operator r(f) is of trace class. The Haar measures being chosen as in [G-J-1, §§6-7] we assume more over that vol (Z1 · Z(F)\Z(A)) = I. Then tr r(j) is the sum of the expressions (i)-(vii) below (we assume thatfis a tensor product of local functions/.) (cf. [L, §8]). (i)
I;
zEZ! (F)\Z(F)
vol (Z1 • G(F)\G(A)) · /(z),
(ii) where Iff is a set of representatives of the conjugacy classes of elliptic elements (i.e., whose eigenvalues are not in F) taken modulo Z1(F), and e (r) is 1- (resp. 1) if the equation ()- 1 ro = zr has (resp. has not) a solution in z E Z1(F) - { 1 } . (iii) where D o is the set of pairs r; = (f.-1, ).!) of characters of A x jp such that f.-1).! in duces w on Zh where M(r;) and JC� are defined in [G-J-1, §4], and with JC�(f) = JzM
(v)
where ).0 and other notations are defined in [G-J-1, §7-B]. The remaining terms will not be written as in [G-J-1] since they are there expressed by noninvariant local distributions. For example the local distribution t A 1 (r, f. )
=
-
.:J(r)
) log
J lxvl>l ( fvK
a (a - b)x. o b
jx. l dx.
where r = (g 2), .:J .(r) = j (a - b)Zjab j �12 and ff(g) = JK. f.(k- 1gk) dk, can be written A 2(r, f.) + A 3(r , f.) where A 2(r, f.) is invariant, and A 3 fits with other terms to provide an invariant expression. More precisely one takes for a nonarchimedean place v : A 2(rJ.)
=
log j (a
- b) f a i .F(r J.) +
J
.:J.(r)f,K(z) lxvl>l log jx. l dx. - l afb l 1 12 · 1 w .l
(z (� �)), =
126
P. GERARDIN AND J. P. LABESSE
where F(r. f.) = �Mr) JA.1c. fv(g- 1rg) for F• . If v is archimedean one takes A2(r , f.)
=
log I I -
� I.
dg
F(r. J.) -
This yields the term (vi)
and
w.
is a uniformizing parameter
�'g : �:�2 J v
z.N.,G. J.
(g - 1 znog) dg .
w*- v
It can be checked that r .... A3(r. J.) extends to a continuous map on A (F.) and one sees that the terms 6.34 in [G-J-1] minus our (v) plus 6.35 minus our (vi) yield the term This can in turn be transformed by a kind of Poisson summation formula to
One has to add the term 6.36 of [G-J-1] minus our (iv) to get the final term : (vii) 2. The twisted trace formula. Here we shall use definitions and results of the Kott witz lecture (see [K] in these PROCEEDINGS) . As above we follow closely [L, §8]. Let L2 (Z(AE)\G(AE), w) = l2 where w = (t) 0 NE l F· The Galois group r = Gal(E/F) acts on £ 2 by up(x) = p(xu) for p E £ 2, X E G(AE) and fJ E F. Let Rd denote the restriction of the natural representation of G(AE) in £ 2 to the discrete spectrum l5 EB L;P. The projection commutes with the action of the Galois group ; hence Rd can be extended to a representation R� of r !>( G(AE). Let ¢ E 0'�(Z(AE)\G(AE), w- 1 ) ; then Ri
tr(R�(fl)Ri
=
J
Z (AE) \G (AE)
K(
Assuming, as usual, that ¢ is a tensor product : ¢ = @¢., one can proceed as in [G-J-1] to compute this integral ; if fJ -:f. 1 it is the sum of the following terms
(1)-(7) : (1)
I; vol(Z(A)G�(E)\G�(AE)) •
J
z�q�w�
where the sum runs over the (]-conjugacy classes of elements o such that N(o) is central, and G� is the (]-centralizer of o (cf. [K]). (2)
BASE CHANGE PROBLEM FOR
GL{2)
1 27
where C(j is a set of representatives of the a-conjugacy classes that are not a-con jugate to a triangular matrix, taken modulo Z(E) in G(E), and e(o) is t or I accord ing as the equation -.-qo-r = zo has or not a solution in Z(E) with z ¢ Z(£)1- u. (3)
where r;v = (v, f-l) if r; = (f-l, v) is a pair of characters of Aj;/E x with f-lV = w. The representation 7r:7J is realized in a space of functions on G(AE), the action of a E r defines an operator n�(a) from the space of 7r:7J to the space of 7r:11" ; then ME( 117J) intertwines 7r:17" and n••• but 177Jv = r;. Hence the product M(117J) n�(a) nkp) is a well defined operator in the space of n.7J., and the above expression is meaningful. (4)
where ij = (f-l o NE1F, v o NE 1F) if r; = (f-l, v). There are / 2 elements ij giving rise to the same ij. The reader should be aware that our notation ij has not the same meaning as in [L]. Ao TI v 017(0, ifJv),
(5)
where 017(0, r/Jv) = L(l , l v)-1 J J J ifJv(k-11t-11n-qn0ntk)t-2P dn dt dk, with k the standard maximal compact subgroup of GL2(E ® Fv), t =
(� �),
�-2p =
1 � r-l
and flo
=
E
Kv,
(� D
such that trE; F z = I . The integration is on Kv x Z(Ev)\A(Ev) x N(Fv)\N(Ev), where Ev = E ® Fv· (6)
-
;._ 1 A�(o, r/Jv) TI P(o, r/Jw), 1: 1: / OE!F v w *v
where §' = { o E A1-u(E)Z(E)\A(E) iN(o) ¢ Z(F)},
P (o, ifJw) = Llw(r) Jz cE.l A CF.)\G CE.lifJw(g-u og) dg
and r = N(o). An explicit definition of A�(o, r/Jv) will not be given here ; we shall simply say that A�(o, ifJv) = IA 2 (r, fv) if fv is associated to ifJv under the base change correspondence (see [K]). As above the remaining term can be written (7) For a definition and a detailed study of distributions A� and B" the reader is referred to [L, §7] (where the subscript a is omitted) and to [K, Lemma 4]. 3. The comparison.
We assume now on that the function f
0'� (Z1 \ G(A), w-1 ) and the function ifJ = ®ifJv
E
= ®fv
E
0'�(Z(AE)\G(AE), w) are such
that ifJv and fv are "associated" in the sense defined in [K] ; we consider .@ 1 = I tr (R�(a) RiifJ) ) - tr r(f).
I 28
P. GERARDIN AND J. P. LABESSE
PROPOSITION I .
�1 =
� Svo I: (1B"(7Jv• r/Jv) o 2 2
where
V
�
7J= <.u. • .ul
* .u ; •.u.u=w
- I:_
n,.. 7J
)
fJ tr 7r:7Jw(fw) d7J B{7]. , /.) WoFV
tr(M("7J) 7r:�(u) 7r:7J(rp) )
o1 . 2 = I if I = 2, = 0 if I =f. 2 .
The proof amounts to the comparison term by term of the expressions for tr(r(f)) and I · tr(R�(u)R�( rp) ) . For example to prove that 1 · (1) = (i) note that we work there with a sum over elements o e G(E) such that N(o) is central in G{F) ; hence �(E) is the set of F points of a twisted inner form of G. Since we use Tamagawa measures, I vol(Z(A)�( E)\G�(AE)) = I vol(Z(A)G(F)\ G(A) ) = vol(Z1G(F)\G(A)) ;
the expressions to be compared are products of the local analogues, and now, using properties A' and B' in [K] for associated functions and the fact that the number of places where a minus sign occurs is even, we obtain the desired results. To prove I · (2) = (ii) is even simpler since in that case G�(E) = Gr(F), where r = N(o) : the twisting is trivial since Gr is abelian. To compare I · (3) and (iii) is slightly more complicated ; we must distinguish two cases : (a) I =f. 2 ; then "7J = 7Jv implies "7J = 7J = 7Jv and in such a case one has M(7J) = - I . One the other hand one should note that tr 7r:7J.(f.) = Jz.,A. F(a, f.)7] .(a) da and that for associated functions f. and ,P. one has F(a, f.) = F"(b , ,P.) if a = N(b). Moreover tr 7r:�.(U) 7r: 'ij.(rp.) =
J Zv\Av F"(b, r/J.)f;.(b) db,
where z. = Z(E ® F.) and A. = A (E ® F.) . Then tr(1r:�(u) 7r:'ii (rp)) = tr(1r71(f)) i f rp andfare associated and if f; = (f.J. o NEIF• v o NE1F) and 7J = (f.J. , v). This yields I · (3) = (iii). (b) If I = 2, the same arguments apply i f 7J = 7Jv = "7J, but there are other terms corresponding to 7J = (f.J. , v) with f.1. = "v =f. "f.1. and then 1 · (3) - (iii) = -
�
.u *".u ;n=
• .ul ; • l'.u=w tr(M("7]) 7r:�(u) 7r: 7J( rp)).
fi:
To prove that I · (4) = (iv) we use the previous remarks and the fact that mE(fJ)l = fl71,.. 'ij m(7J), where 7J 1-+ ij means that 7J = (f.J. , V) and ij = (f.J. o NEIF• V o NEI F) = (p., P) and by definition : m(7J) = L( I , f.J.- I v) and mE(7]-) = L( I , p.-Ip) . L( l , v lf.J.)
L( I , p.v i)
Now I · (5) = (v) follows from the comparison of orbital and twisted orbital in tegrals on unipotent elements.
BASE CHANGE PROBLEM FOR
GL(2)
1 29
To conclude the proof of Proposition 1 it is enough to show that 1 · (6) = (vi), which in turn follows from the equality :
A 2(r, f.)
A �(o, ¢.) if r
N(o). 4. The main theorem. The representation r is a discrete sum of unitary irreducible representations of G(A) with multiplicity one : r =
=
L:
iE /
11:;
=
=
L: (8)n; . •
i E[
and tr r{ f) = ,E ;E1 TI • tr 71: ;• •(/.) for some set I. Let v be a nonarchimedean place. Given f. in the Heeke algebra of G(F.) then tr 11:; , .( !.) is zero unless 11:; • • is unramified and hence corresponds to the conjugacy class of some semisimple element t;. • e GL2{C), the connected component of the L-group. Any function f. in the Heeke algebra defines a rational class function f'j on GL2(C) such that f';{(t; . •) = tr n; .•{f.). Choose a finite set V of places of F containing archimedean ones and assumef. is in the Heeke algebra for v ¢ V. Then one has LEMMA
I . tr r{f)
= ,E ; TI v E V tr 71:; , .(/.) TI v
The representation Rd can also be written
Rd = L; lli = L; ®Dj . v jEJ
jEJ
V
where lli is an automorphic representation of G(AE) and lli. v a representation of G(F. ® E). Since aRd � Rd and is multiplicity free, (J permutes the lli. If •lli � lli one can restrict the operator R�((J) to the space of lli , and denote this restriction by llj((J) . The nonfixed lli do not contribute to the trace of R�((J)Rd(
L: tr llj((J) lli(
•Dt'"Di
Moreover lli = (8)lli· • and •lli. v � lli. v ; we can define ll}, v ((J) up to lth roots of I . If lli. v is unramified, there is a canonical choice. If ¢. and/. are associated in the Heeke algebras, we choose a semisimple element ti · • e GL2(C) such that tr n;, .((J)llj . v(¢.) = tr nj. .<¢.) = f';{( tj. v),
and then for some big enough finite set V of places of F, we have :
LEMMA 2. tr R�((J)Ri
REMARK. If v is nonarchimedean and split in E, then any f. is associated to some ¢. ; in fact in such a case ¢. may be taken to be fw, ® fw2 ® . . . ® fw, where the W; are the places of E above v, and/. = fw, * fW2 * . . . * fw, can be any smooth function on G(F.) ; the conjugacy class of ti · • is well defined by lli . v · If v does not split in E, is unramified, and if ¢. and f. are associated in the Heeke algebras, one can define a function ¢';{ on GL{2, C) as above ; we have
f';{(t) = ifJ';{(t') for t semisimple in
GL2{C).
In such a case the conjugacy class of the ti. v above is not uniquely defined.
1 30
P. GERARDIN AND J. P. LABESSE
If / #:- 2 let R = /Rd. Assume for a while I = 2 ; if p. is a character of A>t;/ Ex such that 11p.p. = w and 11p. '# p., we consider 1:"' = tc71 , where 7J = (p., 11p.). As was said above, the operat or M(117J)1C�(o") = 1:�(u) maps the space of tc71 into itself. One defines in such a way a representation 1:� of r � G(AE). One should remark that 1:� � ..;w Let us denote by f7 the set of such representations (modulo equivalence) and let R' = IR� $ I; 1:�. � �E$'"
An analogue of Lemma 2 can be stated. We can now state the main theorem (cf. [L, Theorem 9. 1]). THEOREM 1 .
Assume! and ifJ are associated; then tr R'(u)R(ifJ) = tr r(f).
(Recall that the definition of the correspondence ''f and ifJ are associated" de pends on the choice of a 0' E F - {1 }.) The proof ofthe theorem can be carried out as follows : consider the expression tr R'(u)R(ifJ) - tr r(f).
(a)
Thanks to Proposition 1 above, this is equal to
(b) Using properties of some weighted orbital integrals [K, Lemma 4], one can prove that /B(fjv , ifJv) - I; B(7Jv o fv) = 0, 7Jv...,. 7iv
at least when v is nonarchimedean, unramified in E, with ifJv and fv associated and in the Heeke algebras. Hence there is a finite set of places V such that {b) can be written (b') with some nice function {3. Now choose a place v0 � V split in E; then (b') reduces to an absolutely con vergent integral : (b")
b
qis 0) o 0 bq-is J-+oooo o(s)fvv(a
ds
where a, depends on the central character m vo· All we need to know is that o(s) is some continuous, bounded and integrable function on the real line. On the other hand (a) can be written using Lemmas 1 and 2 above (a") with ak e C and tk e GL2(C) semisimple elements corresponding to inequivalent unitary representations of GL2(F0J. The series, as the integral, is absolutely con vergent. Now fv0 is arbitrary in the Heeke algebra (since v0 is split in E) and the
BASE CHANGE PROBLEM FOR GL(2)
131
Heeke algebra separates inequivalent unramified representations. The Stone Weierstrass theorem and easy majorations prove that all a, are zero (which is a stronger statement than Theorem 1). All the desired resu lts can now be xtracted from Theorem I and from some results on the characters of representations of r 11< GLz(E.) (cf. [L, §5]). We shall try to explain some of the steps.
5. Existence of weak liftings. Choose a finite set V of places of F containing all archimedean places and all places ramified in E. Assume that if>v and f. are as sociated and in the Heeke algebras for v ¢ V. One can, using Lemmas I and 2, choose element t,.,. e GLz(C) for v e V and n e N such that
tr R;(u)Riif>) = I; a,.(if>) IT f';j(t,.,.), n
v!tV
n
v$V
n
v!tV
tr ��(u)'t: p.(if>) = E {J,.(rp) IT f';j(t,., .),
tr r(f) = I; r ,.(rp) IT f';/(t,. , .) ;
we may assume moreover they are chosen such that the functions T,. : (r/J.)v!t v ...... IT v !t v f';f(t,.,.) on the product for v ¢ V of the Heeke algebras are distinct. (Recall the remark after Lemma 2.) Let ;;,. = Ia,. + {J,. - r.. ; then the above theorem can be restated in the following form Another use of density arguments, the T,. being distinct, yields PROPOSITION 2.
For all n one has o,.(rp) = 0.
This can be read Ia,. + {J,. = r ,.. Assume that there exist a representation U = ®U., unramified outside V, occurring in [0 such that tr U.(rp .) = f';j(t,.,.) for some n with a,. not zero, and any v ¢ V; then using the strong multiplicity one theorem [C] one concludes that such a U is unique and satisfies U � 119. The fact that L(s, U) is entire allows one to conclude that no other U occurring in [�P or in g- has the property that tr n.(¢>.) = f';/(t,.,.), v ¢ v. Then a,.(if>) = IT V E V tr n;(u)U.(if>.) and {J,.(if>) = 0 ; since Ia,. + {J,. = r.. we conclude that r,.(rp) is not identically zero and hence there exists (at least) one 11: in L� ® L� such that 11: = ®n. and tr n.(f.) = f';f(t,.,.) for v ¢ V and then n. is P the lifting of n. for v ¢ V. We shall say that U is a weak lifting of 11: if n. is a lifting of n. for almost all v. We then have proved
If n is a cuspidal automorphic representation of GLz(AE) such t1 n, then n is the weak lifting of some automorphic representation 71: of
THEOREM 2.
that n � GL2(A).
A direct study of L�P allows one to prove :
PROPOSITION 3. Any 11: occurring in L�P lifts to a U in [�P and all u-fixed represen tations in [�P are obtained in this way.
In the case I = 2, assume that U = n(p., 11p.) occurs in g- and that p.. is unramified for v ¢ V; one can show using Proposition 2 and results on L-functions on GL2 x
1 32
P. GERARDIN AND J.P. LABESSE
GL2 that ll is the weak lifting of 11: = n(p) where p = lnd�� p. and that there exists n E N such that This can be used to show that at all places of F: tr ll� (O')ll.(f/J.) = tr n.{f.), and hence THEOREM 3. Let the fields E and F be either local or global. Then n(p., "p.) is the lift ing ofn(p) where p = Ind�� p..
6. Any cuspidal n has a weak lifting. Assume for a while that some 11: occurring in � L has no weak lifting. Let V be a finite set of places of F including archimedean places and those where E or 11: are ramified. Let / = ®f. be associated to some ifJ and such that f. is in the Heeke algebra for v 1/: V. Consider the 11:k in L� EB L';P such that tr nk .• (f.) = tr n.(f.) if v 1/: V. Since the 11:k have no weak lifting, Proposition 2 shows that the sum of the 11:k gives a zero contribution to tr r(f) :
IT tr 11:k . vUv) IT tr 11:v(fv) = 0 ; I; k v eV v \EV
hence there is a set V1 c V such that � k D v e v, tr n k . v(f.) = 0. One has to prove that this is impossible unless the sum is empty. The idea of the proof (by induction on the cardinality of V1 ) is that characters of inequivalent representations are linearly independent, but the proof is complicated here by the fact that fv cannot assume all values, since fv must be associated to some f/J. (cf. [L, §9, pp . 25-30]). This yields THEOREM 4.
Any cuspidal n has a weak lifting.
7. Local liftings. To finish the proof of the global theorem on base change for cuspidal representations, one must show that the above weak liftings are liftings at all places. Let ll � "ll, occurring in L�, unramified outside a finite set V chosen as before ; there is an n such that, according to Proposition 2, lan(ifJ) = r.lifJ), which can be written, for some V1 c V, I IT tr n;(O')ll.(f/J.) = I: IT tr 11:k , vUv) , k v e v, v ev,
where the nk = ® nk. v have ll as weak lifting. One then proves [L, §9, pp. 33-34] : LEMMA 3. lf.for some v the representation n. is the lifting of some 11:v then it is the lifting of 11:k , v for all k, that is tr n;(O')ll.(f/J.) = tr 11:k, v (f.).
Then one may assume that V1 does not contain such places. On the other hand existence and properties of local liftings are easy to prove, or are deduced from Theorem 3 above, except for some supercuspidal representations (exceptional ones). Lemma 3 and this remark show that all desired local or global results (cf. part I of this paper) can be deduced from THEOREM 5 [L, §9, PROPOSITION 9.6]). (a) Every supercuspidal nv has a lifting.
(b) lf llv
�
" ll. and is supercuspidal, then n. is a lifting.
Part (a) of this theorem is proved by embedding the local situation in an ad hoc
BASE CHANGE PROBLEM FOR
GL(2)
1 33
global one, where the existence of liftings is known at all places except perhaps at one place, and to use the above equation with V�o reduced to one element. Part (b) then follows from the orthogonality relations of [L, §5]. The last paragraph of Langlands paper [L] is devoted to the proof of the exist ence of lifting for noncuspidal representations, using their explicit description (cf. Appendix C). BIBLIOGRAPHY [B) A. Borel, Automorphic £-functions, these PROCEEDINGS, part 2, pp. 27-61 . [C] W. Casselman. On some results of Atkin and Lehner, Math. Ann. 201 (1973), 301-334. [D) P. Deligne, Formes modulaires et representations de GL(2), Modular Functions of One
Variable. II, Lecture Notes in Math. , vol. 349, Springer-Verlag, Berlin, 1 973. [D-S] P. Deligne and J.P. Serre, Formes modulaires de poids 1 , Ann. Sci. Ecole Norm. Sup. 4 (1 974), 507-530. [G-J-1) S. Gelbart and H. Jacquet, Forms on GL(2) from the analytic point of view, these P Ro CEEDINGS, part 1 , pp. 21 3-25 1 . [G-J-2] , A relation between automorphic forms o n GL(2) and GL(3), Proc. Nat. Acad. Sci. U. S. A. (1 976) . [J-L] H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Math., vol. 1 14, Springer, Berlin, 1 970. [J-PS-S] H. Jacquet, I. I. Piatetskii-Shapiro and J. Shalika, Construction of cusp forms on GL(3), Lecture Notes, Maryland. [J-S] H. Jacquet and J. Shalika, Comparaison des formes automorphes du groupe lineaire, C. R. Acad. Sci. Paris 284 (1 977), 741 -744. [K) R. Kottwitz, Orbital integrals and base change, these PROCEEDINGS, part 2, pp. 1 1 1-1 1 3 . [L) R. P . Langlands. Base change for GL(2), Institute for Advanced Study, Princeton, N. J., 1975 (preprint). [Sa) H. Saito, Automorphic forms and algebraic extensions of number fields, Lectures in Math., no.8, Kinokuniya Book Store Co. Ltd ., Tokyo, Japan, 1975. [S-1] T. Shintani, On liftings ofholomorphic automorphic forms, U.S.-Japan Seminar on Number Theory (Ann. Arbor, 1 975). [S-2] , Two remarks on irreducible characters offinite genera/ linear groups. J. Math. Soc. Japan 28 (1 976), 396-414. [S-3] , On liftings ofholomorphic cusp forms, these PROCEEDINGS, part 2, pp. 97-1 10. [T] J. Tate, Number theoretic background, these PROCEEDINGS, part 2, pp. 3-26. [Tu] J. Tunnell, On the local Langlands conjecture for GL(2), Invent . Math . 46 (1 978), 1 79-1 98.
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UNIVERSITE PARIS
VII
Proceedings of Symposia in Pure Mathematics Vol . 33 (1979), part 2, pp. 1 35-1 38
REPORT ON THE LOCAL LANGLANDS CONJECTURE FOR
GL(2)
J. TUNNELL Let ([J(GL(2)/K) be the set of isomorphism classes of two-dimensional F-semi simple representations of the Weil-Deligne group W� of a nonarchimedean local field K. The purpose of this report is to discuss the following conjecture of Lang lands relating ([J(GL(2)/K) and the set 0(GL(2, K)) of isomorphism classes of irredu cible admissible representations of GL(2, K). Conjecture. For each representation e1 in ([J(GL(2)/K) there exists a representation n = n(CT) in O(GL(2, K)) such that the determinant of e1 and the central quasi character of n are equal and such that L(n ® X) = L(e1 ® X)
and e(n ® X) = e(CT ® X)
for all quasi-characters X of K * . The map e1 ...... n(e1) is a bijection of ([J(GL(2)/K) with O(GL(2, K)). The existence statement in the conjecture was formulated in [4], where it was shown that for a given representation e1 there is at most one representation n satisfy ing the desired conditions. The injectivity statement is equivalent to saying that a two-dimensional F-semisimple representation of W� is determined by its twisted L- and e-factors and determinant. It is straightforward to show directly that redu cible two-dimensional representations are in fact determined by the twisted L factors alone. The known proofs of the injectivity statement for irreducible repre sentations use admissible representation techniques. As discussed in [1, 3.2.3] it follows from the work of Jacquet and Langlands that n(CT) exists when e1 is reducible, and that this establishes a bijection of the set of isomorphism classes of completely reducible (repectively reducible indecompos able) two-dimensional F-semisimple representations of W1c with the set of isomor phism classes of principal series (respectively special) representations of GL(2, K). Let (/J;,,(GL(2)/K) consist of isomorphism classes of irreducible two-dimensional representations of the Weil group WK, and let Dcusp (GL(2, K)) be the set of iso morphism classes of supercuspidal representations of GL(2, K). Members of these sets are characterized by the requirement that their twisted £-factors are all equal to I . To prove the conjecture it is enough to verify that n (CT) exists for all e1 in (/J;,,(GL(2)/K) and to show that this establishes a bijection of (/Ji rr (GL(2)/K) and AMS (MOS) subject classifications (1 970). Primary 1 2B30, 1 2B1 5 ; Secondary 1 0D99. © 1979, American Mathematical Society
1 35
1 36
J.
TUNNELL
Dcusp (GL(2, K) ) . This was first proved for local fields with odd residue characteristic (see §2), which suggested the bijectivity portion of the conjecture in general. The conjecture has been proved for all nonarchimedean local fields except those extensions of Q2 of degree greater than I which do not contain the cube roots of unity. References for the proof are given in the following survey. 1. Existence of n(u). If a is a representation induced from a one-dimensional representation of an index two subgroup of WK there is an explicit construction (due to Weil) of an irreducible admissible representation n(a) of GL(2, K) with the desired properties [4, 4.7]. When K has odd residue characteristic all representations in (/Jirr (GL(2)/K) are induced from proper subgroups [1, 3.4.4], so the construction above applies. For each local field of even residue characteristic there exist irreducible two dimensional representations of the Wei! group which are not induced from a proper subgroup [11, paragraph 29]. There is no explicit construction of n(u) known for such representations. The approach of Jacquet and Langlands to the existence of n-(u) in this case is to imbed the local problem in a global one as follows. Let F be a global field and let p be an irreducible continuous two-dimensional complex re presentation of the Weil group WF. For each place v of F let p. be the restriction of p to the Wei! group of the local field F• .
THEOREM 1 [4, 1 2.2]. If the global L-functions L(s, p ® X) are holomorphic and bounded in vertical strips asfunctions of the complex variable s for all quasi-characters X of A;/ F* then n-(p.) exists. Cases when the hypotheses of this theorem are met have been described in the discussion of Artin's Conjecture in the base change seminar at this conference. A homomorphism of a group to GL(2, C) will be said to be of type H if the composi tion with the quotient map to PGL(2, C) has image isomorphic to H. The results in brief are that Theorem 1 may be applied when p is induced from a proper subgroup (Artin), when F is a field of positive characteristic (Wei!), when p is of A4 type (Langlands-Jacquet-Gelbart), and when F = Q, p is of S4 type and the image of complex conjugation has determinant - 1 (Langlands-Serre-Deligne). THEOREM 2 [10, THEOREM A]. Let K be a nonarchimedean localfield which contains the cube roots of unity if it is a proper extension of Q 2 • Then n-(u) exists for all a in f/J ( GL(2)/K).
This theorem is proved by constructing a global field F, a place v of F such that F. � K, and a representation p of WF satisfying the hypotheses of Theorem 1 such that Pv � a . The nonarchimedean local fields of even residue characteristic which do not contain the cube roots of unity are precisely those for which there exist re presentations of the Weil group of S4 type [11, paragraph 25]. The fields excluded in Theorem 2 are those for which there are two-dimensional representations of the Weil group which are not restrictions of global representations known to satisfy the hypotheses of Theorem 1 . Deligne has indicated that if /-adic representations can be associated to certain automorphic representations related to Shimura varie ties arising from division algebras over a totally real field F, then the hypotheses of Theorem 1 will hold for representations of WF of S4 type with prescribed behavior
THE LOCAL LANGLANDS CONJECTURE FOR
GL(2)
1 37
at the infinite places. Theorem 2 will then be true for K a completion ofF at a prime dividing two ; since each finite extension of Q2 is the completion of some totally real field, the proposition and Theorem 3 of §4 would then prove the conjecture in all cases. 2. Odd residue characteristic. The Plancherel formula for GL(2, K) shows that the supercuspidal representations of the form n'(o) for o in tf>;rr (GL(2)/K) exhaust the supercuspidal representations if K has odd residue characteristic. References [3], [8] and [9] contain treatments of the representation theory of GL(2, K) and related groups when K has odd residue characteristic. The injectivity statement may be proved by examining the explicit formulas for characters of supercuspidal representations in the case of odd residue characteristic which are given in [7] and the references above (at least for SL(2) and PGL(2)). Sup pose that o is induced from a ! -dimensional representation A of the Weil group WE of a quadratic extension E of K. Denote the quasi-character of E * corresponding to A by the same symbol. The restriction of the character function of :n:(o) to a Cartan subgroup isomorphic to E * is given by a formula involving A and its Gal(E/K) conjugate. From the explicit form of the character it can be seen that :n:(o) deter mines A up to Gal(E/K) conjugation, and hence determines the induced represent ation o. 3. Positive characteristic. The discussion of § I shows that :n:(o) exists for all representations of the Weil group of a local field of positive characteristic. The bijectivity assertion of the conjecture seems to have first been proved by Deligne [2] as a consequence of Drinfeld's results relating automorphic representations of GL(2) over global function fields to 1-adic representations. 4. A method is presented in [10] that gives alternate proofs of Langlands' Con jecture for the cases discussed in §§2 and 3 and proves the conjectured bijection for all fields in Theorem 2. PROPOSITION [10, 2.2] . Let o 1 and o2 be two-dimensional representations of the Wei/ group of a localfield, each of which is the restriction of a global representation satisfy ing the hypotheses of Theorem 1 . /f :n:(o 1 ) � :n:(oz) then o 1 � Oz .
The proposition above is proved by an inductive application of base change for GL(2). The assumptions of the hypothesis are necessary because the proofs of the base change results for local fields utilize global methods. The following result holds for any nonarchimedean local field K. THEOREM 3 [10, §§4 AND 5]. There are partitions of t!>irr (GL(2)/K) and Dcusp(GL(2, K)) into finite sets tf>;. and U;. respectively (indexed by a common set A) such that ( 1) If o e l/>1 and :n:(o) exists , then :n:(o) e U;.. (2) Card(tf>;.) = Card(U1) for all A e A.
The partition elements are determined by conditions on the Artin conductor and determinant (resp. conductor and central quasi-character) of elements in tf>irr (GL(2)/K) (resp. Dcusp (GL(2, K))). Since the conductor of a representation is de termined by the twisted e-factors, the first statement follows from the definitions.
1 38
J. TUNNELL
The computation of the cardinality of the sets f/Jl is done by constructing all irredu cible two-dimensional representations of WK as in [11] and counting those with a given Artin conductor and determinant. The sets Dl are studied by utilizing the cor respondence between square-integrable representations of GL(2, K) and admissible representations of the group of invertible elements in the quatemion division algebra over K. Theorem 2 together with the injectivity proposition and the counting results of Theorem 3 show that f/Jl and 0;. correspond bijectively by means of o ,_. n(o) for the local fields in the hypotheses to Theorem 2. This gives the known cases of the conjecture stated in the introduction. 5. Remarks. While the statement of Langlands' Conjecture is purely local, the proofs described above in the case of even residue characteristic utilize global methods (base change and cases of Artin's Conjecture). Only in the case of odd residue characteristic are the proofs described above purely local. Cartier and Nobs have indicated a proof of the conjecture for the field Q 2 which is purely local. They calculate the necessary e-factors for irreducible two-dimen sional representations of Wll:! and match them with the factors of supercuspidal representations of GL(2, Q2) constructed in [6]. The supercuspidal representations are constructed by inducing finite dimensional representations of subgroups of GL(2, K) which are compact modulo the center. In [5] a similar construction of supercuspidal representations for GL(2, K) is given which should allow, in theory, the calculation of e-factors and matching with the factors of representations of WK to be done in general. REFERENCES
1. P. Deligne, Formes modulaires et representations de GL(2), in Modular functions of one variable, Springer Lecture Notes, No. 349, 1 973, pp. 55-1 06. 2. , Review of" Elliptic A-modules" by V. Drinfeld, Math. Reviews 52�1 976), #5580. 3. I. M. Gelfand, M. I. Graev and I. I . Piatetskii-Shapiro, Representation theory and automorphic functions, Saunders, 1 969. 4. H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Springer Lecture Notes, No. 1 14, 1 970. 5. P. Kutzko, On the supercuspidal representations ofGL,. I, II, Amer. J. Math. 100 (1 978), 43-60. 6. A. Nobs, Supercuspidal representations o/GL(2, Qp), including p 2 (to appear). 7. P. J. Sally, Jr. and J. A. Shalika, Characters of the discrete series of representations of SL(2) over a localfield, Proc. Nat. Acad. Sci. U.S.A. 61 (1 968), 1 231-1 237. 8. , The Plancherel formula for SL(2) over a local field, Proc. Nat. Acad. Sci. U.S.A. 63 (1 969), 661- 667. 9. A. J. Silberger, PGL(2) over the p-adics, Springer Lecture Notes, No. 1 66, 1 970. 10. J. Tunnell, On the local Langlands Conjecture for GL(2), Thesis, Harvard University (1 977). 11. A. Weil, Exercises dyadiques, Invent. Math. 27 (1 974), 1 -22. --
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PRINCETON UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 2, pp. 141-163
THE HASSE-WElL �-FUNCTION OF SOME MODULI VARIETIES OF DIMENSION GREATER THAN ONE
W. CASSELMAN Introduction. Let k be a number field of finite degree, G a connected reductive group over k. By restriction of scalars, one may as well assume k = Q. Let KR � G(R) be the product of the center of G(R) and a maximal compact subgroup. Then, in certain circumstances, one may assign to X = G(R)/KR a G(R)-invariant com plex structure ; if r is an arithmetic subgroup of G small enough to contain no torsion then F\X will be a nonsingular algebraic variety. In many cases one can show that it has a model defined over a rather explicit number field, and under certain further assumptions-as Deligne and Langlands may explain-one can choose a canonical model defined over an abelian extension of a special number field E determined, roughly speaking, by G and the complex structure on X. One might expect that the Hasse-Weil �-function of a canonical model is a pro duct of £-functions of the sort Langlands associates to automorphic representa tions of G(A). This turns out to be false (see the Introduction to [22]), but it is suggestive. The first result of this kind is due to Eichler, who showed that when G = GL2(Q) and F= F0(N), then F\X has a model over Q and its Hasse-Weil �-function is, as far as all but a finite number of factors in its Euler product are concerned, a product of £-functions defined in this case by Heeke. This result was extended by others, notably Shimura, Kuga, Ihara, Deligne, and Langlands, to include : (a) other r in this G, (b) other G, (c) �-functions associated to nontrivial locally constant sheaves, and finally (d) factors of the �-function corresponding to primes where the variety behaves badly. With one exception-some unpublished work of Shimura-all this work is concerned with X of dimension one. (Refer-for a sampling-to [11], [28], [14], [6], and [19].) As a consequence of these results one had a generalization of Ramanujan's con jecture, applying the result of Deligne on the roots of the �-function of varieties over finite fields. A further consequence was a functional equation for and analytic continuation of the Hasse-Weil �-function concerned, which turned out-excepting again Shimura's example-to be a �-function associated in a particularly simple way to GL2 or a quaternion algebra. Over the past several years, Langlands has attacked the problem of varieties of dimension > I , and what Milne and I are going to discuss in our lectures is the simplest case he deals with. AMS (MOS) subject classifications (1 970). Primary 1 0D20 ; Secondary 14025, 22E55 . © 1 9 7 9 , American Mathematical Society
141
142
W.
CASSELMAN
To be more precise, let :
F = a totally real field of degree, say, n ; oF = integers in F; B = a quaternion algebra over F;
o8 = a maximal order in B. We recall that this means simply that B is an algebra of dimension four over F such that B ® F = M2(F ), where F is an algebraic closure of F. Since over any locally compact field there exists a unique quaternion division algebra, if v is any valuation of F then B. = B ®F F. is isomorphic either to M2(F.) or to thi s unique division algebra, depending on whether or not B. has 0-divisors. The algebra B is said to be split at v in the first case, ramified in the second. It is in fact split at all but a finite number of valuations ; quadratic reciprocity says this number is even, and another classical result says that for each even set of valuations there is a unique quaternion algebra ramified at exactly those valuations. Thus B itself is a division algebra if and only if it is ramified somewhere. If v is nonarchimedean, the closure of o8 in B. is a maximal compact subring of B.-unique if B is ramified at v, otherwise only unique up to conjugacy by an element of B; . Let G be the algebraic group over Z defined by the multiplicative group of o8 • Thus for any ring R, G(R) = ( o8 ® R)x . In particular, G(Q) � Bx (canonically) and G(R) � GL2(R)1 x (Hx)J (noncanonically), where I is the set of real valua tions of F where B is split, J those where it is ramified. For every rational finite prime p over which B does not ramify,
where the product is over all primes p of F dividing p. Of course the simplest case is B = M2(F), but that is unfortunately the case we will not allow-i.e., from now on we assume B to be a division algebra. Furthermore, we will assume B totally indefinite at the real primes ofF-i.e . , that J = 0 . Langlands himself does not make these assumptions, but acknowledges gaps in the argument unless they hold. Let Ll be a finite set of rational primes containing those over which either F or B ramifies, and let K1 be a compact open subgroup of G(Z1) of the form K4 • f h�<� G(Zp), where K"' is a compact open subgroup of Ti pEd G(Zp). Let Z be the center of G, and ZK = (Z(A1) n K1) · Z(R). Consider ex as embedded in G4(R) : a + b .V - 1
�-+
(ab - ab)
and let KR be the image of (Cx)I in G(R). (This is the connected component of the K8 used before.) Then G(R)/K8 is a product of n copies of C - R. Let K be KR · K1. The set KS(C) = G(Q)\G(A)/K is (as will be explained later)
the union of a finite number of compact complex analytic spaces of dimension n ; it will be nonsingular if (as we assume from now on) K1 is small enough. Milne (in his lectures at this Institute) will show that this space is the set of C valued points on a certain moduli scheme KS which is defined, smooth, and proper over Spec Z[l/d], where d is the product of primes in Ll. He will also discuss the
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THE HASSE-WElL �-FUNCTION
structure of its points over finite fields. What I will do is to identify its Hasse-Weil �-function as one of Langlands' ; the basic idea in doing this will be to use the Selberg trace formula to calculate p-factors of Langlands' L-function, and Milne's results and an unorthodox application of the trace formula to calculate p-factors of Hasse-Weil-both for p not dividing .:::1 . {This will be a lot of trouble, and I should explain at the outset that, although when KS has dimension one it is possible to use a congruence relation to obtain the final result, this will not suffice in general.) One may then apply Deligne's results on the roots of the Frobenius to get a gen eralization of Ramanujan-Petersson ; however, results about a functional equation for Hasse-Weil are incomplete. In everything both Milne and I do we are essentially reporting on Langlands' work. The result we discuss is only a special case of a more complicated result of his involving subgroups of B x . We avoid, in treating this special case, problems of what he calls "L-indistinguishability," which perhaps he himself will talk about. His result is more general in other ways, too ; I include remarks on this in the last section, where I have collected together a number of substantial parenthetical remarks. My main reference is the summary [21] ; complete statement and proofs are in [22]. Also relevant are Langlands' talk in the Hilbert problems Symposium [20} and his talks at this Institute. 1. Cohomology of KS(C) and representations. 1 . 1 . Let v be the reduced norm : G --> Gm . Since B is a division algebra, the coset space G(Q)\G l (A) is compact, where G1(A) is the kernel of the modulus homomor phism I v I : x --> I v (x) 1 . Since the image of Gconn(R), the connected compact of G(R), under l v l is all of Rvos, the set G(Q)\G(A)/Gconn(R) is compact as well. Since K1 is open in G(A1), the set G(Q)\G(A)/Gconn(R)K1 is finite. Therefore there exists a finite set !!£ of elements of G(A) such that G(A) = U G(Q) xGconn(R)K1 (x E !!£). (Strong approximation gives one a better parametrization of !!£, but we won't need that ; see [7].) 1 . 1 . 1 . LEMMA. For any x E G(A), the space G(Q)\G(Q)xGconn(R)K1jK1 as a Gconn(R)-space is isomorphic to Fx\Gconn(R), where Fx is the image in Gconn(R) of G(Q) n Gconn(R) . xK1x-l.
This is because Gconn(R) certainly acts transitively on this space and isotropy subgroup of the coset G(Q)xK1. Let Yl' be the upper half-plane in C. As a consequence of the above :
Fx
is the
1 . 1 .2. PROPOSITION. The Gconn(R)-space G(Q)\G(A)/K1 is isomorphic to a disjoint union of spaces Fx\Gconn(R) (x E !!£). The variety KS(C) is the disjoint union of the Fx\Yf'n.
The same argument as that used to prove Lemma 2 . 1 of [19] may be applied to show that if K1 is only small enough, each rx acts freely and KS(C) is nonsingular.
We assume thisfrom now on. 1 .2. Let do be the space of automorphic forms on G(Q)Z(R)\G(A). It is a direct
sum EBn- of irreducible, admissible, unitary representations of G(A) (an abuse of language since not G(R), but only its Lie algebra g, acts). If g is the Lie algebra of
W.
1 44
CASSELMAN
G (R) = G(R)/Z(R), then in fact the representation on do factors through g. Let t be the Lie algebra of KR KR/KR n ZR. =
1 . 2. 1 . PROPOSITION. The de Rham cohomology H* (KS(C), C) is naturally isomor phic to EBH*(g, f . ?roo) ® n-f! .
The cohomology is the relative Lie algebra cohomology. The sum is over all constituents n- of do (with multiplicity if necessary), which one factors as n- = ?roo ® n-1 where ?roo is an irreducible admissible representation of G(R) (more abuse of language as before) and n-1 one of G(A1). PROOF SKETCH. Let d{ft be the subspace of do of functions fixed by elements of K1. Since g commutes with K1, this is a representation of g which is clearly iso morphic to EB ?r00 ® n-f t . Thus the proposition amounts to the claim that H* (KS(C), C) is isomorphic to H*(g, f , d{ft). Now from 1 . 1 .2 it follows that d{ft is the direct sum of subspaces d{{{ (x e El"), where each d{{{ is the space d(f'"\(jconn(R)) of KR-finite, Z(g)-finite functions on f'"\(jconn(R) (!'" is the image of r" in G). Thus the proof of 1 . 2. 1 reduces to : 1 .2.2. LEMMA .
There exists a natural isomorphism : H* (f'"\ G/K, C)
�
H* (g, f , d(f'"\ G)).
This sort of thing holds in fact for any semisimple Lie group G , cocompact f' c G and maximal compact K. To see it : the cohomology of X = f'"\ G/ K is that ofthe de Rham complex, whose nth term is the space of coo m-forms on X. Now the projection f'"\ G -+ X is a prin cipal bundle, and the bundle ofm-forms is associated to it and the K-space .Am(gf t)t'. Therefore coo m-forms correspond to certain c oo functions from f'"\G to _A m(gj t)t', which by means of an obvious duality may be thought of as K-linear maps from _Am(g/k) to Coo(f'"\G). In short, the de Rham complex on X may be identified with a complex whose mth term is Hom1(.A m(g/ f), Coo(f' "\G)). But this is the mth term of the complex by which H*(g, t , Coo(f'"\G)) is calculated, and it turns out (by an explicit calculation) that the differentials are the same. Therefore 1 .2.2 is true with d(f'"\ G) replaced by Coo(f'\G). To get the final step, roughly, one recalls that ac cording to Hodge theory cohomology classes may be represented uniquely by harmonic classes, and observes that these lift in the above process to elements of d. (See Chapter IV of [3] for details on this and other points.) 1 .3. Although the number of representations occurring in the sum in 1 .2. 1 is infinite, all but a finite number of terms vanish. To be more precise, we must say more about the relative Lie algebra cohomology of admissible representations. First of all, since G(R) � P G4(R) 1 (notation as in the introduction), each ?roo factors as ®n-oo," where each ?roo , , is an admissible representation of PG4(R). One has an easy Kiinneth formula : so that the problem for G(R) is reduced to one for PG4(R). (Note that the Lie algebra of PG4 is the same as that of S4.) For this : let C be the Casimir element in U(�l2), and recall that it lies in the center of U(�l2) and is centralized by the maxi-
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THE HASSE-WEIL �-FUNCTION
mal compact 0(2), hence acts as a scalar on any irreducible admissible representa tion of PGLz(R). 1 . 3. 1 . LEMMA. Let (n-, V) be an irreducible, admissible, unitary representation of PGLz(R). Then
if n- (C)
=f. 0,
� Hom,./A* (�l2/�o2), V) if n- (C) = 0.
The idea of the proof here is that if n- is unitary one can put a natural inner pro duct on the complex Hom,02(A * (�1/�o2), V) , such that n-(C) = dd* + d*d where d* is the adjoint of d. (This is an observation good for all semisimple groups due to Kuga.) The lemma follows immediately. For PGL2{R), there are only three irreducible admissible representations n- with n-(C) = 0 : (a) the trivial representation C; (b) the character sgn(det(g) ) ; (c) a single discrete series representation n-0, which may be identified with the quotient of the 0{2)-finite functions on Pl (R) by the constant functions. Lemma 1 . 5 and an easy calculation concerning the restriction of these to 0(2) give : 1 . 3.2. COROLLARY. If 7r is an irreducible PGL2{R) other than one of these then H * (n-) = (a) when 7r is C or sgn {det), Hm(n-) � C, � 0, � c, (b) when 7r = n-0,
Hm(7r) � 0,
� C + C, � 0,
admissible unitary representation of H * (�12, �o2, n-) = 0. For these : m
= 0,
m m
=
=
m m m
I, 2; = 0, = =
I, 2.
The assumption of unitarity is not in fact necessary (see [3]). Hence if the irreducible admissible representation n-"' of the original G(R) is cohomologically nontrivial, it must be of the form ®n-"'·" where each n-"'· ' is one of the above three representations, and its cohomology may be calculated accordingly. If n- is an irreducible admissible representation of PGLz(R), set
m(n-)
if n- is cohomologically trivial, if n- is either C or sgn(det), I - 1 if n- is n-0.
= 0 = =
If n-"' = ® n-"'·' is an irreducible admissible representation of G(R) define m(n-"') to be f1 m(n-oo,J. If 7r = n-"' ® 7r 1 is an irreducible admissible representation of G(A) where n-"' is trivial on Z(R), then define m(n-, K) = m(n-"') dim n-ft. Thus the size of m (n-, K) is just dim n-ft and its sign reflects the parity of its contribution to co homology. As a final remark let me point out that by applying the strong approximation · theorem one can show that if n- = ®n-. (v over valuations of F) is an irreducible admissible automorphic representation of G(A) and n-. is one-dimensional at a ·
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W. CASSELMAN
place where B is split, then n. is one-dimensional everywhere. In particular, if one factor of n"" is one-dimensional so are all. As a consequence of this one recovers a result of Matsushima-Shimura [23] which says that the interesting cohomology of KS(C) occurs in the middle dimension. 2. The main theorem and some consequences. 2. I . I recall the L-group attached to G. First of all, if G is considered in the most straightforward way as a group over F -i.e., so that for any extension F' of F, G(F') � (B ® F F')x -then its L-group is just the direct product of LGO = GL2(C) and Gal (F/F). This is because it is an inner twisting of GL2(F). Since G as a group over Q is obtained from this one by restriction of scalars, its L-group LGO is the one in some sense induced from this one : it is the semidirect product of Gal(Q/Q) by LG o = GL2 (C)l, where I is the set of embeddings of F into Q and Gal acts on LGO by permutation of factors. (Since Q may be identified with a subfield of C, this I is essentially the same as before.) Without any serious loss for our purposes we may (and will) replace Gal (Q/Q) by Gal (Fnorm/F), where Fnorm is the smallest extension of F normal over Q. It may be helpful if I point out that it is unramified over Q whenever F is. I recall also that if p is a prime of Q unramified in F and f/J is a Fro benius in Gal(Fno rm/F) o ver p, then the local L-group L GQp = LGp may be identified with the subgroup of LGQ whose image in Gal(Fnorm/F) is the cyclic subgroup generated by f/J. Thus one has an exact sequence I ...... GL2(C)l ...... LGp ...... (f/J) ...... I . (Refer to [2] for everything about £-groups.) In order to define £-functions associated to automorphic representations, one must also introduce finite-dimensional representations of LG. There is only one (for each G) that we will be concerned with, and it is defined as follows : the space of this representation p is a tensor product of copies of C 2, one for each element of I; the group LGO = GL2(C)I acts through the standard representation on each factor, and Gal(Q/Q) (or Gal(Fnorm!Q), it makes no difference) acts by permuting the factors. This does indeed define a representation of LG, since it is a semidirect pro duct of these two groups. The dimension of this representation is 2n. When F = Q, for example, LG = GLz(C) and p is just the standard representation itself. This particular choice of p may seem arbitrary, but in fact it Wl;!S motivated ori ginally (in Langlands' formulation) by general considerations about Shimura varie ties (one should refer to the Introduction of [22] for a discussion of this point). 2.2. If X is any smooth, proper scheme over Z[I/d] for some d ;:;; I and p is a prime not dividing d, then the zeta-function of X over Fp is defined to be the func tion Zp(s, X), rational in p- s , such that (at least formally) log Zp(s, X)
� oo
=
I
mp m'
(jj:X(Fpm) ) .
As a consequence of the etale theory, this agrees with the definition in terms of /-adic cohomology : 2-d imX det(I - f/J · r'h• cx, Q1) Zp(s, X) = IT = i O
where f/J is the geometric Frobenius. At least as far as factors other than those divid ing d are concerned, its Hasse-Weil zeta-function Z(s, X) is the product of these Zp(s, X).
THE HASSE-WElL C-FUNCTION
147
The main theorem of Langlands is :
2.2. 1 . THEOREM. Up to prime factors in Ll, the Hasse-Wei/ zeta-function of KS agrees with n L(s - n/2, 1C, p) m (1<,Kl where the product is over a/I re occurring as constituents of d0. Of course one should consider the rc with multiplicities if necessary. In fact, however, because of the theorem in § 1 6 of [16) relating automorphic forms on G with those for GL2 , together with the result for GL2 in §9 of [16], each rc occurs exactly once. In this product, m(rc, K) = 0 for all but a finite number of rc, since after all the cohomology of KS(C) is finite. What is actually to be proven is a purely local result : for p ¢ Ll, the p-factors of Z(s, KS) and ll L(s - n/2, rc, p) coincide. Now each 1C = @rc. with m(rc, K) i= 0 has the property that rcp for p ¢ L1 is unramified, hence corresponds to an element g(rcp) in the local £-group LGp. The p-factor of L (s - n/2, rc, p) is then det(/ - p(g(rcp) ) jp•-,. 1 2)- 1 . Upon expansion, the theorem reduces to a formal equation (p ¢ L1) : This in turn amounts to an equation of coefficients (for p ¢ Ll, m � 1) : (2. 1) This is the form in which the theorem is actually proven. In these lectures I will give a complete proof only in the simplest case, when F is split over p . 2.3. Before beginning the proof, we give an example and some consequences. The case n = 2 is the first interesting one, in the sense that, as already mentioned, the case n = I is an old result and can be (and has been) done more elementarily by means of a congruence relation. So, suppose for a while that F is quadratic over Q, and consider the possible p-factors occurring in L(s, rc, p). There are two cases : (1) when p = p 1 p2 splits in F and (2) when p = p remains prime. We look at (1) first. In this case G(Qp) � GL2(Fp) x GL2 (Fp2) and an unrami fied representation of this must be of the form rc 1 ® rc2 , where each 1C; is an unrami fied representation of GL2(Fp,.), hence corresponds to a pair of unramified charac ters (a;, {3;) of F� . More precisely, 1C; may be the whole principal series representa tion parametrized by the pair when it is irreducible, or the associated one-dimen sional character of GLz(Fp,.) when it is not. (By strong approximation, this last happens only when the global representation at hand is also one-dimensional.) In either case, observe that the local L-group LGP is (because p splits) simply the direct product GL2(C) x GLz (C) and that the corresponding element g(rcp) of LGp is a 1 ( p) az(p) f3 1 ( p) • f32(p) . The representation p is simply C 2 ® C 2, so the p-factor of L(s - I , rc, p) is (1 - a 1 (p) az(p)pi -s)- 1 (1 - a 1 ( p) f3z(p)p 1 -s) - 1 ( 1 - f3 1 (p) az(p)pi -s)- 1 (2.2) · (1 - /31 ( P) /32( p)p-•)- 1 .
((
)(
))
148
W.
CASSELMAN
Next case (2). Here np is a single unramified representation of GL2(F), say cor responding to the characters (a, f3) of F; Jo; . On the other hand, the local £-group is still a semidirect product of GL2(C) x GL2(C) by the Galois group of order 2, since the Frobenius generates Gal. I leave it as an exercise to verify that the element g(np) of LGP is
f/J
((I0 0I ) (a(p)0 f3(p)0 ))f/J. '
If e 1 , e2 are the standard basis elements of cz then g(np) acts as follows on the space of p (recall that interchanges factors) :
f/J
e1 ® e2 ...... a(p)e2 ® eh e 1 ® e 1 ..... a(p)e 1 ® eh , (p)e ez ® e1 ..... f3 1 ® ez ez ® ez ..... f3(p)ez ® ez . Therefore the p-factor of L(s - 1, n, p) is (2.3) {1 - a(p)p i -s)-1(1 - f3(p)p i -s)- 1 (1 - a(p) f3(p)p2-s)-I. For n > 2, the analogous formulae can be rather complicated. Incidentally, the unpublished result of Shimura we have referred to before is precisely that the p factors are of the form (2.2) and (2.3) for certain quadratic fields F. For F = Q (n = 1), one consequence of the expression of the Hasse-Weil zeta function as an £-function associated to automorphic forms is that it has a func tional equation and analytic continuation. For n > 1 , this consequence is not automatic. For n = 2 it is already a difficult question only very recently settled by Asai [1] under some probably unnecessary restriction on F. Nothing seems to be known for n > 2. What is easier is to obtain a functional equation for the zeta function of KS over F itself, at least for certain small degrees. This depends on a generalization of an idea of Rankin due to Jacquet and Shimura (see, for example, [15] and [29]). 3. The analytical trace formula. In this section I will develop a formula for the right side of equation (2. 1 ), in the next one for the left, and in §5 the two will be compared. Beginning in §4 I will assume (although Langlands certainly does not) that p is split in F. This makes things much simpler. The argument will still involve several important ideas, but will avoid what is at once the most complicated and intriguing point of all, the use of paths in a certain Bruhat-Tits building. I hope that what remains is still of some interest. Langlands will presumably say something on this matter in his own talks (see [17]). 3. 1 . The right side of equation (2. 1), (3. 1 ) I; m(n, K) pm n / 2 trace p(g(np)m), 1r
in �o
may be expressed as the trace of a certain operator R1, for some / E C�{G(A)), acting on d0• Recall that do is a discrete direct sum of irreducible, admissible, uni tary representations n = n8 ® nP ® np of G(A), and for any f E C�(G(A) f ZK) one has therefore trace R1 = I; ,.. in .w
1r
I; trace n8(f8) trace nP(fP) trace np(fp)
i n .9/o
149
THE HASSE-WElL C -FUNCTION
if/factors as/8 · fP · /p. Recall that m(1r, K) = m(1r8) dim(1rP) KP. A comparison of (3. 1) with (3.2) suggests choosing the three factors off so that (3.3)(a) for any irreducible admissible representation 11: of G(R)/Z(R) ; (3.3)(b) trace 11:(/P) = dim 11:KP for any irreducible admissible representation 11: of G(A�) ; (3. 3)(c) trace 11:(/�ml) = pmn / 2 trace p(g(11:) m) 11: unramified, =0 otherwise. This is just what will be done. The choice of fP is simple : fP = (meas KP)-1 char KP . That of f�ml is not so explicit but just as simple : according to the Satake isomor phism [4] there exists a uniquef�ml E .Yl'(G(Qp). G(Zp)) satisfying (3.3)(c). But the matter of/8 is more complicated. 3.2. If the representation 11: of G(R) factors as (8)11:, according to G(R) � GL2(R)I then m(11:) = Ti m(11:,). Iff = Ti le accordingly then trace 11:(/) = II trace 11:(J,). Thus the problem of finding/8 reduces to the problem of finding f E C�(PGL2(R)) such that (3.4)
trace 11:(/) = m(11:)
for any irreducible admissible representation 11: of PGLz(R). One can do this in several ways-for example directly from the Paley-Wiener theorem as in [10], and also from considerations of orbital integrals (see [26]). But Serge Lang has pointed out an argument which is in some sense more elementary than either of these and which I present.below. From this point I follow Lang's book [18, Chapter VI, §7] rather closely. The connected component of PGL2(R) is PSL2(R). Now the maximal compact of SLz(R) corresponding to the one of GL2 (R) at hand is S0(2) ; let e : S0(2) � ex be the fundamental character e
(cos -cossin (}) = cos (} + .J - 1 . sm. sin (} (}
(}
(}.
The single discrete series representation 11:0 of PGL2(R) which is cohomologically nontrivial has the property that its restriction to S0(2) is a direct sum EB en (In I � 2, n even). No other discrete series representation has e2 in its S0(2)-decomposition. I recall that the Heeke algebra of all compactly supported bi-S0(2)-finite dis tributions acts on the space of any admissible representation of SL2(R). If D satisfies Lk, Rkz D = em(k 1 )e-n(k2)D for all kh k2 E K, then for any such representation (11:, V) the operator 11:(D) takes all of V into the em-eigenspace and annihilates all but the en-eigenspace. In the special case m = n and D amounts simply to integration . against e-n on K, 11:(D) is the identity on the en-eigenspace. Therefore, by choosing fe C""(SLz(R)) such that
1 50
W. CASSELMAN
and approximating this D closely enough-i.e., choosing f positive, normalized, and with support close enough to K-one may assume that 11:(j) is also the identity on this eigenspace. Apply this to the case m = n = 2, 11: = 11:0 to obtain /1 e C�(SL2(R)) such that (a) /1 (k 1 gkz) = e-2(k1)e- 2(kz)f(g) for all kh k z e S0(2), g e SLz(R) ; (b) 11:o(f1 ) is the identity on the e 2-eigenspace and 0 on any other S0(2)-eigenspace. Because of the remark above about other discrete series representations of PGL2(R) and e2, one even has (3.5)
trace 11:{jj) = I , 0,
11: = 1l:o, 11: is any discrete series representation of PG�(R) other than the 11:0•
At this point recall what Lang calls the Harish transform of an/ e C�(SL2(R) ) :
Here A is the group of all diagonal matrices. This function H1 turns out to lie in C�(A) and even in C�(A)W, where W is the Weyl group. In fact, / � H1 induces an isomorphism of the space of functionsf e C� (S�(R)) hi-invariant under S0(2) with C�(A)W [18, V, §2]. Thus, one may choose /2 e C� (SL2(R)S0(2)) with H1 1 = H�z. Set / = /1 - f2, so H1 = o. Since a discrete series representation possesses no vectors =1- 0 fixed by S0(2), equation (3.5) holds for f instead of fi.. Furthermore, if x is any character of A and 1l:x is the corresponding principal series representation of SL2(R) then (3.7)
trace 11:x(/) =
J H1(a)x(a) da = 0 A
[18, VII, §3]. Since any principal series representation of PGL2(R) restricts to some 1l:
x
on PSL2(R), the same holds for all principal series representations of PGL2(R). Finally, if 11: is any irreducible finite-dimensional representation of PGL2(R) then because its character and that of some unique discrete series representation add up to a principal series representation, trace 11:(/) = - I 11: = C or sgn(det), other finite-dimensional 11:. =0 Define now fa on PGL2(R) to be -f on PSL2(R) and 0 on the other connected component. Equations (3. 5), (3. 7), and (3.8) yield (3.4) for f = fa · 3.3. To recapitulate : the function I = fa . fP . t�m) E coo(G(A)fZK) has the property that trace R1 ldo = I; m(11:, K)p mn / 2 trace p(g(1l:p) m) . (3. 8)
11: in �o
In §5 the trace will also be expressed by means of the Selberg trace formula. I will recall it here in a rather general form for purposes of contrast a little later : Let
THE HASSE-WElL C-FUNCTION
151
W b e any locally compact group, r a discrete subgroup with F\W compact. The conjugacy class of any T E F Will be closed in '§ (a nice exercise) and SO for any F e c:"'(W) the orbital integral J:J"r\!f' F(g 1 rg) dg is defined and finite, where '§7 is the centralizer of r in '§. If F7 = '§7 n F, then clearly F7\W7 is compact also. Under some mild assumption the operator RF is of trace class on L2(F\W) and -
J
meas{F7\W7) r F(g 1 rg) dg !f' \!f'
= 1:
-
4. The algebraic trace. In this and the next section I assume, as mentioned earlier, that p splits completely in F. 4. 1 . There exists a formula for '.Pi KS(Fpm) remarkably similar to that which the Selberg trace formula yields for the right-hand side of equation (2. 1). The set KSCFp) is partitioned naturally by the equivalence relation of isogeny i.e., two points lie in the same class when the abelian varieties-plus-structure that they parametrize are isogenous. The isogeny classes are of two types : (1) Those associated to systems (F',{q;}), where F' is a totally imaginary quadratic extension of F which splits at some prime over p and the q; are certain ideals of F'-if !J I > · · · p, are the primes of F which split in F' then q; is one of the two prime factors of !J; . Two systems (F{, {qu}) and (F2, {q2, ;}) determine the same isogeny class if and only if F{ = F2. and {qi.;}, {q2,;} are either equal or conjugate. (2) A single isogeny class associated to the quaternion division algebra D/F which is ramified precisely (a) where B/F is, (b) at all primes of F over p, and (c) at all real valuations of F. This latter case may be thought of as an amalgamation of all the quadratic im aginary F'/F which do not split over p. If Y = (F', {q;}) or D, the class of points in KS(Fp) associated to it will be denoted KS( Y). It is stable under the Frobenius ; one may describe rather explicitly its struc ture together with this action. (For the parametrization of isogeny classes as well as the structure of each class, refer to Milne's lectures.) To each Y is associated : (I) A certain algebraic group H = Hy defined over Q. If Y = (F', {q;}) then H is (so to speak) the multiplicative group ofF', while if Y = D it is the multiplicative group of D. (2) A certain algebraic group Gy = G defined over Qp . First of all, for each prime p of F over p one has GP = the multiplicative group of F; � (F;) z if Y = (F', { q;}) and p splits in F', = the multiplicative group of D P otherwisei.e., if Y = (F', {q;}) and p does not split in F' or if Y = D. The group (j is n Gpo (3) A coset space X = n Xp , where Xp = Gp(Fp)/Gp(Op). Pn each XP define the transformation (/JP = multiplication by a uniformizing elements of q if Y = (F', { g;}), p splits in F', and q is the q; over p, ,
•
=
multiplication by a uniformizing element of the prime ideal of DP otherwise.
1 52
W. CASSELMAN
Expressing GP as (F;)2 in the first case, with the first factor corresponding to p, lbp is a little more explicitly multiplication by (p, 1). Let X = Xy = TI Xp, lb = l/>y = n lbp · Let H(Q) = H(Q)/H(Q) n ZK, G(Af) (an abuse of language) G(A�) X =
G(Qp)/zK n A'J.
In every case, observe : 4. 1 . 1 . LEMMA . (a) The algebraic group Z is canonically embedded in H ; (b) There exists a canonical class of equivalent embeddings of H(Q) into G(A�)
G(Qp). rendering H(Q) as a discrete subgroup of G(A1).
x
Only the second needs explaining. There clearly exist natural embeddings of
H(Q) into G(A�) and into G(Q p) all equivalent up to inner automorphism. The group H(R)/Z(R) is compact in all cases (no accident) so that since H(Q) is discrete in H(A) = H(A1) x H(R) the group H(Q) is discrete in H(A1)/ZK n A'J, hence in G(A1). Langlands' main result on the structure of KS( Y) is that there is a canonical class ofbijections between its points and the double coset space .
H(Q) \ G(A�)
X
G(Qp)/KP
X
G(Zp)
�
�
H(Q) \ G(AJ)/KP X G(Zp) H(Q) \ G(A�) X X/KP.
The Frobenius on KS( Y) corresponds to (/) acting on this coset space through its action on X. 4.2. The number of points of KS( Y) rational over Fpm is of course the number of fixed points of the mth power of the Frobenius. To express this conveniently requires a digression of some generality. Suppose � to be any locally profinite group, F a discrete subgroup. Then � and hence also C�(�) acts on the space C"'(F\�) through the right regular repre sentation. Explicitly, for every F E C�(�). f E C"'(F\�). RFf(x)
= =
sw F(g)f(xg) dg sw F(x- 1y)f(y) dy JF\W KF(x, y)f(y) dy =
where KF(x, y) = .E rF(x 1 ry). The operator RF (or, sometimes, F itself) will be said to possess a formal trace if KF(x, x) has compact support on F\�. and this trace is then defined to be fF\w KF(x, x) dx. One can presumably relate this to other notions of trace, but all that is important here is that the number of fixed points of a power of the Frobenius can be expressed as such a trace and that one has a formula for it. For each T E F, let �r be the centralizer of T in � and F7 = F n �r· If F lies in C�(�), then its orbital integral fw,,w F(g-1rg) dg is defined in many different circumstances, but the most elementary hypothesis that guarantees this is that F(g-1 rg) has compact support on �r\� ; this in turn is assured by the assumption that the conjugacy class of r is closed in �. 4.2. 1 . PROPOSITION. Let F be in C�(�). Suppose that the T E F such that F(g-1rg) #- 0 for some g E � lie in only a finite number of conjugacy classes in F, and that for -
1 53
THE HASSE-WElL �-FUNCTION
each such r the space F7\<§7 is compact and the conjugacy class of r in <§ is closed. Then F has a formal trace which is given by the formula
where the sum is over all conjugacy classes on r. PROOF. Choose a compact open subgroup '§o £;;; <§ small enough so that r n '§0 = { I } and F is hi-invariant under '§0. Thus r acts freely on '§/'§0• Let £( be a set of representatives, SO that <§ is the disjoint union of the Fx <§o (X E £r).
By one assumption on F, for each x E £( the sum :E rF(x-Irx) may be restricted to a finite number of conjugacy classes in r, hence may be written as :E frl L: rN F(x-Io-1rox) where the first sum is over representatives of conjugacy classes and only a finite number need be taken into account. Because F7\<§7 is com pact and the conjugacy class of r is closed in '§, the function F(g- Irg) lies in C';'(F7\<§); this implies that the above sum is nontrivial for only a finite subset of £(. Thus F does possess a formal trace, which is given by (meas '§0)
·
I: I: I: F(x- I o- I rox)
xE !!t"
! rl Fr\F
where in fact each sum may be restricted to a finite but arbitrarily large subset. Rearranging this it therefore becomes
4.3. Now take F to be fl(Q), <§ to be G(A1)-both corresponding to some isogeny
class Y. Define
FP
Fp(m)
p�m) p < ml
=
=
=
=
(meas KP)-I char KP, (meas Gp( op) ) - 1 char (
· G( o p) ) ,
TI F�m) , FP . p�m) .
4.3. 1 . LEMMA. The hypotheses of 4.2. 1 are satisfiedfor this choice of r, <§, F with m � I .
=
p<mJ
When Y D , this is clearly so since F\<§ is actually compact. Thus let Y = (F', { q1}). The only serious hypothesis to verify is the finiteness of the number of conjugacy classes with elements r such that F(g- 1 rg) =1= 0 for some g E '§. Now in this case H(Q) (F'Y and G(Qp) = TI Gp(Fp) where GP is (a) (F�Y or (b) n; . In either case p�ml (g- 1rg) F(r), and Fj;"l (r) =1= 0 if and only if (a) at each p,. where F' splits r has order m at q,. and is a unit at q,. ; (b) where F' does not split r is a generator of pj!j. Furthermore, in order that FP(g-1rg) =1= 0 for some g E G(A) it is necessary and sufficient that r be a unit at all primes not dividing p. Thus if p < m l (g- I rg) =1= 0 for some g, the order of r is specified at all primes of F'. Any two such r differ by a unit. But the units of F', modulo ZK, are a finite set. The function p < mJ is important because of: =
=
=
1 54
W. CASSELMAN
4.3.2. PROPOSITION. The formal trace of F(ml is the number of fixed points of the mth power of the Frobenius on xS( Y). I leave this as an easy exercise.
5. The comparison.
5. 1 . From §3 one sees that the right-hand side of equation (2. 1) is
(5. 1 )
I: meas(F7\�7)
J
'�·'"
F(g 1 rg) dg -
where F = G(Q)/G(Q) n Zx, � G(A)/Zx , F f11 fP ff'l . Let rp : G( A) -+ � be the canonical projection. It is easy to see that rp(G7(A)}\�r is always compact, so that one may replace �r by rp(G7(A)} in the above formula. But then one may note that rp(G7(A)}\� � GrCA)\G(A). Using the factorization of G(A) each term may be written as the product of =
=
·
•
meas (G7(Q)Zx\Gr(A)} ,
(5.2) (a)
J Gr (Af'J IG (Ap fP(g-l rPg) dg,
(5.2) (b)
JG,(Qp) \G (Qp) J Cml (g-I rpg) dg, JG7 (11) \G (R) f��cg-1r,g) dg. p
(5.2) (c) (5.2) (d)
From §4 one sees that #xS(Fp ) may be expressed as ..
(5.3) where : Y ranges over all systems (F ', {q;}), identifying a system and its conjugate, and the single D ; r ranges over all conjugacy classes of Hy(Q) ; � (}y(At) G(A�) .x Gy( Qp) ; p (m) FP . p�m) as at the end of §4. Just as above, each term here factors as the product of =
=
=
(5.4)(a) (5.4)(b) (5.4)(c)
meas {H7(Q)(Zx n
A j)\G7(A �) GrCQp)),
J Gr <AP ,G <AP FP(g-lrPg) dg, JG7(Qp) IG (Qp) p <ml(g-I rpg) dg. _
p
The proof of the main theorem reduces to a comparison of measures and orbital integrals. 5.2. Even the comparison of measures is not trivial. For the moment let k be an arbitrary local field, rjJ an additive character of k. If H is any finite-dimensional semisimple algebra over k, Tr its reduced trace, then rjJ Tr is an additive character of H. There exists on H a unique measure dx self-dual under the Fourier transform a
THE HASSE-WElL '-FUNCTION
1 55
determined by this character, and this in turn gives rise to an invariant measure dxx = dxjJxJH On H x . If now k is any global field, cp = TI ¢ v a character of Ak/k, and H a semisimple algebra over k, one obtains thus measures on all the local groups H.x as well as on Hx(Ak), which is called the Tamagawa measure determined by cp (§ 1 5 of [16]). Apply this construction in turn to the field F itself, quadratic extensions of F, and quaternion algebras over F (including M2(F)). I will always assume such measures chosen from now on. One classical result is that the global measure of Hx(A)IA'F · Hx(F) is independent of the quaternion algebra H (§ 1 6 of [16], or the lectures [12] of Gelbart and Jacquet). I will also need a comparison of local measures : 5.2. 1 . LEMMA. Let k be any nonarchimedean local field, H the unique quaternion division algebra over k. Then meas ofi = (q - 1)- 1 meas GL2(ok).
PROOF. Changing cp only multiplies both measures by the same constant; choose
cp to be of conductor ok. Letfbe the characteristic function of M2(ok). Then
/(x) =
J
f(y)cf;(Tr(xy) ) dy
M2 (k)
{1
x E M2(o1), = measMz(ok) o: x ¢ Mz(o,), = (meas M2(o1)) · f(x). Therefore meas M2(o1) = 1 . The group GL2(o1) is open in M2(o1) and since x E M2(o1) lies in GL2(o1) if and only if x mod Pr is nonsingular, meas GL2(o1) = :li GL2(Fq)/:li Mz(Fq) = ( 1 - 1/q) Z ( 1 + 1/q). Now let f be the characteristic function of oH · Then /(x) =
J
OH
cf;(Tr(xy)) dy
= meas oH
{1,
0,
X E P/i,
x ¢ P!i ,
or J = (meas oH)(characteristic function of P.ii) .
The Fourier transform of this in turn is J = (meas oH)(meas(P/i))f Since meas(P/i) = q 2 · meas oH, meas oH = q- 1 . Reasoning as above, meas ofi = (1 - 1/q )( l + 1 / q )( l /q ) . The lemma follows. One can similarly compare measures on GL2(R) and ax, but that will prove to be unnecessary. 5.2. Again for a while let k be any local field of characteristic 0, '1l either GL2(k), the multiplicative group Hx(k) of the quaternion division algebra over k, PGLz(k),
1 56
W.
CASSELMAN
or n x ;kx . For any X E � define D(x) = I a - ,8 1 2f ia,81 of the eigenvalues of X (over an algebraic closure of k) are a, ,8. If T is any torus of �' then for every regular t e T andfe C;"(�) define OT(f, t) =
JT\<§ f(g-l tg) dg.
The function F(t) = OT(J, t) satisfies F is smooth on the regular elements of T, (5.5) (a) Dl 1 2 F is locally bounded, (5.5) (b) Fhas compact support on T. (5.5) (c) If 1r: is an irreducible admissible representation of � then the character ch" of 1r: also satisfies (5.5) (a)-(b). For any f e C;"(�), f(g) dg = 1 .E D(t)OT( J, t) dt, 2 !Tl T where the sum is over all conjugacy classes of tori in � ' and if 1r: is an irreducible admissible representation then trace 1r: (f) = 21 I; D(t) ch"(t)OT(J, t) dt. (5.6) !Tl T
J
J
J
(For all this see §7 of [16], and for more about orbital integrals see [12] and [26].) 5.2. 1 . LEMMA. Suppose that � = n x ' F a conjugation-invariant function on �reg whose restriction to any torus T satisfies (5.5) (a)-(c). Jf 1 D(t) cha(t)F(t) dt = 0 2 I; !Tl T
J
for all irreducible admissible representations of� then F
=
0.
Fix a torus T. The set of elements of � conjugate to regular elements of Tis open. If f e C;"(T'•g), there exists h e C;"(�reg) with support in this set and such that OT(h) = f Then
J<§ h(g)F(g) dg = JT D(t)OT(h, t)F(t) dt = J D(t)f(t)F(t) dt. T
But the characters of irreducible representations are complete in the space of cen tral functions on �' so that/ = .E ia · cha (if k is nonarchimedean one can reduce this question to one about finite groups) and hence by hypothesis the above inte gral is 0. In other words, the integral of D · F against any f e C;"(T'•g) is 0, which implies that F itself is 0. One may (and I will) identify conjugacy classes in nx with elliptic and central conjugacy classes in GL2(k). There exists (by § 1 5 of [16] ; see also [12]) a bijective correspondence q +-+ 1r: between (a) irreducible, smooth (hence finite-dimensional) representations q of nx and (b) irreducible, admissible representations 1r: of GL2(k), space-integrable modulo Z, such that
THE HASSE-WElL '-FUNCTION
1 57
ch.. (t) = - ch,.(t)
(5.7)
for all regular elliptic elements t. Representations of nx;kx correspond to repre sentations of PGL2(k). When k = R, the trivial representation of Hx corresponds to the single discrete series representation no of PGL2 . For nonarchimedean k, the one-dimensional representations of Hx correspond to the Steinberg (or special) representations and all others correspond to absolutely cuspidal representations of GL2 . 5.2.2. PROPOSITION. Let fn E C�(PGL2(R)) be as in §3. Then for semisimple x E PGL2(R) = '§,
J
x hyperbolic, 0, = (meas ex jRx ) - 1 , x elliptic, = - (meas Hx /Rx)- 1 , x = l. PROOF. The case of hyperbolic x fell out in the very construction offn · Recall thatfn has the property '!lr\'!1
(g- 1 xg) dg =
trace n Cfn) = 1, n = 1 or sgn(det), = - 1, otherwise. = 0, Consider now the constant function hR on nx ;Rx with the value (meas nx;Rx )- 1 . It satisfies trace a(hn)
1, (J = 1 , = 0, otherwise. Thus by (5. 7) and remarks just afterwards trace n(/8) = - trace a(h8) whenever n and a correspond to one another. This and equation (5.6) now yield
J
cx ;R x
D(t)
ch,.(t) dt
J
cx\PGL2 (R)
=
whenever n and a correspond. version on ex IRx),
J
C x\ PGL2 (R)
=
By
fn(g- 1 tg) dg =
J
-
f8(g- 1 tg) dg
J
cx;Rx
D(t)ch.-(t) dt
J
cx\ H x
h8(g- I tg) dt
(5.7) and 5.2. 1 (or in this case just Fourier in cx\H x
h8(g- 1 tg) dg =
(meas cx ;Rx)- 1
for all t # ± l E Cx/Rx . Since the orbital integrals offn vanish on hyperbolic elements, /8 and - h8 corres pond to one another in the sense of [12] (the section on "matching orbital in tegrals"). Hence 5.3. Now let k be nonarchimedean and local, and for the moment let G = GL2(k),
I 58
W. CASSELMAN
K = GL2{ ok)· For each m � 0 IetJ< m > be the unique function in the Heeke algebra Jlt'(G, K) defined according to the Satake isomorphism by trace 7C(J< m > ) = q m 12(am + {J m) whenever 1C = 1C(a, {3) is the unramified principal series represen tation associated to the characters x ...... (aord(xl , {Jord (xl) of kx . One can exhibitJ < m > rather explicitly. For every i, j E Z recall the Heeke operator T(pi, W) =
(meas K)- 1 char K
( )K 7J ;
7J
j
where 7J is a generator of p. For m � 0 recall also T(pm) = !: T(pi, w) (i, j � 0, = m). There are the more or less classical Heeke operators and satisfy the relation T(p)T(pm) = T(p m+l ) + qT(p, p) T(pn- 1) which may be written formally as (I - T(p) X + q T(p, p)X2) - 1 = I; T(pm)Xm . (5.7) i +j
5.3. 1 . LEMMA {IHARA). One has
J
=
(I
q 1 t2a X) - 1 + (I
2 . I, J
= T(p), J < m> = T(pm)
_
q T(p, p) T{pm-2)
(m � 2).
PROOF. The case m = 0 is clear. And it is well known that trace 1C(T(p)) = q1 12(a + {3) if 1C = 1C(a, {3), which gives the case m = 1 . Formally, then T(p) +-+ q1 12(a + {3) and, also elementary, T(p, p) +-+ a{J. Now _
_
q 1 t2 [J X) - 1 = I; q m 12(am + [Jm) Xm 00
0
+-+
I;J < m > xm
but this expression also equals (I - q1 12a X) + (I - q 1 12[J X) +-+
I (1 - q 1 t2aX) (I - q1 12[JX) The proposition follows from this and (5.2).
2 - T(p) X - T(p) X + q T(p, p) X 2 "
5.3.2. COROLLARY. If
then
f < m > (x)
= =
let
- (meas K)-1(q - I ) , 0,
m = 2/, otherwise.
Continue to let H be the quaternion division algebra over k. From each m � 0 h <m> =
(meas oH)- 1 char(p7j
- p7}+1 ).
5.3.3. PROPOSITION. Let x be a semisimple element of G, m � 1. Then :
(a) Ifx is hyperbolic,
J G,\G J <m> (g-Ixg) dg (b) If x is elliptic
=
=
7J m
( ) e ) mod
1 . if X = I or 0 otherwise.
7Jm
(K n A),
1 59
THE HASSE-WElL �-FUNCTION
(c) Jfx is central, pm>(x) = - h <m>(x). PROOF. Recall (say from [4]) the relationship between the Satake isomorphism and orbital integrals : for fe :Yt'(G, K) /(a) = D(a)l 1 2
J
JA\G f(g-lag) dg,
trace 11:x
By definition of J<m> , then, j<ml (a)
=
qm l 2 ,
a=
( 1 ) or (I r;m)• rr
otherwise.
=0
Hence (a). Claim (c) is immediate from 5.3.2 and 5.2. 1 . Only (b) is difficult. To begin, consider equation (5.6) as 11: ranges over all essentially square-integrable irreducible representations of GL2 : trace 11:(/<m>)
=
JA D(a)ch (a) da JA\G pm> (g-lag) dg + 1 L; J D(t)ch"(t) dt J J <m> (g-ltg) dg 2 {T) T T\G 1 2
n
where now T ranges over all anisotropic tori. Similarly consider (5.6) as a ranges over all irreducible admissible representations of nx :
J
J
trace a(h <m>) = 21 L; D(t) cha(t) dt h<m>(g-ltg) dg. T\G {T} T Applying 5.2. 1 and equation (5. 7), it thus suffices to prove that
J
J
- trace 71:(j'm> ) + 21 A D(a)ch"(a) da A pm> (g-lag) dg = trace a(h <m>) \G whenever a and 11: correspond. For this : (1) no 11: occurring here possesses K-fixed vectors # 0, so trace 11:(j<m>) = 0 always. (2) If 11: is absolutely cuspidal then chn(a) = 0 unless a E A(o k) (first observed in Lemma 6.4 of [19] ; see also [8] and [5]). According to case (a) already done, OA(f) = 0 on A(o), so the whole left-hand side vanishes. But in this case a is not one-dimensional, hence not trivial on o}j, so that the right-hand side vanishes also. (3) Suppose 11: is special, corresponding to the character (J = X 0 NM of nx . Thus a(h Cml) x(r;m) on the one hand and on the other =
1 2
·
sA D(a)ch"(a) da s A\G J<m>(g-1ag) dg
also, by case (a) again.
=
x(r;m)
5.4. Return to earlier notation. It will now turn out that there is a bijective cor respondence between the nonzero terms of the two sums (5. 1) and (5.3), and that corresponding terms agree. This will conclude our p roof.
160
W. CASSELMAN
According to 5.2.2 the orbital integral of fn in (5.2) is 0 unless rn is elliptic or central, so I will assume that to be the case from now on. Suppose first that r E G(Q) is central. The term in (5. 1) corresponding to it is the product of meas(G(Q)ZK\ G( A)), (5.8) (a) (5. 8) (b) J�ml(rp) = ( - l) n (meas Dx(Zp))- 1 , (5.8) (c) fP(rP), (5.8) (d) firn) = ( - l)n (meas Dx(R)/Rx)- 1 . There is exactly one term in (5.7) corresponding to r. among the terms indexed by Y = D. It is the product of meas (Dx(Q)(ZK n Aj)\Dx(AJ)), (5.9) (a) F�ml(rp) = (meas Dx(Zp))- 1 , (5.9) (b) FP(rP). (5.9) (c) These products match since (5.8)(c) and (5.9)(c) are trivially equal and meas (G(Q)ZK\G(A))
= =
meas(Dx(Q)ZK\ Dx(A)) meas(Dx(Q)(ZK n Aj)\Dx(AJ)) meas(Z(R)\Dx(R)).
