Proceedings of Symposia in
PURE MATHEMATICS Volume 33. Part 2
Automorphic Forms, Representations, and L-functions Symposium in Pure Mathematics held at Oregon State University July 11-August 5, 1977 Corvallis, Oregon
A. Borel
W. Casselman
American Mathematical Society
Automorphic Forms, Representations, and L-functions
Proceedings of Symposia in
PURE MATHEMATICS Volume 33, Part 2
Automorphic Forms, Representations, and L-functions A. Borel
W. Casselman Editors
American Mathematical Society Providence, Rhode Island
PROCEEDINGS OF THE SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY
HELD AT OREGON STATE UNIVERSITY CORVALLIS, OREGON JULY 11-AUGUST 5, 1977 EDITED BY
A. BOREL and W. CASSELMAN
Prepared by the American Mathematical Society with partial support from National Science Foundation Grant MCS 76-24539
Library of Congress Cataloging in Publication Data
Symposium in Pure Mathematics, Oregon State University, 1977. Automorphic forms, representations, and L-functions. (Proceedings of symposia in pure mathematics; v. 33) Includes bibliographical references and index.
1. Automorphic forms-Congresses. 2. Lie groups-Congresses. 3. Representations of groups-Congresses. 4. L-functions-Congresses. I. Borel, Armand. II. Casselman, W., 1941- III. American Mathematical Society. IV. Title V. Series. QA331.S937 1977 512'.7 78-21184 ISBN 0-8218-1435-4 (v.1) ISBN 0-8218-1437-0 (v.2)
AMS (MOS) subject classifications (1970). Primary 10D20, 22E50, 12A70; Secondary 20G25, 20G35, 14G10, 22E55, 22E45. Copyright © 1979 by the American Mathematical Society Printed in the United States of America All rights reserved except those granted to the United States Government. This book may not be reproduced in any form without permission of the publishers.
CONTENTS Foreword .............................................................ix
Part 1 1. Reductive groups. Representations Reductive groups ...................................................... 3
By T. A. SPRINGER
Reductive groups over local fields
....................................... 29
By J. TITS
Representations of reductive Lie groups .................................. 71 By NOLAN R. WALLACH
Representations of GL2(R) and GL2(C) ..................................87 By A. W. KNAPP
Normalizing factors, tempered representations, and L-groups ..............93 By A. W. KNAPP and GREGG ZUCKERMAN Orbital integrals for GL2(R) ........................................... 107
By D. SHELSTAD Representations of p-adic groups: A survey .............................III
By P. CARTIER
Cuspidal unramified series for central simple algebras over local fields ....... 157 By PAUL GERARDIN
Some remarks on the supercuspidal representations of p-adic semisimple groups ........................................................... 171
By G. LUSZTIG
II. Automorphic forms and representations
................... 179 Classical and adelic automorphic forms. An introduction ................. 185 Decomposition of representations into tensor products By D. FLATH
By 1. PIATETSKI-SHAPIRO
Automorphic forms and automorphic representations
.................... 189
By A. BOREL and H. JACQUET
On the notion of an automorphic representation. A supplement to the preceding paper ................................................... 203
By R. P. LANGLANDS
Multiplicity one theorems .............................................209 By 1. PIATETSKI-SHAPIRO
Forms of GL(2) from the analytic point of view .........................213 By STEPHEN GELBART and HERVE JACQUET
Eisenstein series and the trace formula .................................253 By JAMES ARTHUR 9-series and invariant theory ..........................................275
By R. HOWE v
CONTENTS
Vi
Examples of dual reductive pairs
......................................287
By STEPHEN GELBART
On a relation between SL2 cusp forms and automorphic forms on orthogonal
groups ........................................................... 297 By S. RALLIS
A counterexample to the "generalized Ramanujan conjecture" for (quasi-) split groups ....................................................... 315
By R. HOWE and I.I. PIATETSKI-SHAPIRO
Part 2 111. Automorphic representations and L-functions Number theoretic background ...........................................3
By J. TATE Automorphic L-functions .............................................. 27
By A. BOREL
Principal L-functions of the linear group
................................. 63
By HERVE JACQUET
Automorphic L-functions for the symplectic group GSp4 By MARK E. NOVODVORSKY
On liftings of holomorphic cusp forms
................... 87
.................................. 97
By TAKURO SHINTANI Orbital integrals and base change ......................................111
By R. KOTTWITZ
The solution of a base change problem for GL(2) (following Langlands, Saito, Shintani) .................................................... 115
By P. GERARDIN and J.-P. LABESSE
Report on the local Langlands conjecture for GL2
....................... 135
By J. TUNNELL
IV. Arithmetical algebraic geometry and automorphic L-functions
The Hasse-Weil ic-function of some moduli varieties of dimension greater than one
......................................................... 141
By W. CASSELMAN
Points on Shimura varieties mod p
..................................... 165
By J. S. MILNE
Combinatorics and Shimura varieties mod p (based on lectures by Langlands)
....................................................... 185
By R. KOTTWITZ
Notes on L-indistinguishability (based on a lecture by R. P. Langlands)
.... 193
By D. SHELSTAD
Automorphic representations, Shimura varieties, and motives. Ein
......................................................... 205
Marchen By R. P. LANGLANDS
CONTENTS
Vii
Varietes de Shimura: Interpretation modulaire, et techniques de
construction de modeles canoniques ..................................247 By PIERRE DELIGNE
Congruence relations and Shimura curves ...............................291 By YASUTAKA IHARA
Valeurs de fonctions L et periodes d'integrales ........................... 313 By P. DELIGNE
with an appendix:
Algebraicity of some products of values of the r function ................. 343 By N. KoBLITZ and A. OGUS
An introduction to Drinfeld's "Shtuka" ................................347 By D. A. KAZHDAN
Automorphic forms on GL2 over function fields (after V. G. Drinfeld) ......357 By G. HARDER and D. A. KAZHDAN
Index Author Index ........................................................381
III AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS
Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 3-26
NUMBER THEORETIC BACKGROUND J. TATE 1. Well Groups. If G is a topological group we shall let Gc denote the closure of the commutator subgroup of G, and Gab = GIG, the maximal abelian Hausdorff quotient of G. Recall that if H is a closed subgroup of finite index in G there is a transfer homomorphism t: Gab --* Hab, defined as follows: if s: H\G -- G is any section, then for g e G,
t(gG`) = 11
x,FH\G
hg,x
(mod H`),
where hg,x E His defined by s(x)g = hg,xs (xg). (1.1) Definition of Weil group. Let F be a local or global field and F a separable algebraic closure of F. Let E, E', denote finite extensions of Fin F. For each such
E, let GE = Gal(F/E). A Weil group for F/F is not really just a group but a triple (WF, p, {rE}). The first two ingredients are a topological group WF and a continuous homomorphism tp: WF --> GF with dense image. Given WF and (p, we put WE = rp 1(GE) for each finite extension E of Fin F. The continuity of tp just means that WE is open in WF for each E, and its having dense image means that tp induces a bijection of homogeneous spaces:
GF/GE - HomF(E, F)
WF/WE
for each E, and in particular, a group isomorphism WFI WE x Gal(E/F) when E/F is Galois. The last ingredient of a Weil group is, for each E, an isomorphism of topological groups rE: CE WE, where CE
The multiplicative group E* of E in the local case, the idele-class group AEIE* in the global case.
In order to constitute a Weil group these ingredients must satisfy four conditions: (W1) For each E, the composed map rE
CE
ab induced by p G11 WE
is the reciprocity law homomorphism of class field theory. (W2) Let w e WF and a = tp(w) e GF. For each E the following diagram is commutative : AMS (MOS) subject classifications (1970). Primary 12A70. © 1979, American Mathematical Society 3
4
J. TATE rE
WE
CE induced
conjugation
by o
by w rEo
WEab
CEe
(W3) For E' c E the diagram WE
CE induced by
transfer
inclusion
E'cE
Wab
rE
CE
E
is commutative. (W4) The natural map
WF-> proj lim{WE/F} E
is an isomorphism of topological groups, where (1.1.1)
WEIF denotes WF/WE (not WF/WE),
and the projective limit is taken over all E, ordered by inclusion, as E This concludes our definition of Weil group. It is clear from the definition that if WF is a Weil group for F/F, then, for each finite extension E of F in F, WE (furnished with the restriction of cp and the isomorphisms rE, for E' E) is a Weil
group for F/E. If WF is a Weil group, then for each F c E' c E the diagram rE
CE (1.2.2)
WE induced by
norm,
NE/E'
inclusion WEcWE' rEi
CE'
. WP
is commutative. This follows from the fact that, when H is a normal subgroup of finite index in G, the composition induced by
Hab
inclusion
- Gab
transfer Hab
is the map which takes an element h e H into the product of its conjugates by representatives of elements of G/H. (In the notation of the first paragraph above, hg,x = s(x) gs(x)-1, if g e H (-- G.) (1.2) Cohomology; construction of Weil groups. Let WF be a Weil group for F/F. Then for each Galois E/F the group WEIF = WFI WE is an extension of WF/WE _ Gal(E/F) by WEI WE x CE. Let aE/F e HZ(Gal(E/F), CE) denote the class of this group extension. For each n e Z, let (1.2.3)
H^(Gal(E/F), Z) -> Hn+2(Gal(E/F), CE)
NUMBER THEORETIC BACKGROUND
5
be the map given by cup product with aE/F. Since CE, -+ CEa1(E/E') is a bijection for
F c E' c E, the property (W3) above implies, via an abstract cohomological theorem (combine the corollary of p. 184 of [AT] with Theorem 12, p. 154, of [S1]),
that an(E/F) is an isomorphism for every n. Moreover, the canonical classes are interrelated by (1.2.4)
infl aE'/F = [E:E']aE,F
and res aE/F = aEIE'
(for the first, use Theorem 6 on p. 188 of [AT]; the second is obvious). Thus, implicit in the existence of Weil groups is all the cohomology of class field theory.
For example, taking n = -1 in (1.2.3) we find Hl(Gal(E/F), CE) = 0. Taking n = 0, we find H2(Gal(ElF), CE) is cyclic of order[E:F], generated by aE/F. Taking n = -2 we find an isomorphism GEIF x CF/NE/FCE which, by (WI), is that given by the reciprocity law. For E/F cyclic, this isomorphism determines aE/F, and it follows that aE/F is the "canonical" or "fundamental" class of class field theory. The same is true for arbitrary ElF as one sees by taking a cyclic E1/F of the same degree as E/F, and inflating aE/F and aE,,F to EE11F, where they are equal by (1.2.4). Conversely, if we are given classes aE/E' satisfying (1.2.4) and such that the maps (1.2.3) are isomorphisms, then we can construct a Weil group WF as the projective limit of group extensions WE/F made with these classes. This construction is abstracted and carried out in great detail in Chapter XIV of [AT]. The existence of such classes aEIE, is proved in [AT] and [CF]. Thus, a Weil group exists for every F; to what extent is it unique? (1.3) Unicity. A Weil group for F/Fis unique up to isomorphism. More precisely: (1.3.1) PROPOSITION. Let WF and WW be two Weil groups for F/F. There exists an isomorphism 0: WF => WW such that the diagrams
WE
WF
GF and CE
are commutative.
For each finite Galois E/F, let I(E) denote the set of isomorphismsf such that the following diagram is commutative
0-> CE --> WE/F -p Gal(E/F) --f 0 id
f
id
0--> CE -' WEIF -' Gal(E/F)-> 0 Since the two group extensions WEIF and WE/F each have the same class, namely
6
J. TATE
the canonical class aE/F, as their cohomology class, I(E) is not empty. Since Hl(Gal(E/F), CE) = 0, an isomorphism f e I(E) is determined up to composing with an inner automorphism of WE/F by an element of CE 7. W. The center of WE/F is CF, and CEICF is compact. Hence I(E), as principal homogeneous space for CEICF, has a natural compact topology. For E1 E, the natural map I(E1) -> I(E) is continuous for this topology, since it is reflected in the norm map NE,/E, once we pick an element of 1(E1). Hence the projective limit proj limE(I(E)) is not empty. An element 0 of this limit gives an isomorphism WF z WW by (W4), and
has the required properties. It turns out (cf. (1.5.2)) that 0 is unique up to an inner automorphism of WF by an element w e Ker (p, but we postpone the discussion of this question until after the next section. (1.4) Special cases. We discuss now the special features of the four cases: F local
nonarchimedean, F a global function field, F local archimedean, and F a global number field. In the first two of these, GF is a completion of WF; in the last two it is a quotient of WF. (1.4.1) F local nonarchimedean. For each E, let kE be the residue field of E and
qE = Card(kE). Let k = UEkE. We can take WF to be the dense subgroup of GF consisting of the elements a e GF which induce on k the map x -> x4F for some n e Z. Thus WF contains the inertia group IF (the subgroup of GF fixing k), and WFIIF -- Z. The topology in WF is that for which IF gets the profinite topology induced from GF, and is open in WF. The map cp : WF -* GF is the inclusion, and the
maps rE: E* -> WE are the reciprocity law homomorphisms. Concerning the sign of the reciprocity law, our convention will be that rE(a) acts as x , XUaUE on k, where IIaIIE is the normed absolute value of an element a e E*. (If ZE is a uniformizer in E, then IIrEIIE = qE1; thus our convention is that uniformizers correspond to the inverse of the Frobenius automorphism, as in Deligne [D3], opposite to the convention used in [D1], [AT], [CF], and [Sl].) (1.4.2) F a global function field. Here the picture is as in (1.4.1). Just change "re-
sidue field" to "constant field", "inertia group IF" to "geometric Galois group Gal(FIFk)", and define the norm IIaIIE of an idele class a e CE to be the product of the normed absolute values of the components of an idele representing the class. (1.4.3) F local archimedean. If F C we can take WF = F*, rp the trivial map, rF the identity.
If F R, we can take WF = F* U jF* with the rules j2 = -1 and jcj-1 = c, where c -. e is the nontrivial element of Gal(F/F). The map cp takes F* to 1 and jF* to that nontrivial element. The map rF is the identity, and rF is characterized by
rF(-1) = jWF, rF(x)
_
WF,
for x e F, x > 0.
(WF is the "unit circle" of elements u e F with II u II = NF/F u = 1.) (1.4.4) F a global number field. This is the only case in which there is, at present,
no simple description of WF, but merely the artificial construction by cocycles described in (1.2). This construction is due to Weil in [W1], where he emphasizes the
importance of the problem of finding a more natural construction, and proves the following facts. The map (p: WF --f GF is surjective. Its kernel is the connected com-
7
NUMBER THEORETIC BACKGROUND
ponent of identity in WF, isomorphic to the inverse limit, under the norm maps NEIE- of the connected components DE of 1 in CE. These norm maps DE -+ DE, are surjective, and rE(DE) = (Ker cp) WE/WE is the image of Ker cp in WE/F. If Ehas rI real
and r2 complex places then DE is isomorphic to the product of R with rI + r2 -1 solenoids and r2 circles. (1.4.5) Notice that in each of the four cases just discussed the subgroups of WF which are of the form WE for some finite extension E of F are just the open subgroups of finite index. Their intersection, Ker (p, is a divisible connected abelian group, trivial in the first two cases, isomorphic to C* in the third, and enormous in the last case. (1.4.6) In each case there is a homomorphism w H II w II of WF into the multipli-
cative group of strictly positive real numbers which reflects the norm or normed absolute value on CF under the isomorphism rF : CF -_ Wpb. By (1.1.2) and the rule II NEi Fa II F = I a II E, the restriction of this "norm" function II w II from WF to a subgroup WE is the norm function for WE, so we can write simply II w II instead of II W II F without creating confusion. In each case the kernel WF of w ', II w II is compact. In the first two cases, the image of w H II w II consists of the powers of qF, and
WF is a semidirect product Z x W. In the last two cases w H IIw1I is surjective, and in fact, WF is a direct product R x W. Let us refer to the first two cases as the "Z-cases" and the last two as the "Rcases". In the Z-cases, rp is injective, but not surjective; in the R-cases, cp is surjective, but not injective.
(1.5) Automorphisms of Weil groups. Let WF be a Weil group for F/F. Let Aut(F, WF) denote the set of pairs (aa, a), where a e GF is an automorphism of F/F, and a is an automorphism of the group WF such that the following diagrams are commutative, the second for all E:
WF
W
GF
Inn(a)
WF
W
GF
CE
'E
induced
CEa
WE
byuaced
rEc
' Wab Ea
Here Inn(a) denotes the inner automorphism defined by a. We shall call an automorphism of WF essentially inner if it induces an inner automorphism on WEB F for each finite Galois E/F.
(1.5.1) PROPOSITION. In the R-cases Aut(F, WF) consists of the pairs Inn(w)), for w e WF.
In the Z-cases, Aut(F, WF) consists of the pairs (a, a,), for a E GF, where aQ denotes the restriction of Inn(a) to WF, viewed as a subgroup of GF via cp. This automorphism ao of WF is not an inner automorphism if as 0 WF, but it is essentially inner in the sense of the definition above. (1.5.2) COROLLARY. The isomorphism 0 in (1.3.1) is unique in the Z-cases, and is
J. TATE
8
unique to an inner automorphism of WF by an element of the connected component WF = Ker cp in the R-cases.
To prove (1.5.1) in the R-cases, we note first that, since cp is surjective, we are reduced immediately to the case of the corollary: We must show that if (1, a) E Aut(F, WF), then a = Inn(w) for some w E W. Going back to the proof of (1.3.1) with WW = WF we find that the group of these a's is given by (1 for norm 1)
proj lim (CE/CF) = proj lim (CE/CF) E
E
= proj lim CE/proj lim CF (by compacity) E
E
= Wo.l/(Z n W°,1)
(existence theorem; 0 for connected component)
= W°/Z as claimed, where Z is the center of W. Suppose now we are in a Z-case. Since cp is injective, i.e., WF c GF, it is clear that Aut(F, WF) consists only of the pairs (a, as). The center of GF is 1, because GFIGE x Gal(E/F) acts faithfully on CE c GE for each finite Galois E/F. Hence, since WF is dense in GF, ao is not an inner automorphism of WF unless a e WF. However, aQ does induce an inner automorphism of WE/F for finite E/F. Since WF is dense in GF it suffices to prove this last statement for a close to 1, say a E GE.
Then ao induces an isomorphism of the group extension 0 --+ CE -' WE/F -' Gal(E/F) -+ 0 which is identity on the extremities, and hence is an inner automorphism by an element of CE, since H1 (Gal(E/F), CE) = 0. (1.6) The local-global relationship. Suppose now F is global. Let v be a place of F and F. the completion of F at v. Let F (resp. F,,) be a separable algebraic closure of F (resp. and let WF (resp. WF) be a Weil group for F/F (resp. for F,/FV). (1.6.1) PROPOSITION. Let i : F - F be an F-homomorphism. For each finite extension E of F in F, let Ev = i(E)F, be the induced completion of E. There exists a continuous homomorphism 8,,: WF,, -> WF such that the following diagrams are commutative WFU
6
WF
GF
induced
by i
GF
E,'
H
E induced
ne
CE
by i,
1
WE
where n maps a e Ev to the class of the idele whose v-component is a and whose other components are 1.
If F is a function field, then By is unique. In the number field case, 0, is unique up to composition with an inner automorphism of WF defined by an element of the connected component WF = Ker cp. The proof of this is analogous to the proof of (1.3.1) and (1.5.1), using the stand-
NUMBER THEORETIC BACKGROUND
9
and relationship between global and local canonical classes, and the vanishing of Hl(Gal(E,/F,), CE). Combining (1.6.1) and (1.5.2) we obtain COROLLARY. The diagram WF,,
GFv
By
WF
.GF
is unique up to isomorphism, and the automorphisms of it are the inner automorphisms of W, defined by elements w eW°, which induce an automorphism of WF,, (i.e., for v nonarchimedean, w fixed by GF;for v archimedean, w a product of an element of W° by an element of (W°)GFv).
2. Representations. Let G be a topological group. By a representation of G we shall mean, in this section, a continuous homomorphism p : G -> GL(V) where V is a finite-dimensional complex vector space. By a quasi-character of G we mean a continuous homomorphism x: G -> C*. If (p, V) is any representation of G, then det p is a quasi-character which we may sometimes denote also by det V. The map V ' -> det V sets up a bijection between the isomorphism classes of representations V of dimension I and quasi-characters. Of course we can identify quasi-characters of G with quasi-characters of Gab. We let M(G) denote the set of isomorphism classes of representations of G, and R(G) the group of virtual representations. A function A on M(G) with values in an abelian group X can be "extended" to a homomorphism R(G) -> X if and only if it is additive, i.e., satisfies A(V) = ,1(V') A (V") whenever 0 --> V' --> V -> V" ---> 0 is an exact sequence of representations of G.
(2.1) Let F be a local or global field, F an algebraic closure of F, and WF a Weil group for F/F. Let (p, V) be a representation of WF. Since WF = proj lim {WE/E} and GL(V) has no nontrivial small subgroups, p must factor through WE/F, for some finite Galois extension E of Fin F. It follows that if a is an essentially inner automorphism of WF in the sense of (1.5), then Va x V. Thus essentially inner automorphisms act as identity on M(WF) and R(WF). By (1.5.1) we can therefore safely think of M(WF) as a set depending only on F, not on a particular choice of F or of Weil group WF for P/F, and the same for R(WF). In this sense, if v is a place of a global F, the "restriction" map M(WF) -* M(WF,,) induced by the map B of Proposition (1.6.1) depends only on v, not on a particular choice of the maps iv and By in that proposition, and the same for R(WF) -* R(WF,,). We shall indicate this map by p pv or V H Vv. (The independence from By results from (1.6.1), and the independence from iv, from (1.5.1).) If E/F is any finite separable extension, we have canonical maps
J. TATE
10
resE/F
R(WF)
restriction
R(WE)
induction indE/F
satisfying the usual Frobenius reciprocity, for we can identify WE with a closed subgroup of finite index in WF. (2.2) Quasi-characters and representations of Galois type. Using the isomorphism CF x WF we can identify quasi-characters of CF with quasi-characters of WF . For example, we will denote by ws, for s e C, the quasi-character of WF associated with the quasi-character c H II c II F, where II C II F is the norm of c e CF. Thus ws(w) _ II w II s in the notation of (1.4).
On the other hand, since gyp: WF -+ GF has dense image, we can identify the set M(GF) of isomorphism classes of representations of GF with a subset of M(WF). We
will call the representations in this subset "of Galois type". Thus, by (1.4.5), a representation p of WF is of Galois type if and only if p(WF) is finite. With these identifications, a character x of GF is identified with the character x of CF to which x corresponds by the reciprocity law homomorphism. (2.2.1) In the Z-cases, i.e., if F is a global function field, or a nonarchimedean local field, then every irreducible representation p of WF is of the form p = a OO co" where 6 is of Galois type. This is a general fact about irreducible representations of a group which is an extension of Z by a profinite group; some twist of p by a quasicharacter trivial on the profinite subgroup has a finite image; see [D3, §4.10]. (2.2.2) If Fis an archimedean local field, the quasi-characters of WF, i.e., of F* WF, are of the form x = z Nws, where z : F ---> C is an embedding and N an integer >_ 0, restricted to be 0 or 1 if F is real. If F is complex, these are the only irreducible representations of F* = WF. If Fis real, WF has an abelian subgroup WF = F* of index 2, and the irreducible representations of WF which are not quasi-characters are of the form p = Indp/F(z N(Os) with N > 0. (For N = 0 this induced representation is reducible: (2.2.2.1)
Indp/Fw, = 0), O+ X-1(01+1
where x: F - C is the embedding of F in C.) (2.2.3) Suppose F is a global number field. A primitive (i.e., not induced from a proper subgroup) irreducible representation p of WF is of the form p = a 0 x where a is of Galois type and x a quasi-character. Choose a finite Galois extension E of F big enough so that p factors through WE/F = WFI WE. Since p is primitive and irreducible, p(WE) must be in the center of GL(V), because WE is an abelian normal subgroup of WE/F. In other words, the composed map WF ? , GL(V) - PGL(V) kills WE and therefore gives a projective representation of Gal(E/F). This projective representation of Gal(E/F) can be lifted to a linear representation ao: GF -> GL(V) (see [S3, Corollary of Theorem 4]). Let a = o cp. The two compositions WF
P
GL(V)
PGL(V)
are equal; hence p = a © x for some quasi-character X. (2.2.4) Note that, in all cases, global and local, the primitive irreducible representations of WF are twists of Galois representations by quasi-characters.
NUMBER THEORETIC BACKGROUND
II
(2.2.5) In the local nonarchimedean case one can say much more about the structure of primitive irreducible representations (see [K]). A first result of this sort is
(2.2.5.1) PROPOSITION. Let F be local nonarchimedean and let V be a primitive irreducible representation of WF. Then the restriction of V to the wild ramification group P is irreducible.
This result is proved in [K] and [B]. The proof depends on the supersolvability of GF/P. (2.2.5.2) COROLLARY. The dimension of V is a power of the residue characteristic p.
Indeed, P is a pro -p-group. (2.2.5.3) COROLLARY. If U is an irreducible (not necessarily primitive) representation of WF of degree prime top, then U is monomial.
Because U is induced from a primitive irreducible V whose dimension is prime to p and a p-power, hence 1. (2.3) Inductive functions of representations. Let F be a local or global field. For a
representation V of WF, let [V] e R(WF) denote the virtual representation determined by V. Let R°(WF) denote the group of virtual representations of degree 0 of WF, i.e., those of the form [V] - [V'], with dim V = dim V'. (2.3.1) PROPOSITION. The group R(WF) is generated by the elements of the form IndE,F[x]for E/Ffinite and x a quasi-character of WE. Similarly, R°(WF) is generated by the elements of the form IndE/F([x] - [x']).
It suffices to prove the second statement, because R(WF) = R°(WF) + Z [1].
Let R*(WF) denote the subgroup of R°(WF) generated by the elements IndE/F([x] - [x']). By the degree 0 variant of Brauer's theorem [D3, Proposition
1.5] we have R°(GF) c R*(WF). The formula Ind(p O Res x) = (Ind p) O x shows that R* x c R° for each quasi-character x of WF. To prove the proposition we must show for each irreducible representation p of WF that [p] - (dim p) [1] e R*(WF). For each p there is a finite extension E of Fand a primitive irreducible representation pE of WE such that p = IndE,F PE. Then [p] - (dim p) [1] is the sum of IndE/F([pE]-(dim pE)[lE]) and (dim pE) (IndE,F[lE]-[E:F] [IF]).
The latter is of Galois type, so by the transitivity of induction we are reduced to the case in which p is primitive and irreducible. But then p = o, ©x with o e R(GF) and x a quasi-character (2.2.4). If n = dim p = dim o
[p] - n[1] = ([o] - n[1]) [x] + n([x] - [1]) and this is in R*(WF) by the remarks above, since [o] -n [1] E R°(GF). (2.3.2) DEFINITION. Let F be a local or global field. Let d be a function which
assigns to each finite separable extension E/F and each V e M(WE) an element 2(V) in an abelian group X. We say A is additive over F is for each E and each exact sequence 0 -+ V' --+ V -+ V" --+ 0 of representations of WE we have A(V) = 2(V')2(V"). When that is so we can define A on virtual representations so that A:
R(E) --+ X is a homomorphism for each E. We say A is inductive over F if it is additive over Fand the diagram
J. TATE
12
R(WE)
IndE/E'
R(WE,)
is commutative for finite separable extensions E/E'/F. We say A is inductive in degree 0 over F if the same is true with R replaced by R°. (2.3.3) REMARK. By (2.3.1) a d which is inductive over F, or even only inductive
in degree 0, is uniquely determined by its value on quasi-characters x of WE (i.e., of CE), for all finite separable E/F. In [D3, § 1.9] there is a discussion, for finite groups, of the relations a function A of characters of subgroups must satisfy in order that it extend to an inductive function of representations. (2.3.4) EXAMPLE. Let a e CF. Put
A(V) = (det V)(rE(a))
for V E M(WE).
Then A is inductive in degree 0 over F. This follows from property (W3) of Weil groups and the rule. det (Ind V) = (det V)
transfer,
for V virtual of degree 0
(cf. [D3, §1]). (2.3.5) EXAMPLE. Suppose v is a place of a global field F, and A is an inductive function over Fv. If we put for each finite 77separable E/F and each V e M(WE)
.lv(V) =
11
A(Vw)
w place of E; wlv
we obtain an inductive function Av over F. If A is only inductive in degree 0, then A, is inductive in degree 0.
Indeed, by a standard formula for the result of inducing from a subgroup and restricting to a different subgroup we have (IndE/ F V )v = +O IndEw/F0 Vw, wlv
because if wo is one place of E over v, then the map or t-+ owo puts the set of double cosets WE\WFI WF0 in bijection with the set of all such places, and for each or we can identify WE,10 with (or WF, 6r-1) (1 WE.
3. L-series, functional equations, local constants. The L-functions considered in this section are those associated by Weil [Wl] to representations of Weil groups. They include as special cases the "abelian" L-series of Hecke, made with "Grossencharakteren" (= quasi-character of CF), and the "nonabelian" L-functions of Artin, made with representations of Galois groups. Our discussion follows quite closely that of [D3, §§3,4, 5] which we are just copying in many places. (3.1) Local abelian L -Junctions. Let F be a local field. For a quasi-character x of F* one defines L(x) e C* U {oo} as follows.
(3.1.1) F . R. For x the embedding of Fin C and N = 0 or 1,
NUMBER THEORETIC BACKGROUND
13
defn L(x Nws) = rR(s) = sit r(s/2).
(3.2. 1) F -_ C. For z an embedding of Fin C and N ? 0, defn
L(Z Nws) = r'(s) = 2(2z)-sr(s). (3.1.3) F nonarchimedean. For r a uniformizer in F,
L(x) = f (1 - x('r)) l 1,
I,
if x is unramified, if x is ramified.
In every case, Lisa meromorphic function of x, i.e., L(xc)s) is meromorphic in s, and L has no zeros. (3.2) Local abelian a -functions. We will denote by dx a Haar measure on F, by d*x a Haar measure on F* (e.g., d*x = Ilxll-1 dx) and by 0 a nontrivial additive character of F. Given 0 and dx, one has a "Fourier transform"
AY) _
f(x) 0 (xy) dx.
The local functional equation (3.2.1)
11(x) w1x 11x) d*x = s(x, 0, dx) f f(x)x(x) d*x
L(x) L(w,x ) defines a number e(x, 0, dx) E C* which is independent off, for f's such that the two sides make sense. If f is continuous such that f(x) and f(x) are O(e °x'l) as I(xll -+ oo, then the two sides make sense naively for x such that x(x) = ll x II ° with 0 <
a < 1, and each side is a meromorphic function of X. One takes the same multiplicative Haar measure d*x on each side. The dependence of a on 0 and dx comes from the dependence of the Fourier transform f (x) on 0 and dx. One finds (3.2.2) (3.2.3)
for r > 0,
e(x, 0, rdx) = re(x, 0, dx), s(x, cb(ax), dx) = x(a)II a II-1 e(x, 0, dx)
for a E F*.
Easy computations carried out in [Ti] and [W2] show that the function a is given by (3.2.2), (3.2.3) and the following explicit formulas:
(3.2.4) F - R. Let x be the embedding of F in C and N = 0 or 1. For 0 _ exp(22rix) and dx the usual measure, 6(X-N(0" 0, dx) = iN. (3.2.5) F C. Let z be an embedding of F in C and
N > 0. For
_
exp(2zi Tro,Rz) and dx = idz A dz (= 2 da db for z = a + bi), s(z Nws, 0, dx) iN.
(3.2.6) F nonarchimedean. Let (9 be the ring of integers in F. Put n(cb) = the largest integer n such that 0(ir no) = 1, a(x) = the (exponent of the) conductor of x (= 0 if x is unramified, the smallest integer m such that x is trivial on units = I (mod rrm) if is is ramified),
c = an element of F* of valuation n(o) + a(x). If x is unramified, (3.2.6.1)
e(x, 0, dx) =
f A.
(In particular, e(x, 0, dx) = 1, if fo dx = 1, and n(cb) = 0 when x is unramified.)
J. TATE
14
For x ramified, defn
e(x, 0, dx)
E fF- X_1(x) 0 (x) dx =nEZ
an 0.
x-J(x) 0 (x) A
(3.2.6.2)
1(x) 0 (x) A.
From these formulas one deduces, for x arbitrary and w unramified (3.2.6.3)
e(xw, ci', dx,) = e(x, cfi, dx)
W0cnc0>+a(X)).
(3.3) Local nonabelian Lfunctions. We owe to Artin the discovery that there is an inductive (2.3.2) function L of representations of Weil groups of local fields such that L(V) = L(x) when V is a representation of degree 1 corresponding to the quasi-character X. The explicit description of L is as follows: (3.3.1) F archimedean. Since L is additive, we can define it by giving its value on irreducible V. For F complex, WF = F* is abelian, and the only irreducible V's are the quasi-characters x, for which L has already been defined. For F real, the only irreducible V's which are not of dimension I are those of the form V = Indp,F x,
where x is a quasi-character of F* = WF which is not invariant under "complex conjugation". For such V we put L(V) = L(x), as we are forced to do in order that L be inductive. (3.3.2) F nonarchimedean. Let I be the inertia subgroup of WF. Let 0 be an "inverse Frobenius", i.e., an element of WF such that 11 0 II = II II F. This condition
determines 0 uniquely mod I and we put L(V) = det(1 - $I VI)-', where VI is the subspace of elements in V fixed by I.
A proof that the "nonabelian" function L defined as above is inductive can be found in [D3, Proposition 3.8] (as well as in [A]). In the archimedean case one uses the relation Pc(s) = I'R(s)PR(s + 1). Technically, in order that L have values in a group, we should view L as a function which associates with V the meromorphic functions F- L(Vws), and take the X in Definition (2.3.2) to be the multiplicative group of nonzero meromorphic functions of s. (3.4) The local "nonabelian" a -function, s(V, cfi, dx). For this there is at present only an existence theorem (see below), no explicit formula.' This lack is not surprising if we recall that the formulas defining a in (3.2) make essential use of the interpretation of x as a quasi-character of F*; if we think of x as a quasi-character of WF we have no way to define e(x, 0, dx) without using the reciprocity law iso-
morphism F* x WF . In fact it was his idea about "nonabelian reciprocity laws" relating representations of degree n of WF to irreducible representations 7r of GL(n, F), and the possibility of defining e(ir, 0, dx) for the latter, which led Langlands to conjecture and prove a version of the following big (3.4.1) THEOREM. There is a unique function a which associates with each choice of
a local field F, a nontrivial additive character cli of F, an additive Haar measure dx on Fand a representation V of WF a number e(V, 0, dx) e C* such that e(V, 0, dx) _ e(x, cji, dx) if V is a representation of degree 1 corresponding to a quasi-character x, and such that if F is a local field and we choose for each finite separable extension E 'Except for Deligne's expression in terms of Stiefel-Whitney classes for orthogonal representations [D5, Proposition 5.2].
NUMBER THEORETIC BACKGROUND
15
of F an additive Haar measure ,uE on E, then the function which associates with each such E and each V e M(WE) the number e(V, c TrE,F, uE) is inductive in degree 0 over F in the sense of (2.3.2).
The unicity of such an e is clear, by (2.3.1); the problem is existence. The experience of Dwork and Langlands indicates that the local proof of existence, based on showing that the e(x, 0, dx) satisfy the necessary relations, is too involved to publish completely. Deligne found a relatively short proof (see [D3, §4]; possibly also
[T2]). It has two main ingredients, one global, one local: (1) the existence of a global e(V) coming from the global functional equation for L(V) (cf. (3.5) below), and (2) the fact that if F is local nonarchimedean and a a wildly ramified quasicharacter of F*, there is an element y = y(a, 0) in F* such that for all quasi-charac-
ters x of F* with a(x) < 1a(a), we have e(xa, 0, dx) = x 1(y)e(a, 0, dx), a rather harmless function of X. Granting the existence of e(V, 0, dx) the following properties of it are easy consequences of the corresponding properties of e(x, 0, dx), via inductivity in degree 0 and (2.3.1). (3.4.2) e is additive in V, so makes sense for V virtual. rdx) = rdim v e(V, 0, dx), for r > 0. In particular, for V virtual of (3.4.3) e(V, degree 0, e(V, cb, dx) = e(V, 0) is independent of A. (3.4.4) s(V, cba, dx) _ (det V)(a) Il a II-dim v e(V, 0, dx), for a e F* (cf. (2.3.4)). (3.4.5) e(Vws, 0, dx) = e(V, 0, dx) f(V)-s 6(cb)-sdimv, where: 3(0) = qF( in the nonarchimedean case and is characterized in the archimedean case by the fact that b(cba) = II a II-1 o(ff), and 6(0) = 1 for 0 as in (3.2.4) and (3.2.5).
f(V) = I in the archimedean case, and = qS(v), the absolute norm of the Artin conductor of V in the nonarchimedean case. This f can be characterized as the unique function inductive in degree 0 such that f(x) = qFW for quasi-characters X. For the well-known explicit formula for a(V) in terms of higher ramification groups, see [Sl] or [D3, (4.5)]. (3.4.6) Suppose F nonarchimedean, W unramified. Then e(V Q W, 0, dx) = e(V, 0, dx)dimw . det W(7ra(V)+dimvn( ))
(3.4.7) Let V* denote the dual of V and dx' the Haar measure dual to dx relative
to 0. Then e(V, 0, dx)e(V*wl, 0(- x), dx') = 1. In particular IE(V, 0, dx)I2 =f(V)(3(0)dx/dx')dimv,
if V*
i.e., if V is unitary. (3.4.8) If E/F is a finite separable extension, VE a virtual representation of degree 0 of WE and VF the induced representation of WF, then e(VF, 0) = e(VE, c o Tr). (3.5) Global L -Junctions, functional equations. Let F be a global field, 0 a non-
trivial additive character of AF/F, and dx the Haar measure on AF such that f AF,Fdx = I (Tamagawa measure). Call cb the local component of 0 at a place v, and let dx = 11, dx be any factorization of dx into a product of local measures such that the ring of integers in F gets measure 1 for almost all v.
J. TATE
16
Let V be a representation of "the" global Weil group WF, and put (3.5.1)
L(V, s) = IT L(Vv cos), v
(3.5.2)
e(V, s) = II s(Vvcos, cv, dxv) V
(3.5.3) THEOREM. The product (3.5.1) converges for s in some right half-plane and defines a function L(V, s) which is meromorphic in the whole s -plane and satisfies the functional equation
L(V, s) = e(V, s)L(V*, I - s)
(3.5.4)
where V* is the dual of V.
For V a quasi-character x this result was proved by Hecke. In the modern version of his proof ([Ti], [W2]) one shows by Poisson summation that for suitable functions f on A
SAf-1_s x-1(x) dx* =
J
x(x) dx*,
the integrals being defined for all s by analytic continuation. Taking f = j] fv and using the local functional equation (3.2.1) (with x replaced by xms) one finds that (3.5.4) holds in the "abelian" case, V = X. At this point, even without having a theory of the local nonabelian e(V,,, cb , dxv)'s, one gets, via (2.3.1), (2.3.5), and the inductivity of the local L's, that L(V, s) is meromorphic in the whole plane for each V, being defined by the product (3.5.1) in a right half-plane, and that L(V, s) is inductive as a function of V. It follows that defn
L(V, s) e (V, s) = L(V*, 1 - s)
is inductive in V and satisfies e'(x, s) _ [j , s(xv(os, 5bv, dxv) for quasi-characters x of A*/F*. It is this fact about the local e(xv, cv, dxv)'s-that their product over all v for a global x has an inductive extension to all global V-that Deligne uses in his "global" proof of the existence of local nonabelian s's. Once their existence is proved, we have s'(V, s) = e(V, s) by the unicity of inductive functions since e(V, s), defined by the product (3.5.2), is inductive in degree 0 by (2.3.5). (3.5.5) Hecke's global function L(x, s) is entire if x is not of the form cos. Artin conjectured (in the Galois case) that L(V, s) is entire for any V which has no constituent of the form ms. Weil proved Artin's conjecture for function fields. Recently
Langlands, using ideas of Saito and Shintani, made a first breakthrough in the number field case, treating certain V's of dimension 2 by base change, using the trace formula. (See The solution of a base change problem for GL(2) (following Langlands, Saito, Shintani), these PROCEEDINGS, part 2, pp. 115-133.) These methods work for all V's of dimension 2 for which the image of WF in PGL(V) is the tetrahedral group. They also work for some octahedral cases, but a new idea will be needed to apply
them in the nonsolvable icosahedral case. However, J. Buhler [B], with the aid of the Harvard Science Center PDP11 and the main result of [DS], has proved the Artin conjecture for one particular icosahedral V of conductor 800, by checking the existence of the corresponding modular form of weight I and level 800.
