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(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 ksubgroup 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 onedimensional 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 padic 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 padic 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 righthand sides we have the L and efactors 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 righthand 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 Egroup 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 Lseries 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 twodimensional 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 efactors 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. PiatetskiiShapiro, 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 2ndimensional 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 kgroup. 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 > 0 such that j t. j Re A � (Nv) a for all v E vl and it E P,. (5) Let G' be a quasisplit inner kform 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 quasisplit over k. We then fix a Borel ksubgroup B of G and view LT as the £group of a maximal ktorus T of G. For a cyclic subgroup D of r,,,,, let Vv be the set of v E vl for which r,. is equal to the inverse image of D in F,. The group U = X* ( T) 0 is then the group of one parameter subgroups of a subtorus S of T such that S/k. is a maximal k.split torus of Gfk. for all v e V0• The group (6) Y = Hom(U, C * ) = Hom(X* (T)D, C * ) (v e V0), is independent of v, and is the Y of §6 for Gfk •. The root datum cp(Gfk.), which is determined by the action of D, is also independent of v e V0.
AUTOMORPHIC £FUNCTIONS
51
Given y e Y, let y0 be a "logarithm" o f y, i.e., an element o f Hom(X* (T) D, C) such that (7) y(u) = NvYo (u) = Nv , for u e XiT) D . This element is determined modulo a lattice, but its real part Re y0 e Hom(U, R), defined by y(u)
(8)
= Nv
is well defined. If y has values in R+, then we choose y0 to be equal to its real part. The space a * is the dual of a = U ® R (the socalled 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 kvroots, their multiplicities, and the indices qa of the BruhatTits 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 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 rightshifts 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 rightinvariant 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 quasicharacter ((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 quasicharacter. (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•+Cn1) 12H(kg, k) dx g = J dk J f/J(k 1 g) j det g j•+en1 ) 12 H(g, k) dxg K Gn = J dk' J f/J(k 1 pk')j det pj •+Cn1) 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(klpk') ® 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 Kfiniteness 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 Kfinite functions r; and r; ' on K such that
JJ f/J(klxk')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(klpk')i det p i •+Cn1) 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(klhlph' k')i det P i s+Cn1) 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[k1pk']Hl (p, k, k')i det p i •+Cn1) 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, mz1, · · · m;1 ; k', k) = ii(pk, k')op112(p). ,
Similarly (2.4.2) gives (2.6.2)
lf!"(mh m2,
···,
m, ; k', k)
=
J f/J1''(k1pk') ® 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(k1pk') J detpJ s+(n1l 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(qs)L(s, n L(s, it) = P(qs)L(s, �), P(O) = P(O) = 1 . Note that P(qs) divides L(s, �) 1 in C[qs]. Moreover 1C) L(s, �)P(qs) (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(qs)s . P(ql+ ) _
_
Since the efactors are units of the ring C[qs , 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) Squareintegrable 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 squareintegrable. Recall that 71: is squareintegrable if it admits a central character w and its coefficients (which transform under w) are squareintegrable modulo the center. A representation 71: is essentially squareintegrable 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 squareintegrable 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 squareintegrable then the poles of L(s, 71:) are in the halfplane 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 squareintegrable 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 squareintegrable, 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 halfplane 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 padic 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" nrb . . · , 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(qs) IT L(s, ..,.) , L(s, if:) = P(qs) 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(qs) 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•Hnl)/2 �(kl ulg)H(g, k) dxg, G. where u is integrated over (] = Up. This is also
JJ
Js
J
JJ l det , 1 det �; �J1 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 •+ 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 ndimensional 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 quasicharacter, 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 halfplane 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( ± 2inx), if F = R, ¢(z) = exp[ ± 2in(z + z)], if F = C, then (/)!\ is still in Y0 ; the lefthand 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 quasicharacter 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 quasicharacter of 70 ; if we differentiate with respect to t the identity (4.3.3)
Z((/), s + ! (n  1),/) =
and then set t
J (/)(xe1)f(x) idet xi •+J, 1  s +
= e(s, �. cp)  1 L( The classical formula
F(x + iy) r(x'  iy)
�
I Y I "'x'
,.  (/)b s + t(n  1) , / ) = 0. (4.5.5) i, j+oo
79
PRINCIPAL LFUNCTIONS
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) = n0). (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) Squareintegrable representations. Suppose A is irreducible. Then n{A) is essentially squareintegrable and conversely. More precisely, if n = 1 then A is a quasicharacter 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 halfplane 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 tfactor from the rfactor. 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(
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 •+Cn11l 12 j det m2 j •+C"21l 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[klzk']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'/[h2m2h21] �(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+ (nJ1)/2 j det m2 j s+ Cnz1)/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
E3
(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 m211s+ (nzl) /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 Afield. (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 quasicharacter 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 halfspace. 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 os. 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 PhragmenLindelof principle. BIBLIOGRAPHY [A.A.] A. Andrianov, Zeta functions of simple algebras with non abelian characters, Uspehi Mat. Nauk 23 (1968), 166 = Russian Math. Surveys 23 (4) (1968), 165. [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), 570 = Russian Math. Surveys 31 (3) (1976), 168. [1.8.A.Z. 2]  , Induced representations of the group GL(n) over a padic field, Funkcional Anal. i Prilozen 10 (3) (1 976), 7475 = Functional Anal. Appl. 10 (3) (1976), 225227. [1.8.A.Z. 3]  , Induced representations of reductive padic groups. II, preprint. [M.E.] M. Eichler, Allgemeine Kongruenzklasseneinteilungen der /deale einfacher A lgebren uber algebraischen Zahlkiirpern und ihrer LReihen, J. Reine Angew. Math. 179 (1 938), 22725 1 . [G.F. 1 ] G . Fujisaki, On the zetafunctions of a simple algebra over the field of rational numbers. Fac. Sci. Univ. Tokyo, Sect. 1 7 (1 958), 567604. [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), 2933 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, SpringerVerlag, 1972. [E.H.) E. Heeke, Uber Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produkten twicklung. I, II, Math. Ann. 114 (1937), 128 ; 3 1 635 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. 381386. [H.J. 2] , Generic representations, NonCommutative Harmonic Analysis, Lecture Notes in Math., vol. 587, SpringerVerlag, Berlin, 1 977, pp. 91101 . [H.J.R.L.) H. Jacquet and R. Langlands, Automorphic forms on GL{2), Lecture Notes in Math., vol. 1 14, SpringerVerlag, 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), 374408. [A.K.G.Z.] A. W. Knapp and G. Zuckerman, Classification theorems for representations of semisimple Lie groups, NonCommutative Harmonic Analysis, Lecture Notes in Math., vol. 587, SpringerVerlag, Berlin, 1977, pp. 1 381 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 nichtanalytischen 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), 132. [G.M. 1) G. N. Maloletkin, Zeta functions of a semisimple algebra over the field of rational numbers, Dokl. Akad. Nauk SSSR 201 (3) (1971), 535537 Soviet Math. Dokl. 12 (6) (1971), 1 6781681 . [G.M. 2) , Zeta /unctions of parabolic forms, Mat. Sb. 86 (1 28) (1971), 622643 = Math. USSRSb. IS (4) (1971), 61 9641. [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), 4787. [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), 461490. [T.T.] T. Tamagawa, On the zeta functions of a division algebra, Ann. of Math. (2) 77 (1963), 387405. [J.T. l) J. Tate, Fourier analysis in number fields and Heeke's zetafunctions, Algebraic Number Theory, edited by J.W.S. Cassels and A. Frohlich, Thompson, 1967, pp. 305347. [J.T. 2) , Number theoretic background, these PROCEEDINGS, part 2, pp. 326. [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), 681691 . [A.W. 1) A. Weil, Fonctions zeta et distributions, Sem. Bourbaki, no. 3 1 2, (196566), 9 pp. [A.W. 2) , Sur certains groupes d'operateurs unitaires, Acta Math. 111 (1 964), 14421 1 . [A.W. 3) , Basic number theory, SpringerVerlag, Berlin, 1967. 

=


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HERVE JACQUET
[A.W. 4] A. Weil, Dirichlet series and automorphic forms, Lecture Notes in Math., vol. 1 89, SpringerVerlag, Berlin, 1 97 1 . [E.W.] E . Witt, RiemannRochscher Satz und �Funktion im Hyperkomplexen, Math. Ann . 110 (1 934), 1 228. [A.Z. 1] A. V. Zelevinskii, Classification of irreducible noncuspidal representations of the group GL(n) over padic 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 padic field, preprint. (Russian) =
 ,
COLUMBIA UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 2, pp. 8795
AUTOMORPHIC LFUNCTIONS 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 Lfunction, 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. PiatetskiShapiro [2]. Here we develop the ideas of [2] for the symplectic group G = G Sp(4) and prove Langlands' conjecture for the standard 4dimensional 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. PiatetskiShapiro's paper Multiplicity one theorems, these PRoAMS (MOS) subject classifications (1 970). Primary lOD lO, 22E55 .
* Supported by NSF Grant MCS76021 60Al . © 1979, American Mathematical Society
87
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MARK E. NOVODVORSKY
CEEDINGS, part 1 , pp. 20921 2) and the standard 8dimensional 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 2dimensional 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 Lfunctions 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 PiatetskiShapiro). 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. PiatetskiShapiro 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 Gpinvariant 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 halfplane 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 halfplane Re s � ; it coincides with the Euler product of Langlands
90
MARK E. NOVODVORSKY
corresponding to the standard 4dimensional 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. PiatetskiShapiro. 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 2dimensional semisimple kalgebra K and a 2dimensional bilinear skewsym 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 Klinear 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 3dimensional commutative horocycle subgroup in G. The algebra K has a unique nontrivial kautomorphism
(2.2) it induces a kautomorphism 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 quasicharacter �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 ksubgroup of G, cf; is a quasicharacter 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 2dimensional 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 2dimensional representations ; such reformulation might be preferable logically but leads to more bulky constructions for Euler factors in case of 2dimen 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. PiatetskiShapiro 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 halfplane Re s � and defines fp( W) as a rational function in qp•. 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 halfplane Re s � , it coincides with Langlands' Euler product corresponding to the standard 4dimensional 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. PiatetskiShapiro. ·
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 infinitedimensional 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 ® wp1 (det) being the contragradient representation to fcP • consists of the functions W*(g)
(3 .4)
W(g) · wp1(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 quasicharacter 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
halfplane Re s � and define .f( W, W, �) and .f( W *, W*, fi>) as rational/unctions in qp•. 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 qs 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 halfplane 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 2dimensional 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, SpringerVerlag, Berlin and New York, 1 970. 2. I. I. PiatetskiShapiro, Euler subgroups, Proc. Summer School BolyaJanos 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 padiques deployes, C. R. Acad. Sci. Paris Ser. A 275 (1 972), pp. 10451048. 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. 42943 1 . 6. H . Jacquet and R . P . Langlands, Automorphic forms on GL(2) , Lecture Notes i n Math., vol . 1 1 4, SpringerVerlag, 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. PiatetskiiShapiro, Generalized Bessel models for the symplectic group ofrank 2, Mat. Sb. 90 (2) (1 973), 246256. 9. M. E. Novodvorsky, On the theorems of uniqueness of generalized Bessel models, Mat. Sb. 90 (2) (1973), 275287. 10. , Theorie de Heeke pour les groupes orthogonaux, C. R. Acad. Sci. Paris Ser. A 285 (1977), 9394. 11. , Fonctions / pour des groupes orthogonaux, C. R. Acad. Sci. Paris Ser. A 280 (1975), 14211422. 12. , Fonction / pour G Sp(4), C. R. Acad. Sci, Paris Ser. A 280 (1 975), 1 91192. 