Now suppose r E G(Q) such that rn is elliptic. Thus F' = F(r) is an imaginary quadratic extension. If this extension splits at no prime of F over p, let Y = Yr = D. Otherwise, Y = Yr is one of the (F', {q;}), and the q; must be specified. Now by the definition ofnmJ as the product ofthefP<mJ and 5.3.3, if the term for r in (5.2) (c) does not vanish then rP is conjugate in GL2(Fp) to
whenever p splits in F'. In other words, rP must have order m at some prime qr of
F' over p and be a unit at the other; set Yr = (F', {qr } ).
In either case the element r clearly gives rise to an element of Hyr(Q), and it turns out to be an easy consequence of 5.2.2 and 5.3.3 that the corresponding terms in (5.2) and (5.4) agree. 6. Supplementary remarks.
6. 1 . When F = Q, the main theorem follows from a congruence relation between Heeke operators and the Frobenius. Although such a relation is likely to hold very generally, it does not yield a formula for the '-function. I want to explain this a bit more carefully. First of all, the integral Heeke operators always define algebraic correspondences on the scheme KS. When F = Q, the congruence relation says that modulo p T(p) = 1/J + 1/J * · T(p, p) where 1/J is the Frobenius and 1/J * its transpose (refer to §7.4 of [28]). An equivalent way of expressing this, as Langlands pointed out to me, is to say that in the ring of algebraic correspondences 1/J is a root of the poly nomial xz - T(p)X + pT(p, p). In general one may consider the polynomial det(X p(g)) as a function of the semisimple element g of LG. By the Satake iso morphism it corresponds to a polynomial whose coefficients are Heeke operators. -
THE HASSE-WElL �-FUNCTION
161
These will in fact be integral, and it looks not very difficult to show that (/J is a root. In more concrete terms, this will imply immediately that the roots of the Frobenius (acting on /-adic cohomology) lie among the roots of this polynomial, or that factors of Zp(X, KS) lie among the factors of n ,. i n -<*'o det (I - g(g(1t'p))) m (1r, KP) . Now when F = Q, one obtains by a further relation which I have always found a little mysterious that in fact the roots coincide, but it is this second step that does not seem to generalize (I refer to 7. 10(2) of [28] ; see also Piatetskii-Shapiro's argument on pp. 333-...:3 36 of [24] for a representation-theoretic analogue). Langlands has remarked that there is in some sense a good reason for this difficulty, inasmuch as when problems of L-indistiguishability arise one may get in some sense only partial coincidence of the roots of the Frobenius with those of the Heeke poly nomial. Incidentally, in his proof of the main theorem in the cases already mentioned, Shimura also used the Selberg trace formula. I might also mention that it is apparently Ihara who first applied the trace for mula to matters of this kind-in [14] he showed, modulo some technical problems, that the Hasse-Weil �-function associated to certain sheaves on SLz(Z)\£ is a product of Heeke L-functions. Both he and Shimura used what one might call a global formulation. The first application involving representation theory and local orbital integrals is Langlands' Antwerp talk [19] ; this earlier work seems to have played a role in Langlands' own further development as well as in Drinfeld's (note the remark in [9] on this point). 6.2. It is very little extra work to extend the proof of the main theorem to allow for nontrivial sheaves as well as a refinement which accounts for a factorization of the �-function according to the direct sum decomposition of d0 • Let me first sketch how to deal with sheaves. Suppose � : G � GLn to be a ra tional representation of G (i.e., rational over Q) trivial on NJ,.1Q , the kernel of the norm map from px to Qx (considered as algebraic groups) which is canonically embedded as scalars in the centre of G. Then one may associate to � a locally con stant Q-sheaf Ee on KS(C) and /-adic sheaves Ee(Q1) on KS (as in a special case in [19]), and consider for each "good" p the p-factor of a �-function
By the Lefschetz trace formula this satisfies
In both these formulas (/J is the Frobenius, and for convenience I have written �;c((/Jm) as the action of(/Jm on the stalk of Ee at the fixed point x of (/Jm . On the other hand one can define numbers m(1t'8, �) analogous to the numbers m(1t'8) in §1 satisfying, for example, m(1t'8, �) = 0 unless Ext � r) (�, 1t'8) "# 0 and set m(l't') = m(1t'8)m(1t'P). The main theorem extends to : Lp(s, K s, �) = . IT L(s - n/2, l't'p , p)m<,.. e) . 1r
1n
do
The proof is almost exactly the same as for � trivial ; one uses an extended version of
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W. CASSELMAN
the trace formula for the trace of R1, / E C�(�), on the representation of � which is £2-induced from � : trace R1
= I: !rl
meas(Fr\�r) Tr Hr)
J
f(g- 1 rg) dg.
(§,\cs
And one also uses a result of Langlands analogous to the one used above which gives not only the structure of xS(Fp) but of E0 on xS(Fp) (the natural guess is correct). Now the second point. As Langlands shows, essentially, in [19], corresponding to the decomposition ..#0 = EB n one has a decomposition H*(xS(C), Q) = EBH:. The only n which occur are those with m(n) #- 0, and for a n = nR ® n1 which does occur the representation n1 may be defined over Q . Corresponding in turn to this decomposition is one of H*(xS, Q1), and hence even of H*(xS x Fp , Q1). Another extension of the main theorem is that for good p 2n Lp(s, n, p) = IT det(l - (tl.l j H�(Q))rs) Hl; . i=O
To prove this : (1) one observes that the subspace H� (Q) is determined in H*(xS,Q) by the effect of operators in Yt'(G(A�), KP) and (2) applies the reason ing given above, but allowingfP to be an arbitrary element of this Heeke algebra. It is this result which Piatetskii-Shapiro is concerned with proving (by means of a congruence relation) in [24], with F = Q. 6.3. It has not escaped me that the result in §5 on the measures on GL2 and nx says, essentially, that the measures given by Jacquet-Langlands are multiples, by the same constant, of what Serre calls in [25] the Euler-Poincare measures on each group. (This is not quite true since strictly speaking the E-P measures on these are zero : but there is a clear relationship between E-P measures on SL2 and N1 . ) Does this observation generalize ? For real groups, I would expect a factor card( WcccJ I Wx) , to play a role, as it does in Harder's work [13]. Indeed, I suspect that Harder's paper does essentially relate inner twisting of measures to Serre's measures in the case when G is the inner twist of an anisotropic group. If G is not such an inner twist then Serre's measure is identically 0; how to describe the rela tionship of measures then? As for comparison of orbital integrals, Drinfeld has a nice result for GLn and division algebras of degree n (see Kazhdan's talk at this Institute). Diana Shelstad had proven in her thesis [27] many pretty results on real groups. REFERENCES I. T. Asai, On certain Dirichlet series associated with Hilbert modular forms and Rankin's method, Math. Ann. 226 (1 977), 8 1 -94. 2. A. Borel, Automorphic L-functions, these PROCEEDINGS, part 2, pp. 27-61 . 3. A. Borel and N. Wallach, Cohomology of discrete subgroups of semisimple groups, Notes,
lnst. Advanced Study, Princeton, N. J., 1 977. 4. P. Cartier, Lectures on representations of '{J-adic groups : A survey, these PROCEEDINGS, part 1 , pp. 1 1 1-1 55. 5 . W. Casselman, Characters and Jacquet modules, Math. Ann. 230 (1 977), 101-105. 6. P. Deligne, Formes modulaires et representations 1-adiques, Sem. Bourbaki 1968/69, no. 355, Lecture Notes in Math. , vol. 1 79, Springer, New York.
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7. P . Deligne, Travaux de Shimura, Sem. Bourbaki 1 97017 1 , no. 389, Lecture Notes in Math . , vol. 244, Springer, New York. 8. , Le support du caractere d'une representation supercuspida/e, C. R. Acad. Sci. Paris Ser. A-B 283 (1 976), A1 55-A1 57. 9. V. G. Drinfeld, Proof of the global Langlands conjecture for GL(2) over a function field, Functional Anal. Appl. 61 (3) (1 977), 74-75 . (Russian) 10. M. Dufto and J-P. Labesse, Sur Ia formule des traces de Selberg, Ann . Sci. Ecole Norm. Sup. 4 (1971), 1 93-284. 11. M. Eichler, Introduction to the theory of algebraic numbers and functions, Academic Press, New York, 1 966. 12. S. Gelbart and H. Jacquet, Forms ofGL(2} from the analyticpoint of view, these PROCEEDINGS, part 1 , pp. 21 3-25 1 . 13. G . Harder, A Gauss-Bonnet formula for discrete arithmetically defined groups, Ann. Sci. Ecole Norm. Sup. 4 (1971), 409-455. 14. Y. Ihara, Heeke polynomials as congruence �-functions in elliptic modular case, Ann. of Math. (2) 85 (1 967}, 2 67 2 95. 15. H. Jacquet, Automorphic forms for GL(2} II, Lecture Notes in Math., vol. 278, Springer, New York. 16. H. Jacquet and R. P. Langlands, Automorphic forms for GL(2), Lecture Notes in Math . , vol. 1 14, Springer, New York, 1 973. 17. R. Kottwitz, Combinotorics and Shimura varieties mod p, these PROCEEDINGS, part 2, pp. 1 85-1 92. 18. S. Lang, SL.(R), Addison-Wesley, 1975. 19. R. P. Langlands, Modular forms and 1-adic representations, Lecture Notes in Math., vol. 349, Springer, New York, 1 973 . 20. , Some contemporary problems with origins in the Jugendtraum, Proc. Sympos. Pure Math. , vol. 28, Amer. Math. Soc. , Providence, R.I , 1 976, pp. 401-4 1 8 . 21. , Shimura varieties and the Selberg trace formula, Canad. J. Math. (to appear} 22. , On the zeta-functions of some simple Shimura varieties (to appear). 23. Y. Matsushima and G. Shimura, On the cohomology groups attached to certain vector-valued differential forms on the product of upper half-planes, Ann. of Math. (2) 78 (1 963), 41 7-449. 24. I. I. Piatetskii-Shapiro, Zeta functions of modular curves, Lecture Notes in Math., vol. 349, Springer, New York, 1 973 . 25. J-P. Serre, Cohomologie des groupes discrets, Prospects in Mathematics, Ann. of Math. Studies, no. 70, Princeton Univ. Press, Princeton, N. J., 1 97 1 . 26. D. Shelstad, Orbital integrals for GL,(R), these PROCEEDINGS, part 1 , pp. 107-1 1 0. 27. , Some character relations for real reductive algebraic groups, Thesis, Yale Univ. , 1 974. 28. G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton, 1971 . 29. , On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. 31 (1 975), 79-98. ---
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UNIVERSITY OF BRITISH CoLUMBIA
Proceedings of Symposia in Pure Mathematics Vol . 33 (1 979), part 2, pp. 1 65-1 84
POINTS ON SIDMURA VARIETIES mod p
J. S. MILNE There is associated to a reductive group G over Q with some additional structure a Shimura variety Sc defined over C. In most cases it is known that Sc has a canonical model SE defined over a specific number field E. For almost all finite primes v of E it is possible to reduce SE modulo the prime and obtain a nonsingular variety Sv over a finite field Fqv" As is explained in [3], in order to identify the Hasse-Weil zeta-function of SE or, more generally, of a locally constant sheaf on SE it is necessary to have a description of (S0(Fq.), Frob) where Sv(Fq.) is the set of points of Sv with coordinates in the algebraic closure Pq. of Fq. and Frob is the Frobenius map Sv(Fq.) -+ Sv(Fq.) which takes a point with coordinates (ah · · · , am) to (aq(• · · · , a'/:{) . To be useful, the description should be directly in terms of the group G. Recently (13] Langlands has conjectured such a description of (Sv(Fq.), Frob) for any Shimura variety S and any sufficiently good prime v. In [12] he has given a fairly detailed outline of a proof of the conjecture for those Shimura varieties which can be realized as coarse moduli schemes for problems involving only abelian varieties, (weak) polarizations, endomorphisms, and points of finite order. (So G(Q) is of the form Aut8(H1 (A, Q), ¢) where B is a semisimple Q-algebra containing an order which acts on the abelian variety A, ¢ is a Riemann form for A whose Rosati involution on End(A) ® Q stabilizes B, and Aut8 refers to B linear automorphisms g of H1 (A, Q) such that cj;(gu, gv) = cj;(u, ,u(g)v) with ,u (g) lying in some fixed algebra F contained in the centre of B and fixed by the Rosati involution ; there is also a Hasse principle assumption.) Earlier [8, Conjecture 1] Ihara had made a similar conjecture when S is a Shi mura curve and had proved it when G = GL2 [9, Chapter 5]. When G = B x , B a quaternion division algebra over Q, Morita [15] proved Ihara's conjecture for all primes p of E ( = Q) not dividing the discriminant of B. Both he and Shimura have obtained partial results for more general quaternion algebras (unpublished). More recently Ihara has proved his conjecture for all Shimura curves and sufficiently good primes (announcement in [11]). While Ihara bases his proof on the Eichler Shimura congruence relations, Morita's method, as described in [10], appears to be quite similar to that of Langlands. In order to give some idea of the techniques Langlands uses in his proof I shall describe it in the case that G is the multiplicative group of a totally indefinite quaAMS (MOS) subject classifications (1 970). Primary 14K1 0 ; Secondary 14010, 14L05. © 1979, American Mathematical Society
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ternion algebra over a totally real number field. In § I it is shown that Sc para metrizes, in a natural way, a family of abelian varieties with additional structure. The following section describes how Artin's representability criteria may be used to prove the existence of a variety Sg over Q which is a canonical model for Sc and which, when reduced mod p, parametrizes a family of abelian varieties (with additional structure) in characteristic p. Thus the problem of describing (Sp(Fp), Frob) becomes one of describing this family. In §5 the Tate-Honda classification of isogeny classes of abelian varieties over finite fields is used to determine the isogeny classes in the family, and in §6 the individual isogeny classes are described. Since this requires the use of p-divisible groups and their Dieudonne modules, these are reviewed in §3. Notation. F is a totally real number field of degree d over Q, B is a quaternion division algebra over F which is split everywhere at infinity, b ...... b * is a positive F-involution on B, and 08 is a maximal order in B. G is the group scheme over Z such that G(R) = ( OYfP (8) RY for all rings R, where Of!P is the opposite algebra to 08. A is the ring of adeles for Q ; A = R x A1 = R x A� x Qj; ; A1 = Z1 (8) Q, Z1 = proj lim ZjmZ; Z1 = Zf x Zp . K is a (sufficiently small) open subgroup of G(Z1) . .::1 is a product of rational prime numbers such that if p % .::1 then p is unramified in F, B is split at all primes of F dividing p, and K = KPG(Zp) where KP = K n G(Af). Sc = KSc is the Shimura variety over c defined by G, K, and the map h ; ex --> G(R) defined in § 1 ; thus its points in C are Sc(C) = G(Q)\G(A)/K=K where K= is the centralizer of h in G(R). If V = V(Z) is a Z-module then V(R) = V ®z R for any ring R. 1. Sc as a moduli scheme. Recall that an abelian variety over a field k is an alge braic group over k whose underlying variety is complete (and connected) ; its group structure is then commutative and the variety is projective. For example, an abelian variety of dimension one is an elliptic curve, and may be described by its equation, which is of the form Y2Z
= X3 +
aXZ2 + bZ3 ,
a, b E k, 4a3 + 27hZ
¥=-
0.
It is impractical to describe abelian varieties of dimension greater than one by equations, but fortunately over C there is a classical description in terms of lattices in complex vector spaces. Let V be a lattice in Cg , i.e., V is the subgroup generated by an R-basis and so V ®z R ""' Cg . Then Cg/ V is a compact complex-analytic manifold which becomes a commutative Lie group under addition. When g = 1 the Weierstrass p-function corresponding to V, and its derivative, define an em bedding z >-> (p(z), p'(z), I ) : Cf V 4 P'f: of Cf V as an algebraic subset of the projective plane. Thus Cf V automatically has the structure of an algebraic variety and so is an abelian variety. This is no longer true if g > 1 for there may be too few functions on Cg/ V to define an embedding of it into projective space. Since any meromorphic function on Cg/ V is a quotient
POINTS ON SHIMURA VARIETIES
mod p
167
of theta functions on 0, eKj V will be algebraic if and only if there exist enough theta functions. By definition, a theta function for V is a holomorphic function 0 on 0 such that, for v e V, {}(z + v) = {}(z) exp (2tci(L(z, v) + J(v))) where L(z, v) is a e-linear function of z and J(v) depends only on v. One shows that L(z, v) is additive in v , and so extends to a function L: eg x eg � e which is e-linear in the first variable and R-linear in the second. Set E(z, w) = L(z, w) - L(w, z). Then (a) E is R-valued, R-bilinear, and alternating; (b) E takes integer values on V x V; (c) the form (z, w) ....... E(iz, w) is symmetric and positive. {The symmetry is equivalent to having E(iz, iw) = E(z, w) for all z, w ; the positivity means E(iz, z) � 0 for all z.) A form satisfying these conditions is called a Riemann form for V and it is known that there exist enough functions to define a projective embedding of eKj V if and only if there exists a Riemann form for V which is nondegenerate (and hence such that E(iz, z) is positive definite). If g = 1 we may always define E(z, w) to be the ratio of the oriented area of the parallelogram with sides Ow, Oz to that of a funda mental parallelogram for the lattice. Since this form always exists, and is unique up to multiplication by an integer, one rarely bothers to mention it. By contrast, if g > 1 , a nondegenerate Riemann form will not usually exist and when it does, it will not be unique up to multiplication by an integer. However since eKj V is com pact the algebraic structure on eKJ V (but not the projective embedding) defined by a Riemann form is independent of the form. Thus, given a lattice in eg for which there exists a nondegenerate Riemann form, we obtain an abelian variety. Conversely, from an abelian variety A of dimension g we can recover a complex vector space W of dimension g and a lattice V in W for which there exists a nondegenerate Riemann form. W can be described (accord ing to taste) as the Lie algebra Lie(A) of A, the tangent space tA to A at its zero element, or as the universal covering space of the topological manifold A( e). The lattice V can be described as the kernel of the exponential exp : Lie(A) � A(e), or as the fundamental group of A(e) which, being commutative, is equal to H1 (A, Z). We shall always regard the isomorphism WJ V � A(e) as arising from the exact sequence,
Since H1 (A, Z) is a lattice in tA; we have H1 (A, R) = H1 (A, Z) ® R � tA . Thus A is determined by H1 (A, Z) and the complex vector space structure on H1 (A, R). A complex structure on a real vector space V(R) defines a homomorphism h : ex � AutR( V(R}), h(z) = (v ....... zv), and the complex structure is determined by h. Thus an abelian variety A is uniquely determined by the pair (H1 (A, Z), h) where h : ex � Aut(H1 (A, R)) is defined by the complex structure on H1 (A, R) = tA . Moreover every pair {V(Z), h) for which there exists a Riemann form arises from an abelian variety. Let V(Z) = H1 (A, Z). A point of finite order on A corresponds to an element of V(R) some multiple of which is in V(Z). More precisely, the group of points of
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finite order on A may be identified with V(Q)/ V(Z) c V(R)/ V(Z). For any integer the group A m(C) of points of order m is equal to m-I V(Z)/ V(Z) � V(ZjmZ) � (Z/mZ)2 d im (Al . We define T1A to be proj limm A m (C) = V(Z1) and, for any prime l, T1A to be proj limmA1m(C) = V(Z1) ; thus T1A = fl 1T1A and T1A
m > 0, �
z1 2d im (A) .
A homomorphism A --+ A' of abelian varieties induces a C-linear map tA --+ tA ' such that H1 (A, Z) is mapped into H1 (A', Z). Conversely, if A and A' correspond respectively to ( V, h) and ( V', h') then a map of Z-modules a : V(Z) --+ V'(Z) extend ing to a C-linear map a ® 1 : V(R) --+ V'(R) (i.e., such that a ® 1 o h(z) = h'(z) o a ® 1 for all z) arises from a map of complex manifolds A( C) --+ A'(C) and the compactness of A( C) and A'(C) implies that the map is algebraic. We write End(A) for the ring of endomorphisms of A and End0(A) for End(A) ®z Q. Since End0(A) has a faithful representation on H1 (A, Q), it is a finite-dimensional Q algebra; it is also semisimple, and its possible dimensions and structures are well understood. To define a homomorphism i : 08 --+ End(A) when A corresponds to (V, h) is the same as to define an action of 08 on V such that h maps ex into Aut080R( V(R)). When such an i is given we say that 08 acts on A provided i(1) = 1 . Such an i induces an injection i : B 4 End0(A). A nondegenerate Riemann form E for A defines an involution a ....... a' of End0(A) by the rule E(az, w) = E(z, a'w) ; this is the Rosati involution, which is known to be positive, i.e., Tr (aa') > 0 for all a # 0 where Tr denotes the reduced trace from End0(A) to Q. Suppose 08 acts on A. We say that two Riemann forms E and E' on A are £-equivalent if there exist nonzero c, c' E OF such that E(u, cv) = E(u, c'v) for all u, v E V(Z), and we define a weak polarization of A to be an £-equivalence class A of nondegenerate Riemann forms. Since F is the centre of B, the Rosati involutions defined by any two elements of su<;h a A induce the same map on i(B). We shall be interested in triples (A, i, A) such that E( i(b)u, v) = E(u, i(b *) v) for u, v E V(R), b E B, E E A, i.e., we require that the Rosati involutions defined by A stabilize i(B) and induce the given involution b ....... b* on B. We next review some notations concerning B. The main involution b ....... b• of B is so defined that under any R-isomorphism B ®F R � M2 (�) , if b corresponds to M = <: ;) then b• corresponds to M' = (�;:) ; thus b + b' = Tr8 1 p(b) and bb' = Nm8 1 p(b). The Skolem-Noether theorem shows that there exists a t E B such that b* = t- l b•t = tb•t-1 for all b E B ; automatically t2 E F and the positivity of b ....... b* implies that t 2 < 0, i.e., t 2 has negative image under all embeddings F 4 R. We fix an isomorphism B ® g R � M2(R) x . . . x M2(R) such that if b +-+ (Ml > ' " • Mn) then b* +-+ (M!r, . . . , M�r) where M:r is the transpose of M1• Since
(
Mtr - 01 _
-
) (
)
1 IM' O - 1 1 0 '
0
-
t maps to an element (c1(� -fi) , . . · , en(� -m with each c1 E R, and t may be chosen so that c1 > 0 . The next lemma implies that if 0 8 acts on a complex manifold cuI V then there is a Riemann form E for V whose corresponding Rosati involution induces b ....... b* on B and any two such forms are £-equivalent, i.e., that there is a unique weak
POINTS ON SHIMURA VARIETIES mod p
1 69
polarization which is compatible with the 08-action and the given involution. LEMMA 1 . 1 . Let V = V(Z) be a free Z-module of rank 4d on which 08 acts. There
is a nondegenerate alternating form cjJ on V(Q) such that : (a) cjJ(u, v) E Z if u, v E V(Z) ; (b) cjJ(ut, u) < O for all u i= 0, u E V(R) ; (c) cjJ(bu , v) = cjJ(u, b*v) for all b E B and u, v E V(Q) ; (d) for any B-automorphism a of V(Q) there exists a f.t(a) E px such that cjJ(au, av) = cf;(u, f.t(a)v) for aU u, v E V(Q). Moreover ' if cjJ' is a second nondegenerate al ternating form on V(Q) satisfying (c) then there exists a c E px such that cjJ(u, cv) = cjJ'(u, v) for all u, v E V.
PROOF. V(Q) has dimension one over B and so, after choosing an appropriate basis vector, we may identify V(Q) with B and V(Z) with a left ideal in 08• Define cjJ(u, v) = Tr81iuv ' t) = Tr81Q(utv*). Then cjJ(u, v) = Tr (utv*) = Tr(vt*u*) = Tr ( v( - t)u*) = - cjJ(v, u), and so cjJ is alternating. (a) is obvious, and cjJ(ut, u) = Tr81Q(ut 2u * ) = TrF1Q (t 2Tr8 1 F(uu* ) ) < 0 for u i= 0, which proves (b) and that cjJ is nondegenerate. For (c) we note that cjJ(bu, v) = Tr(utv*b) = Tr(ut(b*v)* ) = cjJ(u, b*v). Finally, any B-automorphism a of V(Q) = B is multi plication on the right by an element b E B x . Thus cjJ(a u, a v) = Tr (ubb•v•t) = cjJ(u, f.t(a)v) with fl-(a) = Nm81F(b). For the last part, consider the Q-linear map v ...... cjJ'(l , v) : B -. Q. Since Tr81Q : B x B -> Q is nondegenerate, there is a unique b E B such that cjJ'( l , v) = Tr (btv*) for all v E B. Then cjJ'(u, v) = cjJ' ( 1 , u*v) = Tr (btv*u) = Tr (ubt v *) = Tr(ubv•t). We also have cjJ'(l , v) = - cjJ'(v, 1) = - Tr (vbt) = - Tr (t ' b ' v ') = Tr (b ' v' t) = Tr (b•tv*). Thus b = b•, which implies that it is in F, and we may take c = b . For the remainder of this section V(Z) will be 08 regarded as an 08-module and cjJ will be as in the lemma. For any ring R we may identify G(R) with Aut08C2JR ( V(R )) since any 08 ® R-endomorphism of V(R) = 08 ® R is right multiplication by an element of 08 ® R. Define h to be the homomorphism ex -> G(R) = Aut8C2JR ( V(R)) such that h(i) is right multiplication on V(R) = B ® R --""> M2(R) x . . . x M2 (R) by ((<J_1 fi), · · · , ({)_1 fi)) . Thus Koo = {(Mt. · · · , Md)} with M,. of the form (�b �), a, b E R. The form E = cjJ is a Riemann form for ( V(Z), h), e.g., cjJ(iu, iv) = ¢ (uh(i), vh(i)) = cjJ(u, Nm8 ! F(h(i))v) = cjJ(u, v) and cjJ(iu, u) = cjJ(utj - ( - t 2) 1 1 2 , u) > 0 for u -1= 0. Thus ( V(Z), h) defines an abelian variety A. The action of 08 on V(Z) induces a map i : 08 -> End(A) and the Rosati involution defined by the weak polarization A containing cjJ induces b ...... b* on B. Recall that K is an open subgroup of G(Zr). Two isomorphisms ch, cPz: TrA --""> V(Zr) are K-equivalent if there is a k E K such that (PI = k¢2 . For example, if K = Km = Ker ( G(Zr) -. G(ZjmZ)) then to give a K-equivalence class of isomorphisms TrA -> V(Zr) is the same as to give an isomorphism A m (C) --""> V(ZjmZ), i.e., a level m structure. THEOREM 1 .2. There is a one-one correspondence between the set ofpoints Sc(C) = G(Q) \ G(A)/ KooK and the set of isomorphism classes of triples (A, i, (/>) where A is an abelian variety of dimension 2d, i defines an action of 08 on A, and If> is a K equivalence class of 08-isomorphisms TrA --""> V(Zr) ·
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REMARK 1 . 3 . (a) We say that two triples (A, i, �) and (A', i', �') are isomorphic if there exists an isomorphism a : A -+ A' such that a o i(b) = i'(b) o a for all b e 08 and if/ o (T1a) e � for all if/ e �'. (b) Normally when considering families of abelian varieties parametrized by Shimura varieties it is necessary to work with quadruples (A, i, A , �) with A a (weak) polarization. This is not necessary in our case because, as we observed above, A always exists uniquely. PROOF OF .1 .2. We first show how to associate to any g e G(A) a triple (A g , ig , �g). If g = 1 we take (A, i, �) with (A, i) as defined before and � the class of the identity map T1A = V(Z1) .l
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projection V(R) x ( G(A)JK""K) --+ G(A)/K""K. We give G(A)/K""K its usual complex structure and the copy of V(R) over gK""K the complex structure defined by hg. Inside each Vg we have a lattice g V(Z), and these vary continuously with g. Thus we may divide out and obtain a map of complex manifolds .r --+ G(A)/K""K such that the fibre over gK""K is the abelian variety A g . We may now let G(Q) act on both manifolds and divide out again to obtain an analytic family d --+ Sc of abelian varieties. Each fibre A g has the structure defined by (ig, ?>g), and these vary continuously. In fact d --+ Sc is an algebraic family, i.e., d is an algebraic variety and the map is algebraic. If F :F Q the above construction fails because units of F may act on (Ag, ig, cpg) and so the action of G(Q) on .r is not free. However we may "rigidify" the situa tion as follows : consider quadruples (A, i, ?> . e) where A, i, and ?> are as before and e is an injection from the unique weak polarization A to px such that e(if/ ) = ce(if;) if if; ' (u, v) = if;(u, cv) . The isomorphism classes of quadruples are classified by px x Sc(C) which is a disjoint union of copies of Sc(C), one for each element of p x , on which px acts by permuting the copies. px x Sc may be regarded as a scheme over C which is an infinite disjoint union of varieties and the previous process gives an algebraic family of d --+ p x x Sc of abelian varieties with structure. References. The most elegant elementary and nonelementary treatments of abelian varieties over C are to be found respectively in [20] and [17, Chapter I]. Families of abelian varieties parametrized by Shimura varieties were extensively studied by Shimura in the 1960 ' s (see his Annals papers of that period). They are also discussed briefly in [4]. 2. S as a scheme over Z [.::1- 1 ), We shall see shortly that Sc has a model Sg over Q, i.e., that there is a scheme Sg over Q whose defining equations, when considered
over C, give Sc. There is no reason to believe that Sg will be unique but Shimura has given conditions which will be satisfied by at most one model ; such a model (when it exists) is said to be canonical. For example, let F' be a quadratic totally imaginary extension of F which splits B and let A0 be the abelian variety of dimen sion d defined by the lattice OF ' c: F' ® R. Then OF ' acts on A0 and A0 is said to have complex multiplication by F'. Let A = A0 x A0• If we embed F' in B and choose a basis {eh e2 } for B over F' with e 1 o e2 e 08, then we get a map B 4 M2(F') c: M2 (Endo(A0) ) = End0(A) sending 08 into End(A). Also we get a map T1A = ( O F ' ® O F ,) ® Z1 _L 08 ® Z1 = V(Z1) (in the notation of § 1). The triple (A, i, ?>) defines a point of Sc, and hence a point of Sg with complex coordinates. For Sg to be canonical these coordinates must be algebraic over Q and generate a certain explicitly described class field. For the reasons explained in the introduction we would like to have a scheme S defined by equations in Z[Ll- 1 ) which, when regarded over Q, is the canonical model Sg of Sc, and which is such that it is possible to describe explicitly (S(Fp). Frob ) for any p 1 Ll. Such an S will define a functor R ...... S(R) which associates to any ring R in which Ll is invertible the set of points of S with coordinates in R. (More generally, it associates to any scheme T over spec Z[Ll- 1 ) the set S(T) of maps T --+ S. ) Since the functor determines the scheme uniquely this suggests that in constructing S we should write down a functor !/' such that, in particular, !/'(C) = Sc(C) = G(Q)\G(A)/K""K and try to prove that it is the points functor of a
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scheme. After § 1 it is natural to define sP(R) to consist of isomorphism classes of triples (A, i, rp) where each of the three objects is the analogue over R of the cor responding object over C. Thus A is a projective abelian scheme of dimension 2d over R. Intuitively, A can be thought of as an algebraic family of abelian varieties, each of which is defined over a residue field of R. More precisely it is a projective smooth group scheme over spec R with geometrically connected fibres. As before i is to be a homomorphism Os 4 End(A) such that i(I) is the identity map. We assume that A has a polarization whose Rosati involution induces b ...... b* on B. Two problems arise in defining (fi which may be best understood if we write rjJ : T1A -+ V(Z1) as a product IT rp1 : IT T1A -+ IT V(Z1) of maps. Firstly, if p is not invertible in R there will never exist an isomorphism rpp : TpA � V(Zp) ; thus we take rpp to be a map defined only over R[p- 1 ]. Secondly, unless R is an algebraically closed field it is unrealistic to expect there to be an isomorphism rjJ1 : T1A -+ V(Z1) for any I, for this would imply that all coordinates of ali i-power torsion points of A are in R. Instead we assume that K ::::> Km = Ker(G(Z1) -+ G(ZfmZ)) some m, and consider isomor phisms rjJ : A m � V(Z/mZ) defined on some etale covering of R, two such isomor phisms rjJ 1 and ¢2 being K-equivalent if ¢ 1 = krjJ2 , k e K, locally on spec(R), and we take (fi to be a K-equivalence class in this new sense. It is necessary to put one extra condition on the triple (A, i, (/J) : if the R-algebra R' is such that Os ® R' � M2 (0p ® R') then the two submodules of tA I R corresponding to the idempotents (b 8) and (8 �) should be free OF ® R'-modules of rank 1 locally on spec(R'). (This condition holds automatically if R = C; for examples where it fails in an analogous situation in characteristic p, see [18, 1 .29]. ) Having defined our functor sP we now have to see whether it is the points functor of a scheme. Generally speaking this is a very delicate question but M. Artin has given an often-manageable set of criteria for a functor to be the points functor of an algebraic space. An algebraic space is a slightly more general object than a scheme, but for our purposes it is just as good ; it makes good sense to speak of its points with coordinates in a ring, and the proper and smooth base change theorems in etale cohomology, which are the theorems which allow us to compute Hasse-Weil zeta-functions by reducing modulo a prime, �old for algebraic spaces. (In fact, the algebraic spaces we get are almost certainly schemes, and this surely could be proved by using Mumford's methods [16] instead of Artin's.) Consider first the case that F = Q. Then Artin's criteria may be checked and show that there is an algebraic space S, proper and smooth over Z [J- 1 ], such that S(R) = sP(R) for any ring R in which L1 is invertible. In particular S(C) = sP(C) = Sc(C) and S(Fp) = sP(Fp) for any p not dividing .d. The algebraic family .91 -+ Sc constructed in 1 .4(b) is an element of sP(Sc) = S(Sc), and so gives a map Sc -+ S. This induces a map Sc -+ S x spec C which is an isomorphism. Moreover it is known that S0 is the canonical model. When F '# Q, then a slightly weaker result holds, but one which is just as useful to us. Since there are nontrival automorphisms of (A, i, (fi) there can be no algebraic space S with S(R) = sP(R) for all R. However, there does exist an algebraic space S, proper and smooth over Z [J- 1 ], and a functorial map .:f>(R) -+ S(R) which is an isomorphism whenever R is an algebraically closed field. Thus S(C) = sP(C) = Sc(C) as before, and S0 is the canonical model of Sc. To prove these facts one may "rigidify" the moduli problem as in the second paragraph of 1 .4(b), make the
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constructions as in the case F = Q, and then form quotients under the left action by px, or else work directly with stacks. Note that in either case, S (Fp) = f/(Fp) has a description in terms of abelian varieties with additional structure. References. [1] contains a short introduction to Artin's techniques for represent ing functors by algebraic spaces and [2] a more complete one. In [5] and [18] these techniques are applied to a situation which is very similar to ours. (In fact, it is almost identical ; see §7 of the Introduction to [5].) The basic definitions concern ing abelian schemes can be found in [16]. 3. Finite group schemes, p-divisible groups, and Dieudonne modules. In the remain ing sections we shall need to consider the finite subgroup schemes of an abelian variety and so, in this section, we review some of their properties. We fix a perfect field k of characteristic p # 0. Let R be a finite k-algebra (so R is finite-dimensional as a vector space over k) and let N = spec R. For any k-algebra R', a point of N in R' is simply a map of k-algebras R --+ R' ; thus N(R') = Homk_.1g (R, R'). If every N(R') is given the structure of a commutative group in such a way that the maps N(R') --+ N(R") induced by maps R' --+ R" are homomorphisms, then we call N, together with the family of group structures, a finite group scheme over k. As for affine algebraic groups, giving the family of group structures corresponds to giving a comultiplica tion map R !!!... R ®k R. ExAMPLE 3. 1 . (a) Any (commutative) finite group M can be regarded in an ob vious way as an algebraic group over k and hence as a finite group scheme. Indeed, let R be a product of copies of k, one for each element of M, and let N = spec R. Then N, as a set, is equal to M. The group law on M induces a comultiplication on R which, in turn, induces compatible group structures on N(R') for all R'. If R' has no idempotents other than 0 and 1 , then N(R') = M. (b) f.lpn = spec k[T]f(TP" - 1). Then ppn(R' ) = {I: e R' I�P" = 1 } is a group under multiplication for any R', and these group structures make f.lpn into a finite group scheme. Note that ppn(R) = { 1 } if R has no nilpotents and, in particular, if R is an integral domain. (c) ap = spec k[T]/(TP). Then ap(R') = {a e R' laP = 0}. As (a + b)P = aP + bP in any k-algebra, ap(R') is a group under addition, and these group laws make ap into a finite group scheme. Again ap(R') has only one element if R' has no nilpo tents. (d) ZfpZ = spec k[T]f(TP - T). If R' has no idempotents other than 0 and 1 (e.g., R' an integral domain) then (Z/pZ) (R') = Fp , the prime subfield of R', which is a group under addition. This example is a special case of (a), because k[T]/(TP - T) = k[T]/T(T - 1) . . · (T - (p - 1)) � k x . . · x k (p copies). The rank or order of a finite group scheme N = spec R is the dimension of R as a vector space over k. For example the order of the group scheme defined by M in 3. 1(a) is the order of M, while the orders of liP"• ap. and ZfpZ are pn , p and p respectively. A homomorphism from one finite group scheme N1 = spec R 1 to a second N2 = spec R2 is a k-algebra homomorphism R2 --+ R 1 such that the induced maps N1 (R') --+ N2(R') are all homomorphisms of commutative groups.
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From now on we consider only finite group schemes of p-power order. The essential facts are the following. Facts. 3.2.(a) They form an abelian category. Thus we may form kernels, quo tients, etc. exactly as if we were working with a category of modules. (b) When k is algebraically closed the only simple objects are f-lP • ap, ZjpZ. This means that any finite group scheme of p-power order has a composition series whose quotients are f.lp• ap, or Z/pZ. There can be no homomorphism from one simple object to another of a different type. (c) The category is self-dual, i.e., there is a contravariant functor N�-> N ( = Car tier dual of N) which is an equivalence of the category with itself. More precisely, for each N there is a pairing N x N � Gm ( = GL1 ) such that, for any k-algebra R, the pairing induces isomorphisms N(R) � HomR(N, Gm), N(R) � HomR(N, Gm). For example, (ZjpZ)/\ = f.lp and the pairing (ZjpZ)(R) n X f.lp(R) � Gm(R) is (n, ') I-+ ' ; ap = ap and the pairing ap(R) X ap(R) � Gm ( R) is (a, b) � exp(ab) = 1 + ab + · · · + (ab) P- l f( p - 1) ! . (d) Hom (ZjpZ, ZjpZ) = ZjpZ, Hom (f.lp• f.lp) = ZjpZ, Hom (ap, ap) = k. The statement for Z/pZ is obvious, and that for f.lp follows by Cartier duality. The map ap � ap corresponding to c E k is (T ,_. cT) : k[T]j(TP) � k[T]/(TP) on the algebra of ap and (a ,_. ca) : ap(R') � ap(R') on its points. (e) If O � N' � N � N" � 0 is exact then order(N) = order(N')order(N"). Let A be an abelian variety over k. For each n, Apn � Ker(pn : A � A) is a finite group scheme of order (pn)Zdim cAJ , i . e., the order is the same as when p # characteristic(k). The system A p 4 A p2 4 · · · is called the p-divisible (or Barsotti Tate) group A(p) of A. More generally, a p-divisible group of height h is a system of finite group schemes and maps N = (N1 i!., N2 � N3 !2, · · ·) such that Nn has order pnh and in- ! identifies Nn-l with the kernel of (Nn � Nn). For example Qp/Zp = ( Z/pZ � Z/p2Z � · · · ) andppoo = (pp � f.lp2 � f.lp � · · · ) are p-divisible groups of height one. A(p) is of height 2 dim(A). A homomorphism ¢ : N � N' of p-divisible groups is a family of maps ¢n : Nn � N� commuting with the maps in and i�. Exercise 3.3. (k algebraically closed.) For any abelian variety A there are maps ¢n : A pn � Apn such that Ker(¢n) = Ker(¢n+ l ) for all sufficiently large n. Deduce that A has � pdi m A points of order p, and that when equality holds A(p) = (Qp/ Zp) d im CAJ x (ppoo)dim CAl . (Such abelian variety is said to be ordinary.) Let W = Wk be the ring of Witt vectors over k ; it is a complete discrete valua tion ring of characteristic zero whose maximal ideal is generated by p and which has residue field k. There is a unique automorphism a ,_. aCPl of W which induces the pth power map on k. If k = Fp then W is the completion of the ring of integers in the maximal unramified extension Qpn of Qp and a ,_. aCPl is induced by the usual Frobenius automorphism of Qj,n over Qp. Let W[F, V] be the ring of noncommuta tive polynomials over W in which the relations FV = p = VF and Fa = a c p J F, a V = Va CP l , hold for all a E W. There is a contravariant functor, N ,_. DN = Dieudonne module of N, associating to each p-power order finite group scheme a W[F, V)-module which is of finite length as a W-module ; D defines an antiequiv alence of categories. The length of DN as a W-module is equal to the order of N. Thus manipulations with finite group schemes correspond exactly to manipulations with modules over the noncommutative ring W[F, V]. Examples : D(pp) = Wjp W = k ; F acts as 0, V acts as I ;
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D(ap) = k ; F = 0, V = 0 ; D(ZfpZ) = k ; F = I , V = 0. If N is unipotent and pN = 0, then DN = Lie(N) ; the bracket operation on Lie(N} is zero but it has the structure of a p-Lie-algebra and F acts as the "p-power" operation and V acts as zero. More generally, if N is unipotent and killed by p" , then DN = Hom(N, Wn) where WI = Ga = the additive group and wn = the Witt vectors of length n regarded as an algebraic group. There are canonical, nondegenerate, W-bilinear pairings ( , ) : DN x DN � W ® Qp/Zp such that (Fm, n) = (m, Vn)
• • •
• •
• •
•
•
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(as before) as a free Zrmodule of rank 2 dim(A). We write T1A TljA x TpA. Finally we note that to classify p-divisible groups up to isomorphism, it is neces sary to classify the (F, V)-stable lattices in the objects of vtt . References. The best introduction to the subject matter of this section is [6]. 4. S(Fp) as a family of abelian varieties. Fix a prime p not dividing LJ. From §2 we know that points of S( Fp) correspond to isomorphism classes of triples (A, i, (/j) where A is an abelian variety of dimension 2d over Fp, i is an action of 08 on A, and (/j is a KP-equivalence class of isomorphisms cp: TljA ---> V(Zlj) where TljA proj lim ptn A nCFp). (Recall that cpp : TpA ---> V(Zp) is defined only over the ground ring with p inverted, and Fp [p- 1 ] is the zero ring.) The 08 ® Pp-module tA satisfies the following condition : corresponding t o the idempotents ( A 8) and (8 V i n 08 ® Fp (4. 1 ) the subspaces � M2(Fp ) are free OF ® Fp-modules of rank 1 . If A i s defined by equations Za ( i ) TCi l , let A c p J be the abelian variety over Fp defined by the equations Zafil Ti . There is a Frobenius map F FA : A ---> A c pJ which takes a point with coordinates (tl > · · · tJ to (t1, · · · t£) . The map FA is a purely inseparable isogeny of degree p2a , which means that AF, the kernel of F, is a finite group scheme of order p2a with only one point in any field (so that only ap and p.p occur in any composition series for it). As groups, D(A) D(A C P l), but the identity map DA ---> DA c p > is (p)-linear, i.e., am ...... a CP> m for a E W, m E DA. The composite DA J. · · · a,) of P as being the coefficients of the equations defining A. Thus Frob(P) corresponds to (A C P> , ;c p> , cjJ C P l ) where ;c pJ and (/j C P J are such that FA defines a map of triples (A, i, (/j) ---> (A c p > , ;cp > , (/j C P> ). Finally we observe that there are "Heeke operators" acting. Let g E G(A1) and suppose that K' is an open subgroup of G(Z1) such that g- 1 K'g c K; then x >--+ xg : G(A) ---> G(A) induces a map G(Q)\G(A)/Koo K' ---> G(Q)\G(A)/KooK which arises from a map of varieties .9""(g) : K ' Sc ---> KSc. If P corresponds to (A', i', (/j') then .9""(g)P corresponds to (A, i, (/j) if there is an 08-isogeny a : A ---> A' such that =
=
=
,
,
=
=
,
commutes with m some positive integer. When we pass to Sp , only G(Alj) continues to act : if g E G(Alj) and P E K ' S(Fp) and §"(g)P E KS(Fp) correspond respectively to (A', i', (/j') and (A, i, (/j) then there is an isogeny a : A ---> A' whose kernel has order prime to p and a commutative diagram
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with m a positive integer prime to p. This definition is compatible with that over C in the sense that both mappings ff(g) come by base change from a mapping ff(g) : K ' S x spec Zc p l � K S x spec Zc p l where Zc p l = { m/n E Ql(n, p) = 1 } . (Mo �e concretely, this means that if (A, i, �) in characteristic zero specializes to (A, l, (/}) in characteristic p then ff(g)(A, i, �) specializes to ff(g)(A , l, (j).) 5. The isogeny classes. Fix a prime p not dividing L1 and consider pairs (A, i) where A is an abelian variety of dimension 2d over Fp and i is a homomorphism B 4 End0(A) such that i(l) = 1 . We write A - A' if A and A' are isogenous, and (A, i) - (A', i') if the pairs are B-isogenous in an obvious sense . .FP denotes the set of all B-isogeny classes and (A, i) ® Q the class containing (A, i). It will turn out (last paragraph below) that the map (A, i, �) ....... (A, i) ® Q : S(Fp) � .FP is surjec tive and so, to describe S(Fp), it suffices to describe JP and the fibres of the map. The first is done in this section and the second in the next. Note that Frob (and ff(g)) preserves the fibres. We first remark that, as in characteristic zero, there is a unique weak polarization on A inducing the given involution on B, and that it gives an F-equivalence class of pairings A n x A n � Gm for all n (cf. [17, §23]). In turn these pairings give an equivalence class of skew-symmetric pairings ¢1 : TIA X TIA � TIGm � zl with nonzero discriminant for each I :I- p, and a similar pairing cp p : DA x DA � W ; this last pairing satisfies the conditions cp p(Fm, n) = cpp(m, Vn)
inv.(E)
=
1-
=
0
_
if v is real, if v j l, I :I- p,
ordv(1r:) - ord.(q) [Q [1r:] • .. Qp] if v I p. Moreover 2 dim(A) = [Q[1r:] : Q] [E : Q[1r:1 ] 1 1 2 and e = [E : Q[7r:]] I I2 is the least common denominator of the inv .(E). The characteristic polynomial PA (T) of 1r: : A � A is m(T)• where m(T) is the minimal polynomial of 1r: over Q. (b) The simple abelian varieties A and A' over Fq are isogenous if and only if there is an isomorphism Q[1r:A] � Q[1r:A '] such that 1r:A >-+ 7r:A ' · (c) Every algebraic integer 1r: which has absolute value q l i Z under any embedding Q[1r:] 4 C arises as the Frobenius endomorphism of a simple abelian variety A" over Fq . (d) For any abelian varieties A and B over Fq and any prime I (including I = p) the canonical map
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Hom(A, B) ® Z1 --> Hom(A(l), B(l)) = Hom( T1A, T1B) is an isomorphism. (If l # p then A(l) and B(l) can be regarded as Gal(Fq (Fq) modules.)
PROOF. The first part of (a) (the Riemann hypothesis) is due to Weil, part (c) to Honda, and the remainder to Tate ; see [17], [21], [7], [22l [23]. For example, if in (c) we take n: p a , q = p2a then we obtain an elliptic curve A !P such that End0(Atp) is a quaternion algebra over Q which is split everywhere except at p and the real prime. Any such elliptic curve is said to be supersingular. It follows easily from (a) that if Q[n:] has a real prime then either A is a super singular elliptic curve or becomes isogenous to a product of two such curves over =
Fq2·
From now on we. let p factor as (p) = Vr · · Vm in OF, where the }1; are distinct prime ideals, and we let d,. be the residue class degree of }1; over p ; thus d = I; d;. 5.2. Let (A, i) be as above. The centralizer of B in End0(A) is either : (a) a quaternion algebra B' over F which splits except at the infinite primes, the
PROPOSITION
primes where B is not split, and the pJor which d; is odd, and there does not split ; or (b) a totally imaginary quadraticfield extension F' ofF which splits B. In thefirst case A ,...., Acrd where A0 is a supersingular elliptic curve and in the second A ,...., A5 where A0 is an abelian variety such that F' c End0(A0).
PROOF. Suppose A ,...., A 0 x AI> r ;?: I , where A0 is a supersingular elliptic curve and Hom(A0, A 1 ) = 0. Then Endo(A) :;:::; M,(E) x End0(A 1 ), where E = Endo(A0), and B embeds into M,(E). Consider F 4 M,(E) ; we must have d/2r, but d = 2r is impossible because F does not split E [19, Theorem 10], and so r = d or 2d. The Skolem-Noether theorem shows that, when composed with an inner automor phism, the map F --> Mr(E) factors through M,(Q). Thus the centralizer C(F) of F in M,(E) is isomorphic to M, 1 a(F) ® E = M,1 a(E ® F). Let C be the centralizer of B in M,(E). Then B ® F C :;:::; C(F) because C(F) and B are central simple alge bras over F [19, §8]. It follows that either rjd = I , C = F, and B = E ® F, or rjd = 2 and C is a quaternion algebra over F such that, in the Brauer group of F, [B] + [ C] = [E ® F]. The first is impossible because B splits at infinite primes while E does not; thus the second holds, and this proves that case (a) of the proposition holds. Next assume that Hom(A0, A) = 0 when A0 is a supersingular elliptic curve, and fix a large subfield Fq of Fp such that A and all its endomorphisms are defined over Fq. From considering A/Fq we get a Frobenius endomorphism n: E Endo(A), and the assumption implies that there is no homomorphism Q[n:] --> R. Consider
B -B [n]-B ®F C - E I I I F- F [n:]- C I I Q - Q [n:]
where E is Endo(A) and C is the centralizer of B in E. Clearly, F[n:] = F would contradict our assumption. On the other hand we must have [F[n:] : F] ;;£ 2 and F [n:] = C for otherwise E would contain a commutative subring of dimension
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> 4d = 2 dim( A) over Q, which is impossible by 5. 1(a). Let F' = C = F[11:] ; it is a quadratic extension of F and can have no real prime because that would contradict our assumption. It splits B because, for any finite prime l :1- p, ( T1A) ®z1 Q 1 is free of rank 2 over F; = F' ® Q1 , from which it follows that B ® F; � M2(F;), and we are assuming that B splits at any infinite prime or prime dividing p. Let e be an idempotent :1- 0, 1 , in (B ® F') n End(A). Then A0 = eA is an abelian variety such that A "' A0 x A0 • Since elements of F' commute with e, F' c End0(A0). REMARK 5.3. In case (b) of 5.2, A0 is isogenous to a power of a simple abelian variety, A0 "' Ar, because the centre of E = End0(A) is a subfield of the field F'. It follows that, for any pair (A, i) as above, A is isogenous to a power of a simple abelian variety and hence End0(A) is a central simple algebra over the field Q[11:]. Let (A, i) and (A ', i') be such that there exists an isogeny a: A -+ A'. The Skolem Noether theorem shows that the map B _!_. End0(A) � End0(A'), where a * (r) = ara-1, differs from i' : B -+ End0(A') by an inner automorphism (r ...... f3 r f3 - 1 ) of End0(A'). Thus {3a is a B-isogeny A -+ A' , and we have shown that A "' A' im plies(A, i) "' (A', i'). We now consider in more detail the situation in 5.2(b). Let ll h · · · , lJ1, 0 � t � m, be the primes of F dividing p which split in F' and write lJ; = q,q; for i � t. Since OF n End(A0) and Zp both act on A0(p), their tensor product does, and the splitting F ® Qp � Fp , x · · · x FPm induces an isogeny A0(p) "' A0(lJ 1 ) x · · · x Ao(llm) and an isomorphism D 'A0 � D'Ao(ll 1 ) x · · · x D'Ao(llm). Clearly Ao(ll;) has height 2d, and so D' (Ao(lJ,)) has dimension 2d, over W'. Since cjJp(am , n) = cjJp(m , an) for a e F the decomposition of D'A0 is orthogonal for cf;p, and cjJp restricts to a non degenerate form on each D' (Ao(ll;)) . This implies that the set of slopes {A h il2 , · · · } of D' (Ao(ll;)) is invariant under il �-+ 1 - il. Fix an i � t. As F;; � F�i x F�:' acts on D'Ao(ll;), Ao(ll;) splits further : A0(ll;) "' A0(q,) x A0(q ;) , D'Ao(lJ,) � D' (A 0(q ,) ) x D ' ( A0 (q ;)) . Since F�; c End0(A(q,)) has degree d; = height(A0(q,)) over Qp, A(q,) is isogenous to a power of a simple p-divisible group : we may write D' A(q,) = Dk;ld;, 0 � k, � d,. Correspondingly, D' A(q ;) Dk �ld; with k; + k; = d,. Fix an i > t. The [F�; : Qp] 2d, = height Ao(ll;) and so, as above, Ao(ll;) is isogenous to a power of a simple p-divisible group and we may write DA0(lJ;) = D• 1r. Since sfr 1 - sfr we must have sfr = d,f2d,. We write k, d,/2. Note that for some i, 1 � i � m, we must have k, :1- d;/2 for otherwise all slopes of A(p) would equal f. Then (see §3 and 5 . 1 (a)) l 1t'fq 1 12 1 1 for all primes v of Q[11:] and so some power of it would equal one. On replacing Fq by a larger finite field we would have 11: = q 1 12, and this would imply that A is isogenous to a power of a supersingular elliptic curve, i.e., we would be in case (a). This means that t � 1 -at least one prime lJ; splits in F'. =
=
=
=
=
THEOREM 5.4. fp contains one element for each pair (F ' , (k;)v;;, ; �m ) where F' is a totally imaginary quadratic extension of F which splits B and is such that at least one lJ; splits in it ; if p, splits in F' then k, is an integer with 0 � k, � d, and otherwise k, = d,/2 ; /or at least one i, k, :1- d; - k,. When p, splits in F' we regard k; and k; as being associated to q, and q;, and we do not distinguish between two pairs (F', (k;)) and (F', (k;)) which are conjugate over F. There is one additional "supersingular" element.
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For example, if F = Q then there is the supersingular isogeny class and one class for each quadratic imaginary number field F' which splits B and in which p splits. If p splits completely in F then there is the supersingular class and one class for each totally imaginary quadratic extension F' of F of the right type and choice of one out of each pair of primes dividing a lJ; which splits in F' ; one family of choices is not distinguished from the opposite family. PROOF OF 5.4. We first construct an isogeny class (A, i) ® Q corresponding to (F', (kh . · km)) . As before we let lJ;, 1 � i � t, be the primes dividing p which split in F'. Consider the ideal in OF'• .
,
where f = 2d1 dm . For some h, ah is principal, say ah for the nontrivial F-automorphism ofF' then 12:1t e F and . . •
= (12:) .
If we write a f-+
ii
Thus 12:1t = upfh with u a unit in OF . If u is a square in F then we may replace 12: by 12:/u 1 1 2 and obtain an equation 12:1t = q with q = pfh. If u is not a square then we replace 12: by 12:2/u and obtain a similar equation with q = p2fh. Note that the condi tion k ; #- k; for some ; implies that 12: ¢ F and hence that F' = F[12:]. Under any embedding F' 4 C, F maps into R. Thus complex conjugation on C induces a f-+ ii on F'. In particular 1t is the complex conjugate of the complex number 12: and so 12:1t = q implies that l1rl = q 1 1 2. Let A" be the abelian variety corresponding, as in 5. 1(c) to 12:, and let E = End0(A"). For any prime v of Q[1r] invv(E)
= = =
if vlp, (k,-/d;) [ Q[7r] 0 : Qp] if v IP and q;Jv, (k;/d;) [Q[7r]0 : Qp] if v jp and q;J v. 0
Let e0 be the denominator of invv(E) (when it is expressed in its lowest terms) and let e be the least common multiple of the e0 • Then 2 dim(A") = re where r = [Q[1r] : Q] . Clearly evlfF; : Q[7r]0] for any q j v, vjp, which implies (by class field theory) that F' splits E and (trivially) that e divides [F' : Q[1r]] = 2dfr. As [M2d ; re(E) : Q[1r]] = (2dfre) 2e2 = [F' : Q[1r1] 2, F' embeds into M2d l re(E) [19, Theorem 10]. Let A0 = Aidlre . The characteristic polynomial PA.(T) of 12: on A,. is c,.(T)• where c,.(T) is the minimal polynomial of 12: e Q[1r] over Q (5. 1(a)). Thus PAo(T) is c,.(T)2d l r which equals the characteristic polynomial of 12: e F' over Q. Corresponding to the splitting F; = F;1 x F�; x . . , we have A0(p) "' A0(q1) x A0(qi) x . . . and PAo(T) = P1(T)P{(T) . . . where P,(T) (resp. P;(T)) is the char acteristic polynomial of the image 12:; of 12: in F;; (resp. 12:; of 12: in F�'.) over Qp. Thus (see §3) A(q;) has slopes equal to ordq{7r;) = (fh/d;)k,/fh = k;d; ana A(q;) has slopes equal to k;/d,. Thus A = A0 x A0, regarded as an abelian variety over Fp. and the map i induced by B 4 M2(F') represent an isogeny class corresponding to (F', (kl > . . . , km)) . (A �d, i), where A 0 is a supersingular elliptic curve, represents the supersingular class. .
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Obviously if (A, i) (A', i') then both represent the supersingular class or correspond to the same pair (F', (kl> km)). It remains to show that if (A, i) and (A', i') both correspond to (F', (k l > . . . , km)) then (A, i) - (A', i'). By considering A and A' to be defined over some finite subfield of Fp we get elements n = nA e F' and n' = nA' e F'. The assumption implies that ordq(tr:) = ordq(n') for all q l p, q a prime ofF', and 5. 1 (a) then shows that l n/n' lv = 1 for all primes of F'. Thus n and n' differ by a root of 1 and so, after extending the finite field, we may take them to be equal. It follows that A and A', being isogenous to powers of the same abelian variety A,., are themselves isogenous, and 5.3 com pletes the proof. The proof that any class in ..Fp is represented by an element of S(Fp) requires the following lemma. "'
· · · ,
5.5. Let T c T1A be such that T1A/T is finite ; then there exists an isogeny A such that T1a maps T1A' isomorphically onto T.
LEMMA
a : A'
�
PROOF. The finiteness of T1A/T means that, for all n :> 0, the cokernel N of TfnT � T1A/nT1A is independent of n. Thus there is a map A n = T1A!nT1A .1� N. Define A' to be the co kernel of a ...... (ifJ (a), a) : A n � N x A, and a : A' � A to be (b, a) ...... na ; then (T1a)(T1A') = T. Let (A, i) represent a class in ..Fp and let 0' = B n End(A) ; it is an order in B. Regard (A, i) as being defined over a large finite field Fq ; then End(A) ® Z1 � Endp .(T1A) and 0' ® Z1 = (B ® Z1) n Endp 0(T1A). For almost all l, 0' ® Z1 will equal 08 ® Z1 and we take T1 = T1A ; for the remaining l we may choose a Tz of finite index in TzA which is stable under OB, i.e., such that Endp .(Tz) n (B ® Z1) = 08 ® Z1• Note that D(Tp/PTp) = M is a W[F, V]-module of finite length over W. We may choose Tp such that MfFM satisfies (4. 1). Let A' correspond to T = fi 1T1 as in the lemma. Then A' together with the obvious i and some (fi lies in S(Fp) and represents (A, i) ® Q. 6. An isogeny class. It remains to describe the set Z = Z(A, i, ifJA) of elements (A', i', (fi') of S(Fp) such that (A', i') is isogenous to a given pair (A, i). An 08-isogeny A' � A determines an injective map T1a : T1A' � T1A whose image A satisfies the following conditions : (a) A is 08-stable. (b) T1A/A is a finite group scheme. (More precisely, Coker(A/nA � T1AjnT1A) = Coker(A� � A n) = Ker(A' � A) for n :> 0.) (c) D(A/pA)/FD(A/pA) satisfies (4. 1). (For A/pA = Af, and so D(A/pA)/FD(A /pA) =
DA/F(DA).)
Consider all subobjects A of T1A satisfying (a), (b), (c). Any such A may be written A = AP x Ap with AP a Z�-lattice in T�A (in the usual sense of modules over Z�) and Ap c TpA. We let Y be the set of pairs (A, (/J) with A as above and (fi a K-equivalence class of isomorphisms AP � V(Z�). Since (A, (/J) is determined by a pair ((AP, (/J), Ap), we may write Y = YP x Yp . By 5.5, every (A, (/J) e Y arises from a triple (A', i', (fi') e S(Fp) equipped with an isogeny a ; A' � A. Thus we have a surjective map Y -� Z c S(Fp). For n a positive integer we set n(A, (/J) = (nA, n(/J), and we define Y ® Q to be the set of pairs (y, n) with y e Y and n e Z>o• where (y, n) and (y', n') are identi-
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fled if n'y = ny'. Then we write Y ® Q = ( YP ® Q) x ( Yp ® Q) where YP ® Q may be identified with the set of 08-stable lattices A in ( T1A) ® Q equipped with a K-equivalence class of isomorphisms A � T1A. There is an action of H(Q) = End0iA)x on Y ® Q: for a e H(Q) we choose a positive integer m such that rna is an isogeny of A and define a(A , qi , n) = ( T1(ma),
ifJ T1(ma)-l, mn). z.
LEMMA
6. 1 . The map Y -+ Z described above induces a bijection H(Q)\ Y ® Q �
PROOF. (A, qi , n) and (A', qi ', n') map to the same element of S(Fp) if and only if there exist D8-isogenies a
A' ==:' A and an DB-isomorphism f/Jo : T1A '
-+
a
V(Z1) such that
n (( T1a) T1A ' , ifJ0(T1a)) = (A, qi) and n' (( T1a' ) T1A ' , ifJ0( T1a ')) = (A', qi').
Then a ' a-1 makes sense as an element of End0(A) and a ' a-1 (A, qi , n) = (A', qi' , n'). LEMMA 6.2. The map G(A,) -+ YP ® Q, g �--+ (g ( T1A ) , ifJAg- 1 ), induces a bijection G(A ,)/K -+ YP ® Q. PROOF. Obvious. LEMMA 6.3. There is a one-one correspondence between Yp ® Q and the set X of W[F, V]-submodules M of D' A which are free of rank 4d over W, DB-stable, and
such that M/FM satisfies (4. 1).
PROOF. p: A -+ A induces maps in : AjpnA 4 Afpn+lA which define a p-divisible group A(p) = (AfpnA, in). The exact sequence 0 -+ A -+ TpA -+ N -+ 0 (N finite) gives rise to 0 -+ N -+ A(p) -+ A(p) -+ 0. On applying D we get 0 -+ DA -+ DA(p) -+ DN -+ 0. Since DN is torsion, we may identify D'A(p) with D'A. To (A, n) e YP we associate
n- 1 (DA(p)) e X. THEOREM
6.4. With the above notations,
Z(A, i, qi) � H(Q)\G(A,)
x
X/KP .
Frob acts by sending M e X to FM; the Heeke operator .r(g), g e G(A,), "acts" by multiplication on the right on G(A,). This simply summarizes the above. It remains to give a more explicit description of X. Note that, corresponding to the splitting D'A � D'A(p 1) X X D 'A ( P m) , we have X � x X Xm . 1 X It is convenient to write G;(Zp) = Aut08(A(p;)) and G;(Qp) = End08(A(p;)) x = End08(D' A(p;)) x . In the simplest cases G;(Q p) acts transitively on the lattices M c D' A(p;) which belong to X;, and in this case X; � G;(Qp)/G;(Zp). (To say that G;(Qp) acts transitively means that each A'(p;) is isomorphic to A(p;) and not merely isogenous ; cf. [6, p. 93].) PROOF.
. . •
• • •
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183
EXAMPLES 6.5. (a) F = Q , F ' is a quadratic extension o f Q , (p) = qq ' i n F', and (A, i, �) is in the isogeny class corresponding to (F', (0)). Then A(p) � (Qp/Zp)2 x (f.'poo)2 with 08 ® Zp = M2(Zp) acting in the obvious way on each factor. Thus G(Zp) = {(3 g) Ia, e Zp } and G(Qp) {(3 g) Ia, b e Q; } . In this case X = G(Qp)/G(Zp). Frob acts as (ij �). (b) As above, except (A, i, �) corresponds to (F', (1)). Then A(p) � (f.'poo)2 x (Qp/Zp)2 (i.e., in the splitting Fp = F� x F�,, F� now corresponds to the f.'poo factor). G(Zp), G(Qp) and X are as before but Frob acts as (� �). (c) F = Q, (A, i, �) is in the supersingular class. Then D'A(p) � D l / 2 x D l / 2 and End(D l 1 2) = Bp, the unique division quater nion algebra over Qp. B acts through the embedding B ® Qp � M2(Qp) � M2(Bp). Thus G(Qp), the centralizer of B ® Qp in M2(Bp) is (B�)x . Moreover G(Zp) may be taken to be o x where 0 is the maximal order in Bp. In this case X � G(Qp)/G(Zp) · Frob acts as multiplication by w, a generator of the maximal ideal of 0. (d) F arbitrary, p splits completely in F, (p) = lh · · · lJd , (A, i, �) corresponds to
b
.
(F', (k l > · · · , kd)) . Then X � xl
X • • • X xd where X; is as in case (a) if lJ; splits in F' and k; = 0, as in case (b) if lJ; splits and k; = 1 , and as in case (c) otherwise. (e) The general case. For a statement of the result, see [14]. (This case is treated in detail in : J. Milne, Etude d'une classe d'isogenie, Seminaire sur les groupes reductifs et les formes automorphes, Universite Paris VII ( 1977-1978).) Added in proof (November 1 978). The outline of a proof in [12] of the conjec ture for those Shimura varieties which are moduli varieties is less complete than appeared at the time of the conference. The above proof (completed in the report referred to in 6.5(e)) for the case of the multiplicative group of a quaternion algebra differs a little from the outline in that it depends more heavily on the Honda-Tate classification of isogeny classes of abelian varieties over finite fields. The complete seminar referred to in 6.5(e), which redoes in greater detail much of the material in this article and [3], will be published in the series Publications
Mathematiques de l'Universite Paris 7.
BIBLIOGRAPY 1. M. Artin, The implicit function theorem in algebraic geometry, Colloq. in Algebraic Geometry, Bombay, 1 969, pp. 1 3-34. 2. , Theoremes de representabilite pour les espaces algebriques, Presses de l'Universite de Montreal, Montreal, 1973. 3. W. Casselman, The Hasse- Wei/ �-function of some moduli varieties of dimension greater than one, these PROCEEDINGS, part 2, pp. 141-163. 4. P. Deligne, Travaux de Shimura, Seminaire Bourbaki, 1 970/71 , no. 389. 5. P. Deligne and M. Rapoport, Les schemas de modules de courbes elliptiques, Antwerp II, Lecture Notes in Math. , vol. 349, Springer, New York, pp. 143-3 1 6. 6. M. Demazure, Lectures on p-divisible groups, Lecture Notes in Math., vol. 302, Springer, New York, 1972. 7. T. Honda, Isogeny classes of abelian varieties over finite fields, J. Math. Soc. Japan 20 (1 968), 83-95. --
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8. Y. Ihara, The congruence monodromy problems, J. Math. Soc. Japan 20 (1 968), 107-121. 9. On congruence monodromy problems, Lecture Notes, vols. 1 , 2, Univ. Tokyo, 1 968, �� ,
1969. 10. Non-abelian class fields over function fields in special cases, Actes Congr. Internat. Math., vol. 1, Nice, 1 970, pp. 381-389. 11. Some fundamental groups in the arithmetic of algebraic curves over finite fields, Proc. Nat. Acad. Sci . U.S.A. 72 (1 975), 3281 -3284. 12. R. Langlands, Letter to Rapoport (Dated June 1 2, 1 974-Sept. 2, 1 974). 13. Some contemporary problems with origins in the Jugendtraum, Proc. Sympos. Pure Math. , vol. 28, Amer. Math. Soc., Providence, R. I., 1 976, pp. 401 -418. 14. Shimura varieties and the Selberg trace formula, Canad. J. Math. 29 (1 977), 1 2921 299. 15. Y. Morita, Ihara's conjectures and moduli space of abelian varieties, Thesis, Univ. Tokyo (1 970). 16. D. Mumford, Geometric invariant theory, Springer, 1 965. 17. Abelian varieties, Oxford, 1 970. 18. M. Rapoport, Compactifications de l 'espace de modules de Hilbert-Blumenthal, Compositio Math. (to appear). 19. J. P. Serre, Expose 7, Seminaire Cartan 1 950/5 1 . 20. H . Swinnerton-Dyer, Analytic theory of abelian varieties, L.M.S. lecture notes 14 (1 974). 21. J. Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1 966), 1 341 44. 22. Classes d'isogenie des varietes abeliennes sur un corps fini, Seminaire Bourbaki, 1968-1 969, no. 352. 23. W. Waterhouse and J. Milne, Abelian varieties over finite fields, Proc. Sympos. Pure Math. , vol. 20, Amer. Math. Soc., Providence, R.I., 1 971 , pp. 53-64. ��,
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UNIVERSITY OF MICill GAN
Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 2, pp. 1 85-1 92
COMBINATORICS AND SHIMURA VARIETIES
mod p (BASED ON LECTURES BY LANGLANDS)
R. E. KOTTWITZ One of the major problems in the study of Shimura varieties is that of expressing their Hasse-Weil zeta-functions as products of L-functions associated to auto morphic forms. In [1] this problem has been solved for a certain class of Shimura varieties : those associated to an algebraic group G over Q which is the inverse image under ResnQ(B*) � ResF1Q(F*) of some connected Q-subgroup of ResF1Q(F*), where B is a totally indefinite quaternion algebra over a totally real number field F. In this paper we will discuss only the case G ResF1Q(B*). In this special case, the zeta-functions of the corresponding Shimura varieties can be expressed as products of automorphic L-functions associated to the group G itself (in general, groups besides G are needed as well). This case has already been discussed in [2] and the purpose of the present discussion is to provide further details on the com binatorial exercise referred to on the last page of that paper. This is worked out in detail in §4 of [1] for all of the subgroups of ResF1Q(B*) mentioned previously, but even so it is interesting to carry out the exercise in our situation (with further sim plifying assumptions added later) so that the main ideas can be understood more easily. First we sketch the arguments of [2] which lead to the combinatorial exercise. Let K be a compact open subgroup of G(A1) (where A1 is the ring of adeles of Q having component 0 at oo ). Let SK be the Shimura variety obtained from G and K as in [2]. The variety SK is defined over Q since B is totally indefinite, so its zeta function Z(s, Sx) is a product over the places v of Q of local factors Zv(s, Sx) . Let F' be a finite extension field of F which is Galois over Q. For a E Gal(F'/Q)/Gal(F'/F), let H" GL2(C) and let V" be the standard representa tion of H(f on C 2 . Then LGO = n (f H(f has a natural representation r on v 0" V(f , which may be extended to a representation of the L-group LG Gal(F'JQ) 1>< LG o by putting r (r) (@" v") = ®" w" for 1: E Gal(F'/Q) where.J:V" v,-1 " . The result of [2] is that Z(s, Sx) = fLr L(s - d/2, n, r)m C1r, K l up to a finite number of local factors, where d [F: Q], n runs over the representations of G(A)/Z(R) (where Z is the center of G) which occur in V ( G(Q)Z(R)\G(A)), and m(n, K) is an integer which is associated to n and K in a way that is described in [2]. The product over n is actually finite since m(n, K) turns out to be 0 for all but a finite number of n. The precise result is =
=
=
=
=
=
AMS (MOS) subject classifications (1 970). Primary 05C05, 14Gl0, 22E50, 22E55. © 1 9 7 9 , American Mathematical Society
1 85
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R. E. KOTIWITZ
Zp (s, SK)
(1)
= n "'
L(s - d/2, 1Cp , r)m C n , K l
where 11:P is the local factor of 11: at a finite rational prime p which satisfies : (A) F is unramified at p ; (B) B splits at every prime ofF above p ; (C) K = KpKP where Kp i s a maximal compact subgroup of G(Q p) and KP i s a compact open subgroup of G(A�), A� being the ring of adeles of Q with component 0 at p and oo . For the rest o f this paper p will denote a fixed rational prime satisfying (A)-(C). Because of these assumptions about p, SK has good reduction at p, and m(11:, K) = 0 unless 1T:p is unramified. Take the logarithm of each side of (1), and use the power series expansion of log(l - X). Comparing the coefficients of the powers of p-s in the two sides, we see that (1) is equivalent to (2) Card SK (Fpi) = � m(11:, K)pid 1 2 tr(r(g")i) "
for all positive integers j, where 11: runs over all the representations of G(A)/Z(R) which are unramified at p and which occur in V(G(Q)Z(R)\G(A)), and where g"P is an element of the semisimple conjugacy class of LG associated to 11:p · For every place v of Q choose a Haar measure dg. on G(Q.) such that the measure of G(Z.) is 1 for almost all finite places v . Use dg. to define the action of L l (G(Q.)) on representations of G(Q.). Choose a Haar measure dzoo on Z(R), and use the quotient measure dgoo/dzoo to define the action of V(G(R)/Z(R)) on representations of G(R)/Z(R). LetfP be the characteristic function of the subset KP of G(A�) divided by the measure of KP (with respect to 11 "*P· dg.). It is pointed out in [2] that there exists f E ceca( G(R)/Z(R)) such that m(1C, K) = m(1C)tr 1CP{fPfoo) where 1f:P = ®v,p1T:v and m(11:) is the multiplicity of 11: in V(G(Q)Z(R)\G(A)) (it is known that this multiplicity is I , but we will not need to use this fact). By the theory of Heeke algebras, there exists a unique function fPl in the Heeke algebra .?t'(G(Qp). Kp) such that tr 1T:p(fjil) = pid / 2 tr(r(g"P)i) if 1T:p is unramified, = 0 if 11:p is ramified. co
00
So the right side of (2) becomes :E n m (11:) tr 1T:(fPfPfoo), which can be evaluated in terms of orbital integrals by means of the trace formula. For r E G(Q), let dg be the quotient of 11 .dg. on G(A) by some Haar measure dg7 on Gr(A) (where as usual G7 denotes the centralizer of r in G), and let m7 = meas(GiQ)Z(R)\G7(A)) where the measure used is the quotient of dg7 by the measure on Gr(Q)Z(R) � G7(Q) x Z(R) which is the product of the Haar measure on G7(Q) which gives points measure 1 with dzw Applying the trace formula, we find that (2) is equivalent to (3)
where r runs over a set of representatives of the conjugacy classes in G(Q). It turns out that the integral fcrCR) \G (R) foo(r 1 rg) dgoo is 0 unless r is central or elliptic at oo , and so the right side of (3) is
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COMBINATORICS
(4)
where a7 = m7 fGr (R) \G(R) foo(g-1 rg) dg"" and r runs over a set of representatives for the conjugacy classes in G(Q) which are elliptic or central at oo . To g o further we must know more about Card SK(Fpi) · To know this number for all j it is enough (in principle anyway) to know explicitly the set SK(Fp) and the action of the Frobenius (/) on this set. Assumption. To make this description simpler, we assume from now on that p remains prime in F. First of all, for every triple p = (L, m', m") consisting of a totally imaginary quadratic extension L of F which is split at p and can be embedded in B, and in tegers m', m" such that m" > m' ;;; 0 and m' + m" = d, there is an associated
f/J-stable subset Yp of SK(Fp), and SK(Fp) is the disjoint union of the sets Yp and another (/)-stable subset Y0 • Let IP be L * regarded as an algebraic group over Q. It turns out that YP is isomorphic as a Gal{Fp/Fp)-set to lp(Q)\(G(A9) x Xp)/KP for some set Xp on which lp(Qp) and (/) act, the two actions commuting with each other. An embedding L � B gives us a map IP � G, and lp(Q) acts on G(A9) via Jp(Q) � Ip(A9) � G(A9) ; also Ip(Q) acts on Xp via Ip(Q) � lp(Qp), and KP acts on the pro duct G(A9) x XP by acting on G(A9) alone. For any positive integer j and for x E XP , let Tt = {g E lp(Qp) : f/Jix = gx}, and let ot be the characteristic function of Tt. Choose a Haar measure p. on lp(Qp), and for r E IP(Q) let (5)
rp (r) =
1-
-
p.(l,.) ot(r) where I,. is the stabilizer of x in lp(Qp). Choose a Haar measure v on G7(A9). Let v' = TI # p, oo dgv where dgv is the Haar measure on G(Q0) chosen before, and recall that we have defined fP to be the characteristic function of KP divided by v'(KP). It is easy to show that the contribution of YP to Card(SK(Fpi)) is 1:
xEXp mod lp (Qp)
J
-1
"' bT rp (T) G (A� \G(Af} fP(g T g) dv' d"u T f f where b7 = meas(Ip(Q)\G7(A�) x lp(Q p)). The measure used is the quotient of v x p. by the measure on lp(Q) which gives every point measure I . We see that (4) and (6) have roughly the same form, but we must sum (6) over all triples p = (L, m', m"), and it appears at first that there is nothing in (4) which corresponds to the sum over m', m". But in fact we will now see that fPl can be written in a natural way as a sum of functions, and it is this sum which corresponds to the sum over (6)
.f...J
T E [p (Q)
m', m".
The function fPl in the Heeke algebra .Yf = .Yf( G(Qp). Kp) is characterized by the equation tr 1rp(Jp>) = pid / 2 tr(r (g,.P)i) for all unramified representations 1rp of G(Qp). Since p remains prime in F, Fp is a field (of degree d over Qp) and G(Qp) � GL2(Fp ) (since we assumed that B split at every prime of F above p). Regarding GL2(Fp) as the Fp points of GL2 , we get an isomorphism f �--+ JV from .Yf to the algebra of polynomials in a, b, a- 1 , which are symmetric in a, b (a and b are two indeterminates). Regarding GL2(Fp) as the Qp points of ResFJQ GL2, we get an iso morphism f �--+ 1- from .Yf to the algebra of functions on LG o x { (/)} which are
b- 1
188
R . E. KOTTWITZ
obtained by restricting linear combinations of characters of finite dimensional complex analytic representations of the complex Lie group L G. The relation between fV .and /- is · a = !v a 1 0 d x � x ... x J- g1 gd b 1 .� b . We want tofindjpl e Yf such that (Jpl)- (x) = pjd t 2 tr(r(xj)) forallx e L Go x {�} . Let I be the greatest common divisor ofj and d. Then for
(( gJ
( gJ )
((
) (
J)
((
we have tr r(xJ) = tr r a1 ' () aj b .?. b x a2 ' () aj+ l b � b + x 2 1 j l = ((a 1 . . . adV 1 1 + (b 1 . . . bd)j 1 1) 1 . It follows that up))Y((8 g)) = pjd /2 (aj / 1 + bj l l)l. For k' E z with 0 � k' let .fi. k' be the function in the Heeke algebra such that flk' = pjd 12 (ajk' ll bjk"ll + ajk"ll bjk'll) .
)
<
1/2,
(k•)
where k" = I - k'. If I is even, then for k' = k" = 1/2, let.fi. k' be the function in the Heeke algebra such that flk' = pjd 12(L ,)ajk' llbjk'll . By the binomial theorem we � k[1'=/Zlf. have Jl'p(j) - ,i,.,j O J, k' · Now consider a triple p = (L, m', m"), and let r e L - F. Recall that m" is determined by m' (since m' + m" = d). Choose an embedding L --+ B. Then r gives us an element of lp(Q) and of G(Q). We will show that the contribution of r to (6) is 0 unless m' is divisible by djl, in which case it is equal to the contribution of k' = lm' fd and r to the following rewritten version of (4) : wz1 dv' dgp � k'fO aT Gr (A�) \G(A�JP(g-1 r g) dv Gr (Qp) \G(Qp) ./j, k' (g-1 r g) dgT where dgr is the Haar measure on Gr(Qp) corresponding to the Haar measure f-t on Ip(Qp) under the isomorphism lp(Qp) ...::::. Gr(Qp) induced by the embedding L --+ B. It can easily be shown that ar = br with the choice of measures that we have made, so it is enough to show the following : THEOREM. With notations as above, rp < jl (r) = 0 unless dfl divides m', in which case it is Scr (Qp) \G(Qp) .fi. k' (g- 1 rg) (dgpfdgr) where k' = lm' fd. We begin the proof of this theorem by evaluating Scr (Qp) \G(Qp) fj, k' (g- 1 r g) . (dgp/dgr) · This is easy since L splits at p and the hyperbolic orbital integrals of any f e Yf can be read off from JV. Let a , (3 e F! be the two eigenvalues of r and assume without loss of generality that the valuation v(a) of a is less than or equal to the valuation v((3) of (3. Then the integral ofjj, k' over the orbit of r is equal to 0 unless v(a) = jk'/1 and v((3) = jk"/1, in which case it is
J
J
la f3/(a - (3)2 1
�>jd 2 (k)meas(/ )-1) /
'
o
COMBINATORICS
1 89
where /0 is the subgroup of lp(Qp) corresponding to @pP x @pP under Ip(Qp) ....::. x FJ . The final result is that the integral is equal to O unless v(a) = jk'/1 and v(,B) = jk"jl, in which case it is (k, )pik'dlt f.J.(/0)- 1 . We must now evaluate rpi(r). This is the combinatorial exercise referred to in the beginning of this paper. First we will describe XP and the actions of lp(Qp) and r!J on XP . Let Qpn be a maximal unramified extension of Qp containing Fp, let cgpn be its valuation ring, let BIJ' be the set of @'j,"-lattices in the two dimensional vector space Qpn EB Qj," over Qj,", and let BiJ be the set of classes of lattices in Qpn EB Qpn (two lattices Mt. M2 are said to be in the same class if there exists a nonzero element c E Qj," such that M1 = cM2). For any lattice M, let M denote the class of M. The set BiJ is the set of vertices of the Bruhat-Tits building of SL2(Qj,"). This building is a tree. The groups GL2(Qpn) and Gal (Qj,"/Qp) act on BIJ' and BIJ. Let FJ
B = (b
m'
)
0 " pm '
and let (J be the Frobenius element of Gal (Qj,"/Qp). Let XP be the set of sequences {M,.} iEZ of elements M,. of BIJ' satisfying : (A) M; � M,._1 � pM,. for all i E Z; (B) B(JdM; = Mi-d for all i E Z. The action of r!J on XP is given by r!J : {M,.} ...... {M;,} where M; = M,._ 1 . Identify lp(Qp) with A(Fp), where A = m m . The action of an element r E lp(Qp) on XP is given by r{M,.} = {rM,.}. Recall that we are trying to calculate rpi(r) = L: x f.J.(lx)- 1 ot(r), where the sum is taken over x E XP mod lp(Qp)· Let /1 be the subgroup of lp(Qp) = A(Fp) consisting of all matrices of the form where n E Z and c E FJ . As before let /0 = A (@pP) . Then lp(Qp) = /0 x It. Furthermore f.J-(10) = f.J.(lx) [10 : lx] = f.J.(lx) [Ip(Qp) : /1 /x], so rpi(r) is equal to f.J.(I0)- 1 Card S(r, j) where S(r, .i) = {x e XP mod /1 : r!Jix = rx} . Let � be the apartment in BiJ corresponding to the split torus A. Choose a point p0 in �, and let o1t = { x E BiJ : p0 is the point of � nearest x} . Let o!t' be a set of representatives for the lattice classes in o!f, or in other words, let o!t' be a subset of BIJ' such that no two elements of o!t' are in the same class and such that { M: M E A'} = o!t. Then the set of {M,.} E XP which satisfy (C) M0 E o!t' is a set of representatives of XP mod It. The condition r!Ji{ M,.} = r{ M,.} just says (D) r M,. = Mi-i · We can summarize this discussion by saying that S(r,.i) is the set of all sequences of elements of BIJ' satisfying (A)-(D). Any sequence {M,.} in S(r, j) determines a sequence { x,.} = { M,.} of elements of BiJ satisfying : (A') x,. is a neighbor of x,._1 ; (B') B(JdX; = X i-d ; (C') xo E A ; (D') rx,. = x,._i · Let S' (r, j) be the set of sequences { x,.} satisfying (A')-(D'). If the valuation of
1 90
R.