NUMBER THEORETIC BACKGROUND
17
Although the Riemann hypothesis concerning the zeros of L(x, s) has been proved by Weil in the function field case, there seems to be no breakthrough in sight in the number field case. The conjunction of the Artin conjecture for all V and the Riemann hypothesis for all x is equivalent to the positivity of a certain distribution on WF (cf. [W3]). (3.6) Comparison of different conventions for local constants. The modern references for the material we have been discussing are Deligne [D3] and Langlands [L], and we have here followed the conventions of [D3]. Happily, the definition of L-functions, both local and global, in [D3] coincides with that in [L]. But Deligne's local constants s(V, 0, dx), which we will designate in this section by eD instead of just e, differ somewhat from Langlands' e(V, 0) which we will denote by eL here. The relationship is sL(V, 0) = SD(V(0l/2, 0, dxo),
(3.6.1)
where dx0 is the additive measure which is self-dual with respect to 0. The other way around we have (3.6.2)
SD(V, 0, dx) = (dx/dxo)dimI SL(Vw-1/2, )
In the nonarchimedean case the constant dxldx0 is given explicitly by q-n(0) /2 $o dx. Also, in that case if V corresponds to a quasi-character x of F* we have (3.6.3)
eL(x, 0) = x(c)
$ .x 1(u) 0 (u/c) du 0 (u/c) du f
where c is an element of F* of valuation a(x) + n(cb) as in (3.2.6). Langlands puts defn
(3.6.4)
rL(s, V, 0) = eL(Vws-cli2:, 0) = eD(Vws, 0, dxcb)
Then the "constant" e(V, s) in the global functional equation (3.5.4) is given by e(V, s) = jj v eL(s, Vv, cb) for any nontrivial character 0 of A/F, because if dxv is self-dual on Fv with respect to 0, for each place v, then dx = fl, dxv is self-dual on A with respect to 0, and is therefore the Tamagawa measure on A. The behavior of SL under twisting by an unramified quasi-character is given by (3.6.5)
&L(V(0s, 0) = &L(V, 0)
as in (3.4.5), but its dependence on (3.6.6)
f(V)-,
3(Sb)-, aim V
is according to
eL(V, %a) _ (det V)(a)s(V, 0),
instead of as in (3.4.4). If V* is the contragredient of V, then (3.6.7)
eL(V, cb)rL(V*, 0-1) = 1.
Hence, by (3.6.6) (3.6.8)
&L(V, 0) 8L(V*, 0) = (det V)(- 1)
and on the other hand, (3.6.9)
I eL(V, 0)
1,
if V is unitary.
The CL-system has the advantage that it avoids carrying along the measure dx, but it has the following disadvantage: in the nonarchimedean case, if o is a dis-
18
J. TATE
continuous automorphism of C, then V° is again a representation of WF, and 0° an additive character, but EL(V°, 0) is not in general equal to CL(V, ci)° (nor is CL(0, V°, cb°) = eL(0, V, 0)°). The trouble is that the absolute value (in (3.6.3)) may not be preserved by o,, and/or that the self-dual measure dxo in (3.6.1) may involve .,/p, and hence may not be preserved by a. If one does wish to eliminate the measure dx, it is probably preferable to define, say, e1(V, 0) = eD(V, 0, dxi),
(3.6.10)
where dxl is the measure for which 9 gets measure I in the nonarchimedean case, and is the measure described in (3.2.4) and (3.2.5) in the archimedean case. This convention has the minor disadvantage that the e(V) in the global functional equation is not equal to the product of the local el(V,,, 0,)'s, but is, rather, a-' times that product, where a is the square root of the discriminant for a number field, and is qg-1 for a function field of genus g with q elements in its constant field. But the e1(V, 0) has the advantage that in the nonarchimedean case we do have 61(V°, 06) = el(V, O)° for all automorphisms a of C. This is clear, by unicity (2.3.3) and the formula (3.6.11)
el(x, 0) =
x(c)gn(0)
aEoZan°cx>x(u)(
C
which follows from (3.2.6.1) and (3.2.6.2) where the notation is explained. Thus in the nonarchimedean case we can define, for V and 0 over any field E of characteristic 0 (an open subgroup of I acting trivially on V, and 0 trivial on some 'r P), an el(V, 0) e E*, in a unique way such that e1(Va, cba) = e1(V, ,/.)a for any homomorphism a: E -* E' and such that al is the old el, given by (3.6.10), when E = C. So defined, el(VE, 0 TrE, F) is inductive in degree 0 (2.3.2) for every field of scalars
E of characteristic 0, and el(V, 0) will be given by (3.6.11) if V corresponds to a quasi-character x: F* -> E*. In writing these notes I was tempted to shorten things a bit by using only e1(V, ci) instead of rD(V, 0, dx), but decided against it because (1) the eD-system avoids all choices and is the most general and flexible-any other system, like eL or el can be immediately described as a special case of eD; (2) the dependence of e on dx shows "why" e is inductive only in degree 0, and (3) in case our local field F is nonarchimedean, the like the e1, works over any field E of characteristic 0, as soon as one defines the notion of Haar measure on F with values in E (cf. [D3, eD-system,
(6.1)]).
4. The Weil-Deligne group, 2-adic representations, L-functions of motives. The representations considered in §3 are just the beginning of the story. Those of Galois
type are effective motives of degree 0-which Deligne calls Artin motives in his article [D6, §6] in these PROCEEDINGS-with coefficients in C. We cannot discuss the notion of motive here (cf., e.g., [Dl] and [D6] for this) but we do want to discuss the way in which L-functions and e-functions are attached to motives of any degree. Only very special motives of degree 0 0 correspond to the representations of WF considered in §3, namely, those of type A0, i.e., those which, after a finite extension E/F, correspond to direct sums of Hecke characters of type A0 over E. (A candidate
for a "motivic Galois group" for these is constructed by Langlands in these
19
NUMBER THEORETIC BACKGROUND
PROCEEDINGS [L3].) The simplest motives not of this type are those given by elliptic
curves with no complex multiplication; their L-functions are the "Hasse-Weil zeta-functions" which are not expressible in terms of Hecke's L-functions.
The procedure for attaching L-functions to motives in the form given it by Deligne [D3], [D6] can be outlined schematically as follows: F is a global field. v is a place of F. E is a field of finite degree over Q.
A runs through the finite places of E whose residue characteristic is prime to char(F). a is an embedding of E in C. Motive M over F with cx. multn. by E
System (HA(M)) of A-adic
representations of GF
gd. fld.
1-adic
extension
coho.
restriction
Motive My over F. with cx. multn. by E Hodge theory cf.(4.4)
System (HA(M0)) of A-adic representations of GFv II
4 v archim edean
v nonarchimedean;
cf. (4.2)
Hodge structure Hdg(M0) over Fv with cx. multn. by E Equiv. class V(M0,o) of repns. of the Weil-Deligne group W F, over C, invariant under Aut(C/E)
(cf. (4.4)) 4 Equiv. class V(M,,,Q) of repns. over C of the Weil group WFv
cf. (4.1)
cf. (3.3), (3.4)
L- (i.e. I'-) and s-factors at the archimedean place v
L- and e-factors at the nonarchimedean place v
In the next sections we discuss some of the steps and concepts indicated in the above chart. We begin with the Weil-Deligne group. This is a group scheme over Q, but what counts, its points in and representations over fields of characteristic 0, can be described naively with no reference to schemes. (4. 1) The Weil-Deligne group and its representations. Let F be a nonarchimedean local field and let F, GF = Gal(F/F), WF (Weil group), and I (inertia group) have their usual meaning. For w e WF, let II w II denote the power of q to which w raises elements of the residue field, as in (1.4.6). Thus we have II w II = 1 for w e I, and 110 II = q-1 for a geometric Frobenius element 0. We view WF as a group scheme
over Q as follows: for each open normal subgroup J of I, we view WFIJ as a "discrete" scheme, and we put WF = proj lim (WFIJ), the limit taken over all J. In other words, we have
WF = II On I = nEZ
ll spec A,,,
nEZ
20
J. TATE
where A is the ring of locally constant Q-valued functions on OnI. (4.1.1) DEFINITION [D3, (8.3.6)]. The Weil-Deligne group WW is the group scheme
over Q which is the semidirect product of WF by G0, on which WF acts by the rule wxw 1 = I W II x
Let E be a field of characteristic 0. The group WF(E) of points of WW with coordinates in E is just E x WF with the law of composition (a1, w1)(a2, w2) (a1 + II w1 II a2, w1w2) for a1, a2 e E and w1, w2 e WF.
Let V be a finite-dimensional vector space over E. A homomorphism of group schemes over E
p': WW x Q E - GL(V) determines, and is determined by, a pair (p, N) as in (4.1.2) below, such that, on points, p'((a, w)) = exp(aN) p(w). That is the explanation for the following definition: (4.1.2) DEFINITION [D3, (8.4.1)]. Let E be a field of characteristic 0. A representation of WW over E is a pair p' _ (p, N) consisting of: (a) A finite-dimensional vector space V over E and a homomorphism p : WF ->
GL(V) whose kernel contains an open subgroup of I, i.e., which is continuous for the discrete topology in GL(V). (b) A nilpotent endomorphism N of V, such that p(w)Np(w)-1 = II w II N, for W e W.
(4.1.3) o-semisimplicity. Let p' = (p, N) be a representation of WW over E. Define v : WF - Z by II w II = R-v (w) There is a unique unipotent automorphism u of V
such that u commutes with N and with p(WF) and such that exp(aN)p(w)u v(w) is a semisimple automorphism of V for all a e E and all w e WF-I [D3, (8.5)]. Then p,, = (puv, N) is called the o-semisimplification of p', and p' is called cb-semisimple if and only if p' = p,,, i.e., u = 1, i.e., the Frobeniuses act semisimply. For this it is necessary and sufficient that the representation p of WF be semisimple in the ordinary sense, because p(O) generates a subgroup of finite index in p(WF), and in characteristic 0 a representation of a group is semisimple if and only if its restriction to a subgroup of finite index is semisimple. In his article in these PROCEEDINGS, Borel discusses admissible morphisms WW - LG; when G = GL,,, these are just our 0-semisimple (p, N)'s. (4.1.4) EXAMPLE. Sp(n) is the following representation (p, N) of WW over Q. V=Qn=Qeo+Qe1+...
p(w)e1 = w-(w)e,
(= 11w1l'e),
Net = e1+1
(0 5 i < n - 1),
0.
(4.1.5) Given any (p, N), Ker N is stable under WF. Hence (p, N) is irreducible e= N = 0 and p is irreducible. It is not hard to show that the 0-semisimple indecomposable representations of WW are those of the form p' px Sp(n) with p' irreducible. (The Qx is defined by (p, N) Qx (pl, N1) = (p (D pl, N Qx 1 + 1 (D N1).) (4.1.6) Let (p, N, V) be a representation of WW over E. We put VN = (Ker N)' and define a local L-factor, a conductor, and a local constant by
NUMBER THEORETIC BACKGROUND
Z(V, t) = det (1 - Ot I
VII,1)-I,
21
and L(V, s) = Z(V, q-s), when E c C;
a(V) = a(p) + dim VI - dim VN,
e(V) = e(p)det (- 0 VI/VN), and
e(V, t) =
e(V)ta(v).
Here, for e, the usual 0 and dx are understood, but omitted from the notation. These quantities do not change if we replace V by its 0-semisimplification; but note that they are not additive as functions of V, because VN is not. If N = 0, they
are the same as before. One of the main reasons for introducing the Weil-Deligne group is the fantastic generalization of local class field theory embodied in : (4.1.7) Conjecture. Let F be a nonarchimedean local field and n an integer >- 1. There is a (in fact more than one) natural bijection between isomorphism classes of 0-semisimple representations of WW of degree n, and of irreducible admissible representations of GL(n, F).
For n = I this is local class field theory. For n = 2, it is discussed at length in [D2, (3.2)]. In this conjecture, for any n, the irreducible representations of WW (which are just irreducible representations of WF) should correspond to the cuspidal
representations of GL(n, F). I understand that Bernstein and Zelevinsky have shown that the way in which arbitrary admissible representations of GL(n, F) are built out of cuspidal ones follows the same pattern as the way in which arbitrary 0-semisimple representations of WW are built up out of irreducible ones. Thus the main problem is now the correspondence between irreducibles and cuspidals. A more general conjecture, involving an arbitrary reductive group G rather than just GL(n), relates admissible representations of G(F) to homomorphisms of WF. into the "Langlands dual" of G (see Borel's article in these PROCEEDINGS). This more general conjecture is the nonarchimedean local case of "Langlands' philosophy". (4.2) A-adic representations. Now suppose 1 is a prime different from the residue characteristic p of F and let t1: IF --> Q, be a nonzero homomorphism. (Such a t1 exists and is unique up to a constant multiple, because the wild ramification group P is a pro -p-group, and the quotient I/P is isomorphic to the product H 1r Z1.) We have t1(wa'w I) = Ilwll t1(o,),
for o E I, w E W,
because conjugation by w induces raising to the II w II power in I/P. Let 0 be an inverse Frobenius element (4.1.8). Suppose Ea is a finite extension of Q1. A 2-adic representation of WF is a finite-dimensional vector space VV over EA and a homomorphism of topological groups pA: WF --> GLE,(VA) where GLE,(VA) has the A-adic
topology (i.e., the topology given by the valuation). (4.2.1) THEOREM (DELIGNE [D3, §8]). The relationship VA = V and
p,(Ono') = p(O"o')exp(t1(r) N),
a E I, n E Z,
J. TATE
22
sets up a bijection between the set of A-adic representations (p2, VA) of WF and the set of representations (p, N, V) of WW over EA. The corresponding bijection between isomorphism classes of each is independent of the choice of tt and 0.
To show that every PA is of this form one uses (4.2.2) COROLLARY (GROTHENDIECK). Let (p2, V2) be a 2-adic representation of WF.
There exists a nilpotent endomorphism N of VA such that p2(6) = exp(tt(ou)N) for a in an open subgroup of I.
A proof of the corollary can be found in the appendix of [ST]. Here is a sketch. Since I is compact, p2(I) stabilizes a "lattice" L in VA. Replacing F by a finite extension we can assume that p2(I) fixes L (mod 12). Then p2(I) is a pro-l-group, so is a
homomorphic image of tt(I), since Ker tt is prime to 1. Choose c e Qt such that ctt(I) = Z1. Then there is an a e GL(V2) fixing L (mod 12) such that P2(6) = actt(a) = exp(t1(u)N)
for all or e I, where N = c log a. Conjugating by p2(o) we find p2(0)Np2(0)-1 = q-1N. Thus the set of eigenvalues of N is stable under multiplication by q-1. Since q is not a root of unity in characteristic 0, it follows that the only eigenvalue of N is zero, i.e., N is nilpotent. (4.2.3) COROLLARY. If VA is a semisimple A-adic representation of WF then some open subgroup of I acts trivially on VA, so VA can be viewed as an "ordinary" representation of WF. w 11 N. For any VA the kernel of N is stable under WF because p2(w)Np1(w)-1 So if V2 is irreducible, then N=0, and the statement follows from (4.2.2). A semisimple VA is a direct sum of irreducible subrepresentations. (4.2.4) In view of (4.2.3), s(V2) and a (V2) have meaning if V2 is semisimple. For arbitrary. VA, if (p2, VA) and (p, N, V) correspond as in (4.2.1), we define the L- and e-factors associated to V2 to be those associated to V. These can be expressed di-
rectly in terms of V2 as follows:
Z(V, t) = det(1 - 0I VIA) = Z(VA, t),
a(V) = a(V') + dim(V3s)' - dim VA = a(V2),
s(V) = sMD
det(- 01(V S)') det(- 0 1 VI)
= s(VZ),
and
e(V, t) = s(VA)ta(vz) = e(V2, t),
where V11 is the semisimplification of VA in the ordinary sense. One can define a " O-semisimplification" of VA, analogous to that of V (4.1.3). The quantities on the right do not change if we replace VA by its 0-semisimplification, but they are not additive in V2, because VI is not. (4.2.4) Motives. Suppose now E is a finite extension of Q. Let M be a motive with complex multiplication by E, defined over our nonarchimedean local field F ([Dl], [D6]). Let n be the rank of M. Attached to M will be 1-adic representations H1(M),
NUMBER THEORETIC BACKGROUND
23
vector spaces of dimension n over Q1 on which GF acts continuously, one for each 10 char(F). The field E will act on these, and for each 1 we get a decomposition H1(M) = QQ A11 HA(M), where for each place 2 of E above 1, we put HA(M) =
EA ©E©Q, H1(M), a vector space of dimension m over E, where m is the rank of M over E, given by n = m[E: Q]. For each 10 p and each A above 1 let HH(M) be the representation of W F over EA corresponding to HA(M) by (4.2.1). If our motive M lives up to expectations, the system of A-adic representations HA(M) will be compatible over E in the sense that the system HH(M) is compatible over E in the following naive sense: for any two finite places A, ,u of E not over p and every commutative diagram
EE
the m-dimensional representations of WW over C, HH(M) ©Ex C and HA(M) ®E, C,
are isomorphic (or at the very least, have isomorphic 0-semisimplifications). If so, then the isomorphism class of (the 0-semisimplifications of) these representations depends only on the embedding E c C in the diagram above. We denote this isomorphism class by V(MQ), where or denotes the embedding of E in C and Mo = M OE, O C is the motive of rank m with coefficients in C deduced from the original M, the action of Eon it, and the embedding -a of E in C, cf. [D6, 2.1]. Associated to V(M,) as explained in (4.1.6) are the local quantities a, L, and e which we shall denote by L(MC, s), etc. (4.3) Reduction. Let r be an integer ? 0 and X a projective nonsingular variety over F. In this paragraph we shall restrict our attention to the special motive M = HT(X)given by the r-dimensional cohomology of X, and we shall ignore any com-
plex multiplication. For the moment F can be any field. Put X = X X F F, the scheme obtained by extending scalars from F to F. For each prime 196 char(F) the l-adic etale cohomology group HT(Xet, Q1) is defined, and gives an 1-adic representa-
tion of GF = Gal(F/F) (by functoriality, GF acting on X through F). In the notation of the previous paragraph we have now E = Q, d = 1, H1(M) = HT(Xet, Q,). I do not know to what extent the compatibility of the H1(M)'s is known (assuming now again that F is local nonarchimedean), but the compatibility at least of their 0-semisimplifications is known in one very important case-that of (4.3.1) Good reduction. Let 0 be the ring of integers in F, and k = 0/2c(9 the residue field. The scheme X is said to have good reduction if there exists a scheme 3r projective and smooth over 0 such that X = X x o F. Choosing such an 3r, one calls x o k the reduction of X. Let us denote this reduction by X0. Putting Xo = X0 X k k, where k is the residue field of F, the base-change theorem gives a canonical isomorphism
{*)
H1(M) = HT(X, Q,) - HT(Xo, Q1)
compatible with the action of the Galois groups. Hence H1(M) is unramjed, i.e., fixed by I, and the structure of H,(M) as representation of WF is given by the action
24
J. TATE
of 0. Let rp: Xo -> Xo be the Frobenius morphism, and o: k --> k the Frobenius automorphism. The composition (p x 6 acts on go = Xo x k by fixing points and by mapping f H fq in the structure sheaf. This map induces (a morphism canonically isomorphic to) the identity on the site (X0)et, so the action of the Frobenius morphism (p on Hr(X0, Q1) is the same as that of o-1, which is the one corresponding to our 0 under the isomorphism (*). That is why Deligne calls 0 the geometric Frobenius.
Deligne [D4] has proved Weil's conjecture, that the characteristic polynomial of (p acting on Hr(X0, Q1) has coefficients in Z, is independent of 1, and that its complex roots have absolute value qr/z. From the independence of lit follows in this case of good reduction that the 0-semisimplifications of the H1(M)'s form a compatible system; and the H1(M)'s are known to be 0-semisimple for r = 1. It is natural to say that a motive M over F has good reduction, or is unramiiied if and only if H1(M) = H1(M)I, i.e., if V(M) = V(M),IV. In case M = H1(A), A an abelian variety, this is equivalent to A having good reduction (criterion of NeronOgg-Shafaryevitch in [ST]).
Similarly we say M has potential good reduction a N = 0, and M has semistable reduction if V(M) = V(M)I. Clearly this latter can always be achieved by a finite extension of the ground field. (4.4) F archimedean. Let now M, E, n, m be as in (4.2.4), but take F to be archimedean, instead of nonarchimedean. Let z: F --> C be the embedding of F in C if F is real, or one of the two isomorphisms of F on C if F is complex. Such a z gives us a motive M. over C and M. has a "Betti realization" HB(MZ) which is an ndimensional vector space over Q whose complexification HB(MZ) Qx C = Q Hpq(MZ)
is doubly graded in such a way that the map 1 Qx c (c = complex conjugation)
takes Hpq to HqP. (For example, if M = Hr(X) as in (4.3), then HB(Mz) _ Hr(X:", Q), where Xz° is the complex analytic variety underlying the scheme X X F,z C, and the complexification of this space, Hr(Xz°, C), is doubly graded by Hodge theory.) Let 2 = c. z: F --> C be the map conjugate to z. By transport of structure, there is an isomorphism z: HB(MM) --> HB(MZ) such that z ® c preserves the bigrading on the complexifications; hence z 0 1 carries Hpq(MZ) onto Hgp(MZ). The field E of complex multiplications acts on HB(MZ) preserving the bigradation on the complexification, and z is an E-homomorphism. Let a: E --> C. Putting VZ(MQ) _ HB(Mz) ©E,QC we obtain a bigraded complex vector space of dimension m and a linear isomorphism c 0 1: VV(M,) -> VV(M,) taking VZpg to VZp. There is a natural action of the Weil group WF on these spaces as follows: F complex. z: F x C an isomorphism, WF = F*, and WF acts on Vpg by scalar multiplication via the character z p(z)-q. Clearly, z 0 1 is WF-equivariant, so the two representations V0(Mo) and V ,(M,) are isomorphic. We let V(MQ) denote their isomorphism class. F real. z = z: F --> C is the embedding, and WF = C* LJ jC*. This time M. = M,, so we have only one space, V0(Mv) = VZ(M,), and r 0 1 is an automorphism of it. The action of WF on it is as follows: u e C* acts as multiplication by up(u)_i on V. j acts as ip+g(z Ox 1) on VZg.
Again, let V(M,) denote the equivalence class of this representation.
NUMBER THEORETIC BACKGROUND
25
Notice that the representations obtained from motives via Hodge theory are very special, in that the p and q are integers. Finally, define L(MQ, s) and s(M0, s) to be the L- and s-factors associated to the representation V(M,) as in §3. For a table making these explicit see [D6, 5.3]. (4.5) F global. Let F be a global field and M a motive with complex multiplication
by E, defined over F. For each place v of F, let M denote the restriction of M to F. Let o-: E -+ C. The product L(MC, s) = r] v L(MV,Q, s) converges in a right halfplane. It is conjectured that it is meromorphic in the whole s-plane and satisfies the functional equation L(MQ, s) = s(Mo, s) L(MM ,
1 - s)
with M* = Hom(M, Q) and s(M0, s) = ]T s(M0,,, s). In the function field case this conjecture has been proved by Deligne. Let q be the number of elements in the constant field k of F. Grothendieck proved that for any given A-adic representation V of GF which is unramified at all but a finite number of places v, the corresponding L-function L(V, s) = JI v L(VV, s) is a rational function of q-S (even a polynomial if VO = 0 and VG = 0, where G is the geometric
Galois group, i.e., the kernel of the map of GF to Gk), and satisfies a functional equation of the form L(V, s) = s(V, s)L(V*, 1 - s) with an s which is a monomial Later, Deligne showed that Grothendieck's in q -s of degree s(V, s) is equal to the product of the local s(V,,, s)'s if V = Vo is a member of a family (V2)2E, of A-adic representations of GF for some infinite set of places 2 of a number field E, and the family is compatible in the following weak sense: for each A, ,u E Y there is a finite set S of places of F such that for v 0 S, the representations VA and VF, are unramified at v and the characteristic polynomals of 0. acting on VA and V,, have coefficients in E and are equal. Deligne's method is to prove that Grothendieck's a is congruent to the product of the local s's modulo A for all 2 s. and is therefore equal to that product. By (4.3) any d-adic representation coming from 1-adic cohomology, i.e., from a motive, is a member of a system which is weakly compatible in the above sense.
When dim(V) = 2, then by Jacquet-Langlands (resp. Weil), Springer Lecture Notes 114 (resp. 189), these results show that L(V, s) comes from an automorphic representation of (resp. modular form on) GL2(AF). On the other hand, Drinfeld has recently shown that automorphic representations of GL2 give rise to systems of l-adic representations occurring as constituents in tensor products of those coming from 1-dimensional l-adic cohomology, hence from motives. Thus for GL2 over function fields, the equivalence between motives, compatible systems of 1adic representations, and automorphic representations is pretty well established. In this connection it should be mentioned that Zarhin [Z] has proved the isogeny theorem over function fields : if two abelian varieties A and B over a global function field F give isomorphic 1-adic representations, then they are isogenous; more precisely,
Q1 x0 HomF(A, B) = HomoF,(VI(A), VI(B)).
Over number fields our knowledge is not nearly so advanced. For Artin motives of rank 2, Langlands has made a beginning with the theory of base change (see the remarks (3.5.5)). For elliptic curves M over Q, it is not even known whether
26
J. TATE
L(M, s) has a meromorphic continuation throughout the s-plane, or whether the isogeny theorem is true. For a more detailed account of our ignorance, as well as of a few things which are known, see [S4]. REFERENCES
[A] E. Artin, Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren, Abh. Math. Sem. Univ. Hamburg 8 (1930), 292-306 (collected papers, pp. 165-179). [AT] E. Artin and J. Tate, Class field theory, Benjamin, New York, 1967. [B] J. Buhler, Icosahedral Galois representations, Ph.D. Thesis, Harvard Univ., (January 1977). [CF] J. W. S. Cassels and A. Frbhlich, Algebraic number theory, Academic Press, London, 1967. [DI] P. Deligne, Les constantes des equations fonctionnelles, Seminaire Delange-Pisot-Poitou 1969/70, 19 bis. [D2] , Formes modulaires et representations de GL(2), Antwerp II, Lecture Notes in Math., vol. 349, Springer-Verlag, 1973, pp. 55-106. [D3] , Les constantes des equations fonctionnelles des fonctions L, Antwerp II, Lecture Notes in Math., vol. 349, Springer-Verlag, 1973, pp. 501-595. [D4] , La conjecture de Weil. I, Inst. Hautes Etudes Sci. Pub]. Math. 43 (1974), 273-307. [D5] , Les constantes locales de l'equation fonctionnelle de la fonction L d'Artin dune representation orthogonale, Invent. Math. 35 (1976), 299-316. [D6] , Valeurs defonctions L et periodes d'integrales, these PROCEEDINGS, part 2, pp. 313-346. [DS] P. Deligne and J.-P. Serre, Formes modulaires de poids 1, Ann. Sci. Ecole Norm. Sup. (4) 7 (1974), 507-530. [K] H. Koch, Classification of the primitive representations of the Galois groups of local fields, Invent. Math. 40 (1977), 195-216. [LI] R. P. Langlands, On Artin's L -Junctions, Complex Analysis, Rice Univ. Studies No. 56, 1970, 23-28; also the mimeographed notes: On the functional equation of Artin's L functions. [L2]
, Modular forms and 1-adic representations, Lecture Notes in Math., vol. 349,
Springer-Verlag, 1973, pp. 362-498. [L3] , Automorphic representations, Shimura varieties, and motives. Ein Miirchen, these PROCEEDINGS, part 2, pp. 205-246.
[Sl] J.-P. Serre, Corps locaux, Hermann, Paris, 1962. [S2] , Facteurs locaux des fonctions zeta des varietes algebriques (Definitions et conjectures), Seminaire Delange-Pisot-Poitou, 1969/70, no. 19. , Modular forms of weight one and Galois representations, Proc. Durham Sympos. on [S3] Algebraic Number Fields, Academic Press, London, 1977, pp. 193-267. [S4] , Representations 1-adiques, Algebraic Number Theory, Papers for the Internat. Sympos., Kyoto 1976 (S. Iyanaga, Ed.), Japan Soc. for the Promotion of Science, Tokyo, 1977, pp. 177-193. [ST] J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968),
492-517.
[T1] J. Tate, Fourier analysis in number fields and Hecke's zeta functions, Thesis, Princeton Univ., (1950) (published in [CF], 305-347). [T2] , Local constants, Proc. Durham Sympos. on Algebraic Number Fields, Academic Press, London, 1977. [WI] A. Weil, Sur la theorie du corps de classes, J. Math. Soc. Japan 3 (1951), 1-35. [W2] , Basic number theory, Springer, New York, 1967. , Sur les formules explicites de la theorie des nombres, Izv. Akad. Nauk SSSR Ser. [W3] Mat. 36 (1972), 3-18. [Z] Ju. G. Zarhin, Endomorphisms of abelian varieties over fields of finite characteristic, Izv. Akad. Nauk SSSR Ser. Mat. 39 (2) (1975) = Math USSR-Izv. 9 (1975), 255-260. HARVARD UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 27-61
AUTOMORPHIC L-FUNCTIONS A. BOREL This paper is mainly devoted to the L-functions attached by Langlands [35] to an irreducible admissible automorphic representation 7c of a reductive group G over a global field k and to local and global problems pertaining to them. In the context of this Institute, it is meant to be complementary to various seminars, in particular to the GL2-seminars, and to stress the general case. We shall therefore start directly with the latter, and refer for background and motivation to other seminars, or to some expository articles on this topic in general [3] or on some aspects of it [7], [14], [15], [23].
The representation 7c is a tensor product ir = &z, over the places of k, where 1r is an irreducible admissible representation of G(kv) [11]. Accordingly the L-functions associated to 7c will be Euler products of local factors associated to the 2cv's. The definition of those uses the notion of the L-group LG of, or associated to, G.
This is the subject matter of Chapter I, whose presentation has been much influenced by a letter of Deligne to the author. The L-function will then be an Euler product L(s, 'c, r) assigned to ' and to a finite dimensional representation r of LG. (If G = GL,,, then the L-group is essentially GL (C), and we may tacitly take for r the standard representation r,, of so that the discussion of GL,, can be carried out without any explicit mention of the L-group, as is done in the first six sections of [3].) The local L- and e-factors are defined at all places where G and 7r are "unramified" in a suitable sense, a condition which excludes at most finitely many places. Chapter II is devoted to this case. The main point is to express the Satake isomorphism in terms of certain semisimple conjugacy classes in LG (7.1). At
this time, the definition of the local factors at the ramified places is not known in general. For GL,, and r,,, however, there is a direct definition [19], [25]. In the general case, the most ambitious scheme is to associate canonically to an irreducible admissible representation of a reductive group H over a local field E a representation of the Weil-Deligne group WW of E into LH, and then use L- and e-factors associated to representations of WW [60]. This problem is the main topic of Chapter III. The L-function L(s, 7c, r) associated to z and r as above is introduced in § 13. In fact, it is defined in general as a product of local factors indexed by almost all places of k. It converges absolutely in some right half-plane (13.3; 14.2). Some of the main AMS (MOS) subject classifications (1970). Primary 10D20; Secondary 12A70, 12B30, 14G10, 22E50. © 1979, American Mathematical Society
27
28
A. BOREL
conjectural analytic properties (meromorphic continuation, functional equation), and the evidence known so far, are discussed in §14. From the point of view of [35], a great many problems on automorphic representations and their L-functions are special cases of one, the so-called lifting problem or problem of functoriality with respect to L-groups. It is discussed in Chapter V. It is closely connected with Artin's conjecture (see §17 and the base-change seminar [17]). In §18 brief mention is made of some known or conjectured relations between automorphic L-functions and the Hasse-Weil zeta-function of certain varieties, to be discussed in more detail in the seminars on Shimura varieties [8], [40].
Thanks are due to H. Jacquet and R. P. Langlands for various very helpful comments on an earlier draft of this paper. CONTENTS
1. L-groups
....................................................................................28
1. Classification ...........................................................................29 2. Definition of the L-group .........................................................29 3. Parabolic subgroups ..................................................................31 4. Remarks on induced groups .........................................................33 5. Restriction of scalars ...............................................................34 II. Quasi-split groups; the unramiied case .............................................35 6. Semisimple conjugacy classes in LG ............................................. 35 7. The Satake isomorphism and the L-group. Local factors in the
unramified case ........................................................................38 III. Weil groups and representations. Local factors .....................................39
8. Definition of O(G) .....................................................................39 9. The correspondence for tori ......................................................41 10. Desiderata ..............................................................................43 11. Outline of the construction over R, C .............................................46 12. Local factors .
........................................................................48
IV. The L -Junction of an automorphic representation .................................49 13. The L-function of an irreducible admissible representation of GA.........49 14. The L-function of an automorphic representation ...........................52
V. Lifting problems ...........................................................................54 15. L-homomorphisms of L-groups ...................................................54 16. Local lifting ........................................................................... 55 17. Global lifting ........................................................................... 56 18. Relations with other types of L-functions .......................................58
References ....................................................................................... 59 CHAPTER 1. L-GROUPS.
k is afield, k an algebraic closure of k, ks the separable closure of k in k, and T,, the Galois group of k5 over k. G is a connected reductive group, over k in 1.1, 1.2, 2.1, 2.2, over k otherwise. §§ 1, 2 will be used throughout, §3 from Chapter III on. The reader willing to take on faith various statements about restriction of scalars need not read §§4, 5.
AUTOMORPHIC L-FUNCTIONS
29
1. Classification. We recall first some facts discussed in [58].
1.1. There is a canonical bijection between isomorphism classes of connected reductive k-groups and isomorphism classes of root systems. It is defined by associating to G the root datum O(G) = (X *(T), cp, X*(T), cpv) where T is a maximal torus of G, X*(T) (X*(T)) the group of characters (1-parameter subgroups) of T and 0 (0v) the set of roots (coroots) of G with respect to T. 1.2. The choice of a Borel subgroup B T is equivalent to that of a basis 4 of 0(G, T). The previous bijection yields one between isomorphism classes of triples (G, B, T) and isomorphism classes of based root data OO(G) = (X*(T), 4, X*(T), 4v). There is a split exact sequence (1)
(1) -p Int G - Aut G -p Aut OO(G) - (1).
To get a splitting, we may choose x,, e G,, (a e 4) and then have a canonical bijection (2)
Aut OO(G)
' Aut (G, B, T, {xa}aE4).
Two such splittings differ by an inner automorphism Int t (t E T). 1.3. Given r e f'k there is g e G(ks) such that g rT g I = T, g rB g I = B, whence an automorphism of cbo(G), which depends only on r. We let uG: f'k -+ Aut OO(G) be the homomorphism so defined. If G' is a k-group which is isomorphic to G over k (hence over ks), then uc = G, G' are inner forms of each other. 1.4. Let f: G -* G' be a k-morphism, whose image is a normal subgroup. Then f induces a map 0(f): cb(G) -> cb(G') (contravariant (resp. covariant) in the first (last) two arguments). Given B, T c G as above, there exists a Borel subgroup B' (resp. a maximal torus T') of G' such thatf(B) c B', f(T) c T', whence also a map cbo(f ): Sbo(G) - Oo(G').
2. Definition of the L-group. 2.1. The inverse system ? to the based root datum ?o = (M, 4, M*, 4v) is O v = o (M*, 4v, M, 4). To the k-group G we first associate the group LG° over C such that cfio(LG°) = cio(G)". We let LT°, LB° be the maximal torus and Borel subgroup defined by co , and say they define the canonical splitting of LG°. Let f be as in 1.4. Then f also induces a map cLio(f) : cio(G')v -> 00(G)v. An algebraic group morphism of LG'° into LG° associated to it will be denoted Lf °. Given
one, any other is of the form Int t°Lf°°Int t' (t e LT°, t' E LT'°), and maps LT'° (resp. LB'°) into LT° (resp. LB°). 2.2. EXAMPLES. (1) Let G = GL,,. Then LG° = GL,,. In fact, let M = Zn with
{x;} its canonical basis. Let {et} be the dual basis of M* = Z. Then (M, 4, M*, Jv) with 4 = {(x1 - xi+1), 1 < i < n}, 41 = {(ei - ei+1), 1
< n},
hence 00 = 95'S
(2) Let G be semisimple and 7f1'0(G) = (M, 0, M*, 0v). As usual, let P(0) M Qx Q be the lattice of weights of 0 and Q(0) the group generated by 0 in M. Define P(0'') and Q(0") similarly. As is known G is simply connected (resp. of adjoint type) if and only if P(O) _ M (resp. Q(O) = M). Moreover P(0) = {2 E M 0 QI
c Z},
P(OV) _ {A e Mx p Qk<2, 0> E Z}.
30
A. BOREL
Therefore: G simply connected o:> LG° of adjoint type; G of adjoint type ..T-> LG° simply connected.
(3) Let G be simple. Up to central isogeny, it is characterized by one of the types A,,,
..., G2 of the Killing-Cartan classification. It is well known that the map
Oo(G) -> cbo(G)v permutes B. and C and leaves all other types stable. Thus if G = Sp2s (resp. G = PSp2,,), then LG° = S02n+1 (resp. LG° = In all other cases, G '-> LG° preserves the type (but goes from simply connected group to adjoint group, and vice versa). (4) Let again G be reductive and let f : G -> G' be a central isogeny. Let
N = coker f*: X*(T) X*(T') N' = cokerf*: X*(T') --> X*(T).
(T' = f(T)),
Then N and N' are isomorphic and ker Lf ° = Hom(N, C*)
N. In particular,
Lf ° is an isomorphism if and only if f is one.
(5) Let f : G -> G' be a central surjective morphism, Q = ker f, and Q° the identity component of Q. Then ker Lf ° - Q/Q° If Q is connected, then T" = T (1 Q is a maximal torus of Q, and the injectivity of Lf ° follows from the fact that the exact sequence I --> T" --> T -> T' --> 1 necessarily splits. If Q is not connected, then r : H = G/Q° -> G' is a nontrivial separable isogeny, with kernel Q/Q°. Lf° factors through Lr° and, by the first part and (4), ker Lf ° = ker Lr ° = Q/Q°. In particular, if we apply this to the case where G' = Gad is the adjoint group of G, and use (2), we see that the derived group of LG° is simply connected if and only if the center of G is connected. As an example, let G = GSp2" be the group of symplectic similitudes on a 2n-dimensional space. Then the derived group of LG° is isomorphic to Spin,,+i In fact, we have LG° = (GL1 x Spines+1)/A where A = {1, a} and a = (a1, a2), with a1 of order two in GL1 and a2 the nontrivial central element of Spin2n+1. If n = 2, then Spin2n+1 = Sp2n. It follows that if G = GSp4, then LG° = GSp4(C). 2.3. We have canonically Aut ?I = Aut To. Therefore we may view UG as a homomorphism of 1'k into Aut co. Choose a monomorphism (1)
Aut co =- Aut (LG°, LB°, LT°)
as in 1.2(2). We have then a homomorphism ,uG: 1'k --> Aut (LG°, LB°, LT°).
The associated group to, or L-group of, G is then by definition the semidirect product (2)
L(G/k) = LG = LG° x 1'k,
,u is well defined up to an inner automorphism with respect to by an element of LT°. The group LG is viewed as a topological group in the obvious way. The canonical splitting of LG° (2.1) is stable under Tk. We have a canonical projection LG -* rk with kernel LG°. The splittings of the exact sequence (3)
1 --> LG° _> LG
vc
1'k ----> 1
AUTOMORPHIC L-FUNCTIONS
31
Aut(LG°, LB°, LT°, {xj) are called defined as in 1.2 via an isomorphism Aut ) admissible. They differ by inner automorphisms Int t (t e LT°). Note that if G splits over k, then rk acts trivially on LG° and LG is simply the direct product of LG° and
r.-
2.4. REMARKS. (1) So far, we can in this definition take LG° over any field. We have chosen C since this is the most important case at present, but it is occasionally useful to use other local fields. (2) There are various variants of this notion, which may be more convenient in certain contexts. For instance we can divide rk by a closed normal subgroup which acts trivially on ffV, hence on LG°, e.g., by rk' if k' is a Galois extension of k over which G splits. Then rk is replaced by Gal(k'/k), and LG is a complex reductive Lie group.
We can also define a semidirect product LG° x 2?, for any group 2 endowed with a homomorphism into rk, e.g., the Weil group of k, if k is a local or global field. In that case, we get the "Weil form" of LG. (3) Let G' be a k-group which is isomorphic to G over k. Then G and G' are inner forms of each other if and only if LG is isomorphic to LG' over rk. In fact, the first condition is equivalent to ,uo = UG', and the latter is easily seen to be equivalent to the second condition. In particular, since two quasi-split groups over k which are inner forms of each other are isomorphic over k, it follows that if G, G' are quasisplit and LG LG' over rk then G and G' are k-isomorphic.
2.5. Functoriality. Let f: G -> G' be a k-morphism whose image is a normal subgroup. Then fro: cbo(G) - cio(G) clearly commutes with rk, hence so does foo : cbo(G')v -+ cbo(G)v and Lf ° : LG'° -j LG°. We get therefore a continuous homomorphism Lf : LG' -). LG such that
LGi LfLG VG
Y
Al LC
rk
is commutative, which extends Lf ° 2.6. Representations. For brevity, by representation of LG we shall mean a continuous homomorphism r: LG -+ whose restriction to LG° is a morphism of complex Lie groups.
Clearly, ker r always contains an open subgroup of rk, hence r factors through LG° > rk-Ak, where k' is a finite Galois extension of k over which G splits. The group
LG° x rk, is canonically a complex algebraic group and r is a morphism of complex algebraic groups. 3. Parabolic subgroups.
3.1. Notation. We let q(G/k) denote the set of parabolic k-subgroups of G, and write q(G) for 9(G/k). Let p(G/k) be the set of conjugacy classes (with respect to G(k) or G(k), it is the same) of parabolic k-subgroups, and p(G) = p(Glk). Let p(G)k be the set of conjugacy classes of parabolic subgroups which are defined over k (i.e., if P e o e p(G)k, then rP e o for all r e rk). In particular p(Glk) c. p(G)k. There is equality if G is quasi-split/k. 3.2. We recall there is a canonical bijection between p(G) and the subsets of J.