PURDUE
UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 2, pp. 971 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 Lfunctions. 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 padic 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 padic 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
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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 atwistedly 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 atwistedly conjugate to x in GF. The set of atwisted 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"1z · · · 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 twodimensional ktorus. We call a conjugate class in Gk regular if it consists of regular elements. A atwisted 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 atwisted 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 padic 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') = (zz', 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 atwistedly 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 gfixed 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 atwisted 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 kalgebra) 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 quasicharacters 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 quasicharacter 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 quasicharacter 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 quasicharacter w of K x, denote by w the qu isicharacter 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 JJadic 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(gxg1) 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 f0 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 Cvalued 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 vcomponent 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• = 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 padic 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 kalgebra. 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 kalgebra 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
� 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(gaxg) = gl(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(gl 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) : g17og = 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 aconjugate 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(gu 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(gI 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 complexvalued compactly supported functions on G(F) which are hi invariant under Kp. Let YfE = Yf(G(E), KE)· Saito [2] introduced a Calgebra 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 Calgebra 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 Calgebra homomorphism from the algebra of regular functions on DE to the algebra of regular functions on DF, and hence also a Calgebra 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. 971 10). RICHLAND, WASHINGTON
Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 2, pp. 1 1 5133
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. §§ 15 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 twodimensional 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], [S1], [S2], [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 [S2] 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 pfield, W� is the WeilDeligne 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 efactors. 2. Base change for GL( l ) . From abelian classfield 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 onedimensional representations of WF are given by the quasicharacters of Wflb = CF ; by composing a quasicharacter 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 quasicharacters of CF in the set d(E) of quasi characters of CE · The group r acts on the set of quasicharacters 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 quasicharacters 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 Lgroups. 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 quasisplit, 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 efactors 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 twodimensional 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 Finvariants in l/J(GE 1 F). (b) In the following cases, the lifting is given by
(fl 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 fl• the representations which have the same lifting are the il' EB 1l'' 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 Finvariant 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 ainvariant distribution on G(E), i.e., invariant under aconjugation : z �> yt�zy, y, z E G(E). Let us state some properties of the aconjugation. 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 aconjugacy 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.llll' and vI v' in F; (d) w"EIF = (w")E I F • (11: ® X)E I F = 11:E I F ® XEI F for any onedimensional 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 quasicharacters), (twisting by a quasicharacter), (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 twodimensional 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 D4the socalled mattress groupis 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 onedimensional 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 Lfunction : s ..... L(s, p). According to JacquetLanglands [JL, Theorem 1 1 . 1, p. 3 50], the Lfunction 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 Lfunctions. From [JL, pp. 404407], 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 twodimensional 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 threedimensional 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 threedimensional representation is induced by a onedimensional 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 onedimensional represen tation () of the stabilizer of one of these lines (obtained by restriction of Ap) ; but this stabilizer is the pullback of the dihedral group D4 c �4 in WF, which is the subgroup WE : Ap = Ind �� (}, From [JPSS), to such a threedimensional 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 [GJ2]. 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 [JS] 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 Lfunction 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 [JS], 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 Lfactors are then equal. If v does not split in E and is unramified for E and p, the two local Lfunctions are those associated to the ninedimensional 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 Lfactor. 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 DeligneSerre [DS], 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 twodimensional 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 twodimensional 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 WeilDeligne 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 twodimensional 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 onedimensional representations of WF defined by f.1. and v ; f.1. (£) v � v (£) p ; det(p (£) v) = pv ; (ft (£) v) ® X (f!.X) EEl (vx) ; (p E9 v)v = f..l. 1 E9 v  1 The L and efunctions 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) . ( ft v EEl v.) (if F is global).
(if Fis global),
(2) Dihedral type : r = Ind�f (), where () is a quasicharacter 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 quasicharacter 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 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 ufixed 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 Lfunctions 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 . 2530]). 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. 3334] : 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. 2761 . [C] W. Casselman. On some results of Atkin and Lehner, Math. Ann. 201 (1973), 301334. [D) P. Deligne, Formes modulaires et representations de GL(2), Modular Functions of One
Variable. II, Lecture Notes in Math. , vol. 349, SpringerVerlag, Berlin, 1 973. [DS] P. Deligne and J.P. Serre, Formes modulaires de poids 1 , Ann. Sci. Ecole Norm. Sup. 4 (1 974), 507530. [GJ1) S. Gelbart and H. Jacquet, Forms on GL(2) from the analytic point of view, these P Ro CEEDINGS, part 1 , pp. 21 325 1 . [GJ2] , A relation between automorphic forms o n GL(2) and GL(3), Proc. Nat. Acad. Sci. U. S. A. (1 976) . [JL] H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Math., vol. 1 14, Springer, Berlin, 1 970. [JPSS] H. Jacquet, I. I. PiatetskiiShapiro and J. Shalika, Construction of cusp forms on GL(3), Lecture Notes, Maryland. [JS] 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 11 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. [S1] T. Shintani, On liftings ofholomorphic automorphic forms, U.S.Japan Seminar on Number Theory (Ann. Arbor, 1 975). [S2] , Two remarks on irreducible characters offinite genera/ linear groups. J. Math. Soc. Japan 28 (1 976), 396414. [S3] , On liftings ofholomorphic cusp forms, these PROCEEDINGS, part 2, pp. 971 10. [T] J. Tate, Number theoretic background, these PROCEEDINGS, part 2, pp. 326. [Tu] J. Tunnell, On the local Langlands conjecture for GL(2), Invent . Math . 46 (1 978), 1 791 98.



UNIVERSITE PARIS
VII
Proceedings of Symposia in Pure Mathematics Vol . 33 (1979), part 2, pp. 1 351 38
REPORT ON THE LOCAL LANGLANDS CONJECTURE FOR
GL(2)
J. TUNNELL Let ([J(GL(2)/K) be the set of isomorphism classes of twodimensional Fsemi simple representations of the WeilDeligne 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 quasicharacters 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 twodimensional Fsemisimple representation of W� is determined by its twisted L and efactors and determinant. It is straightforward to show directly that redu cible twodimensional 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) twodimensional Fsemisimple 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 twodimensional 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
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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 onedimensional 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 twodimensional 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 Lfunctions L(s, p ® X) are holomorphic and bounded in vertical strips asfunctions of the complex variable s for all quasicharacters 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 (LanglandsJacquetGelbart), and when F = Q, p is of S4 type and the image of complex conjugation has determinant  1 (LanglandsSerreDeligne). 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 twodimensional 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 quasicharacter 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 1adic 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 twodimensional 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 quasicharacter) 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 efactors, 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 twodimensional 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 squareintegrable 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 efactors for irreducible twodimen 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 efactors 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. 551 06. 2. , Review of" Elliptic Amodules" by V. Drinfeld, Math. Reviews 52�1 976), #5580. 3. I. M. Gelfand, M. I. Graev and I. I . PiatetskiiShapiro, 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), 4360. 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 2311 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 padics, Springer Lecture Notes, No. 1 66, 1 970. 10. J. Tunnell, On the local Langlands Conjecture for GL(2), Thesis, Harvard University (1 977). 11. A. Weil, Exercises dyadiques, Invent. Math. 27 (1 974), 1 22. 
=