E.
KOTTWITZ
det(r) is not equal toj, then conditions (A) and (D) are incompatible, and S(r, j) is empty. So from now on we assume v(det r) = j. In this case, any element of S'(r, j) can be uniquely lifted to an element of S(r,j) (to check this one needs to use the fact that v(det B) = d) . So Card S(r, j) = Card S'(r, j) if v(det r) = j. Let r, s e Z be such that rj + sd = I. Conditions (B') and (D') together are equivalent to the two conditions : (E') qdi/1 x; = B-j / 1 rd/1 x; ; (F') r' B•q•dx; = X i-1 · We will now show that S'(r, j) is empty unless B-i 1 1rd 1 1 e /0• Suppose S'(r, j) is nonempty, and let {x;} e S'(r, j). Then p0 is the point of � closest to x0, so ?:Po is the point of 1:� closest to ?:Xo, where 1: = q-dj / 1 B-i l l rd l l, But ?:Xo = Xo by (E'), and 1:� = � since q� = �. B � = � and r� = �. So 1: fixes p0• Since q-di 1 1 fixes p0, we conclude that B -i l lrd l l fixes p0. But B-i l lrd l t is diagonal and its determinant is a unit (since we are assuming that v(det r) = j), so that the fact it fixes Po im plies it belongs to /0• So we have shown that S'{r, j) is empty unless B-i 1 1ra 11 e /0. Let a, f3 e FJ be the diagonal entries of r. so that r = {3 Z). Then B-i11rd1 1 e lo if and only if v(a) = jm' Id and v(f3) = jm"Id. In particular we see that q;i(r) = 0 for all r unless jm' ld and jm"ld are integers. Recalling that I = (j, d) and that m' + m" = d, we see thatjm'fd, jm"/d e Z if and only if dll divides m', andthat if dll does divide m', then rpi(r) is still zero unless v(a) = jm'ld = jk'/1 and v(f3) = jm"/d = jk"I I. Comparing this with the orbital integrals we have computed, we find that all we have left to show is the following lemma. LEMMA. Suppose B-i 1 1 rd l l e /0• Then Card S'(r, j) = (L,)pik' dll where k' = lm'ld. To prove this we first look more closely at conditions (E') and (F'). Let PJ� = {x e PJ : 1:(x) = x}, where 1: is the automorphism of PJ defined previously. Then (E') simply says that X; e (14� for all i e Z. Let Qj, be the completion of Q[,n. There exists a unique continuous action of GL2(Qj,) on PJ which extends the action of GL2{Q'f,n) on PJ (PJ is given the discrete topology). Each element of GL2(Qp) acts by a simplicial automorphism of the tree PJ. Using the fact that B-i 1 1rd 11 e /0, it is easy to show that there exists a diagonal matrix o with diagonal entries in Qj, - {0} such that o"jd/l ()- 1 = B-;ttrdtl. A simple calculation shows that PJ� = oPJ"djll where P4"di/l denotes the set of fixed points of qdi 1 1 in PJ. The subtree PJ"dj/l of PJ can be canonically identified with the Bruhat-Tits building of SL2 over the fixed field of Q[,n under qdi l t. So PJ"di/l is a tree in which every vertex has exactly pdi 1 1 + 1 neigh bors. Since (14� is simply the translate of PJudill by o, it too is a tree in which every vertex has exactly pdi l l + 1 neighbors. A further consequence of B-i 1 1rd l l e /0 is that PJ� contains �. The reason for this is that qdi 1 1 fixes � pointwise, as does every element of /0. In (F') the automorphism r' B•qsd of PJ appears. We will need to know the effect of this automorphism on �. The automorphism qsd acts trivially on �. The automorphism B acts on � by translation by m" - m'. The automorphism ra acts on � in the same way that Bi does, since B-i l lrd l l e /0, and therefore r acts on � by translation by (m" - m')jld. So r'Bsqsd acts on � by translation by I - 2k'. This discussion has reduced our problem to that of proving the following lemma. LEMMA. Let T be a tree in which every vertex has q + 1 neighbors, and let � be a
191
COMBINATORICS
subtree of T in which every vertex has 2 neighbors. Let p0 e �- Let I and k be nonnega tive integers such that I - 2k is positive. Let � be an automorphism of T which acts on � by translation by I - 2k. Let N(l, k) be the set of sequences of length 1 + 1 ofpoints x0, · · · , x1 in Tsatisfying : (i) X; is a neighbor of X;+l for i = 0, · · · , I - 1 ; (ii) Po is the point of � closest to xo ; (iii) XI = exo. Then N(l, k) = (i)q k .
Let us agree to call a sequence of points x0 ,
I joining x0 to x1• Then
· · ·,
x1 satisfying (i) a path of length
N(l, k) = .E Card {paths oflength I joining Xo to exo}, XQE./1
where A is the set of x0 e T which satisfy (ii). Let a, b e Z with a � 2b � 0, and let x, y be two vertices of T at distance a 2b apart. Then the number of paths of length a joining x to y is independent of the choice of x and y, and will be denoted by M(a, b). The figure below shows that the distance between Xo and exo is equal to I - 2k + 2n where n is the distance from x0 to p0. Xo
n
So N(l, k) = .E �=o Card {x0 e A : distance from x0 to Po is n} M(l, k - n). The number of points x0 at distance n from p0 such that p0 is the point of � closest to x0 is (1 - q-1)q n if n > 0 and is 1 if n = 0. Therefore N(l, k)
=
M(l, k) + ( 1
- q-1) .E �=l q n M(l, k - n).
The function M(l, k) on the set of pairs of integers (/, k) such that l � 2k � 0 is characterized by the three properties : (i) M(l, 0) = 1 for 1 � 0 ; (ii) M(l, k) = (q + I)M(l - 1 , k - 1) for l = 2k > 0 ; (iii) M(l, k) = M(l - 1 , k) + qM(l - 1 , k - 1) for I > 2k > 0. Condition (i) and the fact that {i), (ii), (iii) determine M uniquely are both obvious, and (ii), (iii) become obvious if it is observed that to give a path of length l from x to y is the same as to give a neighbor x' of x and a path of length l - 1 from x' to y. For l, k e Z, let m(l, k) be the coefficient of XI Yk in the formal power series ex pansion of f(X, Y) = (I - q Y) (1 - Y)-1 (I - X - qXY)-1 . We will show that M(l, k) = m(l, k) for I � 2k � 0. It is enough to show that conditions (i)-(iii) hold with M replaced by m. From the definition off(X, Y) we get (1 - X - qXY) · f (X, Y) = (1 - q Y) (I - Y)-1 . Comparing coefficients of X1 Y k on the two sides we see that m(l, k) - m(l - 1, k) - qm(l - 1 , k - 1) = 0 if / > 0, which shows
1 92
R.
E.
KOTTWITZ
that m satisfies (iii). Setting Y = 0 in the definition off(X, Y) we get !: l=o m(l, O)X1 = (I - X)-1 = 1 + X + X2 + · · . , which shows that m satisfies (i). We still have to show that m satisfies (ii). Let g(X, Y) = ( I - X - qXY)-1 and let n(l, k) be the coefficient of Xt Yk in the formal power series expansion of g(X, Y). We have ( 1 - X - qXY)-1 =
� (X + qXY)I = 1� to (k)xl-k(qXY)k = I; (kl )qkX I Yk, t ;;,O; ;;,O
k and therefore n(l, k) = (f.)qk. In particular for k > 0 we have n(2k - 1 , k) = k k 1 q k = 2 � qk-1 q = qn(2k - 1 , k - 1).
( f l)
)
e
This shows that the coefficient of X2k-l Yk in (1 - q Y)g(X, Y) is zero. But (I - q Y) · g(X, Y) is equal to (l - Y)f(X, Y), so the coefficient of X2k 1 P in (l - Y) f(X, Y) is zero, which means that m(2k - 1 , k) = m(2k - 1 , k - 1) for all k > 0. (7) We have already seen that if I > 0, then m(l, k) = m(l - 1 , k) + qm(l - 1 , k - 1). Take I = 2k with k > 0. Then m(2k, k) = m(2k - 1 , k) + qm(2k - 1 . k 1). Combining this with equation (7) we see that m satisfies condition (ii). Going back to our discusion of N(l, k), we see that -
-
00
N(l, k) = m(l, k) + (1 - q-1) I; q nm(l, k - n) n=1 since m agrees with M whenever M is defined and m(l, k - n) is 0 for n > k. We may use the equation above to extend the definition of N(l, k) to all integers I and k. With this convention we find that ,E N(!, k)X1 Yk is equal to 00
I; m(l, k)X I Yk + (1 - q-1) I; q n I; m(l, k - n)X I Yk l, k n =l l, k = f(X, Y) + - q-1) I; qn yn I; m(l, k - n)Xt p-n l, k n =l = f(X, + (1 - q-1) q n yn = f(X, Y)(1 - Y)(l - q Y)-1 = g(X, Y). 1 We have already computed the formal power series expansion of g(X, Y) . Looking back at the answer, we find that N(l, k) = (f.) q k. This proves the lemma.
Y) [1
(1
00
�
]
REFERENCES 1. R. P. Langlands, On the zeta-functions of some simple Shimura varieties, Notes, Institute for Advanced Study, Princeton, N. J., 1 977. 2. Shimura varieties and the Selberg trace formula, Canad J. Math. (to appear). -- ,
UNIVERSITY OF WASIDNGTON, SEATTLE
Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 193-203
NOTES ON L-INDISTINGUISHABILITY
(BASED ON A LECTURE OF R. P. LANGLANDS) *
D . S HELS TAD These notes are intended as a brief discussion of the results of [4]. Although we consider essentially just groups G which are inner forms of SL2, we emphasize for mulations (cf. [7]) which suggest possible generalizations. We assume that F is a field of characteristic zero. In the case that F is local there are only finitely many irreducible admissible representations of G(F) which are "L-indistinguishable" from a given representation. We structure this set, an L-packet, by considering not the characters of the members but rather sufficiently many linear combinations of these characters. In the case that F is global we consider certain L-packets of representations of G(A) and describe the multiplicity in the space of cusp forms of a representation 1C = ®v1Cv in terms of the position of the local representations 1Cv in their respective L-packets. 1. �(T), 'll( T) and C(T). Suppose that G is a connected reductive group defined over F, any field of characteristic zero, and that T is a maximal torus in G, also defined over F. Fix an algebraic closure F of F. Then we set �{T) = {g e G(F) : ad g- 1 /Tis defined over F} and 'll( T) = T(F)\�(T)fG(F). If g E �(T) then O" � g" = O"(g)g- 1 is a continuous 1-cocycle of @ = Gal(F/F) in T(F). The map g � (O" � g") induces an injection of 'll( T)into a subgroup C(T) of H 1 {®, T(F)) defined as follows. Let T.c be the preimage of T in the simply-connected covering group Gsc of the derived group of G. Then C(T) is the image of the natural homomorphism of H l (@, T.c(F)) into H l (@, T(F)). If H 1 {®, G.c(F)) = 1 and so, in particular, if F is local and nonarchimedean then 'll{ T) coincides with C{T). L-indistinguishability appears when G contains a torus T such that 'll ( T) is non trivial. 2. Groups attached to G (local case). Assume now that F is local. Fix a finite Galois extension K of F over which T splits. We replace F by K and @ by @KIF = Gal(K/F) in the definitions of the last section. An application of Tate-Naka yama duality then allows us to identify C(T) with the quotient of {A. e X* (T.c) : .E a eQJK/F O"A = 0 } by
{A
E
X* (T.c) : A = 1:: O"fl-11 aE I/!JK /F
- fJ-11 , fl-11 E
}
X* (T) ,
AMS (MOS) subject classifications (1 970). Primary 22E50, 22E55.
*This work has been partially supported by the National Science Foundation under Grant MCS76-0821 8 . © 1 979, American Mathematical Society
1 93
1 94
D. SHELSTAD
X* ( ) denoting Hom(GLr. ). A quasi-character " on X* (Tsc) trivial on this latter module defines, by restriction, a character on &( T). In [7] there is attached to each triple (G, T, K) a quasi-split group over F (there denoted H). We will pursue this just in the case that G is an inner form of SL2 • 3. Groups attached to an inner form of SL2 (local case). Suppose that G is an inner form of SL2 and that F is local ; we will continue with this assumption until § 14. We have then two groups to consider : S4 and the group of elements of norm one in a quaternion algebra over F. We may take PGL2(e) as LGo (notation as in [1]) and the diagonal subgroup as distinguished maximal torus LT o . Fix a maximal torus T in G, defined over F. A quasi-character " from the last section is just a @K 1 rinvariant quasi-character on X* (T). We fix an isomorphism between X* (T) and X * (LT0) = Hom(LT0, ex) as follows. If G is SL2 and T the diagonal subgroup we use the map defined by the pairing between X * (T) and X * (Lro) ; if T is arbitrary in SL2 we choose a diagonalization and compose the in duced map on X * (T) with that already prescribed. If G is the anisotropic form we may still regard T as a torus in SL2 and proceed in the same way. Using this isomorphism between X* (T) and X* (LT0) we transfer " to a quasi character on X * (Lro) ; using the canonical isomorphism between Hom(X * (LT0), ex) and LTo we then regard ,. as an element of LT0• At the same time we transfer the action of @K I F on X* (T) to X * (Lr o) and LT0, writing O"r for the new action of O" E @KIF · Here are the possibilities. If T is split then @K 1 F acts trivially and " is an arbitrary element of LT0• If T is anisotropic, suppose that T is defined by the quadratic ex tension E of F. We shall assume that K is some fixed large but finite Galois exten sion of F containing, in particular, E; @K 1 F acts on T through @E 1 F · Let 0"0 be the nontrivial element of @E I F and aV be a coroot for Tin G. Then 0"0av = - av so that (�>(aV)) 2 = 1 . Since aV generates X* (T) there are then just two possibilities for " · The nontrivial " defines the element c-o �) * of LTO ; here, and throughout these notes, we use (� �) * to denote the image in PG4(C) of the matrix (� �) in GL2 (e). The action of O"r on L To is described as follows : if O" e @K I F maps to the trivial element in @E l F under @K I F -+ @ElF then O"r acts trivially and if O" maps to 0"0 then O"r acts by
As an element of LT0, " is O"rinvariant, O" e @K I F · We define LHo to be the con nected component of the identity in the centralizer of " in LGo. Whatever T, if K is trivial then LH o = LG o and if " is nontrivial then LH o = LT0• Let O" e @K I F · Then since both LH o and LTo are invariant under O"r we may multiply O"r by an inner automorphism of LH o to obtain an automorphism O"n stabilizing LTo and fixing each root, if any, of LTo in LH0• The collection {O"n, O" E @K 1 F} defines a semidirect product LH = LH o XI WK I F where WK I F• the Weil group of K/F, acts through @K 1 F · In duality, we obtain a quasi-split group H over F. Specifically : (a) . If ,. is trivial (whatever T) then LH = LG = LGo x WKI F and SL2. (b) If T is split and " nontrivial then L H = LT o x WK l F and H = T. PROPOSITION.
H
=
195
L-INDISTINGUISHABILITY
(c) lf T is anisotropic and " nontrivial then L H = LTo ><J WK 1 F where w E WKl F acts trivially on L To if w maps to 1 under WK I F ---> @K I F ---> @E1F, and w acts by (g g)* ---> CS �)* if w maps to a o (E, (J 0 as before) ; and H = T.
To indicate that H is defined by (T, tc) we write H = H(T, K). Note that the choice (of diagonalization) made in defining the isomorphism between X*(T) and X*(L To) does not affect H(T, tc) . 4. Embedding L-groups. For later use we specify an embedding cH of L fl in LG. We refer to the proposition above. In (a) cH is to be the identity, in (b) cH is the inclusion and in (c) cH extends the inclusion of L ro in LGo by : if w E WKIF
(� �t maps to 1 under
X
W ---4
(� �t
X
WKI F -> @3KIF -> @E l F '
W
and
if w maps to a 0 • As remarked earlier there are analogues of these groups "H" for any connected reductive group G over F. In general, it need not be that L H embeds in LG; [7] indicates how to accommodate this (see Lemma I and the last several paragraphs). 5. Orbital integrals (normalization). We will transfer certain integrals from G to H. For this, normalizations are required. Fix a pair (T, tc) . We choose Haar measures dt and dg on T(F) and G(F) respectively. If /E C�(G(F) ) , o E 'l)(T), and r E T(F) is regular in G then we set r/J"Cr. .f) =
_ Jh-IT(F)h\G (F) f(g-1h-1rhg) ____!jg_ (dt)h
where h E %f(T) represents o and (dt)h is the measure on h-IT(F)h obtained from dt by means of ad h. For o E �(T) - 'l)(T) we set (/)0( ,f) = 0. Recall that %f(T), �(T), �(T) were defined in § 1 . In the case that " i s trivial (whatever T) we define (/)T 1 •( , f) = f/JT I I( , f) = c(G) 1::; (/Jii( , f) /jEC (T)
where s(G) = I if G = SL2 and s(G) = - I if G is anisotropic. We call r/JT II ( , ) a "stable" orbital integral. If T is split and " nontrivial then we define (/)T I •( r ' f)
=
l r 1 - rz l I; tc(o)r/Jii(r ) ,/ l n rzl 1 1 2 oEc crJ
where r �> rz are the eigenvalues of T · Suppose that T is anisotropic and that ,. is nontrivial. Then our normalization depends on two choices. First choose a nontrivial additive character cf;F on F. If E is the quadratic extension of F attached to T then we define J...(E/F, r/JF) as in [5]. Also choose a regular element r o of T(F). Let tc ' denote the quadratic character of px attached to E. Then we define
196
D.
rpr l •(r .J)
=
SHELSTAD
(
).(E/F, i/JF)tc' r; - r� 7 1 - rz
) Ilrn1 r-z l 1r1z2l
.r; tc(o)f/la(r . J)
i}E.f (T)
where the order on the eigenvalues T I > rz of r and r�. rg of r o is prescribed by fixing a diagonalization of T. A different choice of 1/JF or r o causes at most a sign change in the normalizing factor. 6. Transferring orbital integrals. The integrals rpr 1 •( , f) can be transferred to stable orbital integrals on H = H(T, tc) in the following sense : LEMMA. If f E C�(G(F)) then there existsfH E C�(H(F)) such that
rpr 1 1(r , fH) = rpr l •(r , f)
and
(i) ifG is anisotropic and tc trivial (so that H is SL2) then (/J T'Il(
, JH)
=
0 for T' split ;
(ii) if G is SL2 and K trivial (so that H is SL2 again) then for all T', provided that the Haar measures dt' are chosen consistently.
This is proved (in [4]) by a case-by-case argument. Note that if H is T then
rpr 1 1( , JH) is justfH itself.
For general (real) groups a formalism for transferring the (/JT 1 •( , f) to H is devel oped in [10]. Some progress towards obtaining a result as above for any real group is made in [9] and [10].
7. Stable distributions and a map. We define the space of stable distributions on G(F) to be the closed subspace, with respect to simple convergence, of the space of all distributions on G(F) generated by stable orbital integrals, that is, by the distri butions f f/JT 1 1(r , f), where r is a regular semisimple element of G(F) and T denotes the torus containing T · The transfer of the rpr 1 •( , f) to H establishes a correspondence (f, fH) between C�(G(F)) and C�(H(F)). Dual to this correspondence there is a well-defined map from the space of stable distributions on H(F) to the space of invariant distributions on G(F) ; that is, if 6lH is a stable distribution on H(F) then we may define a (con jugation-) invariant distribution 6l on G(F) by the formula 6l (f) = eH(fH), f E C� ( G(F) ) This map, 6lH ..... 6l, will be central to our study of L-indistinguish ability. We denote by X" the character of (the infinitesimal equivalence class of) an irre ducible admissible representation of G(F) ; we regard X" as a function on the regular semisimple elements of G(F), using the same normalization of Haar measure as in §5. There is a simple way to determine whether X"' as distribution, is stable. Let G be GL2 in the case that G is SLz , or the full multiplicative group of the underlying quaternion algebra in the case that G is anisotropic. Then a linear combination X of characters is stable if and only if X is invariant under G(F) . . . or (of relevance for generalizations (cf. [9])) if and only if X is invariant under &(T), T c G.
.....
.
L-INDISTINGUISHABILITY
197
8. Local L-packets and some stable characters. We assume that the Langlands correspondence has been proved for G ; in fact enough has been proved in each of the cases being considered. In the following definitions we also allow G to be a torus. We denote by f/J(G) the set of equivalence classes of admissible homomor phisms of W];. into LG o XI W];. (notation and definitions as in [1]). To each {rp} e f/J(G) there is attached a finite collection Drrp> of irreducible admissible represen tations of G(F), an L-packet. Two irreducible admissible representations of G(F) are said to be L-indistinguishable if they belong to the same L-packet. In the case that G is an inner from of SL2 there is a simpler definition : tr 1 and tr2 are L-indis tinguishable if and only if there exists g e G(F) such that trz is equivalent to tr1 o adg. We set X rrpl = .E "Enr.> X " · Clearly : PROPOSITION.
X frpl is stab/e.
Since the Langlands correspondence has been proved for any (connected reduc tive) real group [6] we can define characters Xrrpl in that case. In general X rrpl need not be stable ; however if the constituents of Dr"'> are tempered (cf. [3]) then Xr"'> is stable [9]. From § 12 on we will not need to distinguish in notation between an admissible homomorphism rp : W];. --+ LGo XI Wf. and its equivalence class ; we will then denote both by rp and write U"' , X"'' etc. 9. Character identities (introduction). We return to the map of stable distributions on H(F) to invariant distributions on G(F). If {rpH} e f/J(H) then, as we have observed, X rrpnl is stable. Its image in invariant distributions on G(F) is repre sented by some function X on the regular semisimple elements of G(F). This func tion X is computed by the Weyl integration formula. We write : X rrpnl � X · Recall the embedding cH of LH in LG (§4). We could have used W];. in place of WK l F in defining LG, LH, and tH · With these modifications, suppose that 'PH is an admissible homomorphism of w;. into LH. Then rp = tH o 'fJH maps Wf. to LG. Suppose that rp is admissible ; recall that (local} admissibility imposes the condition that the image of rp lie only in parabolic subgroups of LG which are "relevant to G" (cf. [1]). Then we say that {rp} e f/J(G) factors through {rpH} e f/J(H). Suppose that { rp} factors through { rpH }. Then linear combinations of the charac ters of the representations in Dr"'> make natural candidates for X ( � X rrpn> ).
10. "S0\S". We introduce a useful group. It is easier to work with a homomor phism rp : Wf. --+ LG rather than an equivalence class {rp}. We exclude the case of sp(2) [1] and corresponding special representation of G(F), and consider just an admissible homomorphism rp : WKIF --+ LG where K, LG (and LH, tH) are as earlier. We define S"' to be the centralizer in LGo of the image of rp ; s; will be the con nected component of the identity in S"' . If rp' is equivalent to rp then there exists g E LGO such that s"', = gS"'g- 1 and s;, = gs;g- 1 . Suppose that rp = tH o 'PH where 'PH is an admissible homomorphism of WK I F into LH and H = H(T, ti:). We regard " as an element of LT0• Recall that L H o is the connected component of the identity in the centralizer of " in L G o . By definition (§3), (JH fixes " ' (J e @K 1 F · It follows then that " lies in the center of the image of LH in LG. Therefore " centralizes the image of 'PH( WK 1 F) in LG; that is, " centralizes
I 98
e
D. SHELSTAD
q>( WK; F) · We have then that " Srp. We define srp(�>), or just s(�>) when qJ is under stood, to be the coset of " in s;\srp. Recall that this quotient appeared in [3]. Suppose that qJ' is equivalent to qJ and that qJ' = &H o f/J'H where ({Ju : WKIF � L H and H = H(T', �>'). Then ,., E srp' · Write (/)1 as ad g 0 (/), g E LG0• Then g-l ,.'g E Srp. We define srp(�>') to be the coset of g- 1 ,.'g in S;\Srp. We will see that s;\Srp is abelian. Therefore srp(�>') is independent of the choice for g. We continue with the same qJ, qJ' . The conjugation ad g induces an isomorphism behyeen s;\Srp and s;.\Srp' which carries srp(�>) to srp,(�>). Because both groups are abelian this isomorphism is independent of the choice of g in the equivalence qJ ' = ad g o f/J · We may therefore regard S;\Srp and the elements srp(�>) as attached to {qJ}. 11. Calculations. We compute explicitly s;\Srp and the elements s(�>) = srp(�>). We need take just one homomorphism qJ : WK l F � LG from each equivalence class. If we write as x WKIF • then the homomorphism qJ1 : WKIF � LG o = PGLz(C) lifts to a two dimensional representation ip 1 of WK l F [7, Lemma 3]. (i) Suppose that ip 1 is reducible. Then ;p 1 factors through ..- : WKIF � F x and is defined by a pair (fl. , lJ) of quasi characters on Fx . We take
qJ(w) qJ1(w) w, w e
iflt(w) = (fl.(z-6w)) lJ (z-�w))) '
If (fJ./lJ) 2 # I then srp = s; = LTO and if fl. = )) then srp = s; = LG0• However, if fl.flJ has order two then Srp = L To ><1 ( l ,(n) * ) ; recall that ( ) * denotes the image of( ) in PGL2(C). Hence S;\Srp = Z/(2). Whatever fi./lJ, qJ factors through L T, T a split torus. The corresponding " ((6 �) * , some z # I) lies in s; and s(�>) is trivial. Note that (this or any) qJ factors through H when H = H( , 1) ; s(l) is trivial also. Suppose that fl.flJ has order two. In (ii) we will show that a homomorphism equivalent to qJ factors through LT, where T is a torus defined by the quadratic extension E of F attached to fl.flJ, and that the associated s(�>) = srp(K) is the coset of (� 6) * . (ii) Suppose that ip 1 factors through a representation ip2 = Ind( WE l F • Ex, ()) of WEIF where E c K is a quadratic extension of F and () is a quasi-character on Ex . Then f/JI factors through qJ2, the projective representation defined by ip2. Let (1 ° be the n ontrivial element of @EI F and define O(x) = 0(Xa0), X E Ex . We realize WEIF explicitly as {x x p ; x Ex, p e { I , q0 } }, with multiplication rule (x x p)(x' x p' ) = x(x') P ap p' x pp ' where ap, p' = I unless p = p ' = (1 ° and aaa, aa is some (chosen) element a of Fx NmE 1 FEx . Then we may assume that ()(x) 0 qJ2(x x I) = 0 O(x) , and
e
,
_
(
)
-
199
L-INDISTINGUISHABILITY
where z is a c�osen square root of 8(a). Thus _
cp2(x
x
1)
=
(8(x) 0 ) O (x)
0
and cpz(l
x
*
,
x e EX,
(� �) . .
C1°) =
Let T be a torus attached t o E. Then cp factors through LT; indeed we chose the embedding c T of L T in LG (§4) so as to insure this. Recall that T = H(T, .t) where .t, as element of L T0, is (-A �) * ; s(.t) is the coset of (-A �) * in s;1s"'. To compute S"' , suppose first that (8/0) 2 #- 1 . Then
so that s;\S"' = Z/(2) ; we have already recovered the nontrivial element of s;\S"' as an s(.t) . Suppose now that 8/0 has order two. Then S"' coincides with cp 1 ( WKIF) ; that is, Therefore s;\S"' = Z/(2) (£) Z/(2). We have recovered only (-A �) * (and I) as an ( ) To recover the remaining elements of s;\S"' we recall homomorphisms cp', cp" : WKIF -+ LG which are equivalent to cp and factor through other LH. Let E0 be the quadratic extension of E defined by 8/0. Then pxfNmEoJFEt = Z/(2) (£) Z/(2) so that there are two distinct quadratic extensions E' and E" of F which are distinct from E and contained in E0• We pick a quasi-character 8' on (E'Y such that 8' o NmEotE' = 8 o NmEotE and 8' /0' is the quadratic character attached to E0/E' (O'(x) = 8'(x"'), 1 #- a ' e @E'tF) ; we pick 8" similarly. Then iiJ2 = Ind( WE'tF• (E'Y , 8') and ip� = Ind ( WE"tF• (E") x , 8") define representations fiJi, ip'{ of WKIF• each equivalent to iph and hence homomorphisms cp', cp" : WKI F -+ LG, each equivalent to cp. We define cpi, cp2 and cp'{, cp� as we did CfJh cp2 ; we realize cp2 and cp� as we did cp2. Then cp 1 ( WK 1 F), cpi( WK1F), cp'{( WKIF), S"' , S"', and S"'n all coincide and equal
s .t .
As with cp earlier, cp' ( . . . cp") factors through L(T') ( . . . L(T")), where T' ( . . . T") is some torus attached to E' ( . . . E"). Suppose T' = H(T', .t ') and T" = H(T", .t " ) . Then, writing cp = ad g ' o cp' = ad g " o cp", g ' , g " e LGo , we have ( ')
s .t
=
( ') =
s"' .t
g
and ( ")
s .t
=
( ")
s"' .t
=
g
"
'
( - 1 0) 0 1
( - � �)*
* g
g
' -1
"-1 .
200
D. SHELSTAD
An elementary argument shows that none of the conjugations ad g', ad g", ad(g ' )- 1 g " may fix (-0 �) * and so we conclude that {s(�e'), s(�e")} =
{ (� �t · ( _� �t } ·
Thus we have recovered each element of s;\s'P as an "s(�e)". We have one further case to consider : 0 = 0. We pick a quasi-character p. on px such that 0 = p. NmEI F · Then cp is equivalent to the homomorphism of type (i) above attached to the pair (p., 'p.) where ' is the quadratic character of px defined by the extension E/F. We now call that homomorphsim cp' . We had that s;,\s'P, has two elements and that (� ij)* is a representative for the nontrivial coset. We now recover this coset as an s'P,(�e). Let T be a torus attached to E. Then cp factors through LT and the associated ,. is (-0 �) * If cp ' = ad g cp, g E LG0, then -1 0 - - 0 1 g 0 l *g 1 = 1 0 * and we are done. (iii) Suppose that ip 1 is of tetrahedral or octahedral type. Then S'P = s; = 1 . This is a straightforward exercise. In summary : PROPOSITION. (1) s;\S'P is one of I , Z/(2) or Z/(2) Ef) Z/(2). (2) For each element s of s;\s'P there is a pair (T, �e) such that s = s(�e). 12. Character identities (continued). From now on we will not distinguish in notation between an admissible homomorphism cp: WK 1 F --+ LG and its equivalence class ; that is, we denote both by cp . . as we have noted, s;\S'P can be regarded as attached to the equivalence class. Suppose that cp factors through CfJH• where H = H(T, �e). Let s = s(�e). Then, in the notation of §§8, 9 : a
(
.
)
( )
a
.
LEMMA .
There exist integers (s, n), 7r: E ll'P, such that X'Pn � I: (s, n) Xn· 1CE U�
The proof [4] is again a case-by-case argument. Here is a summary of the ex plicit results. (A) G = S4. We consider cp as in (i), (ii), (iii) ofthe last section. (i) If (p./v) 2 ¥- I or if p. = v then U'P contains one element which we denote by n ; s;\S'P = 1 and ( 1 , n ) = 1 . I f p./v has order two then U'P has two elements which we denote by n1 and n2 . Recall that S;\ S'P = Z/(2) ; (s, n1) and (s, n2 ) are the two characters in s. These characters depend on the choices we made in normaliz ing orbital integrals (§5) ; that is, for some choices < , n1 ) is the trivial character and for others < n2 ) is the trivial one. (ii) We have already considered the case 0 = 0. If 0/0 is not of order two then U'P has two elements and s;\S'P = Z/(2) ; the result is as in (i). Suppose that 0/0 has order two. Then we had that s;\S'P is Z/(2) ® Z(2) ; U'P has four elements, too. Suppose U'P = {n,, i = I, . . . , 4} . Then we may (and do) normalize orbital integrals so that the ( , n;) are the four characters on s;\s'P. ,
20 1
L-INDISTINGUISHABILITY
(iii) Here Drp has one element, say n ; s;\Srp = 1 and ( 1 , n) = 1 . (B) G anisotropic. We have only to consider qJ as in (ii) with () #- 0 , and (iii). Suppose F = R. Then s;\Srp = Z/(2) since we exclude (iii) also. However Drp has only one element, say n; (s, n) is a character in s, trivial or nontrivial according to our choice of normalizing factors for orbital integrals. Suppose that F is nonarchimedean. Then Drp has the same number of elements as the corresponding L-packet in SL2, and the result is as there, except when qJ is of type (ii) with ()j[J of order two. Then the corresponding packet in SL2 has four elements ; Drp has only one element, say n. We must set ( 1 , n) 2 and (s, n) = 0, s #- 1 ; in particular, ( , n) is not a character. As for generalizing these identities we will report some progress for real groups in a forthcoming paper ; in the case that K is trivial, so that H is a quasi-split inner form of G, an appropriate identity is known provided that the constituents of nrp are tempered [9]. 13. Structure of local £-packets. From the results of the last section we conclude : PROPOSITION. If G is the quasi-split form (SL2) then the pairing ( , ) identifies nrp (noncanonically) as the dual group of s;\srp. If G is anisotropic we remark only the existence of the functions ( , n), n E Urp. With some qualifications, we can expect an analogue of the proposition for a general quasi-split real group (cf. [3]). 14. A global application. Suppose now that F is a global field and that G is SL2 . There are analogues for the groups H of the local case ; again H may be a maximal torus in G, defined over F, or G itself. We assume that K is some large but finite Galois extension of F and consider just homomorphisms qJ : WK 1 F ---+ LG defined by representations of the form Ind( WK 1 F• WK 1 E • ()) , where E is a quadratic extension of F contained in K and () is a grossencharacter for E not factoring through NmE I F· If Tis a torus attached to E then (/J factors through L T (as earlier, provided that L T is correctly embedded in L G). We define nrp to be the set of representations n = ®vnv of G(A) which are irreducible, admissible (as in [1]) and such that """ E nrp. for each place v . The set Hrp is an L-packet in the following sense. We define two irreducible admissible representations n1 ®vn� and n2 = ®vn� of G(A) to be £-indistinguishable if for all places v, n� and n� are £-indis tinguishable and for almost all v, n� and n� are equivalent. An L-packet is an equivalence class for this relation. We have singled out the packets Hrp for the following reason. As is proved in [4], two representations from such a packet may appear with different multiplicities in the space of cusp forms for G. One may appear, with multiplicity one, and the other not appear. For the remaining £-packets the members do appear with equal multiplicity (conjectured to be either zero or one). In [4] there is a formula for the multiplicity with which a representation from nrp appears in cusp forms ; it is this we wish to discuss. As in the local case, we define Srp to be the centralizer of qJ( WK l F) in LG o and s; to be the connected component of the identity in Srp. As in (ii) of § 1 1 we have that either s;\srp = Z/(2) or s;\Srp = Z/(2) EB Z/(2). For each place v, Srp embeds =
=
202
D. SHELSTAD
naturally in S'P. and s; in s;, There is then a natural map of s;\S'P into s;.\s'P, In notation we will not distinguish between an element of S�\S'P and its image in s;.\s'P, Recall that the local characters < , �v> depend on the choices we made when normalizing orbital integrals (§§5, 1 2). We chose a nontrivial additive character cf; F. on Fv and regular element r o = r� of T(Fv ) for each torus T anisotropic over Fv . Instead, we now choose a nontrivial additive character cj; on F\A and for each torus T anisotropic over F a regular element r o in T(F). At each place v where T does not split we use cf; v and r o to specify (])T 1 K( , ), " nontrivial. Fix s E s;\s'P and � E n'P. Then we have (s, �v> = 1 for almost all v. Therefore we may define (s, �> = Dv(s, �v> · Then [4] :
PROPOSITION. (i) (s, �) is independent of the choices madefor cj; and r0• (ii) < ' > induces a (canonical) surjection of n'P onto the dual group of s;\s'P. Also : THEOREM. The multiplicity with which � E n'P appears in the space of cusp forms is : 1 (s, �> . [s'Po\S'P] SESI;;\s. That is, those � for which < , �) is the trivial character appear with multiplicity one and those � for which < , �> is nontrivial do not appear. 15. Afterword. More generally, and of relevance also for Shimura varieties (cf. [8]), we can consider inner forms of a group G such that Res£ 1 FSL2 c G c ResE;FGL2 with E some finite Galois extension of F, and develop a local and a global theory in the same way. The analogous multiplicity formula is [4] :
__5p___
I; < s' � > 'P where d'P is defined as follows. If q; , q;' : WF -+ LG are admissible homomorphisms (cf. [1, § 1 6]) call q; and q;' locally equivalent everywhere if (/Jv is equivalent to q;� for each place v. Call q; and q;' weakly globally equivalent if q;' is equivalent to wq; where w is a continuous 1 -cocycle of WF with values in the center of LG o such that the restriction of w to each local group is trivial. Then d'P is the number of weak global equivalence classes in the everywhere local equivalence class of q;. In [8] L-indistinguishability plays a role in relating the zeta-functions of certain Shimura varieties to automorphic L-functions. Briefly, in the case discussed in [2], where G is the full multiplicative group of some quaternion algebra over a totally real field and there is no L-indistinguishability, functions L(s, �. p) appear (� an automorphic representation of G(A), p a certain representation of LG, fixed as in that lecture). In the general case of [8], where G is a subgroup of the full multi plicative group, we must consider at least some of the L-packets n'P where different multiplicities in cusp ( . . . automorphic) forms occur. Then q; : WF -+ LG factors through ty : LT ..... LG, with T a nonsplit torus of G. If q; = Cy 0 qJy, �T E n'PT and Pr = p · tr then L(s, �. p) = L(s, �r. py). There is a natural decomposition Pr = p} EfJ p�, where p}, p� may be reducible, and it is the functions L(s, �r. p�) which are relevant.
[S;\S'P]
s E S 0 \S0
L-INDISTINGUISHABILITY
203
REFERENCES 1. A. Borel, Automorphic L-functions, these PROCEEDINGS, part 2, pp. 27-61 . 2. W. Casselman, The H asse-Wif!il �-function 'of some moduli Ivarieties ofdimension greater than one, these PROCEEDINGS, part 2, pp. 141-163. 3. A. W. Knapp and G. Zuckerman, Normalizing factors, tempered representations and L-groups, these PROCEEDINGS, part 1 , pp. 93-105 . 4. J-P. Labesse and R. P. Langlands, L-indistinguishability for SL(2) (to appear). 5. R. P. Langlands, On Artin's L-functions, Rice Univ. Studies, vol. 56, no. 2, pp. 23-28, 1 970. 6. , On the classification of irreducible representations of real algebraic groups (to appear). 7. , Stable conjugacy ; definitions and lemmas (to appear). 8. , On the zeta functions ofsome simple Shimura varieties (to appear). 9. D . Shelstad, Characters and inner forms of a quasi-split group over R, Compositio. Math. (to appear). 10. , Orbital integrals and a family of groups attached to a real reductive group, Ann. Sci . Ecole Norm. Sup. (to appear). ---
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COLUMBIA UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol . 33 (1 979), part 2, pp. 205-246
AUTOMORPIDC REPR'ESENTATIONS, SmMURA VARIETIES, AND MOTIVES. EIN MARCHEN
R.
P.
LANGLANDS
1. Introduction. It had been my intention to survey the problems posed by the study of zeta-functions of Shimura varieties. But I was too sanguine. This would be a mammoth task, and limitations of time and energy have considerably reduced the compass of this report. I consider only two problems, one on the conjugation of Shimura varieties, and one in the domain of continuous cohomology. At first glance, it appears incongruous to couple them, for one is arithmetic, and the other representation-theoretic, but they both arise in the study of the zeta-function at the infinite places. The problem of conjugation is formulated in the sixth section as a conjecture, which was arrived at only after a long sequence of revisions. My earlier attempts were all submitted to Rapoport for approval, and found lacking. They were too im precise, and were not even in principle amenable to proof by Shimura's methods of descent. The conjecture as it stands is the only statement I could discover that meets his criticism and is compatible with Shimura's conjecture. The statement of the conjecture must be preceded by some constructions, which have implications that had escaped me. When combined with Deligne's conception of Shimura varieties as parameter varieties for families of motives they suggest the introduction of a group, here called the Taniyama group, which may be of impor tance for the study of motives of CM-type. It is defined in the fifth section, where its hypothetical properties are rehearsed. With the introduction of motives and the Taniyama group, the report takes on a tone it was not originally intended to have. No longer is it simply a matter of for mulating one or two specific conjectures, but we begin to weave a tissue of surmise and hypothesis, and curiosity drives us on. Deligne's ideas are reviewed in the fourth section, but to understand them one must be familiar at least with the elements of the formalism of tannakian categories underlying the conjectural theory of motives, say, with the main results of Chapter II of [40]. The present Summer Institute is predicated on the belief that there is a close re lation between automorphic representations and motives. The relation is usually AMS (MOS) subject classifications (1 970). Primary 14010, 14F99. © 1979, American Mathematical Society
205
206
R.
P. LANGLANDS
couched in terms of L-functions, and no one has suggested a direct connection. It may be provided by the principle of functoriality and the formalism of tannakian categories. This possibility is discussed in the highly speculative second section. However there is a small class of automorphic representations which are certainly not amenable to this formalism. I have called them anomalous, and in order to make their significance clearer I have discussed an example of Kurokawa at length in the third section. Although the anomalous representations form only a small part of the collection of automorphic representations, they are frequently encoun tered, especially in the study of continuous cohomology, and so we have come full circle. Our long divagation has not been in vain, for we have acquired concepts that enable us to appreciate the global significance of the local examples described in the seventh section, which deals with the second of our original two problems. At all events, I have exceeded my commission and been seduced into describing things as they may be and, as seems to me at present, are likely to be. They could be otherwise. Nonetheless it is useful to have a conception of the whole to which one can refer during the daily, close work with technical difficulties, provided one does not become too attached to it, but takes pains to ensure that it continues to con form to the facts, and is. prepared to abandon it when that is called for. The views of this report are in any case not peculiarly mine. I have simply fused my own observations and reflections with ideas of others and with commonly accepted tenets. I have also wanted to draw attention to the specific problems, on which expert advice would be of great help, and I hope that the report is sufficiently loosely written that someone familiar with continuous cohomology but not with arithmetic can turn to the seventh section, overlooking the first few pages, and see what the study of Shimura varieties needs from that theory, or that someone familiar with Shimura varieties but not automorphic representations or motives will be able to find the definition of the Serre group in the fourth section and the Taniyama group in the fifth, and then turn to read the sixth section. Finally a word about what is not discussed in this report. The investigations of Kazhdan [26] and Shih [46] on conjugation of Shimura varieties are not reviewed ; their bearing on the problem formulated here is not yet clear. L-indistinguishabil ity is not discussed. Little is known, and that is described in another lecture [44]. Problems caused by noncompactness are ignored. There is a tremendous amount of material on compactification and on the cohomology of noncompact quotients, but no one has yet tried to bring it to bear on the study of the zeta-functions. The omission I regret most is that of a discussion of the reduction of the Shimura varieties modulo a prime [36]. Here there is a great deal to be said, especially about the structure of the set of geometric points in the algebraic closure of a finite field, starting with the work of Ihara on curves. I hope to report on this topic on another occasion. 2. Automorphic representations. Our present knowledge does not justify an attempt to fix a language in which the relations of automorphic representations with motives are to be expressed, Nonetheless that of tannakian categories [40] appears promising and it might be worthwhile to take a few pages to draw attention to the
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problems to be solved before it can be applied in the study of automorphic repre sentations. We first recall the rough classification of irreducible admissible representations of GL{n, F), F being a local field, and of automorphic representations of GL(n, AF), F being a global field, reviewed in some of the other lectures ([3], [5], [35], [48], cf. also [7], [31]). In either case the representation n has a central character w and z � l w(z) l may be extended uniquely from Z(F), or Z(AF), to a positive character v of GL(n, F), or of GL(n, AF). If F is a local field then n is said to be cuspidal if for any K-finite vectors u and v, in the space of n and its dual, v- l (g) (n (g)u, v) is square-integrable on the quotient Z(F)\GL(n, F). To construct an arbitrary irreducible admissible repre sentation one starts from a partition {nh · · ·, n,} of n and cuspidal represen tations n 1 o · · ·, n, of GL(n;, F). If w; is the central character of n;, there is a real number s; such that lw;(z) l = lzl•; if z lies in the centre of GL{n;, F), a group isomor phic to Fx . Changing the order of the partition, one supposes that s 1 � • • • � s,. The partition defines a standard parabolic subgroup P of GL{n) and o = ®n; a representation of M(F), because the Levi factor M of P is isomorphic to GL(n 1) x · · · x GL(n,). The representation o yields in the usual way an induced represen tation /" of G(F). /" may not be irreducible, but it has a unique irreducible quotient, which we denote n 1 E8 · · ·EEl n,. Every representation is of this form and n 1 E8 · · ·EEl n, � n :i E8 · · · E8 n; if and only if r = s and after renumbering n; � n;. Thus every representation can be represented uniquely as a formal sum, in the sense of this notation, of cuspidal representations. The representation n 1 E8 · · · E8 n, is said to be tempered if all the S; are 0. We can clearly define, in a formal manner, the sum of any finite number of representations. We can in fact formally define an abelian catefory H (F) whose collection of objects is the union over n of the irreducible, admissible representations of GL(n, F). If n = n 1 E8 · · · E8 n" n' = n:i E8 · · · E8 n;, with n; and nj cuspidal, we set Hom(n, n')
=
{i, jl1t:; -"jl
EB
C.
The composition is obvious. The tempered representations form a subcategory H0 (F) . If F is a global field then n is said to be cuspidal if n is a constituent of the representation of GL(n, AF) on the space of measurable cusp forms rp satisfying (a) rp(zg) = w(z) rp(g), z E Z(AF), (b) Jz (AF)G (F)\G (AF) V -2(g) j rp(g) j 2 dg < 00 . I f n 1 o · · , n, i s a partition of n and n 1 o · · · , n, cuspidal representations of GL{n;, AF) we may again change the order so that s 1 � · • � s, and then construct the induced representation /" . Every automorphic representation is a constituent of some /" . For an adequate classification, one needs more. The following statement may eventually result from the investigations of Jacquet, Shalika, and Piatetskii Shapiro, but has not yet been proved in general. A. If n is a constituent of /" and of lu' then the partitions {nh · · · , n,} and {n:i, - · · , n;} have the same number of elements, and, after a renumbering, n; = n; and n; n;. •
•
�
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R . P. LANGLANDS
If 7C; = (8).7C;(v) then one constituent of I" is the representation 7C = ®v 7C(v) with local components 1e(v) = 1e 1 (v) [±) · · · [±) 1e,(v). This representation will be denoted 1e = 1e 1 [±) · [±) 7C" but the notation is not justified until statement A is proved. The representations of this form will be called isobaric and can again be used to define an abelian category D(F). We agree to call 7C = 1e 1 [±) [±) 7C, tempered if each of the cuspidal re presentations 7C; has a unitary central character, that is, if each of the s; is 0. How ever the language is only justified if we can prove the following statement, the strongest form of the conjecture of Ramanujan to which the examples of Howe Piatetskii-Shapiro and Kurokawa allow us to cling. B. If 7C = (8)1e(v) is a cuspidal representation with unitary central character then each �{the factors 1e(v) is tempered. The tempered representations form a subcategory 0°(F) of D(F). It is clear that we have tried to define the categories H(F) and H0(F) in such a way that if v is a place of F there are functors H(F) --+ H(F.) and Do(F) --+ 0°(F.) taking 7C to its factor 1e(v). However without a natural definition of the arrows, there is no unique way to define Hom (7C, 1e') --+ Hom (1e(v), 1e'(v)) . If I" is not irreducible it will have other constituents in addition to 1e 1 [±) · · [±) 1e,. These automorphic representations will be called anomalous. Although the principle of functoriality may apply to them, there is considerable doubt that they can be fitted into a tannakian framework. Observe that statement A and the strong form of multiplicity one imply that if 1e is any automorphic representation there is a unique isobaric representation 1e ' such that 1e(v) - 1e'(v) for almost all v. If F is a global field the purpose of the tannakian formalism would be to provide us with a reductive group over C whose n-dimensional representations, or rather their equivalence classes, are to correspond bijectively to the isobaric automorphic representations of GL(n, Ap). This hypothetical group will have to be very large, a projective limit of finite-dimensional groups. We denote it by GucFJ · The category Rep(GocFJ ) of the finite-dimensional representations over C of the algebraic group GucFJ would certainly be abelian, but in addition it is a category in which tensor products can be defined. Moreover there is a functor to the category of finite-dimensional vector spaces over C. If (rp, X), consisting of the space X and the representation rp of GucFJ on it, belongs to Rep (GocFJ) one sim ply ignores rp. The tensor product satisfies certain conditions of associativity, com mutativity, and so on, and the functor, called a fibre functor, is compatible with tensor products and other operations of the two categories. A theorem of [40], but not the principal one, asserts that, conversely, an abelian category with tensor products and a fibre functor is equivalent to the category of representations of a reductive group, provided certain natural axioms are satisfied. Thus it appears that if we are to be able to introduce GocFJ we will have to as sociate to each pair consisting of a cuspidal representation 7C of GL(n, A) and a cuspidal representation 1e' of GL(n', A) an isobaric representation 7C (g) 1e' of GL(nn', A). In general, if 7C = 1e 1 [±) · · · [±] 1e, and 1e' = 1ei [±] · · · [±) 1e; we would set 7C (g) 7C' = ffi (7C; (g) 7Cj). · ·
· · ·
·
{,
j
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
209
In addition, we will have to associate to each isobaric representation 7T: of GL(n, A), n = I , 2, , a complex vector space X('IT:) of dimension n, together with isomorphisms . . ·
There are a large number of conditions to be satisfied, among them one which is perhaps worth mentioning explicitly. Suppose 7T: and 'IT:' are cuspidal and 7T: � 'IT:' = ffi�=1 'JT:; with 'IT:; cuspidal. Then the set {'IT:;, · · · ,'IT:,} contains the trivial representation of GL(I, A) if and only if 'IT:' is the contragredient of 'IT:, when it contains this repre sentation exactly once. At the moment I have no idea how to define the spaces X(7T:) ; indeed, no solid reason for believing that the functor 7T: � X(7T:) exists. Even though the attempt to introduce the groups Gn(F) may turn out to be vain the prize to be won is so great that one cannot refuse to hazard it. One would like to show, in addition, that if 7T: and 'IT:' are tempered then 7T: � 'IT:' is also, and thus be able to introduce a group Gna
X
Dp.
One will also wish to introduce, by a similar process, groups Gn
)
l wl- 1 1 2
X
W.
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P . LANGLANDS
Notice that according to these classifications there are homomorphisms of algebraic groups Gn cFJ --)- Gal(F/F) and G0ocFJ --)- Gal(F/F), the group on the right being a projective limit of finite groups. The principle of functoriality cannot be valid unless there are similar homomorphisms when F is a global field. Let Dp , Dt, D�, be the groups of multiplicative type whose modules of rational characters are, respectively, the module of all characters of px , or Fx\IF if F is global, of all positive characters, or of all unitary characters. Then DF = Dt x D� and there will be homomorphisms Gn cFJ
--)-
Dp,
Gno cFJ -)- D � .
If F is nonarchimedean and local we may also define Dun• D;;n , and D�n' by re placing the modules of characters of various types by the modules of unramified characters of the same type. The groups Dum Dtm and D�n contain a distinguished point over C, the Frobenius f/J, which is simply the image of a uniformizing parameter in px . In any case the formalism will certainly allow us to introduce for any representation a of the algebraic group Gn cFJ over C an L-function L(s, a) and if a corresponds to the representation n of GL(n, Ap) then L(s, a) = L(s, n) . But the reasons for wishing to introduce the groups GncFJ and Gno cFJ and the associated formalism are not simply, or even primarily, aesthetic. There are prob lems which will be difficult to formulate exactly without them. Suppose, for example, that n = ®vnv is an isobaric representation of GL(n, A) and each of the factors nv is tempered. For almost all v, nv is unramified and associated to a conjugacy class {gv} {g(nv)} in GL(n, C). Since nv is supposed tempered this class meets the unitary group U(n) and I may, as I prefer, regard it as a conjugacy class , in U(n). The general analytic analogue of the Tchebotarev theorem or the Sato Tate conjecture would be a theorem or conjecture describing the asymptotic distribution of the classes {gv} · Suppose the formalism existed and n were associated to a representation a of GnocFJ · The image a(GnocFJ (C)) would be a reductive subgroup H(C) of GL(n, C) with a maximal compact subgroup KH. For almost all v, av would factor through Gno (F,) -> D�n and av(f/Jv) would be defined. Since its conjugacy class in H(C) would meet KH, it would define a conjugacy class in KH, which we denote {av(f/Jv) } . Of course {avCf/Jv)} � {gv} and the asymptotic distribution of the classes {gv} can be inferred from that of the classes {avCf/Jv)} . There is a natural probability measure on the space of conjugacy classes in KH. If X is a set of conjugacy classes and if the set Y = U x E X x is measurable in KH, one takes meas X = meas Y. It is natural to suppose that it is this measure which defines the asymptotic distribution of the classes {avCtPv)} . To verify the supposition, it will be necessary to establish that if p is any representation of G0ocFJ over C then the order of the pole of L(s, p) at s = 1 is equal to the multiplicity with which the trivial representa tion of Gno cFJ occurs in p . If the existence of G0o cFJ were established, it would be easy enough to deduce this from the recent results of Jacquet and Shalika =
[25] .
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
21 1
Within this formalism, the principle of functoriality asserts that if F is a local field and G a reductive group over F then any L-packet of representations of G(F) is associated to a homomorphism q> : GucF) --+ LG of algebraic groups over C for which GucFl .'!.... LG '\. _ /
Gal(F/F) is commutative. If GucFl is replaced by Guo cFl • the L-packet should be tempered. If F is global, some caution will have to be exercised. If the L-packet H consists of 1C = (8)1e. for which the 1Cv are always tempered, it should correspond to a q> : Guo (F) --+ LG. Otherwise this may not be so, for a reason which will perhaps be clearer after an example of Kurokawa [29] is discussed in the next section. If there is a representation cj; : LG --+ GL(n) x Gal(F/F) and if the image cf;ill) of n given by the principle of functoriality is not isobaric then n can be associated to no q>. One may nonetheless hope to prove, both locally and globally, that to each q> : GucFl --+ LG is associated an L-packet, provided q> commutes with the homo morphisms to the Galois group. If G is not quasi-split the local behaviour of q> with respect to parabolic subgroups will also have to be taken into account [3]. For archimedean fields one recovers the usual classification. The few examples studied [44] suggest that questions about the multiplicity with which the elements of n occur in the space of automorphic forms will have to be answered in terms of q>. The principal reason for wishing to define the group Gn cFl is that it provides the only way visible at present to express completely the relation between auto morphic forms and the conjectural theory of motives [40]. The category of motives over F, a local or a global field, is Q-linear and tannakian, but it does not always possess a fibre functor over Q and seldom a single naturally defined one. Thus tan nakian duality associates to it not a group over Q but a group-like object, a "gerbe" in the rustic terminology which has become so popular in recent years. Over C this object becomes a group GMotCFl • and the relations between motives and automorphic representations will probably be adequately expressed by the ex istence of a homomorphism pF : Go cFl --+ GMot CFl defined over C. The field F can be local or global. The local and global homomorphisms are to be compatible with each other and with the formation of £-functions. Both the image and the kernel of PF will probably be rather large when F is a number field (cf. C.6.2 of [43]) but rather small when Fis local. 3. Anomalous representations. Since the anomalous representations cause some difficulty in the stu dy of the zeta-functions of Shimura varieties, it will be useful to acquire some feeling for them before going on. The brief remarks of the previous section suggest that an automorphic representation 1C of the reductive group G, or rather the L-packet H containing it, should be called anomalous if for some homomorphism cf; : LG --+ GL(n, C) x Gal(F/F) the principle of functoriality takes
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R. P . LANGLANDS
1C or U to an anomalous representation of GL(n, A). It may be that the counter examples to the Ramanujan conjecture of Howe-Piatetskii-Shapiro [19] are anom alous in this sense. Since their paper is not available to me as I write, I have to test this suggestion on another example, discovered by Kurokawa and quite explicit. The group G is to be the projective symplectic group in four variables over Q. The L-group is then the direct product of LG0, the symplectic group in four variables and the galois group Gal(Q/Q). Since all the groups with which we shall deal in this section will be split, we may ignore the factor Gal (Q/Q). Let G1 be the product of PGL(2) over Q with itself, so that LG� = SL(2, C) x SL(2, C) and let G2 be GL(4) over Q. Define cp1 : LG� --+ LG to be the homomor phism
(;: ��) (;: �:) �1 X
-->
0
and let cp 2 be the standard imbedding of LGo in LG� which is GL(4, C). In order to analyze Kurokawa's example one must formulate his statements representation-theoretically. I state the facts necessary to this purpose, but have to ask that the reader understand the discrete series sufficiently well to verify them for himself. There is nothing to prove. It is simply a question of writing down explicitly for the special case of concern to us here some of the results of [17] and and [18], and some definitions from [31]. An automorphic representation 1C = ®1Cv of G(A) is associated to a holo morphic form of weight k in the classical sense of [39] if and only if 7C"' is a member of the holomorphic discrete series and lies in an L-packet Dq,ckl • where the restric tion of if;(k) to e x s;; WeiR has the form Z -->
with A = (2k - 3)/2, p. = t . Since G is the projective group, k must be even. Notice that zlz-l is to be calculated as z2l(zz)-l = z- 2.1 (zz)l. We will eventually take k = 10. On the other hand an automorphic representation 1C = ®1Cv of PGL(2, A) is associated to a holomorphic form of weight 2k - 2 if and only if 7C "' belongs to the discrete series and to an L-packet Drp w , where cp(k) : z -->
(
zz .l
--
.l
z- · z-·
)
with A as above. In particular cp1 takes the £-packet Drp Ckl ® Drpc2l , consisting in fact of a single representation, to Uq, Ckl . We want to apply the principle of functoriality to cp 1 and a very special auto-
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
213
morphic representation "" of G 1 (A). "" must be a tensor product 1r' ® 1r" of two representations of PGL(2, A). "";, and ""� will both be members of the discrete series, the first in Drp
lxl-1 / Z
of GL(l, A), and then construct the induced representation of GL(2, A) as in the preceding section. Any constituent it" of the induced representation factors through a representation it" of PGL(2, A). We so choose it" that ""� E Drp
f>
=
{Ad g h l g E G(R)} 0
and it will be best simply to let h denote an arbitrary element of f>. Recall that the pair (G, h) is subject to three conditions [10, § 1 .5) : (a) If w is the diagonal map GL(l) -+ GL(l) x GL(l) then the homomorphism h o w is central. (b) The Lie algebra @ of G(C) is a direct sum @ = p + f + p and if(zh z2) E �(C) then
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R . P. LANGLANDS
X E p, = z11z2 X, = X, X E r, XE p. = zl zil X, In fact the summands p, f , and p vary with h, and when it is useful to make the depend ence on h explicit we write ph, th, and Ph · (c) The adjoint action of h(i, - i) on the adjoint group is a Cartan involution. Since G(R} acts on the real manifold .p by conjugation every element of @ defines a complex vector field on .p. Let Xh be the value of the vector field associated to X at h E .p. The complex structure on .p is so defined that the holomorphic tangent space at h is {Xhl X E p} and the antiholomorphic tangent space is {Xhl X E p}. If A1 is the ring of adeles over Q whose component at infinity is 0 and K is an open compact subgroup of G(A1) then G(A1)/K is discrete and Xx = .p x G(A1)/K is a complex manifold on which G(Q) acts to the left. If K is sufficiently small then any r E G(Q} with a fixed point lies in Z(Q) n K and thus fixes the whole manifold. We shall always assume that K is sufficiently small and then ad h(z)(X)
is a complex manifold, proved by Baily-Borel to be the set of complex points on an algebraic variety Shx = Shx(G, h) = Shx(G, ,P) over C. Deligne anticipates that Shx will often be a moduli space for a family of motives over C. This is sometimes so, the motives then being those attached to abelian varieties, but can certainly not yet be proved in general. Nonetheless there is a good deal to be learned from a rehearsal of the considerations that suggest such an interpretation of Shx. In essence one observes that Shx(C) is the parameter space for a family of polarized Hodge structures ; the difficulty is to show that these Hodge structures all arise from motives. A real Hodge structure V is a finite-dimensional vector space VR over R together with a decomposition of its complexification Vc = E9 p, q EZ vP• q , satisfying yq, p = VP, q . The collection of real Hodge structures forms a tannakian category over R, whose associated group is /?4. Indeed to a real Hodge structure V, one associates the representation rJ of 1?4 defined by (4. 1) The relations yq ,p = VP . q imply that· rJ is defined over R. Conversely each represen tation of 1?4 that is defined over R yields a Hodge structure, the elements of yp , q being defined by (4. 1 ) . The real Hodge structure Vis said to be of weight n if VP ,q = 0 whenever p + q #- n. Certainly any Hodge structure is a direct sum V = ffi n Vn with vn of weight n. We are however interested in the category of polarized rational Hodge structures. A rational Hodge structure V is formed by a finite-dimensional vector space VQ over Q, a direct sum decomposition VQ = E9 V(j, and real Hodge structures of weight n on Vfi = VJ ® Q R. There is a distinguished object of weight - 2, the Tate object Q( l ) , in the category of rational Hodge structures. The underlying rational vector space is Q(l)Q = 2�iQ 5:::; C and, by definition, Q( l) - 1 .- 1 = Q( l )c .
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES , AND MOTIVES
21 5
It seems to be customary to identify the underlying vector space of Q(n) = Q(l)®" with (2n-i)"Q and Q(n)8 with (2n-i)"R !;;;;; C. The factors 2n-i have been chosen for reasons which need not concern us . It is no trouble to carry them along. If V is a rational Hodge structure and u the associated representation of f!lt let C be u( - i, i) acting on V8. If V is of weight n, a polarization of V is a bilinear form P : V x V -+ Q ( - n) satisfying : (a) For all u and v in Vc and all r e R(C) P(u(r)u, u(r)v) = u(r)P(u, v). Thus the form is compatible with the Hodge structures. (b) P(v, u) = ( - 1)"P(u, v). (c) The real-valued form (2n-i)"P(u, Cv) on V8 is symmetric and positive-definite. A rational Hodge structure is said to be polarizable if each of its homogeneous components admits a polarization, a polarization of the full structure being defined by polarizations of the homogeneous components. The category Jf(!}!!&(Q) of polarizable Hodge structures is tannakian, with a natural fibre functor WHod : V -+ V0 and an associated group GHod • reductive but overwhelmingly large. It does have factor groups of manageable size. If V is a polarizable rational Hodge structure, one may take the tannakian cate gory generated by V and Q(l) and the repeated formation of duals, sums, tensor products, and subobjects. The associated group is called the Mumford-Tate group of V and denoted by .Itff( V). It is finite-dimensional and reductive, and there is a surjection GHod -+ .Hff(V) defined over Q. If u is the representation of f!lt attached to V then .Hff( V) is simply the smallest subgroup of the group of automorphisms of the rational vector space underlying V which contains u(f!lt) and is defined over Q [37]. It is consequently connected. The polarizable rational Hodge structures for which .ltff(V) is abelian play a particularly important role in the study of Shimura varieties. They are said to be of CM type. The second description of the groups .ltff( V) shows that the category of such Hodge structures is closed under sums and tensor products and thus is a tan nakian category. The associated group has been studied at length in [41] and is often called the Serre group. At the risk of making a comparison with [41] difficult, for Serre himself employs a different notation, we shall denote the group by !7'. It is not difficult to describe !7'. Let Q be the algebraic closure of Q in C and let t e Gal(Q/Q) be complex conjugation. To construct X*(Y), the module of rational characters of !7', we start with. the module M of locally constant integral-valued functions on Gal(Q/Q). The Galois group acts by right translation and X*(Y) = {A e Mi (u l)(t + 1 )). = (t + 1)(u - 1)). = 0 'flu e Gal(Q/Q)} . In particular if ). e X*(Y), -
(u
1)).( 1)
+
(u
l) ).(c ) = 0 because the left side is (t + l)(u - 1)).(1). The lattice of rational characters of f!lt is canonically isomorphic to Z E9 Z and the homomorphism ho : f!lt -+ [!' dual to the homomorphism X*(Y) -+ X*(f!lt) which sends ). to ().(1 ) , ).(c)) is defined over R. The composite h o w : GL(l) -+ !7' is dual to the homomorphism X*(Y) -+ Z taking ). to ).(1) + A(t) and is defined over Q. -
-
216
R . P . LANGLANDS
To verify that the group !I' just defined in terms of its module of characters is the Serre group defined in terms of Hodge structures is easy enough. The existence of the two homomorphisms h0 and h0 o w implies that every representation of !I' defined over Q defines a rational Hodge structure. It is enough to show that these are polarizable when the representation is irreducible. To obtain the irreducible representations, one takes a A E X*(!I') and defines the field F by Gal(Q/F) = {a E Gal(Q/Q) I aA = A}. The underlying space of the representation is the vector space over Q defined by F, and the representation r = r1 is that defined symbolically by r1(s) : x E F -+ A(s)x. The weight of the associated Hodge structure is - (A( I) + A(c)) = n. There is an a E F such that c(a) = ( - I) na and ( - I)l <el j na is totally positive. A possible polarization is P(u, v) = (2n:i)-n Trnguac(v).
Conversely suppose one has a rational Hodge structure V whose Mumford-Tate group vltff(V) is abelian. Since there is a homomorphism � -+ vltff( V), the coweight GL(I) -+ � of the group � defined by z -+ (z, 1) also defines a coweight v V of Jtff( V). The lattice Y* of coweights of Jtff( V) is a Gal(Q/Q) module gen erated by v v . We define an injective homomorphism of its lattice of rational charac ters V* into X*(!l') by sending v E Y* to the element A given by a E Gal(Q/Q). The dual homomorphism !I' -+ vHff( V) is surjective and V is defined by a rational representation of !1'. We return to the Shimura varieties ShK(C) and show how one attaches families of rational Hodge structures to ShK(C). Let G0 be the largest quotient of G such that (a - l)(c + l)v V = (c + l)(a - l)v V = 0 for every coweight of the centre of G0. Let � be a rational representation of G on V which factors through G0• To each x = (h, g) in XK we associate a triple ( Vx , t x , q;"). vx is a rational Hodge structure whose underlying space is V� = Vg, the Hodge structure being defined by the representation � o h of �. The third term q;" is an isomorphism Vj1 -+ VA1 and is given by v -+ �(g)-l v. It is only defined up to composition with an element of �(K). The homogeneous components of V� are independent of x and may be written V0. It follows from Lemma 2.8 of [11] that if x E XK there is at least one collection p x of bilinear forms P� : V0 x V0 -+ Q( - n)g which is invariant under the derived group of G and is a polarization of vx . Let \13" denote the collection of all such polarizations. If g E G(Q), px E \13", and x' = rx then the collection px' given by P�' : (u, v) -+ P�(�(r-l)u, �(r-l)v) lies in \13"'. We must also verify that the family { V"} over XK is a family of rational Hodge structures in the sense of [9]. Otherwise it could not possibly be attached to a family of motives. There are two points to be verified. Let v;, q be the subspace of Ve of type p, q and set Vp = E{Jp';;;p Vp', q ' · The space v; £; Vc must be shown to vary holomorphically with x. In other words if v(x) is any local section of v; and Y any antiholomorphic vector field then Yv(x) also takes values in v;. The condition of transversality must also be established, to the effect that for the same v(x) and any holomorphic vector field Y the values of Yv(x) lie in V�1 •
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
217
There is certainly no harm in supposing that v(x) takes values in VP, q· Then one has to show that if X0 (h0, g'j) is fixed and Y is the vector field defined by XE @ then Yv(x0) lies in Vp when X E lJho and in v;_1 when X E f:iho · Let Kho be the stabi lizer of h o in G(R). We represent XK as the quotient G(A)/Kho K and lift v(x) to a function on G(A) which we write as =
The function u takes values in the constant space Vfq· Moreover Yv(xo ) �(X) u( I , g/) + Xu( l , gf) . The second term lies in Vfq for all X E @. If X E lJ then .;(X) Vfq � V�1, q_1 and if X E fj then .;(X) vrq � Vt1 . q+l · If r E G(Q) and x' = rx then v ---+ .;(r) v provides an isomorphism between ( Vx .� .., rpx) and ( Vx', �x', rpx'). Thus, if s E SK(C), any two elements of {(Vx, � .., rp"') I x ---+ s } are canonically isomorphic, and we may take ( V•, �s, rps) to be any one of them, and redefine our family as a family of rational Hodge struc tures, with supplementary data, over the base SK(C). The locally constant sheaf Fe(Q) of rational vector spaces underlying this family is the quotient of VQ x XK by the action r : (v, x) ---+ (.;(r)v, rx) of G(Q). For this quotient to be well defined, the group K must be sufficiently small, for G(Q) n KhK is then contained in the kernel of .; for all h (cf. II.A.2 of [41]). It is here that the condition that .; factors through G0 intervenes. When dealing with motives, one does not need to introduce the polarizations explicitly as part of the moduli problem, but in order to introduce a Hodge struc ture on the cohomology groups of the sheaves Fe(Q) one must verify that on each connected component of SK(C) a locally constant section s ---+ p s can be defined. Let G0(R) be the connected component of G(R) and choose X0 (h, g1) in XK. If =
=
X�
=
{(Ad g o h, gfk) j g E G0(R), k E K}
then the image of X� in SK(C) is open. If x and x' lie in X� and x' rx then r lies in (4.2) All we need do is find a collection P = { Pn } such that P E � .. for all x E X� and Pn (�(r)u, .;(r)v) Pn (u, v) if r lies in the group (4.2). Choose any P in � ..0• Then P E � .. for all x E X�. There are certainly homomorphisms An of G into the general linear group of vn such that =
=
The eigenvalues of A(r) are positive if r lies in G0(R)Kh. Moreover A is trivial on the derived group of G and factors through G0• It therefore follows from the results of II.A.2 of [41] that each An is the identity on all of (4.2). Although we have defined the families ( Vs, �s, rps) for any G, it is clear that SK(C) is not going to appear as a moduli space unless G is equal to G0, and so for the rest of this section we assume this. The moduli problem is best formulated completely in the language of tannakian categories. We can drop the polarizations and retain only the pairs ( Vs, rp5), or ( Vx, rpx), but we now have to emphasize that ( VX, rpx) is
21 8
R.
P . LANGLANDS
defined for every (finite-dimensional) representation � of G over Q, and so we write (�, V(�)) for the representation and ( V"(�), «p"(�)) for the pair ( Vx , «p"). On the category f]£gfJ'(G) of finite-dimensional representations of G we have the natural fibre functor (L)Rep (G) : (�, V(�)) --> V(�)Q and rt : (�. V(�)) --> Vx(�) is a ®-functor from :JllgfJ'(G) to .Yf(!!Ef?(Q) which satisfies (L) Hod o r;x = (L)Rep (Gl · Since .Yf(!!Ef?(Q) and f]£g£37!(GHod) are the same categories, r; x defines a homomorphism [40, 11.3.3.1] «px : GHod --> G and r;x may be defined by (�, V(�)) --> (� o «pX, V(�)). When we emphasize r;X. «p x appears as an isomorphism of two fibre functors However, when we emphasize «pX, as we shall, then these two fibre functors are the same, for they are both obtained from (L)Hod o r;" = (L)Rep (Gl by tensoring with A 1 , and «p x may be interpreted as an isomorphism of (L)�[p CGl · Such an isomorphism is given by a g- 1 E G(A1) [40, 11.3]. This is the g appearing in x = (h, g). Only the coset gK !";:::; G(A1) is well defined. We have arrived, by a rather circuitous route, at the conclusion that XK parame trizes pairs («p, g), «p being a homomorphism from GH od to G defined over Q, and g in G(A1) being specified only up to right multiplication by an element of K. In addition, «p is subject to the following constraint : H. The composition of «p with the canonical homomorphism :Jll --> GH od lies in � If r E G(Q) the pairs («p, g) and (ad r o «p, rg) will be called equivalent. The variety ShK(C) parametrizes equivalence classes of these pairs. One of the important tannakian categories is the category vlf(!! f7(k) of motives over a field k. It cannot be constructed at present unless one assumes certain con jectures in algebraic geometry, referred to as the standard conjectures [40]. It is covariant in k, and rational cohomology together with its Hodge structure yields a ®-functor h8 H : vlf{!!f7(C) --> .Yf(!!Ef?(Q). vlf(!! .oT(C) together with the fibre functor (L)Mot (c) of rational cohomology also defines a group GMot (c) over Q and h8 H is dual to a homomorphism hJH : GHod --> GMot (c) defined over Q. Implicit in Deligne's construction is the hope that any homomorphism «p' : GH od --> G satisfying H is a composite «p' = «p o h�H · According to the Hodge con jecture, «p would be uniquely determined [40, VI.4.5] and ShK(C) would appear as the moduli space for pairs («p, g), with g as before, but where «p is now a homo morphism from GMot Ccl to G defined over Q and satisfying : H'. The composition of «p with the canonical homomorphism fJ£ --> GMot CCJ lies in � This may be so but it will not be a panacea for all the problems with which the study of Shimura varieties is beset. So far as I can see, we do not yet have a moduli problem in the usual algebraic sense, and, in particular, no way of deciding over which field the moduli problem is defined. We can be more specific about this difficulty. Suppose 1: is an automorphism of C. Then 7:-1 defines a ®-functor r;(1:) : vlf{!! .oT(C) --> vlf{!! f7(C). Let GfJot CCJ be the group defined by vlf{!! .oT(C) and the fibre functor (L)Mot CcJ o 7](7:). The dual of 7](7:) is then an isomorphism over Q : «p(7:) : GMot (C) --> GfJot (C) · The homomorphism «p has a dual, a ®-functor r; : fl£
AUTOMORPHIC REPRESENTATIONS , SHIMURA VARIETIES , AND MOTIVES
rp• :
G' · "' , and rp ' = rp' 0 rp(z-) :
GK!o t (C) -->
GMot (C)
219
--> G'· "' ·
Moreover the two fibre functors (l)d61 ccJ and (l)d61 ccl o r;(z-) are canonically isomor phic. As a consequence, there is a canonical isomorphism G(A1) --> G•· "'(A1). Let g' be the image of g. The pair (rp ', g') seems once again to define a solution to our moduli problem. The difficulty is that G'· "' may not be the group G or, even if it is, the composition of rp ' with the canonical homomorphism may not lie in .p. One of the purposes of the next two sections is to discover what G' · "' is likely to be. 5. The Taniyama group. There is one type of Shimura variety which is very easy to study, that obtained when G is a torus T. Then the set .p reduces to a single point {h}. For each open compact subgroup U of T(A1) the manifold Shu(C) con sists of a finite set and Shu = Shu(T, h) is zero-dimensional. In general a special point of (G, ,P) will be a pair (T, h) with T c:;; G and h E .p. If U = K n T(A 1) then Shu(C) is a subset of ShK(C), the points of which have traditional!y been referred to as special points, and I shall continue this usage. But it is best to give priority to the pair (T, h) rather than to the points of ShK(C) it defines. There are a number of un solved problems about Shimura varieties and their special points that I want to describe in the next section. To formulate them some Galois cocycles have to be defined. Deligne has shown me that my original construction gave, in particular, a specific extension of Gal(Q/Q) by the Serre group, Y', an extension I venture to call the Taniyama group and denote by .r. Since the cocycles needed are, as Rapoport observed, often easily defined in terms of .r, I begin by constructing it. The group Y' is an algebraic group over Q, and .r will also be defined over Q. Thus we will have an exact sequence Y' --> .r --> Gal(Q/Q) --> 1 . Recall that X*(Y') i s a module of functions on Gal(Q /Q), and that the Galois action on X*(Y') giving the structure of Y' as a group over Q is defined by right translation. We are still free to use left translation to define an algebraic action of Gal( Q/Q) on Y', and it is this action which is implicit in (5. 1). The extension will not split over Q but it will be provided with canonical splittings over each l-adic field Q�> Gal(Q /Q) --> .r(Q1), which will fit together to give Gal{Q /Q) --> .r(A1). Rather than attempting to work directly with Y', I choose a finite Galois exten sion L of Q, let ff' L be the quotient group of Y' whose lattice of rational characters consists of all functions in X*(.9") invariant under G(Q/L), and define extensions (5. 1)
(5. 2)
I
I
__.
__.
ff'L
__.
.r L
__.
Gal(£ab fQ)
-->
I,
afterwards lifting to Gal(Q /Q), and then passing to the limit. To motivate the construction we suppose that the extension is defined and that there is a section r --> a{r) of _r L --> Gal(Labf Q) with a(z-) E _r L(L). Let a(z- 1 )a(z- 2) = d,1, ,2a(z-1r2) with d,1, ,2 E ff'L(L), and with (5. 3)
Observe that the elements of the Galois group play two different roles. They are first of all elements of a quotient group of _rL , and secondly they are automorph isms of Lab and thus act on _r L(£ab) , since _r L is defined over Q. In the first role
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R . P. LANGLANDS
they will be denoted by
p or a.
We have p(a (1:))
(5.4)
1:,
perhaps with a subscript added, and in the second by
= cp (1:) a(1:) with cp (7:) cp .- (a(1:))
=
E f/'L(L). Certainly p(c.-(7:)) cp (7:) .
In addition (5.5)
Conversely if we have collections {cp(7:)} and {d�1. <'2} satisfying (5. 3), (5.4), and (5.5), we can construct f7L over Q, together with the section a . Any splitting Gal(LabJQ) -+ f7(A1) will be of the form 1: -+ b(1:) a(1:) with b(1:) E f/'L ( A1 (L)). In order that it be a splitting, we must have b(7: 1 )7: 1 (b(7:2))d�!o <'2 = b(1:1 1:2). (5.6) If the b(1:) a(1:) are to lie in f7L(A 1) we must have p(b(1:)) cp (1:) = b(1:). (5.7) Again any collection {b(1:)} satisfying (5.6) and (5.7) defines a splitting, and it is our task to construct {b(1:)}, { cp (7:)}, and {d��o �J · The group !/' is a quotient of GHod and thus is provided with a canonical homo morphism h : � -+ !/'. Over C the group fJ.f is canonically isomorphic to GL(l) x GL(l). Restricting h to the first factor we obtain a coweight fl. of !/', the canonical coweight. If ). E X*(!/') then ()., fl. ) = ).(1). Since f/'L is a quotient of !/', fl. also defines a coweight of f/'L, which for convenience will also be denoted by fl. · If v is any coweight of f/'L and x any invertible element of L or of Aj(L) then x" will be the element of f/'L(L), or f/'L ( A1(L)) , satisfying ).(x") = x for all ). E X*(f/'L). Recall that we have two actions of Gal(L/Q), or Gai(LabfQ), on X*(f/'L) or on X* (f/'L) = Hom (X * (f/'L), Z), the lattice of coweights. That defined by right translation we write v -+ av, and that defined by left translation we write v -+ v1:. Thus ()., afl. ) = ).(a- 1 ) while ()., f1.7:) = ).(1:) , because an inverse intervenes in the action by left translation, and f1.7:- 1 = 7:f1.. Moreover u(x�<) = u (x)u" while 1:(x�<) = x�<� . These aspects of the notation have to be emphasized because at some points our convention of distin guishing between p , a on the one hand, and 1: on the other, fails us. For the study of Shimura varieties, it is best to take Q to be the algebraic closure of Q in C, and we shall do this. Thus we provide ourselves with an extension of the infinite valuation on Q to Q. There is a property of the Weil groups that will play a prominent role in our discussion. Let v be a valuation of Q and hence of Q, and let Q. and Q. be the completions of Q and Q with respect to v . Eventually v will be defined by the inclusion Q s;:; C, and Q. will be R and Q. will be C. In any case, the data provide us with imbeddings (5.8)
Gal(FvfQ.) 4 Gal(F/Q), if F is any finite Galois extension of Q in Q. The local and global Weil groups WF,to, and WFI" are defined as extensions (5.9)
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
and 1
� CF � WFtQ � Gal(F/Q) �
221
1.
We may imbed the arrows (5. 8) and (5.9) in a commutative diagram 1 1
� �
Fvx
l
CF
�
WF,tQ.