32
A. BOREL
Then p(G)k corresponds to the rk-stable subsets of J and p(Glk) to those rkstable subsets which contain the set JO of simple roots of a Levi subgroup of a minimal parabolic k-group. In particular we have p(G/k) = p(G)k if G is quasisplit over k. Given P e g(G), we let J(P) be the subset of J assigned to the class of P Since two conjugate parabolic subgroups whose intersection is a parabolic subgroup are identical, we see in particular that if P is defined over k, P' P, and the class of P' is defined over k, then P' is defined over k. 3.3. Parabolic subgroups of LG. A closed subgroup P of LG is parabolic if rG(P) _ rk and P° = LG° A P is a parabolic subgroup of LG°. Then P = NLG(P°). In other words, a parabolic subgroup is the normalizer of a parabolic subgroup P° of LG°, provided the normalizer meets every class modulo LG°. We say P is standard if it contains LB. The standard parabolic subgroups are the subgroups (1)
LP° > rk,
where LP° runs through the standard parabolic subgroup of LG° such that J(LP°) C
Jv is stable under rk. Every parabolic subgroup of LG is conjugate (under LG or, equivalently, LG°) to one and only one standard parabolic subgroup. We let .1(LG) be the set of parabolic subgroups of LG and p(LG) the set of their conjugacy classes. The given bijection J H Jv yields then, in view of 3.2, a bijection (2)
p(G)k - p(LG).
We shall say that a parabolic subgroup of LG is relevant if its class corresponds to one of p(Glk) under this map. We let L,A(LG) be the set of relevant parabolic subgroups and Lp(LG) the set of their conjugacy classes, the relevant conjugacy classes of parabolic subgroups. Thus, by definition (3)
p(G/k) 4 -* Lp(LG).
Thus, if G and G' are inner forms of each other, p(LG) and p(LG') are the same, but Lp(LG) and Lp(LG') are not. If G' is quasi-split, then Lp(LG') J- p(LG'); hence we have an injection (4)
Lp(LG) - Lp(LG') = p(LG').
If 2G is anisotropic over k, then Lp(LG) consists of G alone. 3.4. Levi subgroups. Let P be a parabolic subgroup of LG. The unipotent radical
N of P° is normal in P and will also be called the unipotent radical of P. Then
P/N- P°lN x rk. In fact, it follows from (1) that P is a split extension of N, and is the semidirect product of N by the normalizer in P of any Levi subgroup M° of P°. Those normalizers will be called the Levi subgroups of P. Let P e 91(Glk), M a Levi k-subgroup of P. Let LP be the standard parabolic subgroup in the class associated to that of P (see (3)). Then LM may be identified to a Levi subgroup of LP. In fact if M corresponds to (X*(T), J, X*(T), Jv), then LM° corresponds to (X*(T), Jv, X *(T), J) and LM° x rk is equal to LM by definition and is a Levi subgroup of LP, as defined above. A Levi subgroup of a parabolic subgroup P of LG is relevant if P is.
AUTOMORPHIC L-FUNCTIONS
33
For the sake of brevity, we shall sometimes say "Levi subgroup in G" for "Levi subgroup of a parabolic subgroup of G." Similarly for LG. 3.5. LEMMA. The proper Levi subgroups in LG are the centralizers in LG of tori in _9(LG°), which project onto 1'k.
Let M be a proper Levi subgroup in LG. It is conjugate to a subgroup _T(S)° x I'k, where S c LT° is the identity component of the kernel of a subset J 5i dv stable under 1'k. Let then S' be the one-dimensional subtorus of S l 2(LG°) on which the remaining simple roots are all equal. It is clear that 9(S')° = Y(S), and that S' is pointwise fixed under I'k. We have then M = -T(S'). Let now S be a nontrivial torus in _9(LG°) such that T(S) meets every connected component of LG. Fix an ordering on X*(S). There is a proper parabolic subgroup P° of LG° of the form 1(S)° U, such that the weights of S in the unipotent radical U of P° are the roots of LG° with respect to S which are positive for this ordering. U is normalized by _T(S); hence _T(S) U is a proper parabolic subgroup P of LG, and then s(S) is a Levi subgroup of P. 3.6. PROPOSITION. Let H be a subgroup of LG whose projection on 1'k is dense in 1'k. Then the Levi subgroups in LG which contain H minimally form one conjugacy class with respect to the centralizer of H in LG°.
Let C be the identity component of the centralizer of H in _9(LG°), and D a maximal torus of H. If D = {1}, then, by 3.5, H is not contained in any proper Levi subgroup in LG, and there is nothing to prove. So assume D 0 {1}. Let r' be a normal open subgroup of I'k which acts trivially on LG°. It is then normal in LG, and H. F' projects onto 1'k. Since 9(D) contains H. I", it projects onto Fk, hence is a proper Levi subgroup by 3.5. Let M be a Levi subgroup containing H. By 3.5, M = Z(S), where S is a torus in Y(LG°). Then S c C, there exists c e C such that c S c-1 c D, hence c M M. c-1 = t(S') Y(D). 4. Remarks on induced groups. (To be used mainly to discuss restriction of scalars in §5 and 6.4.) 4.1. Let A be a group, A' a subgroup of finite index of A and E a group on which A' operates by automorphisms. Then we let (1)
IndA,(E) = IA,(E) = {f : A ---> EI f(a'a) = a' f(a) (a e A; a' eA')}.
It is a group (composition being defined by taking products of values). It is viewed as an A-group by right translations: (2)
r,,f(x) = f(xa)
(x, a e A).
For s E A'\A, let (3)
ES = {f(=- IA,(E) If(a) = O if a s}.
Then ES is a subgroup, IA,(E) is the direct product of the ES's (s e A'\A), and these subgroups are permuted by A. The subgroup E. is stable under A' and is isomorphic to E as an A' module under the map f Hf(e). The product of the Es's (s e A'\A, s 0 e) is also stable under A'. We have therefore canonical homomorphisms
34
A. BOREL
(4)
E >a A'- I(E) x A' ->E x A'
whose composition is the identity. 4.2. Let B be a group, p:B -+ A a homomorphism. Let B' = u 11(A') and assume
that p induces a bijection: B'\B -, A'\A. Let E be a group on which A' operates by automorphisms, also viewed as a B'-group via p. Then f p -f induces an isomorphism
p':IA,(E)
(1)
le(E),
whose inverse is p-equivariant. This follows immediately from the definitions.
4.3. Let A, E be as before, C a group and v : C -> A a homomorphism. Let ep: C -> E x A be a homomorphism over A. The map 0: C -+ E such that ep(c) (ci(c), v(c)) (c e C) is a 1-cocycle of C in E and ep '- 0 induces a bijection H1(C; E)
cOA(C, E),
where, by definition, (PA(C, E) denotes the set of homomorphisms cp:C -* E x A over A, modulo inner automorphisms by elements of E. 4.4. Let A, A', B, B' and E be as in 4.2. We have a commutative diagram with exact rows
1 - IA ,(E) ----> IA ,(E) A A
A
1
(1) T
A' - 1
T
T
1 ---> E ---> E v A' where the vertical maps are natural inclusions (4.1).
Let cp: B -+ IA,(E) x A be a homomorphism over A. Using 4.1(4), we get by restriction a homomorphsim ip : B' --, E x A' over A'.
4.5. LEMMA. The map cc F- 0 of 4.4 induces a bijection ccA(B, IA.(E)) epAB', E). We have, using 4.2, 4.3: (1)
(2)
cOA(B, JA
(E)) = H1(B; I A (E)) = H1(B; I B' (E)), OA,(B', E) = H1(B'; E).
By a variant of Shapiro's lemma, contained, e.g., in [4, 1.29] :
H1(B; Ia,(E))
H1(B'; E),
and it is clear that the isomorphisms (1), (2) carry this isomorphism over to cp H gyp.
5. Restriction of scalars. In this section, k' is a finite extension of k in k,, G' is a connected k'-group, and G = Rk,ik G'.
5.1. The Galois group Tk' of k, over k' is an open subgroup (of finite index) of Tk and X`k', k = Tk,\Tk may be identified with the set of k-monomorphisms of k'
into ks. We have, in the notation of 4.1 (with A = Tk, A' = Tk') (1)
IT\rk°G'(k) G(k) = I k,(G'(k)) = ,.Fk
35
AUTOMORPHIC L-FUNCTIONS
Assume G' to be reductive. Then we see easily that O(G) = (M, cp, M*, gyp") is related to ci(G') = (M', (p', M'*, gyp'") by
M = I; k, (M'),
(2)
cp = U (p'. a. aEA'\A
Similarly, if J' is a basis of rp', then
4=
(3) is one for (p.
From this it follows that we have a natural isomorphism Ip (LG'°)
LG°
(4)
We have then a commutative diagram LG'°
I (5)
LG' = LG° x
I
rk' -> rk' -> 1
I
1 -> LG° = Irk,(LG'O) --+LG = LG° ) rk
I
rk ---> 1
5.2. The map P' Rk',k P' induces a bijection between g(G'/k') and 9(G/k). Moreover P' is a Borel subgroup of G' if and only if Rk'/k P' is one of G. Hence G' is quasi-split/k' if and only if G is quasi-split over k (see [5, §6]).
Since G(k) -, G'(k'), we also get a bijection p(G'lk') -, p(Glk).
If J' c 4' is stable under rk' then J = UaEA'\Ara(J') is stable under rk. This map is easily seen to yield a bijection between rk'-stable subsets of J' and rk-stable subsets of J, whence also canonical bijections p(LG')
p(LG),
Lp(LG') --. Lp(LG).
CHAPTER II. QUASI-SPLIT GROUPS. THE UNRAMIFIED CASE.
In this chapter, G is a connected reductive quasi-split k-group. From 6.2 on, G is assumed to split over a cyclic extension k' of k, and a denotes a generator of Gal(k'lk). 6. Semisimple conjugacy classes in LG.
6.1. Assume B and T to be defined over k. Then the action of rk on X *(T) or X*(T) given by uG coincides with the ordinary action. The greatest k-split subtorus Td of T is maximal k-split in G, and its centralizer is T; in particular, Td contains regular elements of G. Hence any element w e W which leaves Td stable is completely determined by its restriction to Td. It follows that kW may be identified with the subgroup of the elements of W which leave Td stable or, equivalently, with the fixed-point set of Pk in W. If we go over to the L-group and identify canonically W with W(LG°, LT°), then kW is also the fixed-point set of rk in W, and it operates on the greatest subtorus S of LT To which is pointwise fixed under rk. The group S always contains regular elements; hence any element of k W is determined by its restriction to S. We let kN be the inverse image of k W in the normalizer N of LT° in LG°. 6.2. LEMMA. Every element w e kW has a representative in kN which is fixed under 6.
36
A. BOREL
Write Jv = D1 U .. U Dm, where the Di's are the distinct orbits of T(k'/k) in Jv. Let 3i be the common restriction to S of the elements of Di and S. the identity component of the kernel of oi. Then kW, viewed as a group of automorphisms of S, is generated by the reflections si to the S. (1 < i < m), and it suffices to prove the lemma for w = S. (1 5 i < m). [The "reflection" si is the unique element 0 1 of W which leaves S stable, fixes Si pointwise and is of order two.] We let Lie(M) denote the Lie algebra of the complex Lie group M. For a G Jv, let, as usual (1)
ga = {X e Lie(LG°)IAd t(X) = a(t) . X (t E LT°)}.
It is one-dimensional. Fix i between 1 and m. By construction of LG, we can find nonzero elements ea e ga (resp. e_a e g_a (a e D.)) which are permuted by a. We have then [ea, e-a] = c a,
(2)
where c is # 0, and independent of a e Di since f'(k'/k) is transitive on Di. Here X*(LT°) p Cis identified with Lie(LT°), and a with a OO 1. The element
.f±i = E e±a,
(3)
aEDi
is fixed under a. Moreover, since the difference of two simple roots is not a root, we have (4)
hi = [.fi, f-j] _aED; E [ea, e-a] = c E a. a"EDi
Using (3) and (4), we get (5)
e±p.
[hi, f± ] = c a, QED;
By the transitivity of Gal(k'lk) on Di, the number (6)
d =aeD; E (a, (3>
is also independent of (3 E D.; therefore (7)
[h, .f+i] = c d .f+i
We claim that d 0 0, in fact that d = 1, 2. The irreducible components of Di are permuted transitively by Gal(k'lk) and have a transitive group of automorphisms. Therefore they are of type Al or A2. Then, accordingly, d = 2 or d = 1. It follows that hi, f, and f_1 span a three-dimensional simple algebra pointwise fixed under a. Then so is the corresponding analytic subgroup G. of LG°. The group Gi centralizes Si and S n Gi is a maximal torus of G1, with Lie algebra spanned by hi. Then the nontrivial element of W (Gi, s n Gi) is the required element. REMARK. An equivalent statement is proved, in a different manner, in [35, pp. 19-22].
6.3. We let Y = L(Td)°. The group X*(Td) may be identified to the fixed-point set of rk in X*(T). The inclusion of X*(Td) = X*(Y) into X* (T) = X*(LT°) induces a surjective morphism LT° -+ Y, to be denoted v. The map A : t H t-1 °t is an endomorphism of LT°, whose differential dA at I is
(da - Id). Let
37
AUTOMORPHIC L-FUNCTIONS
U = (ker A)°,
(1)
V = im A.
Then U is pointwise fixed under a, the Lie algebra of U (resp. V) is the kernel (resp. image) of dA. Since dA is semisimple, they are transversal to each other; hence
LT° = U. V,
(2)
and u n V is finite.
Moreover,
V = ker v,
(3)
v(U) = Y.
In the rest of this chapter, we let LG stand for the "finite Galois form" LG° x Gal(k'/k) of the L-group. We now want to discuss the semisimple conjugacy classes in LG° x a with respect to LG°. We have (4)
g1 (h x a). g= g1 h °g x a
(g, h e LG°)
therefore LG°-conjugacy in LG° x a is equivalent to a-conjugacy in LG°. 6.4. LEMMA. Let v' : LT° x a --> Y be defined by v' (t x 6) = v(t) (t e LT°). Then
v' induces a bijection Y/kW. v : (LT° x a)/Int kN Let n e kN. By 6.2, we may write n = w s with w = °w and s e LT °. Then the LT°-component of n -1(t x a) n is
(1)
s-1 w 1.t.wos = s-1.0s.(w-1.t.W) e hence
v' (n I (t x a) n) = v(w 1 t w) = w-1 v(t) w = w-1 V (t x v) w.
Thus v' is equivariant with respect to the projection kN -> k Wand therefore induces a map of the left-hand side of (1) into the right-hand side of (1), which is obviously w for some surjective. Let t, t' e LT ° and assume that v'(t x a) = w-1 v'(t' x w E kW. Then we have v(t) = v(w I t' w), where w is a representative of w fixed
under u, whence t = v (w-1 t'. w), with v e V. We can write v = s-1 Us for some s e LT °, and get t x or = n 1(t' x 6.5. LEMMA. Let (LG° x the map ,u
with n = ws.
be the set of semisimple elements in LG° x or. Then
: (LT° x a)/IntkN -' (LG° x a)s,/Int LG°,
induced by inclusion is a bijection.
By results of F. Gantmacher [12, Theorem 14], ,u is surjective. Let now s, t e LT°
and g e LG° be such that g-1 (s x a) g = t x u, i.e., such that g-1 s °g = t. Using the Bruhat decomposition of LG° with respect to LB°, we can write uniquely
g = u n v, with u, v in the unipotent radical of LB° and n in the normalizer N of LT°. These groups are stable under a, and normalized by LT°. We have then s.Qu.vn.ov =
u.n.t.(t-1.v.t);
hence °n s-1 = n t. Therefore the connected component of n in N is stable under
38
A BOREL
a, i.e., n represents an element of kW; hence n e kN, and (t X u) and (s x a) are conjugate under kN. REMARK. This proof was suggested to me by T. Springer. 6.6. If M is a complex affine variety, we let C[M] denote its coordinate algebra. The algebra C[Y] may be identified with the group algebra of X*(Y) = X*(Td). The quotient Y/kW is also an affine variety (in fact isomorphic to an affine space) and C[Y/kW] = Let Rep(LG) c C[LG] be the subalgebra generated by the characters of finite dimensional holomorphic representations. Its elements are constant on conjugacy classes. In particular, they define by restriction functions on (LG° >a 6), /Int LG°. C[Y]kW.
6.7. PROPOSITION. The map
a = ,u °
L-I
: Y/kW -, (LG° x Q),,/Int LG°
is a bijection, which induces an isomorphism of C[Y/k W] onto the algebra A of restrictions of elements of Rep(LG).
REMARK. We shall use 6.7 only when k is a nonarchimedean local field. In that case 6.7 is proved in [35, pp. 18-24]. PROOF. That a is bijective follows from 6.4, 6.5. We prove the second assertion as in [35]. Let p be a finite dimensional holomorphic representation of LG and fp the
function on LT° defined by ff(t) = tr p(t x a). It can be written as a finite linear combination f = ZcAA of characters A G X*(LT°). Since tr p is a class function on LG, we have fP(s-1 t °s) = ff(t) for all s, t e LT °. By the linear independence of charac-
ters, it follows that if cz 0 0, then A is trivial on V (cf. 6.3(1)), hence is fixed under a, i.e., may be identified to an element of X*(Y). Thus we may view fp as an element of C[ Y]. But invariance by conjugation and 6.4 imply that f e C[Y/kW], whence a map 3 : A -+ C[Y/kW], which is obviously induced by a. There remains to see that /i is surjective. Note that C[Y/kW] is spanned, as a vector space, by the functions (1)
(PA = E
wE*W
where A runs through a fundamental domain C of k Won X*(Td). But it is standard that we may take for C the intersection of X*(Td) with the Weyl chamber of Win X*(T) defined by B. Therefore every d e C is a dominant weight for LG° with respect to LT°. It is then the highest weight of an irreducible representation it of LG°. Since it is fixed under a, the representation 17rA : g H zz(°g) is equivalent to irz. From this it is elementary that ira extends to an irreducible representation kA of LG of the same degree as ira. The highest weight space is one-dimensional, stable under U. Let c be the eigenvalue of a on it. Then the trace gives rise to a function equal to c (PA modulo a linear combination of functions cps, with a < 2, in the usual ordering. That im contains rpx (A e C) is then proved by induction on the ordering.
7. The Satake isomorphism and the L-group. Local factors. 7.1. We keep the previous notation and conventions. We assume moreover k to be a nonarchimedean local field, k' to be unramified over k, and a to be the image of a Frobenius element Fr in Tk. Let Q be a special maximal compact subgroup of G(k) [61]. We assume Q n T is the greatest compact subgroup of T(k) and Q contains representatives of kW. Let
39
AUTOMORPHIC L-FUNCTIONS
H(G(k), Q) be the Hecke algebra of locally constant, Q-bi-invariant, and compactly supported complex valued functions on G(k). The Satake isomorphism provides a canonical identification H -, C[Y/kW], hence also one of Ylk W with the characters of H [6]. By 6.7, we have now a canonical isomorphism of H with the algebra A of restrictions of characters of finite dimensional representations of LG to semisimple LG°conjugacy classes in (LG° x o), hence also a canonical bijection between characters of H(G(k), Q) and semisimple classes in LG° x a. Furthermore, each such class can be represented by an element of the form (t, o,), with t E LT° fixed under 6 (and is determined modulo the finite group u n V, in the notation of 6.3). 7.2. Local factors. Assume now that U is hyperspecial [61]. Let 0 be an additive character of k. Let (pr, U,,) be an irreducible admissible representation of G(k) of class 1 for Q and r a representation of LG. Then the space of fixed vectors of Q in U,r is one-dimensional, acted upon by H via a character x, To the latter is assigned by 7.1 a semisimple class SX in LG° x u. We then put (1)
L(s, 9G, r) = det(1 - r((g >4 a)), q-s)-,
e(s, ir, r, yb) = 1,
where q is the order of the residue field, and (g, aa) any element of SX. CHAPTER III. WEIL GROUPS AND REPRESENTATIONS. LOCAL FACTORS.
In this section, k is a local field, Wk (resp. Wk) the absolute Weil group (resp. WeilDeligne group) of k. If H is a reductive k-group, then II(H(k)) is the set of infinitesimal equivalence classes of irreducible admissible representations of H(k). G denotes a connected reductive k-group.
The main local problem is to define a partition of II(G(k)) into finite sets Il,,,c or II,p indexed by the set O(G) of admissible homomorphisms of WW into LG, modulo inner automorphisms (see §8 for O(G)), and satisfying a certain number of conditions. So far, this has been carried out for any G if k = R, C [37], for tori over any
k [34] and (essentially) for G = GL2 (cf. 12.2). §9 recalls the results for tori; §10 describes some of the conditions to be imposed on this parametrization; §11 summarizes the construction over R or C. Such a parametrization would allow one to assign canonically local L- and s-factors to any it e II(G(k)) and any complex representation of LG. Two elements z, z' in the same set ff. would always have the same local factors, and are hence called L-indistinguishable. In the case of GL however, local factors have been defined in an a priori quite different way, so that the parametrization problem becomes subordinated to one concerning L- and e-factors. This is discussed in §12. 8. Definition of c(G). 8.1. Jordan decomposition in W. If k = R, C, then Wk = Wk and, by definition, every element of Wk is semisimple. Let k be nonarchimedean. Then x e WW is said to be unipotent if and only if it belongs to Ga; the element x is semisimple if either e(x) # 0 or x is in the inertia
group. Here e: Wk - Z is the canonical homomorphism Wk -+ Wk -> k* -> Z. Every element x E Wk admits a unique Jordan decomposition x = x, x,, with xs semisimple, x,, unipotent and xsx = [60].
40
A. BOREL
8.2. The set 0(G). We consider homomorphisms a : WW
LG over Tk, i.e., such
that the diagram a
-LG
W, 1 d
Tk
is commutative, and which satisfy moreover the following conditions: (i) a is continuous, a(G0) is unipotent, in LG°, and a maps semisimple elements into semisimple elements (in LG: x = (u, r) is said to be semisimple if its image under any representation (2.6) is so). (ii) If a(WW) is contained in a Levi subgroup of a parabolic subgroup P of LG, then P is relevant (3.3). Such a's are called admissible. We let 0(G) be the set of their equivalence classes modulo inner automorphisms by elements of LG°. If we write a(w) = (a(w), v(w)) with a(w) e LG° then w '- a(w) is a 1-cocycle of Wk (acting on LG° via w; -+ Pk) in LG°. It follows that
0(G) c Hl(W,; LG°).
(1)
Let H be a subgroup of W. Then a: Wk -+ LG is said to be trivial on H if v(H) acts trivially on LG° and a(H) _ {1}. Note that if v(H)l, acts trivially on LG°, then a1H is a homomorphism. 8.3. Assume G' is an inner quasi-split form of G. Then 0(G) c 0(G').
(1)
In fact LG '' LG' and Lp(LG')
Lp(LG); therefore a e 0(G)
a c 0(G').
8.4. PROPOSITION. Let k' be a finite separable extension of k; let G' be a connected reductive k-group and G = Rk,,kG'. Then there is a canonical bijection 0(G) -, 0(G').
We consider the situation of 5.2, 5.4 with A = Tk, A' = I'k', B = Wk, B' = Wk,, E = LG'°. We have the injections (8.2):
0(G) c H'(W,; LG°),
0(G') c Hl(W,,; LG'°).
Moreover LG° = Irrk,(LG'°) (see 5.1); whence, by Shapiro's lemma and 5.2: k
Hl(Wk; LG°)
H1(WW,; LG'°).
But it is clear that this isomorphism maps 0(G) onto 0(G'). 8.5. Let ZL = C(LG°). If a: Wk -+ ZL and b: Wk -- LG° are 1-cocycles, then ab: w H a(w)b(w) is again a 1-cocycle of Wk in LG°. If a is continuous and b corresponds to rp e 0(G), then ab corresponds to an element of 0(G). We get therefore maps (1)
H'(Wk; ZL) x Hl(WW ; LG°) - Hl(WW ; LG°),
(2)
H1(Wk ; ZL) x 0(G) - 0 (G),
which define actions of the group H'(Wk; ZL) on the sets Hl(WW; LG°) and 0(G).
AUTOMORPHIC L-FUNCTIONS
41
8.6. PROPOSITION. Let cp: WW -- LG be an admissible homomorphism. Then the Levi subgroups in LG which contain cp(WW) minimally form one conjugacy class with respect to the centralizer of cp(WW) in LG°.
Since cp(WW) projects onto a dense subgroup of rk by definition, this follows from 3.6.
REMARK. Formally, this also applies to the archimedean case, but the proof in that case is simpler [37, pp. 78-79]. In fact, the argument there applies in all cases to admissible homomorphisms of the Weil (rather than Weil-Deligne) group because cp(Wk) is always fully reducible. In this case, the Levi subgroups which contain cp(Wk) minimally are those of the parabolic subgroups which contain cp(Wk) minimally. Those parabolic subgroups form therefore one class of associated groups. 9. The correspondence for tori.
9.1. Let T be a complex torus. A continuous homomorphism cp: T --> C* is described by a pair of elements 2, p e X*(T) Qx C such that 2 - p e X*(T), by the rule cp(t) = tutu. Similarly, a continuous homomorphism cp:C* ---* T is given by p, v e X*(T) Qx C
such that p - v c X*(T); we have cp(z) = zlzy, meaning that, for any 2 e X*(T), A ° o: C* - C* is given by A(co(z)) = z« '
> y«, v>
This can also be interpreted in the following way: identify X*(T) O C with the Lie algebra Lie(T(C)). Then the exponential map yields an isomorphism (X*(T) QC)/2iriX*(T) = T(C). Then p, v e Lie(T(C)) are such that cp(eh) _ e C). 9.2. Let G = T be a k-torus, and 1 = dim T. Any cp e 0(G) is trivial on G,,; hence (1)
LT°) = He'.t(Wk; X*(T) O C*),
0(G) =
where HH refers to continuous cocycles. On the other hand (2)
II(G) = Hom((X*(T) Qx k*)rk, C*).
We have canonically [34, Theorem 1]
II(G) _ 0(G).
(3)
In fact, LT and Wk are replaced in [34] by a finite Galois form LT° X Tk,,k and a relative Weil group Wk-/k, where k' is a finite Galois extension of k whose Galois group acts trivially on LT°; this is easily seen not to change 0(G). The proof then consists in showing first that the transfer from Wk,,k to k'* yields an isomorphism (4)
HI(Wyik; X*(T))
HI(k'*; X*(T))rk'ik . (k'* (9 X*(T))rk'ik,
and second that the pairing (5)
H t(Wk'ik; LT°) x HI(Wk'ik; X*(T)) --> C*,
associated to the evaluation map (t, 2) H A(t) (t e LT°; 2 e X*(T)) yields an
42
A.BOREL
isomorphism of the first group onto the group of characters of the second group, which is then (3) by definition. For illustrations, we discuss some simple cases.
9.3. k = C. Then Wk = C* and O(G) = Hom(C*, LT°). The correspondence follows from 9.1 since both Hom(C*, LT°) and Hom(T, C*)are canonically identified with {(A, pu)M,1, u e X*(T) © C, A - u e X*(T)}. 9.4. k = R. We have (1)
WR = C* x {a} with r2
1, r - z -r -l = z (z e C*).
Put C* = S x R+, with S = {z e C*, z z = 11. Then Int z is the identity on R+, the inversion on S. Write (p(z) = (a, a'), where a is determined modulo v-conjugacy, hence may be
assumed to be fixed under or (6.3(3)). We have then (p(- 1) = a2. Let p, v be the elements of X*(T) p C such that (2)
(p(z) = zu z1'
(z e C*), u - v e X*(T),
(see 9.1). We have (p(z) = v((p(z)) (z e C*); hence v = v(u). Fix h e X*(T) Q C such that a = exp 2irih. Then the character' associated to (p is given by (3)
ir(ex) = exp()/2 (x e X*(T) (9 C)
[37, p. 27]. Here - denotes the complex conjugation of Lie(T(C)) = X*(T) p C with respect to X*(T) Q R; hence x -- a - x is the complex conjugation with respect to Lie(T(R)). It follows that ex e T(R) if and only if x - a x e 2ici X*(T). EXAMPLES. (a) Let T be anisotropic over R. Then a = - 1 and we may assume a = 1, h = 0. We have ex e T(R) if x is purely imaginary and then (3) yields it = U. The fact that(p(- 1) = I shows that p e X*(T), confirming that ff(T(R)) = X*(T). (b) Let T be split over R. Then o = 1, ,u = v, (p(z) = (z z),,, a2 = 1 and h e X*(T)/2. We have ex e T(R) if and only if x - x e 2zi X*(T). It is then easily checked that r is given by ,u on the connected identity component of T(R), while its restriction to the torsion subgroup of T(R) is the character naturally, defined by h.
9.5. The unramified case. Let k be nonarchimedean, and assume T to split over an unramified extension k' of k. A character x of T(k) is said to be unramified if it is trivial on the greatest compact subgroup °T(k) of T(k). On the other hand, (p e 0 (T) is unramified if it is trivial (see 6.2) on the inertia group. The bijection 0(T) 11(T) induces a bijection between the sets Ounr(T) and Hunr(T) of unramified elements [34]. In view of its importance, we describe it in more detail.
Given t e T(k'), let v(t) a Hom(X*(T), Z) be defined by v(t)(m) = ord m(t) (m e X *(T)). It is well known, and easily deduced from Hilbert's Theorem 90, that H'(1'k'lk; ok*.) = 0, where ok is the group of units in the ring ok, of integers of V. Since T splits over k', it follows that H'(J'k,ik; OT(k')) = 0. By Galois cohomology, this implies that (T(k')/°T(k'))rk = T(k)/°T(k), therefore t -' v(t) yields a bijection (1)
T(k)/OT(k)
Hom(X*(T), Z)rk = X*(T)rk = X*(7'e),
where Td is the greatest k-split torus of T (this can also be expressed by saying that
AUTOMORPHIC L-FUNCTIONS
43
the inclusion Td c T induces an isomorphism Td(k)/°Td(k) - T(k)/°T(k), cf. [6]). We have then (2)
]I,,,,r(T(k)) = Hom(X*(T)I'k, C*) = Y.
The group 1'k operates on LT ° via the cyclic group rk,ik which is generated by the image a of a Frobenius element Fr. An unramified rp is completely determined by rp(Fr), which can be written rp(Fr) = (t, Fr), where t e LT° is determined up to conjugacy by LT°. Thus Oanr(T) = (LT° >4,o,)/Int LT°; and elementary special case of 6.4 provides a canonical isomorphsim of the latter set onto Y, whence the desired isomorphism.
10. Desiderata. In order to formulate them, we need two preliminary constructions. 10.1. The character x,, of C(G) associated to rp a 0(G) (cf. [37, pp. 20-34]). We want to associate canonically to rp e 0(G) a character of the center C(G) of G. Let CSrad be the greatest central torus of CSI. Then Grad --* G yields a surjective homomor-
phism LG -> LGrad, whence a map 0(G) -> O(Grad) In view of 7.2, this allows us to associate to (p e 0(G) a character XP of Grad. Thus, if C(G(k)) c Grad(k), our problem is solved. In the general case, G is enlarged to a bigger connected reductive Gl generated by
G and a central torus, whose center is a torus. One shows that 0(GI) -a 0(G) is surjective. Using the previous step, we get a character of C(G1), hence one of C(G) by restriction. It is shown to be independent of the choice of GI (loc. cit.), and is xw by definition. The map 'p H x9 is compatible with restriction of scalars [37, 2.11]. 10.2. The character ira associated to a e HI(Wk; ZL) [37, pp. 34-36]. We recall that ZL denotes the center of LG° (8.5). We can always find a k-torus D such that HI (Pk; D) = 0, and a k-group G isogeneous to G x D such that there is an exact sequence (1)
I
D ---a G -a G ---a 1.
Since Hl(Pk; D) = 0, the map p: G(k) --a G(k) is surjective. Let G, be the universal covering of the derived group.9G of G. We have a commutative diagram
A. BOREL
44
Going over to L-groups, we get
LD° I
LGsr<-LG°(_L(G/Gsc)°
1
Since LGs° is of adjoint type (2.2), we see that ZL = ker La°. Moreover, it is easily seen that (2)
ZL c--- ker
L01.
This yields a map (3)
Hl(WW; ZL) --- ' ker{Hl(WW; L((g/G5°)°) - H'(WW; LD°)}.
This allows us to associate to a e HI(WW; ZL) a character o of (C/GS°)(k) which is trivial on D(k), hence a character za of G(k) = C(k)/D(k). It can be shown to be independent of the choice of D. The map a z,, is compatible with restriction of scalars [37, 2.12] and satisfies: (4)
xaw = 2a . xw
(a e H1(Wk; ZL), p e O(G))
10.3. Conditions on the sets llg,. (1) If c e IIg,, then ir(z) = xg,(z) Id (z e C(G)). (2) If rp' = a cp (gyp, cp' e Q(G), a E H'(WW; ZL)) (see 6.5), then III, _
E 11g,) .
(3) The following conditions on a set 1Iw are equivalent: (i) One element of Il, is square-integrable modulo C(G). (ii) All elements of 1I, are square-integrable modulo C(G). (iii) rp(W,) is not contained in any proper Levi subgroup in LG. (4) Assume cp(GQ) = {1}. The following conditions on a set II, are equivalent: (i) One element of Il,, is tempered. (ii) All elements of 1l, are tempered. (iii) rp(Wk) is bounded.
(5) Let H be a connected reductive k-group and 77: H - G a k-morphism with commutative kernel and cokernel. Let rp e 0(G) and (p' = L7) ° cp. Then any it E Ilw, viewed as an H(k)-module, is the direct sum of finitely many irreducible admissible representations belonging to III-. 10.4. The unramified case. We say that cp e 0(G) is unramified if it is trivial, in the sense of 6.2, on G,, and on the inertia group I. If so, Im p may be assumed to be in
LT. Therefore, if 0(G) contains an unramified element, then G is quasi-split (see 8.2 (ii)).
Assume now G to be quasi-split, to split over an unramified Galois extension
AUTOMORPHIC L-FUNCTIONS
45
k' of k, and let rp e 0(G) be unramified. There exists t E (LT °)rk such that (1)
cp(Fr) = (t, Fr),
(9.5) and we have (2)
,p(w) _ (t, Fr)f(w)
(w e Wk),
where e : Wk -> Z is the canonical homomorphism. The element t defines an unramified character x of a maximal k-torus T of a Borel k-subgroup B of G (9.5). It is then required that II, consists of the constituents of the unramified normalized principal series PS(x) which have a nonzero vector fixed under some hyperspecial maximal compact subgroup. Conversely let (2r, V) be an irreducible admissible representation with a nonzero vector fixed under some hyperspecial maximal compact subgroup. There exists then an unramified character x of T such that (pr, V) is a constituent of PS(x) (and x is determined modulo the relative Weyl group). We have then ('r, V) E II4,, for the unramified cp which maps Fr to (t, Fr), where t represents x (9.5). Note that if U is a special maximal compact subgroup of G(k), then G(k) = B(k) U; hence the fixed-point set of U in PS(x) is at most onedimensional. It follows that PS(x) has at most one irreducible constituent with nonzero fixed vectors under U. This assignment is consistent with 7.2. Namely, if ir e II,, then the semisimple class SX in LG° m a corresponding to the character of the Hecke algebra defined by r is indeed represented by t >4 o,. This follows from [6]. REMARK. Originally, it was thought that 11. should consist of those constituents of PS(x) which had a nonzero fixed vector under some special maximal compact subgroup. However it was pointed out during the Institute by I. Macdonald that such representations may belong to the discrete series. If so, this condition would contradict 10.3(3). Upon a suggestion of J. Tits, this has led to the restriction to hyperspecial maximal compact subgroups made above. Those cannot belong to the discrete series, so that 10.3(3) and 10.4 are consistent. 10.5. EXAMPLE. Assume that k = R and that G is semisimple, possesses a Cartan subgroup T which is anisotropic over R, and is an inner form of a split group. Then LG is the direct product of LG° and Tk, the Weyl group W contains - Id and G(R) has a discrete series. We want to describe the parametrization of the latter in terms of 0(G). As the notation implies, we shall view LT as the L-group of T. Let cp e 0(G). It is given by a continuous homomorphism rp' : WR --> LG°. We may assume that Im (p' is contained in the normalizer of LT°. Let n = gp(r) and let w E W be the element of W represented by n. Then w2 = 1. Let p, v E X*(T) Q C be such that (1)
g(z) = zu z"
(z e C*), ,u - v e X*(T)
(see 9.1). We have (2)
go(Z) = n ' rp(z) . n-1 = zw'u . Zw'Y
hence IC = w v, v = w ,u. Assume now that Im cp is not contained in any proper Levi subgroup in LG or, equivalently, that Im (p' is not contained in any proper Levi subgroup in LG°. Then w = - Id and ,u is regular: in fact, the proper Levi subgroups in LG° are the centralizers of nontrivial tori. This implies first that w does
46
A. BOREL
not fix pointwise any nontrivial torus in LT°, hence w = - Id; if now p were singular, then the centralizer ofu(C*)would contain a semisimple subgroup H {1} stable under Int n, the latter would leave pointwise fixed a torus S # {I } of H, and Im cp' would be contained in the centralizer Z(S) of S, a contradiction. Since v = w ,u
,u, we have
2((p(- 1)) _ (- 1)(21 2>, for all A e X*(T).
(3)
Let 3 be half the sum of the roots a of G with respect to T such that
> 0.
Then, Lemma 3.2 of [37] implies in particular
A(rp'(- 1)) _ (- 1)(z6, a', for all ,1 e X*(T).
(4)
It follows that ,u E 5 + X *(T).
(5)
Therefore p is among the elements of X*(T) O Q which parametrize the discrete series in Harish-Chandra's theorem. We then let 14 be the set of discrete series representations of G(R) with infinitesimal character x.. If G(R) is compact, then /4 consists of the irreducible finite dimensional representation with dominant weight p - J. In that case, no proper parabolic subgroup of LG is relevant; hence 0(G) consists of the (p considered here.
10.6. Let G = GL,,, k nonarchimedean. Let 0 be an admissible representation of W. If it is irreducible, then Sb(Ga) = 1. If it is indecomposable, then it is a tensor product p ® sp(m), where m divides n, p is irreducible of degree n/m, and sp(m) is m-dimensional, trivial on 1, maps a generator of the Lie algebra of Ga onto the nilpotent matrix with ones above the diagonal, zero elsewhere, and w e Wk onto the diagonal matrix with entries a(w)i (0 < i < n) [9, 3.1.3]. If x is a character of Wk (hence of k*), and (p = x O sp(n), then ll. consists of the special representation with central character determined by X. In fact, the Weil-Deligne group came up for the first time precisely to fit the special representations of GL2 into the general scheme (see [9]). 11. Outline of the construction over R, C. We sketch here the various steps which yield the sets II4 when k = R. For the proofs see [37]. We note first that we may always assume rp(Wk) c N(LT°), and we can write (9.1) rp(z) = z'2zy
(z e C*;,u, v e X*(T) (D C, It - v e X*(T)).
11.1. LEMMA. Let cp e 0(G). Assume cp(WR) is not contained in any proper Levi subgroup in LG. Then (i) G has a Cartan k-subgroup C such that (-?G n S)(R) is compact [28, 3.1]. (ii) g is regular;
The group LC° may be viewed as a maximal torus of LG°; hence there is an isomorphism LC ->- LT defined modulo an element.of W. Therefore (P defines an
orbit of W in 0(C), hence, by 9.2, an orbit X, of W in X(C(R)). [Note that W, which is defined in G(C), operates on C(R), since C(R) n 9G is compact, hence on X(C(R)).] 11.2. Let Go = C(R)((_9G)(R))°. Let A0 be the set of representations of Go which
AUTOMORPHIC L-FUNCTIONS
47
are square-integrable modulo the center, and have infinitesimal character XA (A e X.). The induced representations it = Ico(s)(iro) (Iro a Ao) are irreducible [37, p. 50]. By definition, 17. is the set of equivalence classes of these representations [37, p. 54]. 11.3. Let cp e 0(G). Let LM be a minimal relevant Levi subgroup containing Im cp. It is essentially unique (8.6). We assume LM : LG; we may view cp as an ele-
ment of 0(M). By 11.2, there is associated to it a finite set of IIP,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 ^G/k). Then U = X*(T) p R = X*(LT°) ® R. Let V be the subspace of elements of U which are orthogonal to roots of LM, and fixed under 1'k. It may be identified with the dual ap of the Lie algebra of a split component A of P. Let be the character of C(M) defined by the elements of II,,,M. We may assume that e C1(a**+). Let P1 be the smallest parabolic k-subgroup containing P such that lCl, when restricted to ap,, is an element of the Weyl chamber a** . Let M1 =
z(ap) and P = P n M,. Then P is a parabolic subgroup of M1. Moreover the restriction of IeI to the split component M1 n AP of P' is one; therefore, for each p e IIp,M, the induced representation IndP;(p) is tempered. Let 1I', be the set of all constituents of such representations. Then by definition, ]I, is the set of Langlands quotients J(Pl, o) with a e ]I,, (cf. [37, p. 82]). 11.4. Complex groups. Assume now k = C. Then Wk = C*, and 0(G) may be identified to the set of homomorphisms of C* into LT°, modulo the Weyl group W, i.e., to (1)
{(A, ,u), where A, ,u e X*(T) px C, A - ,u e X*(T)}
modulo the (diagonal) action of W. In this case Im ep is in the Levi subgroup LT of LB, which is the LP of 11.3. The set IIIP,M consists of one character of T(cf. 9.1). Choose P1, M, as in 11.3. Since the unitary principal series of a complex group are irreducible (N. Wallach), the set 14, consists of one element. Hence so does ]I,,. Thus
each 17, is a singleton. The classification thus obtained is equivalent to that of Zelovenko. 11.5. Let G = GL,,, k = R. In this case, it is also true that the tempered representations induced from discrete series are irreducible [22]; therefore each set 14, (cf. 9.3) consists of only one element, hence so does II4and we get a bijection between 0(G) and IIG(R). Let n = 2. If cp is reducible, then Im cp is commutative; hence cp factors through (WA)ab = R* and is described by two characters u, v of R*. Then 1I,, consists of a principal series representation ir(u, v) (including finite dimensional representations, as usual). In particular there are three cp's with kernel C*, to which correspond respectively 2r(1, 1), z(sgn, sgn) and 2r(1, sgn), where sgn is the sign character. If co is irreducible, then T(r) may be assumed to be equal to (so, a), where so is a fixed element of the normalizer of LT° inducing the inversion on it. cp(R+) belongs to the center of LG°, and cp(S) is sum of two characters, described by two integers. Then 14, consists of a discrete series representation, twisted by a one-dimensional representation. 11.6. As is clear from these two examples, the main point to get explicit knowl-
A. BOREL
48
edge of the sets II4is 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]. 11.7. Remark on the nonarchimedean case. Langlands' classification [37] is also valid over p-adic fields [57]. In view of 8.6, it is then clear that the last step (11.3) of the previous construction can also be carried out in the nonarchimedean case. Thus, besides the decomposition of tempered representations, the main unsolved problem in the p-adic case is the construction and parametrization of the discrete series.