PRINCETON UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 2, pp. 141163
THE HASSEWElL �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 assumptionsas Deligne and Langlands may explainone 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 HasseWeil �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 HasseWeil �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 exceptionsome unpublished work of Shimuraall this work is concerned with X of dimension one. (Referfor a samplingto [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 HasseWeil �function concerned, which turned outexcepting again Shimura's exampleto 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
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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 0divisors. 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 allowi.e., from now on we assume B to be a division algebra. Furthermore, we will assume B totally indefinite at the real primes ofFi.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� 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 HasseWeil 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 BruhatTits 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 (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> (gIxg) dg (b) If x is elliptic
=
=
7J m
( ) e ) mod
1 . if X = I or 0 otherwise.
7Jm
(K n A),
1 59
THE HASSEWElL �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(glag) dg,
trace 11:x , 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 squareintegrable irreducible representations of GL2 : trace 11:(/<m>)
=
JA D(a)ch (a) da JA\G pm> (glag) dg + 1 L; J D(t)ch"(t) dt J J <m> (gltg) 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>(gltg) 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> (glag) dg = trace a(h <m>) \G whenever a and 11: correspond. For this : (1) no 11: occurring here possesses Kfixed 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 lefthand side vanishes. But in this case a is not onedimensional, hence not trivial on o}j, so that the righthand 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>(g1ag) 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 HASSEWElL �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  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 Zmodule 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 Rbasis and so V ®z R ""' Cg . Then Cg/ V is a compact complexanalytic manifold which becomes a commutative Lie group under addition. When g = 1 the Weierstrass pfunction 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 elinear 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 elinear in the first variable and Rlinear in the second. Set E(z, w) = L(z, w)  L(w, z). Then (a) E is Rvalued, Rbilinear, and alternating; (b) E takes integer values on V x V; (c) the form (z, w) ....... E(iz, w) is symmetric and positive. {The symmetry is equivalent to having E(iz, iw) = E(z, w) for all z, w ; the positivity means E(iz, z) � 0 for all z.) A form satisfying these conditions is called a Riemann form for V and it is known that there exist enough functions to define a projective embedding of eKj V if and only if there exists a Riemann form for V which is nondegenerate (and hence such that E(iz, z) is positive definite). If g = 1 we may always define E(z, w) to be the ratio of the oriented area of the parallelogram with sides Ow, Oz to that of a funda mental parallelogram for the lattice. Since this form always exists, and is unique up to multiplication by an integer, one rarely bothers to mention it. By contrast, if g > 1 , a nondegenerate Riemann form will not usually exist and when it does, it will not be unique up to multiplication by an integer. However since eKj V is com pact the algebraic structure on eKJ V (but not the projective embedding) defined by a Riemann form is independent of the form. Thus, given a lattice in eg for which there exists a nondegenerate Riemann form, we obtain an abelian variety. Conversely, from an abelian variety A of dimension g we can recover a complex vector space W of dimension g and a lattice V in W for which there exists a nondegenerate Riemann form. W can be described (accord ing to taste) as the Lie algebra Lie(A) of A, the tangent space tA to A at its zero element, or as the universal covering space of the topological manifold A( e). The lattice V can be described as the kernel of the exponential exp : Lie(A) � A(e), or as the fundamental group of A(e) which, being commutative, is equal to H1 (A, Z). We shall always regard the isomorphism WJ V � A(e) as arising from the exact sequence,
Since H1 (A, Z) is a lattice in tA; we have H1 (A, R) = H1 (A, Z) ® R � tA . Thus A is determined by H1 (A, Z) and the complex vector space structure on H1 (A, R). A complex structure on a real vector space V(R) defines a homomorphism h : ex � AutR( V(R}), h(z) = (v ....... zv), and the complex structure is determined by h. Thus an abelian variety A is uniquely determined by the pair (H1 (A, Z), h) where h : ex � Aut(H1 (A, R)) is defined by the complex structure on H1 (A, R) = tA . Moreover every pair {V(Z), h) for which there exists a Riemann form arises from an abelian variety. Let V(Z) = H1 (A, Z). A point of finite order on A corresponds to an element of V(R) some multiple of which is in V(Z). More precisely, the group of points of
1 68
J. S. MILNE
finite order on A may be identified with V(Q)/ V(Z) c V(R)/ V(Z). For any integer the group A m(C) of points of order m is equal to mI 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 Clinear 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 Zmodules a : V(Z) + V'(Z) extend ing to a Clinear 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 finitedimensional 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 V(Zp) is defined only over the ground ring with p inverted, and Fp [p 1 ] is the zero ring.) The 08 ® Ppmodule 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 ® Fpmodules 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 08isogeny 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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1 77
with m a positive integer prime to p. This definition is compatible with that over C in the sense that both mappings ff(g) come by base change from a mapping ff(g) : K ' S x spec Zc p l � K S x spec Zc p l where Zc p l = { m/n E Ql(n, p) = 1 } . (Mo �e concretely, this means that if (A, i, �) in characteristic zero specializes to (A, l, (/}) in characteristic p then ff(g)(A, i, �) specializes to ff(g)(A , l, (j).) 5. The isogeny classes. Fix a prime p not dividing L1 and consider pairs (A, i) where A is an abelian variety of dimension 2d over Fp and i is a homomorphism B 4 End0(A) such that i(l) = 1 . We write A  A' if A and A' are isogenous, and (A, i)  (A', i') if the pairs are Bisogenous in an obvious sense . .FP denotes the set of all Bisogeny 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 Fequivalence class of pairings A n x A n � Gm for all n (cf. [17, §23]). In turn these pairings give an equivalence class of skewsymmetric 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) + (ar, a� · · · ) . ) THEOREM 5. 1 . (a) Let A be a simple abelian variety over Fq and let E = End0(A). Then E is a division algebra with centre Q[1r:], 1r: is an algebraic integer with absolute value q l 1 2 under any embedding Q[1r: ] 4 C, and for any prime v of Q[1r:] the invariant of E at v is given by
inv.(E)
=
1
=
0
_
if v is real, if v j l, I :I p,
ordv(1r:)  ord.(q) [Q [1r:] • .. Qp] if v I p. Moreover 2 dim(A) = [Q[1r:] : Q] [E : Q[1r:1 ] 1 1 2 and e = [E : Q[7r:]] I I2 is the least common denominator of the inv .(E). The characteristic polynomial PA (T) of 1r: : A � A is m(T)• where m(T) is the minimal polynomial of 1r: over Q. (b) The simple abelian varieties A and A' over Fq are isogenous if and only if there is an isomorphism Q[1r:A] � Q[1r:A '] such that 1r:A >+ 7r:A ' · (c) Every algebraic integer 1r: which has absolute value q l i Z under any embedding Q[1r:] 4 C arises as the Frobenius endomorphism of a simple abelian variety A" over Fq . (d) For any abelian varieties A and B over Fq and any prime I (including I = p) the canonical map
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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 SkolemNoether 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) = ara1, differs from i' : B + End0(A') by an inner automorphism (r ...... f3 r f3  1 ) of End0(A'). Thus {3a is a Bisogeny 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 pdivisible 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 pdivisible 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 Fautomorphism 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 08isogeny A' � A determines an injective map T1a : T1A' � T1A whose image A satisfies the following conditions : (a) A is 08stable. (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 Kequivalence 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 08stable lattices A in ( T1A) ® Q equipped with a Kequivalence 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 D8isogenies a
A' ==:' A and an DBisomorphism 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 ' a1 makes sense as an element of End0(A) and a ' a1 (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 oneone 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, DBstable, and
such that M/FM satisfies (4. 1).
PROOF. p: A + A induces maps in : AjpnA 4 Afpn+lA which define a pdivisible 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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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 ( 19771978).) 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 HondaTate 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. 1 334. 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. 141163. 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. 1433 1 6. 6. M. Demazure, Lectures on pdivisible 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), 8395. 
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8. Y. Ihara, The congruence monodromy problems, J. Math. Soc. Japan 20 (1 968), 107121. 9. On congruence monodromy problems, Lecture Notes, vols. 1 , 2, Univ. Tokyo, 1 968, �� ,
1969. 10. Nonabelian class fields over function fields in special cases, Actes Congr. Internat. Math., vol. 1, Nice, 1 970, pp. 381389. 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 974Sept. 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 HilbertBlumenthal, Compositio Math. (to appear). 19. J. P. Serre, Expose 7, Seminaire Cartan 1 950/5 1 . 20. H . SwinnertonDyer, 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, 19681 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. 5364. ��,
�� ,
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UNIVERSITY OF MICill GAN
Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 2, pp. 1 851 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 HasseWeil zetafunctions as products of Lfunctions 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 Qsubgroup 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 zetafunctions of the corresponding Shimura varieties can be expressed as products of automorphic Lfunctions 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 Lgroup LG Gal(F'JQ) 1>< LG o by putting r (r) (@" v") = ®" w" for 1: E Gal(F'/Q) where.J:V" v,1 " . The result of [2] is that Z(s, Sx) = fLr L(s  d/2, n, r)m C1r, K l up to a finite number of local factors, where d [F: Q], n runs over the representations of G(A)/Z(R) (where Z is the center of G) which occur in V ( G(Q)Z(R)\G(A)), and m(n, K) is an integer which is associated to n and K in a way that is described in [2]. The product over n is actually finite since m(n, K) turns out to be 0 for all but a finite number of n. The precise result is =
=
=
=
=
=
AMS (MOS) subject classifications (1 970). Primary 05C05, 14Gl0, 22E50, 22E55. © 1 9 7 9 , American Mathematical Society
1 85
1 86
R. E. KOTIWITZ
Zp (s, SK)
(1)
= n "'
L(s  d/2, 1Cp , r)m C n , K l
where 11:P is the local factor of 11: at a finite rational prime p which satisfies : (A) F is unramified at p ; (B) B splits at every prime ofF above p ; (C) K = KpKP where Kp i s a maximal compact subgroup of G(Q p) and KP i s a compact open subgroup of G(A�), A� being the ring of adeles of Q with component 0 at p and oo . For the rest o f this paper p will denote a fixed rational prime satisfying (A)(C). Because of these assumptions about p, SK has good reduction at p, and m(11:, K) = 0 unless 1T:p is unramified. Take the logarithm of each side of (1), and use the power series expansion of log(l  X). Comparing the coefficients of the powers of ps in the two sides, we see that (1) is equivalent to (2) Card SK (Fpi) = � m(11:, K)pid 1 2 tr(r(g")i) "
for all positive integers j, where 11: runs over all the representations of G(A)/Z(R) which are unramified at p and which occur in V(G(Q)Z(R)\G(A)), and where g"P is an element of the semisimple conjugacy class of LG associated to 11:p · For every place v of Q choose a Haar measure dg. on G(Q.) such that the measure of G(Z.) is 1 for almost all finite places v . Use dg. to define the action of L l (G(Q.)) on representations of G(Q.). Choose a Haar measure dzoo on Z(R), and use the quotient measure dgoo/dzoo to define the action of V(G(R)/Z(R)) on representations of G(R)/Z(R). LetfP be the characteristic function of the subset KP of G(A�) divided by the measure of KP (with respect to 11 "*P· dg.). It is pointed out in [2] that there exists f E ceca( G(R)/Z(R)) such that m(1C, K) = m(1C)tr 1CP{fPfoo) where 1f:P = ®v,p1T:v and m(11:) is the multiplicity of 11: in V(G(Q)Z(R)\G(A)) (it is known that this multiplicity is I , but we will not need to use this fact). By the theory of Heeke algebras, there exists a unique function fPl in the Heeke algebra .?t'(G(Qp). Kp) such that tr 1T:p(fjil) = pid / 2 tr(r(g"P)i) if 1T:p is unramified, = 0 if 11:p is ramified. co
00
So the right side of (2) becomes :E n m (11:) tr 1T:(fPfPfoo), which can be evaluated in terms of orbital integrals by means of the trace formula. For r E G(Q), let dg be the quotient of 11 .dg. on G(A) by some Haar measure dg7 on Gr(A) (where as usual G7 denotes the centralizer of r in G), and let m7 = meas(GiQ)Z(R)\G7(A)) where the measure used is the quotient of dg7 by the measure on Gr(Q)Z(R) � G7(Q) x Z(R) which is the product of the Haar measure on G7(Q) which gives points measure 1 with dzw Applying the trace formula, we find that (2) is equivalent to (3)
where r runs over a set of representatives of the conjugacy classes in G(Q). It turns out that the integral fcrCR) \G (R) foo(r 1 rg) dgoo is 0 unless r is central or elliptic at oo , and so the right side of (3) is
1 87
COMBINATORICS
(4)
where a7 = m7 fGr (R) \G(R) foo(g1 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/Jstable 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) .
)
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 BruhatTits 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; = Mid 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,. = Mii · 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 id ; (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
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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; = Bj / 1 rd/1 x; ; (F') r' B•q•dx; = X i1 · We will now show that S'(r, j) is empty unless Bi 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: = qdj / 1 Bi l l rd l l, But ?:Xo = Xo by (E'), and 1:� = � since q� = �. B � = � and r� = �. So 1: fixes p0• Since qdi 1 1 fixes p0, we conclude that B i l lrd l l fixes p0. But Bi 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 Bi 1 1ra 11 e /0. Let a, f3 e FJ be the diagonal entries of r. so that r = {3 Z). Then Bi11rd1 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 Bi 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 Bi 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 BruhatTits 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 Bi 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 Bi 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
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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  q1)q n if n > 0 and is 1 if n = 0. Therefore N(l, k)
=
M(l, k) + ( 1
 q1) .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
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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)xlk(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 � qk1 q = qn(2k  1 , k  1).
( f l)
)
e
This shows that the coefficient of X2kl 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  q1) 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  q1) I; q n I; m(l, k  n)X I Yk l, k n =l l, k = f(X, Y) +  q1) I; qn yn I; m(l, k  n)Xt pn l, k n =l = f(X, + (1  q1) 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 zetafunctions 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. 193203
NOTES ON LINDISTINGUISHABILITY
(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 "Lindistinguishable" from a given representation. We structure this set, an Lpacket, 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 Lpackets 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 Lpackets. 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 1cocycle 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 simplyconnected 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). Lindistinguishability 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 TateNaka 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"fl11 aE I/!JK /F
 fJ11 , fl11 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 MCS760821 8 . © 1 979, American Mathematical Society
1 93
1 94
D. SHELSTAD
X* ( ) denoting Hom(GLr. ). A quasicharacter " 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 quasisplit 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 quasicharacter " from the last section is just a @K 1 rinvariant quasicharacter 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 co �) * 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 quasisplit 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
LINDISTINGUISHABILITY
(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 Lgroups. 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) =
_ JhIT(F)h\G (F) f(g1h1rhg) ____!jg_ (dt)h
where h E %f(T) represents o and (dt)h is the measure on hIT(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 rz 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 casebycase 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 welldefined 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 Lindistinguish 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.
.....
.
LINDISTINGUISHABILITY
197
8. Local Lpackets 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 Lpacket. Two irreducible admissible representations of G(F) are said to be Lindistinguishable if they belong to the same Lpacket. In the case that G is an inner from of SL2 there is a simpler definition : tr 1 and tr2 are Lindis 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 gl ,.'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.(z6w)) 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 > 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] Lindistinguishability plays a role in relating the zetafunctions of certain Shimura varieties to automorphic Lfunctions. 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 Lindistinguishability, 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 Lpackets 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
LINDISTINGUISHABILITY
203
REFERENCES 1. A. Borel, Automorphic Lfunctions, these PROCEEDINGS, part 2, pp. 2761 . 2. W. Casselman, The H asseWif!il �function 'of some moduli Ivarieties ofdimension greater than one, these PROCEEDINGS, part 2, pp. 141163. 3. A. W. Knapp and G. Zuckerman, Normalizing factors, tempered representations and Lgroups, these PROCEEDINGS, part 1 , pp. 93105 . 4. JP. Labesse and R. P. Langlands, Lindistinguishability for SL(2) (to appear). 5. R. P. Langlands, On Artin's Lfunctions, Rice Univ. Studies, vol. 56, no. 2, pp. 2328, 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 quasisplit group over R, Compositio. Math. (to appear). 10. , Orbital integrals and a family of groups attached to a real reductive group, Ann. Sci . Ecole Norm. Sup. (to appear). 