� Gal(F./Q.) � 1
l � � WFtQ IF
l
Gal(F/Q)
�1
Moreover we may so choose the central arrows that they are compatible with field extensions and upon passage to the limit yield I: WQ. � WQ. It is the image of WQ. in WQ that will be fixed, and IF may be changed to w � xiF(w)x-1 where x E CF and xa(x)-1 E F{' for all a E Gal(F./Q.). N ow let v be the valuation given by Q � C. Let F/:, = IT wl vF� . The natural map F/:, � CF is an imbedding, and we sometimes regard F/:, as a subgroup of CF · If we take an element 1: of Gal(Q /Q), lift to WQ, and then project to WuQ we obtain an element w = w(1:) of WuQ which is well defined modulo the connected component, and in particular modulo the closure of L/:,. We choose a set of representatives w,., a E Gal(L/Q), for the cosets of CL in WuQ in such a way that the following conditions are satisfied : (a) w 1 = 1 . (b) If a E Gal(LvfQ.) then w,. E wL.tQ. · (c) If p E Gal(L/Q) and a E Gal(LvfQ.) then WpWu = ap, 11 Wpu with ap.u E L;,. To arrange the final condition we may choose a collection f of representatives r; for the cosets Gal(L/Q)/Gal(L./Q.) and set w'l ,. = w'l w,. if a E Gal(Lv fQ.). We suppose that f contains 1 . With this choice we also have : (d) If {ap.u} is the cocycle defined by wpwu = ap.uWpu then a71,p = 1 for 7J E f and a E Ga1(L.IQ.). If w E WLIQ• let w,.w = c,.(w) w,. , cu ( w) E CL . If w = w(1:) we set bo(7:)
=
n uEGai (L/Q)
c,.(w)U." .
It lies in CL ® X* (9'L), but is not well defined, because w is not. However we can show that it is well defined if taken modulo L(;, ® X* (9'L), and that, in addition, it pehaves properly under extensions of the field L. The ambiguity in w now has no effect, for we are only free to replace w by uw where u = limnun and un lies in the image of L;, U, where U is a subgroup of the group of units of L defined by a strong congruence condition. But [41, II.A.2], n Q a(v)111' = 1 uEGai (Q/ )
for all v E U. Consequently I] cu Cuw)111'
=
(l)a(u)111')(l) cu Cw)111')
is congruent modulo L;, ® X* (9'L) to IT ,. cu(w)111'. Suppose the representatives w,. are replaced by e,.w,., and, hence, ap.u by
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R. P. LANGLANDS
a�, a = epp(ea)e";"Jap.a· If (1 e Gal(L./Q.) then e17 e L'{; and p(e11) e L;,. Since ap,a and a;, a must then both be in L�, we infer that eP = epa (mod L�) when (1 e Gal(L.JQ.). Moreover ca(w) is replaced by c;(w) = ene;;-1c(w) and b0(1:) by
b(kt:)
=
{ l}e:�e;;-17�'}bo(7:) .
The factor may be written Since this change has no effect on b0(1:). In this argument we have denoted the image in Gal(L/Q) of 1: e Gal(Q/Q) by the same symbol, a practice we shall continue to indulge in. If we modify I then w is replaced by xwx- 1 with x e CL and xa(x)- 1 e L: for all a e Gal(L./Q). Then ca(xw.x-l) = a(x)a7:(x-l)ca(w) and ITa(x)-a�'a7:(x)a" = fl a(x)t7C.-'- 1J" .
17
17
Since a(x) = x (mod L;,) if a e Gal(LvfQ.), the same argument as before shows that b0(1:) is unchanged. Finally suppose that L £ L'. Then b (1:) e CL ® X*(,SP L ') and b0(1:) e CL ® X*(,SPL) are both defined, and we must verify that is taken to b0(1:) by the canonical mapping of the first group to the second. Either Lv = R or L. = C, and the two cases must be treated separately. Suppose first that L. = C and hence that Gal(L'/L) n Gal(L�/Q.) = { I }. Since both b0(1:) and b0(1:) are independent of the choices of coset representatives, we may choose those which make it easiest to verify that b0(1:) is the image of b0(1:). Let e be a set of representatives for the cosets Gal(L'/L)\Gal(L'/Q)/Gal(L�/Q.) containing 1 . Every element a of Gal(L'/Q) may be written uniquely as a product (1 = '7Jp, ' e Gal(L'JL), 7J e e , p e Gal(L�/Q.). We may suppose that w; = w�w�w� with w� = 1 and w� e wL�IQv " Let w' be w(1:) with respect to L', and w be w(7:) with respect to L. Then under the canonical map 1r : Wu1g -> Wug the element w' maps to w. If a e Gal(L/Q) it lifts to a unique element of Gal(L' /Q) of the form 7JP · We suppose that W17 is the image of w�w;. Thus if
0
with d7J.p(w') e WL 'IL then ca(w) Gal(L' /Q) then
=
b0(7:)
7r(d7J.p(w')) . On the other hand if a 1
wcd7].p(w')
=
ca,(w')wc.
Consequently, by the very definition of 1r, ca(w)
=
IT C171(w') .
0'!-fl
=
'7JP lies in
AUTOMORPHIC REPRESENTATIONS , SHIMURA VARIETIES , AND MOTIVES
223
It follows immediately that b0(i) is the image of b�('r). If L. = R then X * (sPL) � Z and the Galois group acts trivially. Suppose we replace W11 by e11W11 with e11 E CL . Then cu(w) is replaced by e11e-;;/. Since I1(e11e;,.1 ) 11P u
=
I1(e11e;,.1 )P u
=
I,
this has no effect on b0{'r), and when defining b0(-r) we need not suppose that the collection { w11} is subject to the constraints (a), (b), and (c). If we want to define b�(-r), we may still need to be careful about the choice of the coset representatives w�, a E Gal(L'/Q). However, since we are only interested in the image of b�(-r) in g>L, we may again ignore (a), (b), and (c). We choose a set e of representatives for the cosets Gal(L'/L)\Gal{L'/Q), write a = p'Y), p E Gal(L' /L), 'YJ E e, and take w� = w�w�. If a E Gal(L/Q) is the image of 'YJ• we take W11 = n-(w�). The argument can now proceed as before. Let b(-r) be a lift of b0(-r) to h ® XAYL) = g>L (A(L)) and let b(-r) be the projec tion of b (-r) on g>L ( A 1(L) ) . The element b (-r) is well defined modulo g>L(L) and, as we shall see, this bit of ambiguity will cause us no difficulty. But we have to fix one choice. The first point to verify is that d�1 o �2
=
b (-rl )-r l (b (-rz) ) b(-rl -rz) - 1
lies in g>L(L). When verifying this, we may choose the liftings b(-r1), b(-r2), and b(-r1-r2) in any way we like. We choose liftings cu(w1) and c11(w2) of cu(w1) and C11(w2) to h and take b(-r1)
Since cu{w1w2)
=
=
IT cu (w l ) uP , u
b(-rz)
=
ITciT(wz) uP. u
cu (w1ku�1 (w2), we may take cu {w1w2) to be cu(w1)cu�Jw2). Because -r:t1(b(-rz)) = ITcu (wz)u';- 1�' = ITcuq(wz) uP, u u
the element dq, ,2 will then be I . Finally we have to establish that the elements cp (-r) defined by equation (5.7) lie in g>L(L), for equations (5.4) and (5. 5) will then follow immediately. It will suffice to show that for any w E WuQ and any p E Gal(L/Q) ( 5 . 1 0)
lies in L(;, ® X* (YL). Suppose w = w 1 w2 and w1 projects to -r1 E Gal(L/Q). Then C11(w) = cu(w1)cu�1(w2) and ( 5. 1 0) is equal to
{ I] cu(wl)uPp (cu(wl))-pup}-r1 { I] c11(Wz)uPp (cu(wz))-pup}·
Consequently we need only verify that (5 . 1 0) lies in L/;, ® X* (YL) for w in CL and for w = w�. If w lies in CL then cu(w) = a(w) and Il ua(w)up = Il upa(w)Pul'. The expression ( 5. 1 0) is therefore equal to I . If w = w� then cu (w) = aM and (5. 1 1) However it follows from condition (c) that ap . u = ap, 11, (mod L�). Since (1 + c) · (I - c 1 )p = 0, the right side of (5. 1 1) lies in L;;, ® X (sPL). *
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R. P. LANGLANDS
Since b(r), although defined for -r E Gal(Q/Q), depends only on the image of -r in Gal(La bfQ), the groups f7 L and f7 are now completely defined. The ambiguity in the b(-r) is easily seen to correspond to the ambiguity in the choice of the section a(-r). If E £ Q is any finite extension of Q, we let f7E be the inverse image of Gal(Q/E) in !7. If E £ La b we may also introduce f7k. The group !7f has been introduced by Serre [41], who uses it to formulate some ideas of Taniyama. He makes its arith metic significance quite clear, but his definition is sufficiently different from that given here that an explanation of the reasons for their equivalence is in order. If X is an idele of L then cp(x) = rr Gai (L/Q) a(x)ap is an element of 9"L (A(L)) . By 11.4 of [41] there is an open subgroup U of the group of ideles h such that cp(x) = I if X E u n LX The standard map of IL onto Gal(LabfL) restricts to u and if 'C = "C(x) is the image of x we may take w = w('C) to be the image of x in CL . Then cp(x) is a lifting of b0('C) to 9"L (A (L) ) . However cp(x) depends only on '"· and thus we may take b('C) = cp(x). Then d,h '<2 = I and cp('C) = I if 'C, '"h "C2 lie in the image Gal(LabIF) of U. Here F is the finite extension of L defined by U. The elements cp('C) are in fact 1 for all '" E Gal(LabfL). If we choose a set of representatives e for Gal(LabfF)\Gal(LabfL) and set b(n;) = b(-r)b(r;), '" E Gal(£ab/F), 7J E e, then, in general, d•1. •z will depend only on the images of 'Cb -r2 in Gal(F/L), and !7f may be ob tained by pulling back an extension 0
1
->
9"L
->
f7u
->
Gal(F/L)
->
I
to Gal(LabfL). Here f7 u is the quotient of !7f by the normal subgroup {a("C) i-r E Gal (La b{ F)}. It is the extension f7u that Serre defines directly. He denotes it by the symbol Sm . The map ¢ defines a homomorphism of LX/LX n u � LX U/LX into 9"L(L) and to verify that f7u is the group studied by Serre, we have only to verify that it can be imbedded in a commutative diagram
The right-hand arrow is x -> "C(x)-1. We do not have to pass to the quotient but may define the homomorphism h/LX -> !7 f(L) (5. I 2) directly. The central arrow is then obtained by composing with the projection !7f(L) -> f7u(L) . The homomorphism (5. 1 2) is x
->
{ I] a(x)ap} b('C)-1a('C)-1
ih is the image of x in Gal(LabfL). The composition of our splitting Gal(La bf£) -> !7 f(Q1) with !7f(Q1) -> f7 u (Q1) is either Serre's e1 or its inverse, presumably its inverse, for we are so arranging matters that the eigenvalues of the Frobenius ele ments acting on the cohomology of algebraic varieties are greater than or equal to 1 . The cocycle p -> cp('C) certainly becomes trivial at every finite place, but i s not necessarily trivial at the infinite place, v . Indeed under the isomorphism
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
225
H l (Gal(L.JQ.), ffL(L.)) � H- l (Gal(L./Q.), X* (ffL)) given by the Tate-Nakayama duality it corresponds to the element of the group on the right represented by(l - 7:-l)p. If, as has been our custom, we denote the image of 1: in Gal(L/Q) again by 1: we may suppose that w(1:) = w., for the class of { cp{1:)} depends only on thi� image. According to the discussion of formula (5. 1 1) IT ap,puu<�- 1-l) p = IT (aP• 7J' a-l p, 7J) fl11 (.- 1-l) p = eP (7:) U 7) E f
lies in L;, ® X* (ffL) = ffL(L00). By definition, the classes of {cp(7:)} and {ep(7:)} are inverse to one another in Hl(Gal(L/Q), ffL(A(L)) ). Thus all we need do is calculate the projection of ep(7:) on ffL(L.) for p E Gal(L./Q.). Observe first of all that if we agree to choose coset representatives satisfying (d), then a P • 7J' ap,� = p(a;,�) a P7J · ' = ap 71, , . Again if P'YJ = 1J1p1 then a P 7J • ' = ClqiPb t = 1J l (a Pb ,) a 7JI, Pit a;.� PI = 'YJ l (a Pb ,).
Since aPb , E L�, the term on the right has a projection on L� different from 1 only if 'YJ l = 1 . Then 1J too equals 1 , and so the projection of ep(1:) on ffL(L.) is
in conformity with our assertion. Suppose T is a torus over Q, provided with a coweight It such that (5. 1 3)
(1 + c) (1: - l)p
=
(1: - 1)(1 + c)p
= 0
for all 1: E Gal(Q/Q). Then there exists a unique homomorphism cjJ : !7 --. T such that the composition of cjJ with the canonical coweight of !7 is It · We can transport the cocycles p --. cp(1:) from !7 to T, obtaining cocycles {cp{7:, p)} as well as b(1:, p) E T(A1(L)) . However if T and h : � --. T define a Shimura variety and It is the restric tion of h to the first factor of � the condition (5. 1 3) will not necessarily be satisfied. Nonetheless, we may repeat the previous construction and define b(1:, p) and {cp(7:, p)} for all 1: such that (1 + c)(1:- 1 - l)p = 0. This generalization is necessary for the treatment of those Shimura varieties Sh{G, h) for which G is not equal to G0• It should perhaps be observed that b(1:, p) is not insensitive to the ambiguity in the choice of w = w(1:), although { cp{7:, p)} is. However, if Z is the centre of G the ambiguity all lies in Z(A1) n K, and may be ignored. If E £ C the motives over E whose associated Hodge structure is of CM type are themselves said to be of CM type. They form a tannakian category CM( E) with a natural fibre functor wcM cE» given by rational cohomology. Since one expects that the natural functor CM(Q) --. CM(C) is an equivalence, we may as well sup pose that E £ Q. According to the hopes expressed at the end of the previous section there should be an equivalence 1J : CM(Q) --. �tff[l}J(ff) and an isomorphism WRep (9') o 7J --. WcM (Ql • which would enable us to identify !7 with the group GcMCQl • defined by the category CM(Q} and the functor c.ocMCQl . If E £ Q, one hopes that in the same way it will be possible to identify ffE with GcM(El · There are properties which this identification should have, and it is necessary to describe them explicitly. First of all, ifF £ E the diagram
226
R. [I'
I
--->
P. LANGLANDS §"
1
E
--->
§" F
1
GcM CQJ ---> GcMCEJ ---> GcMCFJ
should be commutative. The other properties are more complicated to describe. Suppose -r E Gal(Q/Q) and -r takes E to E'. lts inverse then naturally defines a ®-functor 7](-r) from CM(E') to CM(E). Let GtMCE' ) be the group defined by the category CM(E') and the functor WcMCEJ o 7](-r). The dual of 7](-r) is then an isomorphism �(-r) : GcMCEJ � GtMCE') · In terms of representations, 7](-r) associates to every representation (.;', V(.;')) of GcM CE'J a representation (�, V(�)) of GcMCEJ · Since WcMCE'J and wcM CEJ o 7](-r) become isomorphic over Q there is a family of homomorphisms, one for each e, ¢(�') : V(e)Q --> V(�)(l, compatible with sums and tensor products. If ¢'(�) is another possible family then there is a t E !YE'(Q), such that cjJ'(e) = cjJ(e)�'(t). Finally since the two functors we"� CE ' ) and we"� CEJ o 7](-r) are canonically isomorphic, arising as they do from the /-adic cohomology, there is a canonical family of isomorphisms ¢4�') : V(�')At --> V(�)Ar On the other hand, suppose a(-r) E !Y(Q) maps to -.. Let a(a(-r)) = ca(-r)a(-r). We may use a(-r) to associate to every representation (.;', V(�')) of !YE' a representation (�, V(�)) of !YE· The space V(�) is obtained by twisting V(�') by the cocycle {�'(cu{-r)- 1 )}. The representation � is t --> e(a(-r)ta(-r)-1 ) . This functor (�', V(�')) --> (�, V(�)) is to be (isomorphic to) that obtained from 7](-r) by identifying !Y E and GcMCEJ and !Y E ' and GcMCE'J · Moreover one possible choice for ¢(�') is to be the isomorphism ¢0(�') : V(e)Q --> V(�)Q implicit in the definition of V(�). If b(-r)a(-r) is the image of -r under the canonical splitting Gal(Q/Q) __:. !Y(A1), then ¢A/e) is to be ¢0(�') o �'(b(-r)) - 1 . If -r E Gal(Q/E) then E' = E and 7](-r) : CM(E') --> CM(E) is the identity functor. Thus the functor (.;', V(e)) --> (�, V(�)) from Rep(TE,) = Rep(TE) to Rep(TE) must be canonically isomorphic to the identity. A final property, which seems to be independent of the preceding ones, is that this isomorphism should be given by �, --> � and ¢0(�') o �'(a(-r)) : V(e) --> V(�). Observe that a(-r) is now in !YE and these transformations are defined over Q. I assume that all this is so, just to see where it leads, and especially to see what it suggests about the groups Gr,rp introduced in the previous section. But there are some lemmas to be verified first. I conclude the present section by describing a property of the Taniyama group whose significance was pointed out to me by Casselman. It will be needed to show that the zeta-functions of motives, and especially abelian varieties, of CM type can be expressed as products of the L functions associated to representations of the Weil group. The point is that there is a natural homomorphism � of the Weil group WQ of Q into !Y(C) and thus for any finite extension F of Q a homomorphism �F : WF --> !YF(C). To define it we work at a finite level, defining WuQ --> ;YL(C), and after wards passing to the limit. Fix for now a set of coset representatives w,. which satisfies (a), (b), and (c). If w E WuQ we define b0(w) = fluE Gai CL!Q) c,.(w) 11�'. If -r is the image of w in Gal(LabjQ) then b0(w) = b0(-r) (mod L{;, ® X *(Y'L)), and we may lift b0(w) to b(w) in Y'L(A(L)) in such a way that the projection of b(w) in Y'L(A1(L)) is b(-r). However a simple
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
227
calculation shows that if..- I> z-2 are the images of wi > w2 then b(w1 )-r 1 (b(w2))b(w1 w2)- 1 E f/L(L). Since its projection on SL(Aj(L)) is equal to dr�> rz• it is itself equal to dr�o rz· If b.(w) is the projection of b(w) in f/L(L.) = f/L(C), we may define rp by rp : w � b.(w)a(z-) . If the coset representatives W17 are changed then rp is replaced by rp ' = ad a o rp, a E f/L(C), but this is of no importance. If GwF is the group over C defined by the tannakian category of contiauous, finite-dimensional, complex, semisimple representations of WF then lfJF is the com posite of the imbedding Wp � GwiC) and an algebraic homomorphism cpp : GwF � !TF· Moreover if the principle of functoriality is valid, there is a surjection GucFJ � GwF and we can expect to have a diagram GU(F)
1
PF
__,
GwF ---;;;-;-'
whose two composite arrows differ by ad s, s e f/(C) s;;; !Tp(C). 6. Conjugation of Shimura varieties. The principal purpose of this section is to formulate a conjecture about the conjugation of Shimura varieties, a conjecture whose first justification is that it is a simple statement which implies what we need for the study of the zeta-functions at archimedean places and is compatible with all that we know. Some lemmas are necessary before it can be stated, and we shall see that these lemmas together with the hypothetical properties of the Taniyama group suggest an answer to the question that arose at the end of the fourth section. This answer in its turn throws new light on the conjecture, so that we can weave a consistent pattern of hypotheses, and our task will be ultimately to show that it has some real validity. We need a construction, which we make in sufficient generality that it applies to all Shimura varieties and not just those associated to motives. Suppose the pair (G, ,))) defines a Shimura variety, T and f' are two Cartan subgroups of G defined over Q, and h : f!ll � T, h � f!ll � f' both lie in .)). Let f-t and f1 be the coweights of T and f' obtained by restricting h and h to the first factor of f!/l, and choose a finite Galois extension L which splits T and f'. Let -r E Gal(Q/Q) ; then the coweights (1 + c )(z-- 1 - 1)p and (1 + c ) (z-- 1 - 1)(1 are both central and they are equal. Choose w = w(z-) as before, and set b o(Z", p) = n c�(w)""�' , bo(z- , fl ) = n cu(w)"".U. 17
17
Let b (z- , p) and b(-r, fl ) be liftings to T(A(L)) and T(A(L)) and let B(-r) = B(z- , f-t• fl) be the projection of b(-r, fl)- 1 b(z- , p) on G(Aj(L)) . Although we may not be able to define the cocycle {cp (-r , p) } , we can define { cp(-r, /-tad)} if /-tad is the composition of f-t with the projection to the adjoint group. It may be as well to check that B(z-) is indeed independent of the choice of w and of the coset representatives w.,., provided the usual conditions (a), (b), and (c) are satisfied. If w.,. is replaced by e.,.w.,. then, apart from a factor in L'{;, ® X*(T), b0(-r, p) is multiplied by n 1)E f e�Cl+c) (r-1-ll l', and bo(Z", fl) by n 1)E f e�Cl+c) (r- 1-l)Ji. Since these two terms are central and equal by assumption, the change has no effect on B(z-). If w is replaced by xwx- 1 with xa(x- 1 ) e L{; for a e Gal(L.IQ.), then b0(-r, p) is modified
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R. P. LANGLANDS
by the product of an element in L/:o @ X* ( T) and TI 'I x'� CHtl C<- 1-lJ ,u r: . Since b0(r:, fl) undergoes a similar modification, B(r:) is not affected. One can also show easily that B(r:) is not changed when L is enlarged ; the argument is once again basically the same as that used to treat b(r:). B(r: ) does depend on the choice of b(r:, f.J.) and b(r: , f1). These choices made, we will use them consistently to define cp(7:, f.l.ad), cp(r:, /1ad), and, when (1 + c)(r:- 1 - l ) f.J. = (1 + c)(c 1 - I)fl = 0, cp(r:, f.J.), cp(r:, p ). Thus cp(r: , f.l.ad) is to be the proj ection of p (b(r:, f.J.)-1) b (r:, f.J.) in G.d (Aj(L)) . With these conventions, the ambiguity in B(r:) will cause no harm in the construction to be given next. Let G< . .u and G'· ii be the groups obtained from G by twisting with the cocycles { cp(r:, f.l.ad )-1 } and { cp(r: , flad)-1 } . We are going to verify the following : FIRST LEMMA OF COMPARISON. (i) If Cp = cp(7:, f.i.ad ) then rP = B(r:) ad cp-1 (p(B(r:)- 1 ))
lies in G< . .u(L). (ii) The cocycle {rp} in G< . .u(L) bounds. (iii) If ( I + c)(r:- 1 - l )f.J. = ( I + c)(r;- 1 - l)p = 0 then 7P =
cp(r:, p)-1 cp(r:, f.J.).
The third assertion is clear ; it is the other two with which we must deal. The element r p can be obtained by projecting (6. 1) on G(A1(L)) . This makes it perfectly clear that rP is not affected if w = w("t"j-is--re· placed by xw(r:) with x E CL . Thus we may assume that w = wn where 1: is here also used to denote the image of 1: in Gai(L/Q). Then we have to show that b(r: , p )- 1 p (b(r:, p ))
=
b(r:, f.J.)- 1 p (b(r: , f.J.)) (mod G(L"') G(L)) .
It follows easily from (5. 1 1) that if iiP· 'I is a lift of aP·'I to congruent to
h,
then both sides are
The desired equality follows. To prove the second assertion we shall apply Hasse's principle, but for this we need a group G whose derived group Gde r is simply connected. Let Gsc be the simply connected covering of Gde r · The Cartan subgroup T defines Tder and Tsc · We have an imbedding X* (T.J -> X* (T) and Gde r = G.c if and only if the quotient is torsion free. If we can construct a diagram of Gal(L/Q)-modules
in which P is torsion-free and P -> M is a surjection whose kernel is a free
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
229
Gal(L/Q)-module, then we can use it to define a central extension G' of G with --> G(R) will be surjective. To construct the diagram, we choose an exact sequence of Gal(L/Q)-modules X*(T') = Q. We will have Gde r = G�c = G,0 and G'(R)
0 -> N -> P -> M -> 0,
with P torsion-free and N free. Then Q is the set of all (x, p) in X*(T) EEl P for which x and p have the same image in M. We lift ,u to ,u' = (,u, v) with v E P. If f1 = ad g o ,u and g is the image of g ' , we set p ' = ad g' o ,u'. If the assertion is valid for ,u', p ' , and G', it is valid for ,u, p, and G. Consequently we may suppose that Gde r is simply-connected. There are two types of ambiguity in B('r). lt can be changed to tB('r) with t E t(L). Then {rp} is replaced by {(trp ad cp-1 (p(t)- 1 )}, and its cohomology class is not affected. We can also change B('r) to B('r)t with t E T(L) . Then { cp} is replaced by { c�} with c� = p(t;l)cptad • tad being the image of t in Gad • and {rp} is replaced by {r�}, with r� = rp ad cp-1 (p(t- 1))t. o ad c;1 (p(o- 1 )) then r� = (ot)ad- 1 c� (p(ot)- 1 ) . Consequently the ambiguity i n B(z) has n o effect o n the assertion (ii). Rather than prove (ii) directly for a given choice of the pair ,u. p, we want to prove it for a succession of pairs. For this one should first check that the validity of (ii) defines an equivalence relation. If ,u and f1 are interchanged then {r p} is replaced by {r;1 }, and if rp = o ad cp-1 (p(o- 1 )) then r;1 = o- 1 ad cp-1 (p(o)) . To show the transitivity, we introduce a new notation, denoting rP by rp(fl , ,u), and cP by cp(,t.t). Suppose rp(,uz , ,t.t3) = t ad c;1 (,t.t3) (p(r - 1 )), rp(,t.tr . ,t.tz) = s ad cp-1 (,u z)(p(s - 1 )) .
If rP
=
Observing that rp(,t.tz , ,t.t3) ad cp-1 (,t.t3)(p(s- 1 ))rp(,t.tz, ,t.t3)- 1 = ad cp-1 (,u2)(p(s- 1 )) and that rp(,u r . ,u3) = rp(,t.t r . ,u2)rp(,u2, ,u3), one deduces with little effort that r/,u r . ,t.t3) = st ad cp(,u3)(p(st)- 1 ) . Transitivity established, we return to the original notation. If f1 i s conjugate to ,u under G(Q), say f1 = ad x o ,u then rP = x ad cp- 1 (x- 1 ) = x ad c;1 (p(x- 1 )),
and certainly bounds. In general f1 and ,u are not conjugate under G(Q), but they are conjugate under G(R). Since G(Q) is dense in G(R) we may take advantage of the transitivity and assume that they are conjugate under Gde r(R). It is now that the assumption that Gde r is simply connected intervenes. If we are
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R. P. LANGLANDS
careful in our choice of b(z-, p.) and b(z-, p.), defining them by liftings of cu(w) to h, then B(z-) and the TP will lie in GJ.fr· Moreover the cocycle { rp} in GJ.!r(L) certainly bounds at every finite place. Since we are applying Hasse's principle, we need only verify that it bounds at infinity as well. One begins with a calculation similar to the one made while studying the cocycle {cp(t·)}. If p E Gal(L/Q), set and define ep(-c, p.) in a similar fashion. The projection of { rp} on G�·�'(L,.,} is co homologous to ep(z-, p.)ep(z-, p.)-1. If p E Gal(Lv/Qv) the projection of ep(z-, p.) on G�·�'(Lv) is Thus all we need do is show that/p('t", p.)/p('t", p.)-1 bounds in Gr·P(Lv)· Recall that the cocycle defining G�·P(Lv) is For brevity, we write oio. ar,�} = {ap(v)}. If X E G(R) we write x(p.) = ad X
x (p.) = p. then
0
P. ·
If
x/p(z-, p.) f,(z-, p.)-1 ad hp(p(x-1)) = ap((xz-- 1 ,x- 1 - z- -1)p.), because p(x) = x and fp(z-, p.)-1 ad hp(x-1) = x-1 /p(z-, p.)-1. If w lies in the normalizer of Tde r in Gder(Lv) then (6.2) However {wp(w-1)} is a cohomology class in Tde r(Lv) or TJ
(6.3) wp. = z--1p.. Then (1 + e)(z--1 - l)p. = 0 and {c;1(z-)} = {c;1(z-, p.)} is defined. It bounds in G(L). Once again it is enough to verify this when Gder is simply connected, although this time the modifications made to arrive at a simply-connected derived group would be different . It is no longer important that P � M be surjective, or that its kernel be free, but there must be a v e P which maps to p. and is fixed by Gal(Q/E). The set of all z- which satisfy (6.3) for at least one w form a group. Let E = E(G, h) = E(G, ,P) be its fixed field. Then E £ L. In order to apply Hasse's principle we have to arrange that {cp(z-)} lies in Gde r(L) and that it obviously bounds in Gde r (A1(L)) . Since cp(z-) only depends on the image of z- in Gal(L/Q), we may cal culate it when w = w� . If @i is a set of coset representatives for Gal(L/Q)/Gal(L/E), then
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
231
We write ava. r = v(au. �)av.a � a;, � and vap. = vp. + v(a - l)p.. Now for a e Gal(L/E), v(a - l)p. is a weight of the derived group and f L Ti a a��;-I J .u may be lifted to Tde r (A(L)) . On the other hand Ti v Ti a a�;,� a;,"f = 1 . If a = Ti a aa.r then a lies in CE and lifts to a' in IE. Moreover Ti v v(a)v.u lifts to Ti v ))(a') v.u which lies in T(A). If iivtM is a lifting of ava . r to IL then we so choose cp(-c) that (6.4) modulo T(L"'). It remains to verify that the {cp-1('c)} so defined bounds at the infinite place. The projection of the right side of (6.4) on CL ® X*(Tde r) is IT pvacr -1 -I J .u IT aiXICD'� -1 -IJ .u v, D' ap, va • aEGa!(LIQ) p, Thus, restricting {cp-1 (z-)} to Gal(L/Q.) and projecting on TdelL.) , we obtain a class cohomologous to {ap ((z--1 - I)p.)} = {ap ((c.o - l)p.) } . We have observed already that it is shown in [45] that the right side bounds in GdelL.) . Although there is one more consequence to be derived from (6.3), there are some things that must first be said about the general (T, h), or (T, p.). We drop the assumption (6.3) for a while, and return to it later. We have as sociated to the pair (T, p.) and z- a twisted form a� ... of G. The twisting of T in G is trivial, and T� = T is a Cartan subgroup of a� . .u. Let p.� be z-- lp.. There is a unique homomorphism h� : fJll --+ T� whose restriction on the first factor is p.� and which is defined over R. The pair (G� . .u, h�) defines a Shimura variety. The roots { r } of T in G are the same as the roots of P in a� ... . However the classification into compact and noncompact differs for the two pairs. The root r of Tin G is compact or noncompact according as ( - 1 )
•
�(g) : ShK(C) --+ ShK1 (C). It is algebraic and is called a Heeke correspondence. If z- is an automorphism of C, we set Shk(G, h) = Shk = ShK ®.- 1 C, the Shimura varieties being at the moment only defined over C. Then �·(g) : Shk --+ Shh If G = T is a torus and K = U then Shu is 0-dimensional. Let z- also denote the element of Gal(Q/Q) defined by z-. If we set U• = U then
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R. P. LANGLANDS
Shu(C) = T(A1)/ U = T<(A1)j U< = Shu, · This gives us an isomorphism Shu = Shu(T, h) ---> Shu = Shu,(T<, h<). In addition there is the natural map r from complex points of Sh&(T, h) to complex points of Shu(T, h). Define rpr = rpr(U, T, h) by the commutativity of the diagram Shu(T, h)
,
Shb(T, h) / �� r Shu,(T , h<)
In general Gr . p and G are different. But G< . p is defined by the cocycle {c;1(r, ,U ad) } and in G.d(A1(L)) Thus
g -> gr = ad b(r, ,Uad)-1(g) defines an isomorphism of G(A1) with G<·P(A1). Conjecture. There is a family of biregular maps rpr = rpr (K, G, h) , K s; G(A1) defined over C, taking Shk- (G, h) to Shx,(G<·P, h<), and rendering the following diagrams commutative : Shu(T, h) 4 Shx(G, h) r rr r (a) Sh[r(T, h) 4 Shk(G, h)
�' 1
1�,
Shu,(T<, h<) 4 Shx,(G<·P,h<) Here U = T(A1) n K and the rpr in the left column is rpr(U, T, h). (b) The conjecture in this form refers to a specific T and a specific h factoring through T. If we choose another pair (f, h) we obtain another group G<· P. and an other collection {tl>r = rpr (K; G, h)} . The conjecture is inadequate as it stands, and must be supplemented by a statement relating rpr and tP r · Observe that there can be at most one family {rpr} satisfying the conditions (a) and (b). We have already associated to the two pairs (T, h) and (f, h) a cocycle { rp } which bounds in G<·P(L). Let rP = u ad c;1 (p(u- 1 )). Then g ---> ugu- l defines an isomorphism of Gr ·�'(Q) with G<· P.(Q) or of G< . P(A1) with G<· P.(A1). Set uK r = uK
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
233
If this statement is true when u is replaced by a v in G(L.) = G(C) which also trivializes { rp} restricted to Gal(L./Q.) then it is true for u. An examination of the proof of the second property of {rp} shows that we can take v to lie in x- 1 wT(L.), x and w being as in that proof. Since x- 1 w(..-- 1 ,u) = ..-- 1 p, we have ad v o h• = lz•. We infer that ad u carries �· to .f)• and hence defines a bijection G• . .u(Q)\�· X G•·I'(At)f K• ----> G•· P(Q)\,P• X G•· P(Af)/"K• and an isomorphism ¢ : ShK.( G• · .u, h•) ----> Sh.uK< (G•· P, h•).
Since u and B(..-) both trivialize { rp} in G• . .u (AJ
----+ ShKr( G•· P' h•) rp
are commutative. The conjecture as it stands certainly implies that the conjugate of a Shimura variety is again a Shimura variety. Together with its supplement, it implies the usual form of Shimura's conjecture [10]. To verify this one applies the Weil criterion [49] for descent of the field of definition. For this we need families of isomorphisms /p : ShHG, h) ----> ShK(G, h) defined for automorphisms p of C over E(G, h) and satisfying f"P = /p/C. Choose a Cartan subgroup T and an h which factors through it. We know that when ..- fixes E(G, h) the cocycle {c;1(..-, ,u)} is defined and bounds in G(L). Let c;1(..-, ,u) = vp(v- 1 ). Then g � vgv- 1 is an isomorphism of G(Q) with G• . .u(Q). Methods which we have already used show easily that The composite ad v o h is conjugate under G•·.u(R) to h•. Consequently ad v defines an isomorphism � � �· and then, as before, we obtain from an isomorphism On the other hand mutativity of
zv
sK(G, h) ____. s.xeo• . .u, h•). = b(..-, ,u)- 1 with z E G• . .u(A1). We define f. by the com
234
R. P. LANGLANDS
I omit the calculations, lengthy but routine, by which it is deduced from the con jecture and its supplement that.f.. does not depend on the choice of T and h and that the cocycle conditionf"P = /p iC is satisfied. Up to now the SK have been taken as varieties over C, but by the criterion for descent we may now define them over E(G, h) in such a way that the .f.. are simply the identity maps. It has to be verified that the models thus obtained are canonical, but the construction is clearly such that only the case that G is a torus T need be considered. Let a be the transfer of w = w, to CE and a' a lifting of a to IE. The proof that in this case cp(r, p.) is trivial shows in fact that we may take it to be 1 and b = b('r, f-l) to be n Gal {L/Q) /Gal {L/E) l.l(a')"l'. We take V to be 1 and Z to be b- 1 = b('r, p.)- 1 . The composition .- 1
Shu(T, h) ----> Shu(T, h)
ff (b)
---->
Shu(T, h)
is then the identity for all 1: fixing E(T, h), and this is just the condition that Sh{T, h) be the canonical model. Suppose E(G, h) � R. If we take the canonical model for ShK then the complex conjugation defines an involution 8 of the complex manifold ShK(C). It is necessary to have a concrete description of this involution in terms of the representation ShK(C)
= G(Q)\.P
X
G(At)f K,
and one purpose of the conjecture is to provide it. Choose some special point (T, h). If E(G, h) � R then we may define b(c, f-1.) and {cp(c, p.)} . However the condition (c) on the coset representatives w" used to define b(c, p.) implies that b(c, p.) = 1 . We may also take cp(c, p.) to be 1 , and then v may be taken to be 1 as well. It follows that h and h' are conjugate in G(R). Since Kh, = Kh, r; : ad g o h' --> ad g o h is a well-defined map of .p to itself. Consequence of the conjecture. The involution 8 may be realized concretely as the mapping (h, g) --> (r;(h), g) of G(Q)\.P x G(A1) /K to itself. Since we are comparing two continuous mappings which commute with the lJ(g), g E G(A1), it is enough to see that they coincide on the point in Su(T', h') represented by (h', 1). Since /, is the identity and v = z = 1 , q;, takes this point to the point in ShK,(G'• I', h') = ShK(G, h) represented again by (h', 1). It follows immediately from condition (a) of the conjecture that c applied to the point re presented by (h', 1) is (h, 1). For each T and f-1. let l.l(f-1.) be the fibre functor from f?lUff!JJ(G) to �CfJ'(G'·�') which takes (�', V(�')) to (�, V(�)) , with � = �', and with V(�) being the space obtained from V(�') by changing the Galois action to (J : X --> e(ca(7:, p.)- 1) fl(X). lf (t, h) is another special point, and if l.l (p., p) is defined by the diagram
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
235
then, according to the first lemma of comparison, the two fibre functors WRep(G<·PJ and WRep (G•·Pl o v{,u, fi) are isomorphic. On the other hand, we associated, at the end of the fourth section, to each pair (q/, g) a group G�· "' and a pair (�', g'). The groups G and G�· "' are associated to the same tannakian category, and there is thus an equivalence of categories v(�) : BU&J(G) --+ BU&J(G�·'�'), determined up to isomorphism. If � factors through the Serre group, then it can be factored through a Cartan subgroup T of G and defines a coweight p of T. The hypothetical properties of the Taniyama group imply that a�, rp may be taken to be a�. ;; with v(�) being v(p). Thus we have a diagram fl,ftff&J (G�·P) • < 11. rpl
� / fJfC&J(G) / " (rp)
fJftff&J(G�. "') which is commutative up to isomorphism of functors. Moreover the two fibre functors WRep (Gr·P) and WRep (G•·�J o v(,u, �) are isomorphic. If we choose an isomor phism between them, We obtain [40, 11.3.3] an isomorphism over Q, G� .rp --+ G�•P, Composing with �· we obtain a homomorphism �" : GMot (C) ---+ a� . P. Since there are so many special points, it is not unreasonable to hope that v(,u, �) exists for all �' and that WRep(G•·P) and wRep (G•·�J o v(,u, �) are always isomorphic. The second lemma of comparison in conjunction with the hypothetical properties of the Taniyama group implies that the composition of �" with the canonical homomorphism fJf --+ GMot(c) lies in s;,� when � factors through the Serre group, and once again we may surmise or hope that this will be so in general. If � � is the biregular map appearing in the conjecture then the composition � � o -.-1 defines a map from the set of complex points on ShK(G, h) to the set of com plex points on ShK.(G� . ,.. , h�). The idea is that �� o -.-1 will take (�, g) to a pair (�", g"), by a process which can be defined within the moduli problem. We have just seen how to obtain �", at least at the hypothetical level at which we are working. To obtain g" we observe that we have two homomorphisms (6.5) One is obtained from the chosen isomorphism over Q by extending scalars to A1. The other is obtained by a lengthy composition. The g from which we start pro vides an isomorphismwtlct
x --+ f (b(-r, ,u)-1)x
of
V(e')
with V(.;). Composing
236
R . P. LANGLANDS
(6.6) and (6.7) we obtain a second isomorphism between the two fibre functors figuring in (6.5). According to general principles, it can be obtained by composing the first with an element (g")- 1 in G� ·�-'(A1) [40, §II]. If one can establish, in some way or another, that the map ¢� : (q>, g) -+ (q>", g") is really defined, then to prove the conjecture and its supplement one will only need to verify that the composite ¢ � o 1: is complex analytic. However our purpose here has been to see how the wheels mesh, not to find the mainspring. 7. Continuous cohomology. If G = G0 then, according to the principles of the fourth section, we should be able to attach to each point of ShK(C) an equivalence class of pairs (q>, g). Here q> is a homomorphism from GMot(c) to G defined over Q and if r; is the associated 0-functor fJU&J(G) -+ .A(!} !7( C), then g defines an isomorphism (J)AJ M
ot (C) 0
'Yl -+ "I
(J)Af
Rep (G) •
In general we have mappings ShK(G, h) -+ ShK 0(G0, h0), and by pulling back we can associate to each point of ShK(C) a pair (q>, g) where g is again in G(A1), but q> now takes GMot (C) to Go. If (.;, V(.;)) is a representation of G0 over Q or, what is the same, a representation of G factoring through G0, then to each point s of ShK(C) we may associate the motive M(s, .;) defined by .; o ¢, together with the isomorphism wt!c, t (c) (M(s, �))
�
V(.;)A1
defined by g. The variety ShK(G, h) should be defined over E = E(G, h) and so should this family of motives. Suppose now that the variety ShK(G, h) is proper. In the best of all possible worlds, one might be able to form the cohomology groups M', 0 � i � 2 dim ShK, of the family M( · , .; ) , which would again be motives, now over E, and thus correspond to representations ai of GMo t (E) · If the formalism of the second section were established, one could compose ai with PF to obtain represen tations pi = ai ® PF of Gn<El · Then the basic problem would simply be to describe the image pi(Gn(E)) . But we do not have all this formalism, and one of the principal reasons for studying Shimura varieties is the hope that by grappling with the specific arithmetic problems they pose we will obtain an insight that will help us with its construction. Informed by the general principles and hypotheses we are attempting to establish, we can try to formulate questions that are, at least in part, tractable and which if answered will confirm or, if the answer is other than expected, perhaps refute these principles. In the present context we can first observe that even if the Mi remained unde fined, the zeta-function Z(z, Mi) can be defined directly in terms of the data at our disposal. It is a product over the places of E, II vZvCz, Mi) . At a nonarchimedean place it can be defined by the /-adic representation of the Galois group on the ith cohomology group of the /-adic sheaf F�(Q1) associated to .;, as in the papers [30] and [34]. Since our principal concern now is with the factors for the archimedean places, we need not enter into details. The field E is contained in C, and the archimedean places are obtained by ap plying automorphisms of C, or of Q, to E. We first define the factor Z0(z, Mi) for the place v given by E � C. We have seen that we can associate to .; a locally con-
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
237
stant sheaf Fe(Q) over ShK(C). Moreover, we have an analytic family of polarized Hodge structures on Fe(C) = Fe(Q) ® C. By a construction of Deligne [12], [50] this defines a Hodge structure on the cohomology groups Hi = Hi (ShK (C) , Fe(C)). In accordance with the ideas of Serre [42] the factor Zv(z, M•) will be defined as L(s, pi) where pi is a representation of the Weil group Wc!Ev on Hi . The Hodge structure on Hi defines a representation of �(R). Since ex = �(R), this can be used to define pi on ex � WctEv If Ev is equal to C, this defines pi completely. If E. = R then to define pi completely we also have to define pi(w), if w is the element of the Weil group which projects to c and has square - 1 . Since ShK is defined over E, c also defines an involution on ShK(C} which we denoted by (). What we need is a map of order two ¢ : (J *F ---+ F,
F = Fe(C),
such that on each fibre .q (J *FsP, q = pPO(s)
---+
psq, p
•
The associated map c* on cohomology takes H P.q to H q , p and we set pi(w) equal to ( - l )Pc*.
To define ¢ we have to assume that the consequence of the conjecture which was described in the previous section is valid. It can be proved directly in several cases. Then () can be obtained by taking the map (h, g) -+ (r;(h), g) and passing to the quotient. Given (h, g), the fibres at the image of (h, g) and (r;(h), g) may both be identified with V(�)c and ¢ is simply the identity map. If we replace the imbedding E � C by z--1 : E -+ C, -r being an automorphism, then the complex manifold ShK(C} is replaced by Shk(C), which we may identitfy by means of 'P r with ShK,(C), the manifold associated to ShK,(Gr ,p, hr). Thus the factor of the zeta-function defined by the place associated to z--1 : E = E(G, h) -+ C can be calculated by replacing G by G� ·�', h by h�, and E by z--1(£) = E(G� ·�', h) with the place defined by its inclusion in C. The space V(�) has also to be twisted by the cocycle {�(c11(-r, ,u)-1)}. The function Z(z, Mi) defined, the immediate problem is to show that it can be expressed as a product of L-functions associated to automorphic representations (7 . 1 )
Z(z, Mi)
= ij L(z - ai, J
ni, ri) .
Here ai e C is a translation, ni is an automorphic representation of some group Hi, and ri is a representation of the L-group LHi. The first step is to decide which Hi, which ni, and which ri intervene in the product. The first step is to use the theory of continuous cohomology to compute the cohomology groups of the sheaves Fe(C) together with their Hodge structure, and thus to compute Zv(z, Mi), v being again defined by E � C. Using this together with an analysis of the L-packets of automorphic representations of G [44], one searches for an identity (7. 1) which is at least valid when both sides are replaced by their factors at v. An example is discussed in detail in [34]. The identity found, it must be verified for the local factors at the other places v' . If v' is an archimedean place, then the theory of continuous cohomology will allow us to compute Zv,(z,Mi) in terms of the automorphic representations of a G� ·P, a group which
238
R. P. LANGLANDS
differs from G by an inner twisting. To make the comparison it will be necessary to have established the principle of functoriality for the pair G ano G � . ,., and to have understood in detail how it manifests itself. This is bound up with the study of L packets and is primarily an analytic problem, for we expect that the trace formula will give us a good purchase on it [23]. At the finite places the identity (7. 1) is difficult to treat as it stands, and for reasons familiar from topology one replaces the left side by (7.2)
Z(z)
=
IT Z(z, Mi) C-I l ' i
modifying the right side accordingly. The right side is then analyzed by the trace formula, at least if there is no ramification or, at worst, a mild sort [8], [14], [30] . I do not see at the moment any general way of dealing with a truly nonabelian situation, although a rather curious method has been discovered by Deligne for treating the group GL(2) rt3]. If there is no ramification, the factor of (7.2) at a finite place can be analyzed by the fixed point formulae of /-adic cohomology. Apart from combinatorial difficul ties [28] the critical factor is to have a reasonably explicit description of the set of geometric points on ShK(G, h) in itp, the algebraic closure of the residue field at a prime p of E, together with the action of the Frobenius on it [32]. This is an idea first applied by Ihara [20], who has since intensively studied the structure of this set for Shimura curves [21], [22]. Not much has been done when there is ramification. The first thing is to analyze in reasonably simple cases the manner in which the variety reduces badly. Some interesting discoveries have been made for curves [8], [14], [15], but higher dimen sional varieties behave in a more complicated manner. However tools are available for studying their reduction, and it is time to begin. None of these steps will be easy to carry out. The study of L-packets is in an em bryonic stage, and even the combinatorial problems will demand considerable ingenuity in their solution [27]. There is still a great deal to be learned from the study of specific examples. The theory of continuous cohomology is itself in its infancy, and my purpose in this section is to draw attention to some problems which arise in the study of Shimura varieties and which the mature theory should resolve. I begin by introducing a representation r of the L-group LG which will play a fundamental role in the discussion. The group LG is a semidirect product LGo x Gal(Q/Q). If T is a Cartan subgroup of G over Q then there is an isomorphism of XiT), the lattice of coweights of T, with X * (Lro), the lattice of weights of Lro, defined up to an element of the Weyl group. In particular if {T, h) is a special point then fl. defines an orbit e in X*(LT0), and e is independent of{T, h) . Let r 0 be the representation of LGo whose set of extreme weights is e. The group Gal(Q/Q) acts on the weights of LTo and preserves the set of dominant weights. The group Gal(Q/E) fixes the set e. Theus Gal(Q/E) fixes the dominant element p.v in e, and we may extend r 0 to LGo x Gal(Q/E) in such a way that Gal{Q/E) acts trivially on the weight space of p.v . The extended representation will also be called r 0 , and we define r to be the induced representation
AUTOMORPHIC REPRESENTATIONS , SHIMURA VARIETIES , AND MOTIVES
r = Ind(LG, LG o
x
239
Gal(Q/E), r 0) .
Led d be the dimension of ShK(G, h). One expects that the function (7.2) will be equal to a product of functions (7.3) L(z - d/2, n, p). Here n is an automorphic representation of one of the groups H attached to G in [33]. There is, in general, an imbedding (/) : LH c... LG and p is a subrepresentation of r o f/J· If w is any place of Q, let rw be the restriction of r to the L-group LGw, which equals LGo x Gal(QwfQw). Implicit in this notation is an imbedding Q !;;;;; Qw. Then the double cosets in Gal(Q/E)\Gal(Q/Q)/Gal(Qw/Qw) parametrize the places of E dividing w, and rw = EBvlw r0, with rv = Ind(LGw, LG o x Gal(Qw/E.), r�) and r�(-r) = r0(u.-ru;1) if O"v is some element in the coset defining v. The repre sentation p will also be a direct sum p = EBvlwPv • and the function (7.3) will be a product IT w iT vlw L(z - d/2, nw, p0). The factor corresponding to the place v is L(z - d/2, 1Cw , p0). Since we shall only be interested in the place v given by E !;;;;; C, we shall write r and p instead of r. and Pv · Moreover we shall write an automorphic representation of G(A) (or of H(A)) as n ® n1, n being a representation of G(R) and n1 of G(A1). Any irreducible representation n of G(R) lies in some L-packet Hrp where f/J is a homomorphism from WetR to LG. If a is the character of We1R obtained by com posing WetR -+ R with the absolute value, let c/JI(n) = a-d 1 2 ® (r o (/) ). Then L(s - d/2, n, r) = L(s, c/JI(n)) . On the other hand, suppose n ® n1 is an automorphic representation of G(A). Let it act on the subspace U ® U1 of the space of automorphic forms. If h e .f) then, according to the principles of continuous cohomology [4], its contribution to the cohomology of ShK(e) with values in F�(C) in dimension i is (7.4) HomKiAigft ® V, U) ® UK1 . Here t is the Lie algebra of Kh and V the dual of V(�). The space U� is the space of vectors in U1 fixed by K. The action of z e ex = �(R) defining the Hodge structure sends f/J ® u to f/J ' ® u with qJ ' (X ® �)
= ({J(r;(z)X ® � (h(z-l)) v) .
Here X -+ r;(z)X is the action which multiplies the exterior product of p holomor phic and q antiholomorphic vectors by z-P :z-q . Thus r;(z) = ad h(z- 1 ) o r;(z) and qJ ' (X® v) = n(h(z-l))({J(r;(z)X ® v) . If E !;;;;; R we may extend this action of ex to an action of WeiR = We/E.. Let n e G(R) be such that ad n o h = h'. Then the element of w which projects to t and has square - 1 sends f/J ® u to f/J ' ® u with qJ ' (X ® v)
= n(n)qJ (ad n-l(X) ® �(n- l)v) .
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R. P. LANGLANDS
In either case the representation of Wc;E. on (7.4) factors as cji(n:) ® I , where cji(n:) acts on HomKh(Ai@jf ® V, U). Let ¢2(n:) be the element in the representa tion ring of Wcm defined by ¢ z(n;) = Efl( - l ) i
Ind( Wc/R • Wc!Ev• cf;•"(n:)) . Let m(n:1 , K) be the dimension o f Ulf. If for all n: we had ¢ z( n:) = m(n:)cf; 1(n:) (7.5) we could expect a relation K (7.6) Z(z) = n L(z - d/2, n:, r ) m (1Coo) m (1Cj, ) . "
Here, as a single exception, we have taken n: to be a representation of G(A), its com ponent at infinity being denoted n:"', and r to be the representation of LG. However (7.5) is not always valid, and it is the true form of the relation between ¢1(n:) and ¢2(n:) that we must discover, for it is the clue to the correct expression of(7.2) as a product of £-functions associated to automorphic representations. Let U(t;) = {n:h · · · , n:r} be the set of discrete series representations with the same central and infinitesimal characters as �- Then U(t;) = n is an £-packet U'P and the representations ¢1 (n:,) are all equal. We denote them by ¢1(0) . The continuous cohomology of the representations n:; is completely understood [4], and it is a simple exercise to prove the following lemma. LEMMA. EBi=1 ¢ z (n:j) = ( - I ) d¢1(0) .
Thus, in this case, the relation (7.5) fails when the £-packet has more than one element. In order to correct (7.6) one has to replace r by a subrepresentation. However r is in general irreducible as a representation of LG, and so we have to introduce the groups L H of [33], and begin the study of £-indistinguishability. If we accept £-indistinguishability, but expect no other difficulties with the cor rection of (7.6), then we have to be prepared to prove that every irreducible com ponent of ¢2(n:) is a component of ¢1(n:) . But we will again be deceived. There is another difficulty. It appears already in the simplest of the examples considered by Casselman [6] and Milne [36], although they had no occasion to draw attention to it. Suppose G is the group associated to a quaternion algebra over Q which is split at infinity but not at p . Let n:1 = n: P ® n;P, and suppose n: ® n:1 is trivial on the centre, n:P is one dimensional and trivial on the maximal compact subgroup Kp of G(Qp), and n: is either one-dimensional or the first element of the discrete series. r; is taken to be trivial. If K = KPKp then L(s - 1-, n: ® n:1 , r) should appear in the zeta-function Z(s, ShK) with the exponent ± m(n:P, KP) . Here m(n:P, KP) is the multiplicity with which the trivial representation of KP occurs in n:�, and the sign is positive if n: is one-dimensional and negative if it is the first element of the discrete series. As Casselman and Milne show in their lectures, this is so locally almost everywhere. One can probably show without great difficulty that the local statement is correct at p as well when n: belongs to the discrete series, for n: then contributes to the co homology in dimension one and n:P = n;((Jp) where (JP is a special representation of the thickened Weil group. In particular
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIFS, AND MOTIVES
L(z - ! ,
n"p ,
r) =
241
I _I e/p• ' l e i = 1 .
However if n- is one-dimensional then n- contributes to the cohomology in dimen sions zero and two and the corresponding local contribution to the zeta-function should be
{ (1 - e/p•)(II - epfp•) }
m (1rP, KP)
·
The factor inside the brackets is not L(z - t. n-p , r). The difficulty is resolved if we realize that when n-, and hence n- ® 1CJ > is one-dimensional we should not be using L(z - f, n- ® n-1, r) at all but rather L(z - t, n-' ® n-j, r) where n-' ® n-j is the one-dimensional representation of G'(A) = GL(2, A) defined by the same character of the idele-class group as n- ® n-1. Since G'(Qv) - G(Qv) and n-� - n-v for almost all places v, the error of using n- ® n-1 instead of n-' ® n-{ is not detected when one only considers the local zeta function almost everywhere. The significance of the considerations of the second and third sections begins to appear. The representation n- ® n-1 and the representation n-" ® n-j of G'(A) associated to it by the principle of functoriality are anomalous, because n-; is one dimensional for almost all places v while n-; is infinite-dimensional. The isobaric representation equivalent to n-" ® n-j almost everywhere is n-' ® n-/. It was implicit in the discussion of the second section that anomalous representations would have nothing to do with motives, and so it should come as no surprise now that we must discard n- ® n-1 and replace it by n-' ® n- / . In this example n- itself was not changed for G(R) - G'(R) and n- - n-'. However in general we must expect that n- itself will have to be modified. Thus the proper factor will not be L(z - d/2, n- ® n-1, r) but L(z - d/2, n-' ® n-/, r), where n-' ® n-/ is an automorphic representation of a group G' obtained from G by an inner twisting. Again n-� will have to be equivalent to n-v almost everywhere. Since at the moment we are primarily interested in the infinite place. we simply ask whether it is possible to find a candidate for n-' or, rather, for an L-packet {n-'} = U'. There are apparently two conditions to be satisfied, the first arising from the compatibility of functional equations. (a) Let n- E U"' and let {n-'} = Hrp' · For any additive character ¢ of R and any re presentation q of LG = LG', is equal to
e ' (z, q o cp, ¢) = e(z, q o cp, ¢) L(l - z, q o cp) L(z, q o cp)
�
e ' (z, q o cp' , ¢) = e (z, q o cp', ¢) L( - z, q (z, (J cp
0,
0
j')
.
(b) It is possible to find a summand ¢0(n-') of ¢ 1 (n-') which is such that ¢2(n-)
a¢o (n-'), a E Z.
=
These conditions are only tentative, and may have to be modified in the course of time, but they will serve for the explanation of our problem.
242
R . P . LANGLANDS
The first condition involves only rp and rp' and we begin by constructing some pairs that satisfy it. Fix an element w of Wc18 that projects to c and satisfies w2 = - 1 . We may suppose that rp(w) = a x c with a in the normalizer of LTo in LG0• Then rp(w) also normalizes LT0• Let rp(c) denote the transformation of X * (LT0) or of X* (LT0) defined by rp(w). We may also suppose that rp takes ex to L TO and that rp(z) = zA z"'CeJ A with A e X* (LT0) ® C and A - rp(c)A e XiL ro). The representation rp' will be defined in a similar way. Thus rp'(w) = a' x c with a' in the normalizer of LT0, and rp'(z) = zA z'I'' Ce l A . Notice that A is to be the same for rp' as for rp. However a, which is given, is replaced by a', which we must now define. We suppose that rp{c) sends every root to its negative, and choose A in X* (LT0) such that AV(a) = e2ni<J., J.v> for any weight Av of LTo which is orthogonal to all roots. We shall take a' to lie in L To and to be such that AV(a') = e2ni<J.'J.v> when Av is orthogonal to all roots. Here ).' is still to be defined. If we also denote by a the operator on X* (LT0) ® C defined by a and if we let q be one-half the sum of the positive roots then ).' is to be given by the equation ).' = 1 ; a A (1 - a)( + rp'(c)) A + i .
�
_
Observe that the action of rp'{c) is the same as that of c . We are assuming that rp is a given, well-defined homomorphism, and hence [31] that
In order to show that rp' is also well defined we must verify that (7.7) We begin with the equations a rp'(c) = rp'(c)a = rp(c) and a {l + rp(c)) = 1 + rp(c), remarking also that the square of both rp(c) and rp'(c) is the identity. We infer that the left side of (7. 7) is equal to {1 + rp'(c)) ( l + rp(c)) A 2
_
( 1 - a) ( l + rp'(c)) A + 4
�
= q. The sum is in turn congruent to - rp(c) A _ (1 - a) {l + rp'(c)) A = 1 - rp'(c) A . 2 4 2
because {1 + rp'(c)) q/ 2 1
Consequently the homomorphsim rp' can be constructed whenever rp is defined and rp(c) sends every root to its negative. The following lemma is valid in this generality. LEMMA. For any representation u of the Wei/ form of LG and any nontrivial char acter ¢ ofR, e ' (r , u o rp , ¢) = e ' (-r, u o rp', ¢ ) .
243
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
The proof is a computation based on the proof of Lemma 3.2 of [31] and on Chapters 5 and 6 of [24], but it is rather lengthy, and not worth including here. Observe that we could have started with rp', defined by an a ' in LT o , and, reversing the process, passed to ). and a . More generally if LM and LM' are two parabolic subgroups of LG, and rp : We1R --+ LM has an image which lies in no proper par abolic subgroup of LM, then we can use the process to pass to a rp" whose image lies in the minimal parabolic ofLM, and thus of LG or LM', and afterwards reverse it to pass from rp" to a rp' : We1R --+ LM' whose image lies in no proper parabolic subgroup of LM'. My intention now is simply to show, by means of a few examples, how for a given 11: in some n"' one can choose one of the rp' just described so that the condition (b) is satisfied for the elements 11:' of n"'. . Of course the problem is to decide if such a choice is always possible. Without more examples or a general theorem, we cannot be at all confident that this is so. If all the continuous cohomology of ' ® 11: is zero there is no difficulty satisfying (b). We take rp' = rp and a = 0. The simplest nontrivial example is obtained by taking ' trivial and 71: trivial. Let 71: E n"'. Then rp(z) = zqztp C tl q , with q equal again to one-half the sum of the positive roots. If LM is the parabolic subgroup of LG correspondi ng to the minimal parabolic of G over R, then the image of rp lies in LM and rp(t) takes every root of L To in LM to its negative. Define rp ' as above, with a ' e LT0• G' can be taken to be the quasi-split form of G over R. The continuous cohomology of 11: is all in even dimensions and all of type (p, p) for some p. To compute it one observes that it is the same as the cohomology of the compactdual, which can be computed by using Schubert cells. One verifies without difficulty that for 11:' e n"', the representati on (/JI(11:'), which depends in reality only on rp ' , is equivalent to (/J2(7C). If G is not quasi-split over R then rp' is different from rp. If G is not quasi-split over R then it is certainly not quasi-split over Q, and the trivial representation of G(A) is anomalous. Once again we see that the passage from 11: to 11: ' is the local expression of the passage from an anomalous representation to one which is not anomalous. Other interesting examples are the representations 11: = J;, j of PSU(n, 1) dis cussed in Chapter XI of the notes of Borel-Wallach [4]. Take � trivial. In this case ¢z(7C) is ( - I )i+i times a representation induced from ex ' the representation of ex used having the weights z-i z-i, z-:-i-1 z -i-1 , . . . , z- Cn-j l z - Cn -i l . (7.8) Here 0 ;;;;; i + j ;;;;; n - I and 0 ;;;;; i, j. Borel and Wallach lapse into vagueness at one point, and it may be that the roles of i and j should here be reversed, but that is of little consequence. The group LGo is SL(n + 1 , C) and L T0 may be taken to be the group of diagonal matrices. The representation r o is the standard representation of SL(n + 1 , C). It is easy enough to deduce from [4] that if 11: e n"' then
with A being equal to (n/2
-
i, n/2, n/2
-
1,
· ··,
n/2
-
i + 1 , n/2
-
i
-
1,
244
R. P. LANGLANDS
- n/2 + j + I , - n/2 + j - I , · · · , - n/2, - n/2 + j). The numbers occurring here are n/2, n/2 - 1 , · · · , - n/2, but the order is somewhat unusual. The transfor mation g>(t) is given by
We are of course using the obvious representation of the elements of X* (LT0) ® C as sequences of n + I complex numbers whose sum is 0. Suppose, to be definite, that i � j. We will choose g>' to be such that the trans formation g>'(t) takes (xi> · · · , Xn+I ) to ( - Xn+ l > - Xm · · · , - Xn-i+ l > - Xn-i • · · · , - xi+Z• - Xn-i+ I > · · · , - Xn-i • - xi + I > • · · , - x 1 ). The indices within the gaps decrease or increase regularly by one. If rc' E n'P' then the representation (h(rc') is induced from a representation of ex with weights
(7.9)
z -i- I z-i-1 , . . . , z -
• • .
z-
.•
- (n-i) z-i ; 'z
· , z-nz-n , z- (n-j) =- (n-i l .
Happily the set (7.8) i s a subset of (7.9) and the condition (b) i s satisfied. It should be observed that the representation ¢0(rc') that is chosen to satisfy (b) will have to be, except for some degenerate values of i and j, a proper subrepresen tation of ¢ 1 (rc'). This phenomenon will, I hope, be taken into account by L-in distinguishability. For example if e is the element of LTo with diagonal entries _______. ..___
j
�
I, - 1, - 1,· · ·, - 1, I, · · ·, I, - 1, · · ·, - 1, I
then e commutes with g>'( Wc1R) and ¢0(rc') may be taken to be the restriction of ¢ 1 (rc') to the + I eigenspace of r(e). REFERENCES 1. J. Arthur, Eisenstein series and the trace formula, these PROCEEDINGS, part I , pp. 253-274. 2. W. Baily and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (I 966), 442-528. 3. A. Borel, Automorphic £-functions, these PROCEEDINGS, part 2, pp. 27-6 I . 4 . A . Borel and N . Wallach, Seminar o n the cohomology of discrete subgroups of semi-simple groups, Institute for Advanced Study (1 976/77). 5. P. Cartier, Representations ofp-adic groups : A survey, these PROCEEDINGS, part I , pp. 1 1 I-I 55. 6. W. Casselman, The Hasse- Wei/ �-function of some moduli varieties of dimension greater than one, these PROCEEDINGS, part 2, pp. I4I-I63. 7. --, GL(n), Algebraic Number Fields (£-Functions and Galois Properties), A. Frohlich , ed., Academic Press, New York, I 977. 8. I. V. Cherednik, Algebraic curves uniformized by discrete arithmetical subgroups, Uspehi Mat. Nauk 30 (1 975), 1 83-1 84. (Russian)
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
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P. Deligne, Travaux de Griffiths, Sem. Bourbaki (1 970). Travaux de Shimura, Sem. Bourbaki (1971). 1 1 . -- , La conjecture de Wei/ pour les surfaces K3, Invent. Math. IS (1 972), 206-226. 12. -- , Theorie de Hodge. III, Inst. Hautes Etudes Sci. Pub!. Math. 44 (1 974), 5-78. 13. -- , Letter to Piatetski-Shapiro. 14. P. Deligne and M. Rapoport, Les schemas de modules de courbes elliptiques, Modular Functions of One Variable. II, Lecture Notes in Math., vol. 349, Springer, New York, 1 972, pp. 1 43-3 1 6. 15. V. G. Drinfeld, Coverings of p-adic symmetric spaces, Functional Anal. Appl. 10 (1 976), 29-40. (Russian) 16. J. Giraud, Cohomologie non-abe/ienne, Springer-Verlag, Berlin and New York, 1 97 1 . 17. Harish-Chandra, Representations of semi-simple Lie groups. VI, Amer. Math. J . 7 8 (1 956), 564-628. 18. --, Discrete series for semi-simple Lie groups. I, Acta Math. 113 (1 965), 241-3 1 8 ; II, 1 1 6 (1 966), 1-1 1 1 . 19. R . Howe and I . Piatetski-Shapiro, A counterexample to the "generalized Ramanujan coniec ture " for (quasi-) split groups, these PROCEEDINGS, part 1 , pp. 3 1 5-322. 20. Y. Ihara, Heeke polynomials as congruence (,-functions in elliptic modular case, Ann. of Math. (2) 85 (1 967), 267-295. 21. --- , On congruence monodromy problems, vols. 1, 2, Lecture notes from Tokyo Univ. , 1 968, 1 969. 22. --- , Some fundamental groups in the arithmetic of algebraic curves over finite fields, Proc. Nat. Acad. Sci . 72 (1 975). 23. S. Gelbart and H. Jacquet, Forms of GL(2) from the analytic point of view, these PROCEED INGS, part 1 , pp. 21 3-25 1 . 24. H . Jacquet and R . P . Langlands, Automorphic forms on GL(2), Lecture Notes in Math., vol. 1 1 4, Springer, New York, 1 970. 25. H. Jacquet and J. Shalika, A non-vanishing theorem for zeta-functions of GL(n), Invent. Math. 38 (1 976). 26. D. Kazhdan, On arithmetic varieties, Lie Groups and Their Representations, Halsted, New York, 1 975. 27. R. Kottwitz, Thesis, Harvard Univ., 1 977. 28. -- , Combinatorics and Shimura varieties mod p, these PROCEEDINGS, part 2, pp. 1 85-192. 29. N. Kurokawa, Examples of eigenvalues of Heeke operators in Siegel modular forms, 1 977 (preprint). 30. R. P. Langlands, Modular forms and 1-adic representations, Modular Forms of One Variable. II, Lecture Notes in Math., vol . 349, Springer, New York, 1 973, pp. 361-500. 31. -- , On the classification of irreducible representations of real algebraic groups, Institute for Advanced Study, Princeton, N. J., 1 973. 32. ---, Some contemporary problems with origins in the Jugendtraum, Proc. Sympos. Pure Math., vol. 28, Amer. Math. Soc., Providence, R. I., 1 976, pp. 401-4 1 8 . 33. --- , Stable conjugacy : definitions and lemmas, Canad. J. Math. (to appear). 34. -- , On the zeta-functions of some simple Shimura varieties, Canad. J. Math. (to appear). 35. --- , On the notion of automorphic representation, these PROCEEDINGS, part 1, pp. 203-207. 36. J. Milne, Points on Shimura varieties m od p , these PROCEEDINGS, part 2, pp. 1 65-1 84. 37. D. Mumford, Families of abelian varieties, Proc. Sympos. Pure Math., vol. 9., Amer. Math. Soc., Providence, R. I . , 1 966, pp. 347-35 1 . 38. I . Piatetski-Shapiro, Multiplicity one theorems, these PROCEEDINGS, part 1 , pp. 209-212. 39. H. Resnikoff and R. L. Saldana, Some properties of Fourier coefficients of Eisenstein series of degree two, Crelle 265 (1974), 90-109. 40. N. Saavedra Rivano, Categories tannakiennes, Lecture Notes in Math., vol. 265, Springer, New York, 1 972. 41. J.-P. Serre, Abelian 1-adic representations and elliptic curves, Benj amin, New York, 1 968. 42. --- , Facteurs locaux des fonctions zeta des varietes algebriques (definitions et conjectures), Sem. Delange-Pisot-Poitou, 1 970. 43. -- , Representations 1-adiques, Algebraic Number Theory, Tokyo, 1 977. 9.
10. -- ,
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44. D. Shelstad, Notes on £-indistinguishability, these PROCEEDINGS, part 2, pp. 1 93-203 . 45. , Orbital integrals and a family of groups attached to a real reductive group, 1 977 (preprint). 46. K.-Y. Shih, Conjugations of arithmetic automorphic function fields, 1 977 (preprint) . 47. J . Tate, Number theoretic background, these PROCEEDINGS, part 2, pp. 3-26. --
48. N. Wallach, Representations of reductive Lie groups, these PROCEEDINGS, part 1 , pp. 71-86. J. Math. 78 (1 956), 509-524. 50. S. Zucker, Hodge theory with degenerating coefficients. I, 1 977 (preprint).
49. A. Wei!, The field of definition of a variety, Amer. THE INSTITUTE FOR ADVANCED STUDY
Proceedings of Symposia in Pure Mathematics Vol. 33 ( 1 979), part 2, pp. 247-290 ,
,
,
VARIETES DE SHIMURA : INTERPRETATION MODULAIRE, ET TECHNIQUES DE CONSTRUCTION DE MODELES CANONIQUES
PIERRE DELIGNE SOMMAIRE Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Rappels, terminologie et notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 1. Domaines hermitiens symetriques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 1 . 1 Espaces de modules de structures de Hodge . . . . . . . . . . . . . . . . . . . . . . . . 25 1 1 .2 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 1 . 3 Plongements symplectiques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 2. Varietes de Shimura . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 2.0 Preliminaires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 2.1 Varietes de Shimura . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 2.2 Modeles canoniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 2. 3 Construction de modeles canoniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 2.4 Loi de reciprocite : preliminaires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 2. 5 Application : une extension canonique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 2.6 La loi de reciprocite des modeles canoniques . . . . . . . . . . . . . . . . . . . . . . 284 2.7 Reduction au groupe derive, et theoreme d'existence . . . . . . . . . . . . . . . . 285 Bibliographie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Introduction. Cet article fait suite a [5], dont nous utiliserons les resultats essen tiels (ceux des paragraphes 4 et 5). Dans une premiere partie, nous tentons de motiver les axiomes imposes aux systemes (G, X) (2. 1 . 1) a partir desquels sont definies les varietes de Shimura. On montre que, grosso modo, ils correspondent aux espaces de modules de structures de Hodge x+ du type suivant. (a) x+ est une composante connexe de l'espace de toutes les structures de Hodge sur un espace vectoriel fixe V relativement auxquelles certains tenseurs t 1 . . tn sont de type (0, 0). Le groupe algebrique G est le sous-groupe de GL(V) qui fixe les t; et X est l'orbite G(R) · x+ de x+ sous G(R). ·
4 WS (MOS) subject classifications (1 970). Primary 10D20, 14099 ; Secondary 200 1 5 , 20030. © 1 979, American Mathematical Society
247
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PIERRE DELIGNE
(b) La famille de structure de Hodge sur V parametree par x+ verifie certaines conditions, verifiees par les families de structures de Hodge qui apparaissent na turellement en geometrie algebrique : pour une structure complexe convenable (et uniquement determinee) sur x+, c'est une variation de structure de Hodge polarisable. L'espace x+ est automatiquement un domaine hermitien symetrique ( = espace hermitien symetrique a courbure < 0). Les domaines hermitiens symetriques peuvent tous etre ainsi decrits comme des espaces de modules de structures de Hodge ( 1 . 1 . 1 7 ) , et je crois cette description tres utile. Par exemple : le plongement d'un domaine hermitien symetrique D dans son,dual i5 (une variete de drapeaux) correspond a !'application (une structure de Hodge) >-+ (la filtration de Hodge correspondante). Les descriptions comme "domaine de Siegel de 3e espece" s'inter pretent en disant que, sous certaines hypotheses, si on superimpose a une structure de Hodge une filtration par le poids, on obtient une structure de Hodge mixte, d'ou une application de D dans un espace de modules de structures de Hodge mixtes (cf. les constructions de [1, III, 4. 1]). Ce dernier point ne sera ni mentionne, ni utilise dans l'article. Ce point de vue, et la description de certaines varietes de Shimura comme es paces de modules de varietes abeliennes, sont lies par le dictionnaire : il r evient au meme (equivalence de categorie A >-+ H1(A, Z) ) de se donner une variete abe lienne ou une structure de Hodge polarisable de type {( - 1 , 0), (0, - 1 )} (il s'agit ici de Z-structures de Hodge sans torsion ; par passage au dual (A >-+ Hl(A, Z)) , on peut remplacer {( - 1 , 0), (0, - 1 ) } par {(1 , 0), (0, 1) } ) . Polariser la variete abelienne revient a polariser son H1 . Avec parametres, de meme, il revient au meme de se donner un schema abelien polarise sur une variete complexe lisse S, ou une varia tion de structures de Hodge polarisee de type {( - 1 , 0), (0, - 1)} sur l'espace analy tique s•n. Une famille analytique de varietes abeliennes, parametree par s•n, est automatiquement algebrique (ceci resulte de [3]). Pour interpreter des structures de Hodge de type plus complique, on aimerait remplacer les varietes abeliennes par des "motifs" convenables, mais il ne s'agit encore que d'un reve. Au numero 1 .2, nous donnons une description commode basee sur le formalisme precedent de la classification des domaines hermitiens symetriques, en terme de diagrammes de Dynkin et de leurs sommets speciaux. Au numero 1 .3, nous clas sifions un certain type de plongements de domaines hermitiens symetriques dans le demi-espace de Siegel. Les resultats sont paralleles a ceux de Satake [11]. Une application de l'astuce unitaire de Weyl, pour laquelle nous renvoyons a [7], ra mene la classification a la connaissance d'un fragment de la table, donnee par exemple dans Bourbaki [4], donnant !'expression des poids fondamentaux comme combinaison lineaire de racines simples. Le lecteur ctesireux d'en savoir plus sur les variations de structure de Hodge, et la far;on dont elles apparaissent en geometrie algebrique, pourra consulter [6] ( dont no us ne suivons pas les conventions de signes) ; certains faits, enonces dans [6], sont demontres dans [7]. Aux numeros 2. 1 et 2.2 nous definissons, dans un langage adelique, les varietes de Shimura KMc(G, X) (notees KMc(G, h) dans [5], pour h un quelconque element de X), leur limite projective Mc(G, X) et la notion de modele canonique. Je renvoie au texte pour ces definitions, et dirai seulement qu'un modele canonique de
VARIETES DE SHIMURA
249
M0(G, X) est un modele de M0(G, X) sur le corps dual (2.2. 1) E(G, X), i.e. un schema M(G, X) sur E(G, X) muni d'un isomorphisme M(G, X) ® E < G .Xl C .:::... Mc(G, X), cette fayon de definir Me(G, X) sur E(G, X) ayant des proprietes convenables ( G(Af) equivariance, et proprietes galoisiennes des points speciaux (2.2.4)). On definit
aussi la notion de modele faiblement canonique (meme definition que les modeles canoniques, avec E(G, X) remplace par une extension finie E c C). Ils jouent un role technique dans la construction de modeles canoniques. Les differences ap parentes entre les definitions de 2. 1 , 2 . 2 et celles de [5] proviennent d'un autre choix de conventions de signes (action a droite contre action a gauche, loi de reciprocite en theorie du corps de classes global. . .). Pour une description heuristique, je renvoie a }'introduction de [5]. Pour une breve description, sur des exemples, de comment on passe du langage adelique a un langage plus classique, je renvoie a [5, 1 .6-1 . 1 1 , 3 . 1 4--3 . 1 6, 4. 1 1-4. 16]. Dans [5], nous avions systematise les methodes introduites par Shimura pour construire des modeles canoniques. Dans Ia seconde partie du present article, nous perfectionnons les resultats de [5]. Au numero 2.6, nous determinons l'action du groupe de Galois Gal(Q/E) sur }'ensemble des composantes connexes geometri ques d'un modele faiblement canonique (suppose exister) de M0(G, X) sur E, sans supposer, comme dans [5], que le groupe derive de G est simplement connexe. Le point essentiel est la construction, donnee au numero 2.4, d'un morphisme du type suivant. Soient G un groupe reductif (connexe) sur Q, p : G --+ G le revetement universe} de son groupe derive Gd•r et M une classe de conjugaison, definie sur un corps de nombres E, de morphismes de Gm dans G. On construit un morphisme qM du groupe des classes d'ideles de E dans le quotient abelien G(A) /pG(A) · G(Q) de G(A). Ce morphisme est fonctoriel en {G, M), et, si F est une extension de E, le diagramme C(F)
l�
G(A) /pG(A) · G(Q)
C(E)
est commutatif. Si G n'a pas de facteur G' sur Q tel que G'(R) soit compact, on dectuit du theoreme d'approximation forte que
n0 ( G(A) /pG(A)G(Q)) = n0 ( G(A) / G(Q)) , et qM fournit une action sur n0 (G(A)/ G(Q)) de n0 C(E), le groupe de Galois rendu
abelien Gal(Q/E)•h d'apres la theorie du corps de classes global. La seconde idee nouvelle-en fait un retour au point de vue de Shimura-est !'observation suivante : les resultats de 2.6 permettent de reconstruire un modele faiblement canonique ME(G, X) de M0(G, X) a partir de sa composante neutre M8(G, X+) (une composante connexe geometrique, dependant du choix d'une com posante connexe x+ de X), munie de l'action (semi-lineaire) du sous-groupe H de G(A!) x Gal(Q/E) qui Ia stabilise. Soient Z le centre de G, G•d le groupe- adjoint, G•d(R)+ Ia composante neutre topologique de G•d(R), et G•d(Q)+ = G•d(Q) n G•d(R)+. L'adherence Z(Q)- de Z(Q) dans G(Af) agissant trivialement sur M0(G, X), l'ac-
250
PIERRE DELIGNE
tion de H sur M8(G, X+) se factorise par H/Z(Q)-. On peut en fait faire agir un groupe un peu plus gros, extension de Gal(Q/E) par le complete de G•d(Q)+ pour Ia topologie des images des sous-groupes de congruence de Gd•r(Q) . A isomorphisme unique pres, cette extension ne depend que de G•d , Gd er et de Ia projection x+•d de x+ dans G•d (2. 5). On Ia note tfE(G•d , Gder , x+•d) . La composante neutre M8(G, X+) est Ia limite projective des quotients de x+•d par les sous-groupes arithmetiques de G•d (Q)+, images de sous-groupes de congruence de Gd•r(Q). On verifie enfin que les conditions que doivent verifier son modele M8(G, X+) sur Q, et I' action de tfE(G"d , Gd•r, x+•d), pour correspondre a un modele faiblement canoni que, ne dependent que du groupe adjoint G•d , de x+•d , du revetement Gd•r de G•d et de I' extension finie E (contenue dans C) de E(G•d , x+•d). Ces conditions definis sent les modeles faiblement canoniques (resp. canoniques, pour E = E(G•d , x+•d)) connexes (2.7. 10). Le probleme de !'existence d'un modele canonique ne depend done, en gros, que du groupe derive. Cette reduction au groupe derive est une version beaucoup plus commode de Ia maladroite methode de modification centrale de h dans [5, 5. 1 1 ] .