12. Local factors. 12.1. Let 7z e II(G(k)) and r be a representation of LG (2.6). Assume that 7z e III
for some (p e 0(G). For a nontrivial additive character 0 of k, we let (1)
L(s, 7r, r) = L(s, r ° (p),
e(s, 7r, r) = e(s, 7r, r, 0) = s(s, r ° (p, (l1),
where on the right-hand sides we have the L- and s-factors assigned to the representation r ° (p of Wk [60]. In the unramified situation of 10.4, 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. 12.2. Let now G = GL,,. In this case there are associated to 7z e 1I(G(k)) local factors L(s, 7r) and s(s, 7z, 0) 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 12.1, where r = r is the standard representation of GL,,, i.e., whether we have equalities (1)
L(s, 7r) = L(s, 7r, rn),
e(s, 7r, 0) = e(s, 7z, rn, Y1),
with the right-hand side defined by the rule of 12.1. (a) Let n = 2. It has been shown in [25] that the equivalence class of 7r is characterized by the functions L(s, 7z 0 x), e(s, 7r (& x, 0), where x varies through the
characters of k*. In this case, the parametrization problem and the proof of (1) are part of the following problem : (*) Given o' e 0(G), find 7z = 7z(a) such that (2)
L(s, or ® x) = L(s, 7t ® x),
e(s, a (9 x, 0) = e(s, 7r 0 x, 0)
for all x's, and prove that a H 7r(a) establishes a bijection between 0(G) and 1I(G(k)).
This problem was stated and partially solved in [25]. The most recent and most complete results in preprint form are in [62]; they still leave out some cases of even residual characteristic, although some arguments sketched by Deligne might take care of them (see [63] for a survey). As stated, the problem is local, but, except at infinity, progress was achieved first
mostly by global methods: one uses a global field E whose completion at some place v is k, a reductive E-group H isomorphic to G over k, an element p e 0(H/k) whose restriction to L(H/kv) = LG is a, chosen so that there exists an automorphic representation 7r(p) with the L-series L(s, p) (see §14 for the latter). This construc-
AUTOMORPHIC L-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 [63]. Note also that the injectivity assertion is a statement on two-dimensional admissible representations of WW, namely, whether such a representation 6 is determined, up to equivalence, by the factors L(s, a ° x) and s(s, a ° x, 0). But, so far, the known proofs all use admissible representations of reductive groups [63]. (b) For arbitrary n, (1) has been proved in the unramified case, for special representations, and by H. Jacquet for k = R, C [24]. (c) Local L- and e-factors are also introduced for G = GL2 x GL2 in [21], at any rate for products r x r' of infinite dimensional irreducible representations. Partial extensions of this to GL,,, x GL for other values of m, n are known to experts.
(d) For n = 3, it a II(G(k)) is again characterized uniquely by the factors L(s, 7r OO x) and s(s, r p x, 0) [27], [46]. For n >_ 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 or maybe just 12.3. Local factors have also been defined directly for some other classical groups, in particular for GSp4 by F. Rodier [48], extending earlier work of M. E. Novodvorsky and I. Piatetskii-Shapiro, for split orthogonal groups, in an odd number 2n + 1 of variables by M. E. Novodvorsky [41]. In the latter case LG° = Sp2r,, and in the unramified case, the local factors coincide (up to a translation in s) with those associated by 7.2 to the standard 2n-dimensional representation of the L-group. See also [42]. CHAPTER IV. THE L-FUNCTION OF AN AUTOMORPHIC REPRESENTATION.
From now on, k is a global field, o = ok the ring of integers of k, Ak or A the ring of adeles of k, V (resp. V_, resp. V f) the set of places (resp. infinite places, resp. finite
places) of V. For v e V, k,,, o and Nv have the usual meaning. Unless otherwise stated, G is a connected reductive k-group.
13. The L-function of an irreducible admissible representation of GA. 13.1. Let 7r 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 LG° >4 Tk,,k. We want to associate to r and r infinite Euler products L(s, z, r) and s(s, ir, r), whose factors are defined (at least) for almost all places of k. Let v e V. By restriction, r defines a representation r of L(G/kv) = LG° >4 I'k. On the other hand, 'r = Qx vrc,,, with 'r0 e 1I(G(kv)) [11]. Assume the parametrization problem of Chapter III solved. Then there is a unique cp e O(G/kv) such that zv e ff.,. Then we let (1)
L(s, ir, r) = 1I0L(s, pr,,
(2)
s(s, z, r) = 11vs(s, 7cv, r0, cv),
where 0 is an additive character of ky associated to a given nontrivial additive character of k, and the factors on the right are given by 12.1(1).
50
A. BOREL
The local problem is solved for archimedean v's, and for almost all finite v's (see below) so that the factors on the right are defined except for at most finitely many v e V f. For questions of convergence or meromorphic analytic continuation this does not matter, and we shall also denote such partial products by L(s, ir, r). By 10.4, (p is well defined if the following conditions are fulfilled : G is quasi-split over k,,, G(ov) is a very special maximal compact subgroup of G(kv), k' is unramified
over k, and ,rv is of class one with respect to G(ov). All but finitely many v c Vf satisfy those conditions [61]. 13.2. THEOREM [35]. Let z be an irreducible admissible unitarizable representation of GA and r be a representation of LG (2.6). Then L(s, 'r, r) converges absolutely for Re s sufficiently large.
We may and do view r as a complex analytic representation of LG° >a 1'k,,,, where k' is a finite Galois extension of k over which G splits (2.7). We let V1 be the set of v e Vf satisfying the conditions listed at the end of 13.1. We have to show that
L' =
(1)
L(s, 9Lv, rv),
vEVI
converges in some right half-plane. Let Frv be the Froberiius element of "k,,,/k,, where v' e Vk, lies over v e V1. We have cpv(Frv) = (tv, Frv),
(2)
with tv c LT°
and
L(s, vv, rv) = (det(l - r((tv, Frv))N,
(3)
s))-1.
To prove the theorem, it suffices therefore to show the existence of a constant a > 0 such that (4)
I,ul < (Nv)a
for every v e V1 and eigenvalue p of r((tv, Frv)).
Let n = [k': k]. Since we may assume tv fixed under 1'k, (6.3), we have tv = (tv, Frv)n; hence it is equivalent to show (4) for all eigenvalues p of r(tv). These are of the form t,2, where A runs through the set Pr of weights of r, restricted to LG°. Thus we have to show the existence of a > 0 such that (5)
ltvlRe2 < (Nv)a
for all v e V1 and A e Pr.
Let G' be a quasi-split inner k-form of G. Then LG = LG', and G is isomorphic to G' over kv for all v c V1. We may therefore replace G by G'; changing the notation slightly, we may (and do) assume G to be quasi-split over k. We then fix a Borel k-subgroup B of G and view LT as the L-group of a maximal k-torus T of G. For a cyclic subgroup D of "k,1k, let VD be the set of v e V1 for which T'k, is equal to the inverse image of D in T'k. The group U = X*(T)° is then the group of oneparameter subgroups of a subtorus S of T such that Slkv is a maximal kv split torus of G/kv for all v E VD. The group (6)
Y = Hom(U, C*) = Hom(X*(T)°, C*)
(v c VD),
is independent of v, and is the Y of §6 for G/kv. The root datum O(G/kv), which is determined by the action of D, is also independent of v c VD.
AUTOMORPHIC L-FUNCTIONS
51
Given y c Y, let yo be a "logarithm" of y, i.e., an element of Hom(X*(T)D, C) such that (7)
y(u) = Nvyo(u) = Nv,
for u e X*(T)D.
This element is determined modulo a lattice, but its real part Re yo e Hom(U, R), defined by (8)
y(u) = Nv
is well defined. If y has values in R+, then we choose yo to be equal to its real part. The space a* is the dual of a = U 0 R (the so-called real Lie algebra of S/kv), and is acted upon canonically by kW as a reflection group. We let a*+ be the positive Weyl chamber defined by B. Let p,, be the unramified character of T(kv), given by t H I o(t)Iv, where I I, is the
normalized valuation at v and 5 half the sum of the positive roots. Then its real logarithm po is independent of v e VD. In fact, it is a positive integral power of Nv whose exponent is determined by the k,-roots, their multiplicities, and the indices qa of the Bruhat-Tits theory [61]. But those are determined by the previous data and the action of rk on the completed Dynkin diagram [61], which is also independent of v e VD. We write po instead of p,,,o. We have po e a*+.
The representation 7cv 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 x,,,0 to be contained in the closure rel(a*+) of a*+. Since 1r is unitary, the associated spherical function is bounded, and hence Re x,,,o is contained in the convex hull of kW(po), i.e., we have (9)
<po - xv,o, A> ? 0, for all A E a*+.
(See remark following the proof.) For A e X*(LT°), let 2' be the restriction of 2 to X*(TD). In view of 10.4 and our conventions, we have then (10)
I A(tv) I = Nv.
Let ,l = k W(A') n W/(a*+). Since Re xv,o e lel(a*+), we have (11)
Nv < Nv
Combined with (9), this implies (12)
I A(tv) I < Nv<po,
A>
If now A runs through P there are only finitely many possibilities for ,l, whence (4), with a = sup<po, A> (A E P,), for v e VD. Since Vl is a finite union of such sets, this proves (4). REMARK. The relation (9) is proved in [35, pp. 27-29] for the split case. For a general semisimple simply connected group, see I. Macdonald, Spherical functions on a group of p-adic type, Pub]. Ramanujan Institute 2, Madras, Theorem 4.7.1, or H. Matsumoto, Lecture Notes in Math., vol. 590, Springer-Verlag, Berlin and New York, Proposition 4.4.11. In fact, we have used it for a general connected reductive
52
A. BOREL
group but the reduction to the case of simply connected semisimple groups is easily carried out by going over to the universal covering of the derived group. 13.3. COROLLARY. Let P be a parabolic k-subgroup of G, P = M N a Levi decom-
position over k of P. Assume that ir is a constituent of a representation Ind PA (o) induced from a unitarizable irreducible admissible representation o of MA, viewed as a representation of PA trivial on NA. Then L(s, 'r, r) is absolutely convergent in some right half-plane. We view LM as a subgroup of LG (3.3). Let r' be the restriction of r to LM.
Let v e Vr be such that the conditions listed at the end of 13.1 are satisfied by M, G, ov and irv. Then, by the transitivity of induction, it follows that there exists xv as in the above proof such that ov (resp. icv) is the constituent of class 1 with respect to M(ov) (resp. G(ov)) of the principal series PS(xv) for M(kv) (resp. G(kv)). Then L(s,'v, r) = L(s, o,,, r') (7.2, 10.4). This being true for almost all v's, we are reduced to 13.2. 14. The L-function of an automorphic representation. 14.1. A smooth representation of GA is automorphic if it is a subquotient of the regular representation of GA in Gk\GA. It is cuspidal if it consists of cusp forms. If
so, it is unitary modulo the center. We let W(G/k) denote the set of equivalence classes of irreducible admissible automorphic representations of GA. By Proposition 2 of [39], every 7c e 21(G/k) is a constituent of a representation induced from some cuspidal o e %(Mlk), where M is a Levi k-subgroup of a parabolic k-subgroup of G. Combined with 13.3 this yields the 14.2. THEOREM (LANGLANDS). Let 2r e W(G/k) and r be a representation of LG. Then L(s, z, r) is absolutely convergent in some right half-plane.
The L-function of an irreducible admissible automorphic representation will also be called an automorphic L-function. 14.3. There are several conjectures on the analytic character of L(s, Z, r) for automorphic z, all checked in some special cases, going back to the work of Hecke on L-series attached to Grossencharaktere and to modular forms. (a) If' E W(G/k), then L(s, z, r) admits a meromorphic continuation to the whole complex plane. (b) Assume that r and G are such that the local solution to the local problem yields factors L and e at all places. It is then conjectured that there is a functional equation L(s, ir, r) = e(s, zc, r) L(1 - s, k, r), where k is the contragredient representation to 7r. (c) In a number of cases, it has been shown that: (*) If ir is cuspidal, r irreducible nontrivial, then L(s, ir, r) is entire. Here and there, conjectures to the effect that this should be a general phenomenon have been stated. However, there are counterexamples. Heuristically, one sees this is likely to happen if c is lifted from a cuspidal representation of a reductive group H (in the sense of V below) and the restriction of r to LH contains the trivial representation.
14.4. (a) Let G = GL and r = r be the standard representation of Then 14.3(b), (c) are proved in [25] for n = 2, in [19] for n >_ 2, if L and e are de-
AUTOMORPHIC L-FUNCTIONS
53
fined to be the products of the L- and e-factors mentioned in 12.4. As recalled in 12.4, these are the same as those considered here at almost all places, and for n = 2, at all places.
(b) If G = GL2 x GL2 and r = r2 Q r2, similar results are established by Jacquet in [21].
(c) Let G = GL2. If r: GL2(C) --> GL3(C) is the adjoint representation, then 14.3(b), (c) are announced in [16]. This extends results of Shimura [54]. If r = Sym3(r2), Sym4(r2), then 14.3(b) is stated in [15], in the context of the global lifting (see V); for Sym3(r2), it is also proved in [51], in the framework of 14.5 below.
(d) Let k be a function field, G = GLm x GL and r = rm Qx r,,. Let it (resp. yr') be a cuspidal automorphic representation of the first (resp. second) factor. By the methods of [19], [26], [27], one can define L and a, and (Jacquet dixit) show 14.3(b), and also the holomorphy, except when m = n and r is contragredient to zr'. These methods also yield further examples for other groups and for other representations. It is expected that similar results hold over number fields. (e) 14.3(a) has also been checked when G = PSp(4) in some cases in [1], and, in general, in [42]. A functional equation is also established. 14.3(a), (b) are announced in [41] for orthogonal groups in an odd number of variables over functional fields, for the local factors mentioned in 12.3. For a survey and earlier references, see [43]. See also [44].
14.5. We describe some cases in which 14.3(a) has been verified in [33] (see also [18] for a survey). Let C be a split k-group, of adjoint type, endowed with its canoni-
cal u-structure. Fix a Borel subgroup B of C and a maximal torus T of B defined over o. Let P be a maximal proper standard parabolic subgroup and P = M N its standard Levi decomposition. Since C is adjoint, it is easily seen that C(M) is a torus. The group M/C(M) is semisimple, split over k, of adjoint type, of rank equal to rk(C) - 1. We let G = M/C(M). The group LG° is simply connected (2.2(2)). We have a natural inclusion LG _ LM, and LM is the Levi subgroup of a standard parabolic subgroup LP = LM. U with unipotent radical U (3.3). Let A be the split component of P in T, and LA° the split component of LP° in LT°. The group LA° acts on the Lie algebra u of U and its eigenspaces are irreducible LG°-modules. We let Fp denote the set of contragredient representations to these LG°-modules. The L-functions considered in [33] are of the form L(s, ir, r) with r e Fp and Z an irreducible cuspidal automorphic representation of G. A number of examples are given in which L(s, g, r) admits a meromorphic continuation. This is deduced from the results of [32]: let m be the length of a composition series of u with respect to M. Then, for suitable numbering of the elements of FP and strictly positive integers ai, there is a relation (1) M(s) L(ais, z, ri) L(sai + 1, z, ri)-1, 15ism
where M(s) is the intertwining operator occurring in the theory of Eisenstein series with respect to P, and is known to have a meromorphic continuation to the complex plane [32]. If r = 1, this and 13.2 yield the meromorphic continuation. In general, if we have the analytic continuation for all ri's except one, (1) gives it for the remaining one. 14.6. The converse problem is to what extent automorphic representations can
be characterized by analytic properties of their L-functions, or to give analytic
54
A. BOREL
conditions on a given L-function which will insure that it is automorphic. The first main result was Hecke's characterization of the Mellin transform of a parabolic
modular form. Then came Weil's extension of this theorem to congruence subgroups [64], [65], its generalization in the context of representations in [25], and the extension to GL3 [46], [27]. In those results, conditions are imposed on the L-func-
tions of ir and of the twists ir Qx x of ir by characters. However, the analogous statement is false from n = 4 on [46]. It may remain true if one imposes conditions on the twist i p p of r by representations of GLi_1 or only of For results in that direction, over function fields, see [45]. Note however that in the general problem outlined here, one wishes rather to turn things around and deduce the analytical properties of some given L-series by showing directly that it is automorphic (see the seminars on base change and on zetafunctions of Shimura varieties [17], [8], [40]). 14.7. Other problems. (1) One "representation theoretic" form of "Ramanujan's
conjecture" is the following: if ir = &7r, is an irreducible nontrivial admissible cuspidal automorphic representation (and G is simple), then each it is tempered. It is now well known to be false for certain orthogonal or unitary groups, and even for one split group [20].
(2) Let x be a unitary irreducible representation of GA. If G = GL2, then its multiplicity in the space of cusp forms °L2(G(k)\G(A)) is at most one, "multiplicity one theorem" [25]. In fact there is even a "strong multiplicity one theorem" [38] : given vv for almost all v's, there is at most one constituent r of the space of cuspforms with those local factors. The multiplicity one theorem has been proved for GL [52] and the strong form for GL3 [28]. It is unknown whether it is true for SL2. On the other hand, there are counterexamples for some inner forms of SL2 [31]. CHAPTER V. LIFTING PROBLEMS.
Although the problems on automorphic L-functions discussed in §14 are only partially solved, the solutions provide practically all cases in which an L-series (automorphic or not) has been proved to have meromorphic or holomorphic analytic continuation with functional equation. This suggests trying, given an L-series and a reductive group G, to see whether G has an automorphic representation with the given L-series. Many instances of such questions can be viewed more precisely as special cases of the "lifting problem" or of the "problem of functoriality with respect to morphisms of L-groups." There is also a local version. For the sake of exposition, we shall start with the latter, but it should be borne in mind that the motivation and requirements stem from the global one, and that local and global are at present inextricably linked in many proofs. These questions were raised by Langlands in [35]. 15. L-homomorphisms of L-groups. 15.1. Let E be a field and H, G connected reductive E-groups. A homomorphism u: LH -> LG over rk is said to be an L-homomorphism if it is continuous and if its restriction to LH° is a complex analytic homomorphism of LH° into LG°. Let E be local and G quasi-split. If rp e O(H), then u ° (p e O(G). In fact, condition 8.2(i) is clearly satisfied, by u ° cp, and so is 8.2(ii) because every parabolic subgroup of LG
AUTOMORPHIC L-FUNCTIONS
55
is relevant, G being assumed to be quasi-split. Therefore (p H u ° (p defines a map 0(H) -- 0(G), to be denoted 0(u). 15.2. Let E = k be a global field. For v e V, the Galois group 7'k, is a subgroup of Tk; hence the L-group of G viewed as a k, -group, to be denoted L(G/kv), is a
subgroup of LG = L(Glk). Thus, in particular, the L-homomorphism u of 15.1 defines by restriction an L-homomorphism uv: L(Glkv), hence also a map 0(u,): O(Hlk,,) - O(Glk,,) (v e V). The "lifting problem" is, roughly speaking, whether such maps are mirrored by maps of representations in the local case, or of automorphic representations in the global case. 15.3. EXAMPLE: BASE CHANGE. Let H be a split over E, F a finite Galois extension
of E, and G = RF/EH. Then LG° is a product of copies of LH°, indexed and permuted by FF/E (5.1). There is then a natural L-homomorphism u which is the identity on f'E and the diagonal map on LH°. If E is a local field, then WF is an open normal subgroup of WE, and the map 0(u) may be viewed as given by the restric-
tion to WF. 16. Local lifting.
16.1. Let k = E be a local field, G quasi-split over E, H a connected reductive E-group and u: LH -> LG an L-homomorphism. The problem of local lifting is, roughly, to establish a correspondence 11(u) : 11(H(k)) -> l1(G(k)) which preserves L- and e-factors. If the local parametrization problem of III is solved, then 17(u) is the map between indistinguishable classes which assigns 11,,,,,G to IIP,H
((p e 0(H)). The element 11 e 17(G(k)) is said to be a lift of 'r e 11(H(k)) if 11 e 17uoq,G, where (p e 0(H) is such that ir e 1111H. We have then (1)
L(s, 17, r) = L(s, 7r, r ° (p),
e(s, 17, r, (b) = e(s, 7r, r ° u,
for every representation r of LG. 16.2. The local lifting is thus viewed as a map between classes of L-indistinguishable representations rather than one between representations. However it is pos-
sible to single out one lifting under assumptions which, in the global case, are satisfied almost everywhere : assume H, G to be quasi-split, split over an unramified extension F of E, endowed with an OE-structure such that H(OE) and G(oE) are very special maximal compact subgroups, and v of class one with respect to H(oE). Then (p such that v e 11,,H, and the set c are well defined. Moreover, lluov, G contains exactly one element of class one (with respect to G(oE)), to be called the natural lift of nr.
16.3. A full solution of the local parametrization problem does not seem to be in sight, and it is conceivable that it may require proving at the same time global results such as Artin's conjecture. Meanwhile, one wants to settle some approximations to it, notably to be able to prove some cases of Artin's conjecture. Note that if G = GL,,, then the sets Il,,o are either known or conjectured to consist of one element (12.2, 12.3). Such a lifting problem can then be stated as one of constructing a map u*: i1(H(k)) = 17(G(k)) satisfying certain conditions. So far, there are two examples: (a) Base change (cf. 15.3) when H = GL2 and F is cyclic of prime degree over E [17], [38], [49], [56]. Besides some naturality conditions and 16.3, the main require-
A. BOREL
56
ments relate the characters of rr and of the hypothetical u*(7r). The results also describe the fibres and the image of u*. [Note that the results of [38] on this problem are used in [62], so that we cannot invoke the solution of the local parametrization problem (12.4) for GL2 just to use the map 1f(u) of 16.1. If we could, then the local questions [38] would be mainly to relate the characters of Z and If(u)(rr).]
(b) H = GL2, G = GL3, and (1)
u: LH° = GL2(C)
LG° = GL3(C)
is given by the adjoint representation of LH° (see [16]). In this case, 1f = u*(ir) must be trivial on the center of LG° and be such that the L- and e-factors of u*(rr) OO x (x character of E*) are certain given functions. There is at most one such 1f (12.4(d)). In [16], His stated to exist, except possibly if E has even residual characteristic and r is "extraordinary." 16.4. In 16.3(a), the lifting problem was connected with the existence of relations between characters. This is a direct connection between H(H) and H(G), which is of great importance for the use of the trace formula in proving or using the local or global lifting. We now mention two other examples of such relations. Assume that G is a quasi-split inner form of H. There is then an isomorphism u: LH =. LG and
an embedding 0(u): 0(H) c 0(G). If f: H -' G is a k, -isomorphism such that f-1 rf is an inner automorphism of G for every r c 1'k, then f establishes a bijection between conjugacy classes which are stable under f'k. Using results of Steinberg [59], one then sees easily that maximal k-tori in H are isomorphic over k to maximal k-tori in G. This allows one in some cases to assign regular semisimple classes in G(k) to such classes in H(k), so that it makes sense to compare values of characters of H(k) and of G(k) on such classes. (a) Let k be either R or nonarchimedean with odd residual characteristic. Let G = GL2 and H be the group of invertible elements in the quaternion algebra over k. The sets A. are singletons, 0(u) assigns to a (finite dimensional) irreducible representation rc of H(k) a discrete series representation rr' of G(k). In this case, the semisimple classes of H(k) correspond to the elliptic classes in G(k). It is proved in [25] that the characters of 7r and ir' differ only by a sign on those classes.
(b) Let k = R. For cp e 0(H), 0(G), let xw be the sum of the characters of the elements in Ifs. Choose D e 0(H) such that 14 consists of tempered representations. Then xP and x,,.. are equal on the regular semisimple classes of H(k), up to a sign depending only on H and G [53, 6.3].
16.5. We could also take the Weil forms of the L-groups. In that case an Lhomomorphism, restricted to WE, is assumed to satisfy the obvious analogue of 8.2(i). Take in particular the case where H = {I }. Then u is just an element of 0(G). The lifting problem in this case is part of the local problem of III. 17. Global lifting. 17.1. Assume G to be quasi-split. Let H be a reductive k-group and u: LH -* LG
an L-homomorphism. Let u,,: L(H/kv) -> L(G/Q and 0(u,): 0(H/kv) -* 0(G/kv) be the associated maps (v E V) (see 15.1). Let 7r = ©, it (resp. 1f = (&,H,,) be an irreducible admissible representation of HA (resp. GA). Then His said to be a lift of 2V if ff is one of 2r for every v e V (16.1). If that is the case, then, for every representation r of LG, we have (1)
L(s, H, r) = L(s, z, r ° u),
s(s, H, r) = e(s, iv, r ° u).
AUTOMORPHIC L-FUNCTIONS
57
It is also usually requested that Ilv be the natural lift (16.2) of jr for almost all v's. The question is then whether every automorphic 'v has a lift, which is automorphic,
or, somewhat more ambitiously, whether there is a map u*: 91(H/k)
W(G/k)
with reasonable properties, which sends r e 1(H/k) onto a lift of 2c. One also wants to describe the fibres and the image of u,,. In that degree of generality, the problem appears to be inaccessible at present. However, there are many results, old and recent, which are very striking illustrations of this principle, some of which will be extensively discussed in various seminars. Here, for orientation, and to give an idea of the scope of the problem, I shall list briefly some special cases, referring to the literature or to other seminars for more details. REMARK. Let r be a representation of LH of degree n. Then it defines an L-homo-
morphism u : LH , LGL = GL (C) x rk in the obvious way. A positive answer to the lifting problem would imply in particular that if 9r is an automorphic representation of H, then L(s, n, r) = L(s, 11, where 11 is an automorphic representa-
tion of GL and r the standard representation. This would therefore to a large extent reduce the study of automorphic L-functions to those of GL,,, with respect to the standard representation. 17.2. Let H = { 11, G = GL,,. Then an L-homomorphism u is just a continuous complex n-dimensional representation of 1'k. The question is then whether the Artin L-series L(s, u) is an automorphic L-series of GL (with respect to the standard representation of which should be cuspidal if u is irreducible. In view of known results on GL (cf. 14.4) this would imply Artin's conjecture. For n = 1, a positive answer is given by class-field theory. For n = 2, 3, a positive answer is equivalent to Artin's conjecture, since there are converses to Hecke theory [25], [65], [27], [46]. For n = 2, it has been proved for dihedral or tetrahedral representations of 1'k, and for some others over Q (see [38], [17], [15]).
17.3. Let k' be a Galois extension of k, n the degree of k' over k. Take H = Rk,,kGLI, G = GL,,. There is a natural homomorphism f: LH° x 1'k,ik into the normalizer of a maximal torus LT° of LG°. Since the former group is a quotient of LH, and LG = LG° x 1'k, we can define an L-homomorphism u: LH ---> LG by u(h, r) = (f(h), r) (h e LH°, r e 1'k). An automorphic representation of H is a Grossencharakter x of V. The problem is then whether the Artin L-series L(s, x) is the L-series of an automorphic representation of G. If n = 2, k = Q, and k' is imaginary, this was proved by Hecke; 7c is associated
to a cuspidal holomorphic automorphic form. If n = 2, k = Q, and k' is real quadratic, this was established by H. Maass. it is then associated to a nonholomorphic automorphic form. For n = 3, this is proved in [26], [27]. 17.4. Base change. This is the global counterpart to 16.3(a). Let k' be a finite Galois extension of k. Assume H to be k-split and G = Rk.,kH. There is again an L-homomorphism u: LH - LG whose restriction to LH° is a diagonal map. In this case G(A) and G(k) are canonically isomorphic to H(Ak') and H(k'); therefore the problem is to associate an automorphic representation of H(Ak) to an automorphic representation of H(Ak). Again, it should be a counterpart to the restriction to Wk, of homomorphisms Wk .+ LH° If H = GL2 and k' is cyclic of prime degree, the lifting map u* for representations is constructed in [38], which also gives a description of its image and fibres.
58
A. BOREL
This extends work of Doi-Naganuma, Jacquet [21] (on the quadratic case) and of Saito [49], Shintani [55], [56] (cf. [17]).
17.5. Let G be quasi-split, and H an inner form of G. Then LH = LG and O(H/kv) c O(G/kv) for all v's (8.3). Moreover, for almost all v's, H and G are isomorphic over kt,; hence O(H/kv) = and ]I(H(kv)) = The question is then, given = is there an automorphic representation II = Qx,jII of G such that IIv = 2rv for almost all v's? If G = GL2 and His the group of invertible elements of a quaternion algebra D over k, a positive answer is given by Jacquet-Langlands [25]. Note that, in that case, because of the "strong multiplicity one theorem," at most one 11 may be associated to a given v in this way. The possible Xs are in fact the cuspidal automorphic representations for which II,, belongs to the discrete series for all v's over which D does not split (loc. cit.). 17.6. If G = GL2, G = GL3 and u is given by the adjoint representation, as in 13.4, the global lifting problem has been solved by Gelbart-Jacquet [16], the "local lifting" being the one of 16.2(b). 17.7. Let M be a Levi k-subgroup of a parabolic k-subgroup P of G. Then LM imbeds naturally into LG (3.3), whence an L-homomorphism u: LM --> LG. If ' is cuspidal, then the analytic continuation and residues of Eisenstein series [32] are known to yield a unitary u,(2r) in many cases, and, conjecturally, in general.
18. Relations with other types of L-functions.
18.1. In 17.2, the lifting problem amounts to identifying an Artin L-function with an automorphic L-function on GL,,. One can also include in this problem more general representations of Weil groups if one passes to the Weil form of the L-groups. For simplicity, let us limit ourselves to relative Weil groups Wk.,k, where k' is a finite Galois extension of k over which H and G split. An L-homomorphism u: LH° >a Wk',k -> LG° x Wk,,k is then a continuous homomorphism compatible with the projections on Wk,,k, whose restriction to LH° is a complex analytic homomorphism into LG°, and such that, for w e Wk,/k, u(w) = (u'(w), w) with u(w) semisimple (cf. 8.2(i)). If H = {1}, an L-homomorphism is said to be an admissible homomorphism of Wk,/k into LG. In analogy with the definition of O(G) in the local case, we can consider the set Ok./k(G) of equivalence classes of such homomorphisms, modulo inner automorphisms of LG°, and then pass to a suitable limit O(G) over k'. The lifting problem asks in this case to associate to any q E O(G) an automorphic representation z, such that, for any representation r of LG, L(s, z, r) is equal to the Artin-Hecke L-series of r ° u. In particular, is every Artin-Hecke L-series that of an automorphic representation of GL,,, with respect to the standard representation? If G is a torus, then [34] provides a positive answer. In fact, in this case the irreducible admissible automorphic representations of G are the characters of G(k)\G(A), and [34] gives a homomorphism with finite kernel of Ok',k(G) onto the set of such characters. 18.2. In the same vein, it is natural to ask whether Hasse-Weil zeta-functions (or even L-functions of compatible systems of 1-adic representations of Galois groups)
can be expressed in terms of automorphic L-functions. For elliptic curves over function fields, it is a theorem. That it should be the case for elliptic curves over
AUTOMORPHIC L-FUNCTIONS
59
Q is the Taniyama-Weil conjecture; it has been checked in a number of special cases (see [2], [14] for surveys from the classical and representation theoretic points of view respectively). Apart from that, this problem has been pursued mostly for Shimura curves and certain Shimura varieties; we refer to the corresponding seminars for a description of the present state of affairs.
Finally, one may ask whether it is possible to characterize a priori those automorphic representations whose L-series have an arithmetic or algebraico-geometric significance. A necessary condition if k is a number field is that for an infinite place v, 7rU should be associated to a representation 6v of Wk. whose restriction to
C* is rational, C* being viewed as real algebraic group, i.e., be of type A0 in [3, 6.5]. If the L-series of it is to be an Artin L-series, then i should even be of type A00 (loc. cit.), i.e., 6v should be trivial on C*. Let k = Q. Then there are three possibilities for 1rc (11.5). If ir_ = 7r(1, sgn), then 7r corresponds to 2-dimensional representations of 1'Q with odd determinant by the theorem of Deligne-Serre [10], [50]. Modulo the Artin conjecture for such representations, the correspondence is bijective. However, I am not aware of any result for the other two possible values of 7rc. A positive answer would involve nonholomorphic automorphic forms. In [36], it is shown in many cases for GL2 over Q that the L-series of a representation of type A0 is that of a compatible system of 1-adic representations of 1'Q. Over a function field, there is no condition such as A0. In fact, for GL2, Drinfeld has shown that all irreducible admissible automorphic representations are associated to 1-adic representations (see the lectures on his work by G. Harder and D. Kazhdan). REFERENCES
1. A. N. Andrianov, Dirichlet series with Euler product in the theory of Siegel modular forms of genus two, Trudy Mat. Inst. Steklov 112 (1971), 73-94.
2. B. J. Birch and H. P. F. Swinnerton-Dyer, Elliptic curves and modular functions of one variable. IV, Lecture Notes in Math., vol. 476, Springer, New York, 1975, pp. 2-32. 3. A. Bore], Formes automorphes et series de Dirichlet (d'apres R. P. Langlands), Sem. Bourbaki, Expose 466, (1974/75); Lecture Notes in Math., vol. 514, Springer, New York, pp. 189-222. 4. A. Bore] and J.- P. Serre, Theoremes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1964), 111-164. 5. A. Borel et J. Tits, Groupes reductifs, Inst. Hautes Etudes Sci. Publ. Math. 27 (1965), 55-151. 6. P. Cartier, Representations of reductive P-adic groups, these PROCEEDINGS, part 1, pp. 111-155.
7. W. Casselman, GL,,, Proc. Durham Sympos. Algebraic Number Fields, London, Academic Press, New York, 1977.
8. -, The Hasse- Weil l; function of some moduli varieties of dimension greater than one, these PROCEEDINGS, part 2, pp. 141-163.
9. P. Deligne, Formes modulaires et representations de GL,, Modular Functions of One Variable, Lecture Notes in Math., vol. 349, Springer, New York, 1973, pp. 55-106. 10. P. Deligne and J.-P. Serre, Formes modulaires de poids 1, Ann. Sci. Ecole Norm. Sup. (4) 7(1974), 507-530. 11. D. Flath, Decomposition of representations into tensor products, these PROCEEDINGS, part 1, pp. 179-183. 12. F. Gantmacher, Canonical representations of automorphisms of a complex semi-simple group, Mat. Sb. 5 (1939), 101-142. (in English) 13. S. Gelbart, Automorphic functions on adele groups, Ann. of Math. Studies, no. 83, Princeton Univ. Press, Princeton, N. J., 1975. 14. , Elliptic curves and automorphic representations, Advances in Math. 21 (1976), 235292.
60
A. BOREL
15. S. Gelbart, Automorphic forms and: Artin's conjecture, Lecture Notes in Math., vol. 627, Springer-Verlag, Berlin and New York, 1977, pp. 241-276. 16. S. Gelbart and H. Jacquet, A relation between automorphic forms on GLZ and GL Proc. Nat. Acad. Sci. U.S.A. 73 (1976), 3348-3350. 17. P. Gerardin and J.-P. Labesse, The solution of a base change problem for GLZ (following Langlands, Saito, Shintani), these PROCEEDINGS, part 2, pp. 115-133.
18. R. Godement, Fonctions automorphes et produits euleriens, Sem. Bourbaki (1968/69), Expose 349; Lecture Notes in Math., vol. 179, New York, 1970, pp. 37-53. 19. R. Godement and H. Jacquet, Zeta-functions of simple algebras, Lecture Notes in Math., vol. 260, Springer, New York, 1972. 20. R. Howe and I. I. Piatetski-Shapiro, A counterexample to the generalized Ramanujan conjecture' for (quasi-) split groups, these PROCEEDINGS, part 1, pp. 315-322.
21. H. Jacquet, Automorphic Forms on GL(2). II, Lecture Notes in Math., vol. 278, Springer, New York, 1972. 22. , Generic representations, Non-Commutative Harmonic Analysis, Lecture Notes in Math., vol. 587, Springer, New York, 1977, pp. 91-101. 23. , From GLZ to GL,,, U.S.-Japan Seminar on Number Theory (Ann Arbor, Mich., 1975).
24.
, Principal L -Junctions of the linear group, these PROCEEDINGS, part 2, pp. 63-86.
25. H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Math., vol. 114, Springer, New York, 1970.
26. H. Jacquet, I. Piatetskii-Shapiro and J. Shalika, Construction of cusp forms on GL Univ. of Maryland Lecture Notes, no. 16, 1975. 27.
, Hecke theory for GL Ann. of Math. (to appear).
28. H. Jacquet and J. Shalika, Comparaison des representations automorphes du groupe lineaire, C. R. Acad. Sci. Paris Ser. A 284 (1977), 741-744. 29. A. Knapp and G. Zuckerman, Classification of irreducible tempered representations of semisimple Lie groups, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), 2178-2180. 30. , Normalizing factors, tempered representations and L-groups, these PROCEEDINGS, part 1, pp. 93-105. 31. J. P. Labesse and R. P. Langlands, L-indistinguishability for SLZ, Institute for Advanced Study, Princeton, N. J. (preprint). 32. R. P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Math., vol. 544, Springer, New York. 33. , Euler products, Yale Univ. Press, 1967. 34. , Representations of abelian algebraic groups, Yale Univ., 1968 (preprint). 35. , Problems in the theory of automorphic forms, Lectures in Modern Analysis and Applications, Lecture Notes in Math., vol. 170, Springer, New York, 1970, pp. 18-86.
36. -, Modular forms and 1-adic representations, Modular Functions of One Variable. II, Lecture Notes in Math., vol. 349, Springer, New York, 1973, pp. 361-500. 37. , On the classification of irreducible representations of real algebraic groups (preprint). , Base change for GLZ: The theory of Saito-Shintani with applications, Notes, Inst. 38. Advanced Study, Princeton, N. J., 1975. 39. , On the notion of an automorphic representation, these PROCEEDINGS, part 1, pp. 203207. , Automorphic representations, Shimura varieties, and motives. Ein Marchen, these 40. PROCEEDINGS, part 2, pp. 205-246.
41. M. E. Novodvorsky, Theorie de Hecke pour les groupes orthogonaux, C. R. Acad. Sci. Paris Ser. A, 285 (1975), 93-94. 42. , Automorphic L -functions for the symplectic group GSp(4), these PROCEEDINGS, part 2, pp. 87-95.
43. M. E. Novodvorsky and I. I. Piatetskii-Shapiro, Rankin-Selberg method in the theory of automorphic forms, Proc. Sympos. Pure Math., vol. 30, no. 2, Amer. Math. Soc., Providence, R. I., 1977, pp. 297-301. 44. I. Piatetski-Shapiro, Euler subgroups, Lie Groups and Their Representations, Proc. Summer School, Budapest, 1971.
AUTOMORPHIC L-FUNCTIONS 45. 46. 47.
61
, Zeta-functions of GL,,, Notes, Univ. of Maryland, College Park, Md. , Converse theorem for GL,, Notes, Univ. of Maryland, College Park, Md. , Multiplicity one theorems, these PROCEEDINGS, part 1, pp. 209-212.