COLUMBIA UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol . 33 (1 979), part 2, pp. 205246
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 zetafunctions 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 representationtheoretic, but they both arise in the study of the zetafunction 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 CMtype. 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 Lfunctions, 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. Lindistinguishabil 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 zetafunctions. The omission I regret most is that of a discussion of the reduction of the Shimura varieties modulo a prime [36]. Here there is a great deal to be said, especially about the structure of the set of geometric points in the algebraic closure of a finite field, starting with the work of Ihara on curves. I hope to report on this topic on another occasion. 2. Automorphic representations. Our present knowledge does not justify an attempt to fix a language in which the relations of automorphic representations with motives are to be expressed, Nonetheless that of tannakian categories [40] appears promising and it might be worthwhile to take a few pages to draw attention to the
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
207
problems to be solved before it can be applied in the study of automorphic repre sentations. We first recall the rough classification of irreducible admissible representations of GL{n, F), F being a local field, and of automorphic representations of GL(n, AF), F being a global field, reviewed in some of the other lectures ([3], [5], [35], [48], cf. also [7], [31]). In either case the representation n has a central character w and z � l w(z) l may be extended uniquely from Z(F), or Z(AF), to a positive character v of GL(n, F), or of GL(n, AF). If F is a local field then n is said to be cuspidal if for any Kfinite vectors u and v, in the space of n and its dual, v l (g) (n (g)u, v) is squareintegrable on the quotient Z(F)\GL(n, F). To construct an arbitrary irreducible admissible repre sentation one starts from a partition {nh · · ·, n,} of n and cuspidal represen tations n 1 o · · ·, n, of GL(n;, F). If w; is the central character of n;, there is a real number s; such that lw;(z) l = lzl•; if z lies in the centre of GL{n;, F), a group isomor phic to Fx . Changing the order of the partition, one supposes that s 1 � • • • � s,. The partition defines a standard parabolic subgroup P of GL{n) and o = ®n; a representation of M(F), because the Levi factor M of P is isomorphic to GL(n 1) x · · · x GL(n,). The representation o yields in the usual way an induced represen tation /" of G(F). /" may not be irreducible, but it has a unique irreducible quotient, which we denote n 1 E8 · · ·EEl n,. Every representation is of this form and n 1 E8 · · ·EEl n, � n :i E8 · · · E8 n; if and only if r = s and after renumbering n; � n;. Thus every representation can be represented uniquely as a formal sum, in the sense of this notation, of cuspidal representations. The representation n 1 E8 · · · E8 n, is said to be tempered if all the S; are 0. We can clearly define, in a formal manner, the sum of any finite number of representations. We can in fact formally define an abelian catefory H (F) whose collection of objects is the union over n of the irreducible, admissible representations of GL(n, F). If n = n 1 E8 · · · E8 n" n' = n:i E8 · · · E8 n;, with n; and nj cuspidal, we set Hom(n, n')
=
{i, jl1t:; "jl
EB
C.
The composition is obvious. The tempered representations form a subcategory H0 (F) . If F is a global field then n is said to be cuspidal if n is a constituent of the representation of GL(n, AF) on the space of measurable cusp forms rp satisfying (a) rp(zg) = w(z) rp(g), z E Z(AF), (b) Jz (AF)G (F)\G (AF) V 2(g) j rp(g) j 2 dg < 00 . I f n 1 o · · , n, i s a partition of n and n 1 o · · · , n, cuspidal representations of GL{n;, AF) we may again change the order so that s 1 � · • � s, and then construct the induced representation /" . Every automorphic representation is a constituent of some /" . For an adequate classification, one needs more. The following statement may eventually result from the investigations of Jacquet, Shalika, and Piatetskii Shapiro, but has not yet been proved in general. A. If n is a constituent of /" and of lu' then the partitions {nh · · · , n,} and {n:i,  · · , n;} have the same number of elements, and, after a renumbering, n; = n; and n; n;. •
•
�
208
R . P. LANGLANDS
If 7C; = (8).7C;(v) then one constituent of I" is the representation 7C = ®v 7C(v) with local components 1e(v) = 1e 1 (v) [±) · · · [±) 1e,(v). This representation will be denoted 1e = 1e 1 [±) · [±) 7C" but the notation is not justified until statement A is proved. The representations of this form will be called isobaric and can again be used to define an abelian category D(F). We agree to call 7C = 1e 1 [±) [±) 7C, tempered if each of the cuspidal re presentations 7C; has a unitary central character, that is, if each of the s; is 0. How ever the language is only justified if we can prove the following statement, the strongest form of the conjecture of Ramanujan to which the examples of Howe PiatetskiiShapiro 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 ndimensional 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 finitedimensional groups. We denote it by GucFJ · The category Rep(GocFJ ) of the finitedimensional 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 finitedimensional 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 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£
f7u
>
Gal(F/L)
>
I
to Gal(LabfL). Here f7 u is the quotient of !7f by the normal subgroup {a("C) ir 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 righthand 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 TateNakayama 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 ad g o h is a welldefined 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 f1. let l.l(f1.) 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 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 onehalf 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, welldefined 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 onehalf 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 quasisplit 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 quasisplit over R then rp' is different from rp. If G is not quasisplit over R then it is certainly not quasisplit 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 BorelWallach [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 zi zi, z:i1 z i1 , . . . , z Cnj 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 · · · ,  Xni+ l >  Xni • · · · ,  xi+Z•  Xni+ I > · · · ,  Xni •  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 zi1 , . . . , z  + 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 Zstructures 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 demiespace 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 .61 . 1 1 , 3 . 1 43 . 1 6, 4. 1 14. 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 nouvelleen fait un retour au point de vue de Shimuraest !'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 (semilineaire) du sousgroupe 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
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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 sousgroupes 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 sousgroupes arithmetiques de G•d (Q)+, images de sousgroupes 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 cidessus 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
semisimple 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 semisimple 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 Falgebre, 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 = zp:zqv 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 ft) !'action de Gm sur Vc definie par ft(z) v = zPv 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 sousanneau de R tel que A ® Q soit un corps (en pratique, A = Z, Q ou R), une Astructure de Hodge est Amodule 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 Zstructures 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 celleci 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 Zstructure 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 Zstructure 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. 501597) et !'inverse de celle de [6]. Elle est justifiee d'une part par l'exemple 1 . 1 .4 cidessus, 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 Zmodules de type fini, tel que V = Vz ® R. On parlera alors de variation de Zstructure 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 Zstructures de Hodge, en rempla 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 Gmorphismes 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 sousgroupe 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 sousgroupe 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) cidessus. (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 sousvariete 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 Gmodules, 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 ha savoir le sousespace 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 sousespace 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 cidessous. 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 Cpo/arisation d'une representation V de G est une forme bilineaire ?f! Ginvari ante, telle que ?f!(x, Cy) soit symetrique et defini > 0. Pour tout g E G(R), on a alors ?f!(x, gCgly) = ?f!(glx, Cgly) : Ia notion de Cpolarisation 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 Cpo/arisable ; (iii) G admet une representation reel/e Cpolarisable fide/e. On notera que la co ndition 1 . 1 . 1 5(i) entraine que GO est reductifpour 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 sousgroupe 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 sousgroupes 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 sousgroupe
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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 whnote west 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 sonsgroupe in variant connexe tel que G soit quotient de w(Gm) x G2 par un sousgroupe central fini. II est unique : engendre par le groupe derive et le plus grand soustore 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 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 sousgroupe compact ouvert de G(Ar) , on a G(k)pG(A8) = G(k)p(G(k) · G(krs)
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(krs) X K) c rs c pG(As)· L'intersection de G(k)pG(As) avec le sousgroupe ouvert pG(krs) 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 Sunites. D'apres un theoreme de Chevaelly, tout sousgroupe d'indice fini de U est un sousgroupe de congruence (voir [12] pour une demonstration elegante). II en resulte que si T' > T est une iso genie, l'image d'un sousgroupe de congruence pour T' est un sousgroupe 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 Sunites de Z(k) il existe un sousgroupe 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 Scongruence, done peut se relever un petit element z de Z(k), et r = 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 sousgroupe distingue, avec u n quotient abelien. De meme, posant x 0n(G) = n0n(G)/n0 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, n0 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 Melimite 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 sousgroupes 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 sousgroupe 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 sousgroupe 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 sousgroupe 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 cidessus 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 sousgroupes d'indice fini du groupe des unites ; il re E(G", X"). 2.2.2. Soient T un tore, E un corps de nombres, et ft 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 ft• on obtient RE1o(ft) : 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(ft) : E * � T. Nous le noterons simplement NRE(ft), ou meme NR(p,). Si E' est une extension de E, ft 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(fth) : 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 demiespace 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 sousgroupe 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 soustore 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 Qsimple 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. Celleci 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 souscorps de KD fixe par le sousgroupe 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 cidessousdeduite de 1 . 3.9nous 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 extremitecorrespondant a une racine courte pour les types B, Cd'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 : Souscas 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!. Souscas Df. L'unique 9' verifiant (b), (c), (d) est !'ensemble des {s} pour s
272
PIERRE DELIGNE
l'extremite "gauche" d'un D • . Le revetement G(9') de G est de la forme RngG*, pour G * le revetement double de G forme de S0(2l), cf. 1 .3 . 1 0. Type D4• Rempla�ons 9', verifiant (d), par S = {s l {s} e 9'} . La condition (b) sur 9' devient : S est contenu dans l'ensemble E des extremites de D, et S n I(X) = 0 . Pour Ia definition de I(X) voir 2.3.5. La partie de E stable par Gal(Q/Q) maximale pour cette propriete est Ie complement de Gal(Q/Q) · .2'( X). Bile rencontre chaque D. en 0, 1 ou 2 points. Dans Ie premier cas, il n'existe pas 9' verifiant (b), (c). Dans Ie second (resp. 3e), elle (resp. son complement) est l'image d'une section r de X + I, invariante par Galois ; r(J) est disjoint de (resp. contient) I( X). Ap pelant r( v) le sommet "gauche" de D., on retrouve Ia situation de D1 (I ;;;;; 5) : Souscas 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. Souscas 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 audessus 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 audessus de Inc · Le resultat principal de ce numero est la