En 2 . 3 , nous construisons une provision de modeles canoniques a I' aide de plon gements symplectiques, en invoquant [5, 4.21 et 5.7]. Les resultats de 1 . 3 nous per mettent d'obtenir les plongements symplectiques desires avec tres peu de calculs. En 2.7, nous expliquons Ia reduction au groupe derive esquissee ci-dessus et nous deduisons de 2.3 un critere d'existence de modeles canoniques qui couvre tous les cas connus {Shimura, Miyake et Shih). Dans l'article, nous utilisons }'equivalence entre modeles faiblement canoniques et modeles faiblement canoniques connexes pour transporter a ces derniers des resultats de [5) (unicite, construction d'un modele canonique a partir d'une famille de modeles faiblement canoniques). II eut ete plus naturel de transposer les demon strations, et de transposer de meme Ia fonctorialite [5, 5.4], et le passage a un sous groupe [5, 5.7] (evitant par la la sybilline proposition [5, 1 . 1 5]). Le manque de temps, et la lassitude, m'en ont empech6. J'ai recemment demontre qu'on pouvait donner un sens purement algebrique a Ia notion de cycle rationnel de type (p, p), sur une variete abelienne A (sur un corps de caracteristique 0). On peut retrouver a partir de Ia le critere d'existence 2. 3 . 1 d e modeles canoniques, e t donner une description modulaire des modeles obtenus avec son aide (cf. [9]). Cette description ne se prete malheureusement pas a la reduc tion modulo p. Cette methode evite le recours a [5, 5.7], (et par la a [5, 1 . 1 5]) et fournit des renseignements partiels sur Ia conjugaison des varietes de Shimura. 0. Rappels, terminologie et notations. 0. 1 . Nous aurons a faire usage du theoreme d'approximation forte, du theoreme d'approximation reelle, du principe de Hasse et de Ia nullite de H 1 (K, G) pour G
semi-simple simplement connexe sur un corps local non archimedien. Des indica tions bibliographiques sur ces theoremes sont donnees dans [5, (0. 1 ) a (0.4)]. Signa Ions en outre I' article de G. Prasad (Strong approximation for semi-simple groups over function fields, Ann. of Math. (2) 105 (1 977), 553 - 572) prouvant le theoreme d'approximation forte sur un corps global quelconque. Soit G un groupe semi simple simplement connexe, de centre Z, sur un corps global K. Nous n'utiliserons
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le principe de Hasse pour H l (K, G) que pour les classes dans l'image de H l (K; Z). En particulier, les facteurs £8 nous indifferent. 0.2. Groupe reductif signifiera toujours groupe reductif cormexe. Un revetement d'un groupe reductif est un revetement connexe. Groupe adjoint signifiera groupe reductif adjoint. Si G est un groupe reductif, nous noterons G•d son groupe adjoint, Gder son groupe derive, et p : G -> Gd er le revetement universe! de Gd er . Nous noterons souvent Z (ou Z(G)) le centre de G, et (conflit de notation) Z celui de G. 0.3. Nous noterons par un exposant 0 une composante connexe algebrique (par exemple : zo est Ia composante neutre du centre Z de G). L'exposant + designera une composante connexe topologique (par exemple : G(R)+ est Ia composante neutre topologique du groupe des points reels d'un groupe G). On notera aussi G(Q)+ Ia trace de G(R)+ sur G(Q). Pour G reductif reel, no us noterons par un indice + I' image inverse de G•d(R)+ dans G(R). Meme notation + pour la trace sur un groupe de points rationnels. Pour X un espace topologique, nous notons 1r0(X) !'ensemble de ses composantes connexes, muni de Ia topologie quotient de celle de X. Dans I' article, l'espace 7ro(X) sera toujours discret, ou compact totalement discontinu. 0.4. Un domaine hermitien symetrique est un espace hermitien symetrique a cour bure < 0 (i.e. sans facteur euclidien ou compact). 0.5. Sauf mention expresse du contraire, un espace vectoriel est suppose de di mension finie, et un corps de nombres est suppose de degre fini sur Q. Les corps de nombres que nous aurons a considerer seront le plus souvent contenus dans C; Q designe la cloture algebrique de Q dans C. 0.6. On pose i = proj lim Z/nZ = Ti p ZP • A f = Q ® i = TI P Qp (produit restreint) et on note A = R x A f l'anneau des adeles de Q . On designera parfois encore par A l'anneau des adeles d'un corps global quelconque. 0.7. G(K), G ®F K, GK : pour G un schema sur F (par exemple un groupe alge brique sur F) et K une F-algebre, on designe par G(K) I'ensemble des points de G a valeur dans K, et par GK ou par G ® F K Ie schema sur K deduit de G par extension des scalaires. 0.8. Nous normaliserons l'isomorphisme de reciprocite de Ia theorie du corps de classes global ( = choisirons lui ou son inverse) 1roA"t:/E* ___::::__. Gal(Q/E)•b de telle sorte que Ia classe de l'idele egal a une uniformisante en v et a 1 aux autres places corresponde a un Frobenius geometrique (l'inverse d'une substitution de Frobenius) (cf. 1 . 1 .6 et Ia justification loc. cit. 3.6, 8 . 1 2). ·
1. Domaines hermitiens symetriques. 1 . 1 . Espaces de modules de structures de Hodge. 1 . 1 . 1 . Rappelons qu'une structure de Hodge sur un espace vectoriel reel V est une bigraduation Vc = ffi Vpq du complexifie de V, telle que VPq soit le complexe conjugue de VqP . Definissons une action h de C* sur Vc par la formule (1 . 1 . 1 . 1) h(z)v = z-p:z-qv pour V E Vpq . Les h(z) commutent a Ia conjugaison complexe de V0, done se deduisent par ex-
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tension des scalaires d'une action, encore notee h, de C* sur V. Regardons C comme une extension de R, et soit S son groupe multiplicatif, considere comme groupe algebrique reel (autrement dit, S = RciR Gm (restriction des scalaires a Ia Weil)) ; on a S(R) = C*, et h est une action du groupe algebrique S. On verifie que cette construction definit une equivalence de categories : (espaces vectoriels reels munis d'une structure de Hodge) ..... (espaces vectoriels reels munis d'une action du groupe algebrique reel S). A !'inclusion R* c: C* correspond une inclusion de groupes algebriques reels Gm c: S. No us noterons wh ( ou simplement w) la restriction de h- 1 a Gm , et l'appel lerons le poids w : Gm ..... GL( V). On dit que V est homogene de poids n si Vpq = 0 pour p + q f; n, i.e. w(i\) est l'homothetie de rapport i(n . Nous noterons /-lh (ou simplement f-t) !'action de Gm sur Vc definie par f-t(z) v = z-Pv pour v E Vpq . C'est un compose Gm ..... Sc ..... h GL( Vc). La filtration de Hodge Fh (ou simplement F) est definie par FP = EB r ":; p v rs . On dit que V est de type rff c: Z x Z si Vpq = 0 pour (p, q) ¢ rff . Plus generalement, si A est un sous-anneau de R tel que A ® Q soit un corps (en pratique, A = Z, Q ou R), une A-structure de Hodge est A-module de type fini V, muni d'une structure de Hodge sur V ®A R. EXEMPLE 1 . 1 .2. L'exemple fondamental est celui oil V = Hn (X, R), pour X une variete kahlerienne compacte, et oil VPq c: Hn (X, C) est l'espace des classes de cohomologie representees par une forme fermee de type (p, q). D'autres exemples utiles s'en deduisent par des operations de produits tensoriels, passage a une struc ture de Hodge facteur direct, ou passage au dual. Ainsi, le dual Hn(X, R) de Hn (X, R) est muni d'une structure de Hodge de poids - n. L'homologie, ou Ia coho mo1ogie, entieres fournissent des Z-structures de Hodge. ExEMPLE 1 . 1 .3. Les structures de Hodge de type {( - 1 , 0), (0, - 1)} sont celles pour lesquelles !'action h de C* = S(R) est induite par une structure complexe sur V; pour V de ce type, Ia projection pr de V sur v- 1 • 0 c: Vc est bijective, et verifie pr (h(z) v) = z pr( v) . EXEMPLE 1 . 1 . 4. Soit A un tore complexe ; c'est le quotient L/F de son algebre de Lie L par un reseau F. On a F ® R ....::::. L, d'oil une structure complexe sur F ® R. Considerons celle-ci comme une structure de Hodge (1 . 1 .3). Via l'isomorphisme r = H1 (A , Z), c'est celle de 1 . 1 .2. EXEMPLE 1 . 1 . 5. La structure de Hodge de Tate Z(l) est Ia Z-structure de Hodge de type ( - 1 , - 1) de reseau entier 211: iZ c: C. L'exponentielle identifie C* a CjZ(l), d'oil un isomorphisme Z(l) = H1 (C * ). La structure de Hodge Z(n) = dfn Z(l)®n (n E Z) est Ia Z-structure de Hodge de type ( - n, - n) de reseau entier (271: i) nZ. On note . . . (n) le produit tensoriel de . . . par Z(n) (twist a Ia Tate). 1 . 1 .6. REMARQUE. La regie h(z)v = z -Pz -q v pour v E vpq est celle que j'utilise dans (Les constantes des equations fonctionnelles des fonctions L, Anvers II, Lecture Notes in Math., vol. 349, pp. 501-597) et !'inverse de celle de [6]. Elle est justifiee d'une part par l'exemple 1 . 1 .4 ci-dessus, d'autre part par le desir que C * agisse sur R(1 ) par multiplication par Ia norme (cf. Joe. cit., fin de 8 . 1 2). 1 . 1 .7. Une variation de structure de Hodge sur une variete analytique complexe S consiste en (a) un systeme local V d'espaces vectoriels reels ;
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(b) en chaque point de S, une structure de Hodge sur la fibre de V en s, variant continument avec s. On exige que la filtration de Hodge varie holomorphiquemen.t avec s, et verifie l'axiome dit de transversalite : la derivee d'une section de FP est dans FP- 1 • II sera souvent donne un systeme local Vz de Z-modules de type fini, tel que V = Vz ® R. On parlera alors de variation de Z-structure de Hodge. De meme pour Z remplace par un anneau A comme en 1 . 1 . 1 . REMARQUE 1 . 1 . 8. Regardons S comme etant une variete reelle, dont l'espace tangent en chaque point est muni d'une structure complexe, i.e. d'une structure de Hodge de type {( - 1 , 0), (0, - 1)}. L'integrabilite de la structure presque complexe de S s'exprime en disant que le crochet des champs de vecteurs est compatible a la filtration de Hodge du complexifie du fibre tangent : [T0 .- 1 , ro.- 1 ] c ro.- 1 . De meme, l'axiome des variations de structure de Hodge exprime que la derivation (fibre tangent) ®R (sections coo de V) --> (sections coo de V) ( ou plut6t le complexifie de cette application) est compatible aux filtrations de Hodge : ovFP c FP pour D dans T0 ,- 1 (holomorphie) et ovFP c FP- 1 pour D arbitraire (transversalite). PRINCIPE 1 . 1 .9. En geometrie algebrique, chaque fois qu'apparait une structure de Hodge dependant de parametres complexes, c'est une variation de structure de Hodge sur l'espace des parametres. L'exemple fondamental est 1 . 1 .2, avec para metres : si f : X --> S est un morphisme pro pre et lisse, de fibres Xs kahleriennes, les Hn(X., Z) forment un systeme local sur S, et la filtration de Hodge sur le com plexifte Hn(X., C) varie holomorphiquement avec s et verifie l'axiome de transver salite. 1 . 1 . 10. Une polarisation d'une structure de Hodge reelle, de poids n, V est un morphisme If! : V ® V --> R( - n) tel que la forme (2n i)n 1/f(x, h(i)y) soit symetrique et definie positive. De meme pour les Z-structures de Hodge, en rempla<;ant R( - n) par Z( - n), . . . . Puisque 1/f(h(i)x, y) = 1/f(x, h( - i)y) (h(i) est trivial sur R( - n) ) , et que h( - i)y = ( - I) nh(i)y, la condition de symetrie revient a : If! symetrique pour n pair, alterne pour n impair. Les structures de Hodge qui apparaissent en geometrie algebrique sont des Z structures de Hodge homogenes et polarisables. Exemple fondamental : les theo remes de positivite de Hodge assurent que fin(X, Z), pour X une variete projective et lisse, est polarisable (noter que h(i) est !'operation notee C par Weil dans son livre sur les varietes kahleriennes) . 1 . 1 . 1 1 . Soient des espaces vectoriels reels ( V;);"' 1 et une famille de tenseurs (si)J EJ dans des produits tensoriels de puissances tensorielles des V; et de leurs duaux. On s'interesse aux families de structures de Hodge sur les V;, pour lesquelles les s1 sont de type (0, 0). Pour interpreter cette condition "type (0, 0)" dans des cas particu liers, noter que f : V --> W est un morphisme si et seulement si, en tant qu'element de Hom( V, W) = V* ® W, il est de type (0, 0). Soit G le sous-groupe algebrique de T1 GL( V;) qui fixe les s1. D'apres 1 . 1 . 1 , une famille de structures de Hodge sur les V; s'identifie a un morphisme h : S --> T1 GL(V;). Pour que les si soient de type (0, 0), il faut et il suffit que h se factorise par G : il s'agit de considerer les morphismes algebriques h : S --> G. On peut regarder G, plutot que le systeme des V; et s1, comme 1' objet primordial : si G est un groupe algebrique lineaire reel, il revient au meme de se donner h : S -->
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G, ou de se donner sur chaque representation V de G une structure de Hodge, fonctorielle pour les G-morphismes et compatible aux produits tensoriels (cf. Saavedra, [10, VI, §2]). Les morphismes wh et fi.h de 1 . 1 . 1 deviennent des mor phismes de Gm dans G, et de Gm dans Gc respectivement. 1 . 1 . 12. La construction 1 . 1 . 1 1 amene a considerer les espaces de modules de structures de Hodge du type suivant : on fixe un groupe algebrique lineaire reel G, et on considere une composante connexe (topologique) X de l'espace des mor phismes ( = homomorphismes de groupes algebriques sur R) de S dans G. Soit G1 le plus petit sous-groupe algebrique de G par lequel se factorisent les h E X : X est encore une composante connexe de l'espace des morphismes de S dans G 1 . Puisque S est de type multiplicatif, deux elements quelconques de X sont conjugues : l'espace X est une classe de G1 (R)+-conjugaison de morphismes de S dans G. C'est aussi une classe de G(R)+-conjugaison, et G 1 est un sous-groupe invariant de la composante neutre de G. 1 . 1 . 1 3. Vu 1 . 1 .9 et 1 . 1 . 10, nous ne considerons que les X tels que, pour une famille fidele V,. de representations de G, on ait (a) Pour tout i, la graduation par le poids de v,. (de complexifiee la graduation de V;c par les V,1- = EB fi+q =n vr) est independante de h E X. Conditions equiva lentes : h(R*) est central dans G(R)O ; Ia representation adjointe est de poids 0. (f3) Pour une structure complexe convenable sur X, et tout i, la famille de struc tures de Hodge definie par les h E X est une variation de structure de Hodge sur X.
(r) Si V est la composante homogene d'un poids n d'un V,., il existe 1/f : V ® V --+ R( - n) qui, pour tout h E X, soit une polarisation de V. PROPOSITION 1 . 1 . 14. Supposons w!rifie (a) ci-dessus. (i) II existe une et une seule structure complexe sur X telle que les filtrations de Hodge des V,. varient holomorphiquement avec h E X. (ii) La condition 1 . 1 . 1 3(f3) est verifiee si et seulement si Ia representation adjointe est de type {( - 1 , 1), (0, 0), (1, - 1)}. (iii) La condition 1 . 1 . 1 3(r) est w!rifiee si et seulement si G 1 (defini en 1 . 1 . 12) est reductif et que, pour h E X, l'automorphisme interieur int h(i) induit une involution de Cartan de son groupe adjoint. (i) Soit V Ia somme des V,.. C'est une representation fidele de G. Une structure de Hodge est determinee par Ia filtration de Hodge correspondante (plus la graduation par le poids si on n'est pas dans le cas homogene) : en poids n = p + q, on a ypq = FP n f'q. L'application cp de X dans la grassmannienne de Vc : h r-+ la filtra tion de Hodge correspondante, est done injective. Nous allons verifier qu'elle identifie X a une sous-variete complexe de cette grassmannienne ; ceci prouvera (i) : Ia structure complexe sur X induite de celle de la grassmannienne est la seule pour laquelle cp soit holomorphe. Soient L l'algebre de Lie de G et p : L --+ End( V) son action sur V. L'action p est un morphisme de G-modules, injectif par hypothese. Pour tout h E X, c'est aussi un morphisme de structures de Hodge. L'espace tangent a X en h est le quotient de L par l'algebre de Lie du stabilisateur de h-a savoir le sous-espace LOO de L pour la structure de Hodge de L definie par h. L'espace tangent a la grassmannienne en cp(h) est End( Vc)/F0 (End( Vc)) . Enfin, dcp est le compose
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L/£00 __!____. End( V)/End( V)OO
]'
]'
Lc/FlLc__i__. End( Vc)/FlEnd( Vc) Puisque p est un morphisme injectif de structures de Hodge, drp est injectif; son image est celle de LcfFOLc, un sous-espace complexe, d'oil l'assertion. (ii) L'axiome de transversalite signifie que l'image de drp est dans p- 1 End( Vc)/£0End(Vc), i.e. que Lc = F- 1 Lc. Pour prouver (iii), nous ferons usage de [7, 2.8], rappele ci-dessous. Rappelons qu'une involution de Cartan d'un groupe algebrique lineaire reel (non necessaire ment connexe) G est une involution u de G telle que Ia forme reelle G11 de G (de conjugaison complexe g --+ u(g)) soit compacte : G11(R) est compact et rencontre toutes Ies composantes connexes de G11(C) = G(C). Pour C E G(R) de carre central, une C-po/arisation d'une representation V de G est une forme bilineaire ?f! G-invari ante, telle que ?f!(x, Cy) soit symetrique et defini > 0. Pour tout g E G(R), on a alors ?f!(x, gCg-ly) = ?f!(g-lx, Cg-ly) : Ia notion de C-polarisation ne depend que de la class de G(R)-conjugaison de C. Rappel. 1 . 1 . 1 5 [7, 2.8]. Soient G a/gebrique reel et C E G(R) de carre central. Les conditions suivantes sont equivalentes : (i) Int C est une involution de Cartan de G ; (ii) toute representation reel/e de G est C-po/arisable ; (iii) G admet une representation reel/e C-polarisable fide/e. On notera que la co ndition 1 . 1 . 1 5(i) entraine que GO est reductif-pour avoir une forme compacte. Elle ne depend que de Ia classe de conjugaison de C. Prouvons 1 . 1 . 14(iii). Soit G2 le plus petit sous-groupe algebrique de G par lequel se factorisent les restrictions des h E X a Ul c: C*. Pour qu'une forme bilineaire ?f! : V ® V --+ R( - n) verifie 1 . 1 . 1 3(r ), il faut et il suffit que (277:i) n ?J! : V ® V --+ R soit invariant par les h(U 1 )-donc par G2-(ceci exprime que ?f! est un morphisme), et une h(i)-polarisation. D'apres 1 . 1 . 1 5, 1 . 1 . 1 3(r) equivaut a : int h(i) est une involu tion de Cartan de G2• On en deduit d'abord que G1 est reductif: G2 l'est, pour avoir une forme com pacte, et G 1 est un quotient du produit Gm x G2. Puisque G2 est engendre par des sous-groupes compacts, son centre connexe est compact : il est isogene au quotient de G2 par son groupe derive. L'involution 0 = int h(i) est done une involution de Cartan de G2 si et seulement si e'en est une de son groupe adjoint, et on conclut en notant que G1 et G2 ont meme groupe adjoint. Les conditions en 1 . 1 . 14 ne dependant que de (G, X), on a le COROLLAIRE 1 . 1 . 1 6. Les conditions 1 . 1 . 1 3(a), (�), (r) ne dependent pas de lafamil/e fidele choisie de representations V;. CoROLLAIRE 1 . 1 . 17. Les espaces X de 1 . 1 . 1 3 sont les domaines hermitiens symetriques. A. Prouvons X de ce type. On se ramene successivement a supposer : (1) Que G = G1 : remplacer G par G1 ne modifie ni X, ni les conditions 1 . 1 . 1 3. (2) Que G est adjoint : par(l ), G est reductif, et son quotient par un sous-groupe
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central fini est le produit d'un tore T par son groupe adjoint G•d. L'espace X s'identifie encore a une composante connexe de l'espace des morphismes de SfGm dans G•d : si un tel morphisme se releve en un morphisme de S dans G, avec une projection donnee dans T, le relevement est unique. Les conditions enoncees en 1 . 1 . 14 restent par ailleurs verifiees. (3) Que G est simple : decomposer G en produit de groupes simples G; ; ceci decompose X en un produits d'espaces X; relatifs aux G;. So it done G un groupe simple adjoint, et X une G(R)+ -classe de conjugaison de morphismes non triviaux h : SfGm --+ G, verifiant les conditions de 1 . 1 . 14(ii), (iii). Le groupe G est non compact : sinon, int h(i) serait trivial (par (iii)), Lie G serait de type (0, 0) (par (ii)) et h serait trivial. Soit h E X. D'apres (iii), son centralisateur est compact; il existe done sur X une structure riemannienne G(R)+-equivariante. D'apres (ii), h(i) agit sur l'espace tangent Lie(G)/Lie(G)OO de X en h par - 1 : l'espace X est riemannien symetrique. On verifie enfin qu'il est hermitien symetrique pour Ia structure complexe 1 . 1 . 14(i). II est du type non compact (courbure < 0) car G est non compact. B. Reciproquement, si X est un espace hermitien symetrique, et que x E X, on sait que Ia multiplication par u (lui = 1) sur l'espace tangent T:x a X en x se prolonge en un automorphisme m:x(u) de X. Soient A le groupe des automorphismes de X, et h(z) = m(z/ t) pour z E C*. Le centralisateur A x de x commute a h, et Ia condition de 1 . 1 . 14(ii) est done verifiee : Lie(A:x) est de type {0, 0), et T:x = Lie(A)/Lie(A:x) de type {( - 1 , 1), (1, - 1)}. Enfin, on sait que A est Ia composante neutre de G(R), pour G adjoint, et que l'espace riemannien symetrique X est a courbure < 0 si et seulement si Ia symetrie h(i) fournit une involution de Cartan de G (voir Helgason [8]). 1 . 1 . 1 8 . Indiquons deux variantes de 1 . 1 . 1 5 (cf. [7, 2. 1 1]). (a) On donne un groupe algebrique reel reductif (0.2) G, et une classe de G(R) conjugaison de morphismes h : S --+ G. On suppose que wh-note w-est central, done independant de h (condition 1 . 1 . 1 3(a)), et que int h(i) est une involution de Cartan de Gfw(Gm ). Puisque G est reductif, w(Gm) admet un supplement G2 : un sons-groupe in variant connexe tel que G soit quotient de w(Gm) x G2 par un sous-groupe central fini. II est unique : engendre par le groupe derive et le plus grand sous-tore compact du centre. II contient les h(U1 ) (h E X), et int h(i) en est une involution de Cartan. Si V est une representation de G, sa restriction a G2 admet done une h(i)-polarisa tion f/J. Si V est de poids n, w(Gm ) agit par similitudes, done G de meme : pour une representation convenable de G sur R, f/J est covariant. Pour cette representation, R est de type (n, n) ; ceci permet de faire agir G sur R(n), de fa<;:on compatible a sa structure de Hodge, et de voir Iff = (27Ci)-"f/J comme une forme de polarisation G invariante : V ® V --+ R( - n) (b) Supposons que G se deduise par extension des scalaires a R de G0 sur Q, et que w soit defini sur Q. Le groupe G2 est alors defini sur Q, car c'est l'unique supplement de w(Gm ), et tout caractere de G/G2 est defini sur Q, car ce groupe est trivial ou isomorphe, sur Q, a Gm . Si une representation (rationnelle) V de G0 est de poids n, les formes bilineaires G-invariantes V ® V --+ Q( - n) forment un espace vectoriel F sur Q. L'ensemble de celles qui sont de polarisation (rei. h E X) est Ia trace sur F d'un ouvert de FR, et cet ouvert est non vide d'apres (a). II existe done des formes de polarisation 7/f : V ® V --+ Q(n) G-invariantes. .
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On prendra garde que les formes en (a) et (b) ne sont pas toujours de polariza tion pour tout h' E X: si h' = int(g)(h), la formule 1/f(x, h'(i)y) = gl/f (g-lx, h(i)g-ly) montre que la forme (271:i) nlf! (x, h'(i)y) est symetrique et definie-mais definie posi tive ou negative selon I' action de g sur R( - n) . 1 .2. Classification. Dans la suite de ce paragraphe, nous utilisons la relation 1 . 1 . 1 7 entre do maines hermitiens symetriques et espaces de modules de structures de Hodge, pour reformuler certains resultats de [1] et [8], et donner quelques complements. 1 .2. 1 . Considerons les systemes (G, X) formes d'un groupe algebrique reel simple adjoint G, et d'une classe de G(R)-conjugaison X de morphismes de groupes al gebriques reels h : S ---> G, verifiant (les notations sont celles de 1 . 1 . 1 , 1 . 1 . 1 1 ). (i) La representation adjointe Lie(G) est de type {( - 1 , 1), (0, 0), (1, - 1) } (en particulier, h est trivial sur Gm c S) ; (ii) int h(i) est une involution de Cartan ; (iii) h est non trivial ou-ce qui revient au meme (cf. 1 . 1 . 17)-G est non compact. D'apres 1 . 1 . 1 7, les composantes connexes des espaces X ainsi obtenus sont les domaines hermitiens symetriques irreductibles. L'hypothese (ii) assure que les involutions de Cartan de G sont des automor phismes interieurs, done que G est une forme interieure de sa forme compacte (cf. 1 .2.3). En particulier, G, etant simple, est absolument simple. La classe de G(C)-conjugaison de f.l-h : Gm ---> Gc ne depend pas du choix de h E X. Nous Ia noterons Mx. PROPOSITION 1 .2.2. Soit Gc un groupe algebrique complexe simple adjoint. A chaque systeme (G, X) forme d'une forme reelle G de Gc, et de X verifiant 1 .2. 1 (i), (ii), (iii) associons Mx · On obtient ainsi une bijection entre classes de Gc(C)-con jugaison de systemes (G, X), et classes de Gc(C)-conjugaison de morphismes non triviaux fl. : Gm ---> Gc verijiant Ia condition suivante. ( * ) Dans Ia representation ad fl. de Gm sur Lie(Gc), seuls apparaissent les caracteres z, 1 et z-1 .
Pour verifier 1 .2.2, nous utiliserons la dualite entre domaines hermitiens syme triques, et espaces hermitiens symetriques compacts : 1 .2.3. Soient G une forme reelle de Gc, X une classe de G(R)-conjugaison de mor phismes de S/Gm dans G, et h E X. La forme reelle G correspond a une conjugaison complexe (J sur Gc ; definissons G* comme la forme reelle de conjugaison complexe int (h(i)) (J : G*(R) = {g E (C) I g = int(h(i) ) (J(g) } .
Le morphisme h est encore defini sur R, de SfGm dans G* : on a h(C*JR*) c G *(R) ; definissons X* comme la classe de G*(R)-conjugaison de h. La construction (G, X) ---> (G*, X*) est une involution sur !'ensemble des classes de Gc(C)-conjugai son des systemes (G, X) formes d'une forme reelle G de Gc, et d'une classe de G(R) conjugaison de morphismes non triviaux de S/Gm dans G. Elle echange les (G, X) comme en 1 .2.2, et les (G, X) tels que G soit compact et que X verifie 1 .2. 1(i). On sait que les formes reelles compactes de Gc sont toutes conjuguees entre elles. Puisque si g E Gc normalise une forme reelle G, on a g E G(R) (ceci parce que G est adjoint), la dualite ramene 1 .2.2 a I' en once suivant :
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LEMME 1 .2.4. Soit G une forme compacte de Gc. La construction h --+ /-lh induit une bijection entre (a) classes de G(R)-conjugaison de morphismes h : S/Gm --+ G, verifiant 1 .2. l(i), et (b) classes de Gc(C)-conjugaison de morphismes p : Gm --+ Gc. verifiant 1 .2.2(•). Soient T un tore maximal de G, et Tc son complexifie. On verifie d'abord que !'application h --+ ph : Hom(S/Gm , T) --+ Hom(Gm , Tc) est bijective. Si W est le groupe de Weil de T, on sait que Hom(U l , T)/ W ____:____. Hom(U l , G)/G(R) et L'application h --+ /-lh induit done une bijection Hom(S/Gm , G)/G(R) ____:____. Hom(Gm , G)/GJC), et, pour que h verifie 1 .2.l(i), il faut et il suffit que p,. verifie 1 .2.2(•). 1 .2.5. Soit G un groupe algebrique complexe simple adjoint. Nous allons enume rer les classes de conjugaison de morphismes non triviaux p : Gm --+ G verifiant 1 .2.2(•), en terme du diagramme de Dynkin D de G. Rappelons que ce dernier est canoniquement attache a G en particulier, les automorphismes de G agissent sur D-On peut identifier ses sommets aux classes de conjugaison de sous-groupes paraboliques maximaux. Soient T un tore maximal, X(T) = Hom(T, Gm), Y(T) = Hom(Gm , T) (le dual de X(T) pour l'accouplement X(T) x Y(T) --+ o Hom(Gm , Gm) = Z), R c X(T) !'ensemble des racines, B un systeme de racines simples, a0 l ' oppose de la plus grande racine et B+ = B U {a0} . Les sommets de D sont parametres par B, et ceux du diagramme de Dynkin etendu n+ par B+. Une classe de conjugaison de morphismes de Gm dans G a un unique representant p E Y(T) dans la chambre fondamentale (a, p) � 0 pour a E B. Il est uniquement determine par les entiers positifs (a, p) (a E B) et, G etant adjoint, ceux-ci peuvent etre prescrits arbitrairement. La condition 1 .2.2( * ) , pour p non trivial, se recrit (ao , p) = 1 . -
-
Ecrivons la plus grande racine comme combinaison lineaire de racines simples, = 0, avec n(a0) = 1, et appelons speciaux les sommets de n+ tels que, pour la racine correspondante a E B+, on ait n(a) = 1 . On sait que le quotient du groupe des copoids par celui des coracines agit sur n+, et de fa<;on simplement transitive sur ]'ensemble des sommets speciaux. Les sommets speciaux soot done les conjugues sous Aut(D+) du sommet correspondant a a0, et leur nombre est l ' in dice de connexion l n- 1 (G) I de G (cf. Bourbaki [4, VI, 2 ex 2 et 5a)]). La condition ( *) se recrit (•)" Pour une racine simple a E B correspondant a un sommet special de D, on a (a, p) = 1 . Pour les autres racines simples, (a, p) = 0. 1 .2.6. Au total, les classes de Gc(C)-conjugaison de systemes (G, X) comme en 1 .2.2 soot parametrees par les sommets speciaux du diagramme de Dynkin D de Gc. En particulier, pour G une forme reelle donnee de Gc, X est determine par le I:aEB+n(a)a
'
VARIETES
DE
SHIMURA
259
sommet special s(X) correspondant ( G(R) c Gc(C) est en effet son propre normali sateur). Le sommet correspondant a x- 1 = {h- 1 1h E X} est le transforme de s(X) par !'involution d'opposition. Dans 1 .2.3, G et G* sont des formes interieures l'un de l'autre. S'il existe X verifiant 1 .2. 1 (i), (ii), (iii), G est done une forme interieure de sa forme compacte. En d'autres termes, Ia conjugaison complexe agit sur le diagramme de Dynkin de Gc par !'involution d'opposition. PROPOSITION 1 .2.7. Soit G un groupe algebrique reel simple adjoint, et supposons qu'il existe des morphismes h : C*/R* -+ G verifiant 1 .2. l {i), (ii), (iii). L'ensemble de ces morphismes a alors deux composantes connexes, echangees par h �--+ h- 1 . Chacune a pour stabilisateur Ia composante neutre G(R)+ de G. L'hypothese (ii) assure que le centralisateur K de h(i) est un sous-groupe compact maximal de G(R). En particulier, 7Co(K) ..::. 1CoG(R). II a meme algebre de Lie que le centralisateur de h. Ce dernier est un groupe algebrique connexe-en tant que centralisateur d'un tore-et compact-en tant que sous-groupe du centralisateur de h(i). II est done topologiquement connexe et Centr(h) = K+ = K n G(R)+. Le centre de K+ est de dimension 1 : le complexifie de K+ est le centralisateur de /1h done, d'apres ( * )", un sous-groupe de Levi d'un sous-groupe parabolique maximal. On peut aussi le deduire de ce que Ia representation de K+ sur Lie(G)/Lie(K+) est ir reductible (cf. [8, preuve de V, 1 . 1]). Le morphisme h est done un isomorphisme de SfGm avec le centre connexe de K+, et, K+ determine h au signe pres. A fortiori, h(i) determine h au signe pres. Des lors (a) L'application h �--+ h(i) est 2 : 1 . (b) Elle envoie isomorphiquement l'orbite G(R)+jK+ de h sous G(R)+ sur !'en semble G(R)/K de toutes les involutions de Cartan dans G(R). La proposition en resulte. CoROLLAIRE 1 .2. 8 . Soit (G, X) comme en 1 .2. 1 , et s le sommet correspondant du diagramme de Dynkin de Gc. (i) Si s n'est pas .fixe par /'involution d'opposition, G(R) et X sont connexes. (ii) Si s est fixe par /'involution d'opposition, G(R) et X ont deux composantes connexes; les composantes de X sont echangees par h �--+ h- 1 , et par les g E G(R) - G(R)+. Signalons que le cas (i) est encore caracterise par les conditions equivalentes (i') le systeme de racines relatif de G est de type C (plutot que BC) ; (i") X est un domaine tube. 1 .3 . Plongements symplectiques. 1 . 3. 1 . Soit V un espace vectoriel reel, muni d'une forme alternee non degeneree 1/f. Le demi-espace de Siegel s+ correspondent admet la description suivante : c'est l'espace des structures complexes h sur V, telles que 1/f soit de type ( 1 , 1 ) (pour !'identification (1 . 1 . 3) entre structures complexes et structures de Hodge de type {( - 1 , 0), (0, - 1)}) et que la forme 1/f(x, h(i)x) soit symetrique et definie positive. Si on remplace "defini positif" par "defini", le double demi-espace de Siegel S± obtenu est une classe de conjugaison de morphismes h : S -+ CSp(V) (CSp = similitudes symplectiques ; dans [5], ce groupe est note Gp). 1 .3.2. Soient G un groupe algebrique reel adjoint (0.2) et X une classe de conjugai son de morphismes h : S -+ G. On suppose verifiees les conditions (i), (ii) de 1 .2. 1 , et o n remplace (iii) par
260
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(iii') G est sans facteur compact. Le systeme (G, X) est done un produit de systemes (G,, X,) comme en 1 .2. 1 , et X, correspond a un sommet special du diagramme de Dynkin de G,c (1 .2.6). Considerons les diagrammes oil G est le groupe adjoint du groupe reductif Gh et oil X1 est une classe de G 1 (R)-conjugaison de morphismes de S dans G1 • On dispose d'une section G � Gh de sorte que V est une representation de G . Notre but est Ia determina tion 1 .3 . 8 des representations complexes irreductibles nontriviales W de G , qui est essentiellement equivalent a figurent dans Ia complexifiee d'une representation ainsi obtenue. Ce probleme celui resolu par Satake dans [11]. LEMME 1 . 3 . 3 . II suffit qu 'existe (Gh X1 ) � (G, X), comme ci-dessus, et une repre sentation lineaire ( V, p) de type {( - 1 , 0), (0, - 1)} de Gh telle que W figure dans Vc. Remplacant G1 par le sous-groupe engendre par le groupe derive G;_ et par I' image de h, on se ramene a supposer que int h(i) est une involution de Cartan de G 1 fw(Gm)· II existe alors sur V une forme de polarisation W ( 1 . 1 8(a)) , telle que p soit un mor phisme de (Gh X1 ) dans ( CSp(V), S±) . 1 .3 .4. Considerons le systeme projectif {H,.)n E N suivant : N est ordonne par divisibilite, H,. = Gm • et le morphisme de transition de H,.d dans Hn est x � xd (lim proj H,. est le revetement universel-au sens algebrique-de Gm). Un morphisme fractionnaire de Gm dans un groupe H est un element de lim inj Hom(H,., H). De meme pour le groupe S. Pour p. : Gm � H fractionnaire, defini par p." : H,. = Gm � H, et V une representation lineaire de H, V est somme des sous-espaces V11 (a E ( 1/ n)Z) tels que, via p. " , Gm agisse sur Va par multiplication par x" 11 • Les a tels que V11 =1- 0 sont les poids de p. dans V. De meme, un morphisme fractionnaire h : S � H determine une decomposition de Hodge fractionnaire p, s de V (r, s E Q). LEMME 1 . 3.5. Pour h E X, soit ilh le relevement fractionnaire de fl. h a Gc. Les re presentations W de 1 . 3.2 sont celles telles que p.h n 'ait que deux poids a et a + 1 . La condition est necessaire : Relevant h en h 1 E Xh on a fl.h, = p.h · v, avec v cen tral. Sur V, f1.h1 a les poids 0 et 1 . Si - a est !'unique poids de v sur W irreductible dans Vc, les seuls poids de P.h sur W sont a et a + 1 . Pour W non trivial, I' action de Gm via p.Hn assez divisible) est non triviale (car Gc est simple), done non centrale, et les deux poids a et a + 1 apparaissent. La condition est suffisante : Prenons pour V l'espace vectoriel reel sous-jacent a W, et pour groupe G 1 le groupe engendre par !'image de G , et par le groupe des homotheties. Pour h E X, de relevement fractionnaire ii a G, soit h l (z) = h(z)z-az l -a. Si Wa et Wa+l sont les sous-espaces de poids a et a + 1 de W, iz agit sur W11 (resp. Wa+ l ) par (z/z)a (resp. (zfz) l+ a), et h 1 par z (resp. z) : h 1 est un vrai mor phisme de S dans Gh de projection h dans G, et V est de type {( - 1 , 0) (0, - 1)} rei. h 1 . II ne reste qu'a appliquer 1 .3.3. 1 .3.6. Traduisons la condition 1 . 3.5 en terme de racines. Soient T un tore maximal de Gc, t son image inverse dans Gc, B un systeme de racines simples de T, et p. E Y(T) le representant dans Ia chambre fondamentale de Ia classe de conjugaison de fl.h (h E X). Si a est le poids dominant de W, le plus petit poids est - -r (a), pour ..-
26 1
VARilhES DE SHIMURA
!'involution d'opposition. II s'agit d'exprimer que (f-t, {)) ne prend que deux valeurs a et a + 1 , pour {3 un poids de W. Ces poids etant tous de la forme (a + une com binaison Z-lineaire de racine), et les (f-t, z ) , pour r une racine, etant entiers, la con dition s'exprime par (f-t, - z"(a)) = (f-t, a) - 1 , soit (f-t, a + -.(a)) = I . Determinons les solutions de (1 .3.6. 1). Pour tout poids dominant a, (f-t, a + -.(a)) est un entier, car a + -.(a) est combinaison Z-lineaire de racines. Si a =1- 0, il est > 0, sans quoi f-t annulerait tous les poids de la representation correspondante. Un poids dominant a verifiant (1 .3.6. 1) ne peut done etre somme de deux poids : (1 .3.6. 1)
LEMME 1 .3.7.
Seuls les poids fondamentaux peuvent verifier 1 .3.6. 1 .
1 .3.8. D'apres 1 .3. 7 , les representations W cherchees se factorisent par un facteur simple G, de G, et leur poids dominant est un poids fondamental ; il correspond a un sommet du diagramme de Dynkin D, de G,c. La condition necessaire et suf fisante (1 .3.6. 1) ne depend que de la projection de f-t dans G,c ; celle-ci correspond a un sommet special s de D, ( 1 . 2.6), et s a racine simple a5• Le nombre (f-t, m), pour m un poids, est le coefficient de as dans !'expression de m comme combinaison Q lineaire de racines simples. Pour m fondamental, ces coefficients sont donnes dans les tables de Bourbaki [4]. lis sont donnes par la table suivante, oil sont enumeres les diagrammes de Dynkin munis d'un sommet special (entoure). Chaque sommet correspond a un poids fondamental m , et on l'a affecte du nombre (f-t, m). Les sommets correspondant aux poids qui verifient (1 .3.6. 1) sont soulignes. TABLE 1 . 3.9. Dans la table, . . . indique une progression arithmetique. A p+ q- l (sommet special en pieme position)
.
. .
+ q) j(p + q) pqj(p + q) qj(p 1-- . .. . .... . . . ... . .... . .... -B1- . . . . . .. . . ......... . ...p. .. --:2._ •
•
•
1 /2 1 1 � ··················································· � . . . l/2 1 /2 1--- ··················································· �
i- . . . ..
D{!+ 2
:. :. .:. . . . . . . . . . . -<::: k ( : � r: : � k
... .... . . . .. . ....
�� ·�;, 2
·
5
T3 13
/2
;, �
5 /?
4/2 I
+ 1 /2
3 /2 -G)
+2
�
5)
262
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REMARQUES 1 .3. 1 0. (i) Pour G simple exceptionnel, aucune representation W ne verifie 1 .3.2. (ii) Pour G simple classique, sauf le cas lYf (I � 5), les representations W de 1 .3.2 forment un systeme fidele de representations de G. Pour IJll , on obtient seulement une representation fidele d'un revetement double de G (a savoir, Ia composante connexe algebrique du groupe des automorphismes d'un espace vectoriel sur H muni d'une forme antihermitienne non degeneree-une forme interieure de S0(2n)) .
2. Varietes de Shimura. 2.0. Preliminaires. 2.0. 1 . Soient G un groupe, r un sous-groupe et q; : r --+ L1 un morphisme. Sup
posons donnee une action r de L1 sur G, qui stabilise r, et telle que (a) r(q;(r)) est l'automorphisme interieur int7 de G ; (b) q; est compatible aux actions de L1 , sur F par r, e t sur lui-meme par automor phismes interieurs : q;(r(o)(r)) = int,;(q;(r)). Formons le produit semi-direct G ><1 L1. Les conditions (a), (b) reviennent a dire que !'ensemble des r · q;(r)- 1 est un sous-groupe distingue, et on definit G * rL1 comme Ie quotient de G ><1 L1 par ce sous-groupe. On notera que les hypotheses entrainent que Z = Ker(q;) est central dans G, et que Im(q;) est un sous-groupe distingue de L1. Les !ignes du diagramme o ___. z ___. r --- L1 ___. L1/r --- o (2.0. 1 . 1)
II
!'
!'
II
0 ---> Z ---> G --> G * r L1--+ L1/F --> 0
soot exactes, d'ou un isomorphisme (2.0. 1 .2) F\ G -"'---. L1\ G * r L1 et, mise en evidence, une action a droite de G * rL1 sur F\G. Pour cette action, G agit par translations a droite, et L1 par I' action a droite r - 1 . Si G est un groupe topologique, que L1 est discret, et que !'action r est continue, la groupe G * r L1, muni de la topologie quotient de celle de G ><1 L1, est un groupe topologique, G/Ker(q;) en est un sous-groupe ouvert, et !'application (2.0. 1 .2) est un homeomorphisme. La construction 2.0. 1 garde un sens dans la categorie des groupes algebriques sur un corps. Si G est un groupe reductif sur k, on a un isomorphisme canonique G = G *z eal Z(G) (pour l'action triviale de Z(G) sur G). 2.0.2. Soient G un groupe algebrique sur un corps k, et G•d le quotient de G par son centre Z. L'action par automorphismes interieurs de G sur lui-meme (x, y) --+ xyx- 1 : G x G --+ G est invariante par Z x { e} agissant par translation, done se factorise par une action de G•d sur G. Prendre garde que !'action de r E G•d(k) sur G(k) n'est pas necessairement un automorphisme interieur de G(k) (Ia projection de G(k) dans G•d(k) n'est pas toujours surjective). Un exemple typique est !'action de PGL(n, k) sur SL(n, k). De meme, !'application "commutateur" (x, y) = xyx-1y -1 : G X G --+ G est invariante par Z x Z agissant par translation, et se factorise par une application "commutateur" ( , ) : G•d x G•d --+ G.
VARIETES DE SHIMURA
263
Tout ceci, et le fait que ces "commutateurs" et "automorphismes interieurs" verifient les identites usuelles se voit au mieux par descente, i.e. en interpretant G comme un faisceau en groupes sur un site convenable, et G•d comme le quotient de ce faisceau en groupes par son centre. En caracteristique 0, si on s'interesse seule ment aux points de G sur des extensions de k, il suffit d'utiliser Ia descente galoi sienne-cf. 2.4. 1 , 2.4.2. Variante. Pour G reductif sur k, les groupes G et G ont meme groupe adjoint, et les constructions precedentes pour G et G sont compatibles. En particulier, !'ap plication commutateur ( , ) : G x G -+ G a une factorisation canonique ( ' ) : G X G ---+ G•d X G•d ---+ (} ---+ G. On en deduit que le quotient de G(k) par le so us-groupe distingue pG(k) est abelien. 2.0.3. Soient k un corps global de caracteristique 0, A l'anneau de ses adeles, G un groupe semi-simple sur k et N = Ker(p : G -+ G). Soient S un ensemble fini de places de k, As l'anneau des S-adeles (produit restreint etendu aux v ¢ S) et posons Fs = pG(A5) n G(k) (intersection dans G(A5)). C'est le groupe des ele ments de G(k) qui, en toute place v ¢ S, peuvent se relever dans G(kv) (se rappeler que p : G(A) -+ G(A) est propre). La suite exacte longue de cohomologie identifie G(k)/pG(k) a un sous-groupe de H 1 (Gal(k/k), N(k)), et F5/pG(k) aux elements localement nul, en les places v ¢ S, de ce sous-groupe. En particulier, F5/pG(k) est contenu dans le sous-groupe H�(Gai(k/k), N(k)) des classes dont Ia restriction a tout sous-groupe monogene _est triviale (argument et notations de [12]). Si Im Gal(k/k) est l'image de Galois dans Aut N(k), on a. H�(Gal(k/k), N(k)) = H�(Im Gal(k/k), N(k)) (loc. cit.) ; en particulier, F5/pG(k) est fini. PROPOSITION 2.0.4. (i) Fs ne depend que de /'ensemble des places v E S oil le groupe de decomposition Dv c Im Gal(k/k) est non cyclique. En particulier, il ne change pas si on ajoute a
s les places a l'infini.
(ii) F5/pG(k) s 'identifie au sous-groupe du groupe fini H�(Im Gal(k/k), N(k)) forme des classes de restriction nulle a tout sous-groupe de decomposition Dv, v ¢ S. En particu/ier, pour S grand, on a F5/pG(k) = H�(lm Gal(k/k), N(k)). La restriction d'un element de H�(Im Gal(k/k), N(k)) a un groupe de decom position cyclique est automatiquement nulle, d'oi:l (i). Pour (ii), on peut supposer que S contient les places a l'infini. Le principe de Hasse pour G (pour des classes venant du centre) assure alors que tous les elements du groupe (ii) sont effective ment realises comme classe d'obstruction. COROLLAIRE 2.0.5. Tout sous-groupe de S-congruence assez petit de G(k) est dans Fs .
Si U est un sous-groupe de S-congruence, UfU n pG(k) est fini : !'obstruction a relever dans G(k) meurt dans une extension galoisienne de degre et de ramification bornees, done est dans H 1 (Gal, N(k)) pour Gal un quotient fini de Gal(k/k). Des conditions de S-congruence permettent alors de passer de ce H 1 a F5/pG(k), cf.
[12].
REMARQUE
2.0.6. On notera par ailleurs que si G verifie le theoreme d'approxi-
264
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mation forte rei. S, tout sous-groupe de S-congruence sur F8/pG(k).
U
c Fs
de G(k) s'envoie
CoROLLAIRE 2.0.7. Pour toute place archimedienne v, un sous-groupe de S-con gruence assez petit U de G(k) est dans Ia composante connexe topologique G(kv)+ de G(kv).
Puisque G(k.) est connexe, on a G(k")+ = pG(k.) et (2.0.4 et 2.0.5).
U
c Fs = Fsu tvl c G(k.)+
CoROLLAIRE 2.0.8. Le sous-groupe G(k)pG(A8) de G(A8) est ferme, topo/ogique ment isomorphe a pG(As) *rs G(k) (i.e. pG(A8) en est un sous-groupe ouvert).
C'est un sous-groupe parce que, vu 2.0.2, pG(A8) est distingue dans G(A8), avec un quotient commutatif. Soit T :=> S assez grand pour que G(k) soit dense dans G(Ar) (approximation forte). Notons kr- s le produit des kv pour v E T - S. Pour K un sous-groupe compact ouvert de G(Ar) , on a G(k)pG(A8) = G(k)p(G(k) · G(kr-s)
x
p- 1K) c G(k)(pG(kr- s) x K) .
D'apn!s 2.0. 5, pour K assez petit, on a dans G(Ar) : G(k) n K c Fr , d'ou dans G(As) : G(k) n (pG(kr-s) X K) c rs c pG(As)· L'intersection de G(k)pG(As) avec le sous-groupe ouvert pG(kr-s) x K est done contenue dans pG(A8), et le corollaire en resulte. COROLLAIRE 2.0.9. Si (} verifie /e theoreme d'approximation forte rei. S, /'adherence de G(k) dans G(A8) est G(k) · pG(A8).
2.0. 1 0. So it T un tore sur k et S un ensemble fini de places contenant les places archimediennes. Soit U c T(k) le groupe de S-unites. D'apres un theoreme de Chevaelly, tout sous-groupe d'indice fini de U est un sous-groupe de congruence (voir [12] pour une demonstration elegante). II en resulte que si T' --> T est une iso genie, l'image d'un sous-groupe de congruence pour T' est un sous-groupe de con gruence pour T. 2.0. 1 1 . Soient G reductif sur k, p : (} --> Gder le revetement universe I de son groupe derive, et zo Ia composante neutre de son centre Z . Voici quelques corollaires de 2.0. 1 0 (on suppose que !'ensemble fini S de places contient les places archimedien nes). CoROLLAIRE 2.0. 1 2. Pour U d'indice fini dans le groupe des S-unites de Z(k) il existe un sous-groupe compact ouvert K de G(As) te/que
G(k) n (K . Gder( As) ) c Gder (k) . U. Appliquons 2.0. 10 a l'isogenie zo --> GJGder : pour K petit, un element r de G(k) dans K · Gd•r(As) a dans (GJGd•r)(k) une image petite, pour Ia topologie des sous groupes de S-congruence, done peut se relever un petit element z de Z(k), et r =
z.
CoROLLAIRE 2.0. 1 3 .
Le produit d'un sous-groupe de congruence de Gder et d'un sous-groupe d'indice fini du groupe des S-unites de Z0(k) est un sous-groupe de S congruence de G(k).
VARIETES DE SHIMURA
265
COROLLAIRE 2.0. 1 4. Tout sous-groupe de S-congruence assez petit de G(k) est contenu dans Ia composante neutre topologique G(R)+ de G(R). Appliquer 2.0. 1 3, 2.0.7 a Gd•r , et 2.0. 10 a zo. 2.0. 1 5 . On sait que Gder(k)pG(A) est ouvert dans G(k)pG(A) (car image inverse de {e} c le sous-groupe discret GfGder(k) de (G/Gd•r)(A)) . D'apres 2.0.8, G(k)pG(A) est done un sous-groupe ferme de G(A). On pose
(2.0. 1 5 . 1 )
n-(G) = G(A)/G(k)pG(A).
L'existence de commutateurs 2.0.2 montre que !'action de G•d (k) sur n-(G), deduite de !'action 2.0.2 de G•d sur G, est triviale. 2. 1 . Varietes de Shimura. 2. 1 . 1 . Soient G un groupe reductif, defini sur Q, et X une classe de G(R)-conjugai son de morphismes de groupes algebriques reels de s dans GR . On suppose verifies les axiomes suivant (les notations sont celles de 1 . 1 . 1 et 1 . 1 . 1 1 ) : (2. 1 . 1 . 1) Pour h e X, Lie(GR) est de type {( - 1 , 1), (0, 0), ( 1 , - 1) } . (2. 1 . 1 .2) L'involution int h(i) est une involution de Cartan du groupe adjoint GRad · (2. 1 . 1 . 3) Le groupe adjoint n'admet pas de facteur G' defini sur Q sur lequel la projection de h soit triviale. L'axiome 2. 1 . 1 assure que le morphisme wh (h e X) est a valeurs dans le centre de G, done est independant de h. On le note Wx, ou simplement w. Quelques simpli fications apparaissent lorsqu'on suppose que : (2. 1 . 1 .4) Le morphisme w : Gm --+ GR est defini sur Q. (2. 1 . 1 . 5) int h(i) est une involution de Cartan du groupe (G/w(Gm)) R· D'apres 1 . 1 . 14(i), X admet une unique structure complexe telle que, pour toute representation V de GR , Ia filtration de Hodge Fh de V varie holomorphiquement avec h. Pour cette structure complexe, les composantes connexes de X sont des domaines hermitiens symetriques. La preuve de 1 . 1 . 17 montre aussi que si l'on decompose G'Ji en facteurs simples, h se projette trivialement sur les facteurs com pacts, et que chaque composante connexe de X est le produit d'espaces hermitiens symetriques correspondant aux facteurs non compacts. L'axiome 2. 1 . 1 . 3 peut en core s'exprimer en disant que G•d (resp. G, cela revient au meme) n'a pas de facteur G' {defini sur Q) tel que G'(R) soit compact, et le theoreme d'approximation forte assure que G(Q) est dense dans G(Af). 2. 1 .2. Les varietes de Shimura KMc (G, X)-ou simplement KMc-sont les quo tients K Mc(G, X) = G(Q)\X x (G(Af)/K) pour K un sous-groupe compact ouvert de G(Af). D'apres 1 .2.7, et avec les notations de 0.3, 1 'action de G(R) sur X fait de n-0 (X) un espace principal homogene sous G(R)/G(R)+ . Puisque G(Q) est dense dans G(R) (theoreme d'approximation reel), on a G(Q)/G(Q)+ ...::. G(R)/G(R)+ , et, si x+ est une composante connexe de X, on a K Mc(G, X) = G(Q) +\X+ x (G(Af)/K) .
Ce quotient est Ia somme disjointe, indexee par !'ensemble fini G(Q)+\G(Af)/K de doubles classes, des quotients Fg\X+ du domaine hermitien symetrique x+ par les images Fg C:: G•d(R)+ des sous-groupes r; = gKg- l n G(Q)+ de G(Q)+ . Les Fg sont des groupes arithmetiques, d'ou une structure d'espace analytique sur Fg\X+ .
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L'article [2] fournit une structure naturelle de variete algebrique quasi-projective sur ces quotients, done sur KMe(G, X). Si rg est sans torsion (tel est le cas pour K assez petit) ; il resulte de [3] que cette structure est unique. Plus precisement, p our tout schema reduit Z, un morphisme analytique de Z dans Fg\ X+ est automatique ment algebrique. 2. 1 .3. On a
n-0 KMe = G(Q) \ n-0(X)
(G(Af)/K) = G(Q)\G(A)/G(R)+
K = G(Q)+\G(A!)fK. Puisque G(Af)/K est discret, on peut remplacer G(Q)+ par son adherence dans G(Af). La connexite de G(R) assure que pG(Q) c G(Q)+ . Par le the o reme d'approxi mation forte pour G, pG(Q) est dense dans pG(Af), et G(Q)+ ::::> pG(Af). Des lo rs, x
?ro KMe = G(Q)+pG(Af)\G(Af)/K = G(A!)fpG(Af) · G(Q)+ . K
(2. 1 .3.1)
x
= G(Af)/G(Q)+ . K,
puisque pG(Af) est u n sous-groupe distingue, avec u n quotient abelien. De meme, posant x 0n-(G) = n-0n-(G)/n-0 G(R)+ , on a ?ro K Me = G(A)/pG(A) = n-(G)/G(R)+
(2. 1 .3.2)
G(Q) · G(R)+ x K x K x0n-(G)/K. ·
=
En particulier, n-0 KMe ne depend que de l'image de K dans G(A)/pG(A). 2. 1 .4. Pour K variable (de plus en plus petit), les KMe forment un systeme pro jectif. II est muni d'une action a droite de G(Af) : un systeme d'isomorphismes g : KMe � g -!Kg Me. II est commode de considerer p1utot le schema Mc(G, X) -ou simplement Me-limite projective des KMe. La limite projective existe parce que les morphismes de transition sont finis. Ce schema est muni d'une action
a droite de G(Af), et il redonne les KMe : KMe = Me iK.
Nous nous proposons de determiner Me et sa decomposition en composantes connexes. DEFINITION 2. 1 .5. Fixons une composante connexe x+ de X. La composante neutre M8 de Me est Ia composante connexe qui contient /'image de x+ x {e} c X X G(Af). DEFINITION 2. 1 .6. Soient G0 un groupe adjoint sur Q, sans facteur G� defini sur Q tel que G�(R) soil compact et G1 un revetement de G0 . La topologie -r(G1) sur G0(Q) est cel/e admettant pour systeme fondamental de voisinages de /'origine /es images des sous-groupes de congruence de G1(Q). Nous noterons 11 (rei. G1), ou simplement 11, Ia completion pour cette topologie. Soit p : G0 --> G1 ]'application naturelle, notons - ]'adherence dans G1(Af), et posons r pG0(A) n G1(Q). Puisque G0(R) est connexe, r c G1(Q) + . On a
=
(2.0.9, 2.0. 14)
Go(Q)11 (rel. G1) = G1(Q) - *c1 cQl Go (Q) pGo(A!) *r Go( Q), Go (Q)+II(rel. G1) = GI (Q)+ *c1 CQl + Go(Q) + pGo (Af) *r Go( Q) + . PROPOSITION 2. 1 .7. La composante neutre M8 est Ia limite projective des quotients F\X+ , pour r un sous-groupe arithmetique de G•d(Q) + , ouvert pour Ia topo/ogie -r(Gd•r).
(2. 1 .6. 1 ) (2. 1 .6.2)
=
=
267
VARIETES DE SHIMURA
D'apn!s 2. 1 .2, c'est Ia limite des F\X 0 , pour F l'image d'un sous-groupe de con gruence de G(Q)+ · Le Corollaire 2.0. 1 3 permet de remplacer G par Gd•r. 2. 1 .8. La projection de G dans G•d induit un isomorphisme de X+ avec une classe de G(R)+-conjugaison de morphismes de S dans G'k_d et, d'apres 2. 1 .7, M8(G, X) ne depend que de G•d, Gd•r et de cette classe. Formalisons cette remarque. Soient G un groupe adjoint, x+ une classe de G(R)+-conjugaison de morphismes de S dans G(R) qui verifie (2. 1 . 1), (2. 1 .2), (2. 1 .3) et G1 un revetement de G. Les varietes con nexes de Shimura (rei. G, G1, X+) sont les quotients F\X+, pour F un sous-groupe arithmetique de G(Q)+ , ouvert pour Ia topologie -r(G1) . On note M8(G, Gh x+) leur limite projective, pour r de plus en plus petit. On notera que !'action par transport de structure de G(Q)+ sur M8(G, Gh x+) se prolonge par continuite en une action du complete G(Q)+A (rei. G1). Avec les notations de 2. 1 .7, et !'identification ci-dessus de x+ avec une classe de G(R)+-conjugaison de morphismes de S dans G'k_d , on a M8(G, X)
= M8(G•d, Gd•r, x+).
2. 1 .9. Soient Z le centre de G, et Z(Q)- !'adherence de Z(Q) dans Z(A f). D'apres Chevalley (2.0. 10), c'est le complete de Z(Q) pour Ia topologie des sous-groupes d'indice fini du groupe des unites ; il re<;oit isomorphiquement !'adherence de Z(Q) dans n:0Z(R) x Z(Af). Pour K c G(A f) compact ouvert, on a Z(Q) · K Z(Q)- · K (dans Z(A!)), et K Mc = G(Q)\X
=
G(Q) Z(Q)
=
x
(G(A!) fK) =
\x
x
CoROLLAIRE
x
=
G(Q) Z(Q)
\X
x
-
(G(A!) fZ(Q) ) .
2. 1 . 1 1 . Si les conditions 2. 1 .4 e t 2. 1 . 5 sont verifiees, o n a Mc(G, X)
G(Q)\X X G(A!).
Dans ce cas, Z(Q) est discret-dans Z(A!) et Z(Q)-
COROLLAIRE
(G(A!) fZ(Q) · K)
( G(A!)fZ(Q)-) est propre. Ceci permet le pas
2. 1 . 10. On a Mc(G, X)
x
( G(At) f Z(Q)- . K) .
L'action de G(Q)/ Z(Q) sur X sage a Ia limite sur K: PROPOSITION
���j \X
2. 1 . 12. L'action
a droite de
=
= Z(Q) .
G(A!) se factorise par G(A!)JZ(Q)-.
=
2. 1 . 1 3 . Soient G•d(R)1 !'image de G(R) dans G•d (R), et G•d(Q)1 G•d(Q) n G•d(Rk L'action 2.0.2 de G•d sur G induit une action (a gauche) de G•d(Q)1 sur le systeme des K Mc i nt(r) : K Mc � rKr-1 Mc,
et a la limite sur Me. Pour r E G•d(Q)+, cette action stabilise la composante neutre (done toutes les composantes, cf. ci-apn!s) et y induit !'action 2. 1 .6. Convertissons cette action en une action a droite, notee · r · Si r est !'image de
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PIERRE DELIGNE
o e G(Q), l'action · r coincide avec l'action de o , vu comme element de G(Af) : pour u E Me image de (x , g) E X X G(AI), u . r est image de (r- l (x) , int71 (g)) = (o-t(x) , o- lgo) ,..., (x , go) mod G(Q) a gauche.
Au total, nous obtenons ainsi une action a droite sur Me du groupe
(2. 1 . 1 3. 1)
G(A ! ) G•d(Q) 1 - G(Af) * G•d(Q)+ . Z(Q) - G (Q)+IZ(Q) Z(Q) - * G (Q)IZ(Q) _
PROPOSITION 2. 1 . 14. L'action a droite de G(Af) sur 7eoMefait de 7eoMc un espace principal homogene sous son quotient abelien G(A!)JG(Q)+ = 1r o7e( G). Cela resulte aussitot par passage a Ia limite des formules 2. 1 .3. 2. 1 . 1 5. Puisque G•d(Q) agit trivialement sur 1e(G) (2.0. 1 5), et que G•d(Q) + sta bilise au moins une composante connexe (2. 1 . 1 3), le groupe G•d(Q) + les stabilise toutes. Pour !'action 2. 1 . 1 3 du groupe (2. 1 . 1 3 . 1 ) sur Me. le stabilisateur de chaque composante connexe est done
(2. 1 . 1 5. 1)
��S��
* G (Q)+IZ(Q) G•d(Q)+ =
2 . o. 1 3
G•d(Q) +A(rel. Gd•r).
Resume 2. 1 . 16. Le groupe G(AI)jZ(Q)- * G CQ) IZ CQ) G"d(Q)1 agit a droite sur Me. L 'ensemble profini 7eoMc est un espace principal homogime sous taction du quotient abelien G(AfjG(Q)+ = 1r o7e(G) de ce groupe par /'adherence de G•d(Q)+ . Cette ad herence est le complete de G•d(Q)+ pour Ia topologie des images des sous-groupes de congruence de Gder(Q). L 'action de ce complete sur Ia composante neutre, une fois convertie en une action a gauche, est /'action 2. 1 .8. 2.2. Mode/es canoniques. 2.2. 1 . Soient G et X comme en 2. 1 . 1 . Pour h E X, le morphisme /-th (1 . 1 . 1 , com plete par 1 . 1 . 1 1) est un morphisme sur C de groupes algebriques definis sur Q : /-th : Gm � Gc. Le corps dual ( = reflex field) E(G, X) c C de (G, X) est le corps de definition de sa classe de conjugaison. Si x+ est une composante connexe de X, on le notera parfois E(G, X+). Soient (G', X') et (G", X") comme en 2. 1 . 1 . Si un morphisme f : G' � G" envoie X' dans X", on a E(G', X') => E(G", X"). 2.2.2. Soient T un tore, E un corps de nombres, et f-t un morphisme, defini sur E, de Gm dans TE. Le groupe E * , vu comme groupe algebrique sur Q, est Ia restric tion des scalaires a Ia Weil RE10(Gm). Appliquant .RE1o ·a f-t• on obtient RE1o(f-t) : E * � REIQ TE . On dispose aussi du morphisme norme NEto : RE1oTE � T (sur les points ra tionnels, c'est Ia norme, T(E) � T(Q)) . D'ou par composition un morphisme NE1o o RE1o(f-t) : E * � T. Nous le noterons simplement NRE(f-t), ou meme NR(p,). Si E' est une extension de E, f-t est encore defini sur E', et (2.2.2. 1)
2.2.3. Soient en particulier T un tore, h : S � TR et X = {h } . Si E c C contient E(T, X), le morphisme /-th est defini sur E, d'ou un morphisme NR(f-th) : E * � T. Passant aux points adeliques modulo les points rationnels, on en deduit un homo morphisme du groupe des classes d'ideles C(E) de E dans T(Q)/T(A), et, par pas sage aux ensembles de composantes connexes, un morphisme
VARIETES DE SHIMURA
269
noNR(p.h) : no C(E) -+ n0(T(Q)/T(A)). La theorie du corps de classe global identifie n0 C(E) au groupe de Galois rendu abelien de E. Le groupe n0 (T(Q)\ T(A)) est un groupe profini, limite projective des groupes finis T(Q)\T(A)/T(R)+ x K pour K compact ouvert dans T(Af). C'est n0T(R) x T(Af)/T(Q)-. Les varietes de Shimura KMc(T, X) sont les ensembles finis T(Q)\{h} x T(Af)/K = T(Q)\ T(A!)fK. Leur limite projective T(A!)fT(Q)-, calculee en 2. 1 . 10 est le quotient de n0(T(Q)\T(A)) par n0 T(R). Nous appellerons morphisme de reciprocite le morphisme rE(T, X) : Gal(Q/E) a b -+ T(Af)/T(Q)- inverse du compose de l'isomorphisme de la theorie du corps de classe global (0.8), de n0 NR(p.h) et de Ia projection de n0 T(A)/T(Q) sur T(A!)fT(Q)-. II definit une action rE de Gai(Q/E)ab sur les KMc(T, X) : Q ..... Ia translation a droite par re (T, X) (Q). Le cas universe! (en E) est celui oil E = E(T, X) : il resulte de (2.2.2. 1) que !'ac tion rE de Gal(Q/E) est Ia restriction a Gal(Q/E) c Gal(Q/E(T, X)) de rE c r . x l . 2.2.4. Soient G et X comme en 2. 1 . 1 . Un point h e X est dit special, ou de type CM, si h : S -+ G(R) se facto rise par un tore T c G defini sur Q, On notera que si T est un tel tore, !'involution de Cartan int h(i) est triviale sur !'image de T(R) dans le groupe adjoint, et que cette image est done compacte. Le corps E(T, {h}) ne de pend que de h. C'est le corps dual E(h) de h. No us transporterons cette terminologie aux points de K M0( G, X) et de Mc(G, X) : pour x e KMc(G, X) (resp. Mc(G, X)), classe de (h, g) e X x G(A!), Ia classe de G(Q)-conjugaison de h ne depend que de x. Nous dirons que x est special si h l'est, que E(h) est Ie corps dual E(x) de x, et que Ia classe de G(Q)-conjugaison de h est le type de x. Sur !'ensemble des points speciaux de KMc(G, X) (resp. de M0(G, X)) de type donne, correspondant a un corps dual E, nous allons definir une action r de Gai(Q/E). Soient done x e KMc(G, X) (resp. Mc(G, X)), classe de (h, g) e X x G(Af), T c G un tore defini sur Q par lequel se factorise h, Q e Gai(Q/E), et r(Q) un representant dans T(Af) de rE(T, {h})(Q) e T(Af)/T(Q)-. On pose r(Q)x = classe de (h, r(Q)g) . Le lecteur verifiera que cette classe ne depend que de X et de (1 . L'action ainsi definie commute a !'action a droite de G(A!) sur M0(G, X). 2.2. 5. Un modele canonique M(G, X) de M0(G, X) est une forme sur E(G, X) de M0(G, X), muni de !'action a droite de G(Af), telle que (a) les points speciaux sont algebriques ; (b) sur !'ensemble des points speciaux de type -. donne, correspondant a un corps dual E(-r), le groupe de Galois Gal(Q/E(-r)) c Gal(Q/E(G, X)) agit par !'action 2.2.4. Par "forme" nous entendons : un schema M sur E(G, X), muni d'une action a droite de G(Af), et d'un isomorphisme equivariant M ® E cc . x l C ...::::. M0(G, X). Soit E c C un corps de nombres qui contient E(G, X) . Un modele faiblement canonique de M0(G, X) sur E est une forme sur E de M0(G, X), muni de !'action a droite de G(At), qui verifie (a) et (b * ) meme condition que (b), avec Gal(Q/E(-r)) remplace par Gal(Q/E(-r)) n Gal(Q/E). 2.2.6. Dans [5, 5.4, 5. 5], nous inspirant de methodes de Shimura, nous avons
270
PIERRE DELIGNE
montre que Mc(G, X) admet au plus un modele faiblement canonique sur E (pour E(G, X) c E c C), et que, lorsqu'il existe, il est fonctoriel en (G, X). 2.3. Construction de modeles canoniques. Dans ce numero, nous determinons des cas oil s'applique le critere suivant, demontre dans [5, 4.2 1 , 5.7], pour construire des modeles canoniques. Critere 2.3. 1 . Soient (G, X) comme en 2. 1 . 1 , V un espace vectoriel rationnel, muni d'une forme alternee non degeneree l/f, et S± le double demi-espace de Siegel cor respondant (if 1 .3.1). S'i/ existe un p/ongement G 4 CSp( V), qui envoie X dans S±, alors Mc(G, X) admet un modele canonique M(G, X).
PROPOSITION 2.3.2. Soient (G, X) comme en 2. 1 . 1 , w = wh (h e xr et ( V, p) une representation fidele de type {( - 1 , 0), (0, - 1)} de G. Si int h(i) est une involution de Cartan de GRfw(Gm), il existe une forme alternee l/f sur V, te/le que p induise (G, X) 4 {CSp( V), S±) .
Par hypothese, Ia representation fidele V est homogene de poids 1 . Le poids w est done defini sur Q, et on prend pour l/f une forme de polarisation comme en 1 . 1 . 1 8(b). -
COROLLAIRE 2.3.3. Sojent (G, X) comme en 2. 1 . 1 , W = Wh (h E X), et ( V, p) une representationfidele de type {( - 1 , 0), (0, - 1)} de G. Si le centre zo de G se dep/oye sur un corps de type CM, il existe un sous-groupe G2 de G, de meme groupe derive et par lequel sefactorise X, et uneforme alternee l/f sur V, telle que p induise (G2, X) 4 {CSp( V), S±)) .
L'hypothese sur zo revient a dire que le plus grand sous-tore compact de ZB est defini sur Q. On prend G2 engendre par le groupe derive, ce tore, et l'image de w, et on applique 2.3.2. 2.3.4. Soit (G, X) comme en 2. 1 . 1 , avec G Q-simple adjoint. L'axiome (2. 1 . 1 .2) assure que GR est forme interieure de sa forme compacte. Exploitons ce fait. (a) Les composantes simples de GR sont absolument simples. Si on ecrit G comme obtenu par restriction des scalaires a Ia Weil : G = RF!0G• avec G• absolument simple sur F, cela signifie que F est totalement reel. Posons les notations : I = }'ensemble des plongements reels de F, et, pour v E I, G. = G•®F. v R, D. = dia gramme de Dynkin de Gvc · On a GR = TI G., Gc = IJ G.c, et le diagramme de Dynkin D de Gc est Ia somme disjointe des D.. Le groupe de Galois Gal(Q/Q) agit sur D et I, de facon compatible a la projection de D sur /. (b) La conjugaison complexe agit sur D par l'involution d'opposition. Celle-ci est centrale dans Aut(D). Des lors, Gal(Q/Q) agit sur D via une action fidele de Gal(Kn/Q), avec Kn totalement reel si l'involution d'opposition est triviale, quad ratique totalement imaginaire sur un corps totalement reel sinon. 2.3.5. On a X = IJ x., pour X. une classe de G.(R)-conjugaison de morphismes de S dans G• . Pour G. compact, x. est trivial. Pour G. noncompact, x. est decrit par un sommet s. du diagramme de Dynkin D. de G.c (1 .2.6). Quelques notations : Ic = I' ensemble des v E I tels que G. so it compact, Inc = I- I., De (resp. Dnc) = Ia reunion des D. pour v E Ic (resp. v E Inc), Gc (resp. Gnc) = le produit des G. pour v E Ic (resp. v E Inc) ; de meme pour les revetements univeG sels ; enfin, 2(X) = I' ensemble des s. pour v E Inc · La definition 2.2. 1 donne : ·
VARIET!�S
DE
SHIMURA
27 1
PROPOSITION 2.3.6. Le corps dual de (G, X) est le sous-corps de KD fixe par le sous-groupe de Gal(KD/Q) qui stabilise l,'(X).
2.3.7. Supposons qu'il existe un diagramme
(2.3.7. 1) Le revetement universe! G de G se releve dans Gh ce qui permet de restreindre Ia representation V a G. Le quotient de G qui agit fidelement est par hypothese le groupe derive de G1 . Appliquons 1 .3.2, 1 .3.8 au diagramme On trouve que les composantes irreductibles non triviales de Ia representation Vc de Gncc se factorisent par l'un des G.c (v e Inc), et que leur poids dominant est fondamental, de l'un des types permis par Ia Table 1 .3.9. L'ensemble des poids dominants des composantes irreductibles de Ia representation Vc de Gc est stable par Gal(Q/Q). Puisque Gal(Q/Q) agit transitivement sur I, et que Inc -:f. 0 , on trouve que (a) Toute composante irreductible W de Vc est de Ia forme ®vET w., avec w. une representation fondamentale de Gvc ( v e T c I), correspondant a un sommet 't"(v) de D •. Nous noterons 9'( V) !'ensemble des 't"( T) c D pour W c Vc irreducti ble. (b) Si S E 9'( V), S n Dnc est vide ou reduit a un seul point ss E Dv (v E Inc), et, dans Ia Table 1 .3.9 pour (D., s.), s8 est l'un des sommets soulignes. (c) 9' est stable par Gal(Q/Q). On a 9' rt. { 0 } . S i un ensemble de parties 9' de D verifie (b) et (c), nous noterons G(S")c le quotient de Gc qui agit fidelement dans Ia representation correspondante de Gc. La condition (c) assure qu'il est defini sur Q. Le cas le plus interessant est celui oil (d) 9' est forme de parties a un element. Si 9' verifie (b), (c), !'ensemble 9'' des {s} pour s E s E 9' verifie (b), (c), (d), et G(S"') domine G(S"). Dans Ia table ci-dessous-deduite de 1 . 3.9-nous donnons La liste des cas oil il existe 9' verifiant (b), (c). D'apres 1 .3. 10, ce ne peut etre le cas que si G est de l'un des types A, B, C, D, et ces types seront successivement passes en revue. L'ensemble 9' verifiant (b), (c), (d) maximal, et le groupe G(S") correspondant (il domine tous Ies G(S"), pour 9' verifiant (b), (c)). TABLE 2 . 3.8 . Types A, B, C. Le seul S" verifiant (b), (c), (d) est !'ensemble des {s}, pour s une extremite-correspondant a une racine courte pour les types B, C-d'un dia gramme D. (v E I). Le revetement G(S") est le revetement universe!. Type D1 (I � 5). Pour qu'il existe 9' verifiant (b), (c), il faut et il suffit que les (G., X.) (v e Inc) soient ou bien tous de type Df, ou bien tous de type Dfl. Dis tinguons ces cas : Sous-cas Df. Le 9' verifiant (b), (c), (d) maximal est !'ensemble des {s} pour s a l'extremite "droite" d'un D •. Le revetement G(S") est le revetement universe!. Sous-cas Df. L'unique 9' verifiant (b), (c), (d) est !'ensemble des {s} pour s
272
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l'extremite "gauche" d'un D • . Le revetement G(9') de G est de la forme RngG*, pour G * le revetement double de G forme de S0(2l), cf. 1 .3 . 1 0. Type D4• Rempla�ons 9', verifiant (d), par S = {s l {s} e 9'} . La condition (b) sur 9' devient : S est contenu dans l'ensemble E des extremites de D, et S n I(X) = 0 . Pour Ia definition de I(X) voir 2.3.5. La partie de E stable par Gal(Q/Q) maximale pour cette propriete est Ie complement de Gal(Q/Q) · .2'( X). Bile rencontre chaque D. en 0, 1 ou 2 points. Dans Ie premier cas, il n'existe pas 9' verifiant (b), (c). Dans Ie second (resp. 3e), elle (resp. son complement) est l'image d'une section -r de X -+ I, invariante par Galois ; -r(J) est disjoint de (resp. contient) I( X). Ap pelant -r( v) le sommet "gauche" de D., on retrouve Ia situation de D1 (I ;;;;; 5) : Sous-cas D�. II existe une section -r de X -+ I avec -r(I) => .Z(X). Cette section est alors unique, et Ia situation est Ia meme qu'en Df, l � 5. Sous-cas Df. On donne une section -r de X -+ I avec -r(J) n .2'(X) = 0 . Si on n'est pas dans le cas D�. cette section est unique, et l'unique 9' verifiant (b), (c), (d) est l'ensemble des { s } pour s l'extremite "gauche" d'un D• . Le revetement G(9') de G est de Ia forme RngG•, pour G• un revetement double de G• qui se decrit en terme de -.. Pour Ia suite de ce travail, il nous sera commode de redefinir le cas Df comme excluant D�. Avec cette terminologie, il existe 9' verifiant (b), (c) si et seulement si (G, X) est de l'un des types A, B, C, DR, DH et, sauf pour Ie type DH, il existe 9' verifiant (b), (c), (d) tel que G(9') soit le revetement universel de G. 2.3.9. Nous aurons a considerer des extensions quadratiques totalement imagi naires K de F, munies d'un ensemble T de plongements complexes : un au-dessus de chaque plongement reel v E I• . Un tel T definit une structure de Hodge hT : S -+ K3 sur K (considere comme un espace vectoriel rationnel, et sur leque1 K * agit par multiplication) : si J est !'ensemble des plongements complexes de K, on a K ® C = Cl, et on definit hT en exigeant que le facteur d'indice ff E J soit de type ( 1 , 0) pour ff E T, (0, - 1) pour if E T, et (0, 0) si ff est au-dessus de Inc · Le resultat principal de ce numero est la
-
PROPOSITION 2.3. 10. Soit (G, X) comme en 2. 1 . 1 , avec G Q-simple adjtJint, et de l'un des types A, B, C, DR, DH. Pour toute extension quadratique totalement imagi naire K de F, munie de T comme en 2.3.9, il existe un diagramme
pour lequel (i) E(G1, X1) est le compose de E(G, X) et de E(K * , hT). (ii) Le groupe derive G� est simplement connexe pour G de type A, B, C, DR , et le revetement de G decrit en 2.3.8 pour le type DH.
Soit S le plus grand ensemble de sommets du diagramme de Dynkin D de Gc tel que { {s} 1 s E S} verifie 2.3.7(b), (c). Nous l'avons determine en 2.3.8. Le groupe de Galois Gal(Q/Q) agit sur S, et on peut identifier S a Hom(K5, C), pour Ks un produit convenable d'extension de Q, isomorphes a des sous-corps de Kv puisque Gal(Q/K0) agit trivialement sur D, done sur S. En particulier, Ks est un produit de corps totalement reels ou CM. A la projection S -+ I correspond un morphisme F -+ K5• Pour s E S, soit V(s) la representation complexe de Gc de poids dominant le
273
VARIETES DE SHIMURA
poids fondamental correspondant a s. La classe d'isomorphie de la representation EB V(s) est definie sur Q. Ceci ne suffit pas a ce q u 'on puisse la definir sur Q ; l' obstruc tion est dans un groupe de Brauer convenable. Toutefois, un multiple de cette representation peut toujours etre defini sur Q. Soit done V une representation de G sur Q, avec Vc EB V(s) n , pour n convenable. No us noterons V5 !'unique facteur de Vc isomorphe a V(s) n . Ces facteurs sont permutes par Gal(Q/Q) de fa<;on com patible a !'action de Gal(Q/Q) sur S, et la decomposition Vc EB v. correspond done a une structure de K8-module sur V: sur v• . Ks agit par multiplication par l'homomorphisme correspondant de Ks dans C. Notons G ' le quotient de G qui agit fidelement sur V. C'est le revetement de G considere en (ii). Soit h E X, et relevons h en un morphisme fractionnaire (1 .3.4) de S dans G�. On en deduit une structure de Hodge fractionnaire sur V, de poids 0. Soit s E S, et v son image dans I. Le type de la decomposition de v. est donne par la Table 1 .3.9 : (a) si v E I" V, est de type (0, 0) ; (b) si v E Inc• V, est de type {(r, - r), (r - 1 , 1 - r)} oil r est donne par 1 .3.9 : c'est le nombre qui affecte le sommet s de n., muni du sommet special qui definit "'
"'
x• .
On definit une structure de Hodge h2 de V, en gardant V, de type (0, 0) pour Ic, et, pour v E Inc• en renommant la partie de type (r, - r) (resp. (r - 1, 1 - r)) de V, comme etant de type (0, - 1) (resp. ( - I , 0)) . Si G2 est le sous-groupe algebri que de GL( V) engendre par G ' et Kf, la structure de Hodge h2 est un morphisme S --+ G2R . Notant X2 sa classe de G2(R)-conjugaison, on dispose de (G2, X2) --+ (G, X), et E(G2, X2) = E(G, X). Munissons K ® F V de la structure de Hodge h produit tensoriel de celle de V et de celle de K (2.3.9). On a (K ®F V ) ® R = EevEI(K ®F.v R) ® R ( V ®F. v R). Cette decomposition est compatible a la structure de Hodge, et sur le facteur cor respondant a v E Ic (resp, v E Inc), la structure de Hodge est le produit tensoriel d'une structure de type {( - 1 , 0), (0, - 1)} sur K ®F.v R (resp. V ®F.v R) par une de type {(0, 0)} sur V ®F.v R (resp. K ®F.v R). Au total, h3 est de type {( - I , 0), (0, - I )} . Si G3 est le sous-groupe algebrique de GL(K ®F V) engendre par K* et G2, la structure de Hodge h3 est un morphisme S --+ G3R . Si X3 est la classe de conjugaison de h3 , on dispose de (G3, X3) --+ (G, X). Le groupe derive de G3 est G ', et E(G3, X3) est le compose de E(G2, X2) = E(G, X) et de E(K*, hT) · Pour obtenir (G 1 o X1 ) cherche, il ne reste plus qu'a appliquer 2.3.3 a ( G3, Xg) et a sa representation lineaire fidele K ® F v. REMARQUE 2. 3. I l . La construction donnee se generalise pour fournir un dia gramme (2.3.7. 1) oil 9'( V) est n'importe quel ensemble de parties de D verifiant 2.3.7(b), (c) . En gros : (a) si 9' verifie 2.3.7(b), (c), on definit Ky par Hom{Ky, C) = 9', on construit une representation V de G telle que 9'( V) = 9', et Ia decomposition Vc = EB Vs (S E 9') fournit sur V une structure de Ky-module ; (b) la structure de Hodge fractionnaire de Vs est de type (0, 0) pour S au-dessus de Ic, de type {(r, - r), (r - I, 1 - r)} avec r decrit---c omme ci-dessus-par le point de S au-dessus de Inc sinon ; (c) on convertit {(r, - r), (r - 1 , I - r)} en {(O, - 1), ( - I , O)} comme plus haut ; vE
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(d) pour convertir le (0, 0) en {( - 1 , 0), (0, - 1 )}, on tensorise V, sur Kfl', avec K'g, de type CM muni d'une structure de Hodge convenable h. Par cette methode, on obtient pour (G�o X1) un groupe derive G(S"), et un corps dual compose de E(G, X) et E(K:S.* , h). Noter que, meme pour .9' verifiant (b), (c), (d), Ia conversion indiquee de (0 , 0) est plus generate que celle de 2.3. 10. REMARQUE 2.3. 1 2. Pour les types A, avec l,'(X) fixe par l'involution d'opposition, B, C et D R, le corps dual E( G, X) est le sous-corps (totalement reel) de Kv fixe par le sous-groupe de Gal(Kv/Q) qui stabilise Ic. Si Ic = 0 , c'est Q. Si Ic (resp. Inc) est reduit a un element v, c'est v(F) . Les E(K, hr) sont des extensions de E(G, X). REMARQUE 2.3. 1 3 . Pour ces types, et DH, les V, de 2.3. 1 0, pour v -e Inc• sont de type {( - t , !) , ( t , - !) } . Ceci permet, dans 2.3.10, de remplacer G2 par le sous groupe de GL( V) engendre par F* et G' . Si lc = 0 , on peut meme le remplacer par le sous-groupe de GL( V) engendre par Q * et G ', et le critere 2.3.2 s'applique direc tement a ce groupe, d'ou (Gt. X1 ) avec E(Gt. X1 ) = E(G, X) ( = Q hors le cas DB). 2.4. Lois de reciprocite : preliminaires. Les constructions de ce numero nous permettrons, au numero 2.6, de calculer Ia loi de reciprocite des modeles canoniques, i.e. l'action du groupe de Galois sur l'ensemble des composantes connexes geometriques. Bien qu'elles s'exprini.ent mieux dans le langage de la descente fppf, nous les avons exprimees dans celui de Ia descente galoisienne, le croyant plus familier aux non-geometres. Ceci expose a quelques redites et inconsequences, et introduit des hypotheses parasites de separabilite ou de caracteristique 0. Soit G un groupe reductif sur un corps global k. Avec la notation de 2.0. 1 5, notre but est de construire des morphismes canoniques des deux types suivant. a. Pour k' une extension finie (qu'on supposera separable) de k, et G' deduit de G par extension des scalaires a k', un morphisme norme
(2.4.0. 1) b. Pour T un tore, et M une classe de conjugaison, definie sur k, de morphisms de T dans G, un morphisme
(2.4.0.2)
qM
:
""(T) --+ ""(G).