48. F. Rodier, Les representations de GSp(4, k), oti k est un corps local, C. R. Acad. Sci. Paris 283 (1976), 429-431. 49. H. Saito, Automorphic forms and algebraic extensions of number fields, Lecture Notes in Math, vol. 8, Kinokuniya Book Store Co., Ltd., Tokyo, Japan, 1975. 50. J.-P. Serre, Modular forms of weight one and Galois representations (prepared in collaboration with C. J. Bushnell), Proc. Durham Sympos. Algebraic Number Fields (London), Academic Press, New York, 1977, pp. 193-268. 51. F. Shahidi, Functional equations satisfied by certain L functions, Compositio Math. (to appear). 52. J. A. Shalika, The multiplicity one theorem for GL,,, Ann. of Math. (2) 100 (1974), 171-193. 53. D. Shelstad, Characters and inner forms of a quasi-split group over F (to appear). 54. G. Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. (3) 31 (1975), 79-98. 55. T. Shintani, On liftings of holomorphic automorphic forms, U.S.-Japan Seminar on Number Theory, (Ann Arbor, Mich., 1975). 56.
, On liftings of holomorphic cusp forms, these PROCEEDINGS, part 2, pp. 97-110. 57. A Silberger, The Langlands classification for reductive p-adic groups (to appear). 58. T. A. Springer, Reductive groups, these PROCEEDINGS, part 1, pp. 3-27. 59. R. Steinberg, Regular elements of semi-simple algebraic groups, Inst. Hautes Etudes Sci. Publ. Math. 25 (1965), 49-80. 60. J. Tate, Number theoretic background, these PROCEEDINGS, part 2, pp. 3-26. 61. J. Tits, Reductive groups over local fields, these PROCEEDINGS, part 1, pp. 29-69. 62. J. B. Tunnell, On the local Langlands conjecture for GL(2), Thesis, Harvard Univ.
63. -, Report on the local Langlands conjecture for GL,, these PROCEEDINGS, part 2, pp. 135-138.
64. A. Weil, Ueber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967), 149-167. 65. , Dirichlet series and automorphic forms, Lecture Notes in Math., vol. 189, Springer, New York, 1971. THE INSTITUTE FOR ADVANCED STUDY
Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 63-86
PRINCIPAL L-FUNCTIONS OF THE LINEAR GROUP HERVE JACQUET
Regard the group G = GL(n) as an algebraic group over some local or global field F. Then LGn = GL(n, C). Let rn denote the natural representation of this last
group on Cn; the L-functions attached to r play a central role in "Langlands philosophy". We review certain aspects of their theory, taking into account recent results on classification of representations. 1. Local nonarchimedean theory. We let Fbe a local nonarchimedean field. When this does not create confusion, we write Gn for Gn(F) and use a similar notation for any F-group. (1.1) Let r be an admissible representation of Gn on a complex vector space V;
we denote by k the representation contragredient to 'r, V the space on which it operates, and < , > the canonical invariant bilinear form on V x V. The representation Tr is admissible, and irreducible if r is. Moreover (n)- = ir. The functions (1.1.1)
g F--->
(v E V, v E V)
and their linear combinations are the coefficients of 7r. Clearly if f is a coefficient of 7r, then the function f v defined by (1.1.2)
Mg) _ f(g1)
is a coefficient of k. Let M(p x q, F) be the space of matrices with p rows, q columns and entries in F; denote by 9(p x q, F) the space of Schwartz-Bruhat functions on M (p x q, F).
If f is a coefficient of x, 0 is in ,"(p x q, f), and s in C set (1.1.3)
Z(0, s,.f) = f 0(g) Idetgjs.f(g) dxg.
The integral is extended to the group G. and dxg is a Haar measure on this group. Below (Propositions (1.2), (1.4)) we state again the results of [R.G.-H.J.]. (1.2) PROPOSITION. Suppose 'r is irreducible.
(1) There is so such that the integrals (1.1.3) converge absolutely in the half-plane Re(s) > so. (2) If the residual field of F has q elements, then the integrals represent rational AMS (MOS) subject classifications (1970). Primary 12A85, 22E50, 22E55, 12A80, 12B35. © 1979, American Mathematical Society 63
HERVHJACQUET
64
functions of q-s; as such, they admit a common denominator which does not depend on
for0. (3) Let 0 0 1 be an additive character of F. There is a rational function r(s, Z, 0) such that for all coefficients f of z and all 0
Z(0^, 1 - s + 2 (n - 1),fv) = 7"(s,', 0) Z(0, s,f), where OA denotes the Fourier transform of 0 with respect to . In (3) On is defined by (1.2.4)
OA(x) =
J
0(y) 0 [tr(yx)] dy,
where dy is the self-dual Haar measure on M(n x n, F). On the other hand the left-hand side in (3) has a meaning by (1) and (2) applied to n. Besides we could formulate Proposition (1.2) for the pair (pr, Tr) rather than for 7r, the situation being symmetric in z, k; this symmetry will be apparent at each step of the proof anyway. EXAMPLE (1.2.5). Suppose z is cuspidal. This condition is empty and therefore
always satisfied if n = 1; in this case it is just a quasi-character of F" and we know Proposition (1.2) from [J.T. 1] or [A.W. 3]. If n > 1, the coefficients of ,r are compactly supported modulo the center of G,,; we can then exploit this fact to obtain Proposition (1.2), the proof being essentially the same as in the case n = 1 ([R.G.-H.J.], [H.J.1]) (1.3) The proof of Proposition (1.2) will be given in §2. For the time being, we derive some simple consequences of Proposition (1.2). If f and 0 are as above and h is in G,,, then the functionsf1 and 01 defined by (1.3.1)
.fi(g) = f(gh), 01(x) = O(xh)
are functions of the same type. Moreover if we assume (1.2.1), (1.2.2), then: (1.3.2)
Z(01, s,f1) = Idet hl-s Z (0, s, f).
It follows that the subvector space I(2r) of C(q S) spanned by integrals (1.3.3)
Z($, s + 2(n - 1), f )
is in fact a fractional ideal of the ring C[q ,, q>]. Furthermore if we take f such that
f(e) 0 0 and 0 with support in a small enough neighborhood of e, we find that (1.3.3) is actually independent of s and 0 0. Thus I(ir) contains the constants; in other words it admits a generator of the form P(q S)-1 with P e C[X]. We will normalize P by demanding that P(0) = 1 and will set (1.3.4)
L (s, ,r) = P (q-)-1.
EXAMPLE (1.3.5). If ir is supercuspidal then L(s, 2r) = 1 unless n = 1 and ir(x) _
Ixit. Then (loc. cit.) L(s, z) = (1 - q-1-t)-1. Assume now Proposition (1.2) for (7r, 3c). Because I(z)
0, the factor r is unique
(which justifies the notation). Set (1.3.6)
Then (1.2.3) reads
r(s, ir, 0) = r(s, z, O)L (s, ir)/L(1 - s, k).
65
PRINCIPAL L-FUNCTIONS
Z (0^, 1 - s + 2(n - 1), f")/L(1 - s, k) (1.3.7)
= e(s, ir, ci) Z (o, s + 21(n - 1), f)/L(s, 2r).
In view of the definition of the L-factors, this implies that both e(s, Z, 0) and its inverse are in C[qs, q']. Thus e(s, z, 0) is a monomial in q -s. Let also co be the central quasi-character of 7r, that is to say the homomorphism w : F" --> Cx such that
z(a) _ z(a -
(1.3.8)
w(a)ly
for a e F".
If the assertions of (1.2.3) are true for one choice of 0, they are true for any other
choice 0'. In particular (1.3.9)
if cb'(x) = %(xb).
e(s, 7r, 0') = w(b) I bI (s-1/2)n e(s, 7r, 0)
By exchanging r and k we find also (if Proposition (1.2) is true for (pr, k)) :
e(1 - s, k, c,)e(s, Z, 0) = w(- 1).
(1.3.10)
Moreover let us denote, for any quasi-character x, by 7r ® x the representation g H 7r(g)Z(det g) of Gn on V. Let also aF or a be the module of F. Then :
L(s,7r®ai)=L(s+t,,v),
(1.3.11)
e(s, ir (9 at, 0) = e(s + t, Z, 0).
(1.3.12)
The 7--factor satisfies similar identities.
2. Induced representations. Again F is a local nonarchimedean field.
(2.1) Let P be an F-parabolic subgroup of G. To be specific we take P to be standard of type (n1, n2,
, nr), so that the matrices p in P have the form ui1
m1
m2
(2.1.1)
P=
mi e Gn.. 0
Mr
)
We also set
(2.1.2)
U = Up (unipotent radical of P),
M=MP=P/UP.
Most of the time we identify M to the subgroup of p e P such that ui, = 0. Then, in
a natural way, (2.1.3)
M = fl Gn.
(1 5 i 5 r).
If a-i, 1 < i :!- r, is an admissible representation of Gn;, we can form the representa-
tion a = x a'i of M = P/ U, regard it as a representation of P trivial on U, and induce it to G. The representation (2.1.4)
= I(G, P; a) = I(G, P; c , 0'2, ... ar)
66
HERVEJACQUET
that we obtain is admissible. Its contragredient & is equivalent to
= I(G, P; d) = I(G, P; a1, a2, "', ar)
(2.1.5)
More precisely let W be the space of Q. Then the space V of a consists of all functions F from G to W which are smooth and satisfy (2.1.6)
F(pg) = OP(P)1'2F(g),
where Op is the module of P. The group G. operates by right-shifts on V. The space
V' of e' is defined similarly in terms of the space W of d. For F E V, F e V' the function rp(g) = satisfies (2.1.7)
(p(Pg) _ OP(P)(P(g)
Denoting by (2.1.8)
$(g) dg
p
(P
a positive right-invariant form on the space of continuous functions satisfying (2.1.7), we may set
= f
(2.1.9)
F'(g)> dg
P\G
and this is a pairing which allows us to identify V' to V and ' to Let RF be the ring of integers of F and let
.
K = K = GL(n, RF).
(2.1.10)
Since G = P K,,, we may take (2.1.8) to be cp p-,
(2.1.11)
fK (p (k) A.
(2.2) It will be convenient to have a description of the coefficients of e in terms of those of o. So let f 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(g1, g2) = (F(gl), F(g2)> satisfies the following conditions: (2.2.1)
H(ulmgl, u2mg2) = SP(m)H(g1, 92)
for g1 e G., u, e UF, m e MP;
(2.2.2)
for any g1, g2 the function m I --> H(mg1, g2) is a coefficient of u O 6112;
(2.2.3)
H is K x K. finite on the right.
Moreover f is given by (2.2.4)
f(g) = $P\G H(hg, h) dh = IK H(kg, k) A.
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 ir. The coefficient f v of & is then given by
67
PRINCIPAL L-FUNCTIONS
(2.2.5)
f V(g) = f'\G H(hg, h) dh where II(gl, g2) = H(g2, g1)
and H satisfies (2.2.1)-(2.2.3) with d instead of (2.3) Before formulating the main theorem of this section we remark that if IV is an admissible representation which is perhaps not irreducible but admits a central quasi-character ((1.3.8)), then the assertions of (1.2) make sense for (iv, k), although they may fail to be true. Below we assume that each of admits a central quasicharacter; thus a admits also a central quasi-character. (2.3) PROPOSITION. With the notations of (2. 1) suppose the assertions of (1.2) are true for each pair (6i, 6i). Then they are true for (e, ). Moreover:
I (0 = III(o'i), r(S, , 0) = IIr(s, oi, 0)i i The proof will occupy (2.4)-(2.6). (2.4) Let f be given by (2.2.4). Then, exchanging the order of integrations, we get
Z(o, s + z(n -1 ),.f) = =
(2.4.1)
J K dk J
c fi(g)
I det g I S+(n-1)'2H(kg, k) d"g
f dk SG 0(k-1 g) I det g I'd-(n-1)/2 H(g, k) dxg
= f dk dk' f O(k-lpk')I det
pls+(n-1)12H(pk', k) d1p.
If moreover we define (p being as in (2.1.1)) (2.4.2) (2.4.3)
?1(m1, m2, ... m,; k, k') = f
0(k-lpk') (D dui;,
h(ml, m2, ..., m,; k, k') = H(pk', k)5PV2(p),
then, after integrating in the variables ui;, integral (2.4.1) can be written as
(2.4.4)
f dk dk' f ?If(ml, m2, ..., m,; k, k')h(ml, ..., m,; k, k') rI Idet
mils+(na-1)12 Qx dxmi.
i
Because of the K-finiteness of the functions involved, for 0 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', ?11(ml, m2i .. , m,; k, k') is a finite sum of products Iji 'i(mi) with
, m,; k, k') is a finite sum of products jji f (mi) where fi is a coefficient of o'i ((2.2.2)). Thus (2.4.1) is a finite sum of products ?1f e .9'(n1 x ni, F) ; similarly h(ml, m2,
(2.4.5)
IIZ(01, s + i(ni - 1),J ), i
where Oi is in b(ni x ni, F) and f, is a coefficient of ai. So we have proved (2.1.1), (2.1.2) for , and even the inclusion I(g) c Ij i I(6i). (2.5) Now we prove the reverse inclusion. Starting with an expression (2.4.5) we can certainly find 0 e .9'(n x n, F) so that with the notations of (2.1.1) :
HERVEJACQUET
68
(2.5.1)
f 0(p) Ox du1, = 1101(m1).
Moreover there are two K-finite functions
and r' on K such that
ff 0(k-lxk')j(k)i'(k') dk dk' = 0(x).
(2.5.2)
Then (2.4.5) is equal, for large Re s, to (2.5.3)
f 0(k-lpk')Idet ply+(n-1)12 jl ji(mi)312(p)i2(k)iy'(k') dip 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)
f A dh' f 0(k-lh-lph'k') I det pI,+(n-1) /2 j1 f (m1)o'2(p)V(hk)i2'(h'k') dip dk dk'. i
Now change p to hph'-1 and write h, h' in the form (2.1.1) with h, E Kni, h; E K. instead of m;. We see that (2.5.4) is equal to (2.5.5)
f
0[k-lpk']H1(p, k, k')Idet pl s+(n-1)/2 d1p dk dk'
where
H1(p, k, k') = ff r1(hk)r)'(h'k') f
jf,.(hi .
h ' - 1 ) 61 2(p) dh dh'.
r
It is easily verified that there is a function H satisfying (2.2.1)-(2.2.3) such that
H(pk,k') = H1(p,k,k')
(pEP,kEK, k' e K').
If f is the corresponding coefficient of e ((2.2.4)), then, comparing (2.5.4) with (2.4.1), we find that
HZ(0 s + 4(n; - 1), f=.) = Z(0, s + (n - 1), f ). So we have proved that Ij;I(C;). (2.6) Now we pass to the functional equation. In (2.4.1), let us replace f by fv and 0 by 0^. Then His replaced by H ((2.2.5)) and the identity (2.4.3) gives: (2.6.1)
h(mi 1, mz 1, ..., m;1; k', k) = H(pk, k')o P i2(p)
Similarly (2.4.2) gives (2.6.2)
A(ml, m2, ..., m.; k', k) =
f OA(k-lpk') Ox du;1,
where ?^ denotes the Fourier transform of ?IT with respect to each one of the variables m;. Instead of (2.4.1) and (2.4.4) we have now for Re s large enough
PRINCIPAL L-FUNCTIONS
69
Z(01", s + -21(n - 1),f")
= f dk dk' f OA(k-1pk') l det pl s+(n-1)12H(pk', k) dip (2.6.3)
=
$dkdk'$(mi,m2, , m,.; k', k) h(mi 1, m21, ..., mr 1; k', k) 11 ldet mils+(ni-1)12 © d>mi. i
The functional equations for the representations of and the remarks we made on h and 0' imply now
Z(O^,1 - s + I (n - 1), f v) = IT r(s, oi, O)Z(1, s + 1(n - 1), f). 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, oi),
L(s, t) = fIL(s, di). i
i
Then the last assertion of (2.3) reads (2.7.2)
i
Let z be an irreducible component of e; then any coefficient of 7r is a coefficient of and Tr is an irreducible component of . It follows that Proposition (1.2) is true
for (n, fl. Moreover (2.7.3)
I(z) c I(e), I(mo) - I(), r(s, Z, 0) = r(s, , 0)
Thus there are two polynomials P, P in C[X] such that
L(s, 2r) = P(q s)L(s, ), L(s, Tr) = P(gs)L(s, ), Note that P(q5) divides L(s, )-1 in C[q s]. Moreover (2.7.4)
P(O) = P(O) = I.
L(s, ir)
L(s, se)P(q s) a(s, 7r, ) = r(s, Z, 0) L(1 - s, it) = r (s, e, 0) L(1 - s, )P(ql+s) (2.7.5)
= E(s, ") P(qs) Since the e-factors are units of the ring C[qs, qs], we find that (2.7.6)
P(X) = 11(1 - a;X),
P(X) = f](1 - a;1gX).
(2.8) In general, if r is an arbitrary irreducible admissible representation of G, it is a component of an induced representation of the form (2.1.4) where the of are cuspidal. Since (1.2) is true for each of ((1.2.5)), it is true for (e, ) ((2.3)) and thus for (7r, -c). So (1.2) is completely proved.
3. Computation of the L-factor. We have established (1.2) for any irreducible admissible representation z of Gn; we also know r(s, x, 0) ((2.7.2)). It remains to compute the L-factor.
HERVBJACQUET
70
(3.1) Square-integrable representations. We have seen that if r is cuspidal then
L(s, i) = I unless n = 1 and i = at, in which case (3.1.1)
L(s,z) = (1-q s-1)-1.
We now compute L(s, i) when ir is essentially square-integrable. Recall that i is square-integrable if it admits a central character a) and its coefficients (which transform under w) are square-integrable modulo the center. A representation r is essentially square-integrable if it has the form z = io Q at where io is squareintegrable 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 G. of type (j, j, , j). Let also r be a cuspidal representation of Gr. Set for 1 < i S r, ui = r ® as-1. Then the induced representation (3.1.2)
= I(G, P; 61, a2, ... 6r)
admits a unique essentially square-integrable component z. All essentially squareintegrable representations ir are obtained in this way and r, r are uniquely deter-
mined by i. PROPOSITION (3.1.3). With the above notations L(s, 'r) = L(s, r).
Indeed suppose L(s, r) = 1. Then L(s, ai) = 1 and by (2.7.1) L(s, ) = 1. By (2.7.4) we have then L(s, 2r) = 1. Suppose now L(s, r) 0 1. Then j = 1, r = n, r = a' and our assertion is nothing but Proposition (7.11) of [R.G.-H.J.]. REMARK (3.1.4). If 7r is square-integrable then the poles of L(s, 2r) are in the
half-plane Re(s) <_ 0. This follows from (3.1.3) or from Proposition (1.3) of [R.G.-H.J.]. (3.2) Tempered representations. Let again P be a parabolic subgroup of type (n1, n2, ..., nr). Let now a,: be a square-integrable representation of G,,.. Then it is well known that the induced representation (3.2.1)
i = I(G, P; a1, a2, ..., 6r)
is irreducible (cf. for instance [H.J. 2]). The irreducible representations of this type are precisely the tempered ones. Note that if it is equivalent to another representation (3.2.2)
n' = I(G, F; a, aZ, ... 6r)
where the ai are square-integrable, then P' and P are associate. More precisely r = r', P = MUp, P = M' UP,, and there is an inner automorphism of G taking M to M' and a = x ai to o' = x a'; in other words one passes from (M, (ai)) to (M', (a=)) by a permutation of the diagonal blocks of M and M'. Conversely if P' and the a= are related to P and the ai in this way, then z' is equivalent to i. By (2.3),
L(s, 7r) = ]1L(s, a.), L(s, k) = JJL(s, 6i), (3.2.3)
i
i
s(s, i, ) _ IT E(s, ai, Sb)
PRINCIPAL L-FUNCTIONS
71
REMARK (3.2.4). It follows from (3.2.3) and (3.1.4) that the poles of L(s, Ir) for z tempered are contained in the half-plane Re(s) < 0.
A representation jr is said to be essentially tempered if it has the form it = no O at where 'ro is tempered and t is real. Then L(s, z) = L(s + t, Jro).
(3.2.5)
(3.3) Langlands construction. We now review the work of Silberger and Wallach which extends the results of Langlands to the p-adic case. Let Q = MQUQ be a parabolic subgroup of type (PI, p2i , Pr) and for each i, 1 <_ i S r, ri an irreducible essentially tempered representation of G,,,;. Set v = x r j. Then r; = r;,o ® at' where r;,o is tempered and t, real. We assume that
tl > t2 > ... > t,.
(3.3.1)
Then the induced representation Yf = I(G, Q; r1, r2, ..., r,)
(3.3.2)
has a largest proper subrepresentation 77' (possibly {0}); the irreducible representation 2V = v/72' is noted (3.3.3)
7C = J(G, P; rI, r2, ..., r,).
Every irreducible representation 7r has the form (3.3.3) where P (standard) and the r1 are uniquely determined. The space V' of ij' may be described explicitly. Let W be the space of r and W the space of i. Then V' is the space of Fin the space V of e such that (3.3.4)
f du = 0,
v e W, g E G,,.
The integral is absolutely convergent and extended to U = UQ = UQ, where Q = IQ is the parabolic subgroup opposed to Q. It easily follows that the coefficients of 'r can be obtained by integrals similar to (2.2.4). Namely let H: G x G ---> C be a function satisfying the following properties: (3.3.5)
H(u1mgl, u2mg2) = H(g1, g2),
u1 e UQ, u2 E UQ, M E MQ;
(3.3.6)
for any g1, g2 the function m --> H(mg1, g2) is a coefficient of r O 51/2;
(3.3.7)
H is K x K finite on the right.
Then the function f defined by the convergent integral (3.3.8)
f(g) =
H(hg, h) dh = f SM\G
uXx
H (ukg, k) dk du
is a coefficient of 'r and all coefficients of r 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
HERVEJACQUET
72
(3.3.9)
7r' = J(G, Q; T1, f2, ..., fr)
of the induced representation (3.3.10)
7)' = I(G, Q; 'fl, f2, ..., fr)
is defined. Its coefficients are given by integrals (3.3.8) where H satisfies (3.1.5)(3.1.7) with (Q, z) instead of (Q, z). In particular suppose f is the coefficient of 7r defined by (3.3.8) where H satisfies (3.3.5)-(3.3.7) for (Q,,r). Then f V(g) = fM\G H(hg, h) dh,
(3.3.11)
with H(g1, g2) = H(g2i g1). But H indeed satisfies (3.3.5)-(3.3.7) for (Q, z). Thus f v is a coefficient of 7r' and
7r' _ .
(3.3.12)
Of course Q is also conjugate to the standard parabolic subgroup Q' of type , n1) so that
(nr, nr_1i
(3.3.13)
J(G, Q'; zr, fr-1)
zi).
(3.4) THEOREM. Let the notations be as in (3.3). Then
L(s, k) = II L(s, zi),
L(s, 7r) = II L(s, zi),
e(s, 7c, 0) = fle(s, zi, 0) It is enough to prove the assertion relative to L(s, 7r) ; indeed the one for L(s, Ir) can be obtained by exchanging 7r and 7c and the last assertion follows then from (2.7) and (cf. (2.3)) r(s, V, 0) = jj; r(s, z;, 0). 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, 7) = P(qS) II L(s, zi)
L(s, 7r) = P(q5) II L(s, zi),
It follows from (3.2.4) and (3.3.1) that L(s, z1)-1 and r[ L(1- s, f;)-1 are relatively prime. Thus P(q5) is prime to L(s, rl)-1 ((2.7.5)). So if P 1 then (3.4.1)
T1 L(s, a;) 2siSr
is not in I(7r). Therefore it will suffice to show that (3.4.1) is indeed in I(7r). (3.5) The identity (3.5.1)
L(s, 7r) = fl L(s, z;) 1 isr
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 I(7r).
Set n1 = pl, n2 = P2 + (P2, (3.5.2)
+ Pr. Let Q' be the parabolic subgroup of type
, pr) in G,+2. Then the representation 92 = J(G, , Q'; z2, ..., Zr)
is defined. Set also u1 = Tl, 6 = 61 x 62 and let P be the parabolic subgroup of type (n1, n2) in G. Then 7r is actually a quotient of
73
PRINCIPAL L-FUNCTIONS
(3.5.3)
= I(G, P; 61i 0'2).
More precisely it is easy to see that the coefficients of z are given by the integrals (3.3.8) where H is any function satisfying (3.3.5)-(3.3.7) with (P, 6) instead of
(Q, z) If f is a coefficient of x defined in this way, then
Z(0, s + 2(n - 1), f) (3.5.4)
= J f dk du
fGn
gl`+(n-1)/2 0(k-1u Ig)H(g, k) d"g,
Idet
where a is integrated over U = U. This is also
dk dk' f Idet (3.5.5)
1 i2
1 .
5(0
M2)-
(f
0[k-1 (ml
JJ
mils+(nl-1)/2 Idet
H[(0
m2ls+(m-1)/2
m°)k', k ] d"mi d"mz 1
\xml m2 Y+
z
j k'] A dy.
Here m; ranges over Gn.. We are going to see that given 02 a bo(n2 x n2, F) and a coefficient f2 of 62 there are H and 0 so that Z(02, s + 2'(n2 - 1), f2) is equal to (3.5.5). Since, by the induction hypothesis, L(s, (72) is equal to (3.4.1), it will follow that (3.4.1) belongs to I(7r) which will conclude the proof. There is a coefficientfl of 7r1 and a 01 in 9'(n1 x n1, F) with support in a compact neighborhood of e such that (3.5.6)
Z(01, s + 2I i(n1
- 1),fl) =
1.
There are also 012 e S°(n1 x n2, F), 021 e 91(n2 x n1, F), with supports in a neighborhood of 0 so that the function 0 e ,9'(n x n, F) defined by (3.5.7)
O(x 1 m2) = 01(m1)012(Y)021(x) 02(m2)
satisfies (3.5.8)
S
Y ]dxdy = 01(m1)02(m2) 0[ml xm1 m2 + xy
There are also two K-finite functions and ie' on K such that (3.5.9)
if 0(k-lzk') (k)ie'(k') dk dk' = 0(z).
Then :
Z(02, S + I(n - 1),f2)
= f f Idet
m1is+(nl-1)
2 Idet
m2ls+(,2-1)/2 A(m1)A(m2)
(3.5.10)
dxmldxm2 if JS
k-1(xm1 m2 +
dk dk' xy) k'] dx dy.
HERVEJACQUET
74
The proof is then finished as in (2.5). Namely let dh and dh' be the normalized Haar measures on K n P and K n P respectively. For h e K n P, k e K n P set
hl 0l
h,_ hi *l
\* h2/'
\0 hz/'
h_
then (3.5.10) is also equal top
ff dk dk' J f Idet
m2)-1i2Hlr/0 1
6401
(3.5.11)
f
m2I'+(nz-Ui2
mils+cn1-1>i2 Idet
k-i(xml
m2), k' k dxml dXmz
m2 + xy)
J
k'1 dx dy
where
H1[\0
m0 1
k, k'
2/'
(3.5.12)
oP\0
J )1 izf l
rr
1 mz
f e(hk)e'(h'k').fi(hlmlhi-1)f2(h2m2hz 1) A dh'.
There is H satisfying (3.3.5)-(3.3.7) for (P, u) so that
H[\0
(3.5.13)
k] 1 m) z k',
0
H1[\01 M2) '
k, k'].
Comparing (3.5.11) to (3.5.5) we obtain our assertion.
(3.6) Unram jed representations. Suppose ir is unramified, that is contains the unit representation of K. Then B denoting the parabolic subgroup of type (1,1,1,
, 1), there are u; e C so that r is the unique unramified component of
(3.6.1)
= I(G, Bn;,ul, ,u2, ..., ,un),
,ui = a"i.
u,,. Changing notations we may write the
We may even assume u1 >_ u2 n-tuple of the ii in the form
(11... ''l, v2, , vnl, v1, v2, 1
2
2
2 K2' ...
r 1 i' v2,
r , vnr
where n = n1 + n2 + + nr, v;' = atiAI, AJ is a character, t1 is real and t1 > t2 > > tr. Then z; = I(Gn1, B"j; vi, vz, , vn j) is an essentially tempered unramified representation of G,,; and in fact (3.6.2)
i = J(G, Q; z1, z2, ..., Zr),
if Q is the parabolic subgroup of type (n1, n2,
, nr). This follows from an explicit
computation: if H satisfies the conditions (3.3.5)-(3.3.7) for (Q, x z;) and H(k, k') = 1 then (3.6.3)
f H(u, e) du 0 0.
Thus the "J-component" of , = I(G, Q; zl, Z2, , ar) is unramified; since e = 7) we arrive at (3.6.2).
PRINCIPAL L-FUNCTIONS
75
Thus L(s, 7c) = II L(s, ,Ui), i
(364) e(s, 7c,
]Te(s, pi, cb) i
L(s, k) = II L(s, ,u; 1),
(= I if the exponent of 0 is zero).
For more precise results see [R.G.-H.J., §6]. REMARK (3.6.5). Let the notations be as in (2.3). Suppose ai is irreducible unramified. Then a admits a unique unramified component 7c. It easily follows from (3.6.4) that L(s, 7c) = n L(s, o;) with similar relations for L(s, k) and e. REMARK (3.6.6). Let 7c be a tempered representation of G,,. Let p _
I(G,,, B,,; 1, 1, , 1) and S. be the spherical function attached to p. Then any coefficient of 7c is majorized by a multiple of En. Let the notations be now as in (3.3). Then the coefficients of zi are majorized by a multiple of 5,,; ® a'i. Moreover if H satisfies (3.3.5)-(3.3.7) for (Q, z) then IHI S cHo where Ho satisfies (3.3.5) and Ho(mk, k') = 5112(m) I I_n,(mi) i
Thus as far as convergence is concerned we may replace zi by p,,, Qx a' and H by Ho > 0. Together with the next lemma, this takes care of all problems of convergence.
LEMMA (3.6.7). If 0 is a bounded set of .9'(n x n, F), then there is 0 > 0 in Y(n x n, F) so that 00(k1xk2) = Oo(x) for ki e K and 101 < Oo.
We may assume that 0 contains also all functions x H 0(k1xk2) with 0 e Q. Then there is 01 >_ 0 so that 101 S 01 for all 0 in 0 [A.W. 1, Lemma 5]. It suffices to take Oo(x) = $$01(kixk2) dk1 dk2 where dki is the normalized Haar measure on K. (3.7) According to "Langlands' philosophy" there should be a "natural bijection" a -+ 7c(a) between the n-dimensional semisimple representations of the WeilDeligne group WW and the irreducible admissible representations of G (F). Moreover, one should have for 7c = 7c(a) L(s, 7c) = L(s, a),
7L(a)' = ii(U),
e(s, 7c, cl) = e(s, a,
7C(a O x) = 7C(a) O x.
It is clear that if the map a 7c(a) could be defined for the irreducible representations of the Weil group, then the conjecture would be proved (cf. §5 below). 4. Local archimedean theory. In §4, 5 the ground field F is local archimedean. (4.1) We let K be O(n, R) if F = R and U(n) if F = C. We donote by the Lie algebra of the real Lie group G (F). We consider only admissible representations of the pair (C3,,, (as in [N.W.]) although we will often allow ourselves to speak of a
representation of the group in any case, we assume the reader to be thoroughly familiar with the relation between representations of the group G (F) and of the pair (03,,,
Let 7r be an admissible representation of (( ,,, then one can define the contragredient representation Tc, the coefficients of 7c, and its central quasi-character, even though 7c is not a representation of the group (cf. [H.J.-R.L., §§5, 6]). Again if f is a coefficient of 7t then f v ((1.1.2)) is a coefficient of k.
HERVEJACQUET
76
As in the nonarchimedean case let 1(p x q, F) be the space of Schwartz func-
tions on M(p x q, F). We also introduce the subspace go = 9' (p x q, F) of functions 0 of the form (P being a polynomial),
0(x) = P(x;;)exp(--zEx)
if F = R,
0(z) = P(z1t, z,;)exp( - 27r E z;121;),
if F = C.
We can consider, for a given z, the integrals (1. 1.3) where 0 is in ,"(p x q, F) and f is a coefficient of i. (4.2) PROPOSITION. Suppose (@., Ku).
r is an irreducible admissible representation of
(1) There is so so that the integrals (1.1.3) converge absolutely in the half-plane Re(s) > so. (2) For 0 in So0(n x n, F), the integrals are meromorphic functions of s in the whole complex plane. More precisely they can be written as polynomials in s times a fixed meromorphic function of s.
(3) Let 0 0 1 be an additive character of F; there is a meromorphic function r(s, z, 0) so that, for all coefficients f of n and all 0 e , "0(n x n, F),
Z(OA, 1 - s + 2(n - 1),f") =
?-(S'
i, 0)Z(0, S,f).
Again 0' is defined by (1.2.4). If 0 is in .0(n x n, F) and 0 is defined by
if F = R, O(z) = exp[±2i'c(z + 2)], if F = C,
O(x) = exp(± 2i7cx), (4.2.4)
then On is still in 9'o; the left-hand side of (4.2.3) has then a meaning by (4.2.2) applied to k. If 0 is not given by (4.2.4) then some adjustment has to be made, that we leave to the reader (cf. (1.3.11), (1.3.12)). From now on, we assume 0 is given by (4.2.4).
EXAMPLE (4.2.5). If n = 1, then 7c is just a quasi-character of F" 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 > so is actually uniform
for 0 in a bounded set; thus, for Re s > so, (4.3.1)
0 '-' Z(0, s, f )
is a distribution, depending holomorphically on s. Since .So0(n x n, F) is dense in So(n x n, F), it follows that if f 0 0 then there is at least a 0 e go so that Z(0, s, f)
00.
Suppose (4.2.2) has been proved. Let L(s) be a meromorphic function of s so that, for all 0 in .©o, Z(0, s + Mn - 1), f)/L(s) is a polynomial; let also Ibe the subvector space of C[s] spanned by those ratios. By what we have just seen I : {0}. Moreover if 0 is in go, so is the function 0' defined by (4.3.2)
0'(x) = at O(Xe t)I`-o.
77
PRINCIPAL L-FUNCTIONS
Let co be the central quasi-character of ir; if we differentiate with respect to t the identity (4.3.3)
Z(cL, s + 21(n - 1),f) = f o(xe t).f(x)Idet
xIs+(n-1)i2dxx
0)(e t)exp[- is - 2 (n - 1)] and then set t = 0, we arrive at a relation of the form
(4.3.4) (as + b) Z(o, s + Z(n - 1), f) + Z(O', s + Z(n - 1), f) = 0,
a 0 0.
This shows that I is an ideal. Let P0 be a generator of I and L(s, 7c) = Po(s) L(s). We see that the ratios Z(Q, s + I (n - 1), f)/L(s, z) are again polynomials, but this time they span C[s] as a vector space. Up to multiplication by a constant, these properties characterize L(s, rr). Assume now (4.2) proved for the pairs (7r, Tc). Then r is uniquely determined and one can define a factor e as in (1.3.5) ; then the functional equation (4.2.3) reads as in (1.3.6). In view of the definition of the L-factors, this implies that both e(s, 'r, 0) and its inverse are in C[s]. Thus e(s, 7r, 0) is just a constant if 0 is given by (4.2.4); otherwise it is an exponential factor ((1.3.9)). REMARK (4.3.5). For the time being the L- and e-factors are defined up to multiplication by constants; of course these constants are related since r is intrinsically defined. For n = 1, one may take the factors to be those given in [J.T.1, 2]. REMARK (4.3.6). Relations (1.3.9)-(1.3.12) apply to the archimedean case. (4.4). Let the notations be as in (2.1), the ground field F being now R or C; it goes without saying that ai is now an admissible representation of (03 i, K i). The induced representation e ((2.1.4)) can still be defined [N.W.] and its coefficients are given by the rule of (2.2), the only difference being that in (2.2.2) gi must be
in K and in (2.2.3) H must be C°° on G x G,,. The remarks made before (2.3) apply and, as in (2.3), we have :
PROPOSITION (4.4.1). With the notations of (2.1) suppose that each ai admits a
central quasi-character and the assertions of (4.2) are true for each pair (ai, ai). Then they are true for t). Moreover, r(s, 0) = rT ir(s, a=, 0) and one can take L(s, SC) = IIL(s, ai), i
L(s, ) = 11L(s, ai), i
e(s, , 0) = Ile(s, ori, 0). i
The proof is similar to the proof of (2.3). It suffices to observe that if 0 is in .9'o(n x n; F) the functions x F- 0(klxk2), ki e K,,, span a finite dimensional vector space of Yo and for each k, k' in K the functions (2.4.2) belong to the space
Yo(ni x ni, F). Notations being as in (4.4.1), let it be an irreducible component of . Then any coefficient of it is a coefficient of e and ?c is a component oft. Thus (4.2) is true for (ir, Ic). More precisely there are polynomials R and R such that (4.4.2)
L(s, 7r) = R(s)r[L(s, ai), L(s, Tc) = R(s)r[L(s, ai) i
i
HERVEJACQUET
78
while (4.4.3)
r(s, 7r, 0) = HAS, Ui, 0) i
Since e(s, z, 0) and e(s, o, 0) are constants R and R have the form
R(s) = fl(s - si), R(s) = cf j(1 - s - si).
(4.4.4)
i
i
Finally any given irreducible admissible representation z of G is a component of some induced representation a as in (4.4.1) with ni = 1. Since (4.2) is then true for each (6i, di), it is true for (ic, k) and (4.2) is proved. (4.5) PROPOSITION. For all s, L(s, r) 0 0.
PROOF. Let z be an irreducible admissible representation of G so that i is an irreducible representation of some induced representation as in (4.4.1) with ni = 1. We first show that for any 0 in 9(n x n, F) the ratio
Z(O, s + I(n - 1), f)/L(s, ir)
(4.5.1)
continues to an entire function of s. Indeed let so be such that for Re(s) > so (resp. Re(s) < 1 - so) the integrals
Z(O, s + Z(n - 1), f) (resp. Z(O, 1 - s + Z(n - 1), fv)) converge absolutely. Select sI, s2 such that so < sI < s2. Let also Q be a polynomial such that Q(s)L(s, ) has no pole in the strip 1 - 2 5 Re(s) 5 s2. Let of be a sequence of Yo converging to some element 0 of Y. Then, for a given f and a given polynomial R, the following limits are uniform in the domain sI 5 Re(s) 5 s2: (4.5.2)
lim R(s)Z(0i, s + z(n - 1), f) = R(s)Z(O, s + Z(n - 1), f ),
i-+OO
(4.5.3)
=
lim R(s)Z(ci - O;, s + 2(n - 1), f) = 0.
Similarly V; - C, -> 0 so that, uniformly in the domain 1 - s2 (4.5.4)
Re(s) 5 1 - sl,
lim R(s)Z(0= - o; , I - s + 2(n - 1), f v) = 0.
i, i--
Indeed by using repeatedly (4.3.4) one can reduce these assertions to the case R = 1; they are then obvious.
On the other hand, Q(s)Z(Oi - cb , s + 2 (n - 1), f )
= e(s, , 0-1 L(1L(s, e)
Q(s)Z(OZ
- Cj, 1 - s + 2(n - 1), fv)
The classical formula
1'(x+iY)
r(x
IYIxx'
(IyI_'+oo)
iY)
shows that Q(s)L(s, g)/L(l - s, ) is bounded by a polynomial in the previous strip. Thus we also find, uniformly for 1 - 2 5 Re(s) 5 1 - sl, (4.5.5)
lim Q(s)Z(Oi - O;, s + I (n - 1), f) = 0.
79
PRINCIPAL L-FUNCTIONS
Now fix f and let e > 0 be given; choose N so that for i, j >_ N
Q(s)Z($i - O;, s + I (n - 1), f
e,
for Si 5 Re(s) 5 s2 or 1 - s2 < Re(s) 5 1 - sl. But now Q(s)Z(Oi - O;, s + 2(n - 1), f) = Q(s)R(s)L(s, where R is another polynomial. Since L(s, i;) decreases rapidly, on any vertical strips, we find that given (i, j) there is M so that, for IIm(s)I >- M, (4.5.6)
Q(s)Z(bi - ot, s + 2I (n - 1), f
Thus by the maximum principle (4.5.6) is satisfied for i, j N and 1 - s2 5 Re(s) <- s2. It follows that Q(s)Z(c1i, s + 2(n - 1), f) is a Cauchy sequence for the topo-
logy of uniform convergence on the strip 1 - s2 5 Re(s) 5 s2. Its limit Z(s) is holomorphic on 1 - s2 < Re(s) < s2 and coincides with Q(s)Z(A, s + Z(n - 1), f ) on sI 5 Re(s) < s2. Since s2 is arbitrarily large the holomorphy of (4.5.1) follows. Suppose now L(so, ir) = 0 and f 0 0. Then for each 0, the meromorphic function Z(O, s + 2(n - 1), f)vanishes at so; but if 0 has compact support contained
in G then Z(o, s + 2(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 = so and 0 c C, therefore f - 0, a contradiction. Thus L(so, ir)
0.
Q.E.D.
COROLLARY (4.5.8). Suppose is as in (4.4.1) and ir is an irreducible component of . Let R and R be as in (4.4.2). Then if si 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 L-factor; archimedean case. Recall there is a "natural bijection" A H z(2) between the set of classes of semisimple representations of the Weil group WF of degree n and the admissible irreducible representations of G (F) ([R.L.], [A.K.-G.Z], [N.W.]). In particular 2r(A)" = 7r(A). (5.1) THEOREM. For any A of degree n, let 7r be ir(A). Then
L(s, ir) = L(s, 2),
L(s, ;r) = L(s,
e(s, ir, 0) = e(s, 2, Sb)
The proof of this theorem will occupy all of §5. Since the L-factor has not been normalized, it would be more correct to say that one can take L(s, ir) to be L(s, 2) and that r(s, iv, 0) = e(s, A, O)L(l - s, .1)/L(s, A).