PROPOSITION 2.3. 10. Soit (G, X) comme en 2. 1 . 1 , avec G Qsimple 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 souscorps 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
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) compactdone 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)
oG•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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PIERRE DELIGNE
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 cidessus, 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 cidessus, 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 sousgroupe 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 fth• 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 fth (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))
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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 sousgroupe 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 cidessous, 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 sousgroupe 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 cidessus, 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 ) + 7C( T). On en deduit que l'action de Gal(Q/E) n Gal(Q/E(T, h)) sur les points speciaux de type . est compatible a I' action de Gal(Q/E(T, h)) ab = 7C o7C(Gm E ) sur 7C o(Mc(G, X)) deduite, par applica tion du foncteur 7C0, de I' inverse de C o NE (T, h) IQ o q,_, : 7C(Gm E (T, h J ) > 7C(T) > 7C(G).
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 sousgroupes 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 sousgroupe 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) = (ro1, 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 Fequivariante 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+) ;
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(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 sousgroupe G(Q)+A (rei. G') (une action Qlineaire 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 hfixe 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 sousgroupe Gal(Q/E(t")) n Gal(Q/E) de Gal(Q/E) sur !'ensemble des points speciaux de type 1:. Ceuxci 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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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 Qsimple adjoint, G' un revetement de G,
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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), 442528. 3. A. Borel, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, J . Differential Geometry 6 (1 972), 543560. 4. N. Bourbaki, Groupes et algebres de Lie, Chapitres IVVI, Hermann, 1 968. 5. P. Deligne, Travaux de Shimura, Sem. Bourbaki Fevrier 71, Expose 389, Lecture Notes in Math. , vol . 244, SpringerVerlag, Berlin, 1 97 1 . 6 .  , Travaux de Griffiths, Sem. Bourbaki Mai 70, Expose 376, Lecture Notes i n Math., vol. 1 80, SpringerVerlag, Berlin, 1 97 1 . 7 .  , L a conjecture de Wei/ pour les surfaces K 3 , Invent. Math. 1 5 (1 972), 206226. 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. 3473 5 1 . 10. N . Saavedra, Categories tannakiennes, Lecture Notes in Math. , vol. 265, SpringerVerlag, Berlin, 1 972. 11. I. Satake, Holomorphic imbedding of symmetric domains into a Siegel space, Am. J. Math. 87 (1 965), 42546 1 . 1 2 . J. P. Serre, Sur les groupes de congruence des varietes abeliennes, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1 964), 320. 13.  , Cohomologie galoisienne, Lecture Notes in Math., vol. 5, SpringerVerlag, Berlin, 1 965. SGA Sem. de Geometrie Algebrique du BoisMarie. SGA 1 , SGA4 t.3 et SGA4'1' sont parus aux Lecture Notes in Math. nos. 224, 305, 569, SpringerVerlag, Berlin . INSTITUT HAUTES ETUDES SCIENTIFIQUES, BURESSURYVETTE
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 onedimensional 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 padic 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 schemetheoretic 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, CRsystems, § 1). It main ly states that whenever there is an unramified symmetric CRsystem !![ 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
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YASUTAKA IHARA
Here, o is the ring of integers of a ).1adic field k (Np = q), and f7 is the tree asso ciated with PGL2(k) (Main Theorems IIII). 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 oscheme, 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 twodimensional integral oschemes 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 ft : X0 > T be the normalization of the twodimensional integral scheme T, and put f/J; = pr; o ft (i = 1 , 2), where pr; are the projections of T to X;. The system ( 1 . 1 .2) thus obtained will be called a CRsystem 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 CRsystems, 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 kmorphisms 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 Fqzrational 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 twodimensional scheme T. Then the local ring fJr, Y may or may not be normal, depending on x. DEFINITION 1 .4. 1 . An Fqzrational 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 fl · These two components meet at above those y = (x, xq) for which x is special. Therefore, the coincidence of the EulerPoincare 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 aisomor 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 padic 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 Fpzrational points of X be given. Then there exists at most unique unramified CRsystem !:[ 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 wellknown congruence relation is the Kronecker congruence
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YASUTAKA IHARA
relation (Xt. X2 ; T), where XI = X2 is the projective jline 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) jinvariants 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 CRsystem w.r.t. (X, o). The purpose of this section is to establish a fundamental Galoistheoretic 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 CRsystem (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 CRsystem f!{ and consider the disjoint union ( VI\at) U ( V2\at) of two left coset spaces as a pointset. 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 pointset 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 CRsystem 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 CRsystem, we have (for a suitable extension t) : 1 t
 (0p 0) . 
2.8. Now return to an arbitrary CRsystem 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 almostunramifiedness also implies the symmetricity when o = Zp. This definition of almostunramifiedness singles out the class of CRsystems 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 halfplane .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 torsionfree 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 CRsystem, 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 CRsystem 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 Gtorbits Gt · � of those � and that of all ordinary closed points (schemethe 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) ,_. {Fqzconjugacy classes of points of X}, and when !![ is symmetric,
Gp\Pl(L/k) + {Fqconjugacy 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 EgO
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
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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 Gtorbits in Pl(Ljk, [A]) and that of all Fq2conjugacy 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 GPorbits in Pl(Lfk; [A]) and that of all Fqconjugacy 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 Gtorbits 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 ¢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 CRsystem w.r.t. (X, o), and f£ * = {Xf 9>� + Xt + 9'; X: } be another CRsystem 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 omorphisms. Then we shall call (f£ * , /) a .finite etale CRcovering 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* = J1 (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 CRsystem 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 CRcoverings ([![*, 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 Fqzrational 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 CRsystem 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 CRcovering of [![, and (X*,f) be the corresponding finite etale covering of X (§4). Let .J'f' (resp . .J'f'*) be the set of all Fpoints (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 resultsapply to each Shimura curve for almost all !J by Theorem 6.3.3 which is the "pcanonical 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 1dimensional 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 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 ]'autoequivalence M > M ® Z(  1). La cate gorie vft(k) admet vlte f f(k) comme souscategorie 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 autoequivalence 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 autoequival 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' autoequivalence 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 1adique 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 sousjacente 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 1adique. 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 Qstructures 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'),
q hPq, !'autre a I; p;?;q hpq. En particulier, ces dimensions sont egales a celles de sousespaces 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 nforme sur X, definie sur Q et que c est un ncycle 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 sousespace 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 kconsidere comme objet de Jtk,E• un motif M se notera MEet 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 Fdual d'un espace vectoriel sur E avec son Qdual, 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 Fmodule. 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 abelienneprise a isogenie presles foncteurs cidessus 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 Fmodule.
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 audessus 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 Calgebre 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 cidessus, 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 cidessous 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 PROPOSITIONDEFINITION 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 Eespace vectoriel. Sa dimension est le rang sur E de M. L'involution F est Elineaire ; les parties + et  sont done des Esousespaces vectoriels. On note d+(M) et d(M) leur dimension. 00
VALEURS DE FONCTIONS
L
323
Le complexifie de H8(M) est un E ® Cmodule 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 Eespace 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 ® Cmodules 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 Erationnelles 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 Dmodule, 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 cidessous. 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 Eespaces vectoriels" : (a) Se donner un espace vectoriel V sur E revient a se donner, pour toute exten sion etale F de E, un Fespace 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 Dmodule. Pour toute extension etale F de E, et tout Fisomor 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 ® Fmodule 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 nonunicite 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 Dmodules, et T une operation tensorielle. Si les homotheties de L agissent trivialement sur T( W� L), le Fespace 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 Dmodules 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 Dmodules Ht(M) et a H!JR(M). Posons o = o(Hfi(M), HfjR(M)). Le determinant reduit de l'isomorphisme de D ® Cmodules 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 SwinnertonDyer.
4. 1 . Soient A une variete abelienne sur Q, et d sa dimension. La conjecture de Birch et SwinnertonDyer [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 dcycle 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) cidessus. 4.2. La conjecture de Birch et SwinnertonDyer 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 sousgroupe 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, Nvs) (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 sousespaces 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 sousespaces 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 sousespace 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 . 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(  FNvs, 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 semisimplifiees des restrictions des H. = 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( 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 ( 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· nK of surfaces of finite type Mfj· nK, (2) Mfj+l, nK1 ::::J Mfj· nK, and (3) the difference Mfj+l, nK1  Mfj· nK is a union of affine lines. Drinfeld also defines compactifications Mf5· nK of each subspace Mfj· nK, and shows that the birational isomorphism Mfj+l, nK1 "' Mif· nK can be ex tended to a morphism Uf5· n : Mf5+1, nK1 > Mfj· nK such that (a) the composi tion of rc with the natural imbedding Mfj· nK 4 Mfj+l, nK1 is the identity and (b) the restriction of rc to the boundary F/f· n = def Mfj+l, nK1  Mfj+l . nK1 is a radical morphism. Define Mv = proj lim K Mif· nK 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· nK, 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· nK, so Mfj· nK 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· nK is a surface of generic type. It can be proved that the restriction to V, of the canonical cupproduct 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,.
354
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 SchwartzBruhat 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 SB 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 ) to be an extensionthis mapping f/>K must have a onedimensional invariant subspace W c V. Then we may choose a basis vector e 1 E W and e2 E V/ W and we find f/>K(e1 ) = A1 eb A 1 E KFq" = Kn and i = I , 2. f.i.; = NormKn!K (A,) E K, Then we have up to ordering f.!. I> f.l.z : ( 1) f.l. l has a zero at p and is a unit at q . (2) f.l.z has a pole at q and is a unit at p .
(3) The vector space V decomposes in a unique way into a sum of two subspaces V = W EB W', i.e., we can find a complement to W. (4) If L (resp. H1 ) is the subbundle of E induced by W (resp. W') then we can write (P, ¢, a v , aw, f/>) in exactly one way as a left and in exactly one way as a right exten sion :
0 > L > E > H > 0, The proof of this proposition follows from a local analysis of what happens at v and w. We conclude this section with a discussion of the next obvious question. Given L, H or HI> L1 how can we describe the PHextensions 0 > L > E > H > 0, 0 > H1 > E1 > L 1 > 0? We discuss only the first case. Let us fix v, w and a divisor D, still with the stand ing assumption p 1= q, v, w ¢ supp D. Moreover we fix level structures ¢H : H/L(  D)____:::____, f9x/f9x(  D). The extensions of L by H are classified by the elements in H l (X, Hom(H, L)) and the set Hl(X, Hom(H, L)(  D)) = Hl(X, H 1 ® L(  D)) classifies extensions with a splitting along the divisor D. Therefore a vector bundle E which corresponds to an element � E Hl(X, H 1 ® L(  D)) is automatically equipped with a level structure along D. We assume that D has large degree compared to deg(H 1 ® L). We ask for those � which give rise to a vector bundle with a (then unique) PHstructure. This means that we have to get a 0 > L > E > H > 0
"Z �
0 > L(w) > E > H > 0
/ /
0 > L" > E" > H > 0 Here i!JL is the composition U > 1JLL(w  v) 4 L(w) . We get a diagram
GL2
363
OVER FUNCTION FIELDS
H 1 (X, H 1 ® L( D))
X H 1 (X, H 1 ® L(
ri>L
H l (X, H 1 ® L "( D))/ II H 1 (X, H 1 ® L(  D))
 D + W))
and to get an FHsheafwe have to look for the classes � which satisfy i*(�)  til !(�") = 0. PROPOSITION 1 . 6 . 2 . ( 1 ) This equation defines a smooth onedimensional group
scheme over Fq which is isomorphic to Ga. (2) It can be defined over the field of definition of(L, cfJL , tPL) and (H, cfJH). (3) To each pair (L, cfJL), (H, cfJH) we find two affine lines Gfy H, Gff· L of right and left FHextensions with level structure along
D.
The proof of this proposition is again not very difficult : the assertion that the solutions of the above equation form a connected scheme is a little bit tricky. REMARK 1 . The FHsheaves which we constructed as extensions have a particular kind of level structure in the sense that the level structure is adapted to the exten sion. The cjJ : E/E( D) + (f9x/f9x( D))2 maps the L/L( D) to the first component in (f9x/f9x( D))2. One checks easily that any FHextension with an adapted level structure can be obtained in the above way as a point on Gf] H or Gf5· L . 2. On the other hand we have an action of GL2 (f9x/ f9x(D)) on the scheme of FHsheaves with level structure along D (1 .4). Then we may look at the translates of the affine lines GfJ H and Gf5· L and we find some more affine lines on the moduli scheme. The points on the union of these lines correspond to the FHextensions where the level structure is not necessarily adapted. 3. If we drop the assumption that the degree of D should be large, nothing is changed substantially. We can pass to a larger divisor and after that we divide by the corresponding group actions. We find quotients of affine lines by finite groups and this gives again affine lines. 4. If E/ X is an FHsheaf which contains a subbundle F c: C with deg F > ! deg E + 1 then it is easy to see that t/) has to map F" to F', i.e., that E is neces sarily an FHextension. If s G 2 is an integer and if Mfr· wl (s) is the subscheme whose points are given by vector bundles E s. t. deg F � ! deg E s for all line sub bundles F c: E, then Mfl'· wl (s  1)  Mfl'· wl(s) is a union of affine lines of left and right FHextensions, where the subbundle is running over all FHsheaves (resp. sheaves defined over Fq) of degree s and rank 1 . 5. The affine lines GfJ H(resp. Gj5t. L l) where deg L (resp. deg H1 ) is very small will contain a nonempty open piece where the bundles are stable but there will be more and more unstable points of higher and higher degree of instability if deg L (resp. deg H1 ) tends to oo . 

2 . The geometric points o n the moduli space. In this section we want to discuss the main result which establishes a relation between the number of points on cer tain open pieces. of our moduli spaces and the traces of Heeke operators on the space of cusp forms.
364
G.
HARDER AND D. A. KAZHDAN
2. 1 . The fundamental formula. Our standing assumption will be that v, w E X(Fq) and that v and w induce different valuations p and q on K. We consider the schemes Mty· w) t S pec( Fqd) where Fqd is the union of the residue fields of p and q. If E(¢, av , aw , (/)) is an FHextension then we have two exact sequences 0 t L t E t H t 0, and we call the above extension strongly decomposed if deg L > deg H or deg H1 � deg L1 . The nonstrongly decomposed FHsheaves form an open sub scheme UD 4 Mty· w) . On UD we have an action of the group PD of line bundles on X with level structure along D. This group acts by tensorisation on UD and M};v, w) and this action induces an action of the same group on the cohomology with compact supports H i ( DD, Q1) where DD = UD x Fqd Fq and Q1 is an algebraic clo sure of Q1 . Let fJ. c Q1 be· the group of roots of unity. For any finite character w : PD t fJ. we denote by H, ( DD , Q1)"' the corresponding eigenspace in the cohomol ogy. We are interested in the tracei.e., the alternating sum of tracesof the powers Fr m of the geometric Frobenius for dim on the cohomology
trace Frmi H ! ( DD , Ql )w
4
trace Frm i Hf( UD , Q I ) 01• i=O REMARK. The scheme UD t S pec(Fqd) is not of finite type. We have the deter minant map det UD t PD introduced in 1 . 5.2. The fibers U)jl for � E PD(Fq) are quasiprojective schemes and if UH) t u)ja2) . If PD . w i s the kernel of w then
= L: (  l )i
a E
PD then the tensorisation defines a map
H!( UD , Q I ) w c H!( UD , Q I ) PD· "' =
_
_
EIJ 2, , H i ( Ujfl , Q I ) .
e E PDIPD
The information contained in the above trace is therefore equivalent to some in formations on the number of rational points of the Ubel . We now want to relate the above trace of Frm to the trace of some Heeke opera tors acting on a space of automorphic forms. We choose an embedding fJ. 4 C* . Then we can consider the given character w also as a character with values in C* . Let us put .?f"p = GL2(l!!p), .?rq = GL2(l!!q) and %� c IT P '* P. q GL2(l!Jp,) to be the full congruence subgroup defined by D. If Z(A) is the centre of GL2(A) we have an identification z(A)JZ(K) · (Z(A) n .?rp
x
.?rq
x
.?r�)
�
PD
and therefore we may view our character w also as a complex valued character on
Z(A) . We introduce the space
H"' = L�, d isc (GLz(K)\GLz(A)/%�
X
.?f"p
X
.?rq)
where w is the above central character and the index disc means that we restrict our attention to the discrete spectrum [1, §4]. Our central character is unramified at p and q . We introduce some algebras .Yi"p, "'' .Yi"q, "' of Heeke operators. The elements of .Yt'p, w are those functions /: GL2(Kp) t C which are compactly supported modulo the centre, and which satisfy f(xz) = f(x) w (z),
GL2
365
OVER FUNCTION FIELDS
and which are Jt"p hiinvariant. Each such/defines an operator T1 : Hw ..... Hw,
J
TJ (h) (y) = h(yx)f(x 1) dx, GL2 (Kp)/Z(Kp)
where y E GL2(A) and GL2(Kp) 4 GL2(A). In these algebras we have for any n E Z the elements (jj f/'l which are defined as follows : If 1J 5 z. 52 # 0 is a function on GL2(Kp) which satisfies t I 112 1J 5�o 52 1 *z P  J t 1 J P5 1 J t2 J P52 j iL tz l P 0 l
(( )k )
kP E GL2(llJp) and has central character
w,
i.e., w(t) = ltl�1+ 52 then
JZ(Kp)\GLz (Kp) "fl5 52(yx) (jj CJ!l (x1) dx = (NlJ)f n ii2(NlJnsl + ., h
N1Jn 52 )
1Jsh 52 (y)
where NlJ is cardinality of the residue field at lJ. This definition can be expressed in terms of representations. If we have an irreducible representation of class one PP : GL2(Kp) > End(H) which has central character w then p p{ Tq,Cpl) 1J = (N JJ) f n fi2 (NlJ n s l + NjJn s 2) 1J s h s2 n where r; # 0 is a Jt"pinvariant vector and where sh s2 are the parameters of this representation (compare [1, §3, B]). We are now able to state the main equality. Let us denote by dP (resp. dq) the degree of the place lJ (resp. q). The operator (jjm = T q, CPl
mfdp
a
Tq,  m{d Cql : Hw > H w q
is of trace class (compare [1, §2]) and we have trace Frm JHi( DD• Q1) w = trace WmJ Hw  Eisenstein contribution in the trace formula for the operator (jjm · The Eisenstein contribution is the sum of the terms 6.36, 6.37 in [1]. We are here not quite consistent with the notations in [1]. We refer the reader to §2.3. 2.2. Consequences of the main equality. Before discussing the proof of the main equality we shall give some applications. We shall see that it has some geometric consequences for the schemes UD and it implies the Ramanujan conjecture for cusp forms. Let us pick a line bundle r; over X/Fq of degree one. It generates an infinite cyclic group {r;} c PD and PD/{r;} = Po . D is finite. Then it is rather easy to see that
� trace Frm JH;(DD, Q1) w = trace Fr m
roE Po, D
(
_
EB
e EPn;deg e=O, l
)
H ; ( Djp , Q t) .
We express this trace in terms of the eigenvalues a; . • of the Frobenius acting on the cohomology H \( D�el , Q1). Then we obtain L: t= 1 L: tb(  l)i a'f:v for the above trace. Now we look for a similar expression of the trace of Wm on the spaces H00• Here we know that Hro decomposes further,
366
G. HARDER AND D. A. KAZHDAN
Hw = flw, cusp Ef> Hw, l EB Hw ,p Ef> EB C · X p X
where p runs over a finite set of irreducible representations of GL2(A) in the space of cusp forms L'!,, cusp (GL2(K)\GL2(A)) and where x : GL2(A) > C is a function which factors over the determinant X GL2(A) __.det IK > PD __. w ' ft c C* with w' 2 = w.
We compute the trace of (/)m on each of the constituents. The representation p has to be of class one at p and q and therefore O'p is obtained by induction from a representation PsJ. S2 :
(d' P ;2, p)  l t1, P II?I t2, pl�2 1 tl, p/t2, pJV2·
We define �p(p) = q siidp, ��(p) = q s2tdp , etc. Then trace (/)m iHw . p = (dim H;!,p)qm(�p(p)m + ��(p) m) (�q(p)m + ��(p)m) . This follows from the definition of ¢m· The contribution of the onedimensional spaces Cx is also easy to compute. This function X is t * 2 X 1, p > x (t!, p t2, p) = It1, p l sI/ p 2 1 t2, p l s+ p l/2 l t1 , p /t2, p 1 1 / . 0 t2, p We put 7Jp = x(np) 11dP, etc., and get by definition (7];:' + 7}p qm) (7]� + 7}� qm). T�,. X =
)
(
We want to recall that 7] p , 7}q are roots of unity. It can be checked very directly from the trace formula that the Eisenstein contri bution to the trace of(/)m in the trace formula can be written as s
1:
i=O
e;
r;'/'
= Einsenstein contribution in the Selberg
trace formula for the operator (/)m ,
= ± 1 and where the 7}; are algebraic integers of weight 0, 1 , 2, i.e., = 0, 1 ,2 (compare [8, 3 . 1 ]) . If we define a(p) = dim H;!,P then our main equality reads as follows :
with e; q"',
).!;
+ 1: (r;P' X
+
r;pzqm)(r;�
+
r;�q m)
+
s
1:
i=O
e;
lr;; l
=
r;'/'.
Now we are ready to derive some consequences from this equality. First we recall what is known about the a; , ". The a; , " are algebraic integers and we have la;,"l = qj 12 where } is an integer less than or equal to i. This number j is the weight of a,. , ". (Compare [9, 1 4]. ) O n the other hand w e have a simple LEMMA 2.2. 1 . If xi> . . . , Xm x�, . . . , x�,, YI> . . . , Ym• y�, . . . , y�, are complex numbers and if
GL2 OVER FUNCTION n
I; x1
i=l
and if X; =1 yi and x; X; = x;, y; = y;..