Si m e M(k), qM sera Ie morphisme qm induit par m ; la difficulte est de montrer que ce morphisme ne depend pas du choix de m, et de construire qM meme si M n'a pas de representant defini sur k. Les proprietes de fonctorialite de ces morphismes seront evidentes sur leur definition. 2.4. 1 . Nous utiliserons systematiquement le langage des torseurs (que je prefere a celui des cocycles), et celui de la descente galoisienne, sous la forme que lui a donnee Grothendieck (cf. SGA 1, ou SGA 4112[Arcata]). Descente galoisienne : Soit K une extension separable finie d'un corps k. Pour construire un objet X sur k (par exemple un torseur), il suffit de construire (a) pour toute extension separable k' de k, telle qu'il existe un morphisme de k-algebre de K dans k', un objet Xk' sur k' ; (b) pour k" une extension de k', un isomorphisme 'Xk", k' : xk' ® k" � xk, ; et de verifier (c) une compatibilite 'Xk"k"'Xk"k' = 'Xk'"k' · En pratique, cela signifie que pour construire X, on peut .supposer }'existence d'objets auxiliaires qui n'existent que sur une extension separable K de k-a charge
VARIETES DE SHIMURA
275
de montrer Ie X construit ne depend pas-a isomorphisme unique pres-du choix d'un tel objet auxiliaire. REMARQUE 1 . La descente galoisienne est un cas particulier de Ia localisation en topologie etale ; une construction comme en (a), (b), (c) ci-dessus sera souvent introduite par l'adverbe "localement". EXEMPLE 2.4.2. Expliquons le relevement canonique utilise en 2.0.2 de I' applica tion commutateur. L'usage de Ia descente galoisienne-plutot que fppf-nous oblige a supposer que Ia projection de G sur G•d est lisse, et a ne considerer que ( , ) : G•d(k) x G•d(k) --+ G(k), plutot que le morphisme G•d x G•d --+ G. Si x�o x2 E G•d(k), on peut, localement, ecrire X; p(x;)z; avec z; dans le centre de G. L'element X; est unique, a multiplication par un element du centre de G pres. Le commutateur de x1 et x2 ne depend pas de l'arbitraire dans Ie choix des x;, et on pose (x� o x2) x 1 x2x11 x21 . 2.4.3. Pour G un groupe algebrique sur un corps k, un G-torseur est un schema P sur k, muni d'une action a droite de G qui en fasse un espace principal homo gene. Le G-torseur trivial Gd est G muni de l'action de G par translations a droite. On identifie les points x e P(k) aux trivialisations de P (isomorphismes cp : Gd � P) par cp(g) xg. Si f : G1 --+ G2 est un morphisme, et P un Grtorseur, il existe un G2-torseurf(P) muni de f : P --+ f(P) verifiant f(pg) f(p)f(g), et il est unique a isomorphisme unique pres. Nous nous interesserons a Ia categorie [G1 --+ G2] des G1-torseurs P munis d'une trivialisation de f(P). Pour morphismes, on prend les isomorphismes de Grtorseurs, compatibles a Ia G2-trivialisation. On note H0(Gl --+ G2) le groupe des automorphismes de (G1d, e) (c'est Ker(G1(k) --+ G2(k))) et H1 (G1 --+ G2) l'en semble (pointe par (G1d, e)) des classes d'isomorphie d'objets. Chaque x e G2(k) definit un objet [x] de [G1 --+ G2] : le G1-t'orseur trivial G1d, muni de Ia trivialisation x de f(G1d) G2d. Quand cela ne prete pas a confusion, nous le noterons simplement x. L'ensemble des morphismes de [x] dans [y] s'iden tifie a {g e G1(k) if(g)x y} : a g, associer u --+ gu : G1d --+ G l d· Un objet est de Ia forme [x] si et seulement si, en tant que G1-torseur, il est trivial-d'ou une suite exacte
=
=
=
=
=
(2.4.3.1)
1
=
H0(G1 -----+ G2) -----+ G t (k) -----+ G2(k) 1 1 1 -----+ H (G1 -----+ G2) -----+ H (G t ) -----+ H (G2) -----+
(ceci ne decrit pas l'image inverse de p e Hl(G1) ; pour Ia decrire, il faut proceder par torsion, comme dans [13]). 2.4.4. Si f est un epimorphisme, de noyau K, il revient au meme de se donner le G1-torseur P G2-trivialise par x e f(P)(k), ou le K-torseur J- 1 (x) c P : le foncteur nature) [K --+ {e}] --+ [G1 --+ G2] est une equivalence. Plus generalement, si g : G2 --+ H induit un epimorphisme de G1 sur H, et que K; = Ker(G; --+ H), le foncteur nature) est une equivalence [K1 --+ K2] --+ [G1 --+ G2] . 2.4. 5. Si G est commutatif, Ia somme s : G x G --+ G est un morphisme, et on ·definit Ia somme de deux G-torseurs par P + Q s(P x Q). Si G1 et G2 sont com mutatifs, on additionne de meme les objets de [G1 --+ G2], qui devient une categorie .de Picard (strictement commutative) (SGA 4, XVIII, 1 .4).
=
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Tout ce qui precede vaut pour des faisceaux en groupes sur un topos quelconque. 2.4.6. Si k' est une extension finie de k (le cas oil k' fk est separable no us suffit) et G' un groupe algebrique sur k', le foncteur de restriction des scalaires a la Weil Rk'lk est une equivalence de la categorie des G'-torseurs avec celle des Rk'lkG' torseurs. Ceci correspond au lemme de Shapiro H 1 (k', G') Hl(k, Rk'lkG). Si G' se deduit par extension des scalaires de G-commutatif-sur k, on dispose d'un morphisme trace Rk'lkG' --+ G-d'ou un foncteur trace Trk'lk des G'-torseurs dans les G-torseurs. Plus generalement, pour G 1 --+ G2 un morphisme de groupes com mutatifs on trouve un foncteur additif
=
(2.4.6. 1 )
De tels foncteurs sont decrits avec une grande generalite dans [SGA 4 , XVII, 6.3]. Pour k'/k separable, on peut donner une definition simple par descente : locale ment, k' est somme [k' : k] copies de k, [G� --+ G2] s'identifie a la categorie des [k' : k] uples d'objets de [ G 1 --+ G2], et Trk'lk a la somme. Quand les groupes sont notes multiplicativement, on parlera plutot de foncteur norme Nk ' lk· 2.4.7. Soient G un groupe reductif (0.2) sur k, et p : G --+ G le revetement uni versel de son groupe derive. Le cas particulier de 2.4.3 qui nous importe est {G --+ G]. Soient G•d le groupe adjoint de G, Z le centre de G, et Z celui de G. Le morphisme G --+ G•d est un epimorphisme, d'ou une equivalence (2.4.4) (2.4.7. 1 )
[Z ----. Z] ----. [G ----. G].
Puisque Z et Z sont commutatifs, [Z --+ Z] est une categorie de Picard strictement commutative (2.4.5). Utilisant !'equivalence (2.4.7.1), on fait de [G --+ G] aussi une telle categorie. Nous allons calculer [xd + [x2] dans [G --+ G], et les donnees d'as sociativite et de commutativite. On suppose p : G --+ Gd•r separable pour pouvoir proceder par descente galoisienne. Ecrivant (localement) x,- p(g,.)z,- , avec z, central, on a des isomorphismes g,- : [z,-] --+ [x,-] et g1 g2 : [z 1 z2] --+ [x 1 x2], d'ou un isomorphisme
=
(2.4.7.2) Si on change de decomposition : x,.
= p (g;) ;., avec = g;.u,. (u,- E Z) , le diagramme z
g,-
est commutatif. L'isomorphisme (2.4.7.2). (2.4.7.3) est done independant des choix faits. Le lecteur verifiera facilement que, via cet
VARIETES DE SHIMURA
277
isomorphisme, Ia donnee d'associativite est deduite de l'associativite du produit, et que Ia donnee de commutativite est (x 1 o x2) : [x2x 1 ] --+ [x1 x2]. II verifiera aussi que si Y; = p(g,)x; (i = 1 , 2), Ia somme des g,. : [x;] --+ [y;] est
g l + gz = g l int.,1 (gz) : [x l xz] --+ [Y l Yz] , oil int designe I' action de G sur G (definie par transport de structure, ou via l'action 2.0.2 de G•d = (}•d). La categorie [G --+ G] etant de Picard, !'ensemble Hl(G --+ G) des classes d'iso morphie d'objets est un groupe abelien. Les formules ci-dessus montrent que l'injection G(k)/p(G(k)) --+ H 1 (G --+ G) est un homomorphisme. D'apn!s (2.4.3. 1), c'est un isomorphisme si H 1 (G) = 0. Soient k' une extension finie de k, et notons par un ' l'extension des scalaires a k'. L'equivalence (2.4.7. 1) permet de deduire de 2.4.6. 1 un foncteur trace (que nous baptiserons norme)
(2.4.7.4)
PROPOSITION 2.4.8. Si k est un corps local ou global, le morphisme deduit de (2.4.7.4) par passage a /'ensemble des classes d'isomorphie d'objets induit un mor phisme de G(k')fpG"(k') dans G(k)fpG(k) :
G(k)fpG(k)
G(k')/pG(k')
(2.4.8.1)
r
r
Hl(G'---+ G') � Hl (G ---+ G).
Si H l(G) = 0, Ia fleche verticale droite est un isomorphisme, et l'assertion est evidente. Cette nullite vaut pour k local non archimedien. Pour k local archimedien, le seul cas interessant est k = R, k' = C, et le diagramme commutatif Z(C) --+> H 1 (Gc---+ Gc)
lNc!R
(2.4.8.1)
Z(R)
1
Hl(G ---+ G)
montre que est encore defini-a valeurs dans l'image de Z(R) (et meme de sa composante neutre). Pour k global, et x E G(k')/p(G(k')), l'image de Nk'tix) E Hl(G --+ G) dans H 1 (G) est done localement nulle. D'apres le principe de Hasse, elle est nulle. Nous n'utili sons ici le principe de Hasse que pour les classes de cohomologie dans l'image de H l (Z), de sorte que les facteurs £8 ne creent aucun trouble. L'image Nk'tix) est done dans G(k)fpG(k), comme promis. 2.4.9. Pour k local non archimedien d'anneau des entiers V, k' non rami:fie sur k, d'anneau des entiers V', et G reductif sur V, le morphisme 2.4.8 induit un morphisme de G( V')/pG( V) : on le voit en repetant les arguments qui precedent sur V, Ia des cente galoisienne etant remplacee par Ia localisation etale (ici, formellement identi que a une descente galoisienne sur le corps residuel). On peut done adeliser 2.4.8 : pour k global, le produit restreint des morphismes 2.4.8, pour les completes de k, est un morphisme Nk ' tk : G(A')jpG(A') ---+ G(A) /pG(A).
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Divisant par le rnorphisme trace global, on obtient enfin le rnorphisme (2.4.0. 1) Nk'tk : 11: (G ') ---> 11:(G).
De merne que Ia construction du morphisme (2.4.0. 1 ) repose sur celle du foncteur (2.4. 7 . I), celle de (2.4.0.2) reposera sur Ia Construction 2.4. 10. Soient G reductif connexe sur k, p : G � G le revetement
universe/ du groupe derive, T un tore sur k, et M une classe de conjugaison, definie sur k, de morphisme de T dans G . On definira unfoncteur additif qM : [{e} ----> T] ---> [G ----> G]. Nous donnerons de Ia construction deux variantes. I ere methode. Localement, il existe m dans M. Posons X(m) = Z m( T) c G et Y(m) = p- 1 X(m) = Z · {p- 1 m(T))O. Les groupes X(m) et Y(m), extensions de tores par un sous-groupe central de type multiplicatif, soot commutatifs. Ils donnent lieu a un diagramme z___, Y(m) ·
1
1
z___,X(m) <----- T Si g dans G conjugue m en m', il conjugue (l)m en (l)m' (on le fait agir sur G par l'action de (Jad = Gad) . De plus, l'isomorphisme int(g) de (l)m avec ( l )m' ne depend pas du choix de g: si g centralise m, il centralise Z, m(T), Z, ainsi que (p- 1 (T)) O (un tore isogene a un sous-tore de T), done X(m) et Y(m). Deux m dans M etant localement conjugues, ceci permet d'identifier entre eux les diagrammes (1 )m, et d'en deduire un diagramme unique z___, y (I) Z _____, X ,_____ T
1
1
defini sur k. On definit qM cornme etant le foncteur compose ---->
T] ----> [ Y ----> X] . : 4 [Z ----> Z] __::_, [G ----> G]. 2 2eme methode. On suppose p : G � Gde r separable, pour proceder par descente galoisienne. Les objets de [{e} � T] n'ayant pas d'autornorphismes, un foncteur de [{e} --+ T] dans [G � G] est simplement une loi qui a t e T(k) assigne un G torseur G-trivialise qM([t]). Procedons par descente galoisienne. Localement il existe m e M(k). Soit qm le foncteur [t] � [m(t)]. Nous allons definir un systeme transitif d'isomorphismes entre les qm. Ceci fait, nous pourrons definir qM comme etant l'un quelconque des qm. Pour definir tm' , m : qm ...::::. qm'• on choisit g tel que m' = gmg- 1 (nouvelle applica tion de Ia methode de descente) et on pose tm' , m([t]) : [m( t )] � [m'( t )] est (g, m(t)) e G(k). On a bien m'(t) = (p((g, m( t)))m (t ) , et il reste a verifier que (g, m( t )) est independant de g. L'espace C des g qui conjuguent m en m' est connexe et reduit, en tant que tor seur sous le centralisateur du tore m(T). La fonction (g, m(t)) de C dans G a une qM : [{ e }
279
VARIETES DE SHIMURA
projection p((m, m(t))) = m'(t)m(t)- 1 dans G constante. La fibre de crete, elle est constante. La construction est recapitulee dans le diagramme commutatif (2.4. 10. 1)
=
(qm([t])
[m(t)], m'
=
p
etant dis
gmg- 1 ).
Construisons Ia donnee d'additivite de qM. C'est Ia donnee, pour chaque tl> t2 e T(k), d'un isomorphisme qM([t1 tz]) --+ qM([td) + qM([tz]), ces isomorphismes etant compatibles aux donnees d'associativite et de commutativite pour Ia somme dans [G --+ G]. On les definit par descente : pour m e M, et m' = gmg- 1 , le diagramme suivant est commutatif
(fleches obliques 2.4. 10. 1), et definit l'isomorphisme cherche independamment de m. Sa commutativite exprime une certaine identite, dans G , entre commutateurs (2.4.2), et Ia projection de cette identite dans G resulte de ce que les fleches ecrites ont un sens. Pour Ia prouver, on note que localement (descente) c'est Ia projection d'une identite analogue pour G x Z0-de projection vraie dans G x zo, done vraie dans G . La compatibilite a l'associativite et a Ia commutativite se voit par descente, en fixant m ; on utilise que le commutateur 2.4.2 est trivial sur m(T). 2.4. 1 1 . Complements. (i) La construction 2.4. 10 est compatible a I' extension des scalaires : notant par un ' l'extension des scalaires de k a une extension k' de k, on definit de fa<;on evidente un isomorphisme de foncteurs additifs rendant commutatif le diagramme [{e} � T] � [G � G]
1
1
[{e} � T'] � [G' � G']
(ii) On peut repeter Ia construction 2.4. 10 sur une base quelconque ; Ia descente galoisienne est a remplacer par Ia localisation etale. (iii) La construction 2.4. 10 est compatible aux foncteurs normes. Pour definir l'isomorphisme de foncteurs additifs rendant commutatif le diagramme [{e}
�
1
T'] ____!_!i__, [G ' � G']
12.
4. 7. 4
[{e} � T] � [G � G]
(notations de (i), avec k'/k fini separable), et verifier ses proprietes, le plus simple
280
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est de proceder par descente : localement, k' devient une somme k1 de copies de k, -+ G'] devient [G' -+ G]I, Ia norme (trace) devient Ia somme, et tout est trivial. 2.4. 1 2. Pour k local non archimedien, Hl(G) = 0 et, passant aux ensembles de classes d'isomorphie d'objets, on deduit de 2.4. 10 un morphisme qM : T(k) -+ G(k) /pG(k). Pour G reductif sur l'anneau des entiers V de k, T un tore sur V et M sur V, il induit un morphisme de T(V) dans G( V)/pG( V) (2.4. 1 l(ii)) . Pour k archimedien, on peut rencontrer une obstruction dans Hl(G), mais elle disparait pour x dans Ia composante neutre (topologique) T(k)+ de T(k) : par 2.4. 1 1(ii), elle depend continument de x, et est nulle pour x = e. On a done encore un morphisme T(k)+ -+ G(k)fpG(k) . Pour k global, prenant le produit restreint de ces morphismes pour les completes de k, on trouve qM : T(A)+ -> G(A )/pG(A) . (2.4. 12. 1) Si T(k)+ = { x e T(k) I pour v reel, x est dans T(kv)+ } , le principe de Hasse, utilise comme en 2.4.8, fournit qM : T(k) + -> G(k)/pG(k) . · (2.4. 1 2.2) Puisque T(A )+jT(k) + ...:::::. T(A)/ T(k) (theoreme d'approximation reel pour les tores), on obtient finalement par passage au quotient le morphisme (2.4.0.2) promis : [G'
qM : 7t: ( T) -+ 7t: (G). 2.5. Application : une extension canonique.
2.5. 1 . Soit G un groupe reductif sur Q. On suppose que G n'a pas de facteur G' ( defini sur Q) tel que G'(R) soit compact. Le theoreme d'approximation forte assure des lors que G(Q) est dense dans G(Af). Le cas qui nous importe est celui d'un groupe comme en 2. 1 . 1 . Reprenons le calcul de 2. 1 .3. Pour K compact ouvert dans G(AI), (2.5. 1 . 1)
11:0 ( G(Q)\ G(A)/K) = G(Q) \11:0 ( G(R)) x ( G(AI)/K) = G(Q) \ G(A)/ G(R)+ x K = G(Q)+\ G(Af) /K
et on peut remplacer G(Q) (resp. G(Q)+) par son adherence dans G(AI). Celle-ci contient pG(Af), un sous-groupe distingue a quotient abelien de G(Af), et, avec les notations de 2.0. 1 5, (2.5. 1 .2) 7t:o ( G(Q)\ G(A)/K) = 11:(G)jG(R)+ x K = 7t:o7t: (G)/K = G(Af) / G(Q)+ K. Passant a Ia limite sur K, on en deduit que ·
11:0 ( G(Q)\ G(A)) = 7t:o7t: (G) = G(Af) / G(Q)+ -.
Soit n011: (G) le quotient G(Af) / G(Q)+ de n011:(G) par 11:0G(R) + (0 . 3 ). La suite 0 -> G(Q)+/Z(Q) - -> G(A f) /Z(Q) - -> n011:(G) -> 0 (2.5. 1 .3) est exacte. L'action de G•d sur G induit une action de G•d(Q ) sur cette suite exacte. L'existence du commutateur (2.0.2) montre que l'action de G•d(A) sur G(A)fpG(A) est triviale-a fortiori celle de G"d(Q) sur n01r(G). L'application du sous-groupe G(Q) / Z(Q) de G(Q)+/Z(Q)- dans G•d(Q)+ verifie les conditions de 2.0. 1 . Le groupe + G(Q)+/ Z(Q)- * G (Ol +tz
VARIETES DE SHIMURA
28 1
topologie -r(Gd•r) (2. 1 .6). Appliquant de meme la construction * G•d(Q)+ au terme central de 2.5. 1 .3, on obtient finalement une extension (2.5.1 .4) o-G·d(Q)+A(rel. Gd•r) -->
g�;2 *G (Q) +IZ(Q) G•d(Q)+--> 1fon(G)-->O.
2.5.2. Du fait que G•d n'est pas fonctoriel en G, la fonctorialite de cette suite est penible a expliciter. Nous nous contenterons des deux cas suivant : (a) Lorsqu'on ne considere que des groupes extension centrale d'un groupe ad joint donne, et des morphismes compatibles a la projection sur ce groupe adjoint, (2.5. 1 .4) est fonctoriel en G en un sens evident. (b) Soit H c G un tore et, contrairement aux conventions generales, notons H•d son image dans G•d . Tel etant le cas dans les applications, on suppose H•d(R) compact-done connexe puisque H•d est connexe. Cette hypothese assure que H•d (Q) est discret dans H•d (Af), et que H(Q) c G(Q)+ . Posons Z' = Z n H. Le diagramme commutatif G(A!) --> G(Af) /G(Q)+ = 7r 0n(G)
r
r
H(A!) --> H(A!)jH(Q)-
=
7E0n(H)
fournit un morphisme de suites exactes
��{2 * G (Q) +IZ(Q) G•d(Q)+-->lEon(G) -->0 r r H•d(Q) (H)-->0 -->lEon ��;;]- *H (Q) IZ' (Q)
o-G•d(Q)+A (rel. Gd•r) --> (2.5.2. 1)
1
o- H•d(Q)
2.5.3. Soient G comme en 2.5. 1 , E une extension finie de Q, T un tore sur E et
M une classe de conjugaison definie sur E de morphismes de T dans G: M: T---> GE.
Par passage aux n0, le morphisme compose de (2.4.0. 1) et (2.4.0.2) NEIQ qM : n (T) --> n(GE) -------> n(G) fournit un morphisme n0n(T) ---> n0n(G) ---> 7r 0n(G). Nous noterons (G, M) son inverse, et t&'�(G, M) !'extension image inverse par r ( G, M) de !'extension (2.5. 1 .4) :
Soient u : H --+ G un morphisme comme en 2.5.2(a) (H•d = G•d), et N une classe de conjugaison, definie sur E, de morphismes de T dans H. On suppose que u envoie N dans M. Par fonctorialite, u definit alors un isomorphisme du quotient de t&'�(H, N) par Ker(G•d(Q)+A(rel. Gd•r) ---> G•d(Q)+A(rel. Hd•r)) avec t&'�(G, M). Pour H --+ G un tore comme en 2.5.2(b), muni de m : T ---> HE dans M, on trouve u n morphisme d'extensions
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0 -> G•d(Q)+A(rel. Gder) -> C�(G,M) -> rcorc(T) -> 0
J
0 -> H•d(Q)
I
II
-> rc0rc(T) -> 0
2.5.4. Soient E, G, T, M comme ci-dessus, avec G adjoint. Considerons les sys temes (Gb Mb u) formes d'une extension centrale u : G1 � G de G (Gid = G), de finie sur Q, et d'une classe de conjugaison de morphismes M1 de T dans G�o definie sur E, qui releve M. Sur Q, pour m 1 dans M1 d'image m dans M, le centralisateur de m 1 est l'image inverse du centralisateur de m : c'est vrai pour leurs algebres de Lie, ils sont connexes en tant que centralisateurs de tores, et le centralisateur de m 1 contient le centre de G1 . On a done M1 -.:::. M. LEMME 2.5.5. II existe des systemes (Gb Mb u) pour lesque/s G1er est un revetement
arbitrairement prescrit de G.
II suffit de montrer qu'on peut obtenir le revetement universe! G. Sur Q, pour tout m dans M, Ia composante neutre de !'image inverse par m du revetement G de G est un revetement rc : t � T de T. II ne depend pas de m, done est defini sur E, et M se releve en une classe de conjugaison M de morphismes de t dans GE. Par passage au quotient par Ker(rc), on deduit de M x Id : t � GE x t un relevement M{ : T � (GE x T)/Ker(rc). Ce relevement est a valeurs dans un groupe G�. defini sur E, de groupe adjoint GE. II reste a remplacer G� par un groupe defini sur Q. Ecrivons G� = GE * zE Z�. L'idee est de remplacer Z� par le coproduit, sur ZE, de ses conjugues : si on pose Z = RE/Q(Z�)/Ker(TrE1Q : RE;Q(ZE) -> Z)
et G 1 = G * z Z, on a G� c G1£, et M{ fournit le relevement voulu. Construction 2.5.6. A isomorphisme unique pres, /'extension C'( Gb M1) ne depend que de M et G�e r.
Soient deux systemes (G{, M{) et (G�, M;), de meme groupe derive. Conside rons Ia composante neutre G 1 du produit fibre de G{ et G� sur G, et al classe M1 = M{ x M M�. Le diagramme d'extensions C'(G{ , M� ) ,_:::____ C'( G � > M1 ) ____::_____. C'( G�, M;) fournit l'isomorphisme cherche. DEFINITION 2.5.7. Soient G un groupe adjoint, G ' un revetement de G et M une classe de conjugaison, d¢finie sur E, de morphismes de T dans G. L'extension
CE ( G, G ' , M) de rc0rc(T) par /e complete G(Q)+A (rei. G' ) est /'extension C�(G I > M1 ), pour un quelconque systeme (Gb M 1 ) comme en 2.5.4, tel que G�er = G ' .
Pour F une extension de E, la classe M fournit, par extension des scalaires de E a F, une classe de conjugaison Mp de morphismes de TF dans Gp ; I' extension Cp(G, G', M) correspondante est image inverse de CE(G, G ' , M) par la norme NFI E : rcorc(Tp) ..... rc0rc(T) :
0 -> G(Q )+A(rel. G ') -> Cp( G, G ' , Mp) -> rc0rc(Tp) -> O
l N F !E 1 II 0 -> G(Q)+A(rel. G ') -> CE( G, G ' , M) -> rc0rc(T) -> 0
283
VARIETES DE SHIMURA
Les extensions CE(G, G', M) se deduisent toute de CE(G, G , M) par passage au quotient : remplacer G(Q)+" (rei. G) par son quotient G(Q)+A (rei. G'). 2.5.8. Soient H -+ G un tore, avec H(R) compact, et m E M, defini sur E, qui se factorise par H. Pour tout systeme (GI> M1) -+ (G, M) comme ci-dessus, soient H1 Ia composante neutre de l'image inverse de H dans GI> et m1 l'element de M1 au dessus de m (2.5.4.) Prenons l'image inverse par r(H�o {mt}) du morphisme d'ex tensions (2.5.2. 1) : G(Q)+A (rei. G') � CE(G, G', M) � n0n(T)
I
I
I
n0n(T) H(Q) On voit comme en 2.5.6 que, a isomorphisme unique pres, ce diagramme ne depend pas du choix de (G�o M1). Comme en 2.5.7, ce diagramme, rei. un revetement G' de G, se deduit du meme diagramme, rei. G, par passage au quotient. On a aussi Ia meme fonctorialite en E qu'en 2.5.7. En particulier, pour m dans M, defini sur une extension F de E, qui se factorise par H on trouve un morphisme d'extensions
0 � G(Q)+" (rel. G') � CE(G,G',M) � n0n(T) � 0 (2.5.8.1)
1 H(Q)
1
rNFfE
0 n0n(TF) � 0 Nous utiliserons ce diagramme de Ia fac;on suivante : si CE(G, G', M) agit sur un ensemble V, et qu'un point x E V est fixe sous H(Q) c G(Q), il a un sens de de mander qu'il soit fixe "par n0n(TF)'' i.e. par le sous-groupe image de !'extension en 2eme ligne. 2.5.9. Specialisons les hypotheses au cas qui nous interesse. On part d'un systeme (G, X) comme en 2. 1 . 1, avec G adjoint, et on fixe une composante connexe x+ de X. On prend pour E une extension finie, contenue dans C, de E(G, X), et on fait T = Gm, M = Ia classe de conjugaison de f-th• pour h E X. Elle est definie sur E. Le groupe n(T) est le groupe des classes d'ideles de E, et Ia theorie du corps de classe global identifie n0n(T) a Gal(Q/E) ah. Si G' est un revetement de G, l'image inverse par le morphisme Gal(Q/E) -+ Gal(Q/E)ab de l'extension CE(G, G', M) est une extension (2.5.9. 1) O � Gad(Q) +"(rel. G') � &E(G, G', X) � Gal(QfE) � O. -----+
Le cas universe} est celui oil E = E(G, X), et oil G' = G : d'apres 2.5.7, &E(G, G', M) est l'image inverse de Gal(Q/E) c Gal(Q/E(G, X)) dans &E w . x > (G, G', X), et CE(G, G', X) est un quotient de CE(G, G, X). 2.5. 10. Soit h E x+ un point special : h se factorise par H c G, un tore defini sur Q. Puisque int h(i ) est une involution de Cartan, H(R) est compact. On peut done appliquer 2.5.8 a H et a f-th (defini sur !'extension E(H, h) de E(G, X)). Par image inverse, on deduit de (2.5.8. 1) un morphisme d'extensions G(Q)+A (rei . G') � CE(G, G', X) � Gal(Q/E) (2.5.10.1)
I
H(Q)
I
I
Gai(Q/E · E(H, h))
284
PIERRE DELIGNE
2.6. La loi de reciprocite des modeles canoniques. 2.6. 1 . Soient (G, X) comme en 2. 1 . 1 et E c C un corps de nombres qui contient
E(G, X). Supposons que Mrf...G , X) admette un modele faiblement canonique ME(G, X) sur E. Le groupe de Galois Gal(Q/E) agit alors sur !'ensemble profini 7Co(M0(G, X)) des composantes connexes geometriques de ME(G, X). Cette action commute a celle de G(Af), par hypothese definie que E. D'apres 2. 1 . 14, !'action (a droite) de G(Af) fait de 7CoMrf... G , X) un espace principal homogene sous le quo tient abelien 7t o7CG G(Af)/G(Q)f-. L'action de Galois est done definie par un homomorphisme rG , X de Gal(Q/E) dans 7to7C(G), dit de reciprocite. Convention de signe : !'action (a gauche) de a coincide avec l'action (a droite) de rG .x(a) . Ce
=
morphisme se factorise par le groupe de Galois rendu abelien, identifie par la theorie du corps de classe global a 7Co7C(GmE), d'ou (2.6. 1 . 1) 2.6.2. Soit M la classe de conjugaison de fl.h• pour h E X. Puisque E ::J E(G, X), elle est definite sur E. Composant les morphismes 2.4.0, on obtient NE1QqM : 7C (GmE) --+ 7C ( GE) --+ 7C(G). Par passage au 7Co , on en deduit (2.6.2. 1) THEOREME 2.6.3. Le morphisme (2.6. 1 . 1), donnent /'action de Gal(Q/E) sur /'en semble des composantes connexes geometriques d'un modele faiblement canonique ME(G, X) de Mrf...G , X) sur E, est /'inverse du morphisme 7Co NE1Q qM de 2.6.2.
L'idee de la demonstration est que, pour chaque type -. de points speciaux
(2.2.4), on connait !'action d'un sous-groupe d'indice fini Gal r de Gal(Q/E) sur les
points speciaux de ce type (par definition des modeles faiblement canoniques) donc sur !'ensemble des composantes connexes puisque !'application qui a chaque point associe sa composante connexe est compatible a l'action de Galois. Que !'action de Gal, obtenue soit l,a restriction a Gal, de l'action definie par !'inverse de 7Co(NE!QqM) est verifie en 2.6.4 ci-dessous, et il reste a verifier que les Gal, engendrent Gal(Q/E). Un type -. de points speciaux est defini par h E X se factorisant par un tore c : T --+ G defini sur Q. Le sous-groupe Galr correspondant est Gal(Q/E) n Gal(Q/E(T, h)) = Gal(Q/E · E(T, h)) . D'apres [5, 5 . 1 ], pour toute extension finie F de E (G, X), il existe (T, h) tel que I' extension E(T, h) de E (G, X) soit lineairement disjointe de F. Ceci est plus qu'assez pour assurer que les Galr engendrent Gal(Q/E). 2.6.4. Soient T et h comme ci-dessus, et p. = fl.h· Le morphisme p. : Gm --+ T est defini sur E(T, h) et le morphisme 7Co NR(p.h) de 2.2.3 se deduit, par application du foncteur 7C0, de NE (T, h) IQ o q,_, : 7C(G mE
De la fonctorialite de N et de q, il resulte que ce compose est NE (T, h) IQ o qM :
VARIETES DE SHIMURA
285
egal a NE (G, X) IQ 0 qM 0 NE ( T . h ) / E < G . X ) :
Puisque la norme NE < T.h l i E < G .xJ correspond, via la theorie du corps de classe a !'inclusion de Gal(Q/E(T, h)) dans Gal(Q/E(G, X)), on a bien !'action promise. 2. 7. Reduction au groupe derive, et theoreme d'existence. Dans ce numero, schema signifie "schema admettant un faisceau inversible ample". Ceci nous permettra de passer sans scrupules au quotient par un groupe fini. La stabilite de cette condition sera evidente dans les applications, et je ne la verifierai pas a chaque pas. Tout ceci n'est d'ailleurs qu'une question de commodite. 2.7. 1 . Soit F un groupe localement compact totalement discontinu. Nous nous interesserons a des systemes projectifs, munis d'une action a gauche de r, du type suivant. (a) Un systeme projectif, indexe par les sous-groupes compacts ouverts K de F, de schemas SK · (b) Une action p de rsur ce systeme (definie par des isomorphismes PK(g) : sK .::::_. SgKg -l). (c) On suppose que PK(k) est l'identite pour k E K. Pour L distingue dans K, les PL(k) definissent une action sur SL du groupe fini quotient KfL, et on suppose que (K/L)\SL .::::_. SK. Un tel systeme est determine par sa limite projective S = lim proj SK , munie de I' action de r : on a sK = K\S. Nous appelerons s un schema muni d'une action a gauche continue de F. On definit de meme la continuite d'une action a droite par la condition S = lim proj SfK. 2.7.2. Soit n: un ensemble profini, muni d'une action continue de r. On suppose que I' action est transitive, et que les orbites d'un sous-groupe compact ouvert sont ouvertes : pour e E n:, de stabilisateur LJ, la bijection F/L1 --> n: est un homeomor phisme. Si r agit continument sur un schema S, muni d'une application continue equi variante dans n:, la fibre S, est munie d'une action continue de L1 : pour K compact ouvert dans r, K n L1\S, est la fibre en l'image de e de K\S --> K\n:, et S, est la limite de ces quotients. LEMME 2.7.3. Le foncteur s --> s. est une equivalence de Ia categorie des schemas S, munis d'une action continue de F et d'une application continue equivariante dans n:, avec Ia categorie des schemas munis d'une action continue de ,d. Le foncteur inverse est le foncteur d'induction de L1 a F : formellement, ind �(T) est le quotient de r X T par L1 agissant par o(r , t) = (ro-1, at) ; ceci a un sens parce que !'action de L1 sur F est propre ; pour K compact ouvert dans F, on a
286
PIERRE DELIGNE
K\Ind�(T)
= K\F
T divise par i1 11 (rKr- 1 n il)\T. x
rE K\.d=KI1r
La verification detaillee est laissee au lecteur. 2.7.4. Soient E un corps, et F une extension galoisienne de E. Le groupe de Galois Gal(F/E) agit contim1ment sur Spec(F). Plus generalement, si X est un schema sur E, il' agit continument (par transport de structure) sur XF = X x Spec CEJ Spec(F). On a (descente galoisienne) LEMME 2.7.5. Lefoncteur X � XF est une equivalence de Ia categorie des schemas sur E avec Ia categorie des schemas sur F, munis d'une action continue de Gal(F/ E) compatible a /'action de ce groupe de Galois sur F.
2.7.6. Soient E 4 Q un corps de nombres, r un groupe localement compact totalement discontinu, 11: un ensemble profini, muni d'une action de r comme en 2.7.2, sauf qu'on prend ici une action a droite, et e E 11: . On se donne aussi une action a gauche de Gal(Q/E), commutant a l'action de F. Soit F. le stabilisateur de e. Si on convertit l'action a droite de F en une action a gauche, on obtient une action a gauche de r X Gal(Q/E). Le stabilisateur de e, pour cette action, est une extension C de Gal(Q/E) par F,. C Gal(Q/ E) --> 0 0 --> F,
1
_____,
1
I
0 - r - r X Gal(Q/E)-->Gal(Q/E)-->0 2.7.7. Lorsque !'action de F fait de 11: un espace principal homogene sous un quotient abelien n(F) de F, l'action de Galois est definie par un morphisme r : Gal(Q/E) � n(F), tel que O" · x = x · r(O") , C ne depend pas de e : r. est le noyau de Ia projection de F sur n(F), et I' extension C est l'image inverse, par r, de I' exten sion r de n(F) par r.. 2.7.8. Considerons Ies schemas S sur E, munis d'une action a droite continue de F et d'une application Gal(Q/E) et F-equivariante de Sfi dans n . On note S, Ia fibre en e. C'est un schema sur Q, muni d'une action continue (a gauche) de I' exten sion C, et !'action de C sur S, est compatible a son action, via Gal(Q/E), sur Q. Combinant 2.7.3 et 2.7.5, on trouve LEMME 2.7.9. Le foncteur s � s. e�t une equivalence de categories. Le cas qui nous interesse est celui oil les SfK sont de type fini sur E, pour K com pact ouvert dans r, et oil I' application d.e Sfi sur 11: identifie 11: a n0(Sfi). Ces condi tions correspondent a : les K\S., pour K compact ouvert dans r., sont connexes et de type fini sur Q. 2.7. 10. Soient G un groupe adjoint, G' un revetement de G, x+ une G(R)+ classe de conjugaison de morphismes de S dans GR, verifiant les conditions de 2. 1 . 1 , et E c Q une extension finie de E(G, X+). Un modele faiblement canonique ·(connexe) de MO(G, G', X+) sur E consiste en (a) un modele M§ de M0(G, G', x+) sur Q, i.e. un schema M§ sur Q, muni d'un isomorphisme du schema sur C qui s'en deduit par extension des scalaires avec M0(G, G', X+) ;
VARIETES DE SHIMURA
287
(b) une action continue de CE(G, G', X+) (2.5.9. 1) sur le schema Mg , compatible !'action du quotient Gal(Q/E) de CE sur Q, et telle que !'action du sous-groupe G(Q)+A (rei. G') (une action Q-lineaire cette fois) fournisse par extension des sealaires a C 1' action 2. 1 . 8 ; (c) on exige que pour tout point special h e x+, se factorisant par un tore H --+ G defini sur Q, le point de MO(G, G', x+) defini par h-fixe par H(Q)-soit defini sur Q et (en tant que point ferme de M§) fixe par !'image de !'extension en deuxieme ligne de (2.5.10.1) (rei. Jl.h). Lorsque E = E(G, X), on parle de modele canonique (connexe). 2.7. 1 1 . Les proprietes de fonctorialite suivantes soot immediates. (a) Soient des systemes (G,, G;, Xi) comme en 2.7. 10, en nombre fini, et E c Q un corps de nombres contenant les E(G,, X;t"). Si les MS, soot des modeles faible ment canoniques, sur E, des M8(G,, G;, Xi), leur produit est un modele faiblement canonique de M8(TI G,, TI G;, TIX;t-) sur E. (b) Soient (G, G', x+) comme en 2.7. 10, et G" un revetement de G, quotient de G'. Si MS est un modele faiblement canonique, sur E, de M8(G, G', x+), son quo tient par Ker(G(Q)+A (rei. G') --+ G(Q)+A (rei. G")) est un modele faiblement cano nique, sur E, de M8(G, G", x+). 2. 7 . 1 2. Soient G un groupe reductif sur Q, X comme en 2. 1 . 1 , x+ une composante connexe de X et E c Q une extension finie de Q, contenant E(G, X). Si Mc(G, X) admet un modele faiblement canonique ME(G, X) sur E, ce dernier est unique a isomorphisme unique pres [5, 3.5]. L'action (2.0.2) de G•d sur G induit done une action, par transport de structure, de G•d(Q)+ sur ME(G, X). Convertissons cette action en une action a droite. Combinee a !'action de G(Af), elle fournit une action a droite de G(Af) G(At) d Gad (Q)+ - Z(Q) - * G (Q) + IZ (Q) G· (Q)+. Z(Q)- * G (Q) IZ (Q) a
_
Apres extension des scalaires a C, c'est !'action (2. 1 . 1 3). Soit 1r: !'ensemble profini 7r:o(M(l(G, X)) = 1r:0(Mc(G, X)), et e e 1r: la composante neutre (2. 1 .7) rei . x+. Le foncteur 2.7.9 transforme ME(G, X), muni de la projec tion naturelle de M(l(G, X) dans 1r:, en un schema M0(G, X) sur Q, muni d'une action continue de !'extension (2.5.9. 1). PROPOSITION 2.7. 1 3 . L 'equivalence de categories 2.7.9 fait se correspondre les modelesfaiblement canoniques de M(G, X) sur E et les modelesfaiblement canoniques de MO(G•d , Gd •r , X+) sur E.
Dans la definition 2.2. 5 des modeles faiblement canoniques, nous avons impose l'action d'un sous-groupe Gal(Q/E(t")) n Gal(Q/E) de Gal(Q/E) sur !'ensemble des points speciaux de type 1:. Ceux-ci forment une seule orbite sous G(Af), et !'ac tion prescrite commute a I' action de G(Af). Dans la definition 2.2.5, on peut done se contenter d'exiger que pour un point special de type 1: ses conjugues par Galois soient comme prescrit. En particulier, il suffit de considerer les systemes (H, h) formes d'un point special h e x+ se factorisant par un tore H defini sur Q, et, pour chaque systeme de ce type, de prescrire les conjugues sous Gal(Q/E(H{ h})) n Gal(Q/E) de !'image de (h, e) e X x G(Af) dans Mc(G, X). On retrouve ainsi la variante [5, 3. 1 3] de la definition : Mc(H{h}) a trivialement un modele cano-
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PIERRE DELIGNE
nique (c'est un ensemble profini, et on prend le modele sur E (H,{h } ) pour lequel !'action de Galois sur ces points est !'action prescrite), et on impose au mor phisme nature! Mc(H, { h} ) � Mc(G, X) d'etre defini sur E E(H, { h} ). Nous laissons au lecteur le soin de verifier que !'equivalence de categorie 2.7.9 transforme cette condition de fonctorialite en celle qui definit les modeles faible ment canoniques connexes. 2.7. 14. Soient G un groupe algebrique reel adjoint, et x+ une classe de G(R)+ conjugaison de morphismes de S/Gm dans GR. Notons M la classe de conjugaison de fth• pour h e x+ : une classe de conjugaison de morphismes de Gm dans Gc. Si G1 est un groupe reductif de groupe adjoint G, un relevement X( de x+ en une classe de G1 (R)+-conjugaison de morphismes de S dans GR definit un relevement M(Xt) de M: la classe de conjugaison de fth• pour h e X( (cf. 2.5.4). LEMME 2.7. 1 5. La construction X( � M(X() met en bijection les relevements de x+ et ceux de M. La construction h � fth est une bijection de !'ensemble des morphismes h de S dans un groupe reel G avec !'ensemble des morphismes fl. de Gm dans Gc qui com mutent a leur complexe conjugue : on a h(z) = ft(z) p.(z) . Via ce dictionnaire, le probleme devient de verifier que si !J. 1 : Gm � Gc commute a p. 1 Cela resulte de la rigidite des to res : le morphisme int p. 1 (z)(f1. 1 ) coincide avec fl- 1 pour z = 1, et releve fl- 1 pour toute valeur de z. II est done constamment egal a !J. 1 . Ce dictionnaire permet de traduire 2.5.5 en le LEMME 2.7. 16. Soient G, G' et x+ comme en 2.7. 10. II existe un groupe reductifGh le groupe adjoint G et de groupe derive G', et une classe de G 1 (R)O conjugaison Xt de morphismes de S dans G qui releve X+ et telle que E(G, X+) = E(G h X(). 2.7. 17. Ce Iemme, et !'equivalence 2.7. 13, permettent de transporter aux modeles faiblement canoniques des varietes de Shimura connexes les resultats de [5] sur les modeles faiblement canoniques de varietes de Shimura, et etablissent une equi valence entre les problemes de construction correspondant. CoROLLAIRE 2.7. 18. Soient (G, X) comme en 2. 1 . 1 , x+ une composante connexe de X et E c: Q une extension finie de E(G , X). Pour que M(G, X) admette un modele faiblement canonique sur E, ilfaut et il suffit que MO(G•d, Gd•r, x+) en admette un. En particulier, /'existence d'un tel modele ne depend que de (G•d , Gd•r , x+, E). COROLLAIRE 2.7. 19. (Cf [5, 5.5, 5. 10, 5. 10.2]). Soient G, G', X+ et E comme en ·
•
2.7. 10.
(i) MO (G, G', x+) admet au plus un modele faiblement canonique sur E (unicite a isomorphisme unique pres). (ii) Supposons que, pour toute extension finie F de E, il existe une extension finie F' de E dans Q, lineairement disjointe de F, et un modele faiblement canoni que de MO(G, G', X+) sur F' . Alors, il existe un modele faiblement canonique de MO(G, G', x+) sur E. Le corollaire 2. 7. 1 9 et 2.3. 1, 2.3. 10 fournissent de nombreux modeles canoniques. THEOREME 2.7.20. Soient G un groupe Q-simple adjoint, G' un revetement de G,
VARIETES DE SHIMURA
289
et x+ une G(R)+-classe de conjugaison de morphismes de s dans GR, verifiant (2. 1 . 1 . 1 ) , (2. 1 . 1 .2), (2. 1 . 1 .3). Dans les cas suivants, M0(G, G', x+) admet un modele canonique (a) G est de type A, B, C et G' est le revetement universe/ de G. (b) (G, X) est de type D R et G' est le revetement universe/ de G. (c) (G, X) est de type DH, et G' est le revetement 2.3.8 de G.
Appliquant 2.7. 1 1 , 2.7. 1 8, on en deduit le CoROLLAIRE 2.7.21 . Soient G un groupe reductif, X une G(R)-classe de conjugai son de morphisme de S dans GR, verifiant les conditions de 2. 1 . 1 , et x+ une campo
sante connexe de X. Pour que M(G, X) admette un modele canonique, if suffit que, (G•d, x+) soit un produit de systeme (G,, X;+-) du type considere en 2.7.20, et que le revetement Gder de G•d so it un quotient du produit des revetements des G, consideres en
2.7.20.
BIBLIOG RAPHIE
Pour Ia bibliographie des articles de Shimura consacres it la construction de modeles canoniques, je renvoie it [5] . 1. A. Ash, D. Mumford, M. Rapoport and Y. Tai , Smooth compactifications of locally sym metric varieties, Math. Sci. Press, 1 975. 2. W. Baily and A. Borel, Compactification of arithmetic quotients of bounded symmetric do mains, Ann. of Math. (2) 84 (1 966), 442-528. 3. A. Borel, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, J . Differential Geometry 6 (1 972), 543-560. 4. N. Bourbaki, Groupes et algebres de Lie, Chapitres IV-VI, Hermann, 1 968. 5. P. Deligne, Travaux de Shimura, Sem. Bourbaki Fevrier 71, Expose 389, Lecture Notes in Math. , vol . 244, Springer-Verlag, Berlin, 1 97 1 . 6 . --- , Travaux de Griffiths, Sem. Bourbaki Mai 70, Expose 376, Lecture Notes i n Math., vol. 1 80, Springer-Verlag, Berlin, 1 97 1 . 7 . -- , L a conjecture de Wei/ pour les surfaces K 3 , Invent. Math. 1 5 (1 972), 206-226. 8. S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, 1 962. 9. D. Mumford, Families of abelian varieties, Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math., vol. 9, Amer. Math. Soc., Providence, R.I., 1 966, pp. 347-3 5 1 . 10. N . Saavedra, Categories tannakiennes, Lecture Notes in Math. , vol. 265, Springer-Verlag, Berlin, 1 972. 11. I. Satake, Holomorphic imbedding of symmetric domains into a Siegel space, Am. J. Math. 87 (1 965), 425-46 1 . 1 2 . J. P. Serre, Sur les groupes de congruence des varietes abeliennes, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1 964), 3-20. 13. -- , Cohomologie galoisienne, Lecture Notes in Math., vol. 5, Springer-Verlag, Berlin, 1 965. SGA Sem. de Geometrie Algebrique du Bois-Marie. SGA 1 , SGA4 t.3 et SGA4'1' sont parus aux Lecture Notes in Math. nos. 224, 305, 569, Springer-Verlag, Berlin . INSTITUT HAUTES ETUDES SCIENTIFIQUES, BURES-SUR-YVETTE
Proceedings of Symposia in Pure Mathematics Vol. 33 ( 1 979), part 2, pp. 291 -3 1 1
CONGRUENCE RELATIONS AND SHIMURA CURVES
YASUTAKA IHARA Introduction. Nonabelian reciprocity for the one-dimensional function field K over the finite field Fq has been investigated in two directions. One is the recent beautiful work of Drinfeld (realizing a part of Langlands' philosophy), which as sociates to each automorphic representation of GL2(KA) a system of /-adic represen tations of the Galois group over K. The other is older, and is essentially related to automorphic functions in characteristic 0. This is the direction suggested in my lecture notes [Sb] as explicit conjectures (cf. also [Sa, c]) Its aim is a sort of arithme tic uniformization theory for algebraic curves X over Fq by means of discrete sub groups F of PSL2(R) x PGL2(k), where k is a p-adic field with N(p) = q. For example, if XN is the canonical modular curve of level N � 0 (mod p) over Fp2, p a prime, then the space \l3(XN) of all ordinary closed scheme-theoretic points of XN can be expressed as the quotient FN\.Yt', where rN is the modular group of level N over Z[l/p] and .Yt' is the space of all imaginary quadratic subjields M of M2(Q) with (Mjp) = 1 . This simultaneous uniformization \l3(XN) = FN\.Yt' is important, because this naturally describes all the Frobenius elements in the covering system {XN/X1 } . 1 These were proved in [Sb, Chapter 5 of Vols. 1 , 2] by using several ex quisite results of Deuring on the reduction of elliptic curves parametrized by points of .Yt'. A simple characterization of the system {XN/XI} was proved in [6]. The present work started with an observation that these can be proved with only the Kronecker congruence relation T(p) ® Fp = n U 1ll as basis. This is not so sur prising after all, but this observation should be systematically used due to the following reasons. First, Shimura curves also have congruence relations, due to Shimura, for almost all p. Secondly, there may exist curves other than Shimura curves having (unramified) congruence relations. Thirdly, it now seems important to regard congruence relations, not only as relations or theorems, but as categorical objects, because this category is very likely to be equivalent with the other two categories, that of r, and that of X with some additional structures, and the clari fication of this equivalence seems significant. The main purpose of this paper is to outline our theory which gives a systematic study of abstract congruence relations (or equivalently, CR-systems, § 1). It main ly states that whenever there is an unramified symmetric CR-system !![ over o (De finitions 1 . 1 . 1 , 1 .5. 1), there is a satisfactory arithmetic uniformization theory for an algebraic curve X over Fq by means of a discrete subgroup r of PSL2(R) x Aut(Y). .
AMS (MOS) subject classifications (1 970). Primary 14G 1 5, 1 2A90 ; Secondary 14H30, 10D 1 0. 'A brief review of this uniformization is given in §7. 1 . © 1 9 79, American Mathematical Society
29 1
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YASUTAKA IHARA
Here, o is the ring of integers of a ).1-adic field k (Np = q), and f7 is the tree asso ciated with PGL2(k) (Main Theorems I-III). This will then be applied to the case where X is a smooth reduction of a Shimura curve and r is a quaternionic discrete subgroup of PSL2(R) x PGL2(k). Some relations with other works are discussed in §7.2. Some of the proofs, in cluding those which had been sketched in the previous announcement [9a], are totally omitted. The details [9b] will appear shortly. Notations and terminologies. o : a complete discrete valuation ring of characteristic 0 with finite residue field Fq ; ).1 : the maximal ideal of o ; k : the quotient field of o ; Spec o = {7], s } (7] : the generic point, s : the closed point). If Z is an o-scheme, Z� = Z ®o k denotes its fiber over 7J (the general fiber) and z. = Z (8)0 Fq denotes its fiber over s (the special fiber). For any field F, F denotes its algebraic closure. If Z is a proper smooth irreduci ble algebraic curve over F, we write Z = Z (8) F F. If F' is the exact constant field of Z, Z consists of [F' : F] connected components. The genus g of Z is defined by g - 1 = [F' : F](g - 1), where g is the genus of each component of Z. kd c k : the unique unramified extension of k with degree d; od : the ring of integers of kd ; [q] : the Frobenius automorphism of U kd over k. 1. Congruence relations.
1 . 1 . Let X be a proper smooth irreducible algebraic curve over Fq . We do not assume that X is absolutely irreducible. Let n (resp. tll) be the graphs on X X Fq X of the qth power morphisms X -> X (resp. X +- X). Consider ll, 1ll and ll U 1H as closed reduced subschemes of X X F q X. DEFINITION 1 . 1 . 1 . A triple (XJ. X2 ; T) of two-dimensional integral o-schemes is called a congruence relation w.r.t. (X, o), if (i) XI> X2 are proper smooth over o, and have X as the special fiber; (ii) T is a closed subscheme of s = xl X 0 x2 , flat over 0, such that T X s ex x F . x) = n u �n. Let f-t : X0 -> T be the normalization of the two-dimensional integral scheme T, and put f/J; = pr; o f-t (i = 1 , 2), where pr; are the projections of T to X;. The system ( 1 . 1 .2) thus obtained will be called a CR-system w.r. t. (X, o). Since T is the image of X0 in X1 X o X2 , the association (XI> X2 ; T) -> !!£ is invertible. Among the two equivalent notions, congruence relations and CR-systems, we shall use the latter more fre quently. 1 .2. Let Fqc (c � 1) be the exact constant field of X. Then it is easy to see that the exact constant rings of X; (i = 0, I , 2) are oc- Moreover, since T is irreducible, and since the connected components of X1 x o X2 and of X x Fq X correspond bijec tively, n and 1ll must lie on the same component of X x Fq X. This shows that either c = I or c = 2. DEFINITION 1.2. 1 . !!£ belongs to Case 1 if c = I, and Case 2 if c = 2. We denote by g the genus of X = X ®Fq Fq .
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SHIMURA CURVES
1 .3. It follows easily from the definition that X;'l = X; ®o k (i = 0, l, 2) are pro per smooth irreducible algebraic curves over k. Put X;'l = X;'l ®k k, and let g; denote the genus of X;'l (i = 0, 1 , 2). Then g 1 = g2 = g. Put rp;'l = rp; ®o k (i = 1 , 2). Then rp;7J are finite k-morphisms of degree q + 1 . Therefore, the Hurwitz formula gives (1.3.1)
g0
- 1
=
( q + 1)(g - 1) + ! o,
where o is the degree (over k) of the differental divisor ("'Ilifferente") of rp;'l· 1 .4. When x runs over all the Fqz-rational points of X, y = (x, xq) runs over all the geometric points belonging to the intersection II n 1II on T5• Consider y as a point of the two-dimensional scheme T. Then the local ring fJr, Y may or may not be normal, depending on x. DEFINITION 1 .4. 1 . An Fqz-rational point x of X is called a special point if y = (x, xq) is a normal point of T. The special points of X will be denoted by x i > . . . , xH . All other Fq -rational points of X are called the ordinary points. The special fiber Xos of the normalization X0 of T consists of two irreducible components which can be identified with II and 1II via f-l · These two components meet at above those y = (x, xq) for which x is special. Therefore, the coincidence of the Euler-Poincare characteristics of X07J and of Xos gives (1 .4.2)
g0
- 1
=
2(g - 1) + H;
hence by (1 .3. 1 ), we obtain the following formula for the number of special points : (1 .4.3)
H
= (q - l )(g -
l)
+
! o.
By the Zariski connectedness theorem [4, III, 4.3. 1], H is always positive. 1 .5. DEFINITION 1 . 5. 1 . (i) !:l' is called unramified, if rp17J and rp27J are unramified. (ii) !:[ is called symmetric, if X2 = X1 and 1T = T, where 1T is the transpose of T; or more precisely, if there exists a pair (c&'\, t&"2) of mutually inverse a-isomor phisms t&"1 : X1 ...::::. X2, t&"2 : X2 ...::::. X1 which lift the identity map of X and for which (t&"1 x t&"2)( T) = 1T. (It follows easily that such a symmetry (t&"I > t&"2) is at most unique ; [9b, § 1] .) By (1 .4.3), when !:[ is unramified we have H = (q - 1)(g - 1), and conversely. In particular, g > 1 when !:[ is unramified. 1 .6. When o = Zp, the ring of p-adic integers, we can prove, as an application of our previous work [10], the following rigidity of unramified congruence relations. THEOREM 1 .6. 1 . Let X, and a set 6 of Fpz-rational points of X be given. Then there exists at most unique unramified CR-system !:[ with respect to (X, Zp ) which has 6 tis the set of special points. Moreover, when it exists, it is symmetric.
Thus, when o = Zp , !:[ : unramified implies !:[ : symmetric. As for the existence, the question is more difficult. We already know a necessary condition 161 = (q - 1)(g - 1) for the existence of !:l'. But this is not sufficient. The question is essentially connected with a certain differential on a formal lifting of X (see [10], [7]). I hope to discuss the further developments of [10] in the near future. 1 .7. The most well-known congruence relation is the Kronecker congruence
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YASUTAKA IHARA
relation (Xt. X2 ; T), where XI = X2 is the projective j-line over Zp and T is the closed subscheme of XI x zp X2 defined by the modular equation (/Jp(j, j') = 0 of order p. This is symmetric, but not unramified. As is well known, the ordinary (resp. special) points are the singular (resp. supersingular) j-invariants of elliptic curves over Fp, with the exception of the cusp which is also ordinary. For other examples, see §6 and [9a]. 2. The first Galois theory. Let f!£ = {XI 'PI +- X0 --+ 1t'Z X2 } be any CR-system w.r.t. (X, o). The purpose of this section is to establish a fundamental Galois-theoretic property of the system f!£1} = {XI 7J ...fu- Xo1J � X21J} over k . This will be basic for our further studies. 2. 1 . Let K; denote the function field of X; (i = 0, 1, 2). Then each K; is an alge braic function field of one variable with exact constant field kc (cf. § 1 .2) . The mor phisms (/); (i = 1 , 2) induce the inclusions K; 4 K0, and we have K0 = KI K2, [K0 : K;] = q + 1 (i = 1 , 2). DEFINITION 2. 1 . 1 . L is .the smallest Galois extension of K0 such that L/ Kt. L/K2 are both Galois extensions. Call V,- the Galois group of L/K.- (i = 0, 1 , 2). Note that V0 = VI n V2. DEFINITION 2. 1 .2. a; is the subgroup of Aut(L/k) generated by VI and V2• The Krull topologies of V.- will be extended to the topology of a; defined by the characterization " V; are open". 2.2. Suppose for a moment that f!£ is the Kronecker CR-system (see § 1 .7). Then (in the usual sense of notations) KI = Qp(j('�:)), K2 = Qp(j(p7:)), and L is the field generated over Qp by all functions of the formj(pn7: + b) (n e Z, b e Z [ l /p]) This shows that .
and a;
PGLt(Qp). where (in general) PGLt(k) = {g e GL2(k) ; ordP det(g) = O (mod 2)}/k x . S o , i n this case, we already know the structure o f a; relative t o Vt . V2. I n fact, we know the structure of the related double coset ring and, as its consequence, we know such a property of a; that a; is the free product of VI and V2 with amal gamated subgroup V0 [Sb, Vol. 1, Chapter 2, §28], [16]. We can show that these are general phenomena attached to the congruence relations. 2.3. To be precise, return to an arbitrary CR-system f!{ and consider the disjoint union ( VI\at) U ( V2\at) of two left coset spaces as a point-set. Call it .r 0 • Two points VI g, V2g' (belonging to different coset spaces) are called "mates" if VIg n V2g' =f. 0 . Since ( V; : V0) = q + 1 (i = 1 , 2), each point has exactly q + 1 mates. Consider the diagram .r = ff(a; ; Vt. V2) obtained from this point-set ff 0 by con necting each pair of mates by a segment. Then a; acts on .r by the right multipli cations, and the action is effective, due to the minimality of L. =
2.3. 1 . Let f!{ be any CR-system and put .r = .rca; ; Vt. V2). Then (i) is connected and acyclic ; in other words, for any points A, B e ,ro, there exists a
THEOREM
.r
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YASUTAKA IHARA
by t. Then Vo t is well defined, t- 1 Vot =
v1 .
=
Vo , t 2 e Vo , and t- 1 V1t
=
V2, t- 1 V2t
DEFINITION 2.7. 1 . When !!l is symmetric, GP is the group generated by Gt and t. Note that GP is generated by V1 and t, and that (Gp : Gt) = 2. We shall identify ( V1 \Gt) U ( V2 \Gt) with V1 \Gp, by V1g +-+ V1g, V2g +-+ V1tg (g e Gt). Thus, V1g and V1g' are mates if and only if g'g- 1 e V1t V1 . This group GP acts on ff effectively, and Theorem 2.3.l(ii) can be rewritten as : (ii*) If A, B, A', B' e 9""0 are such that /(A, B) = /(A', B'), then there exists g e GP such that A' = Ag, B' = Bg. As a corollary of Theorem 2.3. l(i) we obtain : CoROLLARY 2.7.2. GP is the free product of V1 and V0 U V0t with amalgamated
subgroup V0•
When f£ is the Kronecker CR-system, we have (for a suitable extension t) : -1 t
- (0p 0) . -
2.8. Now return to an arbitrary CR-system f£. When it is unramified, K0fKh K0fK2 are unramified extensions of algebraic function fields, and this implies that L/Ko is also unramified. DEFINITION 2.8. 1 . !![ is called almost unrami.fied if almost all prime divisors of K0fk are unramified in L. By the above remark, !!l: unrami.fied implies !!l: almost unrami.fied. I do not know
whether the almost-unramifiedness also implies the symmetricity when o = Zp. This definition of almost-unramifiedness singles out the class of CR-systems that are related to automorphic functions. There are examples of !![ that are not almost unramified. 2.9. The arithmetic fundamental group F. Let !!l be almost unramified and sym metric. Then to each !!l and an embedding e : k 4 C (C : the complex number field) we can associate a discrete subgroup r of (2.9. 1) determined up to conjugacy, in the following manner. Consider the set Z of all those places �c of L into C U ( oo) such that (i) �c extends e, and (ii) the valuation ring of �c is either L itself or is a discrete valuation ring. Let GP act on Z as �c � g�c where (g�c)(a) = �c(ag) (�c e Z, g e GP, a e L). Then Z carries a natural Gp invariant complex structure (cf. [5b, Vol. I , Chapter 2] or [9b]). Moreover, each connected component Z0 of Z is isomorphic to the complex upper half-plane .p, and GP acts transitively on the set of all connected components of Z. Choose any con nected component Z0 of Z, and put
.
(2.9.2) Then F acts effectively on Z0 � .p, and hence F can be considered as a subgroup of Aut(.f)) � PS�(R). On the other hand, F is naturally a subgroup of GP. By these two embeddings, r will henceforth be considered as a subgroup of PS�(R) x GP. Up to conjugacy in PS�(R) x GP, r does not depend on the choices of Z0 and of the above two isomorphisms. (On the other hand, F depends essentially on e. )
297
SHIMURA CURVES DEFINITION
and e. Put
2.9 . 3. F is called the arithmetic fundamental group belonging to f£ r+ = r n (PSL2(R) L1; = r n (PSLz(R)
(2.9.4)
x
X
Gt), V;)
(i
=
1 , 0, 2).
Then the projections of Ll; to PSL2(R) are fuchsian groups of the first kind, and when f£ is unramified, they are nothing but the universal covering groups of X;r/:i!;>kp.z Therefore, r+ is discrete in the product group (2.9 1), and (in view of (2.9.6) below) moreover it is torsion-free when f£ is unramified. By the same reason, the quotient of the product group (2 9.1) modulo r+ has finite invariant volume, and it is compact when f£ is unramified. The projection of r+ in PSL2(R) is dense, and that in Gt is dense in Gt. Now for F itself: In Case 2, c acts as an involution of k2/k 1 ; hence F = r+. In Case 1 , the symmetry c of K0 can be extended to an element of F; call it c again. Then L10c is well defined, and we have (2.9. 5) .
.
Therefore, F is generated by r + and c. Since Gt = V0F+, and GP we obtain immediately from (2.3 . 2), (2.7 . 2) the following
=
V0F (Case 1),
COROLLARY 2 . 9.6. (i) r+ is the free product of L1 1 and L1 2 with amalgamated sub group L10 • (ii) In Case 2, we have F = r+, and in Case 1 , F is the free product of L1 1 and Llo U L10c with amalgamated subgroup L10•
When f£ is the Kronecker CR-system, we have
r+ = PSLz (Z[1/p]),
and r = {r E GL2(Z [1/p]) ; det r E pZ}f ±pz,
up to conjugacy in PSL2(R)
x
PGL2(Qp). 3. The canonical liftings. Let f£ = { X1 \1'1 +-- X0 --+ 1"2 X2 } be any CR-system w.r.t. (X, o). We shall study in detail the reduction mod p of some special kind of places � of Lfk. Roughly speaking, we consider those � whose stabilizer in Gt is "big" (the condition [A] of §3.3). Our main goal is to outline the proof of Theorem 3.4. 1 , which states that the reduction mod p induces a bijection between the set of all Gt-orbits Gt · � of those � and that of all ordinary closed points (scheme-the oretic points) of X ® Fqz· This is achieved by constructing a canonical lifting x --+ �; of each ordinary geometric point x of X to a geometric point �; of X;11 (i = 1, 2). When f£ is unramified and symmetric, our main results are reformulated in terms of F (§3. 14, Main Theorem I, and §3. 1 5). 'When
c
=
2 , ® is w.r.t. the embedding k
c:.
C determined b y Z0•
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YASUTAKA IHARA
3. 1 . Reduction ofG�-orbits in Pl(L/k). Denote by Pl(L/k) the set of all places of L into k U ( oo) over k. Let Aut(L/k) act on Pl(L/k) by � ---+ ge, where (ge)(a) = e(ag) (a E L, g E Aut(Lfk)). For each e E Pl(L/k), denote by ei its restriction to K; (i = 1 , 2) considered as a point of X;� , and by e;, the unique specialization of �; on X. Since T ® o Fq = n u 1ll, we have ez, = e��± l. Since G� is generated by VI and V2, the repeated use of this fact shows that for any g E G� and 1 � i, j � 2, (ge);, and �is are conjugate over Fq, and that when i = j they are conjugate over Fqz · When f!l' is symmetric, (g�);, and �is are conjugate over Fq for any g E GP . We shall pay attention to the following mappings induced by � ---+ �is (i = 1 , 2) : (3. 1 . 1); Pl(Ljk) ,_. {Points of X}, (3. 1 .2); G�\Pl(Lfk) ,_. {Fqz-conjugacy classes of points of X}, and when !![ is symmetric,
Gp\Pl(L/k) ---+ {Fq-conjugacy classes of points of X}. 1 , 2) they are the transforms of each other by the involution
(3. 1 . 3) ,ymm
As for (3. 1 .2); (i = of Fqz/Fq. 3.2. For each � E Pl(L/k), define its (transcendental) decomposition group Dt and the inertia group It by (3.2. 1)
Dt It
= =
{g e G� ; g� "' � } {g E G� ; g� = � }
= =
{g E Gt ; 8 ( = 8�}, {g E Dt ; g acts trivially on 8�/md,
where - is the equivalence of places, 8� is the valuation ring of �. and m� is its maximal ideal. When f!l' is symmetric, we define D� and Ir; in the same manner, i.e., just by dropping the symbol + in (3.2. 1). Obviously, It (resp. I�) is normal in Dt (resp. D� ) . For each g E G� , and g E GP when f!l' is symmetric, define its degree Deg(g) by (3.2.2)
Deg(g)
=
Min l(A, A g) .
A Eg-O
Then Deg(g) is a nonnegative integer, and it is even if and only if g E Gt. Moreover, Deg(g) = 0 if and only if g is G�-conjugate to some element of V1 or V2. Put (3.2.3)
I� = {g E Jt ; Deg(g)
=
0} .
3.3. We consider the following condition [A] for � E Pl(L/k) : [A] IUorms a subgroup of Jt with infinite index. If [A] is satisfied for �. then also for g� (g E Gt, resp. Gp). When f!l' is unramified, then L/Klf (g E G�) are unramified, so that the usual inertia groups It n g- 1 V;g are trivial. Therefore, I� = { 1 } . Therefore, the condition [A] is equivalent with [A] (!!l' : unramified). It is an infinite group. Return to the general case of !!l', and define Pl(L/k; [A]) as the set of all � E Pl(L/k) satisfying [A]. It is stable under Gt (resp. Gp) · For each � E Pl(Ljk ; [A]), define Deg+(�) (resp. Deg(�) when f!l' : symmetric ) as the minimum value of Deg(g) where g runs over all elements of It - I� (resp. I� - I�). Then Deg+(�) depends only on
299
SHIMURA CURVES
Gt · � . and when GP . �.
?£
is symmetric, both Deg+(�) and Deg(�) depend only on
3.4. Now a main result of §3, in a form still implicit about the canonical liftings, is as follows. THEOREM 3.4. 1 . (i) For each i = 1 , 2, the reduction map � -> �is induces a bijection between the set of all Gt-orbits in Pl(Ljk, [A]) and that of all Fq2-conjugacy classes of ordinary points (see § 1 .4) of X; i. e., red; : Gt\Pl(L/k , [A]) .!!!_, {ordinary closed points of XQ9 Fq2}. Fq
Moreover, we have Deg+(.;) = 2 · Deg(�;s/ Fq2). (ii) When f£ is symmetric, red; (i = 1, 2) induce one and the same bijection between the set of all GP-orbits in Pl(Lfk; [A]) and that of all Fq-conjugacy classes of ordinary points of X; i.e.,
red; : GP \Pl(L/k ; [Al) --!"'. {ordinary closed points of X}. Moreover, we have Deg{.;) = Deg(.;;s/ Fq) . (iii) For any .; E Pl(Lfk ; [A]), I� is a normal subgroup of It such that Jt/I� � Z, and Dtfit is canonically isomorphic to the full Galois group of Be/me over k2 n (Be/me). When ?£ is symmetric, I� is also normal in Ir;, Ir;/I� � Z, and De/Ir; is canonically isomorphic to the full Galois group of Br;/me over k. The proof will be outlined in the rest of §3. The main point is the construction of the inverse maps .;is -> .;, the liftings of .;is to such .; that satisfy [A] (see §§3. 103. 1 1 ) . But before this, we shall give some preparatory materials. 3.5. Rivers on the tree .'T. To look more closely at the Gt-orbits Gt · .; such that .;is are ordinary, the following notion of "river" on .'T = .'T ( Gt ; Vr. V2) is very helpful. DEFINITION 3.5. 1 . A river on .'T is defined when each segment has a direction in such a manner that for each A E .'T 0 and its mates B0, B 1 , . . . , Bq , precisely one of AB; (0 � i � q), say AB0 , is directed outward as AB0 and the rest are all directed inward ; Alir,. · · , � · t
----> 0
____, .
""' /
•
l
0
/ ""'
t
0 <----
. ____,
T
A river on .'T is determined uniquely by an arbitrary infinite flow going down stream o___, e___, o___, •___,
···
300
YASUTAKA IHARA
on ff. It is clear that two such flows on ff determine the same river if and only if they meet somewhere in their downstreams. 3.6. Let � e Pl(L/k). Since �is (i = I , 2) are mutually conjugate over Fq, � l s is ordinary if and only if �2s is so. Call � ordinary when �is are so. Each ordinary element � e Pl(L/k) determines a river on ff, called Riv(,;), in the following manner. Let A = V1 g 1 and B = V2g2 be mates, so that V1 g 1 n V2g2 #- 0 . Take g E V1 g 1 n V2g2 • Let � be the restriction of g� to K0 considered as a point of X0", and �s be its unique specialization on Xos· Since � is ordinary, g� is also ordinary. Therefore, �s does not belong to n n en (the intersection taken on Xos). Therefore, �s lies on just one of n or en. If it is n, give the direction Aii, and if it is en, give it the other way. It is easy to see that this defines a river p = Riv(�) on ff. Now let [A0 --+ A 1 --+ . . ] be any infinite flow in this river, and take g E Dt (resp. De when f£ : symmetric). Then g leaves Riv(,;) invariant, so that the g-transform of the above flow is another flow [A« --+ Af --+ • • ·] of Riv(,;). Therefore, they must meet somewhere in their downstream. Therefore, A� = A;+iJ holds for some o E Z if i is sufficiently large, and o is independent of the choice of [A0 --+ A 1 --+ . . . ]. The mapping g --+ o = o(g) defines a homomorphism o : Dt --+ Z (resp. De --+ Z), and it is easy to see that (3.6. I) g E Dt (resp. De). Deg(g) = io(g) l . .
In particular, 19 is the kernel of ol1t (resp. o l 1eo since I� c Jt) ; hence /� is always a normal subgroup of It (resp. Ie) and we have rtm � (0) or Z {resp. Ie/19 � (0) or Z) whenever � is ordinary. 3.7. The mapping X · Let i = 1 , 2. A point ,;; of X;" is called ordinary if its specialization ,;;s E X is so. Let (X;") o rd denote the set of all ordinary points of X;". The mapping X of X��d U X��d into itself is defined as follows. Let g 1 E .Xy�d (resp. ,;2 E X�d), and let � (resp. �') be the unique point of X0" such that q5 1 (�) = ,; 1 (resp. q52(0 = ,;2) and that its specialization �s (resp. �:) on Xos lies on n (resp. e D). Here, q5; = rp; ® k k.. Then X is defined by x(,; 1 ) = q52(�), x(�2) = q5 1 (�'). Thus, X maps .Xy�d into .X��d, and vice versa. Generically, X is a q-to-one mapping. (An important p-adic analytic property, the rigidity of X• has been investigated by Tate, Deligne and Dwork (ct'. [3]) in the case where f£ is the Kronecker CR-system.) 3.8. Illustration of X · Take any ,; E Pl(L/k) which is ordinary, and put (3.8.1)
(i
=
I , 2).
Then the disjoint union (Gt ,;) 1 U (Gt ,;)2 , which is a subset of points of X1 " can be naturally identified with
U
X2".
(3.8.2) Moreover, the action of X on this set corresponds to the arrows defined from Riv(,;) on ff ; "(Gt ,;) 1
U
(Gt ,;)2 with the action of x "
For example, if I�
=
{ 1 } and It
�
=
{ ff with Riv(�)}/It.
Z, then this quotient looks like (for q
=
2).
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SHIMURA CURVES
(3.8.3)
• • •
• • •
The length of the central cycle is equal to Deg+(.;). 3.9. The schemes T,.i (p 1). For each i, j (1 � i, j � 2) and I � 0 with i - j = l (mod 2), (3.9.1) a closed subscheme Tii(IJ1) of X,. ®. Xi is defined in the following way. Let At. A 2 be the points of g- o corresponding to Vt. V2 , respectively. Fix i, j and I satisfying (3.9. 1), and let B be any point of g- o such that l(A;, B) = I. Then l(A i , B) is even, so that B = A � with some g E Gt- Note that the double coset VigV,. is independent of the choice of B. Consider the ring homomorphism K,. ® k Ki --+ K,. · K� c L de fined by I; ,.u_. ® v,. --+ I; ,.u_.v_.g. Then the kernel defines a closed integral subscheme of X,. x 0 Xi depending only on i,j, I. Call it T,.j(pl). It is easy to check that Ti;(p1) = 1T,.j(p1). When !!£ is symmetric, T;j{p 1) is symmetric and depends only on I. PROPOSITION 3.9.2. Let T;j{p1)5 be the special fiber of T;j{p 1). Then Tii(IJ1) is a s closed subscheme of X x F q X determined by the following two properties : (i) it is locally defined by a single equation, (ii) its irreducible components and their multi
plicities are given by the following formula : Tii(p1)5 = (fll + tffl) + I; q k-l(q 1) (fl 1 -2k + tffl-2k) + e(/)q
_
_
1) Ll,
where ffr is the graph of the qrth power morphism of X, 1ffr is its transposed graph, L1 is the diagonal of X x F q X, and e(l) = 1 (resp. 0) according to 1: even (resp. / : odd).
3. 10. The canonical liftings. Let i = 1 or 2, and I � 1 . For each ordinary point of X with degree I over Fqz, we are going to define its canonical lifting .; .. E x..'f}. Define j = 1 or 2 by the congruence i - j = I (mod 2), and look at Proposition 3.9.2. Observe that the points of X with degree I over Fqz are in a one-to-one cor respondence X <--+ (x, Xq1) With those geometric points of fll n tffl not lying on any other irreducible components of T,.j(p1)s than 01 or 101 . X
PROPOSITION
scheme Tii(pl).
3.10. 1 . If x is ordinary, (x, xq') is not normal on the two-dimensional
This can be proved easily by using a suitable morphism from the normalization
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YASUTAKA IHARA
of T,.j(p1) onto X0• The following lemma, which is crucial to the canonical liftings, is an elementary exercise in two-dimensional local rings : LEMMA 3 . 1 0.2. Let R be any complete discrete valuation ring, and let k (resp . .t) denote its quotientfield (resp. residue field). Put Spec R = {7], s} (s : the closedpoint). Let Z be a two-dimensional integral scheme having a structure of a proper and flat R-scheme, and let z be a .t-rational ordinary double point ofZ, = Z ® R .t which is not normal on Z. Then there is a unique point I;; on Z1J which is not normal and which has z as its specialization. Moreover, I;; is a k-rational ordinary double point ofZ . 1J
Since z is reduced on Z, (being an ordinary double point) and since Z is flat over R, the two-dimensional local ring 8z, z is a Cohen-Macaulay ring. This gives the
existence of 1;; , by the Serre's criterion for normality [4, IV, 5.8.6]. The uniqueness and the last assertion follow from the isomorphism fJz, z � R [[X, Y]]/XY for the completion of 8z, z· Now apply Lemma 3. 10.2 for R = o2, Z = T,.i(p1) ® o o21 and z = (x, xq') to obtain the following
THEOREM 3 . 1 0.3. Let i,j, I and x be as at the beginning of§3. 10. Then there exists a unique k2,-rational point (�,., �j) of T,.j(p1) 7J (�; E .X,.'I , �j E Xi'!) which is not normal and which lifts (x, xq') . Moreover, (�,., �j) is an ordinary double point of T ;j{p1) 7J ® k k2 1 ·
3. 10.4. �; is the canonica/ /ifting of X on X;'l ' 3. 1 1 . The following basic properties of the canonical liftings follow from the uniqueness of (.;,., .;j). First, since Tii(P') = 1 T;iP'>, the definition and the uni queness of (�,., �j) tell us immediately that .;j is the canonical lifting of xq' on Xi'!' DEFINITION
More important properties will be given in the following
THEOREM 3. 1 1 . 1 . Let x be an ordinary point of X. For each i = I , 2, let .;,. be the canonical lifting ofx on .X,.7J . Put d = Deg(x/ Fq) . Then (i) � .. is kd-rational, and is of degree d over k ; (ii) ��q] is the canonical lifting of xq on x,.7] ; (iii) x<� l ) (resp. x<.;2)) is the canonical lifting of xq on .X2 7J (resp . .X17]) ; (iv) �.. is the unique point of .X,.7] which specializes to x and satisfies x2d(�;) = �.. ; (v) when !!{ is symmetric, we have � 1 = .;2.
In the case where !!{ is the Kronecker CR-system, the canonical lifting is the same as the Deuring's lifting of (thej-invariant of) a nonsupersingular elliptic curve over Fp to (the }-invariant of) such an elliptic curve over Qp that has the same endo morphism ring as the former. The above characterization (iv) follows the Dwork's characterization of the Deuring lifting given in [3]. 3 . 1 2. Fix i ( 1 ;;;::; i ;;;::; 2), and consider the canonical lifting x -4 .;; of the ordinary points of X to the points of .X,.'I . By Theorem 3. 1 1 . 1 it induces the lifting of each ordinary closed point (x, xq2 , . . . , xq2'-� (1 = Deg(xjFq2)) of X ®F . Fq2 to a closed point (�;, � �q l 2, . . . , ��ql21-2) of X,.7J ® k k2• Moreover, since ��ql 2 = x 2(�;), the extensions of �... �}ql 2 , . . . to the elements of PI(Lfk) belong to one and the same Gt-orbit Gt · .;. Since X 2'(�;) = �;, Jt contains an element g with o(g) = 2/. From this we can show easily that Gt · � E PI(Lfk ; [A]), Deg+(.;) = 2/, and that Dt/It induces the full automorphism group of 8�;/m�; over k2 n (8�;/m�;) . So, as for Theorem 3.4. 1 , it remains to prove that "these Gt · � exhaust Gt\PI(Lfk ; [A])".
303
SHIMURA CURVES
This can be checked in the following way. First, count the degree of the inter section product T;;(p 21)7] . L17] on x.-7] X k x.-7] by using Proposition 3.9.2. Then count the number of those distinct points of T,-,-(p21)7J L17J that either belong to the Gt-transforms of canonical liftings, or do not correspond to an element of Pl(Lfk ; [A]), by using Riv(e). We can check that the latter number reaches the former, and this gives rise to that "Gt · � exhaust Gt\Pl(Ljk ; [A])". The details [9b] will be published shortly. 3 . 1 3. One can deduce some more conclusions from Theorem 3. 1 1 . 1 . For example, for each � e Pl(Ljk ; [A]), the residue field Be/me is abelian over (Be/me) n k2 (and also over k if PI : symmetric), and its norm group can be determined explicitly. Roughly speaking (i.e., disregarding ramifications), this is determined by the slope of T;;(p21)7J at (�.-. �..). At this stage, we use the work [12] of Lubin-Tate. 3. 14. The First Main Theorem. Now let PI be an unramified symmetric CR-system w.r.t. (X, o). Fix an embedding e : k 4 C, and let F be the arithmetic fundamental group belonging to PI and e (§2.9). In §§3. 14- 1 5, we shall give some reformula tions of Theorem 3.4. 1 and Theorem 3. 1 1 . 1 in terms of F. As in §2.9, take a connected component �0 of �. identify �0 with .p, and con sider r = {r e GP ; r�o = �0} as a group of transformations of .p. For each 'r E .f), let F� denote its stabilizer in F, and put (3. 14. 1) £' = { -r e .f); IF� I = oo } . Obviously, £' is a r-stable subset of .p. The points of £' will be called F-points on •
.p.