(5.2) Square-integrable representations. Suppose A is irreducible. Then i(A) is essentially square-integrable and conversely. More precisely, if n = 1 then A is a quasi-character of Fx, A = iv, and our assertion follows from the definition of the factors L and e for A. Suppose n 0 1; then n = 2 and our assertion has been proved in [H.J.-R.L., Theorem (13.1), Lemmas (13.24) and (5.17)]; actually in this case, the proof is a refinement of (4.4.1).
(5.3) Tempered representations. Suppose A is unitary; then iv is tempered and conversely. More precisely A = Q+ di where Ai is irreducible of degree ni. Set ai
HERVEJACQUET
80
ir(.1=). Then r = I(G, P; 61, 92, (n1, n2, ...
(5.3.1)
, a,) where P is the parabolic subgroup of type
n,). By (4.4.1)
L(s, 7r) = IIL(s, 6r) = IIL(s, A) = L(s, A).
The factors L(s, Ic) and e(s, ir, 0) are computed similarly so our assertion follows again.
The case of the essentially tempered representations follows from (1.3.11), (1.3.12), and the relation (5.3.2)
2r(d xO x) _ 2r(A) xO Z.
REMARK (5.3.3). It follows from (5.3.1) that, when r is tempered, the poles of L(s, 2r) are in the half-plane Re(s) < 0. (5.4) General case. In general, we may write d = ED p;, pi = u;,o 0 ate where ,u;,o is unitary of degree pi, t; real. We set then v,0 = 2r(,u;,o), z= _'r(,u;) so that by (5.3.2) z; = rj,o 0 ati. We may assume (3.3.1) is satisfied. Then if Q is the parabolic subgroup of type (p1, Pz, ..., Pr) the induced representation of (3.3.2) admits a unique irreducible quotient noted as in (3.3.3). That quotient is it = ir(2). By (4.4.1) and (5.3) we already know that
r(s, z, ) = IIr(s, z 0) = e(s, 2, O)L(1 - s, A)/L(s, A). So it will suffice to compute L(s, sr). For exchanging then 2 and A we will get L (s, Tr)
(since k = n(A)) and the e-factor from the r-factor. So it will suffice to show that L(s, 2r) = f j 15;5, L(s, zj). 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, ir) = P(s)HL(s, vi), L(s, Fr) = P(s)F1 L(s, f;) where P and P are polynomials related by (4.4.4). Let so be a zero of order u of P. By (4.5.8) it is a pole of order >-- u of H;L(s, z=) and by (4.4.4) a pole of order >_ u
of fj;L(1 - s, f;). But L(s, z1) and flL(1 - s, z;) 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 Ij1Z2 L(s, z=) = fj,zz L(s, p;) which is not a pole of L(s, z1) 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 - n1, 61 = 2r(u1) = z1, a'z = ir(p2 O+ ... O+ ,u,), O'= 61 x 62. By the induction hypothesis (5.5.1)
L(s, 62) =2IIi L(s, pr). r
Again v is a quotient of (3.5.3) where P = MU is the parabolic subgroup of type (n1, n2). The coefficients of ir are given by the absolutely convergent integrals (5.5.2)
f(g) = fm\G H(hg, h) dh = fKXU_ H(ukg, k) dk
where U = UP, P = IP and H: G x G -+ C is any function satisfying the following properties: (5.5.3)
H(u1mg1, uzmgz) = H(g1, gz),
u1 a U, uz E U, m e M;
PRINCIPAL L-FUNCTIONS
(5.5.4)
81
fork, e K,,, the function m H H(mkl, k2) is a coefficient of o' Qx 3112,
H is C°° and K x K finite on the right.
(5.5.5)
This being so if f is given by (5.5.2) then Z(O, s + I (n - 1), f) is for Re(s) large enough, equal to (3.5.5) (convergence questions are left to the reader since they can be handled by (3.6.6) and (3.6.7)). Conversely let 0 e .9'(n x n, F) and f a coefficient of 6i be given. Set
(5.5.6)
Idetmils+(xil-1)/2 detm2l5+cm i>izfi(ml)
A(0, s,f1,f2) = J
f2(m2) 0[ ( x m 1 m2 +
)111 )1 d<ml d"m2 dx dy. .
This multiple integral converges absolutely if Re(s) is large enough. We are going to show that (5.5.7) For any 0, the quotient A(0, s, f1i f2)/L(s, z) continues to an entire function of s. If 0 is in Soo, it is a polynomial ins.
Assume first 0 is in 9o(n x n, F). Then there are two K-finite functions and ' on K such that (3.5.9) is satisfied. Let dh be the normalized Haar measure on the
compact group K n P= K n P= K n M; for h in this group write h = (01 h2),
(5.5.8)
hi e K,,..
Then there is arrfunction H satisfying (5.5.3)-(5.5.5) such that
H[(01 M2) V, k] (5.5.9)
_
(0 1 mz)
$$fi(himihi_1f2 (hzmzhz
dh dh'.
If f is the coefficient of r corresponding to H by (5.5.2) we get, as in (3.5), (5.5.10)
f).
A(0, s,f1,f2) = Z(0, s + 21(n
This establishes (5.5.7) for 0 in YO. Let us note also that there is for A a relation similar to (4.3.4) where 0' is defined by: (5.5.11)
(y 1 m2)
at (ye e` ` mz)
t=o
and is in .9' if 0 is. Thus the product of A by any polynomial is an integral of the same type with 0 replaced by 01. If O is in 9o so is $1. Moreover 0H 01 is continuous. We are going to see now that there is a functional equation (5.5.12)
B(O^,1 - s + ?(n - 1), fl, f2) = r(s, r, O)A(0, s,ft, f2)
where B is obtained by replacing in the definition of A the pair (P, 6) by the pai (P, 6). It is then clear that (5.5.7) can be proved as the holomorphy of (4.5.1) (cf. proof of (4.5)).
To begin with it is clear that it can be obtained from (P, d) as r is obtained from
HERVEJACQUET
82
(P, oa). (Cf. (3.3).) More precisely if f is a coefficient of z given by (5.5.2) then f v is given by (3.3.11) and ft (loc. cit.) satisfies (5.5.3)-(5.5.5) for (P, 6).
This being so set
B(0, s,.f1,.f2) = f Idet
m1ls+(n,-1)/2ldet
m21s+(n2-1)/2
(5.5.13)
fi(ml)fz(m2) 0[(m1
x yx mmY
2)]dxml dxmz dx dy,
each time f is a coefficient of 6,. Then for 0 e 9o (5.5.7) applies to B with L(s, Tr) instead of L(s, 2c). Now let 0 e $o and a coefficient f of o'i be given; choose , ' 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
ff
0A[k-lzk'] '(k)e(k') dk dk'
V(z)
Moreover H- being given as in (3.3.11) by H(g1, g2) = H(gz, gj), we find HL\01 m2)k k]
_
jJ 2 t\0 1
0
z) $$f'[himih'Jf'[h2m2h']e(h'k')e'(hk) A dh'.
Since f v is given by (3.3.11) we get
B (0^, 1- s,.f v1, .f 2v) = Z(O^, 1 - s + 1(n - 1), .f v). Comparing with (5.5.10) this concludes the proof of (5.5.12) and (5.5.7). (5.6) Let .2/ be the space of functions on X = Gn1 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 topology. 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, x n1, F)} x {M(n2 x n2, F)}. For 01 in Ci°.°(G,, ), 012 in C°°°(M(n, x n2, F)), 021 in COO(M (n2 x n1, F)), and 02 in 9(n2 x n2, F) we may set (5.6.1)
i(m1, x2) = 01(ml) f 021(xm1)012(Y)02(xz + xy) dx dy.
We let co be the subspace of Q/ spanned by these functions; it is easily checked that c/o contains a dense subspace of C°°(X) so is dense in s/. For 0' in d and coefficients fl, f2 of 61, o2 we set (5.6.2)
U W, s, A, f2) = f f
1(m1, m2)
Idet m1I s+(nl-1)/2ldet
m2Is+(n2-1)/2
f1(mt)fz(mz) dxml dxmz
The integral converges absolutely for Re(s) large enough. By (5.5.7) we know that (5.6.3) For .ff in .sago, the ratio U(', s, fl, f2) /L (s, Z) continues to an entire function of s.
On the other hand, we are going to prove that
83
PRINCIPAL L-FUNCTIONS
(5.6.4) For ? in sad, the ratio UQIf, s, fl, f2)/L(s, U2) continues to a holomorphic function of s. As such, it is a continuous function of (1f, s) on d x C. Let us observe that the first assertion of (5.6.4) is trivial for I[I in the (algebraic) tensor product
C-(G,,) Ox '(n2 x n2, F).
(5.6.5)
For if 0' _i x0 W2 the ratio is the product of an absolutely convergent integral by Z(0'2, s + 2(n2 - 1), f2)/L(s, 0`2) which is a polynomial. Moreover assume so is a pole of order u of L(s, U2); then there are fi, f2 and TE
d1 so that the ratio of (5.6.4) has a pole of order u at so. Taking at the moment (5.6.4) for granted, assume so is not a pole of L(s, U1) and is a pole of order < u of
L(s, 2r). Since .moo is dense in d, it follows from (5.6.3) that the distribution U(- , s, fl, f2), which depends meromorphically on s, has a pole of order < u at so. The same is true of the (scalar) meromorphic functions U((', s, f1, f2). By taking T in 41 we get a contradiction. This proves (5.1). It remains therefore to prove (5.6.4). As noted, the first assertion is trivial for V in d1, the ratio being then the product of a polynomial by a function bounded in vertical strips. Now there is a relation (as + b) U(O, s, f1, .f2) + U(P', s, fl, .f2) = 0,
(5.6.6)
a # 0,
where 0' is in sail if 0 is and depends continuously on 0. Next introduce for Re s sufficiently small the integral V(O, s,A,f2)
= f (m1, m2) Idet
mils+(nl-1)i2 Idet
m211-s+(n2-1)12 f1(mi).f2" (m2) d"m1 d"m2.
Then, for 0' E d1, V (V s, fi, f2) = AS, U2, Sb) U(J, S, fI, J 2)
Again V satisfies a relation like (5.6.6). Finally, c1 is dense in sa/. These remarks being made the proof of (5.6.4) is similar to the proof of the holomorphy of (4.5.1). This concludes the proof of (5.1). 6. Global theory. The field F is now an A-field. (6.1) Let r = (3v zv be an irreducible admissible representation of Gn(A); again
r may be a representation of some other object than the group [I.P.S.]. Then Tr=(DCv.Set (6.1.1)
L (s, z) _ ]T L(s, ira), L(s, k) _ ]T L(s, gv). V
V
Suppose that the central quasi-character co of n is trivial on F".Then set (6.1.2)
r(s, n) = He(s, Zv, cv) V
Here 0 = TI Ov is a nontrivial character of A/F; almost all factors in (6.1.2) are one and the product does not depend on 0. (6.2) THEOREM. Suppose r is automorphic. Then the infinite products (6.1.1) converge absolutely in some right half-space. They continue to meromorphic functions of
HERVBJACQUET
84
s in the whole complex plane. As such they satisfy L(s, z) = e(s, z)L(1 - s, k). 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 S) elements, they are rational functions of Q-s.
PROOF. If i is cuspidal this is Theorems 3, 4, 5, VII, §6 of [A.W. 2] for n = 1, and Theorem (13.8) of [R.G.-H.J.] for n > 1. If n > 1 and i is not cuspidal, there is a standard parabolic subgroup of type (n1, n2, , nr), and, for every i, an irreducible automorphic cuspidal representation ai of Gni such that the following conditions are satisfied: for any place v, 7rv is an irreducible component of the induced representation v = I(Gv, P,,; Civ, 62v, .. ", 6rv) (Cf. [R.L.].)Thus, for any place v, L (s, prv) = P,(s) 11L(s, 6i,v),
i
L (s, kv) = P,,(s) J1L(s, 6i,v), i
fl
As, 7cv, cb) _ r(s, 6i,v, yb ),
where Pv is a polynomial in s if Fv = R or C, and a polynomial in qv s if Fv is nonarchimedean with residual field of cardinality qv. Moreover, for almost all v, irv and the 6i,, are unramified so that ((3.6.5)) PV = PV = 1,
e(s, 1cv, Ov) = 1,
almost all v.
Thus
L (s, r) = fJP,,(s) IIL(s, 611), v
i
L(s, k) = 11 Pv(s) fL(s, 61), i v
e(s, 2r) = 11 V
Pv(1 - S) PA(S)
fJe(s, 6i). i
Our assertions are then obvious, except the one on the boundedness of L(s, 2c) in the number field case, which follows from the Phragmen-Lindelof principle. BIBLIOGRAPHY
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Zetafunktionen mit Grossencharakteren and Kugelfunktionen, Math. Ann.
134 (1957), 1-32.
[G.M. 1] G. N. Maloletkin, Zeta functions of a semi-simple algebra over the field of rational numbers, Dokl. Akad. Nauk SSSR 201 (3) (1971), 535-537 = Soviet Math. Dokl. 12 (6) (1971), 1678-1681. [G.M. 2] , Zeta functions of parabolic forms, Mat. Sb. 86 (128) (1971), 622-643 = Math. USSR-Sb. 15 (4) (1971), 619-641.
[A.S.] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. 20 (1956), 47-87. [H.S.] H. Shimizu, On zeta functions of quaternion algebras, Ann. of Math. (2) 81 (1965), 166193.
[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 (1967), 461-490. [T.T.] T. Tamagawa, On the zeta functions of a division algebra, Ann. of Math. (2) 77 (1963), 387-405. [J.T. 1] J. Tate, Fourier analysis in number fields and Hecke's zeta-functions, Algebraic Number Theory, edited by J.W.S. Cassels and A. FrOhlich, Thompson, 1967, pp. 305-347. , Number theoretic background, these PROCEEDINGS, part 2, pp. 3-26. [J.T. 21 [N.W.] N. Wallach, Representations of reductive Lie groups, these PROCEEDINGS, part 1, pp. 7186.
[G.W.] G. Warner, Zeta functions on the real general linear group, Pacific J. Math. 45 (1973), 681-691. [A.W. 1] A. Weil, Fonctions zeta et distributions, Sem. Bourbaki, no. 312, (1965-66), 9 pp. [A.W. 2] , Sur certain groupes d'operateurs unitaires, Acta Math. 111 (1964), 144-211. [A.W. 3] , Basic number theory, Springer-Verlag, Berlin, 1967.
HERVEJACQUET
86
[A.W. 4] A. Weil, Dirichlet series and automorphic forms, Lecture Notes in Math., vol. 189, Springer-Verlag, Berlin, 1971.
[E.W.] E. Witt, Riemann-Rochscher Satz and E-Funktion im Hyperkomplexen, Math. Ann. 110 (1934), 12-28. [A.Z. 1] A. V. Zelevinskii, Classification of irreducible non-cuspidal representations of the group
GL(n) over p-adic fields, Funkcional Anal. i Prilozen 11 (1) (1977), 67 = Functional Anal. Appl. 11 (1) (1977). [A.Z. 2]
, Classification of representations of the group GL(n) over a p-adic field, preprint.
(Russian) COLUMBIA UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 87-95
AUTOMORPHIC L-FUNCTIONS FOR SYMPLECTIC GROUP GSp(4) MARK E. NOVODVORSKY* 0. Introduction. The paper represents a talk delivered at the Summer Institute of the American Mathematical Society at Corvallis in 1977. Let k be a global field, P the set of its normalized valuations, A = II PEP kp the adele ring of k, G a reductive algebraic group over k, GA = jf pEp Gp the group of ideles of G, 7c = ®x pEp 9rp a class of isomorphisms of irreducible admissible representations of GA, p a finite dimensional representation of Langlands' dual group LG. R. P. Langlands [1] defined L-function L(2r, p, s) as a certain Euler product, (0.1)
L(2r, p, s) = pTI L(2vp, p, s),
convergent for all complex numbers s with sufficiently large real part if 7r is unitarizable. He conjectured that if 7r is the class of an automorphic representation, then L(7c, p) can be continued to a meromorphic function on the whole complex plane which satisfies the functional equation (0.2)
L(7r, p, s) = e(zr, p, s)L(2r*, p, 1 - s),
7r* being the contragradient representation to 7r and e(ir, p, s) =P[I e(rp, p, s)
another Euler product whose existence is a part of the conjecture. However, the definition of the local factors L(7rp, 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 7r p of the groups Gp corresponding to nonarchimedean valuations p are of class 1. Therefore, in order to prove Langlands' conjecture one must first construct the Euler factors L(zp) and e(zp). A general approach to this problem was suggested by I. I. Piatetski-Shapiro [2]. Here we develop the ideas of [2] for the symplectic group G = G Sp(4) and prove Langlands' conjecture for the standard 4-dimensional representation p of its Langlands 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 representations z (cf. Piatetski-Shapiro's paper Multiplicity one theorems, these PROAMS (MOS) subject classifications (1970). Primary 10D10, 22E55. *Supported by NSF Grant MCS76-02160A1. © 1979, American Mathematical Society 87
88
MARK E. NOVODVORSKY
CEEDINGS, part 1, pp. 209-212) and the standard 8-dimensional representation p
of its Langlands dual group G Sp(4, C) x GL(2, C) (i.e. p is the tensor product of the standard 4- and 2-dimensional representations of the factors). Our results are complete only for functional global fields k. For number fields k we obtain meromorphic continuations of these automorphic L-functions but, because of certain difficulties with archimedean valuations, not the functional equations (0.2). Since the methods of [2] are based on generalized Whittaker models, we consider generic and hypercuspidal automorphic representations separately (cf. the quoted paper of Piatetski-Shapiro). The results for generic representations of G Sp(4) (§1) can be extended to all split orthogonal groups of Dynkin type B (Sp(4) covers SO(5)) ; in this form they were announced by the author (for char k > 0) in [10] and [11]. The results for hypercuspidal representations of G Sp(4) (§2) were obtained by I. I. Piatetski-Shapiro and the author. The results for G Sp(4) x GL(2) (§3) were announced by the author (for char k > 0) in [12]. 2. In this paper we assume that char k 1. G Sp(4), (generic representations. Let G ti G Sp(4) be the group of similitudes of the bilinear form 0
1 1
-1 -1
0
of four variables over k, Z the subgroup of all upper triangular unipotent matrices from G, 0 a nondegeneratel 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 Won the group Gp which satisfy the equation (1.1)
W(zg) _ O(Z)W(g)
b' z E Zp, g e Gp,
is denoted #'0(G,,); Gp acts in this space by right translations. A class 7cp of isomor-
phisms of irreducible admissible representations of the group Gp is called nondegenerate if it contains the subrepresentation of GP in an irreducible Gp invariant subspace of #"0(Gp); this subspace then is called a Whittaker model of 7cp and denoted by 'o(np). THEOREM (I. M. GELFAND, D. A. KAZDAN [3], F. RODIER [4]). A Whittaker model
of a nondegenerate class ip is unique.
Let z p be a nondegenerate class of isomorphisms of irreducible admissible representations 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, a (1.2)
C9
aek*,
aa
a 'That is, 0 is not trivial on the ideles of any horocycle subgroup of Z.
89
AUTOMORPHIC L-FUNCTIONS FOR GSP(4)
cup can be considered as a character of kp*. The contragradient class ip coincides with,cp O wp(ap) where a is the unique homomorphism of G Sp(4) into k* whose square is determinant (o is the factor of similitude). Therefore, 0-0(z*) coincides with the set of functions (1.3)
W*(g) =
gEGp, WEyv'p(7cp).
We introduce the subgroups a
_- GL(1), 1
1
U=
)EG}
l
1
and consider the integral
a (1.4)
/p(W, s) = f(U-H)p W(uh)jjall
s-I/2d(uh), ha
WE
THEOREM 1. The integral (1.4) is absolutely convergent in a vertical half-plane Re s > and defines fp(W) as a rational function in qps, qp being the number of elements in the residue field of the valuation p. All the functions /p(W), W E #',(Zp), admit a common denominator. There exists a rational in q-ps function r(lrp) such that
/ (W, s) = r(zp, s)/p((/3(W))*, 1 - S), 0
WE
'O(G p),
10 0 1
E GA.
(3=
10 01
0
We denote by Q p(2rp) the polynomial in qp 5 with constant term 1 which is the com-
mon denominator of smallest degree for functions /p(W), W E #-O(zp), and define (1.6)
L(np, s) = [Q(2rp, 9 5)]-I, e(7rp, s) = r(lrp, s) L(2cp, s)/L(2r p, 1 - s).
Let r = Qx pip zp be the class of isomorphisms of an automorphic generic cuspidal representation of the group GA. Then all its nonarchimedean components 2rp are nondegenerate and the Euler product (1.7)
L(2r, s) = IT L(2rp, s), peP\S
S being the subset of all archimedean valuations from P, is absolutely convergent in a vertical half-plane Re s > ; it coincides with the Euler product of Langlands
90
MARK E. NOVODVORSKY
corresponding to the standard 4-dimensional representation of Langlands' dual group LG ^- G Sp(4, C) if k is a functional global field and differs from it by a meromorphic (in the whole complex plane in s) factor if k is a number field. The factors e(lrp) are of the form agpbs and for almost all p e P\S are identically equal to 1; therefore, the Euler product
e(7r, S) =pEP\S H e(lrp, s)
(1.8)
is, in fact, finite and defines e(ir) as an entire function without zeros. THEOREM 2. The function L(7r) admits meromorphic continuation to the whole complex plane. If char k = p > 0, then this continuation is a polynomial in ps and p 5 satisfying the functional equation (1.8)
L(ir, s) = e(x, s)L(n *,1 - s).
2. G Sp(4), hypercuspidal representations. This case has been investigated in cooperation with I. I. Piatetski-Shapiro. For hypercuspidal representations, Whittaker models associated to nondegenerate characters of maximal unipotent subgroups do not work, and one has to consider other subgroups and generalized Whittaker models. It is convenient for our purposes to consider a different realization of the group G = G Sp(4). We choose a 2-dimensional semisimple k-algebra K and a 2-dimensional bilinear skew-symmetric form B in 2 variables over K; G can be realized now as the group of similitudes of the form trK, kB. Now G contains the subgroup (2.1)
G' ^ {g e GL(2, K) : det g e k}
of all K-linear transformations from G; the subgroup of all upper triangular unipotent matrices from G' is denoted by U', the unipotent radical of the normalizer of U' in G is denoted by U; U is a 3-dimensional commutative horocycle subgroup in G. The algebra K has a unique nontrivial k-automorphism (2.2)
X -+ 't;
it induces a k-automorphism of the form trK/k B which is denoted a, z e G. We put
T' = j ( "
x), xe
K* } c G';
obviously, T' is isomorphic to the multiplicative group K* of the algebra K. The groups G', U', and T' are normalized by a. Let p be a nonarchimedean valuation of the field k. We take a nontrivial character x of the factor group U,51 U; and a quasi-character ' of the group Tp whose restriction on the subgroup (e e) e T;, a-' =
e}
is unitary. If ' is v-invariant, we continue it to a character g of the group Tp, (2.3)
T = T' U vT'
If ' is not z--invariant, we put
(v(g') _ ')
AUTOMORPHIC L-FUNCTIONS FOR G sP(4)
T = T',
(2.4)
91
g=S'
In both cases we define (2.5)
Z = T. U,
B(tu) = e(t) x(u) V t e Tp, u e Up;
Z is an algebraic k-subgroup of G, 0 is a quasi-character of the group Zp. As in § 1, we denote by y ' (Gp) the set of all complex locally constant functions W on the group GP satisfying the equation (1.1), and an irreducible admissible subrepresentation of Gp in ,r,, by right translations is called the Whittaker model of its class of isomorphisms lcp and denoted #Y0(zp). THEOREM 3. Every class irp of isomorphisms of irreducible admissible representations of the group GP is either nondegenerate or has Whittaker model #' (icp) for some algebra K, subgroup T, and character SG of the described type; for every fixed 0 this model is unique.
REMARK 1. One class lrp can have Whittaker models corresponding to several of the characters 0 described in §§1 and 2. REMARK 2. The group Tp = Tp U zTp is the stabilizer of x in the normalizer of the group Up in Gp; so, x can be lifted to either a character or a 2-dimensional irreducible representation of the group Tp Up. Theorem 3 can be reformulated as the theorem of the existence and uniqueness of Whittaker models associated to these
1- and 2-dimensional representations; such reformulation might be preferable logically but leads to more bulky constructions for Euler factors in case of 2-dimensional representations. In the case when Z and 0 are defined by the formula (2.3), the uniqueness part of Theorem 3 was proven by I. I. Piatetski-Shapiro and the author [8] and by the author [9]. For supercuspidal representations 7Cp the uniqueness part of Theorem 3 follows from F. Rodier's Theorem 1 [5]. Now we put r
(2.6)
H
and consider the integral
x e k*}
c G' c GL(2, k)
` def
(2.7)
/p(W, S) = $HPWhhdh,
I / II
II x II
THEOREM 4. The integral (2.7) is absolutely convergent in a vertical half-plane Re s > and defines fp(W) as a rational function in qp S. All the functions /p(W), We #(r), admit common denominator.
As in §1, we denote by Qp(tcp) the normalized common denominator of the smallest degree and put (2.8)
L(7rp, s) _ [Q(rp, q
-s)]-I.
THEOREM 5. Euler factor Lp(2rp) does not depend on the choice of the subgroup Z and the character 0 of Zp (particularly, if the class lrp is nondegenerate, the factors L(v p) defined in § § 1 and 2 coincide).
92
MARK E. NOVODVORSKY
Let i = Qx pEP 7rp be the class of isomorphisms of an automorphic cuspidal representation of the group GA. Choosing for every nonarchimedean p a Whittaker model#'O(zp), we obtain Euler factors L(irp) and define
L(z, s) = PEP\S IT L(irp, s),
(2.9)
S being the set of archimedean valuations of k; this product is absolutely convergent in a vertical half-plane Re s >, it coincides with Langlands' Euler product corresponding to the standard 4-dimensional representation of Langlands' dual group LG _- G Sp(4, C) if k is a functional field, and differs from it by a meromorphic (in the whole complex plane in s) factor if k is a number field. THEOREM 6. The function L('r) admits meromorphic continuation to the whole complex plane. If char k = p > 0, then this continuation is a rational in p S function satisfying the functional equation
L(ir, s) = e(2r, s)L(lr*, 1 - s)
(2.10)
where e(7r) is an entire function in s without zeros.
In fact, e(ar) is an Euler product of some factors e(zp) appearing in local functional equations; these equations are rather bulky and will appear in a joint work with I. I. Piatetski-Shapiro. 3. G Sp(4) x GL(2), generic representations. In this section we use a modification of H. Jacquet's treatment of induced representations applied in [7] to automorphic L-functions on GL(2) x GL(2).
The subgroups C and Z of G Sp(4), the character 0 of ZA, the space 'W' (G Sp(4, kp)), p nonarchimedean, Whittaker model 'O(zp) for a nondegenerate class 7rp 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=
(a c
b
g
e G Sp(4), a, b, c, d e k, g e GL(2) } d
))
and also its homomorphisms onto the group GL(2, k) : a b 01:(c g d
g, __>(a
02: Ca c
9
d)
c
d)
We denote by 2 the subgroup of all upper triangular unipotent matrices in GL(2,kp). It is the image of H n Z under the homomorphism 01, and the kernel
93
AUTOMORPHIC L-FUNCTIONS FOR G SP(4)
of this homomorphism considered on the group (H (1 Z)A belongs to its commutator subgroup; therefore, every character of the group ZA defines under 01 a character of ZA which we denote with the same symbol. As in § 1, we denote by r (GL(2, kp)) the space of all locally constant complex functions W satisfying the equation W(zg) = cb(z)W(g),
(3.3)
z E Zp, g e GL(2, kp)
k., a class of an infinite-dimensional irreducible admissible repreand by sentations of GL(2, kp), a subrepresentation of this group in #- (GL(2, kp)) which belongs to kp; such a subrepresentation exists and is unique (cf. H. Jacquet, R. P. Langlands [6]). We denote by cop the restriction of 7Cp on the center of the group GL(2, kp); we consider cop as a character of the multiplicative group k** in view of the canonical isomorphism
GL(2)a(a a)-' aEk*. The space p), 3tp = ffp O Co-1(det) being the contragradient representation to ;p, consists of the functions
W*(g) =
(3.4)
g e GL(2, kp).
For every locally constant complex function 0 on the plane kp +O kp we define its Fourier transform
(x, y) =
(3.5)
J kpE+kp
q(u, v) a(yu - xv) d(u, v)
where a is a fixed nontrivial character of the group A/k of classes of adeles; for every complex quasi-character p of the multiplicative group k** we put
f(h, 0, u) =
(3.6)
J
kp.
h e Hp.
0((0, 002(h)) - p(x) dr,
If the integral (3.6) is absolutely convergent, then the function f satisfies, obviously, the equation
//a
(3.7)
fl I
/a
b
g
h, 0, p) =
p(d-1)f(h, 0, r),
b
g o , h e H.
\
Therefore, we can consider the integral AW, W, 0, p1(s), s) =
I
W(h) W(01(h))f(h, 0, U1) II 02(h) Its dh,
(3.8)
,al(C' S) = II K
and a similar one for /(W *, W*,
112, CO( K) &(K),
,
WE ' O(irp), W E
p2(s), s),
p2(X, S) = II Jr 1125
1(K) C0 1(4
THEOREM 7. The integrals (3.6) and (3.8) are absolutely convergent in a vertical
94
MARK E. NOVODVORSKY
half-plane Re s > and define /(W, W, 0) and f(W *, W *, ) as rational functions in qp'. There exists a rational in qps function r(7rp, kp) such that
AW, W, 0,,1(s), s) = r(lrp, gyp, s) /(W, W*, 0, U2(1 - s), 1 - s) (3. 9)
V W e #'0(7rp), W E
0.
All the rational functions /(W, W, 0) (for different W e 'YY'O(7rp), W e y' (Trp), 0) admit a common denominator.
As in §1, we denote by Q(7rp, 9rp) the normalized common denominator of the smallest degree in qS and define (3.10)
L(7rp Ox kp, s) = [Q(7rp, 7rp, 9 ,)]-t,
O 7rp, s)/L(7rp ©7rp, I - s).
e(7rp O 7Cp, s) = 7-(7rp, 7rp,
If 7r O k is the class of isomorphisms of a cuspidal generic irreducible representation of the group G Sp(4, kp) x GL(2, kp), 7c = Qx pEP irp, 7t = (D PEP kp, then we
put L(n (3 11)
Ox
7r, s) = 11 L(7cp D -kp, s), pEP\S
e(7r Q 7r, s) = f j
pEP\S
'-(7r P O 7t p, S),
S being the set of archimedean valuations of k. As before, the first product is ab-
solutely convergent in a vertical half-plane Re s >. For a functional field k it coincides with Langlands' L-function L(n, p, s) if p is the tensor product of the standard 4- and 2-dimensional representations of the complex groups G Sp(4, C) and GL(2, C); for a number field k these L-functions differ by a factor meromorphic in the whole complex plane. The second product (3.11) is, actually, finite and defines e(7c Ox 5c) as an entire function without zeros. THEOREM 8. The function L(7c OO k) admits meromorphic continuation on the
whole complex plane. If char k = p > 0, L(n Ox Tc) is a rational function in p -s satisfying the equation (3.12)
L(7c 0 k, s) = e(n Ox f, s) L(7c* O Tr*, 1 - s).
REFERENCES
1. R. P. Langlands, Problems in the theory of automorphic forms, Lecture Notes in Math., vol. 170, Springer-Verlag, Berlin and New York, 1970. 2. I. I. Piatetski-Shapiro, Euler subgroups, Proc. Summer School Bolya-Janos Math. Soc. (Budapest, 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, 1971. 4. F. Rodier, Modele de Whittaker des representations admissibles des groupes reductifs p-adiques deployes, C. R. Acad. Sci. Paris Ser. A 275 (1972), pp. 1045-1048. , Le representations de G Sp(4, k) on k est un corps local, C. R. Acad. Sci. Paris Ser. A 5. 283 (1976), pp. 429-431. 6. H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Math., vol. 114, Springer-Verlag, Berlin and New York, 1970.
AUTOMORPHIC L-FUNCTIONS FOR GSP(4)
95
7. H. Jacquet, Automorphic forms on GL(2). Part 2, Lecture Notes in Math., vol. 278, SpringerVerlag, Berlin and New York, 1972. 8. M. E. Novodvorsky and I. I. Piatetskii-Shapiro, Generalized Bessel models for the symplectic group of rank 2, Mat. Sb. 90 (2) (1973), 246-256. 9. M. E. Novodvorsky, On the theorems of uniqueness of generalized Bessel models, Mat. Sb. 90 (2) (1973), 275-287. 10. , Theorie de Hecke pour les groupes orthogonaux, C. R. Acad. Sci. Paris Ser. A 285 (1977), 93-94. 11. -, Fonctions % pour des groupes orthogonaux, C. R. Acad. Sci. Paris Ser. A 280 (1975), 1421-1422.
12. -, Fonction f pour G Sp(4), C. R. Acad. Sci, Paris Ser. A 280 (1975), 191-192. PURDUE UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 97-110
ON LIFTINGS OF HOLOMORPHIC CUSP FORMS TAKURO SHINTANI Introduction. In [10], H. Saito calculated the trace of "twisted" Hecke 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 striking identity between his "twisted trace formula" and the ordinary trace formula for the Hecke 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 characterized the space spanned by lifted forms. In the U.S.-Japan symposium "Applications of automorphic forms to number theory" which was held at Ann Arbor
in June 1975, the author reported a representation theoretic interpretation and generalization of Saito's work (see [13]). Results presented in the author's talk were immediately generalized by Langlands in [8]. Moreover, Langlands discovered an unexpected application of "lifting theory" to the theory of Artin L-functions.
The present paper consists of two sections. In §1, 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" is the group of units of R. We write G(R) = GR = GL(2, R). For a p-adic field k, o(k) = ok is the ring of integers of k. We denote by qk the cardinality of the residue class field of k. For each t e k set d(tx) = ak(t) dx = It Ik dx, where dx is an invariant measure of the additive group of k. For each locally compact totally disconnected group G, Co (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 Nk the norm mapping from F to k. 1.
1.1. Let k be either a finite algebraic number field or a p-adic field. Let F be a commutative semisimple algebra over k of prime degree 1. We assume that the group of automorphisms of F over k contains a cyclic subgroup g of order 1 generated by o,. Then F is isomorphic either to a cyclic field extension of k or to the direct sum of 1-copies of k. Set GF = GL(2, F) and Gk = GL(2, k). We may regard AMS (MOS) subject classifications (1970). Primary lODxx; Secondary 10D15, 10D20. Q 1979, American Mathematical Society
97
98
TAKURO SHINTANI
g as a group of automorphisms of GF (with the fixed point set Gk) in a natural manner. Two elements x and y of GF are said to be o-twistedly conjugate in GF if there exists a g e GF such that y = goxg-1. We denote by XGF, ° the set consisting of all elements of GF which are o-twistedly conjugate to x in GF. The set of o-twisted
conjugacy classes in GF is denoted by 'o(GF). The usual conjugate class in GF containing x is denoted by XGF. For each x e GF, set N(x) = Xo,f-1
.
X,1-2
It is easy to see that N(gU xgl) = gN(x)g1. 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 c'o(GF) into c'(Gk). If F is not afield, the mapping is a bijection.
In the following, for each c E'o(GF), we denote by N(c) E '(Gk) the image of c under the mapping given in Lemma 1. An x r= Gk is said to be regular if the group of centralizers of x in Gk is a two-dimensional k-torus. We call a conjugate class in Gk regular if it consists of regular elements. A o-twisted conjugacy class c in GF is said to be regular if N(c) is regular. We denote by W;(GF) (resp. f'(Gk)) the set of regular a-twisted conjugate classes (resp. regular conjugate classes) of GF (resp. Gk).
1.2. We keep the notation in 1.1 and assume that k is a p-adic field. Let GF be the semidirect product of GF with g. More precisely, GF is the group with the underly-
ing set g x GF with the composition rule given by (T, g)(z', g') = (zz', g"'g') (a, r' e g, g, g' e GF). It is immediate to see (via Lemma 1) that (o, g1) and (o, g2) are conjugate in GF if and only if gl and $2 are a-twistedly conjugate in GF. Denote by Gk (resp. GF) the set of equivalence classes of irreducible admissible unitarizable representations of Gk (resp. GF). For each R e GF and z e g, let Rz be the representation of GF given by RT(x) = R(xv). Then Rz E GF. Thus, the group g operates on GF. Denote by GF the subset of g-fixed elements of GF. A representation of GF is said to be admissible if its restriction to GF is admissible. Let R- be an irreducible admissible unitarizable representation of the group GF. As is well known, the restriction R of R- to GF is either irreducible or a direct sum of 1-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 JQ = R-((v, 1)). Then (1.1)
R(g°) = J,-'R(g)J6
(V g e GF).
Thus R E O. The relation (1.1) characterizes the operator J, up to a multiplication by an lth root of unity. Conversely let R E O. Then there exists a unitary operator of order 1 which satisfies (1.1). Hence, R is extended to an admissible irreducible unitarizable representation R- of GF of the first kind by setting R-'((Um, g)) _ J aR(g). For cp E C o (GF), denote by R-((p) the operator given by
R-(c) = SGFR(, g))ww(g) dg,
ON LIFTINGS OF HOLOMORPHIC CUSP FORMS
99
where dg is the invariant measure of GF normalized so that the total volume of GQ(FI is equal to 1.
It is shown that there exists a locally summable function x(R-) on GF such that trace R-((p) = f GF(P(g)x(R-)(g) dg.
It is proved that x(R")(g) depends only upon the o-twisted conjugate class of g in GF and that x(R") is defined only on W;(GF). For each r e Gk, denote by xr the character of r. Recall that xr(x) depends only upon the conjugate class of x and that xr is defined only on ('(Gk), assume further that the characteristic of the residue class field of k is not equal to 2. THEOREM 1. Let the notation and assumptions be as above.
(1) For each R e GF, there exist an extension R" of R to a representation of GF and an r e Gk such that (1.2)
x(R")(x) = xr(N(x)),
where N is the norm mapping from ' (GF) to c'(Gk) given in Lemma 1.
(2) For each r e Gk, there uniquely exists R e GF whose suitable extension Rr to a representation of GF satisfies (1.2).
For each r e Gk, we call R e GF, which is related to r by (1.2), the lifting of r from Gk to GF. (Jacquet introduced in [4] the notion of lifting from a different viewpoint.) REMARK 1. An analogue of Theorem 1 for finite general linear groups was given
in [14]. Furthermore, an analogue of Theorem 1 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 GF in a concrete manner. If F is isomorphic
(as a k-algebra) to the direct sum of 1 copies of k, GF is isomorphic to the direct product of I-copies of Gk. It is immediate to see that the lifting of r e Gk is given by r© (g 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 1. First, let us recall a description of Gk. For details, see §§3 and 4 of Chapter 1 of [5]. For quasi-characters ul and p2 of kx such that ,ul,u 21 0 a, let p(ui,,u2) be the corresponding irreducible representation of Gk in the principal series. If ,u1,u21 = ak, let o(ui, p2) be the corresponding special representation of Gk. For a quadratic extension K of k and a quasi-character w of Kx such that co' # co (' denotes the conjugation of K with respect to k), let z((o, 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(pl, ,u2), o(ul, p2) and z(c), K). For each quasi-character u of kx, we denote by p the quasicharacter of Fx given by p = u o NF. For a quadratic extension K 0 F of k and a quasi-character co of Kx, denote by w the quasi-character of Lx (L = K. F) given by
w= PROPOSITION 1. The notation and assumptions being as above, the lifting R of r E Gk to GF is given as follows:
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TAKURO SHINTANI
(1) If r = p(,ul,,uz) (resp. uz)), R = P(,ul, fez) (resp. R = o'(,aI, P2))(2) If r = ir(w, K) and K F, R = z (w, L), where L = F K. (3) If r = 2r(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 p-adic field (the residue class field of k may be of characteristic 2). For an x e GF, let Z,(x) be the subgroup of GF consisting of all elements of g of GF satisfying g-Ixg I = x. Normalize invariant measures on GF and ZQ(x) so that total volumes of their maximal compact open
subgroups are all equal to 1. Denote by dg the invariant measure on GF / ZQ(x) given as the quotient of the normalized invariant measure of GF by that of Z4(x). For f e C o(GFA , set (1.3)
A,(f, x) =
f(g °xg I) dg.
JG p/Z,(x)
It is shown that the integral is absolutely convergent. Moreover, for c E'eQ(GF), A,(f, x) = Aq(f, y) for any x, y c c. For each c E'o(G), we put (1.4)
A,(f, c) = AQ(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 f E Co (Gk), set (1.5)
A(f, x) =
J Gk/Z(x)
f(gxg I) dg.
It is known that the integral is absolutely convergent. For each c e "'(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 f e Cp (GF), there exists an f e Co (Gk) which satisfies the following conditions (1) and (2): (1) For each regular c e Co(GF), AQ(f, c) = A(f, N(c)).
(2) For each c e c9'(Gk) - N(V,(GF)), A(f, c) = 0. REMARK 3. In Proposition 2, assume f is the characteristic function of G0(F). Assume further that F is either the unramified cyclic extension of degree 1 of k or is isomorphic to the direct sum of l copies of F. Then one may put f to be the characteristic function of GO(k).