=1
m
I; Y7
i=l
=
n'
I; X�
i=l

367
FIELDS
m'
I; y?,
k
=
=
=
m' and up to order we have
i=l
y� for all i, j then n
n', m
1 , 2, · · · ,
Now we know that �p(p) · �;(p) and r;q(p) · r;;(p) are roots of unity because cv is finite. Moreover it follows from the theory of automorphic forms that I MP) I < q and l�q(p) l < q. This means that the terms (7}p 7Jq) mq2m which occur in the contri bution from C · X are the only terms of weight 4 on the righthand side. The number of these terms is equal to 2 1Po.vl · This is clear since for each cv which is a square we have 2 I Po.viP5, vi solutions of cv' 2 = cv in the group of characters of Pv. On the other hand the terms of weight 4 on the lefthand side come from the top cohomology I:
e. deg (o) =0, 1
trace Frm i H H Vifl , Q l)
=
b4
I: aT. v·
v=l
We shall see in the next section that the V ifl IF9 are all nonempty. The number of � is again equal to 2 1Pvl · Therefore we find the Vifl /F9 are irreducible. On the other hand we see that the terms of weight 3 on the lefthand side occur with a minus sign since they have to come from H r( V .fJl , Q 1 ). But our remark on the Einsenstein contribution shows that there are no terms with minus sign and weight 3 on the righthand side. So there cannot be any such term on the lefthand side. This is easily seen to be equivalent to saying that for any smooth compactifi cation Vifl 4 Y we have H3( f, Q 1 ) = H 1 ( f, Q 1 ) = 0. This absence of terms of weight 3 now implies directly that the terms qm �p(p) etc., have to be of weight 2, i.e., lr;(p)l = 1 , etc. We may summarize : THEOREM 2.2.2. (1) The schemes U jjl I Fqd are absolutely irreducible. (2) The cohomology _groups Hr( V ifl , Q 1) do not contain any classes of weight 3. Moreover we found : ,
Automorphic cusp forms on GL2(A) over a function field satisfy the Ramanujan conjecture, i.e., for any p occurring in H"'· cusp the local components Pp and pq are in the unitary principal series.
REMARK. Drinfeld himself uses another argument to prove (2) in Theorem 2.2.2. It also follows from the existence of the two families of affine lines in U vi F9• 2.3. The proof of the main equality. The proof of the main equality is obtained by a method which is similar to the one used in the theory of Shimura varieties. (Compare [3], [5].) We start from a description of the set of geometric points on the moduli space together with the action of the Frobenius and the Picard group on this set. The set of these geometric points will decompose into 'orbits' ; each of these orbits will give a contribution to the trace of Frobenius. The contribution which comes from a fixed orbit may then be compared to a contribution to the trace of the Heeke operators in the trace formula. 2.3. 1 . The set ofgeometric points. For simplicity we want to assume that our two points v, w e X(Fq) are actually rational ; this means that p and q have degree 1 . This will simplify the exposition but not change the proof substantially.
368
G. HARDER AND D. A. KAZHDAN
In this case the moduli spaces M}J· w) are defined over Fq and since v, w are fixed we also want to drop the superscript ( v, w). The vector bundles with level structure along D over X are in 11 correspondence to the double cosets (compare 1 .2) GL2(Koo)\GL2 (Aoo)/%oo(D) and this tells us that the set M(Fq) = proj lim MD(Fq) v D,
, woesupp (D)
is a subset M(Fq) c GL2(Koo)\GLz(Aoo)/GLz(l1/0) x GLz(l11w). REMARK. For notations compare 1 . 1 . We still consider v, w as valuation of Koo, so K., Kw are completions of Koo and @0, l11w are the rings of integers in K0, Kw. It is clear that exactly those points x E GL2(Aoo) give a point in M(Fq) for which we can solve the equation
where 0' is the mapping induced by the action of the Frobenius on the constants in Koo, where r E GL2(Koo) and where
and where a. (resp. ciw) is the adele which has v (resp. w)component a. (resp. aw) and 1 elsewhere. We observe that r . a., aw are determined by x. If we have two solutions of (* ) : X'n