M AIN THEOREM I. Let l_l3(X) denote the set of all ordinary closed points of X. Then the reduction mod p induces a bijection (3 . 14.2)
ir : F\£'
�
l_l3(X).
This is a direct corollary of Theorem 3.4. l {ii), because there is a canonical bijec tion between F\£' and Gp\Pl(Ljk ; [A]), defined as follows. First, an element of � is algebraic over k if it has a nontrivial stabilizer in GP . Therefore, £' is embedded in Pl ( L/k). Moreover, for any -r e .f) n Pl(L/k) , r� is nothing but the inertia group I� defined in §3.2. Since the condition [A] (for PI : unramified) is equivalent with 11� 1 = oo (§3.3), this implies that .f) n Pl(L/k ; [A]) = £'. Since GP acts transitively on the set of connected components of �. this induces the above bijection. 3 . 1 5. Although the Main Theorem I has a simple form, it does not contain our best knowledge on the reductions and the liftings of geometric points, which will now be described in terms of F as follows. Let L1,- (i = 1 , 2) be the subgroups of F corresponding to V,- (see §2.9). Then L1.-\.P can be identified with the set of points of X,-0 = X,-'1 ®k. C, where ® is w.r.t. the embedding kc 4 C defined by the restric tion of �0 to kc. Let x be a geometric point of X over Fq with degree d, and P = (x, xq , . . · , x qd - 1 ) be the closed point of X containing x. Let £'P denote the set of all those points -. e £' such that the F-or bit containing -. corresponds to P by i r ((3. 14.2)) . Case 1 . In this case, r contains an involution c satisfying (2.9.5), so that there is no distinction between the two choices of i. The quotient L11 \£'P c xlC is illustrated by the diagram :
304
YASUTAKA IHARA
(3. 1 5. 1)
where the arrow is the x-mapping which is q-to-1 , and the central cycle is of length
d which consists of the canonical liftings of xq" (0 � v < d) on ..\\ . Case 2. In this case, the simultaneous o 2-structure for f£ gives mutually con
jugate structures for X1 and X2• Therefore, X1e and X2c are generally noniso morphic. The disjoint union (Lh\Jf'P) U (.:::12\Jf'P) is illustrated by the diagram (3.8.3), where o e.:::1 1 \Jf'P, e.:::12\£P, the arrow is the x-mapping, and the central cycle is of length d (which is even) which consists of the canonical liftings of xq" on X;c (i is distinguished by the parity of v). Finally, in each case, rT is free cyclic and the homomorphism 0 :DT -+ z of §3.6 induces an isomorphism rT � d . z. Therefore, if rT is a generator of rT , the degree d of P over Fq is equal to Deg (r T) , where r T is considered as an element of GP . More over, we can distinguish the two generators of rT by the signs of o (rT) . This sign has also the following interpretation. Let dfdz -+ A · dfdz be the linear transform of the tangent space of .p at 7: induced by r T" Then A belongs to kx (via the embedding e : k -+ C), and ordpA has the same sign as o (rT) . 4. The second Galois theory. The subject of second Galois theory is a system J = (fi, f0, f2) of three finite etale morphisms /; : Xf -+ X, connecting two CR systems f£ = {X1 9'1 +-- X0 -+ 9'2 X2 } and fl'* = {Xf 9'� +- Xt -+9'; Xl} in a com patible way. The purpose of §4 is to present our main results on the two categorical equivalences induced by f -+ f ® C and f -+ f ® Fq . The former is highly nontrivial. 4. 1 . In general, let t1lt = { U1 ¢1 +-� U0 f->¢2 U2} be a system formed of three schemes u, (i = 0, 1 , 2) and two morphisms cp; : U0 -+ u,. (i = 1 , 2). Let •
tflt* = { Ut .!l_ Ut
__£ Ul}
be another such system. We say that a pair (tflt*, /) is a finite hale covering of t1lt, if/ is a triple {fh f0 f2) of finite etale morphisms /;: Uf -+ U,. (i = 0, I, 2) satisfy ing /; cpf = cp; o /o (i = 1 , 2) and Ut � U,.* X U; Uo (canonically ; i = 1, 2). Now let fl' = {X1 9'1 +- X0 -+ 9'2 X2 } be a CR-system w.r.t. (X, o), and f£ * = {Xf 9>� +- Xt -+ 9'; X: } be another CR-system w.r.t. (X*, o), with the common base ring o. Suppose that (f£ * ,f) is a finite etale covering of f£ and that the constituents /; (i = 0, 1 , 2) of f = {.fi., f0 , f2) are o-morphisms. Then we shall call (f£ * , /) a .finite etale CR-covering of f£. In this case, it follows from the definitions that the two finite etale morphisms /;. : X* -+ X obtained from /; (i = 1 , 2) by the base change ®o Fq must coincide. This morphism/;. will be denoted by f It also follows easily that 6* = J-1 (6) , where 6 (resp. 6*) is the set of special points w.r.t. fl' (resp. fl'*). When fl' is unramified, f£* is also unramified, because cp� are the base ,
0
305
SHIMURA CURVES
changes of rp;1 r When !!( is symmetric, we can prove easily by using [4, IV 1 8.3.4] that !!l'* is also symmetric. 4.2. Now let [![ be any unramified symmetric CR-system w.r.t. (X, o), and fix an embedding e : k 4 C. Let F be the arithmetic fundamental group belonging to !!f, e. Then our main result on the second Galois theory reads as follows. MAIN THEOREM I I . The following three categories (i), (ii), (iii), are canonically
equivalent : (i) Finite etale CR-coverings ([![*, j) of !!f. (ii) Subgroups F* of r with .finite indices. (iii) Finite etale coverings f: X* � X, with X* : connected, such that all points of X* lying above the special points xh · · · xH of X are Fqz-rational points of X*. ,
The equivalence functors are as follows ; (i) � (ii) is the functor of taking the arithmetic fundamental group, and (i) � (iii) is the one described in §4. 1 . The proofs are omitted here. The equivalence (i) ,..., (iii) is proved in the same way as in [6]. The proof of (i) ,..., (ii) contains much more delicate points. The main point is our criterion (using some liftings of the Frobenius) for the good reduction of unramified coverings of curves [8]. 5. Simultaneous uniformizations and reciprocity. I n §5, !!( is an unramified symmetric CR-system w.r.t. (X, o), e is a fixed embedding k 4 C, and r is the arithmetic fundamental group belonging to [![, e. 5. 1 . Let F* be a subgroup of r with finite index, ([![*, /) be the corresponding finite etale CR-covering of [![, and (X*,f) be the corresponding finite etale covering of X (§4). Let .J'f' (resp . .J'f'*) be the set of all F-points (resp. F*-points) on .p (§3 . 1 4). - Then .J'f'* = .J'f', because cr� : rn < 00 for any 7: E .p. Let �(X*) denote the set of all ordinary closed points of X* w.r.t. [![*. As we noted in §4, �(X*) is the inverse image of �(X) w.r.t. f By Main Theorem I for !!f*, we have a canonical bijection (5. 1 . 1) ir• : F*\.J'f' � �(X*) defined by the reduction mod p. When X* I X is a Galois covering, and P* is a closed point of X*, the Frobenius automorphism ((X*/X)IP*) is by definition the element a of the Galois group Gal(X*IX) of X* IX that acts on the geometric points of P* as the qdth power map, where d is the degree over Fq of the closed point P of X lying below P*. (Since the action of elements a of the Galois group on points are from the left, and that on functions are from the right connected by f17(e) = f(ae) the geometric and the arithmetic Frobeniuses do not have inverse expressions here.) MAIN THEOREM I l l . (i) The diagram ,
F * \.J'f' � �(X *) (5. 1 .2)
1
F\.J'f'
canon.
----c-----' 'r
1
f
�(X)
is commutative ; (ii) when r* is a normal subgroup of r, the natural action of r 1 r * on F*\.J'f' and the action of Gal(X* I X) on �(X*) correspond with each other through ir• and through the canonical isomorphism FIF * � Gal(X*IX) of Main Theroem II ;
306
YASUTAKA IHARA
(iii) moreover, when F * is normal in r, the Frobenius automorphism of P! = ir· (F * . z-) (z- E £') over X is given by r * . rn where r� is the generator of r� such that o(r�) < 0 (§3. 1 5) ; (5. 1 . 3)
( x;fx)
= r*
.
r� ·
Thus, as a universal expression of the Frobenius automorphism, we may write in a more suggestive way as (5. 1 .4)
( �x)
= r. .
6. The case of Shimura curves. The above results-apply to each Shimura curve for almost all !J by Theorem 6.3.3 which is the "p-canonical version" of the Shimura congruence relation. 6. 1 . Let F be a totally real algebraic number field of finite degree m, and B be a quaternion algebra over F which is unramified at one infinite place oo 1 of F and oo m of F. As usual, the subscripts I, ramified at all other infinite places oo2, A and R indicate the localization, adelization, and the infinite part of the adele, respectively. The reduced norm for B/F (and its localization, etc.) will be denoted by N( * ). For any bn e Bji, N(bn) is positive at oo2, oo m . Let (B�)+ denote the group of all bA e B� with N(bn) : totally positive, and put (Bji)+ = (B�)+ n Bfi. Now let .2 be the function field over F constructed in Shimura [15] for B/F (the case "n = r = 1 "). It is an infinitely generated 1-dimensional extension of F, and is obtained as the total field of arithmetic automorphic functions on B. In [15] Shimura established an isomorphism · · · ,
· · · ,
Aut(-2/ F) � (B�)+ /(Bji)+ P , (6. 1 . 1) where - denotes the topological closure. Recall that the algebraic closure of F in .2 is the maximum abelian extension pa b of F, and that the diagram ·
(6. 1 .2) is commutative, where t is the homomorphism defined by (6. 1 . 1), c is the homo morphism defined by the class field theory, p is the restriction homomorphism, and
N- 1 (bA) = N(bA) -l .
For each open compact subgroup U of Aut(-2/F), let ,2u (resp. (Ph)u) be its fixed field in .2 (resp. pah). Then ,2u is an algebraic function field of one variable over (F"h)u . Let S(U) be the proper smooth curve with function field .2 u . Then S(U) is called the Shimura curve corresponding to U. 6.2. Now let p be a finite place of F at which B is unramified. Consider F: as a (central) subgroup of (B�)+ and let P be the closure of t(F: ) Then P is a central compact subgroup of Aut(-2/F). Let km (resp. k
SHIMURA CURVES
Put (6.2. 1)
307
BxA _ IT ' BxI ' ( * oo, p
(where 1 # oo is the abbreviation for I # oo h · · · , oom), and for each subgroup H of B� (resp. Fi) denote by H- the topological closure in B] (resp. FJ) of the projec tion of H in B] (resp. in Fl). Now look at the canonical isomorphism (6.2.2) Then the groups of (6.2.2) act on £!P in the natural manner. Put (6.2.3) where (F x)+ is the group of all totally positive elements of F. The isomorphism (6.2.2) induces (6.2.4) DEFINITION 6.2.5. For each open compact subgroup UA of BVl f(Fx)-, the fixed field of UA in £!P, with the natural action of B�/F�, will be called the B�fF:-field associated with UA, and denoted by L = L(UA) · Thus, L(UA) is a one-dimensional infinitely generated extension over k
308
YASUTAKA
IHARA
Question 6.2.8. Does there exist a symmetric CR-system El(p, UA) over Op which has _q"(p, UA)TJ as its general fiber?
Shimura's work provides an affirmative answer to this question for "almost all p" in the sense described below, and it has been supplemented by Y. Morita to some results for "individual" (p, UA). But as far as the author knows, this question has not yet been solved for all cases. Note here that by our Main Theorem II (§4) and the last statement of §4. 1 , when (6.2.8) is validfor (p, UA) and when cp;TJ (i = 1 , 2) are unramified, then this is also true for (p, U'J) for all open subgroups U'J of UA.
6.3. Let p be as in §6.2. An open compact subgroup U of Aut (2/F) is said to be coprime with p if U contains the image t( Vp) of a maximal compact subgroup Vp of Bi. In this case, Vp is uniquely determined, and is called the p-component of U. Obviously, U is coprime with almost all p. Let U be the coprime with p. Then (Pb) U c k
Y.
Morita proved among others the following :
THEOREM 6.3.4 (MORITA) . When F = Q, Question 6.2.8 has an affirmative answer for all p at which B is unramified and with which U is coprime. 6.4. Let (13, UA) be such that Question 6.2.8 has an affirmative answer. We shall see how the objects L, GP , r, £', etc. are given explicitly for El = El(p, UA) · Note first that the base ring o = oP is the ring of integers of FP . (i) The field L and the group Gp (§2). The field L for this case is given by L( UA) ® km Fp, and the group Gp is nothing but Bi fFi � PGL2(Fp). (ii) The ramifications and (£', F) (§§2, 3). It is clear that El is almost unramified. Moreover, if B 1!- M2(Q) and UA is sufficiently small, then El is unramified. Let e : Fp 4 C be an extension of oo 1 . Then the group r and its two embeddings are
309
SHIMURA CURVES
given by r = F(UA), proa l ' prp, respectively. To give .?l' explicitly, let Jp(B) denote the set of all such quadratic extensions M of F contained in B that M is totally imaginary and that ((M/F)fp) = 1 . Put (Bx)+ = Bx n (B;4)+. Then (Bx)+ acts on .p through the localization at oo 1 , and for each M e Jp(B) there is a unique com mon fixed point 1:M of Mx on .p. Now .?l' is given by (6.4. 1) Since 7:M # 1:M' for M # M', Yf can be identified with Jp(B) . The corresponding action of F on Jp(B) is then given by M --+ rMr- 1 . (iii) The second Galois theory. The open subgroups Uj of UA correspond bijec tively with the congruence subgroups F* of F(UA) · Our theory (§§2- 5) is valid for any subgroups of F(UA) with finite indices. We do not know whether all such sub groups F * are congruence subgroups, except for the case of B = M2(Q) where it was proved to be valid by Mennicke and Serre. (iv) The reciprocity. For each M E Jp(B), the group r� for 7: = 'l:M is given by F n (Mx/P). Let 'P = p 1 p2 be the decomposition of 'P in M. Then the degree Deg(7:) is the smallest positive integer d such that 'P! is principal ; 'P1 = (a) . When fl£ is unramified, F� is free cyclic and is generated by the class of a (mod Fx). The two generators of r� are distinguished by the choice of p 1 . Choose p 1 to be the restric tion ofp to M and let r� be the generator of F� represented by a (where 'P! = (a)). Then this is the generator describing the reciprocity of Main Theorem III. Thus, we have described explicitly the objects of Main Theorems I, II, III for the case of Shimura curves. A numerical example is given in [9a]. 7. Comments and remarks. 7. 1 . Brief review of the elliptic modular case (cf [5b, Chapter 5 in Vols. 1 , 2] ; [6]). Let X = Spec Z[j] be the affine j-line over Z, and put Xc = X ® C. Let .p be the complex upper half-plane, L1 = PSL2 (Z), and identify LI\.P with Xc via the modular j-function. 3 Let 9R be the set of all imaginary quadratic subfields of M2(Q). For each M e 9R, denote by 7:M the common fixed point of elements of Mx in .p, and put jM = j(7:M) which is a point of Xc. Then jM is determined by the LJ-con jugacy class of M in 9R, and is an algebraic integer. Fix a prime number p, and its extension p in the algebraic closure Q. Put X = X ® Fp . A geometric point of X will be called special (resp. ordinary), if it is the j-invariant of a supersingular (resp. singular) elliptic curve in the Deuring's sense [1]. As is well known, the special points are Fpz-rational. For each M e 9R, denote by jM the geometric point of X obtained by the reduction mod p of jM. Then : (I) 9R 3 M --+ jM E X ® Fp is surjective. (II) If (Mfp) # 1 , thenjM is special. (III) If (Mfp) = 1 , then jM is ordinary ; moreover, when this is so, jM' (M' e 9R) is Fp-conjugate with jM if and only if M' and M are F-conjugate in M2(Q), where (7. 1 . 1)
r = {r E GL2(Z[1/p]) ; det (r) e pZ}f ±p z.
3The constant multiple ofj is normalized in such a way thatj( .; - 1 )
=
1 23•
310
YASUTAKA IHARA
Therefore, if Yf denotes the set of all 7:M with M E 9R and (Mfp) = 1 , then the reduction mod fJ induces a bijection {7. 1 .2) ir: F\ Yf � {Fp-conjugacy classes of ordinary geometric points of X}. Moreover, for each Fp-conjugacy class P = (j, jP, jPd-1) of ordinary geometric points of X, the collection L1\YfP of all points jM such thatjM e P can be illustrated by the diagram (3. 1 5. 1), where the arrow e--+e' is the unique fJ-adic lifting of the pth power map satisfying (e, e') E T(p) ® Q, T(p) being the Heeke correspondence c X x z X defined by the double coset L1(1 p)L1. The mapping X is generically p-to-1 , the central cycle is of length d, and the distance k of jM from the central cycle is the p-exponent of the conductor of M2(Z) n M. Each ordinary geometric point j E X has a unique lifting in the central cycle, called the canonical lijting (or Deuring representative) ofj. These are obtained by the modular reconsideration of the Deuring's results on the reduction of elliptic curves [2]. (IV) Subgroups F* of Fwith finite indices are categorically equivalent with those finite coverings X* --+ X over Fpz satisfying (a) the Igusa's ramification properties, and (b) all points of X* lying above the special points of X are Fpz-rational (cf. · · · ,
[6]). (V) The reciprocity in this system of coverings of X is described as follows. Let
p be as above, and 7: = 7: M E YfP. Let r� be the stabilizer of 7: in r. Then, modulo torsion (which occurs whenj = 0 or 123), r� is isomorphic to Z and has a generator r� represented (modulo pZ) by an element of Mx which generates a positive power of fJ E M. The F-conjugacy class {r� }r is well-defined by P (modulo torsion), and {r� } r is the Frobenius conjugacy class for P in this system of coverings. Here, f is the projective limit of all finite factor groups of F [5b, Vol. 2, Chapter 5]. By (IV), (V), the basic bijection (7 . 1 .2) holds also between F* and X*. 7.2. Remarks on its relations with other works. The chief datum defining a Shimura variety (of GL(2)-type) is a quaternion algebra B over a totally real num ber field F. The dimension of the variety is the number of distinct infinite places of F at which B is split. In the Shimura theory, the totally indefinite case and the one dimensional case are the two extremes. They coincide only when F = Q. (A) Morita's contribution on the Shimura curves over Fp for the case F Q is briefly mentioned in [5c]. (B) Langlands' conjecture on the explicit description of the points of Shimura varieties over finite fields, together with that of the Frobenius action, is (essentially) a higher dimensional generalization of our conjectures and results given in [5b]. Langlands discussed extensively, obtaining definitive results, the case where B is totally indefinite [11]. When F Q, his results essentially coincide with the Morita's solution of our conjecture. 4 When F #- Q, the object varieties are different. (C) With Drinfeld. Let X be a Shimura curve over Fq , r be its arithmetic funda mental group, and consider the system of coverings X* --+ X corresponding to congruence subgroups F* c F. Then the Galois group of this system {X*/X} is an adelic compact group (without infinite and }J factors), and r is the "maximal global subgroup" of this adelic Galois group. Recall (Main Theorem III) that the Fro beniuses in {X* IX} are represented by the global element r� E r. =
=
4There is however a difference about the aspect of supersingular moduli ; mainly because of the (current) difference in the standpoint.
SHIMURA CURVES
31 1
Now let F = Q. Then this adelic presentation of the Galois group gives a system of /-adic representations (not always in GL2{Q1) but in GL2{Q1) when I ID(B)). It is very plausible that there exists an automorphic representation of GL2(KA) (K = Fq (X)) which corresponds with this system via the Drinfeld theorem. On the other hand, if F #: Q, then the reduced norm of r � over F is not usually a power of p ; hence it cannot correspond with an automorphic representation of GL2 (perhaps possible for GL2m (m = [F: Q])). At any rate, this relation would not reduce one theory to another. REFERENCES 1. M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkorper, Abh. Math. Sem. Univ. Hamburg 14 (1941), 1 97-272.
2. --, (a) Teilbarkeitseigenschaften der singularen Moduln der elliptischen Funktionen und die Diskriminante der Klassengleichung, Comment. Math. Helv. 19 (1 947), 74-82. (b) Die Struktur der elliptischen Funktionenkorper und die Klassenkorper der imaginaren quadra tische Zahlkorper, Math. Ann. 124 (1 952), 393-426. 3. B. Dw ork, p -adic cycles, Inst. Hautes Etudes Sci. Pub!. Math. 37 (1 969), 27-1 16. 4 . A. Grothendieck, Elements de geometrie algebrique (EGA) . I - IV, lnst. Hautes Etudes Sci.
Pub!. Math (1 960-1 967). 5. Y. Ihara, (a) The congruence monadromy problems, J. Math. Soc. Japan 20-1 , 2 (1 968), 1071 21 . (b) On congruence monodromy problems, Lecture Notes, Univ. Tokyo, vols. 1 , 2 , 1 968, 1 969. (c) Non-abelian classfields over function fields in special cases, Actes du Congres Internat. Math., Nice, 1 970, Tome 1 , pp. 381-389. 6. -- , On modular curves overfinite fields, Discrete Subgroups of Lie Groups, Proc. Internat. Colloq., Bombay, Oxford Univ. Press, 1 973, pp. 1 61-202.
7. -- , On the differentials associated to congruence relations and the Schwarzian equations defining uniformizations, J. Fac. Sci. Univ. Tokyo Sect. lA Math. 21 (1 974), 309-332. 8. Y. Ihara and H . Miki, Criteria related to potential unramifiedness and reduction of unramified coverings of curves, J. Fac. Sci. Univ. Tokyo Sect. lA Math. 22 (1 975), 237-254. 9. Y. Ihara, (a) Some fundamental groups in the arithmetic of algebraic curves over finite fields,
(b) Congruence relations and Shimura curves. II (to appear in J. Fac. Sci . Univ. Tokyo Sect.
Proc. Nat. Acad. Sci. U.S.A.
72
(1 975), 328 1-3284.
lA Math. 25-3) . 10. -- , On the Frobenius correspondences of algebraic curves, Algebraic Number Theory, Papers contributed for the Kyoto Internat. Symposium, 1 976 ; Japan Soc. Promotion of Sci. , Tokyo, pp. 67-98. 1 1 . R. P. Langlands, (a) Shimura varieties and the Selberg trace formula (to appear). (b) On the zeta functions of some simple Shimura varieties (to appear). 12. J. Lubin and J. Tate, Formal complex multiplication in local fields, Ann. of Math. (2) 81 (1 965), 380-- 3 87. 13. Y. Morita, Ihara's conjectures and moduli space of abelian varieties, Master's thesis, Univ. Tokyo (1 970). 14. G. Shimura, Construction of classfields and zeta functions of algebraic curves, Ann. of Math. (2) 85 (1 967), 58-1 59. 15. --, On canonical models of arithmetic quotients of bounded symmetric domains. I, II, Ann. of Math. (2) 91 (1 970), 144-222 ; 92 (1 970), 528-549. 16. J.-P. Serre, Arbres, amalgames, SL, Asterisque 46, Soc. Math. France, 1 977. UNIVERSITY OF TOKYO
Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 2, pp. 3 1 3-346
VALEURS DE FONCTIONS L ET PERI ODES D'INTEGRALES
P. DELIGNE Dans cet article, j'enonce une conjecture ( 1 . 8, 2.8) reliant les valeurs de cer taines fonctions L en certains points entiers a des peri odes d'integrales. Les fonctions L considerees sont celles des motifs-un mot auquel on n'at tachera pas un sens precis. Ceci inclut notamment les fonctions L d' Artin, les fonc tions L attachees a des caracteres de Heeke algebriques ( = Grossencharakter de type A0), et celles attachees aux formes modulaires holomorphes sur le demi-plan de Poincare, supposees primitives ( = new forms ; on considere toutes les fonctions L, Lk, attachees aux puissances symetriques, Symk , de Ia representation /-adi que correspondante). Cet article doit le jour a D. Zagier : pour son insistance a demander une con jecture, et pour Ia confirmation experimentale qu'il en a donnee, sitot emise, pour les fonctions L3 et L4 attachees a L1 = 1: -r(n)qn (voir [18]). C'est cette confirmation qui m'a donne Ia confiance necessaire pour verifier que Ia conjecture etait com patible aux resultats de Shimura [13] sur les valeurs de fonctions L de caracteres de Heeke algebriques. 0. Motifs. Le lecteur est invite a ne consulter ce paragraphe qu 'au fur et a mesure des besoins. On y rappelle une partie du formalisme, du a Grothendieck, des motifs. Pour les demonstrations, je renvoie a [8]. 0. 1 . La definition de Grothendieck des motifs sur un corps k a Ia forme suivante.
(a) Soit "f/"(k) Ia categorie des varietes projectives et lisses sur k. On construit une categorie additive vl('(k), pour laquelles les groupes Hom(M, N) sont des espaces vectoriels sur Q, munie de a. Un produit tensoriel ®, associatif, commutatif et distributif par rapport a l'addition des objects ("associatif" et "commutatif" n'est pas une propriete du foncteur ®, mais une donnee, soumise a certaines compatibilites-cf. Saavedra [19]) ; {3. Un foncteur contravariant H*, de "f/"(k) dans .lt'(k), bijectif sur les objets et transformant sommes disjointes en sommes, et produits en produits tensoriels (donnee d'un isomorphisme de foncteurs H*(X x Y) = H*(X) ® H*( Y), compa tible a l'associativite et a Ia commutativite). II s'agit, pour l'essentiel, de definir Hom(H*(X), H*(Y)). Pour Ia definition, utiAMS (MOS) subject classifications (1 970). Primary 10D25, 14K10. © 1 9 7 9 , American Mathematical Society
313
314
P . DELIGNE
Iisee par Grothendieck, de ce groupe comme un groupe de classes de corres pondances entre X et Y, voir 0.6. (b) Rappelons (SGA 4 IV 7.5) qu'une categorie additive est karoubienne si tout projecteur ( = endomorphisme idempotent) est defini par une decomposition en somme directe, et que chaque categorie additive a une enveloppe karoubienne, obtenue en lui adjoignant formellement les images des projecteurs. La categorie �e ff(k) des motifs effectifs sur k est l'enveloppe karoubienne de vlt'(k). (c) On definit Ie motif de Tate Z( - I) comme un facteur direct HZ(Pl) convenable de H *(P1) . On verifie que Ia symetrie (deduite de Ia commutativite de ®) : Z( - 1) ® Z( - I) --> Z( - 1) ® Z( - 1) est l'identite, et que le foncteur M --> M ® Z( - 1) est pleinement fidele. (d) La categorie vlt(k) des motifs sur k se deduit de vlte l f(k) en rendant in versible le foncteur M --> M ® Z( - 1). Notons (n) l'itere ( - n)i<me de ]'auto-equivalence M --> M ® Z( - 1). La cate gorie vft(k) admet vlte f f(k) comme sous-categorie pleine, et tout object de vlt(k) est de Ia forme M(n) pour M dans vlte ff(k) et n un entier. Par definition, si F est un foncteur additif de vlt'(k) dans une categorie karou bienne d, il se prolonge a vlte H(k). Si d est munie d'une auto-equivalence A --> A( - 1), et le prolongement de F d'un isomorphisme de foncteurs F(M( - 1)) = F(M)( - 1), il se prolonge a vH(k). ExEMPLE 0. 1 . 1 . Si k' est une extension de k, et F le foncteur H * (X) --> H*(X ® k k') de vlt'(k) dans vH(k'), on obtient le foncteur extension des sea/aires de vlt(k) dans vlt(k'). Si k' est une extension finie de k, on dispose du foncteur de restriction des scalaires a la Grothendieck JL k'lk : "Y (k') --> "Y (k) : (X --> Spec(k')) >-> (X --> Spec(k') --> Spec(k)). On le prolonge en F : H * (X) >--> H * ( JL k'lkX), d'o u un foncteur de restriction des sea/aires Rk'lk : vlt(k') --> vlt(k). ExEMPLE 0. 1 .2. Soit :It une "tbeorie de cohomologie", a valeurs dans une cate gorie karoubienne d, fonctorielle pour les morphismes dans vii ' (k). Le foncteur :It se prolonge a vHe H(k). Si d est muni d'un produit tensoriel, que :It verifie une formule de Kiinneth et que la tensorisation avec :Yl'(Z( - 1)) est une auto-equival ence de d, il se prolonge a vH(k). Ce prolongement est le foncteur "realisation d'un motif dans Ia theorie :It" . No us noterons (n) I' auto-equivalence de d itere ( - n)i<me de Ia tensorisation avec
:Yl'(Z( - 1)). Pour la determination de :Yl'(Z( - 1)) dans diverses theories £', voir 3. 1 . 0.2. Nous utiliserons les realisations suivantes : 0.2. 1 . Realisation de Betti H8. Correspondant a k = C, d = espaces vectoriels que Q, :It = cohomologie rationnelle : X >--> H * (X(C), Q) ; 0.2.2. Realisation de de Rham HDR· Correspondant a k de caracteristique 0, d = espaces vectoriels sur k, :It = cohomologie de de Rham : X >--> H * (X, Ot) ; 0.2.3. Realisation 1-adique H1• Correspondant a k algebriquement clos, de carac teristique #- I, d = espaces vectoriels sur Q1, :It = cohomologie /-adique : X >--> H * (X, Q1). Et leurs variantes : 0.2.4. Realisation de Hodge. k est ici une cloture algebrique de R, et d la cate gorie des espaces vectoriels sur Q, de complexifie V ® k muni d'une bigraduation
VALEURS
DE
FONCTIONS
L
315
V ® k = EB VP . q telle que VqP soit le complexe conjugue de Vpq. Pour theorie de cohomologie, on prend le foncteur X >-+ H * (X(k), Q) muni de Ia bigraduation de son complexifie H * (X(k) , k) fournie par Ia theorie de Hodge. La realisation de
Betti est sous-jacente a celle de Hodge. 0.2.5. Pour k quelconque, et (J un plongement complexe de k, on note Ha(M) Ia realisation de Betti du motif sur C deduit de M par !'extension des scalaires {)' : k --. C. Notons c la conjugaison complexe. On obtient par transport de struc ture un isomorphisme Foo : Ha(M) :::_. Hca(M) , et Foo ® c envoie Hfq sur H£g. Pour (J reel, F est une involution de H a(M), dont le complexifie echange HNM) et 00
HgP(M). La ()'-realisation de Hodge est Ha(M), muni de sa bigraduation de Hodge et, si (J est reel, de !'involution F Pour k = Q, ou k = R et (J le plongement identique, on remplace l'indice (J par B. Pour k = R, et M = H * (X), !'involution F de H�(M) = H*(X(C), Q) est !'involution induite par la conjugaison complexe oo'
00
Foo : X(C) --. X(C). 0.2.6. Realisation de de Rham. La cohomologie de de Rham est munie d'une filtration naturelle, Ia filtration de Hodge, aboutissement de la suite spectrale d'hy percohomologie Efq = Hq(X, QP)
=>
Hf}Rq (X).
De la, une filtration F sur HDR(M) . 0.2.7. Realisation 1-adique. Si X st une variete sur un corps k, de cloture algebrique k, le groupe de Galois Gal(k/k) agit par transport de structure sur H*(Xfi,Q1). Si M est un motif sur k, et qu'on definit H 1(M) comme la realisation /-adique du motif sur k qui s'en deduit par extension des scalaires, ceci fournit une action de Gal(k/k) sur H1(M). 0.2.8. Realisation adelique finie. Pour k algebriquement clos, de caracteristique 0, on peut rassembler les cohomologies /-adiques en une cohomologie adelique H*(X, AI) = ( HH * (X, Z1) ) ® z Q. On peut cumuler les variantes 0.2. 7 et 0.2.8. 0.2.9. Dans les exemples 0.2. 1 , 0.2.2, 0.2.3, et leurs variantes, on peut remplacer Ia categorie d par celle, d*, des objets gradues de d : utiliser la graduation naturelle de H * . Pour que la formule de Kiinneth fournisse un isomorphisme de foncteurs .Yt( M ® N) :::_. .Yt( M) ® .Yt(N), compatible a l'associativite et a la commutativite, il faut prendre pour donnee de commutativite dans d* celle donnee par la regie de Koszul. 0.3. Pour k = C, le tbeoreme de comparaison entre cohomologie classique et cohomologie etale fournit un isomorphisme H*(X(C), Q) ® Q1 :::_. H * (X, Q1). Si X est defini sur R, cet isomorphisme transforme F (0.2.5) en l'action (0.2. 7) de Ia conjugaison complexe. Pour un motif, on a de meme H8(M) ® Q1 :::_. Hr(M). 0.4. Pour k = C , le complexe de de Rham holomorphe sur x•n est une resolution du faisceau constant C. Par GAGA, on a done un isomorphisme 00
H * (X(C), Q) ® C = H * (X(C), C) __:__, .Yt * (X•n , Q *•n) ,____::__ H * (X, {)* ).
Pour un motif, on obtient un isomorphisme (compatible aux filtrations de Hodges 0.2.4 : FP = EBr'" P VP'q' et 0.2.6) H8(M) ® C --. - HDR(M).
316
P.
DELIGNE
Soit plus generalement M un motif sur un corps k, et O" un plongement complexe de k. Appliquant ce qui precede au motif sur C deduit de M par extension des scalaires, on trouve un isomorphisme (0.4. 1) I : H ..(M) Q9 C __:::____, HvR(M) ® k, a C. Notons le cas particulier k = Q, ou on trouve deux Q-structures naturelles sur la realisation de M en cohomologie complexe : l'une, HiM), liee a la description de la cohomologie en terme de cycles, !'autre, HvR(M), liee a sa description en terme de formes differentielles algebriques. 0.5. Ce qu'il advient de ces realisations et compatibilites par extension du corps des scalaires est clair. Pour la restriction des scalaires, les resultats sont les suivants. Soient k' une extension finie de k, M' un motif sur k', et M = Rk' lk (M') (0.5. 1)
H..(M)
= ffi H, (M'), <
ou la somme est etendue a !'ensemble J(O") des plongement complexes de k' qui prolongent O". Cet isomorphisme est compatible aux bigraduations de Hodge, et a Foo •
H DR(M)
(0.5.2)
=
H DR(M') (restriction des scalaires de k' a k).
Cet isomorphisme est compatible a la filtration de Hodge. (0.5.3)
HlM)
=
Ind H1(M') (representation induite de Gal(k/k') a Gal(k/k)).
(0.5.4) Via l'isomorphisme k' ® k . a C = CJ < a > , et l'isomorphisme HvR(M') ® k, u HvR(M') ® k', < C qui s'en deduit, (0.4. 1 ) pour M et O" est la somme des isomorphismes (0.4. 1) pour M' et -r (-r E J(O")) : C = ffirEJ(u)
H..(M) ® C (0. 5. 1)
co.� I) HvR(M )
® k, u C
(0. 5. 2)
0.6. Pour X projectif et lisse sur k, notons za(X) l'espace vectoriel sur Q de base !'ensemble des sous-schemas fermes irreductibles de codimension d de X, et Z�(X) son quotient par une relation d'equivalence R. Pour k de caracteristique 0, une des definitions de Grothendieck des motifs s'obtient en faisant (pour X et Y connexes) Hom(H*( Y), H *(X)) = Zjim
VALEURS DE FONCTIONS
H2d(X, k
Pour k
= C,
X
A!)(d) = H'b'J?(X)(d)
X
317
L H Zd(X, Af)(d).
on appelle encore cycle de Hodge l'image dans H2d(X, k X A!)(d) = H2d(X(C), Q)(d) ® (k
X
A f)
d'un cycle de Hodge. Pour k algebriquement clos, admettant des plongements complexes, un cycle de Hodge abso/u de codimension d sur X est un element de HZd(X, k x A!)(d), tel que, pour tout plongement complexe a de k, son image dans HZd(X ® k C, C x A f)(d) soit de Hodge. On verifie que PROPOSITION 0.8. (i) L ' espace vectoriel sur Q zg.(X) des cycles de Hodge absolu
est invariant par extension des sea/aires de k a un corps a/gebriquement clos k' (admettant encore un p/ongement complexe). (ii) Pour k algebriquement clos de caracteristique 0, et X defini sur un sous-corps a/gebriquement clos k0 de k, admettant un plongement complexe : X = X0 ® ko k, on pose zg.(X) = zg.cx 0) c H2d(X 0, k0
x
At)(d)
c Hzacx, k
x
At)(d).
D 'apres (i), cette definition ne depend pas des choix de X0 et k0• (iii) Pour X defini sur un sous-corps k0 de k : X = X0 ® ko k, le groupe Aut(kfk0) agissant sur H Zd(X, k x A f)(d) stabilise zg.(X). II agit sur zg.(X) a travers un groupe fini, correspondant a une extension finie k'o de ko. On pose zg.(xo) = zg.(X) Aut(klko) .
0.9. Une notion utile de motif s'obtient en faisant (pour X e t Y connexes) Hom(H*( Y), H *(X)) = Zg�m (Y) (X x Y).
Les composantes de Kiinneth de la diagonale de X x X sont absolument de Hodge. Ceci permet de decomposer H*(X) en une somme de motifs Hi(X), de munir la categorie des motifs d'une graduation (avec H i(X) de poids i) et de modi fier la contrainte de commutativite pour ® comme en [19, VI, 4.2. 1 .4]. On verifie que, ceci fait, la ®-categorie des motifs sur k est tannakienne, isomorphe a la ca tegorie des representations d'un groupe proalgebrique reductif. Pour k = C, la conjecture suivante, plus faible que celle de Hodge, equivaut a dire que le foncteur "realisation de Hodge" est une equivalence de la categorie des motifs 0.9 avec la categorie des structures de Hodge facteur direct de la co homologie de variete algebriques (ou deduites par twist a la Tate de tels facteurs directs). Espoir 0. 10. Tout cycle de Hodge l'est absolument. Si X est une variete abelienne a multiplication complexe par un corps quadra tique imaginaire K, avec Lie(X) libre sur k ® K, les methodes de B. Gross [7] prouvent que certains cycles de Hodge non triviaux sont absolument de Hodge. Ajoute sur epreuves : partant de ce cas particulier, j'ai pu verifier 0. 10 pour les varietes abeliennes. J'ai verifie aussi que la ®-categorie de motifs (0.9) engendree par les HA(X), pour X une variete abelienne, contient les motifs H*(X), pour X une surface K3 ou une hypersurface de Fermat. 0. 1 1 . Le principal defaut de la definition 0.9 des motifs est qu'elle ne se prete pas a la reduction mod p. On ignore si un motif 0.9 sur un corps de nombres F fournit un systeme compatible de representations /-adiques de Gal(F/F).
318
P . DELIGNE
0. 12. Nous uti1iserons le mot "motif" de fa9on libre, sans nous preoccuper de faire rentrer les motifs consideres dans le cadre de Grothendieck. L'essentiel pour nous sera de disposer de realisations Yl'(M), pour les theories Yl' considerees en 0.2, et d'avoir pour ces groupes le meme formalisme que pour les Yl'*(X). 1. Enonce de Ia conjecture (cas rationnel).
1 . 1 . Soit M un motif sur Q. Nous admettrons que les realisations /-adiques H1(M) de M forment un systeme strictement compatible de representations /-adiques, au sens de Serre [11, 1. 1 1]. A savoir : il existe un ensemble fini S de nombres premiers, tel que chaque Ht<M) soit non ramifie en dehors de S U {/}, et que, notant par Fp E Gal(Q/ Q) un element de Frobenius geometrique en p (l'inverse d'une substi tution de Frobenius q;p). le polynome det(l - Fp t. H1 ( M)) E Q1[ t ] (p ¢ S U { /}) soit a coefficients rationnels, et independant de /. Notons Zp(M, t) son inverse, et posons Lp(M, s) = Zp(M, p-s). La serie de Dirichlet a coefficients rationnels donnee par le produit eulerien L5(M, s) = TI prEs Lp(M, s) converge pour !?Is assez grand. Pour s quelconque, on definit L5(M, s) par un prolongement analytique (qu'on espere exister). Notre but est d'enoncer une conjecture donnant Ia valeur de Ls(M, s) en certains points entiers, au produit pres par un nombre rationnel. Puisque, pour s entier, p-s est rationnel, le choix de S est sans importance-a ceci pres qu'aggrandir S peut introduire des zeros mal venus. 1 .2. Pour ecrire proprement !'equation fonctionnelle conjecturale des fonctions L, il y a lieu de completer le produit eulerien L5(M, s) par des facteurs locaux en p E S, et a l'infini. La definition des facteurs locaux en p E S requiert une hypothese additionnelle, que nous supposerons verifiee : (1 .2. 1) Soientp un nombre premier, Dp c Gal(Q/Q) un groupe de decomposition en p, lp c Dp son sous-groupe d'inertie et Fp E Dp un Frobenius geometrique. Le polynome det(l - Fpt, H1(M)IP) E Q1 [t] (/ #- p) est a coefficients rationnels, et est independant de /. Posons Zp(M, t) = det (I - Fpt, H1(M)IP)- 1 E Q(t), et Lp(M, s) = Zp (M, p-s). On definit L(M, s) = TI Lp(M, s). (1 .2.2) p
Le facteur a l'infini Loo(M, s) (essentiellement un produit de fonctions F) depend de Ia realisation de Hodge de M-en fait seulement de Ia classe d'isomorphie de l'espace vectoriel complexe H8(M) ® C, muni de sa decomposition de Hodge et de !'involution F Sa definition est rappelee en 5.2. Posant A(M, s) = Loo(M, s)L(M, s), !'equation fonctionnelle conjecturale des fonctions L s'ecrit (1 .2.3) A (M, s) = c(M, s) A(M, I - s) ou M est le dual de M (de realisations les duales des realisations de M) et ou c(M, s), comme fonction de s, est le produit d'une constante par une exponentielle. Sa definition est rappelee en 5.2. Elle depend d'une hypothese additionnelle. DEFINITION 1 . 3. Un entier n est critique pour M si ni Loo(M, s), ni Loo(M, 1 - s) 00 •
n 'ont de pole en s
=
n.
Notre but est de conjecturer Ia valeur de L(M, n) pour n critique, a multiplica-
VALEURS DE FONCTIONS
L
319
tion par nombre rationnel pres. On a L(M(n), s) = L(M, n + s) (3. 1 .2), et de meme pour Loo' Ceci nous permet de ne considerer que les nombres L(M) = dfn L(M, 0). Nous dirons que M est critique si 0 est critique pour M. On verifie que pour que M soit critique, il faut et il suffit que les nombres de Hodge hPq = dfn dim Hpq (M) de M, pour p # q, ne soient non nuls que pour (p, q) dans la partie hachuree du diagramme ci-dessous, et que F agisse sur HP.P, par l'identite si p < 0, par - 1 si p ?;_ 0. 00
(1.3.1)
+
+
Supposons M homogene de poids w : hpq = 0 pour p + q # w, et posons PJ(M) = - w/2. II resulte de la conjecture de Weil que, pour S assez grand, la serie de Dirich let Ls(M, s) converge absolument pour PJ(M) + �(s) > 1 . Pour PJ(M) + PJ(s) = 1 , on est au bord du demi-plan de convergence, et on conjecture (a) que Ls(M, s) ne s'annule pas, (b) que L8(M, s) est holomorphe, sauf si M est de poids pair - 2n, et contient Z(n) en facteur : on s'attend alors a un pole en s = 1 n (une valeur non critique) . L'analogie avec le cas des corps de fonctions, et les cas connus, menent a croire que les facteurs locaux Lp(M, s) (p quelconque, y compris oo) n'ont de pole que pour PJ(M) + PJ(s) � 0. Si tel est le cas, (a), (b) et !'equation fonctionnelle con jecturale {1 .2.3) impliquent que L ( M) # 0, oo pour M critique et PJ( M) # t. Pour PJ(M) = -!, L(M) s'annulle parfois. Notre conjecture (1.8) est alors vide. -
PROPOSITION
1 .4. Soit M un motif sur R. Via l'isomorphisme (0.4. 1)
HnR(M) s'identifie au sous-espace de H8 ( M) ® C fixe par c --+ F00c.
Prenons M = H*(X). La conjugaison complexe sur HnR(M)c = H*(X0, Q*) se deduit par fonctorialite de l'automorphisme antilineaire Foo du schema X0• Remontant la fleche composee (0.4) qui definit /, on l'identifie a !'involution de duite de l'automorphisme (F00, P /:,) de (X(C), Q*•n), puis au compose de FJ, : H * (X(C), C) --+ H* (X(C), C) et de la conjugaison complexe sur les coefficients. Ceci verifie 1 .4.
320
P. DELIGNE
1 .5. Pour M un motif sur R, nous noterons HtlM) (resp. H!J(M)) le sous-espace de H8(M) fixe par Foo (resp. oil Foo = - 1). Posons d(M) = dim H8(M) et d±(M) = dim H�(M). Pour M un motif sur k, et a un plongement complexe de k qui se factorise par R, on note H'/i(M), d'/j(M) et d(M) ces objets pour le motif sur R deduit de M par a. Pour k = Q, on omet la mention de a . Le corollaire suivant resulte aussitot de 1 .4, et de ce que tant Foo que la conjugai son complexe echangent Hpq et flqP. CoROLLAIRE 1 .6. Soit M un motif sur R. Pour Ia structure reelle Hv R(M) de Hs(M) ® C, les sous-espaces Hpq sont definis sur R ; le sous-espace Hjj(M) ® R est reel, et H!J(M) ® R est purement imaginaire. 1 .7. Dans la fin de ce paragraphe, nous ne considererons que des motifs sur Q ; sauf mention expresse du contraire, nous les supposons homogenes. S i leur poids w est pair, nous supposerons aussi que Foo agit sur HPP(M) (w = 2p) comme un scalaire : soit + I , soit - 1 . Cette hypothese est verifiee pour M critique. Puisque Foo echange Hpq et flqP, elle assure que les dimenions d+(M) et d-(M) sont egales l'une a I; p>q hPq, !'autre a I; p;?;q hpq. En particulier, ces dimensions sont egales a celles de sous-espaces F+ et F- figurant dans la filtration de Hodge. Posant HJjR(M) = HvR(M)jF+, on a encore dim HfiR(M) = d±(M). 11 resulte de ce que Foo echange Hpq et flqP que les applications composees (1 .7. 1) sont des isomorphismes. On pose (1.7.2) (1 .7.3)
c±(M) = o(M)
det(J±) , det(J),
le determinant etant calcule dans des bases rationnelles de H� et HJjR (resp. H8 et HvR). La definition de o(M) ne requiert pas les hypotheses faites sur M. D'apres 1 .6, J+ est reel, i.e. induit J+ : H�(M)R -+ HfjR(M)R, tandis que J est purement 'imaginaire. Les nombres c+(M), id - CMl c-(M) et id - (Ml (j(M) sont done reels non nuls. A multiplication par un nombre rationnel pres, il ne depen dent que de M. Les periodes de M sont classiquement les (w, c) pour w E HvR(M) et c E H8(M)v. Par exemple, si X est une variete algebrique sur Q, w une n-forme sur X, definie sur Q et que c est un n-cycle sur X(C), (w, c) = fc w est une periode de Hn(X) . Ex primons c ± (M) en terme de periodes. Le dual de HJjR(M) est le sous-espace F± de HvR(Mv), oil MV est le motif dual de M. Si la base choisie de HJjR(M) a pour duale la base (w;) de F±( HvR(MV)), et que (c;) est la base choisie de H"j(M), la matrice de f± est (w;, ci), et c ±(M) = det ( (w; , ci)). Conjecture 1 .8. Si M est critique, L(M) est un multiple rationnel de c+(M). 2. Enonce de Ia conjecture (cas general).
2. 1 . Jusqu'ici, nous n'avons considere que des fonctions L donnees par des series de Dirichlet a coefficients rationnels. Pour faire mieux, il nous faudra considerer des motifs a coefficient dans des corps de nombres. Voici deux fac;:ons, equivalentes, pour construire la categorie des motifs sur k,
VALEURS DE FONCTIONS
L
321
a coefficient dans un corps de nombres E, a partir de la categorie des motifs sur k. s'agit d'une construction valable pour toute categorie additive karoubienne (O. l (b)) dans laquelle les Hom(X, Y) sont des expaces vectoriels sur Q. A. Un motif sur k, a coefficient dans E, est un motif M sur k muni d'une struc ture de £-module : E --+ End(M). B. La categorie vltk ,E• des motifs sur k, a coefficient dans E, est l'enveloppe karoubienne (cf. 0. 1(b)) de Ia categorie d'objets les motifs sur k-considere comme objet de Jtk,E• un motif M se notera ME-et de morphismes donnes par Hom(XE, YE) = Hom(X, Y) ® E. Passage de B a A. Pour X un motif, et V un espace vectoriel sur Q de dimension finie, on note X ® V le motif, isomorphe a une somme de dim(V) copies de X, caracterise par Hom( Y, X ® V) = Hom( Y, X) ® V ( ou par Hom(X ® V, Y) = Hom(V, Hom(X, Y))). On passe de B a A en associant a ME le motif M ® E, muni de sa structure de £-module naturelle. Passage de A a B. Si M est muni d'une structure de £-module, on recupere sur ME deux structures de £-module : celle deduite de celle de M, et celle qu'a tout objet de vltk , E · L'objet de Jtk .E correspondant a M est le plus grand facteur direct de ME sur lequel ces structures coincident. En detail : l'algebre E ® E est un produit de corps, parmi lesquels une copie de E dans laquelle x ® 1 et 1 ® x se projettent tous deux comme x. L'idempotent correspondant e agit sur ME (qui est un E ® £-module) et son image est l'object de Jtk.E qui correspond a M. Les motifs a coefficient dans E sont le plus souvent donnes sous la forme A. La forme B a l'avantage de rester raisonnable pour E non de rang fini sur Q. Elle est utile pour comprendre le formalisme tensoriel : on peut definir produit tensoriel et dual, pour les motifs a coefficient dans E, par leur fonctorialite et les formules XE ®E YE = (X ® Y)E et (XE)v = (XV)E. Dans le langage A, X ®E Y est le plus grand facteur direct de X® Y sur lequel coincident les deux structures de £-module de X ® Y, et X est le dual usuel de xv, muni de la structure de £-module trans posee. Si on applique ces remarques a la categorie des espaces vectoriels sur Q, plutot qu'a celle des motifs, on retrouve l'isomorphisme du F-dual d'un espace vectoriel sur E avec son Q-dual, donne par 11
w � la forme TrE1Q((w, v)). Nous avons defini en 0. 1 . 1 des foncteurs de restriction et d'extension du corps k des scalaires. Ils transforment motifs a coefficient dans E en motifs a coefficient dans E. On dispose aussi de foncteurs de restriction et d'extension des coefficients : soit F une extension finie de E : Extension des coefficients. Dans le langage A, c'est X � X ®E F ; dans le langage B, c'est XE ...... XF. Restriction des coefficients. Dans le langage A, on restreint a E la structure de F-module. Le lecteur prendra soin de ne pas confondre les roles de k et E. Un exemple type qu'on peut retenir est celui du H1 des varietes abeliennes sur k, a multiplication complexe par un ordre de E. En terme de Ia variete abelienne-prise a isogenie pres-les foncteurs ci-dessus deviennent : extension du cops de base, restriction des scalaires a la Weil, construction ®E F, qui multiplie la dimension par [F: E], restriction a E de la structure de F-module.
322
P. DELIGNE
2.2. Soit M un motif sur Q, a coefficient dans un corps de nombres E. Pour cha que nombre premier /, Ia realisation /-adique H1(M) de M est un module sur le complete /-adique E1 de E. Ce complete est le produit des completes E;,, pour ;t ideal premier au-dessus de /, d'oil une decomposition de H1(M) en un produit de £;,-modules H;,(M). On conjecture pour les H;,(M) une compatibilite analogue a 1 .2. 1 ; si elle est verifiee, on peut pour chaque plongement complexe a de E definir une serie de Dirichlet a coefficients dans aE, convergente pour �s assez grand :
= n Lp(a, M, s), oil Lp(a, M, s) = aZp(M, p-•) avec Zp e E(t) c E;,(t) donne par Zp(M, t) = det{l - Fpt, H;,(M)lP)- l pour ,Up. L(a, M, s)
(2.2. 1)
p
Pour a variable, ces series de Dirichlet se deduisent les unes des autres par con jugaison des coefficients. Nous regarderons le systeme des L(a, M, s) comme une fonction L * (M, s) a valeurs dans Ia C-algebre E ® c, identifiee a cHom (E, C) par (2.2.2)
E ® c� C Hom (E, C) : e ® z 1--+ {z, a(e)) u.
Cette fonction peut aussi etre definie directement par un produit eulerien. On espere comme en 1 . 1 qu'elle admet un prolongement analytique en s. II y a lieu de completer le produit eulerien L(a, M, s) par un facteur a l'infini Loo(a, M, s) dependant de Ia realisation de Hodge de M. Posant A(a, M, s) = Loo(a, M, s)L(a, M, s), !'equation fonctionnelle conjecturale des fonctions L s'ecrit (2.2.3)
A(a, M, s)
= e(a, M, s)A(a, M, 1
-
s),
oil e(a, M, s), comme fonction de s, est le produit d'une constante par une exponen tielle. Les definitions de Loo et e sont rappelees en 5.2. Comme ci-dessus, on re gardera le systeme des A et celui des e, pour a variable, comme des fonctions A * et e * a valeurs dans E ® C. II resulte de 2.5 ci-dessous que Loo(a, M, s) est independant de a, et que Ia fonc tion Loo, pour le motif REtQ M deduit de M par restriction du corps des coefficients (2. 1) est Ia puissance [E: Q]ieme de Loo(a, M, s). Ceci justifie Ia PROPOSITION-DEFINITION 2.3. Soil M un motif sur Q ii coefficient dans E. Un entier n est critique pour M si les conditions equivalentes suivantes sont verifiees (i) l'entier n est critique pour REIQ M; (ii) ni Loo(a, M, s), ni Loo(a, M(l), s) n 'ont de pole en s = n. On dit que M est critique si 0 est critique pour M.
Notre but est de conjecturer Ia valeur de L* ( M) = dfn L* (M, 0), pour M critique, a multiplication par un element de E pres. En d'autres termes, il s'agit de con jecturer simultanement les valeurs des L(a, M) = dfn L(a, M, 0), a multiplication pres par un systeme de nombres a(e) , e e E. 2.4. La realisation H8(M) de M en cohomologie rationnelle est munie d'une structure de E-espace vectoriel. Sa dimension est le rang sur E de M. L'involution F est E-lineaire ; les parties + et - sont done des E-sous-espaces vectoriels. On note d+(M) et d-(M) leur dimension. 00
VALEURS DE FONCTIONS
L
323
Le complexifie de H8(M) est un E ® C-module libre. Identifiant E ® C (2.2.2), On en deduit une decomposition HB(M) ® C = (BH8(u, M), "
CHom (E, C)
a
avec soit HB(u, M) = HB(M)
® E. 11 C.
Les HG· q(M) de Ia decomposition de Hodge etant stables par E, chaque HB(u, M) herite d'une decomposition de Hodge H8(u, M) (BH/i q(u, M). L'involution F"" permute Hpq et HqP, en particulier stabilise HPP, qu'elle decoupe en des parties + et - . On note hpq(u, M) Ia dimension de H�q(u, M), et hPP±(u, M) celle de H�P±(u, M). La proposition suivante permet d'omettre u de Ia notation. =
PROPOSITION
2.5. Les nornbres hpq(u, M) et hPP±(u, M) sont independant de u.
On peut supposer, et on suppose, que M est homogene. Dans ce cas, HPq(M) s'identifie au complexifie du E-espace vectoriel Gr �(HvR(M)) : c'est un E ® C module libre, et Ia premiere assertion en resulte. Pour Ia seconde, on observe que hPP±(u, M) est l'exces de d±(M) sur I:; p>q hpq(u, M). 2.6. Soit M un motif sur Q a coefficient dans E. Dans Ia fin de ce paragraphe, sauf mention expresse du contraire, nous supposons que R81g M verifie les hypotheses de 1 .7. Les espaces F± et HfjR sont cette fois des espaces vectoriels sur E. Les iso morphismes (0.4 . 1 ) et ( 1 .7. 1 ) (2.6.1) ( 2 . 6. 2)
/: H8(M) [± : H"i(M)
®C HvR(M) ® C, ® C _, HfjR(M) ® C, _,
sont des isomorphismes de E ® C-modules entre complexifies d'espaces vectoriels sur E. On pose c±(M) = det (/±) E (E ® C)*, o(M) = det (/) E (E ® C)*, le determinant etant calcule dans des bases E-rationnelles de H"i(M) et HfjR(M) (resp. HB(M) et HvR(M)). La definition de o(M) ne requiert pas les hypotheses faites sur M. A multiplication par un element de E* pres, ces nombres ne depen dent que de M. II resulte a nouveau de 1 .6 que c+(M), id - CM) c-(M) et i d - (Ml o(M) sont dans (E ® R)*. Conjecture 2.7. Posons fJl(M) = - -!w. Si M est critique, (i) L(u, M, s) n 'a jarnais de pole en s = 0, et ne peut s 'annuler en s = 0 que pour
fJl (M) = t.
(ii) La rnultiplicite du zero de L(u, M, s) en s = 0 est independante de u. Pour (i), je renvoie a Ia discussion a Ia fin de 1 .3. Que (ii) soit raisonnable m'a ete suggere par B. Gross. Conjecture 2.8. Pour M critique et L(u, M) =F 0, L*(M) est le produit de c+(M)
par un element de E*.
324
P.
DELIGNE
REMARQUE 2.9. Un motif M sur un corps de nombres k, a coefficient dans E, definit aussi une fonction L*(M, s). Ces fonctions soot couvertes par notre con jecture, vu l'identite L*(M, s) = L*(Rk!Q M, s), oil Rk!Q est la restriction des scalaires de k a Q (appliquer (0.5.3)). Les fonctions L(u, M, s) soot des produits euleriens, indexes par les places finies de k. 11 y a lieu de les completer par les facteurs a l'infini L.(u, M, s), indexes par les places a l'infini, dont la definition est rappelee en 5.2. Ils dependent en general de u. Seul L"'(u, Rk!Q M, s), produit sur v a l'infini des L.(u, M, s), est independant de u. REMARQUE 2. 10. Soient F une extension de E, t le morphisme structural de E dans F et t c son complexifie : E ® C 4 F ® C. On a L*(M ®E F, s) = tcL*(M, s) et c± (M ® E F) = t c c ± (M) . La conjecture est done compatible a !'extension du corps des coefficients. Pour F une extension galoisienne de E, de groupe de Galois G, le Theoreme 90 de Hilbert fl l (G, F*) = 0 assure que ((F ® C)* IF*)G = (E ® C)* IE * : 1 a conjecture est invariante par extension du corps des coefficients. REMARQUE 2. 1 1 . Si E est une extension de F, on a L*(RE; pM, s) = NEI F L*(M, s). Les periodes c± verifiant la meme identite, la conjecture est com patible a la restriction des coefficients. 2 REMARQUE 2. 1 2. Soit D une algebre a division de rang d sur E. Un motif M sur Q, muni d'une structure de D-module, et de rang n sur D, definit une serie de Dirichlet a coefficients dans E dont les facteurs euleriens soot presque tous de degre nd: pour A une place finie de E, HJ.(M) est un module libre sur le complete D;, = D ®E E;,, et on reprend la definition (2.2. 1) en posant
det red (1 - Fp t, HJ. (M) lp) -1 (si D;, est une a1gebre de matrices sur E;,, et e un idempotent indecomposable, le determinant reduit d'un endomorphisme A d'un D.-module H est le determinant, calcule sur E, de la restriction de A a eH; la definition dans le cas general procede par descente). La liberte que nous donne 2.10 d'etendre le corps des coefficients met ces fonc tions L elles aussi sous le chapeau 2.8 : choisissant une extension t : E 4 F de E qui neutralise D, et un idempotent indecomposable e de D ®E F, on a t cL*(M, s) = L*(e(M ®E F), s). On peut aussi definir c+(M) directement dans ce cadre. C'est ce qui est explique en 2. 1 3 ci-dessous. Zp(M,
t)
=
La fin de ce paragraphe est inutile pour Ia suite.
2. 13. Soit D une algebre simple sur un corps E (voire une algebre d'Azumaya sur un anneau . . . ). Pour le formalisme tensoriel, il est commode de regarder les D modules comme de "faux E-espaces vectoriels" : (a) Se donner un espace vectoriel V sur E revient a se donner, pour toute exten sion etale F de E, un F-espace vectoriel VF• et des isomorphismes compatibles Va = Vp ®F G pour G une extension de F. On prend Vp = V ®E F. Par descente, il suffit de ne se donner les VF que pour F assez grand. (b) Soit W un D-module. Pour toute extension etale F de E, et tout F-isomor phisme D ® F ,.., Endp(L), avec L libre, posons Wp, L = Homv® F (L, W ® F) (les produits tensoriels soot sur E). On a un isomorphsime de D ® F-module W®E F = L®p WF, L ·
VALEURS DE FONCTIONS
L
325
Si F est assez grand pour neutraliser D, L est unique a isomorphisme non unique pres ; la non-unicite est due aux homotheties, qui agissent trivialement sur End(L). C'est pourquoi la donnee des WF,L n'est pas du type (a) ; c'est un "faux espace vectoriel sur E". Soient ( W«) une famille de D-modules, et T une operation tensorielle. Si les homotheties de L agissent trivialement sur T( W� L), le F-espace vectoriel T( W� L) est independant du choix de L et on obtient un systeme du type (a), d'ou un espace vectoriel T( W«) sur E. ExAMPLE. Si W' et W" sont deux D-modules de rang n · [D : E] l !2 sur E, on peut prendre T = Hom(;V W�, L• ;\» WF, L) ; on obtient un espace vectoriel o( W' , W") de rang I sur E, et tout homomorphisme f: W' ---> W" a un determinant reduit det red(f) E o( W' , W"). Pour definir c+(M), on applique cette construction aux D-modules Ht(M) et a H!JR(M). Posons o = o(Hfi(M), HfjR(M)). Le determinant reduit de l'isomorphisme de D ® C-modules J+ : Hfi(M)c ---> - HfjR(M)c est dans 1e E ® C -module libre de rang I o ® C. On pose det red(J+) = c+(M) · e pour e une base de o. ,
,
3. Exemple : Ia fonction c.
3. 1 . Pour comprendre les diverses realisations du motif de Tate Z(l), le plus simple est de l'ecrire Z(l) = H 1 (Gm). Le groupe multiplicatif n'etant pas une variete projective, ceci ne rentre pas dans le cadre de Grothendieck, qui demande qu'on definisse plut6t Z(l) comme le dual du facteur direct fl2(Pl ) de H*(P1 ). La realisation en Zrcohomologie de Z(l) est le module de Tate TtCGm) de Gm Z1 (1) = proj lim ,u1n . En realisation de Hodge, on a H8(Z(l)) = H1 (C*) isomorphe a Z (a Q plut6t, en homologie rationnelle). Ce groupe est purement de type ( - 1 , - 1) et Foo = - 1 . En realisation de de Rham, HDR(Z(l)) est le dual de HbRCGm), isomorphe a Q, de generateur la classe de dz/z. L'unique peri ode de H 1 (Gm) est ,( ---z dz = 21/: l.. :r
(3. 1 . 1)
Sur Z1(1), le Frobenius arithmetique rpp (p of. l) agit par multiplication par p. Le Frobenius geometrique agit done par multiplication par p- 1 ; ceci justifie l'identite citee en 1 .3 L(M(n), s)
(3. 1 .2)
=
L(M, n
+
s).
Puisque F agit sur ( - 1)» sur H8(Z(n)), HJJ(Z(n)) est nul pour e = - ( - 1) n , et c•(Z(n)) = I . Pour e = ( - 1)», d'apres (3. 1 . 1), c•(Z(n)) = o(Z(n)) = (211:i) n : pour e = ( - I) n , c•(Z(n)) = (211:i)n (3. 1.3) c•(Z(n)) = I , o(Z(n)) = (211:i)n pour e = - ( - 1)». 00
3.2. La fonction C(s) est la fonction L attachee au motif unite Z(O) = H *(Point). Les entiers critiques pour Z(O) sont les entiers pair > 0, et les entiers impairs � 0. A cause du pole de C(s) en s = I, 0 n'est pas un zero trivial et il serait raisonnable de definir les entiers critiques comme incluant 0. La equation (3. 1 . 3) et les valeurs
3 26
P. DELIGNE
connues de �(n) = L(Z(n)) pour n critique verifient 1 . 8 : �(n) est rationnel pour n impair � 0, et un multiple rationnel de (2n;i)n pour n pair � 0. 4. Compatibilite a Ia conjecture de Birch et Swinnerton-Dyer.
4. 1 . Soient A une variete abelienne sur Q, et d sa dimension. La conjecture de Birch et Swinnerton-Dyer [15] affirme notamment : (a) L(Hl(A), 1) est non nul si et seulement si A(Q) est fini. (b) Soit w un generateur de HD(A, Dtf). Alors, L(H 1 (A), I) est le produit de J A CRl i w l par un nombre rationnel. Le motif Hl(A)(I) est isomorphe au dual H1(A) de Hl(A) : ceci traduit I' existence d'une polarization, autodualite de H 1 (A) a valueurs dans Z( - 1). D'apn':s 1 .7, c+(Hl(A)(I)) se calcule done comme suit : si w 1 , . · ·,wd est une base de HD(A, () 1 ) = F+HJJR(A), et ci> · · ·, c4 une base de H1 (A(C), Q)+, on a (4 . 1 . 1 )
Designant encore par e; un cycle repn!sentatif, on a (w,., ei) = J.j w;. Prenons pour base (e,.) une base sur Z de H1 (A(Rt, Z) c H1 (A(C), Z). Le pro duit de Pontryagin des ei est represente par le d-cycle A(Rt dans A(C), pour une orientation convenable, et, si w est le produit exterieur des w,., le determinant (4. 1 . 1) est l'integrale J A cRJ O w. On a
J
A (R)
lwl
= [A(R) : A(Rt]
IJ
A ( R) O
j
w,
et 1 .8 pour Hl(A)( l) equivaut done a 4 . l (b) ci-dessus. 4.2. La conjecture de Birch et Swinnerton-Dyer donne la valeur exacte de L(Hl(A), I ) ; la description du facteur rationnel en 4. l (b) a partir du motif Hl(A)(l) a la forme suivante : (a) La donnee de M = H 1 (A)( l) equivaut a celle de A a isogenie pres. 11 faut com mencer par choisir A. Cela revient a choisir un reseau entier dans HB(M), dont les /-adifies soient stables sous I' action de Gal(Q/Q). (b) On choisit alors w, par example comme produit exterieur des elements d'une base d'un reseau entier dans HnR(M). Cet w determine la periode, c+(M), et, pour chaque nombre premier p, un facteur rationnel cp(M). Les cp(M) sont presque tous egaux a I, et la formule du produit assure que c+(M) · IT p cp(M) est independant de w. (c) Un autre nombre rationnel, h(M), est defini en terme d'invariants cohomolo giques de A ; le nombre rationnel chercbe est h(M) · IT P cp(M)- 1 . L'invariance de la conjecture par isogenie est par ailleurs un tbeoreme non trivial. 4.3. Pour generaliser 4. l (a) a un motif de poids - 1 M quelconque, il faudrait disposer d'un analogue de A(Q). Le groupe A(Q) peut s'interpreter comme le groupe des extensions de Z(O) par H1 (A), dans la categorie des 1 -motifs [6, § 1 0] sur Q. Ceci suggere de considerer le groupe des extensions de Z(O) par M, dans une categorie de motifs mixtes, paralleles aux structures de Hodge mixtes, mais on ne dispose meme pas d'une definition conjecturale d'une telle categorie ! J'observerai seulement que, dans toutes les theories cohomologiques usuelles, un cycle Y de dimension d, cohomologue a zero, sur une variete algebrique propre et
VALEURS DE FONCTIONS
L
327
lisse X determine un torseur sous H 2d- 1 (X)(d) : on dispose d'une suite exacte de cohomologie 0 ------+ H 2d- 1 (X)(d) ------+ H 2d- 1 (X - Y)(d) ___!____. H¥f(X)(d) ------+ HZd(X)(d), Y definit une classe de cohomologie cl( Y) e H ¥f(X)(d), d'image nulle dans HZd(X)(d), et on prend a- 1 cl( Y). Cette construction correspond a celle qui a un diviseur de degre 0 sur une courbe associe un point de sajacobienne. 5. Compatibilite a l'equation fonctionnelle. Soit E un corps de nombres. Dans (E ® C)*, nous noterons - Ia relation d'equivalence definie par le sous-groupe E*. PROPOSITION 5. 1 . Soit M un motif sur Q, a coefficient dans E. On suppose verifiees les hypotheses de 1 .7. On a alors c+(M) - (271:i)-d - CMl · o(M) · c+(M( I )). Pour L un module libre de rang n sur un anneau commutatif A (A sera E, ou E ® C), posons det L = 1\ n L. On etend par localisation cette definition au cas ou L est seulement projectif de type fini (cette generalisation n'est pas indispensable a Ia preuve de 5. 1). On a des isomorphismes canoniques (5. 1 . 1) det(LV) = det(L)- 1 et, pour tout facteur direct P et L, det(L) = det(P) · det(L/P) (5. 1 .2) (on a designe par un produit tensoriel, et par - 1 un dual). II y a ici des problemes de signe, qu'on resoud au mieux en considerant det(L) comme un module gra due inversible, place en degre le rang de L. Nos resultats finaux etant modulo - , nous ne nous en inquieterons pas. Pour X et Y de meme rang, posons o(X, Y) = Hom(det X, det Y) = det(X)- 1 det( Y). Le determinant de f : X-+ Y est dans o(X, Y). On deduit de (5. 1 . 1) et (5. 1 .2) des isomorphismes o(X, Y) = o( YV, XV) (5. 1 .3) et, pour F c X et G c Y, o(X, Y) = o(F, YJG) · o(G, X/F)- 1 . (5. 1 .4) LEMME 5. 1 .5. Via l'isomorphisme ( 5. 1 .3) , on a det(u) = det(1u). LEMME 5. 1 .6. Soient F et G des facteurs directs de X et Y. On suppose que l'isomor phisme f: X -+ Y induit des isomorphismes fp : F -+ Y/G et fcl : G -+ X/F. Via l'isomorphisme (5. 1 .4), on a det(f) = det(fp) det(fG"1)- 1 . ·
·
La verification de ces lemmes est laissee au lecteur. Pour 5. 1 .6, notons seulement que, pour G 1 = J- 1 (G) et F 1 = f(F), on a X = F Efl G 1 , Y = G Efl F 1 , et que f echange F et F 1 , G et G 1 . Appliquons le Lemme 5. 1 .6 aux complexifies de H -'B(M) c HiM) et de p- c HDR(M). On trouve que, via l'isomorphisme ( 5. 1 .4) , le determinant de I : H8(M) ® C -+ H DR(M) ® C est le produit du determinant de J+ : H"}j(M) ® C -+
328
P. DELIGNE
(HDR(M)jF-) ® C et de l'inverse du determinant du morphisme induit par l'inverse de /: J- : p- ® C -----> (Hn(M) / Hfi(M)) ® C.
Le morphisme J- est le transpose du morphisme I- pour le motif M dual de M. Appliquant 5. 1 .5, et prenant des bases sur E des espaces o(X, Y) en jeu, on obtient finalement que (5. 1 . 7)
On en deduit 5. 1 en appliquant au motif M la formule suivante, consequence de 3 . 1 : (5. 1 . 8)
Notons aussi, pour usage ulterieur, Ia formule analogue o(M) = (2ni)-d CMJ ·n o(M(n)) ,
(5. 1 .9)
5.2. Rappelons la forme exacte de !'equation fonctionnelle conjecturale des fonctions L des motifs ([12], [4]). Nous nous placerons dans le cas general d'un motif sur un corps de nombres k, a coefficient dans un corps E muni d'un plonge ment complexe a (cf. 2.9). La forme generale d'abord : (a) Pour chaque place v de k, on definit un facteur local Lv(fl, M, s) . Pour v fini, la definition de Lv(a, M, s) = flZv (M, Nv-s) (Zv(M, t) E E( t)) depend d'une hypothese de compatibilite analogue a ( 1 .2. 1). On a Zv(M, t) = det(l F"t' H;.(M) f·)-1 . Pour v infini, induit par un plongement complexe -c, on obtient Lv(fl, M, s) en de composant Hr(M) ® E, a C en somme directe de sous-espaces minimaux stables par les projecteurs qui donnent la decomposition de Hodge, et par F pour v reel, en associant a chacun le "facteur F" de la Table (5.3), et en prenant leur produit. Pour v complexe, les sous-espaces minimaux sont de dimension un, d'un type (p, q). Pour v reel, il sont soit de dimension 2, de type {(p, q), (q, p) }, p #- q, soit de dimen sion 1 , de type (p, p), avec F = ± 1 . On note A(a, M, s) le produit des Lv(a, M, s). (b) Soient 1/f un caractere non trivial du groupe A ® kjk des classes d'adeles de k, et 1/fv ses composantes. Soient aussi, pour chaque place v de k, une mesure de Haar dxv sur kv. On suppose que, pour presque tout v, dxv donne aux entiers de kv Ia masse I , et que le produit ® dxv des dx" est la mesure de Tamagawa, donnant la masse I au groupe des classes d'adeles. On definit des constantes locales c.{a, M, s, 1/f" ' dxv), presque toutes egales a I et toutes, comme fonction de s, produit d'une constante par une exponentielle. Soit c(a, M, s) leur produit (independant de 1/f et des dxv). (c) L'equation fonctionnelle conjecturale est A(a , M, s) = c(a , M, s)A(a, M, I - s) . Pour definir Ies cv, une hypothese additionnelle de compatibilite entre Ies H;.(M) est requise. Elle permet d'associer a v, a, M une classe d'isomorphie de representa tions complexes du groupe de Weil W(kvfkv) pour v infini, du groupe de Weil epaissi W(kvfkv) pour v fini, et on prend le c de [4, 8 . I 2. 4], avec t = rs. Pour v infini, ceci revient a decomposer Hr(M) ® E, q c comme en (a), a associer -
oo
00
I
V ALEURS
DE FONCTIONS L
329
a chaque sous-espace de la decomposition un facteur c�, et a prendre leur produit. La table des c�, pour un choix particulier de If!v et de dxv, est donnee en 5.3. Pour v fini, on commence par restreindre les representations H .<(M) a un groupe de decomposition Gal(kvfkv) c Gal(k/k) , puis au groupe de Weil W(kvfkv) . Appli quant [4, 8.3, 8.4], on deduit de H.,.(M) une classe d'isomorphie PJ. de representations de ' W(kvfkv) sur E�.. II est loisible ici, et utile, de la remplacer par sa F-semi-simpli fiee [4, 8.6]. On demande que ces representations, pour i\ variable, soient com patibles, i.e., que si on etend les scalaires de E1. a C, par 6 : E1. --> C prolongeant a, la classe d'isomorphie de la representation obtenue soit independante de i\ et de a. C'est la classe d'isomorphie cherchee. Remontons pour le lecteur les renvois internes de [4]. Une representation de ' W est donnee par une representation p du groupe de Weil dans GL( V) , et par un endomorphisme nilpotent N de V. En terme de la constante locale [4, 4. 1], de p, celle de (p, N) est donnee par c((p, N), s, If!, dx) = c(p Q9 w. , If! dx) det( - FNv-s, VPCil / Ker(N) PCil ) . REMARQUE 5. 2 . 1 .
·
La fonction de s c((p, N) , s, If!, dx)L((p, N) v, 1 - s)L((p, N) , s) - 1 est la meme pour (p, N) et pour (p, 0). Ceci permet d'enoncer l'equation fonction nelle conjecturale des fonctions L en supposant seulement une compatibilite entre les semi-simplifiees des restrictions des H.<(M) a un groupe de decomposition. 5.3. Dans la table ci-dessous, on donne les facteurs locaux, et les constantes, associees aux divers types de sous-espaces minimaux de la realisation de Hodge. Pour les constantes, on a suppose If!v et la mesure dxv choisis comme suit : v reel : lf!.(x) = exp ( 2n ix), mesure dx, v complexe : If! v(z) = exp(2ni Trc/R(z)) , mesure I dz 1\ dz I, soit pour z = x + iy, exp (4n ix) et 2dxdy. On utilise les notations rR(s) = n-s /'2; r(s/2), Fc(s) = 2 . (2n) -s r(s). place
I I
type
I
I
facteur r constante (pour If!, dx ci-dessus)
p I Fc(s - p) I iq Fc(s - p) iq-p+ 1 (p, q ), Foo = ( - l) P+•, c = O ou l iFR(s + c -p) l i •
complexe I (p, q) ou (q, p), p � q reelle {(p, q), (q, p)} , p < q
I
I
Le cas qui nous interesse est celui ou k = Q. On peut dans ce cas prendre If!oo(x) = exp(2nix), If!p(x) = exp( - 2nix) (via l'isomorphisme Qp/Zp = partie p primaire de Q/Z), dxoo = mesure de Lebesgue dx, dxp = la mesure de Haar sur Qp
donnant a Zp la masse 1 . PROPOSITION 5.4. Si M est critique de poids w , on a, modulo un nombre rationnel independant de a,
Loo (a, M(l)) Loo(a, M)- 1 "' (2n)-d- (M)
·
(2n)-wd(MJ I2 .
D'apn!s 2.5, les Loo(a, ) sont independant de a. Ceci nous permet de ne verifier 5.4 que pour a fixe. La formule est compatible a la substitution M ...... M(l) : M(l) est
330
P. DELIGNE
de poids - 2 - w, son d- est d+(M) et abregeant d(M) et d±(M) en d et d±, on a d = d+ + d- et
( - d- - �d ) + ( - d+ -
)
( - 2 - w)d = 0. 2
Ceci permet de ne verifier 5.4 que pour w � - 1 . Pour s entier, on a, modulo Q * , Fis) - (2:n:)-s 1 2 pour s pair > 0, 2 l / :n; S impair, Cl-s pour FR(s) - (2 ) Fc(s) - (2:n:)-s pour s > 0.
(5.4. 1)
Pour w � - 1 , Ia puissance de 2:n: dans Ia contribution de chaque sous-espace de H8(M) ® C, comme en 5.2 (a), est done donnee par {(pq), ( qp)}, p � q (pp) , p pair � O, Foo = - 1 (pp), p impair � 0, F,., = - 1
L,.,(M) p p
2
1 +p -2-
L,.,(M(I)) Loo(M( l))Loo(M)- 1 - l -p - 1 -w -1 -� -
l
-
2
p
-1 -1
;
-�
2
La proposition en resulte aussitot. Posons det M = ;v cM> M (puissance exterieure sur E). PROPOSITION 5.5. s*(M) - s * (det M). Pour des dx. choisis comme suggere en 5.4, nous prouverons plus precisement des equivalences (5. 5 . 1 )
Posant r;.((J, M, Iff.) = s.((J, M, Iff. , dx.) s;1 ((J, det M, Iff., dx.), ceci equivaut a -rr;.((J, M, Iff.) = r;.(-r(J, M, Iff.) (5.5.2) pour tout automorphisme -r de C. II resulte de [4, 5.4] que, si a e Q: est de valeur absolue 1 , et qu'on pose (Iff. · a)(x) = lff.(ax), on a (5.5.3) r;.((J, M, Iff.) = r;.((J, M, Iff. · a) (pour l a l . = 1). Pour -r un automorphisme de C, et v fini, -riff. est de cette forme Iff. · a avec ll a ll . = I ; de meme, W00 = Iff00 ( - 1). Pour v fini, Ia definition de s. est purement algebrique, d'ou -rr;.((J, M, Iff.) = r; .(-r(J , M, -.Iff.) et (5.5.2) resulte de (5.5. 3). Pour v = 00 , on a encore 'ij00((J, M, Woo) = 7)oo(ff, M, Woo) = 7)oo(ff, M, lffoo) ; si on prend Iff00 comme suggere en 5.3, 7)00 est une puissance de i independante de (J, d'ou 7)00((1, M, Iffoo) = ± I , independant de (J, ce qui verifie (5.5.2). •
THEOREME 5.6. Modulo Ia Conjecture 6.6 sur Ia nature des motifs de rang I, Ia Conjecture 2.8 est compatible a !'equation fonctionnelle conjecturale des fonctions L: on a
VALEURS DE FONCTIONS
331
L
L!(M ) c+(M) - e * (M)L! (M(1)) c+( M(l)).