REMARK 4. The correspondence f H fin 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. kA)
ON LIFTINGS OF HOLOMORPHIC CUSP FORMS
101
be the ring of adeles (resp. the group of ideles) of k. Denote by k, (resp. kA,o) the infinite (resp. finite) component of kA. Then kA = k- O+ kA,o and kA = k' x kA,0. Moreover, both groups Gk, and GkAo are embedded into GkA in a natural manner and GkA = Gk_ x GkA, o. Let {oo 1, , con} be the set of infinite places of k. For each g 'e GkA, set g = g,go (gam E G-, g0 E GkA. o) and g, _ (g-,1, ..., g-,n), where g,,1 E GR
is the component of g- corresponding to the infinite place oo,. For g,,, we introduce the following standard parametrization
_ t; g°° _ - ( ± tf)
1 x1/y1 1)(
/yi
,)(_cos Oi sin 01
sin 0 cos B,)
(t, > 0, x. E R, y, > 0, B; a R). Let Q, be the differential operator on GkA given by 2
For 0 = (61,
+
2
z
Y2( aX2
, Bn) E Rn, we denote by K(O) the element of Gk,, whose ith component
is
cos B, sin Bi
-sin Bi cos 6, Let x be a unitary character of the group kllk" 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(oof). Denote by S(x, r, kA) the space of C-valued functions f on GkA which satisfy the following conditions (1.7), (1.8) and (1.9). (1.7) f(g) is bounded, smooth with respect to g, and locally constant with respect
to go. Furthermore, Qif = fr,(r, - 2)/4 (1 < i <- n). (1.8) (1.9)
f(r(z z)g) = x(z)f(g) f (gx(B)) = exp
(V r E Gk, V z e kA),
I E riei)f(g)
(V B n Rn).
Denote by p(k) the set of finite primes of k. For each v e p(k), let kv 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 [j ,Gk,. Normalize the Haar measure of Gk, o so that the volume of f j 0Gov is equal to 1. For each ip E Co (GkA, o),
we denote by T(cp) the linear operator on S(x, r, kA) given by (T(cp)f)(g) = f GkA, o f(gx)ip(x) dx. It is shown that TP is of finite rank. Let F be a totally real cyclic extension of k of prime degree 1. Set x = x o N. Then xis acharacter of the group FAIF". 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 v e g = Gal(F/k) (the Galois group of F with respect to k) and any function f on GFA, denote by JTf the function on GFA given by (Jzf)(g) = f(gr) (g r= GFA). It is easy to see that Jz leaves the space S(x, r, FA) invariant. For each v E p(k), set Fv = k,, ©k F. Let o(FF) 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, Fv is isomorphic to the direct sum of
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TAKURO SHINTANI
1-copies of kv. In both cases, Fv is embedded in FA, o in a natural manner. The group GFA, o is the restricted direct product H VGF,. For each g E GFA, o, we denote by gv E
GF, the v-component of g.
In the following we choose and fix a generator o of g once and for all. Let Sp be an element of Co (GFA,o) of the form'ISo(g) = jj,Sov(gv), where So,E Co (GFv) and
cois Tv the characteristic function of G,,(Fy) except for a finite number of v. For each v e p(k), take cpv c Co (Gkv) which satisfies the equalities (1) and (2) of Proposition 2 for f = So, and f = cpv. We may assume that, except for a finite number of v, cp is the characteristic function of Gvv (see Remark 3 of 1.3). Denote by cp the function on GkA, o given by cp(g) = f j vcpv(g). By class field theory, the group k"NkFA x,_1 be 1 distinct characters of is a subgroup of index l of kA. Let xo (= 1), xt, the group kA/(k"NkFA). We are now ready to present an adelic version of Theorem 5.6 of [10]. THEOREM 2. The notation and assumptions being as above (in particular, rU and cp are related by (1) and (2) of Proposition 1), if r1, r2, , r > 2, 1-1
(1.10)
1 Trace JT((p)I S(x, r, FA) = ZTrace T(cp)I S(xxj, r, kA). =o
REMARK 5. In [10, Theorem 5.6], k is the rational number field and F is a tamely
ramified totally real abelian field of degree 1. Furthermore, Sp is assumed to be unramified. However, the correspondence rll -+ cpv is explicitly described for each v. In that point, Theorem 5.6 of [10] is more precise than Theorem 2. REMARK 6. If 10 2, S(xx=, r, kA) is isomorphic to SQ, r, kA) as Gk,, 0-module.
1.5. Let us consider the representation theoretic meaning of both sides of the equality (1.10) of Theorem 2. Via the right regular representation: g H Tg, the group GFA, o acts on S(x, r, FA). By a theorem of Jacquet-Langlands (see Proposition 10.9 and Proposition 11.1.1 of [5]), the space S(x, r, FA) decomposes into an
algebraic direct sum of irreducible mutually inequivalent GFA, o-submodules. Denote by GFA, o(r, x) the set of equivalence classes of irreducible representations of GFA, o realized on irreducible GFA, o-submodules of S(x, r, FA). For each 7r e GFA, 0(r, x), denote by V(2, r, x) the irreducible GFA, o-submodule of S(x, r, FA) on which 7r is realized. We have S(x, r, FA) _ r, x) (an algebraic direct sum), where the summation with respect to 7r is over GFA, o(r, x). Denote by 7r° the representation of GFA, o given by 7r°(g) = 7r(g°). The obvious relation J°Tgo = TgJ° implies that 7r° E GFA, 0(r, x) for each 7r E GFA o (r, x) and that V(7r°, r, 7r) = J,-' V(7r, r, x). Take a 7r E GFA, o(r, x). It is known that, for each v E p(k), there exists 7rv E OF, such that 7r is equivalent to the restricted tensor product Qx yEP(WCv (except for a finite number of v, 7rv is an unramified representation of GF). Denote by GFA, 0(r, x, g) (g = Gal(F/k)) the subset of GFA, 0(r, x) consisting of all 7r such 7r, J° induces a cyclic permutation among V(7v , r, x) that 7r° = 7r. If 7r° (z E g). Thus the trace of the restriction of JT(cp) to the subspace of S(x, r, FA) spanned by { V(7r, r, x); 7r e GFA, o(r, x) - GFA, o(r, Z, g)} vanishes. If 7r° = 7r, 7rv = 7rv for each v E p(k). There exists a linear operator J°(7rv) of order 1 on the representation space of 7rv such that 7ev(g°) = J°(7Cv) 17rv (g)J°(7rv)
(V g e
GFv).
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ON LIFTINGS OF HOLOMORPHIC CUSP FORMS
Such an operator J,(7r,) is unique up to a multiplication by an lth root of unity. For unramified 7r,, normalize Jo(7rv) so that it fixes 7cv(Gvy)-invariant vectors in the representation space of 7Cv. We may assume that the system of linear operators {Jo(7cv); v E p(k)} is normalized so that JJI V(7r, r, x) ®v-p(k) Jo(7cy). For So = j1 VSOV e Cp (GFA o), set 7Cy(gv) = fGFy Sov(x)7r (x) dx. Then rv(SOv) is a linear oper-
ator of finite rank acting on the representation space of ire,. Moreover Trace JQT((p)I V(7r, r, x) = 11 trace JQ(7Cv)7Gv(S9v). V
Hence, (1.11)
Trace JgT(SO)I S(x, r, FA)
a vEp(k)
trace
JJ(7cy)2Cy((pv),
where the summation with respect to 7r is over all GFA, 0(r, x, g). In a similar man-
ner, we have (1.12)
Trace T(rp)I S(x, r, kA) _ n
r[
trace 7tv((pv)
vEp(k)
for every (p = jl vcpv e Co (GkA, o), where the summation with respect to r = (Dv7rv is over all GkA, o(r, x) and ZAP") = JGk,, cov(x)iv(x) dx.
Recall that in equalities (1.10), (1.11) and (1.12), Soy and cpv 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 rI, ..., r > 2.
(1) For each
= Qx v9Gy E GA,0(r, x) (7r E Gky),
there uniquely exists n =
(97r, E GFA 0(r, x, g) (7ry e GF) such that for each odd place v of k, 2r is the lifting of 7Gv from Gky to GFy. We call 7r the lifting of 7r from GkA 0(r, x) to GFA o(r, x, g).
(2) If 10 2, for each r e GFA, 0(r, x, g), there uniquely exists 2r E GA 0(r, x) such that r is the lifting of n. .
(3) If l = 2, for each 2c E GFA, 0(r, x, g), there exists a z c G kA, 0(r, x) or GkA, 0(r, xXI) (xI is the character of order 2 of kAfk" which corresponds to F in class field theory) such that is is the lifting of 7r. Moreover, rI and '2 E GkA, 0(r, x) have the same lifting to GFA, 0(r, X, g) if and only if 2r1 = 9G2 or 7ZI = 7L2 O xl (det).
2. In this section we expose the proof of Proposition 2. Then we indicate how Theorem 1 and Theorem 3 are made plausible by Theorem 2 (the proof of Theorem 2 is a (more or less obvious) modification of proofs of Theorem 1 and Theorem 5.6 of [10]).
2.1. In the following three subsections we use the notation in 1.1 and 1.3 without further comment. In particular, k is a p-adic field. Assume that F is isomorphic to the direct sum of /-copies of k. Then we may assume that GF is the direct product
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TAKURO SHINTANI
of 1 copies of Gk and that, for each x = (x1, , x1) E GF, x° is given by x° (x1, x1,
, x1_1). For any f c GF, set, for any x c Gk,
f(x) = f(G,)I-lf ((x2, ..., x1)-1 x, x2, ..., x1) dx2 ... dx1.
(2.1)
It is easy to see that f e Co (Gk) and that A,(f, c) = A(f, 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 a generator of the maximal ideal of o(k). Denote by Qk the set of isomorphism classes of two dimensional semisimple algebras over k. For each K E Qk, choose an embedding of K into M(2, k) as a k-algebra. Via the embedding, we identify K with a subalgebra of M(2, k). For each x e K, denote by x' the image of x under the unique nontrivial k-algebra automorphism of K with respect to k. It is well known that ,W'(Gk) = U U(t)Gk (disjoint union), KEQk 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, Gk
,W(Gk) - c9'(Gk) = ZU x{(z z) U z(1 1) } (disjoint union).
Let o(K) be the maximal o(k)-order of K. For/ each nonnegative integer m, we denote by o(K),,, the unique o(k)-order of K such that [o(K), o(K)m] = qk . Let o(K)Mx be the group of units of o(K)m. For each t e o(K)x - o(k)x, there uniquely exists a nonnegative integer i such that t e o(K); - o(K)x1. Set i = i(t) for t e (Gk), set a(z) = f(z) and (3(z) = A(f z(11)) (z e kx) o(K)x - o(k)x. For each f e C -(Gk), (cf. (1. 5)). Then both a and (3 are locally constant, compactly supported functions on kx. Furthermore, for each K e Qk, we denote by (PK a function on (Kx - kx) given by toK(t) = A(f, t) (note that (PK does not depend upon a choice of an embedding of Kinto M(2, k)). Then (PK is a locally constant function on Kx -kx. The next
lemma is a version of Lemma 6.2 of [7] (see also Theorem 2.1.1 and Theorem 2.2.2 of [12]). We include a proof. LEMMA 2. Let the notation and assumptions be as above. Then a triple {tpK; (K E Qk), a, 01 satisfies the following conditions: (1) The support of (PK is relatively compact in Kx.
(2) toK(t) = IoK(t') (V t e Kx - kx). (3) For each z e kx there exists a neighborhood U(z) of z (s.t. z-1 U(z) (-- o(K)x) such that (2.2)
(PK(t) _ {1 - C(K) } a(z) +
C(K)gk(t:-1)-1 ,3(z)
qk - 1
for any t e U(z) - U(z) n kx, where we put C(K) _ [o(K)x, o(K)x]. PROOF. Choose an co e o(K) so that 11, w} is an o(k)-basis of o(K). An o(k)basis for o(K)m is given by {1, 7rmw} (m = 0, 1, 2, ). For each t E K, there uniquely exists
ON LIFTINGS OF HOLOMORPHIC CUSP FORMS
105
such that (tw) = c(t) (').
C (t) = (t2i t2z)
Then the mapping t ' - c(t) is an embedding of K into M(2, k) as a k-algebra. It is known that every proper o(K)m ideal in K is principal (see Proposition 1 of [3]). Thus,
Gk = U G.(k) (1
m) c(KC)
(disjoint union).
M=0
Hence, for any f e Co (Gk) and any t e K" - kx, (2.3)
A(f, t) _ Z CmJ -(e(t)m) M-0
where we set f -(x) = fG (uc) f (uxu 1) du, cm = [o(K) x, o(K)m], c(t)m =
(1
.m) c(t) (1
m)
(cf. the proof of Lemma 7.3.2 of [5]). It is sufficient to prove the lemma for z = 1. Since f is locally constant and compactly supported, there exists a neighborhood U of 1 in o(K) x such that J "(c(t)m) = J -(1
) 1
t12
=
f -(l Z:(1)-m)
(m = 0, 1, 2, ...)
and that 1 7rt(t)+n
f~ 0
1
)= .f(1 2)
(n = 0, 1,
)
for all t e U -u n kx. We note that a(1) = f(12),
3(1) =nRkn(Q 1) and
[o(Kx), o(K)m+1] = 9,[o(K)x, o(K)m]
(m = 1, 2, ...).
Hence we have
(PK(t) _ {1 -
C(K)
a(l) +
C(K)9k(t)-1 j3(I)
J
for anyteU -u nkx. A triple {a, (3, ft; K e Qk} of locally constant compactly supported functions a, (3 on kx and a system {(PK; K e Qk} of locally constant functions ft on Kx - kx is said to be an admissible triple if it satisfies the conditions (1), (2) and (3) of Lemma 2. The next lemma, which is also a version of Lemma 6.2 of [7], follows from Corollary 1.1.4 of [12] and the previous lemma. LEMMA 3. Let {a, (3, ft; K e Qk} be an admissible triple. Then there exists an
f e Co (Gk) such that ft(t) _ !1(f, t) for any t e Kx - kx. Moreover, for such an f, a(z) = f(z 12) and (3(z) _ !1(f, z(1 i)).
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TAKURO SHINTANI
2.3. Let F be a cyclic field extension of prime degree l of k. For each K E Qk, set L = K O k F. Then L is a two-dimensional semisimple algebra over F. A prescribed embedding of K into M(2, k) is naturally extended to an embedding of L into M(2, F). Furthermore, the norm mapping Nk from F to k is naturally extended to the norm mapping NK from L to K. Set LI = {t E Lx, NKt = 1 } and
FI = LI (1 F and L' = {t E Lx; NKt E Kx - kx}. The following description of ,Wo(GF) is due to H. Saito (see Lemma 3.5 and Remark 3.8 of [10]). LEMMA 4.
re,(GF)
(1)
=KEQk U Ut
(t)GF, o,
where the union with respect to t is taken over a complete set of representatives for L'/LI with respect to the action of the automorphism groups of L with respect to F.
(2) If 10 2,
leo(GF) - W,(GF) = U (z.12)GF,aU zEF'1F1
U z(1
1)GF,C.
zEFx/Fl
If 1 = 2, ,Wo(GF)
- 'W(GF) =
Ux 1 Z
zEk
\
1)GF, C U U
zEFll/Fl
z (1 1)GF, °.
For each f r= Co (GF), we define functions a and 0 on kx as follows: For z e Nk Fx, set z = NF z (z e Fx) and a (z) = A ,(f, z 12) and (3(z)
AU(f, z(11)) 111k (cf.
(1.3)). For z e kx - Nk Fx, set (3(z) = 0. If 10 2, set a(z) = 0. If 1 = 2, set a(z)
(9k - 1) A (f' \z 1)).
It follows from Lemma 1 and Lemma 4 that a and 3 are well-defined functions on kx. Moreover, they are both locally constant and compactly supported.
For each K e Qk, we define a function cPK on Kx - kx as follows: For t E (Kx - kx) (1 NK L, set t = NK t (t e Lx) and cDK(t) = A ,(f, t). Fort 0 (Kx -kx) A NKLx, set cgK(t) = 0. Lemma 1 and Lemma 4 again guarantee that cPK is a welldefined function on Kx - kx. The proof of Proposition 2 is now reduced to the following Lemma 5. LEMMA 5. The notation being as above, {a, 0; (PK; K E Qk} is an admissible triple.
For a subgroup V of Kx (K E Qk), set (2.4)
W(V) = V U {t(1
1);
t E V fl kx}.
The next lemma will be applied for the proof of Lemma 5. LEMMA 6. For a given K E Qk and a given compact subset Cl of GF, there exist an open subgroup V of Kx and a compact subset C2 of GF such that g°wg I e Cl for some w e W(V) implies g°g l E C2.
PROOF. There exists an open compact subgroup VI of Kx which satisfies the following conditions:
ON LIFTINGS OF HOLOMORPHIC CUSP FORMS
107
The mapping: x H x' is an isomorphism from VI to V11 whose inverse mapping is given by (2.5)
1/1x H x1" = 9m)(X
1)m
(VI is so small that the series is absolutely and uniformly convergent on VI).
Denote by C3 the image of CI under the norm mapping: x H N(x) (cf. 1.1). Since N(gawg 1) = gN(w)g 1; gawg 1 e C1, for some w e W(VI) (cf. (2.4)) implies
gw'g I e C3. Put C4 = C3 n {gw'g 1; w e W(VI), g e GF}. On C4, the binomial series (2.5) is absolutely and uniformly convergent. Hence the closure C5 of the
image of C4 under the mapping: x H x1 /' is compact. It is easy to see that 9W9-1 = (gw'g I)I"! for w e W(VI). Hence g°wg 1 e CI for some w e W(VI) implies gag -1 e
C1C51. Thus, we may put C2 = C1C5 1 and V = V1.
PROOF OF LEMMA 5. It is easy to see that cOK satisfies the conditions (1) and (2).
For a to e NF Fx, take z e Fx such that to = NF z. Let Co be the support off and choose an open compact subgroup V of Kx which has the property described in Lemma 6 for C1 = z 1Co. Then there exists a compact subset C2 of GF such that zgawg 1 e Co for w e W(V) implies
g°g-1 e C2. (W(V) is given by (2.4).)
Let {yi; i e I} be a complete set of representatives for the double coset G0(F\GFIGk. It follows from Lemma 1 that the mapping: g ,-, gag -1 is a homeomorphism from GF/Gk onto the closed subset {g e GF, N(g) = 1} of GF. Hence, there exists a finite subset Io of I such that (2.6)
zgcwg I e Co for a w e W(V) implies g e UG(DF)yiGk. ielo
Then, for t e V - V n /kx, Z,(t) =/ /Kx if V is small enough and PKltot') _ A ( , zt) _ Ej aiA(fi, t), where u; I is the invariant volume of Gk n y, 1G(oF)yi in Gk, and J. e Co (Gk) is given by .fi(g)
= J c coFf(zu°Y°gy Iu 1) du.
Since Io is finite, it follows from Lemma 2 that there exists a neighborhood VI c V of 1 in Kx such that, for any t e V1 - VI n k and any i e ((Io, 11 A(.fi, t) = {1 - C(K) }f(1) + C(K)gk( 1 A\f`.' ((\1 1//. qk- 1
However, (2.6) implies that
E fi(1)iui = A0(.f z) = a(to),
ielo
Eo iA fi, \1 1// - A ,(f z(1 1// -
11Ik 1 3(to).
There exists a smaller neighborhood V2 c VI such that, for t e V2 - V2 n k', i(t') = ord 1 + i(t). Set U = {tot'; t e V2}. Then U is a neighborhood of to in Kx. We have proved
that, for any t e U - U n kx,
108
(2.7)
TAKURO SHINTANI
C(K)
(PK(t) _ {I _ qk - 1 } a(to) +
C(K)gi(tto'1-113(t0).
Now take a to r= kx NkFx. Set L = F Qk K If 10 2, NKLx fl kx = NkFx Hence there exists a neighborhood U of to in K such that U (1 NKLx = 0. Hence,
(PK(t) = 0 for any t e U - U n V. Since a(to) = (3(to) = 0, the condition (3) is satisfied. Next assume 1 = 2. If K is not a field, there exists a neighborhood U of
to in Kx such that U n NLLx = 0. Hence (PK(t) = 0 for any t e U -u n kx. Since 0(t0) = 0 and C(K) = qk - I, the equality (2.7) is valid for any t F -u - U n
V. Now assume that K is a quadratic extension of k. Then there exists so E Lx such that to = NKso. Furthermore, Z,(so) is isomorphic to the multiplicative group of the division quaternion algebra over k. The proof of the following Sublemma is quite similar to that of Lemma 6. SUBLEMMA. The notation and assumptions being as above, for a given compact subset CI of GF, there exist an open subgroup V of Kx and a compact subset C2 of GF such that g°sovg I E CI for some v c V implies gUsog 1 E C2.
Normalize invariant measures of Z,(so) and Kx so that volumes of their maximal compact subgroups are all equal to 1. It follows easily from the Sublemma that there exists a neighborhood U1 of so in Lx such that the following integral is absolutely convergent for s c U1 and gives rise to a locally constant function on U1: JGF/K< f(gasg 1) dg. If t = NKS kx, the above integral is equal to (PK(t) = A,(f, s). If s = so, the above integral is equal to A,(f, so) X SZ,(so)/Kx dz, where dx is the quotient measure of the invariant
measure of Z,(so) by that of Kx. The volume of Z,(so)/Kx is equal to 1 or 2 according as K is ramified or not. On the other hand, C(K) is equal to qk or qk + 1
according as K is ramified or not. Since a(to) = - (qk - I)A,(f, so) and 0(to) 0, we have shown that there exists a neighborhood U of to such that (2.7) is valid for any t c U - U n V. The proof of Lemma 5 is now complete. REMARK. The proof of Lemma 5 shows the following further relations between
f and f of Proposition 2 when F is a field extension of k. If z = Nkz (z e Fx), .f(z 12) = A (f, z 12) and
A (f
z(1
I l ikAQ(f,
z('
I))
If kx a z 0 NkFx, 11(f, z(1 i)) = 0 and
-(qk -
11
1)11,((
\f, \Z //
if 1 = 2.
1
2.4. When F is isomorphic to the direct sum of 1 copies of k, the correspondence f ---> f of Proposition 2 is made explicit by (2.1). When F is the unramified cyclic extension of k of degree 1, one can make the correspondence explicit if f is bi-Go (F)invariant. In more detail, denote by Lo(Gk, Gock)) the set of all functions f e Co (Gk) which are right and left GO(k)-invariant. Then LO(Gk, GO(k)) becomes a commutative
ON LIFTINGS OF HOLOMORPHIC CUSP FORMS
109
algebra with respect to the convolution product. For indeterminates X and Y, let C[X, Y, X-1, Y-1]0 be the subalgebra consisting of all elements of C[X, Y, X-1, Y-1]
which are symmetric with respect to X and Y. For each f e LO(Gk, GO(k)), Put F(f, k)[X, Y] = E.,,,Z CmnXmYn, where Cmn = qk (m-n)/2J kf (2rm
fin/ \1 1)) dx.
Then it is known that the mapping: f F(f, k) is an algebra isomorphism from LO(Gk, Go (h) ) onto C[X, Y, X-1, Y-1]0 (cf. Theorem 3 of [11]). Let F be the unramified cyclic extension of k of degree 1. For each f E LO(GF, G0(F)), there uniquely exists a 2(f) E LO(Gk, G0(k)) such that F(f, F) (XI, Y')
= F(A(f),k)(X, Y). The mapping: f F-. 2(f) is a C-algebra homomorphism from LO(GF, G0(F)) into LO(Gk, GO(h))
The homomorphism A is discussed in [6] in a more general context. The following
proposition is implicit in Saito [10]. In fact, considerable parts of 3.4-3.13 and 5.1-5.4 of [10] are devoted to the proof of Proposition 3. PROPOSITION 3. The notation and assumptions being as above, f E LO(GF, GD(F))
and 2(f) E LO(Gk, G) are related by (1) and (2) of Proposition 2.
2.5. Let M be a finite subset of 3(k) containing all primes of k which ramify in F. Denote by Co (GFA,0iM) the subspace of Co (GFA,O) spanned by (P = (p (Spv E Co (GF,)) such that, for each v 4 M, Sp E LO(GF,,, GD(F,)). For cp = 11,Spv e C0 (GFA, 0, M), we choose cp = jj (p E Co (GkA, 0) as follows : For v e M, we choose (pt, e Co (Gk) so that Spv and cpv are related by (1) and (2) of
Proposition 2. For v M, if v remains to be prime in F, set rpv = A(Sov), where A is the homomorphism from LO(GF., Gt(Ft)) to LO(Gk,, G,,(k,)) introduced in 2.4. If v splits in F, denote by cpv the function on Gkv given by the right side of (2.1) for f = Spv. In both cases, cp a LO(Gk GO(k)) (v M). For any unramified rv E Gk., denote by iv the lifting of irv from Gky to GF, (see Proposition 1).
It follows from the first part of Proposition 1 that trace
(2.8)
)2cv (pv) = trace 7[v(cpv)
(V V
M).
Denote by GFA, 0(r, X, M, g) (resp. GkA, 0(r, x, M)) the subset of GFA, 0(r, X, g) (resp. Gk,, 0(r, x)) consisting of all n = &x 7rv (resp. ir = Qx 7rv) such that rv (resp. 7rv) is an unramified irreducible representation of GFv (resp. Gk) for any v 0 M. For sp = ijcot E Cp (GFA 0, M), we have, by (1.10), (1.11) and (1.12) that c-1
1E H trace J,(irv) 7rv(cov) = E Ell trace 7rv(cpv),
(2.9)
n
V
j=0 a
v
where the summation with respect to it is over all GFA,0(r, X, M, g) and the summation with respect to ' is over all Gk,, 0(r, xxj, M).
For each 7r = 0 7rv E GFA 0 (r, X, M, g) denote by X(2r) the subset of 0(r, xxj, M) consisting of all 7r = Q iv which satisfy the following
Uj'-1o
=GkA,
condition: For each v
M, the lifting of irv from Gko to GFt, is irv.
TAKURO SHINTANI
110
Then equalities (2.8) and (2.9) together with "the strong multiplicity one theorem" (see Theorem B of [9] and Theorem 2 of [1]) show the following: (2.10)
1 jj trace vaM
Z
II trace ir,,(tpv)
2EX(a) veM
Here I must confess that I was too optimistic at Ann Arbor. I was erroneously convinced that both Theorem 1 and Theorem 3 are immediate consequences of the equality (2.10). Actually, highly nontrivial considerations are necessary to derive these theorems from (2.10). Anyway, far-reaching generalizations of Theorem 1 and Theorem 3 are established in [8]. REFERENCES
1. W. Casselmann, On some results of Atkin and Lehner, Math. Ann. 201 (1973), 301-314.
2. K. Doi and H. Naganuma, On the functional equation of certain Dirichlet series, Invent. Math. 9 (1969), 1-14. 3. Y. Ihara, Hecke polynomial as congruence C -functions in elliptic modular case, Ann. of Math. (2) 85 (1967), 267-295.
4. H. Jacquet, Automorphic forms on GL(2). II, Lecture Notes in Math., vol. 278, SpringerVerlag, 1972.
5. H. Jacquet and R. Langlands, Automorphic forms on GL(2), Lecture Notes in Math., vol. 114, Springer-Verlag, 1970. 6. R. Langlands, Problems in the theory of automorphic forms, Lecture Notes in Math., vol. 170, Springer-Verlag, 1970, pp. 18-61.
7. -, Modular forms and 1-adic representations, Lecture Notes in Math., vol. 349, SpringerVerlag, 1973, pp. 361-500. 8. , Base change for GL(2), the theory of Saito-Shintani with applications, Institute for Advanced Study, Princeton, N.J., 1975. 9. T. Miyake, On automorphic forms on GL2 and Hecke operators, Ann. of Math. (2) 94 (1971), 174-189. 10. H. Saito, Automorphic forms and algebraic extensions of number fields, Lectures in Mathematics, vol. 8, Kyoto Univ., Kyoto, Japan, 1975. 11. 1. Satake, Theory of spherical functions on reductive algebraic groups over p-adic fields, Publ. Math. No. 18, Inst. Hautes Etudes Sci., 1963. 12. J. Shalika, A theorem on semi-simple p-adic groups, Ann. of Math. (2) 95 (1972), 226-242. 13. T. Shintani, On liftings of holomorphic automorphic forms (a representation theoretic interpretation of the recent work of H. Saito), Informal proceedings (available at Univ. Tokyo) for U.S.-Japan Seminar on Number Theory (Ann Arbor, 1975). 14. , Two remarks on irreducible characters of finite general linear groups, J. Math. Soc. Japan 28 (1976), 396-414. 15. , On irreducible unitary characters of a certain group extension of GL(2, C), J. Math. Soc. Japan 29 (1977), 165-188. UNIVERSITY OF TOKYO
Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 111-113
ORBITAL INTEGRALS AND BASE CHANGE R. KOTTWITZ Let F be a local field of characteristic 0 and let E be either F x . . . x F (1 times) or
xF a cyclic extension of F of degree 1, where I is a prime. Embed F in F x x (9F if E _ diagonally. Let OF be the valuation ring of F, and let OE be OF x F x . . x F, and let it be the valuation ring of E if E is a field. If E is a field, let T = Gal(E/F), and let o be a generator of r. If E = F x x F, let or be the automorphism (x1, , x1) '- (x2, , x1, x1) of E, and let r be the group of automorphisms of E generated by o; again ]' is cyclic of order 1.
Let G = GL2. The action of r on E induces an action of I' on G(E). We use the embedding of F in E to identify G(F) with G(E)r. Define a norm map N: G(E) -+ G(E) by putting Ng = g°`-' g°g This map was introduced by Saito [2]. It depends on the choice of o. LEMMA 1. Let g, x e G(E). Then
(i) N(g oxg) = g 1(Nx)g; (ii) (Nx)° = x(Nx)x 1; (iii) det(Nx) = NE,F(det x);
(iv) 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 o--conjugate if
there exists g e G(E) such that y = g cxg. Statements (i) and (iv) together say that N induces a map from 6-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 o-conjugate to y if and only if (a, x) is conjugate to (a, y) in the semidirect product r x G(E). Choose Haar measures dg and dgE on G(F) and G(E) respectively. If F is nonarchimedean, normalize dg, dgE so that meas(KF) = meas(KE) = 1, where KF = G(VF), KE = G((9E).
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 tP E C°° (G(E)) and f E C°°(G(F)). We say that f is associated to tp
(A)
J
Gr(F)\G(F)
f(glrg) dg =
J
tp(g-°6g) ddE
Gr(F)\G(E)
AMS (MOS) subject classifications (1970). Primary 22E50. © 1979, American Mathematical Society III
R. KOTTWITZ
112
whenever N6 = r and r is a regular element of G(F);
f
(B)
f(g-1 rg) dg = o
Gr(F)\G(F)
for every regular element r of G(F) which is not a norm from G(E). In (A) and (B) dt is a Haar measure on Gr(F). REMARKS. (1) For each regular r e G(F) which is a norm, it is enough to check condition (A) of the definition for only one 5.
(2) For any element S e G(E), let Gj,(E) _ {g e G(E): g 0'og = 5). It is not hard to see that if r = NO is a regular element of G(F), then G,67(E) = Gr(F). For r e G(F), define e(r) to be -1 if r is central and is the norm of an element of G(E) which is not a-conjugate to a central element of G(E), and define e(r) to be 1 otherwise. It can be shown that if f is associated to cp, then (A')
fG
f(g1 rg) dg = e(r)
Gr (F(F)
f
P(g° ag) dt E
G b (E) \G (E)
whenever NJ = r belongs to G(F), where dt' is a Haar measure on Ga(E) which depends only on the Haar measure dt on Gr(F);
=0 J f(g1 rg) dg
(B')
Gr(F)\G(F)
for every element r of G(F) which is not the norm of some element of G(E).
LEMMA 2. (i) Let p e C°°(G(E)). There is at least one f e CC°(G(F)) which is associated to cp.
(ii) Let f e C0(G(F)). Then there exists some (p a CA(G(E)) to which f is associated if and only if fGr(F)\G(F) f(g 1 rg) dg/dt = 0 for every element r of G(F) which is not a norm from G(E) (or equivalently, for every regular element r of G(F) which is not a norm from G(E)).
Assume that F is nonarchimedean, and let WF = .rP(G(F), KF) be the Hecke algebra of complex-valued compactly supported functions on G(F) which are biinvariant under KF. Let .*'E = . (G(E), KE). Saito [2] introduced a C-algebra homomorphism b: -WE -+ YeF which we will now describe. A function f in .F gives rise to a function f v on the set DF of isomorphism classes of unramified irreducible admissible representations of G(F) by putting f v(7r) = Tr z(f) for z e DF. The set DF 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 H f v is a C-algebra isomorphism, called the Satake isomorphism, ofF with the algebra of regular functions on the variety DF. This discussion applies to E as well; if E is a field, then DE is again x F, then DE is DF X isomorphic to C* x C* divided by S2, and if E is F x X DF.
There is a map of algebraic varieties from DF to DE; if E = F x x F the map is ir I-+ ,c Ox . ©7r, and if E is a field it is z(z) F--+ n(Res WE z) where z(r) is the unramified representation of G(F) corresponding to the unramified representation z of the Weil group WF of F, and n (Res WE z) is the unramified representation of
113
ORBITAL INTEGRALS AND BASE CHANGE
G(E) corresponding to the restriction of a to the Weil group WE of E. So we get a
C-algebra homomorphism from the algebra of regular functions on DE to the algebra of regular functions on DF, and hence also a C-algebra homomorphism b:
E-
F
REMARK. If E = F x _ A* ... *fl -
x F, then
E
CFO
x®
F, and b(f1 O ... (Df1)
LEMMA 3. If F is nonarchimedean and E is unramified over F (E = F x allowed), then b(tp) is associated to tp for all tp e YeE.
x F is
x F. It was first proved by Saito
This lemma is easy to prove for E = F x
[2] when E is a field; it was subsequently proved by Langlands [1] using the buildings 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 2F on G(F) whose logarithm is the weight factor in these integrals. Let g e G(F) and write g = a(o 1)k with a e A(F) and k e KF. Then AF(g) = 1 if x E OF and 2F(g) = IxI-2 otherwise. The function 2E 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 unramed extension field of F. Then for any 5 E A(E) such that No is regular, and for any tp E YeE,
lf
f
b(co)(g-I(N8)g)log AF(g) da =
q (g°Og)log 2E(g)gE
A(F)\G(E)
A(F)\G(F)
where da is any Haar measure on A(F).
This is proved in §3 of [1]. REFERENCES
1. R. P. Langlands, Base change for GL(2), Notes, Institute for Advanced Study, 1975. 2. H. Saito, Automorphic forms and algebraic extensions of number fields, Lectures in Mathematics, Kyoto University, 1975. , Automorphic forms and algebraic extensions of number fields, U.S.-Japan Seminar 3. on Number Theory (Ann Arbor, 1975). 4. T. Shintani, On liftings of holomorphic automorphic forms, U.S.-Japan Seminar on Number Theory (Ann Arbor, 1975) (see these PROCEEDINGS, part 2, pp. 97-110). RICHLAND, WASHINGTON
Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 115-133
THE SOLUTION OF A BASE CHANGE PROBLEM FOR GL(2) (FOLLOWING LANGLANDS, SAITO, SHINTANI)
P. GERARDIN AND J. P. LABESSE These Notes present a survey of the results on the lifting of automorphic representations of GL(2) with respect to a cyclic extension of prime degree of the groundfield, and of some of its applications to the Artin conjecture, with some sketches of
proofs. §§1-5 are devoted to the definitions and results on the lifting, §6 to the proof of the Artin conjecture in the tetrahedral case. The first part ends up with three appendices describing respectively two-dimensional representations of the Weil group (Appendix A), representations of GL(2) over a local field (Appendix B) and a global field (Appendix Q. Part II gives some indications on the proof of the results on lifting. The main tools are the orbital and twisted orbital integrals, and a twisted trace formula [Sa]. The main references for the lifting are [Sa], [S-1], [S-2], [L]. As a side remark, we would like to point out that the study of the example of the general linear group over a finite field [S-2] is illuminating. 1. DEFINITIONS, THEOREMS, APPLICATIONS
1. Notation. Let F be a local or global field. Then WF is the Weil group of F and, for F a p-field, WF is the Weil-Deligne group of F [T]. We recall that there exists a canonical surjective homomorphism WF -* CF which identifies WF with CF.
In all these notes, E is a Galois extension of F, cyclic of prime degree 1, and r x F 1-times and r is generated by a : (x1, x2, , x1) -' (x2i , xt, x1) is handled easily and is left to the
its Galois group (the "split case", where E = F x F x
reader). For F local, choose a nontrivial character 0 of the additive group of F, and define ybE,F = gb o TrE/F
In the following we shall use systematically the notation given by Borel [B] , and by Tate [T] for Weil groups, the L- and 6-factors.
about L-groups : O(G),
2. Base change for GL(l). From abelian class-field theory, there is a commutative diagram
1 ' WE -* WF I
1 -- CE ° --- CE
1'
1 z
NE/F
CF ---. r
I
AMS (MOS) subject classifications (1970). Primary 10D15, 12A70; Secondary 20E50, 20E55. © 1979, American Mathematical Society 115
P. GERARDIN AND J. P. LABESSE
116
where u is a generator of F. The one-dimensional representations of WF are given by the quasi-characters of WP = CF; by composing a quasi-character of CF with the norm, a map is defined
called the lifting (or more precisely, the base change lift): x H xE/F = x ° NE/F which sends the set d(F) of quasi-characters of CF in the set SI(E) of quasicharacters of CE. The group Facts on the set of quasi-characters of CE by: (re)(Z) = O(Zr),
T E F, Z E CE;
the group f of characters of r can be identified with the set of characters of CF which are trivial on NE/FCE; this group f acts on the set of quasi-characters of CF
by multiplication: x H XC, C e F. Then the exactness of the second line of the above diagram implies the following result: PROPOSITION 1. The lifting x '-' xE/F defines a bijection from the orbits of f in d(F) onto the invariant elements by 1' in d(E); moreover, the following relations hold:
if L(xb),
L(XE/F) E(XE/F)
for /F global,
_ {I .
/
e(XE/F, cbE/F)
AE/F(yb)-1
Fl- E(xb, 0) for F local.
cer 3. Base change for GL(2) on the L-groups.
3.1. Let G be the group GL(2) over F, and GE/F be the group over F defined by restriction of scalars of the group GL(2) over E, so that GE/F(F) = GL(2, E); the group GE/F is quasi-split, and there is a natural map:
LGE/F = GL(2, C)r ) FF;
LG = GL(2, C) X "F
here GL(2, C)r is the set of applications from Fin GL(2, C), and FF = Gal(F/F) acts by permutations of the coordinates [B, §5]. Given any p e 0(G), its restriction to WW defines an element PE/F E O(GE/F); the set O(GE/ F) can be identified with the set 0(G/E) [B], and the above map defines the application
0(G) - O(GE/F) = 0(G/E), p
PEIF
called the base change.
For F local, let FrF be a Frobenius element in FF; when E is unramified, then FrE = FrF e FE is a Frobenius element for E. Moreover, if p is unramified and defined by FrF H s, where s is a semisimple element in GL(2, C), then PE/F is s'. unramified and is given by FrE The properties of L- /and E-factors with respect to induction [T] show that:
L(pE/F) = II. cErL(p (D 0), for Fgl''obal,
E(pE/F) = IT e(p -'I/PE/F, OE/F)
=
AE/F(yb)-Z H E(p
zEr
©S, yb), for F local.
BASE CHANGE PROBLEM FOR GL(2)
117
3.2. From the classification of the two-dimensional admissible representations of WW (see Appendix A), one has the following result: PROPOSITION 2. (a) The lifting p e 0(G) I -. PE/F e O(GE/F) has for image the set of I'-invariants in O(GE/F) (b).In the following cases, the lifting is given by
(,U (D v)E/F =/E/F(BvE/F, /find WK O)E/F = Ind WKEOKE/K for E 0 K, (Ind WK O)E/F = 0 +0 °0
for E = K, 1 # a e I',
(x Ox sp(2))E/F = xE/F (9sp(2) (for F nonarchimedean). (c) Given a nondecomposablep e 0(G), the representations which have the same lifting are the p 0 C for all C E F; for p = 2 ) u, the representations which have the same lifting are the A O uC' for all C, c' e F.