r '  1 X ava w , '
� '
then x and x' correspond to the same point of M(Fq) if and only if x alx'kvkw with a E GL2(Koo) and k. E GLz(l11 .), k w E GLz (l11w). Then we find
=
(Here we take advantage of our assumption that v and w are rational.) The relation r' = arau means that r and r' are O"conjugate and we shall say that a � is O" GLz(l11.)conjugate to a. and a� is O"GL2(l11w)conjugate to aw· This gives us : Each geometric point in M(Fq) gives rise to the following data : (i) a O"conjugacy class {r} in GL2(Koo), (ii) a O"GL2(@.)conjugacy class {a.} in
(iii) a O"GLz(l11w)conjugacy class {aw} in GLz(l11w) ·
(�w 0) GLz(l11w) · 1
l
If these sets of data are given, then we find a point if and only if r is O"conjugate to a. (resp. aw) in GL2{K.) (resp. GLz(Kw)) and if r is 0'conjugate to 1 outside of v and w.
GL2
OVER FUNCTION FIELDS
369
Therefore we find the following strategy to describe the set of geometric points
M(Fq).
(i) Describe the set of aconjugacy classes in GL2(K00). (ii) Given such a class describe the possible classes a., iiw· (iii) Given T• a., iiw describe the set of geometric points which belong to this set Of data, i.e., the solutions of X" = Tlxaviiw With these specific T• a0, iiw · 2 . 3 . 2 . aconjugacy. We start to discuss (i). To do this we make use of the Saito Shintani method of norms. (Compare [7].) If T E GL2(Koo) then we can find an integer s.t. T E GL2(K· Fqn). If this is the case we define Nn(T) = T T" . . . T""1 = To· One can show without difficulties ( I) The conjugacy class of To is defined over K. (2) If we aconjugate T into aTau = T' with a E GL2 (K· Fqn) then aNn(T)a1 = Nn(T'), and we may find an a E GL2(KFqn) , s.t. we have Nn(T') = Nn(aTa") = To E GL2 (K).
The following assertion is slightly less trivial. (3) If NiT) and Nn(T') are conjugate in GL2(Koo) then T and T' are aconjugate. To prove (3) one has to use Galois cohomology and classfield theory. We shall give enough hints for the proof of (3) in the following discussion. We shall always assume that NiT) = To e GL2 (K) and moreover we shall assume that To is semisimple. We are free to do so since we may pass from n to n · P which raises To into the pth power. Therefore we end up with two possibilities : To is central and then we say that the aconjugacy class is central. To is semisimple and no power T3 with k ;:;;; 1 is central. In this case we shall say that the aconjugacy class is not central. If To generates a field we shall call this class nondecomposed. If To sits in a split torus we call the class decomposed. We consider the central conjugacy classes first. If T E GL2(KFqn) and NiT) = To is a central element, then the image f' of T in PGL2(KFqn) defines a 1 cocycle in zl(KFqn/K, PGL2(KFqn)) . Then it follows from Hilbert's Theorem 90 that T and T' are aconjugate if and only if r and r ' define the same cohomology class in Hl(K· Fqn/K, PG1 2(KFqn) ) and then we find that the central aconjugacy classes are in 11 correspondence with the elements of exponent 2 in the Brauer group of K. We shall not say very much about the case of noncentral aconjugacy classes. The only thing we want to mention is that if To = Nn(T) is noncentral, then we have either To E Ex 4 GL2(K) where E/K is a quadratic extension or To E T(K) where T/K is a split torus. In both cases we must have that T is in the centralizer of To• i.e., T E E� or T E T(Koo). 2.3.3. aGL2 ((!).)conjugacy. We come to the discussion of (ii). Let us assume that we have a E GL2(KFqn) ; we ask : Can we solve
and how many aGL2((!).)conjugacy classes for a. do we get? Let us assume in the first case that
370
G. HARDER AND D. A. KAZHDAN
where z is a central element, i.e. z E Kx and e is a unit in K;. Then we may modify r into u.yu;;" with u E K'; s.t. Nn(u. r u;;") = u. u;;u" NnCr) = u. u;;u" Cp z
and we can find a u such that uu"" e = 1 . Then we get x.ua.u"x;;" = uyua and we may change a. > ua.u" since we do not change the aGL2(m.)conjugacy class. This implies that we have Nn(x. a. x;;") = x.N(a.) x;;lln = z E K x .
If we pass to the projective linear group w e find x.N(a.) x;;""
=
1 , i.e.,
N(a .) = x':' x;;1 .
But if we take n large with respect to the order given by divisibility we find x':' x;;1 is close to one and therefore we find an element k. E PGL2(m.) s.t. k�"k;;1 = x�"x;;1 • Changing again a. > k.a.k;;" we find that within the aGL2(m.)conjugacy class of our original a. we may assume Nn(a.) = z. E K.x. It is easy to see that then 21n and
Since we know that H1(KFqn/K, PGL 2 (@ ( Fqn) )) is aconjugate to the element
=
0 it follows that in this case a.
(� '"0).
We ask for which {r} the condition Nn(r) = ro = c p · zP is fulfilled. First of all we observe that this is certainly true if {r} is a central class or if {r} is nondecomposed and the field generated by ro is nonsplit at lJ. In the latter case we just take n to be even and consider the valuation. We say that {r} is nondecomposed at p. Let us assume that ro is noncentral and splits at )J. In this case it is clear since r has to be in the centralizer of ro that ro is conjugate in GL2(Kp) to
and then a. has to be aGL2(@.)conjugate to
( )
?r� 0 0 I .
We shall say that {r} is decomposed at v. Therefore we find that we have only two possible aGL2(@.)conjugacy classes for a. namely
{(�· �)}.
and r has to be aconjugate to one of them at v . 2.3.4. The orbits. Let us assume we have a global aconjugacy class {r} and a., iiw and a solution xg = r 1 x0aviiw · Using approximation we may very well
GL2
x0•
37 1
OVER FUNC TION FIELDS
assume that E GL2(l0.), Xow E GL2(l0w). Then any other solution with the same r, a., iiw is of the form X = Xo Y where y has to satisfy (i) If we write y = y'y. Yw where y ' has component 1 at v_ and w then y' E GL2(Ap, q), i.e., y' is rational outside of p, q . (ii) y.a.v;" = a., Ywawy;;,u = iiw · We define the 0'centralizers Zu(a .) = {Yv E GL2(lV.) jy.a.y;u = a. }, Zu {iiw) = {Yw E GL2(lOw) iYwiiwY";;;u = iiw }• We find that the solutions of xu = rlx a. iiw form an orbit under the action of the group GL2(Ap, q) x Zu(a.) x Zu(iiw). We compute the stabilizer. To do this we observe first that in the nondecomposed case •
where D(K.) x is the multiplicative group of the quaternion division algebra at v and D(K.Y n GL2(lV.) = n m (K.) is the maximal compact subgroup of element whose reduced norm has absolute value one. Then D. = D(K.)x/D m (K.) = Z. In the decomposed case we have
and Zu
where
UP
((�• �)) n GL2(l0.) =
UP
x
UP
is the group of units. In this case we find
Since Zu(a.) n GL2(lV.) acts obviously trivially on the points of the moduli space we find that the set of points in question is an orbit under GL2(Ap, q) X n . X jjw · The 0'Centralizer of r is mapped by a + X()1axo into GL2(Ap, q) X n. X jjw and one sees easily that this 0'centralizer is the stabilizer we are looking for. Therefore our orbit is Zu(r)\GL2(Ap, q) X n. X jjw · Such an orbit Z(r) \GL2(Ap, q) X D. X jjw c M(Fq) is invariant under the action of the Frobenius. To see this we start from We can choose a. = a�, iiw = a� and then it is obvious that for x ' = rxu we have x ' = XoYaviiw and this implies that the Frobenius acts on Zu(r)\GL2(Ap, q) X n. X Dw by multiplication on the right by a.iiw E .D. x Dw .
372
G. HARDER AND D. A. KAZHDAN
We recall that a. =
1
nondecomposed case,
E Z = jj "'
a. = (1,
0) e Z
x
Z = D., decomposed case.
The group of rational line bundles with level structure along D acts on Md (compare 2. 1) and this gives us an action of P = proj limDPD, p · q r/: supp(D), on M. We not ice that P __:::____, /Kf( Up X Uq) · Kx where IK is the adele group, and Up (resp. Uq) are the units at p (resp. q). Since IK is isomorphic to the centre Z(A) c GL2(A) we see that the above action is on the orbits given by Z E Z(A),
X + XZ,
X E Z..(r)\GLz(Ap, q)
X
n.
X
jjw·
The considerations have shown us so far that the aconjugacy class {r} determines the rest of our data and also determines the orbit. But we still have to discuss the question : For which {r} do we get a nonempty orbit, i.e., for which {r} can we solve (* ) ? Let u s assume that our class { r } i s central. Then i t follows from x" = r1xa.aw that r is a boundary at all places different from p and q. Therefore it follows that the quaternion algebra (or better central algebra) defined by r has to be the one which is nonsplit at p and q and split everywhere else. On the other hand we know that there is exactly one central aconjugacy class { r } which corresponds to the unique quaternion algebra which is nonsplit at p and q . For this particular {r} we can solve ( * ) with a. =
(01 l'rv\o)'
and we get exactly one orbit this way. This orbit is called the supersingular orbit. If r is not central then r determines either a quadratic field extension E/ K or a split torus T/K. We shall always assume that E/K (resp. T/K) are embedded into GL2 in such a way that resp. units(T(Kp) ) units(T(Kq) )
= up
=
uq
X
x
up
4
uq 4
GLz(lVp), GL2(lVq) .
Now we get from (* ) that for n sufficiently large we have div (Nn(r)) = np  n q . This implies : If E/K does not split at p and at q then a E Kx , €: E global units in Ex . But then it is clear that {r} is a central class, this implies that those field extensions which are not split at p or q cannot contribute to the nonsupersingular orbits.
GL2
OVER FUNCTION FIELDS
373
On the other hand we shall see that any extension which does split at one of the two places will contribute to the points by one or two orbits. To see this we assume for instance that lJ decomposes in E, lJ = lJilJ2 · Then we can find an element r E Eoo, s. t. the divisor of r is of the form div(r) =  q ' + lJ I  a + a17 where q ' is a prime in E"' which lies above q and a is a positive divisor which does not contain lJ and q in its support. The existence of such a divisor a follows again from Lang's theorem applied to the Jacobian of E. The norm of r will have the divisor div(Nn{r)) = nih  nq1 where q1 = q if q is nonsplit and where q1 is one of the divisors above q in E if q does split in E. Since E/K has a nontrivial automorphism which can be realised as an inner au tomorphism of GL2(K) we see : Each field extension E/K which is split at one of the places lJ or q contributes to the set of orbits. It gives one orbit, if it splits at exactly one of the places ; it gives two orbits if it splits at both places. 2.3.5. The computation of the trace of Frm . We recall that we want to compute the trace Frm iH i ( DD• Q 1 ) ru . To do this we proceed as follows : Our character w contains an element il. E Pn of infinite order in its kernel. We divide the scheme Un/Fq by the action of the infinite cyclic group {il.} . Then we get U U <j} = Z. Un / {il.} = e;eEP v/ {A 2 l
The scheme Z/Fq is quasiprojective. (Compare 2.2.) On this scheme Z/Fq we have an action of the finite group Pn/{il.} which is in duced by the tensorisation with line bundles and H ; ( Dn , Q I) ru = H ; (Z , Q I ) ru c H ; ( Z , Q I ). Therefore we get the information we want, if we are able to compute the traces of Frm o o on H;(Z , Q 1 ) where o runs over the elements of Pn/{il.} . To compute these traces we have to compute the number of fixed points of Frm o o on the set Z(Fq). These fixed points can be counted on the individual orbits, which have been de scribed above. To prove the main equality we shall compare these contributions to the corresponding ones in the Selberg trace formula. We recall the description of an orbit from 2.3.4. It is a set Z11 (r) · {il.}\GL2(Ap, q) x D. x Dw. The element o acts on it by multiplication by an idele (J E Z(A) and Frm is multiplication by (a. iiw) m . Let %� c GL2(Ap, q) be the open compact subgroup defined by D. We define the function 1/fD . m . w as the characteristic function of the set U vE Z %� l"(a. iiw) m · lJ Z(K) where Z(K) is of course the centre of GL2(K). Then we get from very elementary considerations that the number of fixed points on our orbit is equal to ·
·
a conj.
�
classes in Zu (r) IZ (K)
J
_
_
Za (K) \GL2 (Ap, q) x n. x Dw
1/f n . m . o(x I ax) dx
where Za (K) is the centralizer of the element a E GL2(K) in GL2(K). The Haar measure dx has to be chosen in such a way that the volume of the open compact subgroup Jf'_D x {0} x {0} becomes one.
374
G. HARDER AND
D.
A. KAZHDAN
Once we have the above formula for the number of fixed points we can easily derive a formula for the trace of Frm on H1( DD • Q 1 ).,. We define a new function 1/fD . m ., : GL2(Ap, q) x .D. x Dw � C by •
llfD.m . ., (x) = w (z) = 0
if X = k� · z · (a. aw) m , Z E Z(A), k� E .Y('�, otherwise.
Then it is quite easy to see that the contribution of the orbit to the trace of Frm is given by the expression a conj. classes I: in Z,(r)IZ(K)
vol(Za(A)/Za(K) . Z(A)) . ID . m , .,(a)
where ID . m ,.,(a) = Jz.CAJIGLz(Ap. ,J xli.xliw 1/fD . m , ., (x 1 ax) dx and dx is the quotient mea sure on Za (A)\GL2(Ap, q) X .D. X Dw, s.t. dtdx = dx and voldt(Za(A) n .Y('D X {0 } X {0 } ) = I.
At least at this point the reader will realize that the above considerations do not hold for the decomposed orbit. The decomposed orbit will get a special treatment in 2.3.7. We mention that the points on the nondecomposed orbits are contained in U(Fq).
Since the functions 1/fD , m . w are the products of local functions we find that the integral Iv . m . .,(a) will be a product of local integrals. We find the value Iv.m, ., (a) =
(JZa (Ap. 0)\GLz (Ap,
= I�. ., (a) · o(m,
0)
)
x�(x'l ax') dx' · o(a ,
m)
a),
where o(m, a) = op(m , a) . Oq(m , a) and op(m , a) = J Za(Kp) \Zu(«v) l/fD, m, "'· p(xp1 axp) d:xp. Again we choose dxP to be the quotient measure dxP = dxP / dtP with Now we are ready for the comparison to the Selberg trace formula. 2.3.6. Comparison to the Selberg trace formula. We are now in a situation which has been studied by R. Langlands in several different papers. (Compare [3], [5].) The trace of the operator f/Jm on the Hilbert space L�. disc (GL2(K)\GL2(A)/.Yt'v) will be computed by means of the Selberg trace formula. For simplicity we shall assume that our character w is the trivial character. This simplifies the notation considerably, we shall omit all the indices w from now on. The trace formula gives an expression for the trace which is a sum of contribu tions from the different conjugacy classes in GL2(K)/K and some extra terms which are called the Eisenstein terms (compare [1, §6]). The Eisenstein terms can of course be ignored in the proof of the main equality. First of all we consider those terms in the trace formula which come from the identity element and the anisotropic semisimple elements. Each such semisimple element defines a quadratic extension E/K. Then the comparison works out as follows. (i) The supersingular contribution to the trace Frm = contribution of the identity element + contribution of all quadratic extensions which do not split at p and q . (Reminder. These extensions do not give orbits !)
GL2
375
OVER FUNCTION FIELDS
(ii) The sum of the contributions from the extensions which do split at p or at q to the trace of Frm = contribution of the same extensions to the Selberg trace. Let us look at (i). The supersingular orbit is obtained from the quaternion algebra D/K which is not split exactly at the two places p and q. From our discussion in 2.3.4 we get that Z,(r) = D(K)* and Z,(a.) = D(Kp)*, Z,(aw) = D(Kq)*. We have where n m means that the reduced norm should have absolute value one. The orbit is D(K) · D C ll (Kp) · DC ll (Kq)\D(A) = D(K)\GL2(Ap, q) x Z x Z. The conjugacy classes in D(K)* /K* are given by the identity element and the ele ments in E*/K * mod conjugation where E/K runs over those quadratic extensions which are nonsplit at p and q. Therefore the contribution to the trace of Frm is vol{D(K)* · Z(A)\D(A)*) · If!v . m(e) + a sum over the quadratic extensions E/K which do not split at p and q and the contribution of E/K is equal to
j AUT(E) j
I; aE E*/K*
vol(/Efh E*) · l �(a) · o(a, m) .
The contribution in the Selberg trace formula looks very similar : vol{GL2(K) · Z(A)\GLz(A)) · f/Jm (e) ; for the central class and for any extension E/K we get
j AU�(E)j � aE
with eJ(a, f/Jm)
=
K•
vol(/EfhE*) · l [ia) · eJ(a, (/Jm)
eJp(a, f/JIJl · Oq(a, f/J �l) and ep(a,
(/J�l)
=
J
Za (Kp) *IGL2 (Kp)
(/JfJl (xpl axP) dxp .
The decisive point is that we can compare the factors 1/fD . m (e) and (/Jm(e) and o(a, m) and 8(a, (/Jm). Again we look at the central classes first. We observe that vol(D(K) · Z(A)\D(A) *) = (q  1) 2 vol{GLz(K) · Z(A)\GL2(A)) . This equality is equivalent to the fact that the projective group of the division al gebra and the projective linear group have the same Tamagawa numbers. The factor (q  1) 2 is due to the fact that we normalised our measures in such a way that the volumes of %� x D C ll (Kp) x D C l l (Kq) and %� x GL2(@p) x GL2(@q) are both equal to one. On the other hand it is clear that 1/fv . m(e) = 1 if m is even and zero if m is odd. We claim (/JfJl(e) =  (q  1) if m is even, if m is odd. = 0 To verify this assertion we introduce the functions fn E .Yt'p for n
� 0.
376
G. HARDER AND D. A. KAZHDAN
and on the other GL2((!Jp)doublecosets we put In equal to zero. Then one can check rather easily that (/)�P)
=
[n/2]
In  (q  I ) � ln2·v· v=l
This formula is also obtained by inverting the formula in Lemma 3 . 1 in [4]. This implies of course our claim and we get for m even (/)m (e) = (q  1) 2 • This proves the equality of the central contributions. Let us now pick a noncentral element a E D K . We compute op(a, m)
=
J
Za (Kp) *\ D (Kp) *
IffD' m' p (xp1 axp) dxp.
The crucial observation is that the conjugacy class { x;1 axp}xp 0 if this extension is ramified at p. On the other hand we can compute the orbital integrals Op(a, w;,fl) and Oq(a, (/)�l) by using Langlands' formulas in [4, 3.x]. We observe that Lemma 3.5 implies that for u = (3 !),
J
Zu (k •) \GLz (Kp)
(/)�l(xp1 uxP) dxP
=
0.
Now it follows from Lemma 3.6. that in case our element a generates an unramified extension at Op(a, wf;;l ) = 2 if m is even, = 0 if m is odd, and EJp(a, (/)CPl) = I if a generates a ramified extension [4, 3. 7, 3.8]. We get that the corresponding terms o n both sides are equal. At this point we want to say that we did not check the case of characteristic 2. To do this one has to compute the op(a, m) and ep(a, wf;;l ) in the case where a generates an inseparable extension. Let us now look at the other nondecomposed orbits. They come from field ex tensions which are split at at least one of the places p or q. Let a E GL2(K) be an element which generates a field extension which splits at p. We have to compute op(a, m) and Op(a, (/)�l). Since the extension splits at p, we find Dv = Z x Z and the image of a in Dv is given by ( ordp,(a), ordp, (a)) . We have to pick one of the two primes p ' out of the two primes p' , p" which lie above p. Then the action of Frm i s given by (m, 0) E Dv. This means that
GL2 op(a, m)
=
o�(a, m)
=
1 if ordP' (a)  ordP" (a) otherwise. On the other hand it follows from [4, Lemma 3 . 1 ) that =
377
OVER FUNCTION FIELDS
=
0
=
m,
1 if l ordp,(a)  ordP" (a) l = m, otherwise. Let us assume that our extension does not split at q. Then we could also pick p" instead of p" . We get the same orbit. Therefore we get twice the trace of Frm if we work with the function a;(a, m) + a;(a, m) on the lefthand side of the trace for mula. But obviously we have o;(a, m) + o;(a, m) = Bp(a, l/J�l). If we now recall that there is a factor t on the side of the Selberg trace formula and that there is no such factor in the formula for Frm we see that the contributions of the given field on both sides are equal. The argument is very similar in the case that the field extension E/K generated by a is split at p and at q. In that case we have seen in 2.3.4 that we get two orbits. These orbits correspond to the four possible ways of selecting primes above p and q, where the selections conjugate under the action of the Galois group give the same orbit. From here on everything is the same as before and we get the desired equality of the contributions. Now we have compared the contributions coming from the central and elliptic conjugacy classes. There are no contributions from the unipotent class on the side of the Lefschetz fixed point formula. But these contributions do also vanish in the Selberg trace formula. We have already mentioned above that the orbital integrals of (/J�l over the unipotent class vanish. Since this happens at two places it follows from a standard argument (compare [2, p. 523]) that these terms are zero. What remains to be done is the comparison of the terms coming from the decom posed orbit and the split semisimple elements. At this moment we have to recall the definition of UfFq and these considerations deserve a new paragraph. 2.3.7. The decomposed orbit. Let T/K be the standard maximal torus of diagonal matrices in GL2/K. We denote by N+ (resp. N) the unipotent radicals of the Borel subgroups of upper (resp. lower) triangular matrices. Let us choose an element r e K� s.t. div(r) = p  q + a  a If r E K Fq then ro = Nn(r) has the divisor div(ro) = np  nq . We can find an idele x0 in IK00 such that 8p(a, (/J�l)
=
0
u
· .
and we put
( )
.
( )
x0 0 r o xo = 0 I ' r  o 1 · The points in M(Fq) in the decomposed orbit are those which correspond to the elements of the set x0y with y E GL2{Ap, q) x T(Kp) x T(Kq) and the orbit is T(K)\GL2(Ap. q) x (Z x Z) x (Z x Z). Here (Z x Z) = T(Kp)/T((!}p) (resp. T(Kq)/T ((!}q) ) and Frm acts by (m, 0) x (  m, 0) E (Z x Z) x (Z x Z). We have to decide which points of this orbit belong to U(Fq) · To see this we write _
378
G. HARDER AND D. A. KAZHDAN
y = t+(y) · n+(y) · k'+ (y),
resp. y = t( y) · n(y) · k' (y)
with t+(y), t(y) e T(A p, q) x Z4 , n+(y) e N + (A p, q) (resp. n(y) e N(A p, q) ) and k' +(y), k'(y) e %'. These two different ways of writing y correspond to the two different ways of getting the corresponding FHsheaf as an FHextension. To verify this we recall the situation in 1 . 1 . We find that the FHsheaf is given by the lattice Xo ·Y · Mo = Xo · t+(y) · n+(y) · Mo = Xo · t(y) · n(y) · Mo = M.
In both cases we find a subsheaf of rank 1 in M, namely the one generated by the first, resp. second, basis vector. We call these subsheaves L (resp. HI) · Then it is clear that L is an FHsheaf of rank 1 and HI is an invertible sheaf over X/Fq . If we write 0 +(y) = 't O tt '
( )
t
we find (the degree of x0 is zero !)