D'apn!s 5. 1 , 5.3 et 5.5, cette formule equivaut a (2 �i)-d- (M) o(M) - (2�)-a- CM) (2�)-wd(M)/2 e * (det M). Posons D = det M et e = d-(D). On a /J(M) = o(D), d-(M) = e (mod 2), et wd(M) est le poids w(D) de D, de sorte que la formule equivaut encore a e * (D) - (2�)w (motifs d'Artin) et tout motif d' Artin est facteur direct d'un H(X). 6.2. Explicitons cette definition. Soient Q une cloture algebrique de Q, et G le groupe de Galois Gal(Q/Q). Une variete de dimension 0 est le spectre d'un produit fini A de corps de nombres, et la theorie de Galois (sous la forme que lui a donnee Grothendieck) dit que 1e foncteur X = spec(A) .... X(Q) = Hom( A, Q) : 0
0
0
(categorie des varietes de dimension 0, sur Q, et des morphismes de schemas) --+ (categorie des ensembles finis munis d'une action continue de G) est une equivalence de categorie. Le foncteur inverse est I .... spectre de l'anneau des fonctions G-in variantes de I dans Q. Une correspondance d'une variete de dimension 0, X dans une autre, Y, est une combinaisons lineaire formelle a coefficients dans Q de composantes connexes de X x Y . L'application :E a;Z; .... :E a;(fonction caracteristique de Z,{Q) c (X x Y) (Q)) identifie correspondances et fonctions G-invariantes, a valeurs rationnelles, sur (X x Y)(Q) = X(Q) x Y(Q), et la composition des correspondances au produit matriciel. Notons H ie foncteur contravariant X .... espace vectoriel QX @, muni de I' action naturelle de G; (correspondance F: X --+ Y) .... le morphisme F* : Q Y CQJ --+ QX @ de matrice 1F. 11 est pleinement fidele, et identifie la categorie des motifs d' Artin a celle des representations rationnelles de G. 6.3. Dans ce modele, si Q est la cloture algebrique de Q dans C, le foncteur "realisation de Betti" H8 est le foncteur "espace vectoriel sous-jacent". La struc ture de Hodge est purement de type (0, 0), et !'involution Foo est l'action de la con jugaison complexe Foo E G. On a en effet un isomorphisme, fonctoriel pour les correspondances,
332
P. DELIGNE
H*(X(C), Q) = QX Cc) = QXCQ) = H(X). Le foncteur "realisation /-adique" H1 est le foncteur Ht( V) = V ® Q1• On a en effet un isomorphisme, fonctoriel pour les correspondances, H *(X(Q), Q1) QfCQ l = QXCQl ® Q1. Calculous de meme la realisation de de Rham. Pour X = Spec(A), on a HfjR(X) A. Ecrivant que A = Homc(X(Q), Q) = (QX CQJ ® Q)G, on obtient que =
=
(6.3. 1)
La formule (6.3.1) realise HvR( V) comme un sous-espace de V ® Q. Ce sous espace est une Q-structure : on a (V ® Q)G ® Q V ® Q. Apn!s extension des scalaires a Q, H8( V) et HvR( V) sont done canoniquement isomorphes. Etendant les scalaires jusqu'a C, on trouve l'isomorphisme (0.4. 1) (le verifier pour V = ____, -
H(X)).
6.4. Soit E une extension finie de Q. La categorie des motifs d'Artin a coefficient dans E est la categorie deduite de celle des motifs d'Artin comme en 2. 1 . Nous
l'identifierons a la categorie des E-espaces vectoriels de dimension finie, munis d'une action de G. Les motifs d'Artin a coefficient dans E, de rang 1 sur E, correspondent aux caractt::res e : G � E* . Nous allons calculer leurs periodes. Soient done e : G ..... E * etf le conducteur de e : e se factorise par un caractere, encore note e, du quotient (Z/fZ)* = Gal(Q(exp(27d/f))/Q) de G. Notons [e] l'espace vectoriel E" de dimen sion 1 sur E, sur lequel G agit par e. La somme de Gauss g = 1.: e (u) ® exp(2niuff) E [e] ® Q est non nulle, et invariante par G : c'est une base, sur E, de HvR([e]). Le determinant de /: H8([e]) ® C HvR([e]) ® C, calcule dans les bases 1 et g, vaut g- 1 • On sait que, pour tout plongement complexe a de E, on a ag · ifg = f, nombre rationnel independant de a, d'ou o([e]) - 1.: e- 1 (u) ® exp( - 2niu/f) E (E ® C) * . (6.4. 1) PROPOSITION 6.5. Soit D le motif [e](n). C'est un motif sur Q, a coefficients dans E, de rang 1 et de poids - 2n. Posant e( - 1) ( - 1)'7, avec r; = 0 ou 1 , on a ____, -
=
e * (D)
-
(2n) -n i 'l -n o(D).
On sait que la constante de I' equation fonctionnelle de la fonction L de Dirichlet (a plongement complexe de E) est donnee par
L(a, [e], s) = I; ae - 1 (n) ·
n-s
e(a , [e], s)
=
c 'l · js · 1.: ae(u) - 1
exp( - 2niuff).
D'apres (5. 1 .9), on a par ailleurs o(D) = (2ni)n o([e]). On conclut en appliquant (6.4. 1) et en natant que pour s en tier (s = n)Jn est rationnel independant de a. Cette proposition verifie (5.6. 1) pour les motifs [e](n). Pour achever la preuve de 5.6, il ne reste qu'a enoncer la Conjecture 6.6. Tout motif sur Q, a coefficient dans E et de rang 1 est de Ia forme [e](n), pour e un caractere de G = Gal (Q/Q) a valeurs dans les racines de /'unite de E et n un entier.
VALEURS DE FONCTIONS PROPOSITION
333
L
6.7. La Conjecture 2.8 est vraie pour lesfonctions L d'Artin.
Pour les motifs d'Artin, on dispose de !'equation fonctionnelle des fonctions L. De plus, le determinant d'un motif d' Artin, tordu a Ia Tate, est du type predit en 6.6. Les arguments des paragraphes 5, 6 montrent done Ia compatibilite de 2.8 a I' equation fonctionnelle, et il suffit de prouver 2.8 pour les motifs V(n), pour V un motif d'Artin et n un entier � 0. Si V(n) est critique, F"" agit alors sur V par multiplication par - ( - 1) n , et c+ = 1 (cf. 3. 1), et il s'agit de prouver que, pour tout plongement 0' de E dans C et tout automorphisme de C, on a 1:L(O', V(n)) = L (1:0' , V(n)). On le deduit des resultats de Siegel [14]. Voir [2, 1 .2]. 7. Fonctions L attachees aux formes modulaires.
7 . 1 . Posons q = e2niz , et so it f = I; anq n une forme modulaire holomorphe cuspi dale primitive (new form) de poids k ;:;;; 2, conducteur N et caractere s. La serie de Dirichlet .E ann-s admet un developpement en produit eulerien, de facteur local en p % N egal a (1 - ap r• + s(p)pk-1 p-2•)-1 . Soit E Ie sons-corps de C engendre par les an . La forme f doit donner lieu a un motif M(f) de rang 2 a coefficient dans E; de type de Hodge {(k - 1 , 0), (0, k - 1)}, de determinant [e-1 ](1 - k) (notations de 6.4) et de fonction L Ia serie de Dirichlet .E ann-s .
Je n'ai pas essaye de definir M(f) comme etant un motif au sens de Grothendieck. Une difficulte est que M(f) apparait de fa�on naturelle comme facteur direct dans Ia cohomologie d'une variete non compacte, ou encore comme facteur direct dans Ia cohomologie d'une courbe modulaire complete, a coefficient dans I' image directe d'un faisceau localement constant (plutot, un systeme local de motifs !) sur Ia partie a distance finie de cette courbe modulaire. Ceci echappe au formalisme de Gro thendieck, mais permet de definir les realisations du motif M(f). 7.2. Supposons tout d'abord que k = 2, et que e est trivial. Soient X le demi plan de Poincare, N Ie conducteur de f, F0(N) le sons-groupe de SL(2, Z) forme des matrices dont Ia reduction mod N est de Ia forme (
334
P. DELIGNE
noyaux de projecteurs, remplacer r; - an par P(Tn - anLI)* pour P un polyn6me convenable. On sait que M(f) a les proprietes enoncees en 7 . 1 . Dans chacune des theories cohomologiques £' qui nous interessent, I' application (7.2.2) est surjective, et les systemes de valeurs propres des r; sur le noyau n'apparaissent pas dans £ 1 (M). La £-realisation de M(f) est done encore le noyau commun, dans ft'�(M) ® E, des r; - an. Le groupe de cohomologie H}(M, Q) est muni d'une structure de Hodge mixte, de type {(0, 0), (0, 1), { 1 , 0)}, dont fll(M, Q) est le quotient de type {(0, 1), ( 1 , 0)}. En particulier, (7.2.2) induit un isomorphisme sur les sous-espaces F l de Ia filtration de Hodge ; toute forme differentielle holomorphe w sur M definit ainsi une classe de cohomologie a support propre sur M. Ceci peut se voir directement : si .r c (!} est I' ideal des pointes, H.*(M), en cohomologie de de Rham, est l'hypercohomologie sur M du complexe .r � Ql (analytiquement, ce complexe est une resolution du faisceau constant C sur M, prolonge par 0 sur M), et le Hl re�oit H 0 ( M, Ql). Supposons pour simplifier que E = Q, et calculons c+(M(/)(1)). Le motif M(f)(I) est le dual de M{f)et c+ est done une peri ode de M(f) {1. 7) : il faut integrer w1 contre une classe d'homologie rationnelle de .M, fixe par F"'" Relevant M(f) dans ft'�(M), on voit qu'on peut plutot integrer w1 contre une classe d'homologie sans support de M, fixe par F"'" Toutes les integrates non nulles de ce type seront commensurables. Pour calculer Foo, il est utile d'ecrire M comme quotient de X± = C - R par le sous-groupe de GL(2, Z) forme des matrices ayant pour reduction mod N une matrice (t =). La conjugaison complexe est alors induite par z � z, et l'image dans M(C) de iR+ est un cycle sans support fixe par Foo · La formule L(M(f), 1) = - Jboo w1 justifie 1 . 8 pour M{f)(1) : le second membre est, ou bien nul, ou bien - c+(M(/)(1)). REMARQUE 7.3 . On a aussi c± {M{f){l )) ,...,
Jioo
al b
Wt ±
Jioo
-a l b
Wf .
7.4. Pour e (et E ) quelconque, il faut remplacer F0(N) par F1 (N) : le sous-groupe de SL{2, Z) forme des matrices de reduction mod N de Ia forme (/; �). Comme ci dessus, pour calculer Foo, il est plus commode de travailler avec X± et le sous-groupe de GL{2, Z) forme des matrices de reduction mod N de Ia forme (/; =). Par ailleurs, une dualisation apparait, cacbee en 7.2 par Ia symetrie de Ia correspondance Tn . Pour une definition convenable de Tm on a (a) M(f) est le noyau commun des r; - an dans H 1 ( M), M = X/F1 (N). (b) On a 1 T;w1 = anw1, et T;w1 = ti n Wf { noter Ia formule ti n = e(n)-1an), de sorte que w1 est dans Ia realisation de de Rham de M(/ ) = M(J)V( - 1 ) On trouve que c+(M(f){l)) E (E ® C)*/E* E C*Hom (E, cl jE* est donne par le systeme de periodes .
si ce dernier est non nul. Noter que si l'une de ces integrates s'annule, elles s'annu lent toutes. Tel est le cas si et seulement si, dans Ia partie de l'homologie sans support
VALEURS DE FONCTIONS
L
335
tendue par le cycle iR+ et ses transformes par les Tm le systeme de valeurs propres an pour les Tn n'apparait pas. Ceci justifie 2.8, et, partiellement 2.7, pour M{f)(l). REMARQUE 7. S. Les systemesdevaleurs propres des Tn dans Ker(£�(M) --+ £ 1 (M)) sont lies aux series d'Eisenstein, alors que ceux qui apparaissent dans £ 1 (M) sont lies aux formes paraboliques. C'est pourquoi ces ensembles sont disjoints. II en re sulte que Ia structure de Hodge mixte de H�(M, Q) est somme de structures de Hodge : !'extension de Hl(M, Q) par Ker(H:{M, Q) --+ Hl(M, Q)) splitte. Ceci, generalise au cas de n'importe que! sous-groupe de congruence de SL(2, Z), equi vaut au theoreme de Manin [9] selon lequel Ia difference entre deux pointes est toujours d'ordre fini dans la jacobienne (cf. [6, 10.3.4, 10.3.8 et 10. 1 . 3]). La demon stration donnee ne differe d'ailleurs pas en substance de celle de Manin. 7.6. En poids k quelconque, il faut remplacer Ia cohomologie de M par Ia cohomologie de M a coefficient dans un faisceau convenable. Pour decrire ce qui se passe, je remplacerai M et M par Mn et Mm relatifs au groupe de congruence F(n), avec n multiple de N et ;;;; 3. On redescend ensuite a M et M en prenant les invariants par un groupe fini convenable. Soient g : E --+ Mn Ia courbe ellipti que universelle et j !'inclusion de Mn dans Mn · La cohomologie .a considerer est H 1 (M" ' j* Symk-2(Rlg * Q)). Les realisations de M{f) sont facteur direct de ce groupe, calcule dans Ia theorie de cohomologie correspondante. On peut comme precedem ment les relever dans H�(Mm Symk-2(Rlg * Q)). Pour calculer c± du dual de M(f). il faut alors integrer Ia classe dans H� definie par f contre des classes d'homologie sans support de Mn a coefficient dans le systeme local dual de Symk-2(R 1g * Q). Si on prend le cycle image de iR+, muni de diverses sections du dual (une base), on trouve les integrales d'Eichler
iJ0oo f(q) · -dqq · (2 i
)l
n: z
(0 � I � k - 2)
(periodes paires pour I pair, impaires pour I impair)-et on sait comment ecrire L(M(f), n) pour n critique en terme de ces periodes. PROPOSITION 7.7. Soit M un motif de rang 2, a coefficient dans E, de type de Hodge {(a, b), (b, a)} avec a i= b. Alors, d±Symn(M) et c±Symn M sont donnes par les for mules suivantes : (1) si n = 21 + 1 : d±
(2) si n
=
21: d+
=
=
I + 1 , et
I + 1 , d-
=
I, et
c+Symn M = (c+(M) c-(M)) I C I+ll / 2 o(M) HI+ll / 2 , c-SymnM = (c+(M) c-(M)) HI+ll 1 2 o(M) I U-l l / 2 . C'est une question de simple algebre lineaire, que nous traiterons par "analyse dimensionnelle". (a) HiM) est un espace vectoriel de rang 2 sur E, muni d'une involution de Ia forme (A _![) dans une base e+, e- convenable. Les d±Symn M sont les dimensions des parties + et - de Ia puissance symetrique nieme, de bases respectives {e+", e+c• - 2> e-2, · · · } et {e+c• - D e-, e+c•-3> e-S, · · · } -d'oil les valeurs annoncees.
Foo,
336
P. DELIGNE
(b) Soit w, r; une base du dual de HvR(M), telle que w annule F+HvR(M). Le sous-espace F± de HvR(SymnM)V Symn HvR(M)V admet pour base les wn , 1 " ' -d± l d± -1 (V n 7) n + w- , , , et r; n c+Sym M = (wn 1\ (w n- l r;) 1\ . . . , e+" 1\ (e+ c• -2) e-2) 1\ . . . ), (7.7. 1)+ c-SymnM = (wn 1\ (wn- I r;) 1\ . . · , (e+ c•-D e-) 1\ (e+ c• -3) e-3) 1\ . . . ) . (7.7. 1)(c) Posons V = H8(M) ® C HvR(M) ® C; c'est un E ® C-module. Les formules ci-dessus montrent que c±SymnM ne depend que du E ® C-module V, de sa base e+, e-, de (V E V*, et de l'image 7j de 7] dans V*/(w) : le d±-vecteur a gauche du produit scalaire (7 . 7. 1 ) ne change pas si on remplace r; par r; + i\w. Par ailleurs, (w, e+) et (w, e-) sont inversibles, et 7j est une base de V*/(w). Le sys teme ( V, e+, e-, w, ij) est done decrit a isomorphisme pres par les quantites c+ = (w, e+), c- = (w, e-) et a = (w 1\ r;, e+ 1\ e-), dans (E ® C)*. Remplacer e+, e-, w, 7j par ).e+, pe-, w, J)ij/ilp remplace c+, c- et a par ).c+, pc- et J)a. Pour c+ = c- = o = 1 , le second membre de (7.7. 1)± est dans Q*. Dans le cas general c±SymnM est done un multiple rationnel du produit de c+(M), c-(M) et o(M)jc+(M)c-(M) aux puissances respectives les degres auquels figurent e+, e- et 7J dans (7.7. 1). On s'epargne Ia moitie du calcul en notant que remplacer F"" par - F"" echange e+ et e-, done c+ et c-, respecte o, et echange les (7.7. 1)± pour n impair, les conserve pour n pair. n = 21 + 1 : d+ = d- = I + 1 , deg r; dans (7.7. 1)± = 0 + 1 + . . . + I = 1(1 + 1)/2, deg e+ dans (7.7. 1)+ = (21 + 1) + (2/ - 1) + . . . + 1 = ( I + 1) 2 = deg e- dans (7.7. 1)-, deg e+ dans (7.7. 1)- = 21 + (21 - 2) + . . . + 0 = 1(1 + 1 ) = deg e- dans (7.7. 1)+ ; n = 21: d+ = I + 1 , d- = I, �
�
deg r; dans (7. 7. 1)+ =;= 0 + 1 + . . + I = /(/ + 1)/2, deg r; dans (7.7. 1)- = 0 + I + . . . + (I - 1) = 1(1 - 1)/2, deg e ± dans (7.7. 1)+ = 21 + (2/ - 2) + . . . + 0 = /(/ + 1), deg e± dans (7.7. 1) - = (21 - 1 ) + (2/ - 3) + . . . + 1 = f 2 . 7.8. Cette proposition nous fournit une conjecture pour les valeurs aux entiers critiques de L(SymnM(f), s) . Voici les formules : ·
(7.8. 1)
L (a, M( f), s) = .L; aann-s = IT Lp(a , M(f), s) p
oil pour presque tout p Lp (M(f), s)
= (I
_
ap p-s + e(p)pk -1- Zs) -1 = ((1 - a� p-s) ( 1 - a; p-s))- I ,
avec e un caractere de Dirichlet. On a 1\ 2M(f) = [e-1](1 - k), d'ou a (M(f)) (27ri) l -k I; e(u) ® exp ( - 27riu/F), (7.8.2) si e est de conducteur F, et ( 7.8.3) L * (M(f) , m) (27r i) m c ± ( M(f )) , ± = ( - 1) m , pour 1 � m � k - 1 . �
�
·
VALEURS DE FONCTIONS
(7. 8.4)
L
337
L (o-, Symn M(f) , s) = IT Lp (o-, SymnM(f) , s) p
ou pour presque tout p Lp (SymnM( f) , s)- 1 =
n
ij (I t =O
- a'f,a;n-i p-s) ;
conjecturalement, pour m critique, on a L *(SymnM(f) , m) "' (2n i) md±Sym"M(f) c± SymnM(f) , ± = ( - 1)m, ou c± et d± sont donnes en terme de c±(M) et o(M) (caracterises par (7. 8.2), (7.8.3) ) par les formules 7.7. Pour !'evidence numerique en faveur de cette conjecture, voir [18]. 8. Caracteres de Heeke algebriques. Le lecteur trouvera dans [3, paragraphe 5], dont nous utiliserons les notations, les definitions essentielles relatives aux carac teres de Heeke algebriques ( = grossencharaktere de type A0). Conjecture 8. 1 . Soient k et E deux extensions finies de Q, et X un caractere de Heeke algebrique de k a valeurs dans E. (i) II existe un motif M(x) de rang un sur k, a coefficients dans E, tel que, pour toute place A de E, Ia representation A-adique H;.M(x) so it celle definie par X : le ·
Frobenius geometrique en fiJJ premier au conducteur de X et a Ia caracteristique resi duelle l de A agit par multiplication par x(£?1'). (ii) Ce motif est caracterise a isomorphisme pres par cette propriete. (iii) Tout motif de rang 1 est de Ia forme M(x ) . (iv) Decomposons k ® E en produit de corps : k ® E = IT K,., et ecrivons Ia partie algebrique X alg : k* -+ E* de X sous Ia forme X alg(x) = IT NK,tE(x) n1• La decompo sition de k ® E induit une decomposition de HDiM(x)) en les HDR (M(x)),. = HDR(M(x)) ® k@E K,. ; avec cette notation, !a filtration de Hodge est Ia filtration par les EB n;"'P HDR (M(x )) ,. .
L'unicite 8 . I (ii) impose aux M(x) le formalisme suivant ( 8. 1 . 1 ) M(x ' x ") - M(x ' ) ® M(x ' ) . (8 . 1 . 2) Si c : E ----4 E' est une extension finie de E, M(cx) se deduit de M(x) par extension des coefficients de E a E'. Si k' est une extension finie de k, M(x Nk'lk) se deduit de M(x) par exten (8 . 1 . 3) sion des sca1aires de k a k'. a
REMARQUE
8.2. Po sons les notations :
c = la conjugaison complexe, Q = la cloture algebrique de Q dans C, S =
Hom(k, Q) = Hom(k, C), J = Hom(E, Q) = Hom(E, C). La decomposition de k ® E en les K,. correspond a la partition de S x J en les or bites de Gal(Q/Q). Tout homomorphisme algebrique r; : k* -+ E* s'ecrit sous la forme IT NK;IE(x) n1• Si l'orbite sous Gal(Q/Q) de (a-, r) E S x I correspond a K,., i.e., si a- ® r: k ® E -+ C se factorise par K,., nous noterons n (r; ; a-, r) , ou simplement n(o-, r) , l'entier n,.. La fonction n(o- , r) est constante sur les orbites de Gal(Q/Q). Si r; est la partie algebrique d'un caractere de Heeke X · on a de plus L'entier w = n (r; ; a-, r) + n (r; ; co- , r) est independant de a- et r. (8.2. 1)
338
P. DELIGNE
C'est le poids de x (et de M(x)). On ecrira parfois n(x ; a, ..-) pour n(x.1g ; a, ..-) . Reciproquement, un homorphisme r; verifiant (8.2. 1) est presque de la forme Xalg • pour x un caractere de Heeke convenable : (a) une de ses puissances l'est; (b) il existe une extension finie t : E --+ E' de E telle que &7] le soit ; (c) il existe une extension finie k' de k telle que r; o Nk'tk le soit. La regie 8 . 1 (iv) permet de deduire la bigraduation de Rodge de H11M(x) de sa structure de £-module : le facteur direct H11M(x) ®E, r C de H11 M(x) ® C est de type de Hodge (p, q), avecp = n(x ; u, ..-) et q = w - p = n(x ; u, c..-). EXEMPLE 8.3. Soit A une variete abelienne sur k, a multiplication complexe par E. On suppose H1 (A) de rang 1 sur E (type CM). 11 resulte de la tbeorie de Shimura et Tanayama que H1(A) verifie la condition de 8 . 1 (i), pour un caractere de Heeke algebrique x de k a valeurs dans E, et que la partie algebrique Xalg de x se lit sur le k ® £-module Lie(A) : 1 X al g(x) = detE(x ® 1 , Lie(A))EXEMPLE 8.4. Prenons pour k le corps des racines n iemes de l'unite, soit V !'hyper surface de Fermat d'equation projective .E �o xr = 0 et soit M le motif "coho mologie primitive de dimension moitie de V". Soit G le quotient de p';:H par son sous-groupe diagonal. Ce groupe agit sur V par (a;) * (X;) = (a;X;), et sur M par transport de structure. Decomposant M a l'aide de la decomposition de l'algebre de groupe Q[G] en produit de corps, on obtient des motifs verifiant 8 . l (i) pour des caracteres de Heeke algebriques convenables : ceux introduits par Weil dans son etude des sommes de Jacobi. ExEMPLE 8.5. Le motif Z( - 1) verifie 8 . l (i) pour x = Ia norme. Pour x d'ordre fini, un motif d' Artin convenable verifie 8. 1 (i). 8.6. Pour chaque a E S, H11 (M(x)) est de rang 1 sur E. Choisissons une base e11 de chaque H11 (M(x)). Outre sa structure de £-module, la somme des H11 (M(x)) ® C a une structure naturelle de k ® C = cs-module -au total, une structure de k ® E ® C-module libre de rang 1 , pour laquelle e = ,E e11 est une base. 8.7. La realisation de de Rham HvR(M(x)) est un k ® £-module libre de rang 1 . Choisissons en une base w. La somme des 111 : H11 (M(x)) ® C __. - HvR(M(x)) ®k . a C est l'isomorphisme (0.4) de k ® E ® C-module I : EB H11 (M(x)) ® C � HvR(M(x)) ® Q C. 11
Soit w une base du k ® £-module HvR(M(x)), et posons p'(x) = w/I(e) E (k ® E ® C) * . Cette periode depend de X • w et e. Prise modulo (E ® F) * et E*8, elle ne depend que de X · Nous noterons p'(x ; u, ..-)-au simplement p'(u, ..-)-Ia composante d'indice (u, ..-) de son image par l'isomorphisme E ® F ® C __. - csxJ. 8.8. Soit A comme en 8.3 et calculons les peri odes p'(a, ..-), modulo Q*, pour le motif H l (A). On commence par etendre les scalaires de k a Q*, a l'aide de u. Si n(u, ..-) = 1, il existe alors une 1-forme holomorphe w definie sur Q telle que u * w = ..-(u)w pour u E E, et, pour Z E H1(A(C)), on ap'(u, ..-) - Jz w. On peut passer de la au cas general a l'aide de la formule p'(a, ..-) p'(a, c..-) - 2 7C i. 8.9. La conjecture 8 . 1 (ii) affirme en particulier que si deux motifs verifient Ia condition de 8 . 1 (i), ils ont meme periode p. Pour les motifs 8.4, les periodes s'expri-
·
VALEURS DE FONCTIONS
L
339
ment en terme de valeurs de la fonction r et, si on travaille mod Q * , 8.1 suggere la conjecture de B. Gross [7] reliant certaine periodes a des produits de valeurs de la fonction r. La comparaison de 8.4 et 8.5 mene a la Conjecture 8. 1 1 , 8 . 1 3 suivante. Le resultat annonce apres 0. 10 permet de la demontrer. 8. 10. Soient N un entier, k = Q(exp(271:i/N)), f7J un ideal premier de k, premier a N, k9 le corps residue! et q = N&> = l k91 . On notera t !'inverse de la reduction mod fJl!, des racines Nieme de 1 dans k9 a celles de k. Pour a E N- 1 Z/Z, a =f. 0, considerons la somme de Gauss g(fJl!, a, If!) = - 1: t(x- a < q -ll)lf! (x). La somme est etendue a k� et 1/f : k9 -> C * est un caractere additif non trivial. Soit a = l: n(a) oa dans le groupe abelien libre de base N- 1 Z/Z - {0} . Si l: n(a) a = 0, le produit des g(&>, a, 1/f) n < a ) est independant de If! et on pose g(fJl!, a) = g(&>, a, 1/f) n < a ) . Weil [16] a montre que, comme fonction de fJl!, g(&>, a) est un caractere de Heeke algebrique x.. de k a valeurs dans k. Notons (a) le representant entre 0 et 1 de a dans Z/N. Si a = l: n(a) oa verifie Pout tout u E (Z/N)*, on a l:n(a) (ua) = 0, il resulte la determination par Weil de la partie algebrique de x.. que x.. est d'ordre fini ; on note encore x.. le caractere de Gal(Q/k) valant x.,(fJl!) sur le Frobenius geometrique en fJl!. Posons F(a) = F( C* un caractere additif non trivial. Pour a E N- 1 ZJZ - {0}, soient fJl!a. i les ideaux premiers de kH (exp(271:ia)) au-dessus de fJl!. Si ,., est le corps residue! en &>a.i • et que IK' I = q ' , on note g(fJl!a. i • a, If!) la somme de Gauss - l: t(x-a Cq'-ll)lf!(Tr.,1,x) (somme etendue a "' *). Le produit g(&>, a, If!) = TI 1 g(&>a.i • a, If!) ne depend que de l'orbite de a sous H. Soit a = l: n(a) oa , invariant par H, et verifiant l:n(a)a = 0. Pour chaque orbite 0 de H dans N- 1 Z/Z - {0}, on note n(O) la valeur constante de n(a) sur 0. On pose g(fJl!, a) =
a
IT g(fJl!, a, 1/f) n < a ) .
mod
H Comme fonction de fJl!, g(fJl!, a) est un caractere de Heeke algebrique Xa de kH a valeurs dans kH . Pour le prouver, on se ramene par additivite a supposer les n(a) � 0. On applique alors [3, 6.5] pour F = kH , P = Q, k = notre k, et I = une somme disjointes de copies d'orbites H: n(O) copies de 0; pour i E /, d'image afN dans N- 1 Z/Z, on prend pour A; le compose Z(l)p -> (Z/N)(l)p = f.lN(k) _,xa k * ; le caractere de Heeke obtenu est le produit de Xa par le caractere "signature de la
340
P. DELIGNE
representation de permutation de H sur I". Un cas particulier de ce resultat figure deja dans Weil [17]. Le caractere Xa de 8.10 est le compose de Xa ci-dessus avec la norme Nkl kH . Conjecture 8. 1 3. Soient f?lJ un ideal premier de kH , premier a N et Ffl' un F'robenius geomhrique en f?IJ. Si a invariant sous H verifie ( * ) , on a Ffl'r(a) = g(&, a) F(a). ·
En particulier, si le groupe des racines de /'unite de kH est d'ordre N', on a F(a)N' E kH .
On peut de 8 . 1 3 deduire la variante suivante, apparemment plus generale. On remplace la condition ( * ) par ,E n(a)(ua) = k est un entier independant de u E (Z/N)*. (* ') Le caractere de Heeke Ng>-k · g(&, a) est alors d'ordre fini. On l'identifie a un caractere x de Gal(QjkH), et on espere avoir
·
a ((2 1Ci ) -k F(a)) = x (a) ((21Ci) -k F(a) ) .
8 . 1 4. Si k est un corps extension quadratique totalement imaginaire d'un corps totalement reel, soit, comme nous dirons, un corps de type CM, Shimura [13] a determine les valeurs critiques des fonctions L des caracteres de Heeke algebriques de k, au produit pres par un nombre algebrique. Il les exprime en terme de perio des de varietes abeliennes de type CM, a multiplication complexe par k. Dans la fin du paragraphe, nous montrons que son theoreme est compatible a la Conjecture 2.8, qui les exprime en terme de periodes de motifs sur k, de rang I . 8. 1 5. Soient X et M(x) comme en 8. 1 . Excluons le cas 8.5, et supposons que Rk!Q M(x) verifie 1 .7. Le corps k est alors totalement imaginaire, et les nombres de Hodge hPP sont nuls : avec les notations de 8 . 1 et 8 .2, aucun n; n'est egal a w/2. Notre premiere tache est de calculer c+Rk1QM(x) en terme des peri odes p'(x ; a, z-). Rappelons que c'est le determinant de l'isomorphisme de E ® C-module calcule dans des bases definies sur E. Choisissons les e(J de 8.6 de sorte que Fooe(J = ecu · Les e(J ± eC(J forment alors une base de HtfRk1QM(x) c H8Rk!Q Mx = tB uEsHuM(x). Notons au passage que d+ = d- = 1-[k : Q]. Soit S le quotient de S par Gal(CJR). Pour calculer c+ = det(J+), nous utiliserons Ia base (e" + ecu) de Ht. Elle est indexee par S. D'apres 8 . l (iv), la filtration de Hodge de HvRRk1QM(x) = HvRM(x) se lit sur sa structure de k ® £-module : si on note (k ® E)+ le facteur direct de k ® E pro duit des K; tels que n; < w/2, le quotient HfiRRk1QM(x) de HvRR k1QM(x) est le facteur direct correspondant : Soit w comme en 8.7, et utilisons Ia structure de k ® E ® C = csxf-module de HvRM(x) pour decomposer w : w = .E wu. r· On a par definition I(e") = ,E,p'(a, z-)- 1 wu,n et done J+(e" + e,") =
I; n (u, r) <wl2
p'(a, z-)- 1 wu ,r +
�
n (u, r) <wl2
p'(ca, z-)- 1 w,u .r ;
VALEURS DE FONCTIONS
L
7:
341
c'est Ia somme, indexee par 7: E J, de termes egaux a p'(q, )- 1 w... r pour n(a, 7:) < wj2, et a p'(c , 7:) - 1 Wca. r pour n(a, 7:) > wf2. Pour ii E s, de representant (J, posons
q
Wu =
I;
n (a, r)<wl2
Wa, r +
I;
n (ca, �) <wl2
Wca, r ·
Les J+(e.. + ec..) sont des multiples des Wu par des elements de E ® C; les W u for ment done une base de Hi;R. Dans les bases e + eca et wa , Ia matrice de J+ est diagonale ; son determinant det'(J+) e (E x C)* = C *l a pour coordonnees det' (J+) � = n p'(a, 7:) - 1 .
..
n (l1, �) <wl2
.. 11
Soient les applications Es --+ E8 : Ia --+ 1 + I c , pour ii image de a, ES ® C --+ k ® E ® C, deduit de l'isomorphisme de k ® C avec cs, et la projection de k ® E sur (k ® E)+. Par composition, on obtient un isomorphisme de E ® C-modules : Es ® C (k ® E)+ ® C. Nous noterons D(x) son determinant, calcule dans des bases definies sur E des deux membres. Identifiant HDR a k ® E a l'aide de Ia base w , on voit que c'est le deter minant de I' application identique de Hi;R ® C, calcule dans la base wa , a la source, et une base definie sur E, au but. Notant Dr(X) ses composantes dans Cl, on trouve pour c + = det'(J+) · D(x ) la formule suivante. PROPOSITION 8. 16. On a c+Rk!QM(x) = TI p'(a , 7:) -1 · (x ) �e .
____::___.
( n (a, r)<w/2
D. ) J
R EMARQUE 8 . 1 7. Supposons k de type CM, extension quadratique de k0 totale ment reel. Le quotient s de s s'identifie alors a !'ensemble des plongements com plexes de k0, et le diagramme
E8 ® C � k0 ® E ® C .f
.f
Es ® c -. k ® E ® c
-. (k ® E)+ ® c
est commutatif. L'application composee k0 ® E --+ (k ® E)+ est done un isomor phisme, et D(x) est encore le determinant de Es ® C --+ - k0 ® E ® C, deduit par extension des scalaires de Q a E de l'isomorphisme cs --+ - k0 ® C. Ceci fournit pour D( x), bien defini mod E*, un representant dans (Q ® C)* = C * c (E ® C)*, a savoir le determinant de !'inverse de Ia matrice (aa), pour a e S et a parcourant une base de k0 sur Q. L'isomorphisme C 8 --+ k0 ® C transforme Ia forme qua dratique I;x7 en Ia forme Tr(xy). Ceci permet d'identifier (det(aa)) 2 au discriminant de k0 : D( x) - racine carree du discriminant de k0. 8 . 1 8. Notons p"(x ; u, 1:) l'image de p'(x ; u, 1:) dans C*fQ*. Elle ne depend que de X• a et 1:. Si un homomorphisme algebrique r; : k* --+ E* verifie (8.2. 1), une de ses puissance est Ia partie algebrique d'un caractere de Heeke : r;N = Xai g · De plus, si X � lg = x:l g• x ' et x " ne different que par un caractere d'ordre fini et x 'M = x "M pour M convenable. On deduit de (8. 1 . 1) que p"(x M ; a, 1:) = p"(x ; a, 1:) M , et ceci permet de poser sans ambiguite -
342
P. DELIGNE
p(r; ;
0",
-.) = p(x ; O", ,.-) l iN ,
pour r; N = Xalg ·
Ces periodes obeissent au formalisme suivant : (8. 18. 1) p(r;' r;" ; 0", -.) = p(r;' ; 0", -.) p(r;" ; 0", -.) . (8. 1 8.2) p(r; ; O", -.) ne change pas quand on remplace E par une extension E' de E, et ,.- par un de ses prolongements a E'. (8. 1 8.3) p(r; ; O", -.) ne change pas quand on remplace k par une extension k' de k, O" par un de ses prolongements a k', et X par X Nk ' tk · o (8. 1 8.4) Si a est un automorphisme de k, et (3 un automorphisme de E, on a p(r; ; 0", -.) = p((3r;a- 1 , O"a-1 , -.(3 -1 ). (8. 18.5) Le complexe conjugue de p(r; ; O", r) est p(r; ; if, f) . (8. 1 8.6) Pour k = F = Q, p(ld ; Id, ld) = 2Jri. Les formules (8. 1 5. 1) a (8. 1 5.3) resultent de (8. 1 . 1) a (8. 1 . 3), (8. 1 8 .4) et (8. 1 8.5) se voient par transport de structure, et (8. 1 8.6) resulte de (8.5). 8 . 1 9. Soit r; : k* ---> E * un homomorphisme verifiant (8.2. 1). On suppose aussi que n(r; ; O", r) ne vaut jamais w/2, ce qui permet de definir (k ® E)+ comme en 8 . 1 5. Definissons r; * : E* ---> k* par r; * ( y ) = detk(l ® y, (k ® E)+). Cet homomorphisme verifie encore (8.2. 1), et ·
n(r;* ; O", T)
= =
1 si n(r; ; O" , r) < w/2, 0 si n(r; ; O", -.) -> w/2. Si r; * est Ia partie algebrique d'un caractere de Heeke x*, M(x*) est le H 1 d'une variete abelienne sur E, a multiplication complexe par k, dont 1'algebre de Lie, comme k ® £-module, est isomorphe a (k ® E)+. PROPOSITION 8.20. Avec /es hypotheses et notations de 8 . 1 9, prenons pour E un sous corps de C, et no tons 1 /'inclusion identique de E dans C. On a I1
0", 1)
n (<1, 1) <wl2
p(r; ;
= I1 <1
p(r;* ; 1 , O")n C�;,., l) .
Si c : E ---> E' c C est une extension finie de E, le membre de gauche ne change pas quand on remplace r; par c r; (8. 1 8.2). On a (c r;)* = r; * o NE ' tE et, par (8. 1 8.3), le membre de droite ne change pas non plus. Si c : k ---> k' est une extension finie de degre d de k, le membre de gauche est eleve a Ia puissance d quand on remplace k par k', et r; par r; o N k'tk : un plongement complexe de k est induit par d plongements de k', et on applique (8. 18.3). De meme pour le membre de droite, par (8. 1 8.2) et l'egalite (r; o Nk tk)* = cr;*. Ces compatibilites nous ramenent a supposer que E est galoisien et que k est isomorphe a E. Pour chaque isomorphisme w de k dans E, posons n(w) = n (r; ; 1 o w , w). Notant additivement le groupe des homomorphismes de k* dans E*, on a r; = I; n(w) w . Puisque p(r; ; 1 o w , 1) = p(r; o w- 1 ; 1 , 1) (8. 1 5.4), on a '
(8.20. 1) Par
rr
p(r; ; 0", 1 ) =
n (
ailleurs,
rr
n (w) <wl2
p(r; o w- 1 ; 1 , 1) = p
(
1:
n (w) <wl2
)
r; o w-1 ; 1 , 1 .
VALEURS DE FONCTIONS
I;
n(w) <wl2
7J o w- 1 =
I;
n(w J) <w/2; "'2
L
n(w 2)w2 o W11 = I; n(w2) "'2
= I; n(w2) w2 o r; * .
343 I;
n(w1 ) <wl2
w2 o w11
Ceci permet de continuer (8.20. 1) par = IT p(w r;* ; I , l) n ( w ) = IT p(r;* ; I, I w) n (w) = IT p(r; * ; I , o') n(�;O", l) ' "' "' .. 0
0
(nouvelle application de 8 . I 5.4), et prouve 8.20. 8 .21 . Combinant 8. I 6 et 8.20, on trouve pour Ia composante d'indice I de c+ RktQ M( x) e (E ® C)*/E* = C*IJE*, !'expression suivante, mod Q* ct R ktQ M(x) - IT p(x �g ; I , u) -n <x;.., tl . ..
Si x (i.e., M(x)) est critique, Ia Conjecture 2.8 affirme done que L( l o X • 0) "' IT P(X �g ; I , u)-n <x;.., Il (mod Q*) . ..
Pour E assez grand, x�g est Ia partie algebrique d'un caractere de Heeke x * , et les periodes s'interpretent comme periodes d'integrales abeliennes (8. I 9), (8.3). L'enonce obtenu est celui que Shimura a prouve pour k de type CM, ou abelien sur un corps de type CM (avec une restriction sur le poids). REMARQUE 8 .22. Si r; : k* -+ E* verifie (8.2. I), il existe des sous-corps k' de k et c : E' -+ E de E, so it de type CM, so it egaux a Q, et une factorisation 7J = c r; ' Nktk' . On a aiors p (r; ; u, -r ) = p(r;' ; u l k ' , -r l k'). Si maintenant k et E sont de type CM (ou Q), et qu'on note encore c leur con jugaison complexe, on a cu = uc, c-r = -rc, et r; c = cr; , d'oi.t p(r; ; u , -.) - = p(r; ; cu, c-r) = p(r; ; uc, -rc) = p(cr;c-1 ; u, -r) = p (r; ; u , -.) : les peri odes, a priori dans C* JQ*, sont reelles, i.e., dans R* /(Q* n R*). REMARQUE 8.23. Soient G le groupe de Galois de Ia reunion des extensions de type CM de Q dans Q c C, et c e G Ia conjugaison complexe. C'est un element central de G. Si rp est une fonction localement constante a valeurs entieres sur G, nous poserons rp*(x) = rp(xc). Supposons que rp + rp* est constante. Soient G1 un quo tient fini de G tel que rp se factorise par une fonction rp1 sur G1o et k le corps cor respondant. L'hypothese faite signifie que l'endomorphisme 2rp1(u)u de k* verifie (8.2. I). La periode p(l:rp1(u)u ; I , I) ne depend pas du choix de G1 ; on pose P(rp) = p(l:rp 1 (u) u ; I , 1). La fonctionnelle P est un homomorphisme dans C*JQ* du groupe des fonctions localement constante a valeurs entieres sur G qui verifient rp + rp* = constante. Appendix by N. Koblitz and A. Ogus. Algebraicity of some products of values of the r function. Let AN = N-1 Z/Z - {0}, and let UN = (Z/NZ)* operate on AN in the obvious way. Iff: AN -+ C, define
344
P. DELIGNE
PROPOSITION.
HQ is generated by { en . a = n
= I or n is prime and na
=/=
0}.
The orbits of AN under the action of UN correspond to the divisors d of with I < d � N: each orbit can be written uniquely in the form UN(I fd). The stabilizer subgroup Ia of I /d is {u E UN : u = I mod d} and the orbit of Ijd is canoni cally isomorphic to Ua · Thus, a function f on AN is determined by the collection of functionsfa : Ua -4 Q defined by fa(v) = f(vfd). To prove the proposition, we first complexify, so as to be able to work with characters of UN. LEMMA. Iff : AN -4 C and if is a character of UN• then the inner product of (f) X PRooF.
N
and X is given by :
where the sum is taken over those divisors d of N such that X is pulled back from a character X a of Ua· PROOF.
We have c
= =
.r; Cu) x
I; ( vfd )fa C ua v) x (u),
I; I;
diN
.r; < a > tcua) x
EAN uEU N
uEU N
vEUd uEUrv
where ua E Ua is the image of u. Now if we choose a set U� � UN of coset representatives for UN write :
-4
Ud •
we can
Of course, this sum is zero unless x is trivial on Ia , i.e., unless x is pulled back from a character Xa of Ua, in which case it is I; fa(u' v) xaCu' )IIal = I; fa( w) x_ a( w) xi v) I Ial = xiv)IIaiUalxa). u'EUd
weUd
If we substitute this into the above expression for ((f) l x) and note that (v/d)
l v l fd where I vi is the smallest positive representative of v E Ua, we find : ,
C
= I:d
I: l vlxa(v)IIal d-I( fa l xa)
vEUd
= - I; L(O, xa)IIaiUalxa) as claimed. d
=
0
To prove the proposition, we let d1 be the largest divisor of N such that/a =/= 0 ; i t suffices t o prove that any fin H can b e written as g + f', with g a linear combina tion of the e's and dr < d1. Let d = d1 ; we shall show below that/a is a linear com bination of functions h 1 which factor through Ua -4 Uaf{ ± 1 }, and functions hp which factor through Ua -4 Ua1 p for some prime divisor p of d, p =/= d. Any function of the first type is invariant under ± 1 , and hence the corresponding function on the orbit UN(Ifd) is a linear combination of e 1 a s . A function of the second type is invariant under the kernel K of Ua -4 Ua 1 P • which has order p if p divides dfp and order p - 1 otherwise. In the first case, K = {1 + kdfp : k = O, . . . ,p - 1 } , and in .
'
VALEURS DE FONCTIONS
L
345
the second, it is this same set with one element deleted, namely, the value of I + kd/p which is divisible by p. Thus, the set Sp . w l d = (Kw)fd, with the addition of one or two elements in the orbit UN(pfd). Hence it is clear that the corresponding function on the orbit UN(lfd) can be written as a linear combination of ep , a 's and functions supported on the orbit of pfd, and we have obtained our desired decom position f = g + f'. It remains for us to prove the claim about/d. If d is twice an odd number the map Ud -+ Ud 1 2 is an isomorphism, and the claim is trivial. In the other cases, choose a primitive odd character Xd of Ud, and let x be the pull-back of Xd to UN . Since x is nontrivial and (/) is constant, ( (/) lx) = 0. Let us look at the expression for this inner product provided by the lemma ; the only nonzero terms come from those /. such that /, # 0 and I, s.;:; Ker(x). But then Ker(x) contains I. and Id, hence also Idle = Im, where m = g.c.d. (d, e). Since X is primitive, m = d and d divides e, and since f. = 0 for e > d, 0 = ((/) l x) = - L(O, Xd)ildiC!dixd). Since Xd is odd and primi tive, we conclude that (fdixd) = 0, i.e. , /d is orthogonal to every odd primitive char acter of Ud. It follows that fd can be written as a linear combination of characters which are even (factor through Ud/ { ± I }) or imprimitive (factor through some Ud 1p)· When d = p, we should also remark that the constant functions on Up are also invariant under ± I , and hence are already covered by the first case. 0 THEOREM. Suppose that f: AN -+ Q is such that (/) is constant : I: a ( a ) f(ua) = k for all u E UN. Then
f(f)
= def
is an algebraic number.
n-k IT F( ( a)) f < al a
PROOF. The map ( ) : HQ -+ Q sending any f to l: a ( a ) f(a) is evidently linear, and hence f(/ + g) = F(f)f(g), for f and g in HQ . By the proposition, any f in HQ is a Q-linear combination of the e's ; hence nf is a sum of e's for some n. Since f(j) n = F(nf), we are reduced to checking the theorem for the e's described in the proposition. Noting that ( B I . a ) = I and that (ep , a ) = (p + I )/2 if pa # 0, one has only to appeal to the identities :
f(x)f(l - x)
=
n(sin nx)-1,
p-1
1: f
k=l
(
X
+ k p
-
)
=
p l l2-x(2n) < P - ll 1 2 f(px).
0
REMARK. Kubert has recently obtained a much more precise result expressed in a different terminology from which it follows that, if HQ is Z-vaiued, then 2/ is a Z-linear combination of the e's (to appear). BIBLIOGRAPHIE 1 . M. V. Borovoi, Sur /"action du groupe de Galois sur les classes de cohomologie rationnelles de type ( p, p ) des varietes abeliennes, Mat. Sb. 94 (1 974), 649-652. (Russian) 2. J. Coates and S. Lichtenbaum, On 1-adic zeta functions, Ann. of Math (2) 98 (1 973), 498-550. 3. P . Deligne, Applications de Ia formule des traces aux sommes trigonometriques, dans SGA
4 '12 , 1 68-232.
4. -- , Les constantes des equations fonctionnelles des fonctions L, Modular Functions of One Variable. II, Lecture Notes in Math. , vol . 349, Springer-Verlag, New York, 1 973, pp. 501-595. 5. -- , Formes modulaires et representations 1-adiques, Seminaire Bourbaki 355 (Fevrier 1 969), Lecture Notes in Math., vol. 179, Springer-Verlag, New York, 1 39-172.
346
P. DELIGNE
Theorie de Hodge. III, lnst. Hautes Etudes Sci. Publ. Math. 44 (1 974), 5-77. On the periods of abelian integrals and a formula of Chow/a and Selberg, Invent.
6. P. Deligne, 7. B. Gross,
Math. 45 (1978), 1 93-21 1 . 8. Ju. I . Manin, Correspondences, motifs and monoidal transformations. 9. , Points paraboliques et fonction zeta des courbes modulaires, Izv. 36 (1 972), 1 9-66. (Russian) 10. I. I. Piateckii-Shapiro, Relations entre les conjectures de Tate et de Hodge pour les varietes abeliennes, Mat. Sb. 85(4) ( 1 971), 61 0-620 Math. USSR-Sb. 14 ( 1 971), 61 5-625. 1 1 . J. P. Serre, Abelian 1-adic representations and elliptic curves (Me Gill), Benjamin, New York, --
=
1 968. --
12. , Facteurs locaux des fonctions zeta des varietes algebriques (definitions et conjectures), Sem. Delange-Pisot-Poitou 1 969/70, expose 1 9. 13. G. Shimura, On some arithmetic properties of modular forms of one and several variables, Ann. of Math. (2) 102 (1 975), 491-5 1 5. 14. C. L. Siegel, Berechnung von Zetafunktionen an ganzzahligen Stellen, Nachr. Akad. Wiss . Gottingen Math. Phys. Kl. II 10 (1 969), 87-102. 15. J. Tate, On the conjecture of Birch and Swinnerton-Dyer and a geometric analogue, Seminaire Bourbaki 306 (1965/66), Benjamin, New York ; reproduit dans 10 exposes sur Ia theorie des schemas, North-Holland, Amsterdam, 1 968. 16. A. Weil, Jacobi sums as grossencharaktere, Trans. Amer. Math. Soc. 73 (1952), 487-495. 17. , Sommes de Jacobi et caracteres de Heeke, Nachr. Akad. Wiss. Gottingen Math . Phys . Kl. II 1 (1974), 1-14. 18. D . Zagier, Modular forms whose Fourier coefficients involve zeta functions of quadratic fields, Modular Forms of One Variable. VI, Lecture Notes in Math . , vol. 627, Springer-Verlag, New York, 1 977, pp. 1 05-1 69. 19. N. Saavedra, Categories tannakiennes, Lecture Notes in Math . , vol. 265, Springer-Verlag, New York, 1 972. SGA. Seminaire de geometrie algebrique du Bois-Marie. SGA 4. par M. Artin, A. Grothendieck et J. L. Verdier, Theorie des topos et cohomologie etale des schemas, Lecture Notes in Math., vols. 259, 270, 305, Springer-Verlag, New York. SGA 4'''· par P. Deligne, Cohomo/ogie etale, Lecture Notes in Math. , vol. 569, Springer-Verlag, New York, 1 977. --
INSTITUT DES HAUTES ETUDES SciENTIFIQUES, BURES SUR YVETTE, FRANCE
Proceedings of Symposia in Pure Mathematics Vol . 33 (1 979}, part 2, pp. 347-356
AN INTRODUCTION TO DRINFELD'S "SHTUKA"
D. A. KAZHDAN We begin with a formulation of the Langlands conjecture for GL(n) over a local field K of positive characteristic p. Fix a prime l =F p and denote by Q1 an algebraic closure of Q 1• We introduce some notation. For any topological group G with a compact sub group U c G we will consider smooth Q,-representations of G, that is, homomor phisms p : G -+ Aut V, where V is a Q 1-vector space such that the stabilizer of any vector v e Vis open. We say that p is admissible if for every open compact subgroup U c G we have dim vu < oo, where vu is the set of U-invariant vectors. DEFINITION. We denote by G the set of equivalence classes of admissible irredu cible representations of G and by Gn c G (n = 1 , 2, . . . ) the subset consisting of classes of n-dimensional representations. REMARK . It is easy to check that all standard results (see [7]) about admissible C-representations are true for our Q,-representations. Now we can formulate the local unramified reciprocity law. Let KP be a non archimedean local field, o = oP its ring of integers, 1r: e o a prime element, k = o/(7r:) the residue field, q = card k, and v the valuation on Kp such that v(1r:) = 1 . Recall that an irreducible admissible representation of GL{n, Kp) i s unramified if the subspace of GL(n, a)-invariant vectors has positive dimension. REMARK. In this case this dimension is one. The classification of the set Gp, un of equivalence classes of unramified repre sentations of GL(n, Kp) is well known. Let B c GL(n , Kp) be the subgroup of triangular matrices. To any element x = (xh · . . , xn) (x; e Qf) we associate the character X " of B: X "(b) = (ql-nx1)vChu) (q 2- nx2)vChzzJ . . . x�Chnnl where b;; are the diagonal elements of b. By p" we denote the induced representation ind�L (n, Kpl X " (which is realized in the space of locally constant functions on GL{n, Kp) such that f(bg) = x"(b)f(g) for all b e B, g e GL{n, Kp)) . The repre sentation p" is not necessarily irreducible, but it has a finite Jordan-Holder series which contains exactly one unramified component. We denote it by p" . The follow ing statement is well known [1] . THEOREM 1 . (a) For every element s of the symmetric group S"' p" � Ps c"J · (b) The map p : (Qf)"/ Sn -+ Gp, un is an isomorphism. AMS (MOS) subject classifications (1 970). Primary 10H 1 0 ; Secondary 1 0D20, 22E55, 20G35. © 1979, American Mathematical Society
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D . A. KAZHDAN
REMARK. Our parametrization of G p, un differs from the usual one by a shift and does not use fractional powers of q. Denote by KP a separable closure of KP , and by kp the corresponding algebraic closure of kP . We denote by WKp the dense subgroup of Gal(Kp/Kp) consisting of elements which induce on kP the map x -+ xq" for some n E Z, and we denote by hu c WKp the subgroup fixing kP . Thus WKp is the Weil group of Kp and /P the inertia group. A representation of WKp is said to be unramified if it is completely reducible and its restriction to hp is trivial. We can now state and easily prove the local reciprocity law for unramified re presentations. THEOREM-DEFINITION 2. There is a natural 1 - 1 correspondence between Gp, un and the set of equivalence classes ofn-dimensional unramified representations of WKp · REMARK. By an n-dimensional representation we mean a morphism into GL(n, Q1). PROOF-CONSTRUCTION. Let -r: be an unramified semisimple representation of WKp · Since it is unramified, we may consider it as a representation of WKp/hp � Z. It thus corresponds to an element a� in GL(n, Q1) (the image of the Frobenius Fr). Since -r: is semisimple, a� is semisimple, and so the conjugacy class of a� is com pletely determined by the set of eigenvalues x = (xh · · · , Xn) E (Q1)". We associate to -r: the representation p = p, of GL(n, Kp), and we write p -r: . D This construction uses the explicit realization of unramified representations, but there exists a more canonical formulation. It is known (see [1]) that for every natural number m there exists a unique locally constant Q-valued function X m on GL(n, Kp) with compact support such that (a) Xm is bi-GL(n, o)-invariant, (b) for every x = (xh · · · , Xn) E (Q1) n : Tr PiX m) = xT + · · · + x':( . �
Now we can reformulate Theorem 2. THEOREM 2'·. The irreducible unramified representation p ofGL(n, Kp) corresponds to the unrami.fied n-dimensional representation -r: of WKp if and only if Trp(x m) = Tr -r:(Frm) Jor all natural numbers m > 0. Now consider the global case. Let K be a global field, char K = p. We may consider K as the field of rational functions on a smooth absolutely irreducible curve X over the finite field k. Let q = card k. We denote by n the set of all points of K. For every lJ E n we denote by KP the completion of K at p. We denote by A the adele ring of K and consider the Qrspace L of Qrvalued locally constant functions / on GL(n, K)\GL(n, A) which are cuspidal. That is to say for every K-subspace A c Kn , A =F 0, Kn , Su. ckl \u . CAl f(ux) du = 0 where UA is the subgroup of g E GL(n, Kn) such that gA = A and g acts trivially on A and KnfA. The group GL(n, A) acts by right shifts on L. The support of every element f E L is compact mod A * (the center of GL(n, A)) (see, e.g., Godement-Jacquet, Lecture Notes in Math., vol. 260, Springer, p. 142) and, consequently, the representation T of GL(n, A) on L is smooth.
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We will call an irreducible admissible representation of GL(n, A) a cuspidal automorphic representation if it occurs as a subrepresentation of L. We denote by GL(n, A)'i the set of these representations. Recall from, say, [5] that any irreducible admissible representation (p , V) of GL(n, A) may be uniquely expressed as a restricted tensor product ® p� (IJ E U) of local irreducible admissible representations. If S is a finite subset of ll such that p� is unramified for !J ¢; S, we say p is unramified outside S. The next statement (see [10]) is useful for the precise formulation of the Lang lands Conjecture. THEOREM 3. Let p , p ' be two cuspidal automorphic representations of GL(n, A) such that PP = p; for almost all !J E n. Then p = p '.
Now consider the Galois side. Let K be a separable algebraic closure of K, and let WK be the dense subgroup of Gal(K/K) consisting of elements which induce on k the map x -> xq" for some n E Z. For every p E O we have an imbedding WKb c... WK defined up to conjugation in WK (see [12]). DEFINITION. Let 1: be a finite dimensional Qrrepresentation of WK· We say that 1: is unramified at !J if the restriction 7:p of7: to WKp is unramified. REMARK I . It is known that for every such 1: there exists a finite set S c: ll such that 1: is unramified outside S. REMARK 2. If 1: is unramified at p, then 1:(Fr p) is well defined up to conjugation. THEOREM 4 (SEE [12]). Let 1: and 1:' be two irreducible representations of WK• and suppose that , for almost all !J E n, 7:(Fr p) is conjugate to 7:'(Fr !J). Then 7: is equivalent to 1:'. We can now formulate the Langlands Conjecture. There exists a one-one corre spondence 'P : p ie 1: between the set GL(n, A){' and the set ( WK)� which has thefollow ing property : For every cuspidal automorphic p there exists a finite set S c: ll such that for all !J E ll- S the representations p and 1: are unramified outside S and pp ,., 7:p (see Theorem 2).
REMARK 1 . Theorems 3 and 4 imply the uniqueness of such a correspondence (if it exists). REMARK 2. There is another conjecture, in which we ask for a one-one corre spondence between cuspidal automorphic representations and systems of 1-adic representations of WK · It follows from results of Deligne [3] and Drinfeld that this is true for n = 2. REMARK. We can deduce from the global conjecture the following local one. Let M be a local field of positive characteristic. There exists a one-one correspondence between super cuspidal representations PM of GL(n, M) and irreducible n-dimensional representations 7:M of WM · This correspondence is such that for every global field K, for every point !Jo E ll such that M � KPo and for every automorphic representation p = @ pp such that PPo is super cuspidal, the restriction 1:Po of the representation 1: ,., p of WK on WKpo � WM corresponds to PPo ' We can now ask ourselves a rather general question : how can we arrange a one one correspondence between representations of two different groups, say G and W?
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One way is for every representation p of G to define a function Fp on some set X and to do the same for every representation -r of W. Then we can say that p -r iff FP = Fr . If G = GL(2, A) or GL{3, A), W = WK• X is the set of characters of idele class group K * \A * and Fp and Fr are F-functions, this method works very well (see [7], [9]) and permits us for every system of 1-adic irreducible 2 or 3-dimensional representations -r of WK to find the corresponding representation p of GL{2, A) or GL{3, A). But so far this approach does not give us the possibility of going in the opposite direction and constructing a representation -r of WK from a representation p of GL(n, A). Another more explicit approach to producing a correspondence between repre sentations of two groups G and W was first used by H. Weyl [13] who considered the case G = GL(n , R), W = Sm = the symmetric group. He considered the repre sentation of G x W on the tensor power L = y®m of V � Rm . We can decompose L in a direct sum L = EEJL,. of G x W-invariant irreducible subspaces. As is well known, L,. is a tensor product L,. = p; ® -r; where p; are irreducible representations of GL(n , R) and -r,. are irreducible representations of Sm . He found that (for n ;s m) in this way we get a one�one correspondence between representations of Sm and part of the representations of GL(n, R). This means that (a) { -r;} = Sm , (b) if i, j are such that p; � pj then i = j. Drinfeld applies this method to the case when G = GL(2, A), W WK · The first idea would be to construct a Qrrepresentation (T, L) I of GL{2, A) x WK with the following properties. When we consider the decomposition of L = EEJL,. into a sum of G x W-invariant irreducible subspaces and write each L1 as p; ® -r1, p; E GL(l , A)a , 'r; E ( WK)z then (a) {p;} = GL(2, A)� ; (b) for all i, -r1 p,.. But, unfortunately, such a representation (T, L) does not exist. It might be useful to explain why not. To do this we must recall some definitions and results which are well known for representations of finite groups (see [11]) and can be easily restated and reproved for admissible representations. So let G be a group and (p, V) be an irreducible admissible Qrrepresentation of G. We denote by M(p) c Q 1 the field of definition of the equivalence class of p. That is, M(p) is the field which corresponds to the subgroup gp c Gal(Q tf Q1) consisting of elements G E Aut(Q 1 : Q 1) such that p" p. DEFINITION. We say that p is unobstructed if it can be realized over M(p). That means that there exist a vector space V0 over M(p) and a morphism p0 : G -> Aut V0 such that (p0, V0 ® M cpJ Q 1) is equivalent to (p, V). The following two statements are well known (see [11]) : LEMMA 1 . Suppose that (p, V) is a representation of G, H c G is a subgroup and dim V" I . Then p is unobstructed. �
=
�
�
=
LEMMA 2. Suppose that L is the space of a smooth Qrrepresentation of G x W, (p, V) E G, (-r, C) E W are such that Homw(C, L) � V as a G-module. Suppose that p is unobstructed. Then -r is also unobstructed. LEMMA 3. Every irreducible representation of GL(2, A) is unobstructed. 'Because (T, L) is uniquely determined up to isomorphism by its properties, it is natural to expect that it is defined over Q, and not only over Q1•
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PROOF. For every positive ideal a of K we denote by r. c GL(2, A) the subgroup of matrices with integer elements which preserves the vector (0, 1) mod a. It is known (see [10]) that there exists an ideal a such that dim vra = I. So our lemma follows from Lemma I . On the other hand, it is easy to construct a two-dimensional representation 1: of WK which is obstructed. For example, we can take 1: to be in duced from a one-dimensional representation of a subgroup of index two in WK· In this case we can (see [1]) construct p E GL(2, A)� such that p ,.., 1: . So we see that a representation (T, L) with properties (a) and (b) does not exist. What is to be done ? We know that for every irreducible representation 1: of any group H the representation 1: ® f (where f = contragredient of 1:) of H x H is unobstructed. So we can try to realize EB p ® 1: ® f (p E GL(2, A)'i, 1: ,.., p) . This is almost possible. Among other things, we must consider the direct integral instead of the direct sum. To formulate precisely the result of Drinfeld we need some more definitions. DEFINITION. Let (p, V) be a representation of GL(n, A). We say that p is a graded representation if we can write V as a direct sum V = EB tEZ V1 in such a way that for every g E GL(n, A) p(g) Vt
=
vt+v <det
REMARK. It is clear that for any graded representation (p, V) the operators p(f),
f E C�(GL(n, A)), are not of trace class and we cannot define the character Trp·
But sometimes we can define the regularized character
fe q>(GL(n, A)),
where P0 is the projection onto vo . It is not hard to check that the definition of 'frP is independent of the gradation v = EB Vt . Let (p, V) be a graded representation, and let x E A*. We denote by V" the quo tient of V by the subspace generated by { x(det g) v - p(g)v}, v E V, g E GL(n, A). DEFINITION. We say that (p, V) is completely reducible if, for all x E A*, V" is a direct sum of irreducible representations. The following result is easy. LEMMA. Let (p , V), (p', V') be two graded completely reducible representations such that 'frP and 'frp exist and 'frP = 'frp · Then p ,.., p'. '
'
Now we can formulate the "constructive" variant of Drinfeld's result. THEOREM 6. There exists a representation (p, V) ofGL(2, A)
that
x
WK
x
(a) the restriction of p to K* c Center GL(2, A) is trivial; (b) for every x E A*/K*, V" is a direct sum of irreducible representations v"
= EB Vx, i•
(c) GL(2, A)� = { Px .i}x,;; (d) for all (x, i), Px.i "' 't:x,i ·
Vx,i
=
Px.i ® 7:x,i ®f x . .- ;
WK
such
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A . KAZHDAN
Of course, the proof of this theorem consists of two parts : (A) construction of
(p , V) and (B) proof that (p , V) satisfies conditions (a)-(d).
We shall only briefly discuss both parts. First, we consider a geometric interpre tation of WK x WK. Let A be the field offunctions on X x X, A s ::::> A its separable closure, WA c Gal(As /A) a subgroup consisting of elements which induce the map x ---> x q", n E Z, on k. We have the natural map n- : WA ---> WK x WK, which is simply the restriction of any (J E WA c Aut(A s/A) to Ks ® Ks c A s· It is clear that Im n- consists of pairs ((}"', (}"") E WK x WK such that (J ' /k = (J "/k. Drinfeld defines an extension and an epimorphism it : WA ---> WK x WK. To do this consider the group WA of automorphisms r of the algebraic closure A of A such that the restriction of r to the perfect closure A:e�"C2JK ,., c A has the form Fr\"Frq where Frr, Fr2 are partial Frobeniuses on Kver ® Kper · We define W to be WAf {the subgroup generated by the Frobenius} . We have the natural imbedding WA ---> WA, and it gives the extension ( ** ). REMARK. Analogously, we may define groups WK and WK, but in this case we will have WK = WK. Our morphism n- extends to it : WA ---> WK x WK = WK x WK. THEOREM 7 (DRINFELD). (1)
it is an epimorphism.
(2) For every finite group H and every homomorphism q; : WA ---> H, we can write q; as a composition q; = if! o it, where if! is a homomorphism if! : WK x WK ---> H. The proof of this theorem is not difficult, but in order to present it we would have to introduce some new definitions, and this would take too long. So we leave the proof to the reader. Now we can try to imagine the possible construction of the representation (p, V). As we know, we can consider K as a field of functions on a smooth, projective, absolutely irreducible curve X over a finite field k. Let S = X x X. Suppose that we can define a projective system M;, i E I, of algebraic varieties over the generic point r; of S such that : (a) we have the action of GL(2, A) on the projective limit M = proj lim M; (we consider all M; as A-varieties), (b) we have liftings of partial Frobeniuses Frr, Fr2 (as endomorphisms of k(r;)) to the endomorphisms of M;, for all i E I, in such a way that Fr 1 o Fr2 lifts to the Frobenius on M;. Then we can construct a representation (pM, VM) of GL(2, A) x WK x WK · To do this fix any integer r and consider VM = inj lim H�(M;, Qt). 2 Then the group GL(2, A) x WA acts naturally on VM • and WA preserves the images of H�(M;, Q1) in VM for every i E I. It follows now from Theorem 7 that the action of GL(2, A) x WA can be factored through GL(2, A) x WK x W. Drinfeld ' s construction actually follows this scheme. But the varieties M; are not of finite type. To explain why this is so we have to remind ourselves what we want to realize. We want to obtain a representation (p, V) of GL(2, A) x WK x WK 'By H; we mean /-adic cohomology with compact support.
DRINFELD ' S
" SHTUKA "
353
such that its restriction to GL(2, A) will be four times the representation in the space L of cuspidal locally constant functions on GL(2, K)\GL(2, A). We see that in the very definition of L we describe it as a subspace in the bigger space i of all locally constant functions with compact support on GL(2, K)\GL(2, A). So we can expect that the only way to realize the representation (p, V) is to realize it as a subspace of (p, V), and that the restriction of (p, V) to GL(2, A) will be a multiple of i; we can try to realize V as proj lim H�(M;, Q1) and V as a subspace of V. But the representation i is "big". This means that for a compact open subgroup U c GL(2, A) and a character X E (A * /K * )f', the subspace (i,)u of U-invariant vectors in i, is infinite dimensional. This means (see below) that dim Hr(M�, Q1) = oo 3 and M� cannot be a variety of a finite type. After this long explanation of "why the construction cannot be very nice", we describe (but will not present) Drinfeld's construction. For every positive divisor D on X, Drinfeld defines an algebraic space Mv over r;, and for every D' ::::J D he defines a morphism MD' --> MD· This space is the union Mv = U;;o= -ooM?J of connected irreducible components, all the spaces M1; (resp. M�+l), n E Z, are isomorphic. They have the following structure. ( 1 ) Each space M?J is a union M?J = U Mif· n-K of surfaces of finite type Mfj· n-K, (2) Mfj+l, n-K-1 ::::J Mfj· n-K, and (3) the difference Mfj+l, n-K-1 - Mfj· n-K is a union of affine lines. Drinfeld also defines compactifications Mf5· n-K of each subspace Mfj· n-K, and shows that the birational isomorphism Mfj+l, n-K-1 "' Mif· n-K can be ex tended to a morphism Uf5· n : Mf5+1, n-K-1 --> Mfj· n-K such that (a) the composi tion of rc with the natural imbedding Mfj· n-K 4 Mfj+l, n-K-1 is the identity and (b) the restriction of rc to the boundary F/f· n = def Mfj+l, n-K-1 - Mfj+l . n-K-1 is a radical morphism. Define Mv = proj lim K Mif· n-K and M = proj limvMv. Drinfeld shows that the partial Frobeniuses and GL(2, A) act naturally on M. Now we can define "
V
=
_
H2(M, Q1)
def =
_
limD H2(Mf5· n-K, Q1) .
K, n,
To define V we consider the subspace V,
H2(M, Q1) of rational curves on M.
c
V which is generated by classes in
REMARK. It follows from (3) (and the existence of a GL(2, A)-action on M) that we have an infinite number of rational curves on each Mfj· n-K, so Mfj· n-K is an example of a surface which has an infinite number of rational curves on it without having a family. It is very possible that for large D, Mfj· n-K is a surface of generic type. It can be proved that the restriction to V, of the canonical cup-product on H2(M, Q1) is nondegenerate. Drinfeld defines V to be the orthogonal complement of V, in V (or you may consider V as the quotient space Vj V,). Of course, the partial Frobeniuses preserve V,, and as was explained before, we obtain a representation p of GL(2, A) x WK x WK on V. THEOREM 8 .
The representation (p, V) satisfies the conditions of Theorem 6.
We have not presented the actual construction of M. But suppose it given. How 'M; is a connected component of M,-.
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D . A . KAZHDAN
can we prove that it gives us the corresponding representation (p, V) which realizes the Reciprocity Law? The only known way to do this is based on the Trace Formula. We have the following general statement. Let (p, V) be a graded representation of GL(n, A) x WK x WK such that (a) For every Schwartz-Bruhat function on GL(n, A), 'fr(f) exists. (b) For any two places p', p" of K, any number m', m" E z+ and any S-B func tion/ on GL(n, A P' . P"), we have the equality 'fr(xg, ® xg. ® f) X Fr �' X Fr ;:n v = 'fr(Xt X xp,· X f)£, ( !) de f where X;J'(g) = X(:'(g- 1 ). v
Then the representation (p, V) realizes the Reciprocity Law. REMARK. By Fr� ' we denote the measure on WK which can be defined in the following way. First of all, consider the local group WKp' · It contains the inertia group /P ' ' and WKp)lp' """ Z. We denote by fl.':: the /P ,-invariant measure on WKp' which is concentrated on the preimage of m' and satisfies JwKp ' fl.�, = I . We have an imbedding WKp' 4 WK (which is defined up to conjugation), and we denote by Fr� ' the image of fl.�, under this imbedding. Given these remarks, the proof of this statement is rather standard, and we will not present it. To apply it to our case we must first of all define a graded structure on V and secondly explain how to compute both sides. The first part is easy. Our algebraic spaces Mn are disjoint unions of M'[;. The same is true for Mn and M = UMn . So we can write V = ffi V1, where VI = HZ(MZt U MZt+l, Q1) . When you have the definition of M, you will see that the condition p(g) V1 = yt+v (det g) , g E GL(2, A), is satisfied. To find the right side of Tr(X;'f ' X x;t X f), we can apply the Selberg Trace Formula [7] which, fortunately, is known for GL{2, A). To do this we need only (modulo rather complicated computations) to find "orbital integrals" of x;, that is, for very conjugacy class {) c GL(2, Kp) to find the integral J0X;'(w) dw. The answer is well known (see [8]). In fact, Drinfeld has solved the corresponding problem for GL(n, Kp). Let me describe his result, be cause it might be useful for people who try to prove the Reciprocity Law for GL(n). THEOREM 9. (1) If {) is a nonsemisimple conjugacy class in GL(n, Kp), then JoX;' dw = 0. Let {) be a semisimple class, r E D. Then Zc(r) = IT GL(n,., L,.), where L,. => KP are finite extensions of Kp, I; n,.[L,. : Kp] = n. So we can write r = IT r... r.. E Lf . (2) If J0X;' (w) dw =f. 0 then there exists j, such that n,.v(NLjtKp(ri)) = m , and v(NL, :Kp (r;) = 0 for all i =f. j. To define the J0 we have to fix the Haar mea sure on Q. Because {) = Zc (r) \ GL(n , Kp), it is sufficient to fix Haar measure dr on GL(r, L) for all r E Z*, L :::> KP a finite extension. We do this by the condition meas GL(r, oL) = I . (3) Let r E 07 be a semisimple element satisfying the conditions in (2). Then
J
Or
X;:'(w) dw
= ( 1 - q,.)( l - q'f)
· · · (1
- q7;-l)
[1,., k],
355
DRINFELD 'S "SHTUKA "
where I; is the residue field ofL; and q; = card !;.
The proof of this theorem is based on the explicit formula for X;' which Drinfeld found. He proved that (1) X;'(g) :f. 0 � g E GL(n, op) and lJ(det g) = m . (2) If g satisfies (1), then X;'(g) = ( 1 - q) (1 - q 2) . . . (1 - qr- 1 ), where n - r is the rank of the reduction of g mod 1C . 0 To finish, we have to explain how to compute the left side of ( !). Of course, it is sufficient to consider the case when f is the characteristic function of a set U0gU0 for g E G, D a divisor of X, U0 c GL{n, A) the congruence compact open sub group of integer adelic matrices equal to Id mod D. To do this, fix some N � 1 and let NM0 = U n M0r
H2(NMo,
Ql).
The variety N M0 is a surface over the generic point '1J of X x X. For almost all closed points (p', p") of X x X. it has good reduction R over (p', p"). The coho mology of R is isomorphic to H*(N M0, Q1), and we can consider the operator Cg. o = Fr�' ® Frp,:' ® Ag,o on H2(R, Q1). The left side of ( !) reduces to com puting 'fr Cg, Dp · To do this, one applies the Lefschetz Trace Formula (see [8]). It tells us that 1: ( - 1); Tr ( Cg,o i Hi(NM0, Q1) ) =
i=l
1:
xELJ nr g> D
n(x) ,
where L1 c R X R is the diagonal and {' g,O is the reduction of rg.O· First look at the left side. Using the existence of the two systems of rational curves on M, one proves that Hl(N M0, Q1) = H3(NM0, Q1) = 0. It is easy to find n o and fl4, so it rem�ins to compute the right side. To do this, Drinfeld describes the geometric points of the reduction of M over (p', p") as a Frp ' x FrP" x GL(2, A P ' . P")-module. 4 It gives us the description of L1 n rg,O· If R were smooth, we could conclude that for large m ' , m" , n(x) = 1 . Using the fact that R is quasi-smooth, he proves that this is asymptotically correct (for large m ' , m ") , and comparing with the Selberg Trace side, he deduces ( !). REFERENCES 1. P.
Cartier, Representations of a 'jl-adic group : A survey, these
PROCEEDINGS,
part 1 , pp. 1 1 1-
1 55. 2. P. Deligne, Formes modulaires et representations de GL(2), Lecture Notes in Math., vol. 349, Springer-Verlag, Berlin and New York. 3. --, Les constants des equations fonctionnelles des functions L, Lecture Notes in Math., vol. 349, Springer-Verlag, Berlin and New York. 4. --, Cohomologie etale, SGA 4t, Lecture Notes in Math., vol. 569, Springer-Verlag, Berlin and New York.
'A•· , ,.. are adeles without ll' and ll" components.
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5. D. Flath, Decomposition of representations into tensor products, these PROCEEDINGS, part I , pp. 1 79-1 83. 6. I . M. Gelfand, M. I. Graev and I . I. Piatetski-Shapiro, Representation theory and automorphic functions, Saunders, 1 969. 7. H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Math., vol . 1 14, Springer-Verlag, Berlin and New York. 8. R. P. Langlands, Modular forms and 1-adic representations, Lecture Notes in Math., vol . 349, Springer-Verlag, Berlin and New York. 9. I. I. Piatetski-Shapiro, Converse theorem for GL(3), Mimeo Notes No. 1 5, Univ. of Maryland, 1975. 10. Multiplicity one theorems, these PROCEEDINGS, part I , pp. 209-212. 1 1 . J.-P. Serre, Representation lineaires des groupesfinis, Hermann, Paris, 1967. 12. J. Tate, Number theoretic background, these PROCEEDINGS, part 2, pp. 3-26. 13. H. Weyl, The classical groups, their invariants and representations, Princeton Univ. Press , Princeton, 1 939. --- ,
HARVARD UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 2, pp. 357-379
GL2
AUTOMORPHIC FORMS ON
OVER
FUNCTION FIELDS (AFTER v. G. DRINFELD)
G.
HARDER AND D. A. KAZHDAN
Introduction. This is a report on unpublished work of V. G. Drinfeld on automor phic forms over function fields. Our aim is to give a description of the so-called scheme of 'shtuka', to discuss the computation of the trace of Frobenius on the cohomology of this scheme in terms of the Selberg trace formula and the proof of the Ramanujan conjecture. We will not discuss the much more subtle question con cerning the reciprocity law which involves the compactification of the above moduli scheme. We shall not give detailed proofs, but we shall try to give references and hints so that the reader may fill the gaps. 1. The moduli space of FH-sheaves. 1 . 0. Let X/Fq be a smooth projective curve. Let X = X x F q Fq ; Drinfeld introduces some objects over X which he calls 'shtuka'. These objects are vector bundles over X together with some additional structure. These additional structure data involve the Frobenius and some kind of 'Heeke modifications'. Therefore we sug gest to call these objects FH-sheaves. 1 . 1 . Vector bundles. Let X/Fq be our given curve, let K/Fq be its field of mero morphic functions. We denote X= X
X pq
Fq ,
Koo = K · Fq•
The space of adeles of K (resp. Koo) is denoted by A (resp. A00). The geometric points of X/Fq are denoted by v, w e X(Fg). For any v e X(Fq) we denote the completion of Koo with respect to v by K. and its ring of integers by (!} •. The valuations of the field K/Fq will be denoted by p, · · · . Then KP will be the com pletion of K at p and (!}P its ring on integers. If v e X(Fq) induces p on K we write v � p. The points v which lie over a fixed p form an orbit under the action of the Frobenius on X(Fq). A vector bundle of rank d over X is a locally free sheaf of rank d over the struc ture sheaf {!}g. We have an alternate way to describe vector bundles. Let us take a vector space V/Koo together with a basis eh · · · , ed. Let us assume that we have a family M = {M.} vEX (Fq) of (!}. }attices M. c v ® K., s.t. d
M. = EB (!} .e; for almost all i=l
v.
AMS (MOS) subject classifications (1 970). Primary 1 00 1 5 , 14F05, 14H10. © 1 979, American Mathematical Society
357
358
G . HARDER AND D . A . KAZHDAN
We can associate a locally free sheaf E/X to this family of lattices by defining r( V, E) = { x e V I x e M. for all v e V}
where [] c X is an open subset. On the other hand it is very easy to see that every vector bundle can be realized in this form. Let us take for M0 the trivial family Mo,v = (!).e1 Ei3 . . . EB (!)ved for all v e X(Fq). Then we may use an adele x e GLiA�) to produce a new family xM0 = {x.M0 • •}. and it is an easy exercise to check that we have a bijection GLd(K�)\GLd(A oo)/ff� __::.. {set of isomorphism classes of vector bundles of rank d on X} where %� = TI.GLi(!).). 1 .2. Level structures on vector bundles. Let D be a positive divisor on X/Fq , i.e., a divisor which is rational over Fq . We define as usual (!)g ( - D) = sheaf of germs of regular functions f with div(f) � D = sheaf of ideals defining D, and for any vector bundle we define E( - D) = E ® (!)g ( - D). A level structure (along D) on E is an (!)g isomorphism 1/f : E/E( - D) ...::.. ((!)gj(!)g( - D) d) . The isomorphism classes of vector bundles with a level structure along D are given by the elements in the coset space GLiK�)\GLiA�)/Jt"�(D) where Jt"oo(D) is the open congruence subgroup of Jt"� defined by x = I mod D. We are particularly interested in the two cases d = I and d = 2. A vector bundle Ef X of rank 2 with level structure along D is called stable if for any line subbundle L c E we have deg L
<
t(deg E + deg D).
It follows from Mumford's theory that we have a fine moduli scheme .H0 --> Spec(Fq) for the functor of stable bundles with level structure along D. (Compare [6].) If for any v e Z we denote by .Ht;'l the stable vector bundles with deg det E = v then the .Hf5l are smooth quasi-projective schemes over Fq . 1 . 3. Modifications of vector bundles for d = 2. Let E/X be a vector bundle, let v e X(F q). We want to define the notion of a modification of E at v. The fiber of our vector bundle E!X at v is a vector space E. = E ® ((!)vf.H.) = E ® Fq (v) over F q . A point a. e P 1 (E.)(Fq) defines a nonzero linear form a� :E.--> Fq which is unique up to a scalar. We define E(a.) = {sheaf of germs of sections s of E for which a:(s.) a.(s.) = 0} . This give us a new vector bundle on X and we have an exact sequence 0 ---> E(a.) � E ---> Fq ( v) ---> 0, =
i.e., E/E (a.) is a sheaf concentrated on v and the fiber over v is a one dimensional vector space over the residue field Fq = Fq(v). We call E(a.) a lower modification of E at v (in direction a.). We want to give an interpretation of this in terms of lattices. Let us assume that we have a family of lattices M = xM0 with x e GL2(A00). Let us pick an element a. e GL2((!).)
(�· �) GL2 ((!).)
where 1Cv is a uniformizing parameter. Then we put a.
=
(
. . · ,
I , . . · , a., . . . , I . . . ) and
GL2 OVER the bundle defined by M(av) If we observe that the set
=
FUNC TION FIELDS
359
xav · M0 defines obviously a modification of M.
divided by GL2((!7v) on the right is exactly our P1(Ev)(Fq) we see that we have an interpretation of the notion of modification in terms of families of lattices. If w E X(Fq) is another point we define upper modifications at w : For aw E P1(Ew)(Fq) we put E(iiw) = E(aw) ® {!7g(w). In this case we have E 4 E(iiw) -+ Fq(w) and again we have an interpretation in terms of lattices M = xM0 -+ xawMo where iiw = (- · · 1 · · · , iiw, · · · 1 · · · ) and
If v # w are two geometric points and av E P1(Ev)(Fq), aw E P1(Ew)(Fq) then we define E (av, iiw) = (E(av))(iiw). We observe that deg E(av, iiw) = deg E. Another important observation is that we have a canonical identification E(av, iiw) ix-{v)-{ w ) = E i x-{ vi-{w) · This is clear from the definition of E(av, iiw) in terms of locally free sheaves. There fore we know : If D is a positive divisor on X/ Fq and if v, w rt supp(D) then a level structure ¢ on E along D induces a level structure on E(av, iiw) which will be de noted by the same letter ¢. It is very easy to see that the data of such a double modification E -+ E (av, iiw) = E' is equivalent to give a diagram Iff
locally free of rank 2
where deg Iff = deg E + 1 = deg E' + I and where t:j¢.(E) is concentrated at w and