deg (HI) = deg (tz) . deg (L) = deg (tt) =  logq (I tt l), We get a point in U(Fq) if and only if (compare 2. 1) l tt l < i tt I and lt2 1 ;;;; l t1 1 · If 11: is a uniformizing element at a place of K then we have the following identity for all/ > 0 :
(� 71:�) = (��11:�) . (�! �) . (�! �)
and this identity allows us to compute the difference deg(t1)  deg(tz) deg (tt) deg (tt) + 2 · A(n+(y)) where jt (n+(y)) = I; q' *P, q Aq•(nt) and if nt e (!)q ' • Aq•(nt) = 0 = degq,(nt) if nt ¢ (!)q '•


i.e. , if nt = eq ' 11:;/q ' and _[q, > 0 then Aq•(nt) = _[q • . Then we find that x0 y = x0 t+( y) · n+(y) · k'(y) gives a point in U(Fq) if and only if deg(tt) > deg (tt) and if moreover 2 · jt (n+(y) ) ;;;; deg (tt) deg (tt). We introduce the group

•
T C l l (A) =
{(g t�) / l tlft2 1 = 1 }
and we get from a simple counting argument that the contribution of the decom posed orbit to the trace of Frm is equal to I;
aE T (K) !K
2 vol ( T C ll (A)/T(K) Z(A)) . ]�(a)o(m , a)
where i�(a) =

J
:K' xN+ (Ap. ,)
x�(k'nI ank') · A(n) dk'dn.
G L2 OVER FUNCTION
FIELDS
379
Again we compare this to the corresponding term in the Selberg trace formula where the corresponding integral is given by
J ,Jf" x N+ (A)
XD o (/)�l
o
(/) � (k1 n1 ank) · �(n) dk dn.
Here are some differences since in the second integral we are also integrating over the variables np and nq and �(n) = I; Aq'(nq,) + Ap(np) + .A.q{nq) = .A.(n' ) + .A.p(np) + .A.q(nq) . q ':;t:p, q
Now we come to the last point of the proof. We show that the integrals
J
XD o tPf,rl
o
tPff}l (k1 n1 ank).A.p(np) dk dn
vanish and the same is true for q instead of 1J. To see this we observe first of all that we can only get a nonzero contribution if div(a) = ± m1J  ( ± mq). This is clear since a certainly has to be a unit outside of 1J and q. Then the assertion follows from our knowledge about fJp(a, tPf,rl) and 8q(a, tPfi}l) . But if this is so then we shall see that the integral J N+ (Kpl tP':/ (np1 anp) Ap(np) dnP vanishes. This follows from the fact that the integrand itself is zero. To see this we start from the fact that .A.p(np) = 0 implies nP ¢ &p but then one checks easily that np1anP = an;anP ¢ supp ((/)�l ) . This follows for instance from the formulas relating the (/) �l and thefn· Therefore the value of the integral in the Selberg trace formula is ID(a) · 8p(a, (/)�l) · Oq (a, (/)�l) .
There is a factor
t
in front of these terms in the Selberg trace formula, but the
8p(a, tP}/:l) · 8q(a, (/)�l) is four times as often equal to one as o(a, m) and this
proves the equality of the decomposed contributions. This finishes the proof of the main equality. REFERENCES 1. S. Gel bart and H. Jacquet, Forms on GL(2) from the analytic point ofview, these PROCEEDINGS, part I , pp. 21 325 1 . 2 . H. Jacquet and R . Langlands, Automorphic forms o n GL(2), Lecture Notes i n Math . , vol. 1 1 4, SpringerVerlag, New York, 1 970. 3. R. Langlands, Modular forms and 1 adic representations. II, Proc. Summer School Antwerp, Lecture Notes 349, 1 973, pp. 361500. 4. , Base change for GL(2), Mimeographed Notes, Inst. for Advanced Study, Princeton, N.J., 1975. 5. , On the zetafunctions of some Shimura varieties, Mimeographed Notes, lnst. for Advanced Study, Princeton, N.J., 1 977. 6. D. Mumford, Unpublished letter to Harder and Kazhdan. 7. H. Saito, Automorphic forms and algebraic extensions of number fields, Lecture Notes in Math . , Kyoto University, 8, 1975. 8. P. Deligne, La conjecture de Wei/, lnst. Hautes Etudes Sci. Pub!. Math. 43 (1 974), 273307. 9. , Poids dans Ia cohomologie des varietes a/gebriques, Actes du Congres Internat. des Math. , Vancouver, 1974. 


MATHEMATISCHES INSTITUT, BONN HARVARD UNIVERSITY
AUTHOR INDEX Arthur, James, Eisenstein series and the traceformula, p. 253, part 1 Borel, A., Automorphic £functions, p. 27, part 2 Borel, A . and Jacquet, H., Automorphic forms and automorphic representations, p. 1 89, part 1 Cartier, P., Representations of padic groups : A survey, p. I l l , part 1 Casselman, W., The Hasse Wei! (,function of some moduli varieties of dimension greater than one, p. I4I, part 2 Deiigne, Pierre, Varietes de Shimura : Interpretation modulaire, et techniques de construction de mo�es canoniques, p. 247, part 2 , Valeurs defonctions L et periodes d'integrales, p. 3 1 3, part 2 Flath, D., Decomposition of representations into tensor products, p. 1 79, part 1 Gelbart, Stephen, Examples of dual pairs, p. 287, part 1 Gelbart, Stephen and Jacquet, Herve, Forms of GL(2) from the analytic point of view, p. 213, part 1 Gerardin, Paul, Cuspidal unramified series for central simple algebras over local fields, p. I 57, part I Gerardin, P. and Labesse, J.P. , The solution of a base change problem for GL(2) (following Langlands, Saito, Shintani), p. 1 1 5, part 2 Harder, G. and Kazhdan, D. A., Automorphic forms on GL2 over function fields (after V. G. Drinfeld), p. 357, part 2 Howe, R., {}series and invariant theory, p. 275, part 1 Howe, R. and PiatetskiShapiro, I. 1., A counterexample to the "generalized Ramanujan conjecture " for (quasi) split groups, p. 3 1 5, part 1 lhara, Yasutaka, Congruence relations and Shimura curves, p. 291 , part 2 Jacquet, Herve, Principal £functions of the linear group, p. 63, part 2 See Borel, A.  , See Gelbart, Stephen , Kazhdan, D. A., An introduction to Drinfeld's "Shtuka ", p. 347, part 2 , See Harder, G. Knapp, A. W., Representations of GL2(R) and GL2 (C), p. 87, part 1 Knapp, A. W. and Zuckerman, Gregg, Normalizing factors, tempered representa tions, and Lgroups, p. 93, part 1 Koblitz, N. and Ogus, A., Algebraicity of some products of values of the F function (an appendix to Valeurs de fonction L et periodes d'integrales by P. Deligne) p. 343, part 2 Kottwitz, R., Orbital integrals and base change, p. 1 1 1 , part 2  , Combinatorics and Shimura varieties mod p (based on lectures by Lang lands), p. 1 85, part 2 Labesse, J.P., See Gerardin, P. Langlands, R. P., On the notion of an automorphic representation. A supplement to the preceding paper, p. 203, part I 
 ,


, Automorphic representations, Shimura varieties, and motives. Ein Miirchen,
p. 205, part 2
381
3 82
AUTHOR INDEX
Lusztig, G . , Some remarks on the supercuspidal representations ofpadic semisimple groups, p . 1 7 1 , part
1
Milne, J . S . , Points on Shimura varieties mod p, p. 1 6 5 , part 2
Novodvorsky, Mark E . , A utomorphic Lfunctions for the symplectic group GSp4, p . 8 7 , part 2
Ogus,
A.,
See Koblitz, N.
PiatetskiS hapiro, 1.,
Classical and adelic automorphic forms. A n introduction,
p . 1 8 5 , part 1

, Multiplicity one theorems, p. 209, part
, See Howe,
R.
Rallis, S . , On a relation bet ween
SL2 cusp forms and automorphic forms on orthogonal
groups, p. 297, part 1 Shelstad ,
D.,
1
Orbital in tegrals for
GL2(R),
p. 1 07 , part 1
 , Notes on Lindistinguishability (based on a lecture by R. P. Langlands). p . 1 9 3 , part 2 Shintani, Takuro, On liftings of holomorphic cusp forms, p . 97, part 2 Springer, T. A . , Reductive groups, p. 3 , part
1
Tate, J., Number theoretic background, p . 3, part Tits , J., Reductive groups o ver local .fields, p. Tunnell,
29,
2 part 1
J., Report on the lo cal Langlands conjecture for GL2, p.
1 3 5 , part 2
Wallach , Nolan R . , Representations of reductive Lie groups, p. 7 1 , part 1 Zuckerman , Gregg, See Knapp, A. W.
BC DEFGHIJ CM  8 9 8 7 6 5 4 3 2 1 0