4. Base change over a local field. In this section, we denote by Or a generator of
F = Gal(E/F). 4.1. Let 17(G) be the set of classes of admissible irreducible representations of G(F). There is a conjectural bijection 0(G) < 17(G) [B]. The base change map 1l(GE/F). The definition of base change for representations of G(F) will be given in 4.3; since the image of 0(G) is the set of F-invariant elements in O(GE/F), one studies first the admissible irreducible representations of G(E) equivalent to their conjugates by F. 4.2. Let Tr be an admissible irreducible representation of G(E) such that at ^-, i-c; 0(G) ---> O(GE/F) must reflect a map II(G)
then there exists an operator C on the space of Tr such that C-Iit(z)C = k(z°), z e G(E), and C' = Id. This operator is determined up to an Ith root of unity. The mapping Tc' : (6m, z) H Cmk(z) defines an extension k' of k to the semidirect pro-
duct r a G(E). PROPOSITION 3. This representation has a character given by a locally integrable
function Tr t' on r x G(E). On o x G(E), the character Tr k' defines a or-invariant distribution on G(E), i.e., invariant under u-conjugation: z H y °zy, y, z e G(E). Let us state some properties of the a-conjugation. For z e G(E), put NE/F.oZ = Za'-' ...Zo
Zi
or simply N(z) if no confusion can arise. PROPOSITION 4. (a) NE/F,CZ is conjugate in G(E) to an element of G(F);
(b) z H NE/F,Cz defines an injection of the set of u-conjugacy classes of G(E) into the set of conjugacy classes of G(F);
(c) the elliptic classes of G(F) obtained by NE/F,, are those with determinant in NE/FE" ; 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 ' e 11(G) and k
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P. GERARDIN AND J. P. LABESSE
be an irreducible admissible representation of G(E) which is equivalent to its conjugate by a. Then Tr is called a base change lift of 7r, or a lifting of 7r, if either (a) 7C = 7r(,a, v) and k = 7CQ-tE1F, vE/F), or
(b) there exists an extension k' of fr to I' x G(E) such that Tr rr'(o x z) _ Tr7r(x) for z E G(E) whenever NE/F,, z is conjugate in G(E) to a regular semisimple element x e G(F). Some of the notation in the following theorem is explained in Appendix B. THEOREM I (BASE CHANGE FOR GL(2) OVER A LOCAL FIELD). (a) Any 7C E 1(G) has a unique lifting 7CE/F E II(GE/F), and any 7r r= 17(G) fixed by I' is a lifting;
the lifting is independent of the choice of the generator o of ?; (C) ivE/F = ZEIF G 7r' = 7r O b for a b E 1', or 7C = 7r(a, v), 7C' = 7r(/b', V) with /(b)
Pip' and v-1 v' in 1'; (d) WRE/F = ((O)E/F, Or O x)E/F = ICE/F O xE/F for any one-dimensional representation x of F", (7rE/F)V = (70E/F (contragredient representations);
(e) for E
F D k with E and F Galois over k, t(7CE/F) = (r7r)E/F for any
e Gal(E/k), with image r in Gal(F/k); (f) at least for p e i(G) not exceptional, 7r(p)E/F = lr(pE/F) 5. Global base change. 5.1. Let 17(G) be the set of classes of irreducible admissible automorphic representations 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 L-groups should reflect a map from 17(G) to 17(GE/F), the set of irreducible admissible automorphic representations of G(AE) = GL(2, AE) = GE/F(AF); such a map must be compatible with the local data. For any place v of F, put Ev = E OF Fv; it is a cyclic Galois extension of F or a product of 1 copies of F; in this latter case, define the lifting 7rE,/F of 7r E 17(G) by
7t O "' O 7r (1 times). 5.2. Definition of the global base change for GL(2). Let 7r E 17(G), k e 17(GE/F); then k is called a lifting of 7r (or more precisely a base change lift of 7r) if, for every place v of F, itv is the lifting of 7rv.
5.3. The notations used in the following theorem are those of Appendix C. THEOREM 2 (GLOBAL BASE CHANGE FOR GL(2)). (a) Every 7r e II(G) has a unique lifting 7rE/F E H(GE/F),
(b) a cuspidal e 1I(GE/F) is a lifting if and only if it is fixed by r, and then, it is a lifting of cuspidal representations; a cuspidal 7t r= 1I(G) has a lifting which is cuspidal except for l = 2 and 7r = 7r(Ind wE 0) and then 7rE/F = 70, co); (c) for a cuspidal it e 17(G), the representations 7r', which have ZEIF for lifting are
the 7r' =7r xQCwith 1 EI; (d)
W,tE/F = (Wa)E/F (7C O x)E/F = /irE/F O ZEIF
(iE/F)V = /*)E/F
(e) for E D F
(central quasi-characters), (twisting by a quasi-character), (contragredient representations);
k with E and F Galois over k, r(7rE/F) = (ri)E/F for any r E
Gal(F/k) image of r e Gal(E/k) ;
BASE CHANGE PROBLEM FOR GL(2)
119
(f) if 'r = 2r(p) for some p e g5(G), then irE/F = n(pE/F).
REMARK. There are examples of noncuspidal k e II(GE/F), fixed by r which are not liftings (cf. [L, §10]).
6. Artin conjecture for tetrahedral type. Let p be a two-dimensional admissible representation of the Weil group of the number field F; we assume that its image modulo the center: WF -p GL(2, C) -- PGL(2, C) is the tetrahedral group 2C4. This group is solvable: I -> D4 -i 24 - C3 --> 1. The action of the cyclic group C3
on the dihedral group D4-the so-called mattress group-is given by the cyclic permutations of its nontrivial elements. The inverse image of D4 in WF is a normal subgroup of index 3, hence is the Weil group WE of a cubic Galois extension E of F:
1 --> WE ---> WF -> Gal(E/F) --> I
I - D4 ---
W4 -> C3
The restriction PE/F of p to WE has for image the dihedral group D4; it is induced from a one-dimensional representation of a subgroup of index 2 in WE, so that there is a corresponding cuspidal automorphic representation 2r(pE/F) of GL(2, AE). The inner automorphisms of %4 give an action of C3 on D4, and the action of 1' _ Gal(E/F) fixes the class of PE/F, hence also the class of 7r(pE/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 lr(pE/F), which is equal to det PE/F = (det p)E/F; there is only one of them, say 1r, with, the central character det p. The Artin conjecture for the representation p is the holomorphy of the corresponding L-function: s F- L(s, p). According to Jacquet-Langlands [J-L, Theorem 11.1, p. 350], the L-function s H L(s, ir) corresponding to cuspidal it is holomorphic ; hence, the Artin conjecture will be proved for p if we show the equality of these two L-functions. From [J-L, pp. 404-407], this will be done if the following assertion is shown to be true: (co) i = lr(pv) 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 wjv, we have the equations : Ew = Fv, (pE/F)w = pv,
kZ(PE/F))w - ir(pv) = Zv,
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
PEWFv of Pv to WE., and the representation (IrE/F)v = 7r(pEv,Fv) is unramified; since Ev/Fv is unramified this representation is the base change lift of unramified representations. This shows that icv is unramified; call
p,tv a two-dimensional representation of WE,, such that irv = ir(p,,) We have shown that our assertion (co) is equivalent to : (c) if v does not split and is unramified for p and E, pv and p,v are equivalent. The adjoint representation of PGL(2) defines an injection of PGL(2) in GL(3),
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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: (c2) if v is unramified for p and E, the three-dimensional representations Apv and Ap,, are equivalent. In fact, call a (resp. b) the image of a Frobenius in WF, through pv (resp. p,);
if (c2) is satisfied, a e Cxb; but det a = det b, hence a = ±b. If a = -b then, since pv and p.v have the same restriction to WE,,, a3 is conjugate to -a3, 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 Apv is in the tetrahedral group which has no element of order 6; so we have a = b, hence pv = p,,,. The introduction of Ap is motivated by the crucial observation, due to Serre, that this three-dimensional representation is induced by a one-dimensional representation of WE; in fact, the tetrahedral group leaves invariant the set of the three lines joining the middles of the opposite edges of the tetrahedron. This means that Ap is induced by the one-dimensional represen-
tation 0 of the stabilizer of one of these lines (obtained by restriction of Ap); but this stabilizer is the pull-back of the dihedral group D4 c W4 in WF, which is the subgroup WE:
Ap = Ind
0. WE
From [J-PS-S], to such a three-dimensional irreducible monomial representation Ap of WF is associated an irreducible cuspidal automorphic representation ir(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 Air corresponding to it is cuspidal [G-J-2]. To prove(C2)it suffices to show the condition: (c3) the lifting Air is equivalent to 7r(Ap). There is a practical criterion given by [J-S] to prove the equivalence of such repre-
sentations: irI and ire are equivalent if and only if L(s, ir1 x 9r2) has a pole at s = 1, where L is the L-function attached to the representation of GL3(C) x GL3(C) in GL9(C) given by the tensor product.
We shall prove that almost all local factors of L(s, Air x ir(Ap)v) and of L(s, 7r(Ap) x ir(Ap)") are equal; by nonvanishing properties of local factors [J-S], this is enough to prove that L(s, Air x ir(Ap)v) has a pole at s = 1, and hence(C3)will be proved. If v is split, then irv = 9r(pv) and, at least when iv and pv are unramified, (A7r)v = ir(Apv): the local L-factors are then equal. If v does not split in E and is unramified for E and p, the two local L-functions
are those associated to the nine-dimensional representation of WF given by Ap,,, O ((Ap)v) and Apv Ox ((Ap)v ); but we know that Apv = Ind WEv 0v.
Now if U (resp. V) are representations of a group G (resp. a subgroup H) one has U Qx IndH V = Ind H(V (D Res HG U); also recall that the two representations p, and pv have equivalent restrictions to WE,,; hence Ap, Ox (Ap)v is equivalent to (Apv) Qx (Ap)v so that they have the same L-factor.
This concludes the proof of the Artin conjecture for the tetrahedral case. The above proof is taken from a letter of Langlands to Serre (December 1975);
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 (a4 has a normal subgroup W4 of index 2; hence there is a quadratic extension E of F for which the restriction PE/F is of type X14. By the above theorem, 7r(pE/F) exists and, by Theorem 2,
it is the base change lift of two cuspidal admissible irreducible automorphic representations z' and r" 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 (o _1); in such a case the components of 2c' and yr" at the real place verify z'-
z' = 7r(p ,) with p_ = 1 O+ sign; then ir' and '" correspond to holomorphic automorphic forms of weight one.
This situation has been studied by Deligne-Serre [D-S], and they show that z' = z(p') and jr" = z(p") for some representations p', p" of WF in GL(2, C); now, by Theorem 2, Jr(p')E/F ='(p'E/F), and the same is true for 2r"; this shows that p' and p" are the two representations which lift to PE/F; hence either p = p' or p". Thus one concludes that either 'r' = 2r(p) or 2r" = 2r(p), and this gives THEOREM 4. For a two-dimensional representation of WQ which is of octahedral type
and which sends the complex conjugation on (a _°), and such that the above quadratic field E is real, the Artin conjecture is satisfied.
Appendix A. List of the two-dimensional admissible representations of WW [D].
Notations. F is a local (resp. global) field, CF is Fx (resp. the group of ideles classes of F); WF is the Weil-Deligne group of F [T].
For F global, v a place of F, there is an injection WF, -> WF which defines an application p H pv from the two-dimensional admissible representations of WF into
those of WF,; if another representation p' of WF satisfies pv - pv for all but a finite number of places, then p' is equivalent to p, and pv - pv for all v. The twodimensional 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: p (1 v is the sum of the two one-dimensional representations of WF defined by p and v; p +D v - v +D u; det(p +D v) = pv; (p +D v) Dx x = 1 D+ v-1. The L- and e-functions verify: (px) D+ (vx); (p D+ )v = L(p D+ v) = L(p)L(v),
e(p +D v) = s(p)e(v) (ifFis global),
e(p +D v, 0) = e(p, ci)e(v, 0) p +D v = ®x v (pv D+
(if F is local), (if F is global).
(2) Dihedral type: r = IndWK 0, where 0 is a quasi-character of CK, and K a separable quadratic extension of F; r is irreducible if and only if 0 0 cO (1 0 or e Gal(K/F)). For 0 = °B, let x be a quasi-character of CF such that 0 = x . NK/F and let 6 be the character of CF with Kernel NK/FCK. Then IndWK 0 = Ind%0-1; L(r) _ xQ x6; det z = 6 BIcF; Z O Y =IndWK (Ox o Nx/F);
L(B), e(v) = s(B) (F global), Or, 0) = 2K/F(ci)e(O, 0 o TrK/F) (F local); for F global, r = (Dr, with rv = IndWK 0, for K,/F, quadratic, and 'r, = By D+ By either K1 = K2 and 01, 02 conjugate by Gal(K/F), or K1 0 K2, 01"101 -1 and 02"'2021 are of order 2
for Kv = F x Fv. The equivalences are: IndW0 01 - IndWK 02 and B1 o NKIK21KI = 02 o NKIK2/K2.
(3) Exceptional type: the image of the inertia group in PGL(2, C) is W4 (tetrahedral type), C54 (octahedral type), or W5 (icosahedral type); they occur only for F
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P. GERARDIN AND J. P. LABESSE
global or F nonarchimedean local of even residual characteristic; in this latter case, the icosahedral type does not occur.
(4) Special type (occurs only for F nonarchimedean local): x p sp(2) for a quasi-character x of Fx and sp(2) the representation of WW defined by
z e C '-- (0
1)
and we WF H
(I WI
0)
Appendix B. List of admissible irreducible representations of GL(2, F), F local field [J-L].
Notations. G = GL(2, F), I I is the absolute value defined by the dilatation of the Haar measure on F: d(ax) = jal dx, and 0 is a nontrivial character of the additive group of F. Representations. (1) Principal series p(,u, v), where 1u, v are quasi-characters of Fx.
(a) Definition. Let p(,u, v) be the representation of G by right translations in the space of smooth functions for G such that f(ang) = u(u)v(v) IuV 1I1/2 f(g)
for any g e G, a = (o °) a G, n e (I
1).
When this representation is irreducible,
then z(u, v) is p(u, v). When p(u, v) is reducible, there are exactly two irreducible subquotients; one is finite dimensional and z(u, v) is this one. The other one is denoted a(u, v). (b) Equivalences. 2r(u, v) - lr(v, 1u). For F = C, any irreducible admissible representation of G is equivalent to a z(,u, v). (c) Finite dimensional representations. F nonarchimedean: they are one-dimensional, and are the 2r(u, v) for uv 1 = I+1: z(,u, v) (x) _ Idet xj+liuv (det x), x e G: the corresponding representations a(u, v) are called the special representations; F = R: They are the z(u, v) with,w) 1(a) = jaI±(n+1> . sign(a)n for integers n 0; F = C: they are the z(u, v) with uv 1(a) = [an+1(a)m+1]±i for integers n 0, m >_ 0.
(d) Other properties. PI); Restriction to the center: twisting by a quasi-character of Fx: z(u, v) Qx x = z(ux, vx); contragredient representation: z(u, v)v - z(u-1, v-1); local factors: L(r(u, v)) = L(u)L(v), -(z (,u, v), 0) = e(u, ci)e(v, 0). (2) Weil representations. ma(r), v = IndWX 0, with 0 a quasi-character of Kx,
and K a separable quadratic extension of F.
(a) Definition. Let GK be the subgroup of index two in G defined by those elements which have a norm of Kx for determinant. Fix a nontrivial character 0 of the additive group of F. Then 2v(r) is the class of the representation of G induced by the following representation r(0, 0) of GK, in the space of smooth functions f on K" such that f(t-1 x) = 0(t)f(x) fort, x e Kx, NK/F t = 1: (r(0, 0) (0
1)f)(x) =
Sb(uNK/Fx)f(x),
1)f)(x) = dK/F(Sb) (01
J
u e F,
Kxf(Y)0K1F(xY°)d0Y,
BASE CHANGE PROBLEM FOR GL(2)
(r(0, 0)
(U
123
?)f)(x) = 0(b)f(bx) for a = NK/Fb, b e Kx,
where cbK,F = 0 o TrK,F, y°is the conjugate of y by the nontrivial element 6 of Gal(K/F), day is the self-dual Haar measure on K with respect to the character cbK/F and AK,F(ci) is the unitary part of the local factor e(&, 0), where & is the nontrivial character of F" which is trivial on NK/FKx
(b) Equivalences. (1) 7r(IndWK 0) = v(IndWK 10);
(2) 2c(IndWK 0) ^- 2(x, Xe) if 0 = 010 and x is a quasi-character of Fx such that 0(a) = x(NK,F a); (3) Other equivalences: ir(IndWKI 01) = 7r(IndWK2 02) for K1 # K2, if and only if 1 and 02 2021 are of order 2 and satisfy 01 o NKIK2/K1 = 02 0 NK1KZ/K2 (c) Characterization. If a nontrivial character x of Fx fixes a class 7c of irreducible admissible representations of G: Qx x = 7r, then x is of order 2, attached to a
0,1'10
separable quadratic extension K of F and is = 2c(r) where z = IndWK 0 for a suitable 0, and conversely (cf. [L, §5]).
(d) Other properties. For F = R, or nonarchimedean with odd residual characteristic, any irreducible admissible representation is a 7r(A, u) of a 7r(IndWK 0); restriction to the center: con(z) = v OIFx; twisting by a quasi-character of Fx : z(a) OO x = 2c(IndWK 0 . Z . NK/F), contragredient representation: z(i) = rc(IndjK 0-1), local factors: L(,r(z)) = L(0), e(7r(z), ¢) _ AK/F(&(O, SbK/F)
(3) Exceptional representations. They occur only for F nonarchimedean of residual characteristic 2, and, up to twisting by quasi-characters of Fx, their number is 4(121-2 - 1)/3 for F of characteristic 0, infinite for F of characteristic 2. They are supercuspidal (see complements below) (cf. [Tu]). (4) Special representations. They occur only for F nonarchimedean, and are the infinite dimensional subquotient 6(u, v) of the reducible p(u, v), that is for ,UV-1 = a'(v, u), 6(u, v) = xc(I 11/2, 11-1 /2) for x = ul 1-1i2. 1+1; one has o-(a, v) I
Complements.
(1) Representations oo(u, v). When the induced representation p(,u, v) is not irreducible, 6(u, v) denotes any representation equivalent to the unique infinite dimensional subquotient of p(u, v) : for F = R, the representations 6(,u, v) are the representations ir(Ind cx 0), 0 10. (2) For a two-dimensional admissible representation p of WF, there is at most one irreducible admissible representation r of G often denoted 2c(p) when it exists such that co., = det p, L(n 0 x) = L(p (& x) and e(rc (9 x, 0) = e(p (& x, 0) for any quasi-character x of Fx; for p reducible, p = p v, then 7r = z(u, v); for p = x Qx sp(2) then 7c = a-(u, v) with ,u = xI 11/2, v = x 1 1-1/2; for p dihedral p = IndWK 0, then 'r = z(IndWK 0); in the remaining cases, that is when p is exceptional, the existence of v is still not completly settled, but base change techni-
ques were used to prove it in many instances. Conversely, any i should be a 2r(p)
Appendix C. Irreducible admissible automorphic representations of GL(2) [J-L].
Notations. F is a global field, AF its ring of adeles, CF = AFIFx the group of ideles classes, I I the absolute value on AF.
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P. GERARDIN AND J. P. LABESSE
(1) Noncuspidal representations.
(1.1) z(A, u) for ,u Grossencharakters of F: they are the following representations: ir(2, u) = Qx, r(A,, u,,); the one-dimensional representations are the 7r(2, u) for 2p-1 = I 1±1. (1.2) Any noncuspidal irreducible admissible automorphic representation 7r of GL(2, AF) has the following form: there are two Grossencharakters A and 'U of F, and a finite set S of places of F, such that the components irv of ir are given by Zv = r(Av, ji0),
V
S,
rv = 6(Av, i ),
V E S,
denotes the infinite dimensional subquotient of the reducible where o(Av, p(Av,,uv) (Appendix B). (2) Cuspidal representations (examples). (2.1) z(r) with r = IndwK 0 for a separable quadratic extension K of F and a Grossencharakter 0 of K, not fixed under Gal(K/F), is the representation
z(v) = O ir(Indj
0,,)
with IndwKv 9v = 0, O+ 9v for Kv = Fv x F,,.
Properties. (1) Let 2T e 17(G); in order that there exist a nontrivial Grossencharakter X of F such that r 0 x = z, it is necessary and sufficient that there exist a separable quadratic extension E of F and a Grossencharakter 0 of E such that (a) x is the character of CF with Kernel NE/FCE; (b) ir = ir(IndwE 0) (in particular r = 7r(r, rx) if 0 = °9, where r is a Grossencharakter of F which has 0 for lifting to E); either K1 = K2 and 02 = 01 or °91, or K1 :A (2) z(IndwF 91) = 2r(IndWF 02) K2 then 91 °' Oj 1 and 02 °921 are of order 2, and (91) o NK1K21K, = (02) ° NK,K2/KZ.
(3.1) More generally let p be a two-dimensional admissible representation of WF; we say that 7r = (D z, is ir(p) if 2Tv _- 9r(pv) for all v. The existence of such r when p is irreducible is related to the Artin conjecture for the p x0 x, where x is any quasi-character of CF [J-L, §12].
(3.2) Of course there are many other, more complicated, types of cuspidal representations : think of the classical d for example. II. BASE CHANGE FOR GL2, A SKETCH OF THE PROOF.
1. The trace formula. In all the following we shall use notations close to those of [G-J-1] in these PROCEEDINGS.
Let F be a number field, E a cyclic extension of prime degree, put Z1 = NEIF Z(AE), where AE denotes the ring of adeles of E, and Z1(F) = Z1 (1 Z(F); as usual Z is the center of GL2 = G and is identified with the multiplicative group. Since E/F is cyclic one has Z1(F) = NE,FZ(E). Choose a character w of ZI/Z1(F) and consider the space LZ(Z1 G(F)\G(A), co) = L2 of functions on G(F)\G(A), which transform on Z1 according to co: co(zTg) = w(z),p(g),
z e Z1, 7 e G(F'),
and square-integrable on Z1 G(F)\G(A). In such a situation, which is slightly more general than the one studied in [G-J-1]
BASE CHANGE PROBLEM FOR GL(2)
125
(where E = F), one defines in an obvious way the spaces Lo and L L. The restriction of the natural representation of G(A) in L2 to Lo Q+ L p will be denoted by r. If f e c w-1), the space of smooth functions on G(A) compactly supported modulo Z1 which transform according to c o-' on Z1, the operator r(f) is of trace class. The Haar measures being chosen as in [G-J-1, §§6-7] we assume moreover that vol(Z, Z(F)\Z(A)) = 1. Then tr r(f) is the sum of the expressions (i)-(vii) below (we assume that f is a tensor product of local functions fv) (cf. [L, §8]). °(Z1\G(A),
vol(Zl G(F)\G(A)) f(z),
E
(i)
zeZ1 (F) \Z (F)
(ii)
E e(r) vol(Zl
1G,(A)\G(A) f(g 1r g) dg
rE
where & is a set of representatives of the conjugacy classes of elliptic elements (i.e., whose eigenvalues are not in F) taken modulo Z,(F), and e(r) is 1 (resp. 1) if the equation 6-178 = zr has (resp. has not) a solution in z r= Z,(F) - {1}.
-4
(iii)
n
where D° is the set of pairs V = (u, v) of characters of Ax/Fx such that tw induces w on Z1, where M(71) and 7r, are defined in [G-J-1, §4], and with 7r,(f) = f zl\G(A)f(g)2r (g) dg. A Haar measure d77 on D° is defined as in [G-J-1, §7-D] by considering D° as a union of homogeneous spaces under the group of characters of Ax /F< (with the dual Haar measure), acting by x (u, v) = (xu, X-10. This allows us to write the fourth term : (iv)
SDO m 1(i7)
.
m'(7))tr(ir (f))
2
the derivative m' being computed as in [G-J-1, §7-D]. (v)
E
1 'of
zEZ1(F)AZ(F)
v
f
fv(g 1znog) dg
(no
l
L(l, lv)
= (1
0
111
l/1
where A0 and other notations are defined in [G-J-I, §7-B]. The remaining terms will
not be written as in [G-J-1] since they are there expressed by noninvariant local distributions. For example the local distribution
1 Aft, fv)
a (a - b)xv) log Ixvl dxv
4(r) f ixvi>1 f K(0
where r = (o ab°), J ,(r) = 1(a - b)2/ab l ,111 and f K(g) = SK u fv(k-1gk) A, can be
written A2(7, fv) + A3(r, fv) where AZ(r, fv) is invariant, and A3 fits with other terms to provide an invariant expression. More precisely one takes for a nonarchimedean place v:
A2(r,fv) = logl(a - b)/al vF(r,fv) + dv(r)fa (z) f Ixvl>1 log Ixvl dxv- l alb loglwvl - f
f(gznogl) dg
a (Z
0
= (0 a))'
11/2.16)v I
P. GERARDIN AND J. P. LABESSE
126
where F(r, fv) = dv(r) JAv\G, fv(S Irg) dg and co, is a uniformizing parameter for Fv.
If v is archimedean one takes
A2(r,fv) = log
1-b F(r,fv) a
L'(1,1v)2
L (1, 1v)
SZN\G v fv(
og) dg.
This yields the term
E
(vi)
rEZl\A(F);rEZ(F)
12-1 Ei A2(r,fv) 11 F(r,fw) w4 v
v
It can be checked that r '- A3(r, fv) extends to a continuous map on A(Fv) and one sees that the terms 6.34 in [G-J-1] minus our (v) plus 6.35 minus our (vi) yield the term
E - 2-11 E vA3(r,fv) Hw#vF(r,fw)
rEZ,\A(F)
This can in turn be transformed by a kind of Poisson summation formula to ITO
TI tr z w(fw) dry. - E vBI(iv, fv) w*v
One has to add the term 6.36 of [G-J-1] minus our (iv) to get the final term: 2
(vii)
J D°
E B(rlv, fv) F1 tr xr/w(fw) d7) w*v
v
2. The twisted trace formula. Here we shall use definitions and results of the Kottwitz lecture (see [K] in these PROCEEDINGS). As above we follow closely [L, §8]. Let L2(Z(AE)\G(AE), w) = L2 where Co = (o. NE/F. The Galois group r = Gal(E/F) acts on L2 by Up(x) = p(x°) for p e L2, x e G(AE) and a e I'. Let Rd denote the restriction of the natural representation of G(AE) in L2 to the discrete spectrum Lo Q+ L2P. The projection commutes with the action of the Galois
group; hence Rd can be extended to a representation Rd of r x G(AE). Let 0 E c-(Z(AE)\G(AE), C o-'); then Ra(c5) is of trace class and Ra(o) is unitary for a e T. The operator RA(s) can be represented by a kernel K(0, x, y) and then the operator Rd(c)Rd(O) is represented by the kernel K(0, x°, y) and tr(Rd(u)Ra(c)) =
K(0, x°, x) dx. JZ (AE)\G(AE)
Assuming, as usual, that 0 is a tensor product: 0 = Q 0v, one can proceed as in [G-J-1] to compute this integral; if a # 1 it is the sum of the following terms vol(Z(A)Ga(E)\G°s(AE))
(1)
J Z(AE)G,,(AE)\G(AE)
ft °og) dg
where the sum runs over the a-conjugacy classes of elements 6 such that N(S) is central, and Gs is the u-centralizer of S (cf. [K]). (2)
E E(6)vol(Z(A) Ga(E)\Ga(AE)) fZ(AE)G3'(AE)\G(AE) 0(g
deg"
dg) dg
127
BASE CHANGE PROBLEM FOR GL(2)
where 9o is a set of representatives of the Q-conjugacy classes that are not a-conjugate to a triangular matrix, taken modulo Z(E) in G(E), and e(b) is z or 1 according as the equation v °Jz = z3 has or not a solution in Z(E) with z 0 Z(E)1 (3)
4 aE
tr(ME(°V)z,1(6)7r,(O))
where fv = (v, ,u) if 7) = (u, v) is a pair of characters of AE /E" with UV = Co. The representation icr is realized in a space of functions on G(AE), the action of a e I' defines an operator 2c,(oa) from the space of it to the space of 2can; then ME(0'7)) intertwines man and row but o7)V = V. Hence the product M(a72) 7r1(a)7r,)(O) is a welldefined operator in the space of 2V , and the above expression is meaningful. 212 J EDO m61( )mE(YJ) tr(2r (a)1rn(q5)) d71
(4)
where
_ (u o NE/F, v o NE/F) if 72 _ (u, v). There are 12 elements giving rise to
the same . The reader should be aware that our notation V has not the same meaning as in [L]. 0°(0, ov),
20 11
(5)
v
where 001(0, 0v) = L(1, lv)-1 IS I 0v(k-6t-0n anontk)t-2P do dt dk, with k e Kv, the standard maximal compact subgroup of GL2(E ® Fv),
=(0
l
b/'
t
t -2p =
-1
b
and no = (1
z 1)
such that trE/F z = 1. The integration is on Kv x Z(Ev)\A(Ev) x N(Fv)\N(Ev), where Ev = E OO Fv. A2(O, O v)
(6)
77 11v
Fc (o, 0w),
8
where .
= {3 e A1-0'(E)Z(E)\A(E)IN(o) 0 Z(F)},
Fo'(o, Ow) _'w(r)
JZ
\G
0'g) dg
and r = N(o). An explicit definition of A2(a, (b) will not be given here; we shall simply say that A2(5, 0v) = lA2(r, fv) if fv is associated to cv under the base change correspondence (see [K]). As above the remaining term can be written (7)
I2
J
Eon
E
cv) H tr
For a definition and a detailed study of distributions A2 and B° the reader is referred to [L, §7] (where the subscript u is omitted) and to [K, Lemma 4].
3. The comparison. We assume now on that the function f = &x fv e W°(Z1\G(A),w 1) and the function
= (30, e le0(Z(AE)\G(AE), co) are such
that ov and fv are "associated" in the sense defined in [K]; we consider -91 = 1 tr(Rd(a)Rd(q5)) - tr r(f ).
128
P. GERARDIN AND J. P. LABESSE
PROPOSITION 1.
l
B(y1 v, fv) 1
77 11
tr 7[,w(J{ w) dr
31, z
where
ifl=2, =0 if1 2.
51,2=1
The proof amounts to the comparison term by term of the expressions for tr(r(f)) and 1 tr(Rd(o,)Rd(q5)).
For example to prove that 1 (1) = (i) note that we work there with a sum over elements a e G(E) such that N(8) is central in G(F); hence Go' E) is the set of Fpoints of a twisted inner form of G. Since we use Tamagawa measures, I vol(Z(A)Ga(E)\GS(AE)) = 1 vol(Z(A)G(F)\G(A)) = vol(Z1G(F)\G(A));
the expressions to be compared are products of the local analogues, and now, using properties A' and B' in [K] for associated functions and the fact that the number of places where a minus sign occurs is even, we obtain the desired results. To prove I (2) = (ii) is even simpler since in that case G"(E) = Gr(F), where r = N(S): the twisting is trivial since G. is abelian. To compare /. (3) and (iii) is slightly more complicated; we must distinguish two cases: (a) 10 2; then °7) = Vv implies °r] = V = 7)V and in such a case one has M(V) _ F(a, da - 1. One the other hand one should note that tr 7G,U(fv) =
and that for associated functions fv and 0, one has F(a, fv) = F°(b, 0) if a = N(b). Moreover
tr
F°(b, 1Zv\Ao
A,
tr(7r (,f)) if where 2v = Z(E xp F,,) and Av = A(E O Fv). Then tr(7r and f are associated and if _ (u o NE/F, v o NE/F) and 7) = (a, v). This yields 1. (3) = (iii). (b) If 1 = 2, the same arguments apply if
= ,v _ °rj, but there are other terms corresponding to rJ = (u, v) with p = °v # °a and then
1-(3) - (iii) _ -
l
To prove that 1. (4) = (iv) we use the previous remarks and the fact that MAY m(rJ), where F- means that ij = (u, v) and _ (u. NE/F, v. NE/F) _
= jjn
(1, "v) and by definition: M(7))
= L(1' L(1, a v llv) p)
and
mE (71 )
= L(l, p lv)I)
L(1,/i
Now 1. (5) = (v) follows from the comparison of orbital and twisted orbital integrals on unipotent elements.
129
BASE CHANGE PROBLEM FOR GL(2)
To conclude the proof of Proposition 1 it is enough to show that 1-(6) = (vi), which in turn follows from the equality:
if r = N(8).
A2(r,fv) = A1(6, 0v)
4. The main theorem. The representation r is a discrete sum of unitary irreducible representations of G(A) with multiplicity one:
r = ieI E
E Ozi,v
iEI
and tr r(f) = E ieI F1 v tr ii, v(fv) for some set I. Let v be a nonarchimedean place. Given fv in the Hecke algebra of G(F1) then tr it ,v(fv) is zero unless iri,v is unramified and hence corresponds to the conjugacy class of some semisimple element ti,, e GL2(C), the connected component of the
L-group. Any function f in the Hecke algebra defines a rational class function fv on GL2(C) such that fv(ti,v) = tr ii,v(fv) Choose a finite set V of places of F containing archimedean ones and assume fv is in the Hecke algebra for v 0 V. Then one has
LEMMA 1. tr r(f) = EilIvevtr,ri,v(fv) fvevfv(ti,v) The representation Rd can also be written
Rd = E R; _ E Off;,v >ET
>EJ
V
where II; is an automorphic representation of G(AE) and H;,v a representation of G(FV 0 E). Since °Rd Rd and is multiplicity free, or permutes the II;. If 61I; II; one can restrict the operator Rd(oi) to the space of II;, and denote this restriction by II;(a). The nonfixed 11, do not contribute to the trace of Rd(a)Rd(c) (a permutation matrix without fixed point has trace zero) and then
tr Ra(6)Rd(q5) = E tr IIi(o)H;(0) °n;=n,
II;,,; we can define ]I;, v (o) up to 1th roots of Moreover II; = (&]I;,, and °]I;,, 1. If ff,,v is unramified, there is a canonical choice. If 0, and fv are associated in the Hecke algebras, we choose a semisimple element t;,,, c GL2(C) such that tr II;, v(o)R;, v(qv) = tr j7,, v(0v) = f v (t;, ),
and then for some big enough finite set V of places of F, we have :
LEMMA 2. tr Rd(o)Rd(¢) = E; II vEv tr ;, v(o)II;
v
v (t;,v).
REMARK. If v is nonarchimedean and split in E, then any fv is associated to some
0v; in fact in such a case 0, may be taken to be f,,1 0 fw20 ... Of,,, where the wi are the places of E above v, and fv = f,,,, * f,,,2* ... *f,,, can be any smooth function on G(F1); the conjugacy class oft;,v is well defined by ]I;,,.
If v does not split in E, is unramified, and if 0, andf, are associated in the Hecke algebras, one can define a function /v on GL(2, C) as above; we have f v(t) = Ov (tl) for t semisimple in GL2(C). In such a case the conjugacy class of the t;,v above is not uniquely defined.
130
P. GERARDIN AND J. P. LABESSE
If Z 2 let R = IRd. Assume for a while 1 = 2; if u is a character of AE/E'< such that' ap = Co and °,u 0 ,u, we consider r,, _ )c,, where 7) = (u, °,u). As was said above, the operator M(0'y)7r;(c) = r,,(6) maps the space of z,, into itself. One defines in such a way
a representation v,, of r K G(AE). One should remark that i,, denote by
Let us
the set of such representations (modulo equivalence) and let
R' = 1R, p0 TµE
An analogue of Lemma 2 can be stated. We can now state the main theorem (cf. [L, Theorem 9.1]). THEOREM 1. Assume f and 0 are associated; then tr R'(o)R(g5) = tr r(f).
(Recall that the definition of the correspondence "f and 0 are associated" depends on the choice of a o, e I' - {1}.) The proof of the theorem can be carried out as follows : consider the expression
tr R'(o)R(c) - tr r(f).
(a)
Thanks to Proposition 1 above, this is equal to (b)
1JD
(1B(&,
B(ryv, .fv))
II tr icnw(.fw) d91
w#v
Using properties of some weighted orbital integrals [K, Lemma 4], one can prove that lB(&,
B(7)0, f) = 0,
at least when v is nonarchimedean, unramified in E, with ov and fv associated and in the Hecke algebras. Hence there is a finite set of places V such that (b) can be written (b')
$DO
3(7)) vH tr icn,(fv) dd
with some nice function P.
Now choose a place vo 0 V split in E; then (b) reduces to an absolutely convergent integral: (b")
f+ ii(s)fvo(pg sbq_is0) ds
where a, b depends on the central character w o. All we need to know is that 8(s) is some continuous, bounded and integrable function on the real line. On the other hand (a) can be written using Lemmas 1 and 2 above 21 akf,(tk)
k=0
with ak E C and tk e GL2(C) semisimple elements corresponding to inequivalent unitary representations of The series, as the integral, is absolutely convergent. Now f, is arbitrary in the Hecke algebra (since vo is split in E) and the
BASE CHANGE PROBLEM FOR GL(2)
131
Hecke algebra separates inequivalent unramified representations. The StoneWeierstrass theorem and easy majorations prove that all ak are zero (which is a stronger statement than Theorem 1). All the desired results can now be xtracted from Theorem 1 and from some results on the characters of representations of P x GL2(Ev) (cf. [L, §5]). We shall try to explain some of the steps.
5. Existence of weak liftings. Choose a finite set V of places of F containing all archimedean places and all places ramified in E. Assume that 0, and fv are associated and in the Hecke algebras for v 0 V. One can, using Lemmas 1 and 2, choose element tn,,, e GL2(C) for v e V and n e N such that
tr Rd(a)Rd(O) _ E an(O) [I fv (tn.v), v(5V
n
tr
E On(q5) n
tr r(f) =
II fv (tn.v),
v(4V
,,,,((
n
rn(Y') 11 Jv(tn,v); vE4V
we may assume moreover they are chosen such that the functions Tn: (cv)vEv f j vev fv (tn,v) on the product for v V of the Hecke algebras are distinct. (Recall the remark after Lemma 2.) Let bn = Ian + On - rn; then the above theorem can be restated in the following form 0. nEN
Another use of density arguments, the Tn being distinct, yields PROPOSITION 2. For all n one has 8n(0) = 0.
This can be read Ian + On = rn.
Assume that there exist a representation II = Qx llv, unramified outside V, occurring in Lo such that tr ]Iv(gv) = fv (tn,v) for some n with an not zero, and any v 0 V; then using the strong multiplicity one theorem [C] one concludes that such a H is unique and satisfies lI °ll. The fact that L(s, ll) is entire allows one to conclude that no other II occurring in LSP or in 9- has the property that tr HH(cv) = fv (tn,v), v 0 V. Then an(¢) = IIvEv tr H;,(o)Hv(¢v) and On(o) = 0; since Ian + On = in we conclude that 7-n(o) is not identically zero and hence there exists (at least) one v in L2 xQ LSP such that z = Qx zrv and tr iv(.fv) = f (tn,) for v V and then li, is the lifting of 'r,, for v 0 V. We shall say that II is a weak lifting of r if llv is a lifting of irv for almost all v. We then have proved
THEOREM 2. If ff is a cuspidal automorphic representation of GLZ(AE) such that ff _- °ll, then ff is the weak lifting of some automorphic representation i of GL2(A).
A direct study of LSP allows one to prove: PROPOSITION 3. Any z occurring in LSP lifts to a H in LSP and all a-fixed representations in L P are obtained in this way.
In the case 1 = 2, assume that H = Jr(u, °p) occurs in and that ,uv is unramified for v 0 V; one can show using Proposition 2 and results on L-functions on GL2 x
132
P. GERARDIN AND J.P. LABESSE
GL2 that 17 is the weak lifting of
= z(p) where p = IndWE u and that there exists
n e N such that of j tr a r(u)IIv(q ,) = &(0) = rn(qS) = of I tr This can be used to show that at all places of F: tr ilv(U)1Iv(cv) = tr 7rv(fv), and hence THEOREM 3. Let the fields E and F be either local or global. Then ir(u, °,u) is the lifting of ir(p) where p = IndWE a.
6. Any cuspidal ir has a weak lifting. Assume for a while that some ir occurring in Lo has no weak lifting. Let V be a finite set of places of F including archimedean places and those where E or r are ramified. Let f = (&fv be associated to some 0 and such thatfv is in the Hecke algebra for v V. Consider the Irk in Lo (E L P such that tr Ick,v(fv) = tr z,,(fv) if v V. Since the Irk have no weak lifting, Proposition 2 shows that the sum of the nk gives a zero contribution to tr r(f) :
E fi tr zk.v(.fv) II tr tcv(fv) =0; k
vEV
veV
hence there is a set VI c V such that
Llk 11 v, tr lrk,v(fv) = 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 VI) 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 cv (cf. [L, §9, pp. 25-30]). This yields THEOREM 4. Any cuspidal ir 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 IT °]l, occurring in Lo, unramified outside a finite set V chosen as which before; there is an n such that, according to Proposition 2, can be written, for some VI c V, I
jl vEV,
tr II;,(o)1Iv(cv) _ k
II tr Irk,v(Jo),
vEV,
where the lrk = Qrk,v have it as weak lifting. One then proves [L, §9, pp. 33-34] : LEMMA 3. If for some v the representation 1Iv is the lifting of some 1v then it is the lifting of Ik,v for all k, that is tr II;,(o')17v(c5v) = tr 7Ck.v (fv)
Then one may assume that VI 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 9C has a lifting. (b) If il,, °1iv and is supercuspidal, then iv 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)
133
global one, where the existence of liftings is known at all places except perhaps at one place, and to use the above equation with V1, 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 existence of lifting for noncuspidal representations, using their explicit description (cf. Appendix Q. BIBLIOGRAPHY
[B] A. Bore], Automorphic L -Junctions, these PROCEEDINGS, part 2, pp. 27-61. [C] W. Casselman. On some results of Atkin and Lehner, Math. Ann. 201 (1973), 301-334.
[D] P. Deligne, Formes modulaires et representations de GL(2), Modular Functions of One Variable. II, Lecture Notes in Math., vol. 349, Springer-Verlag, Berlin, 1973. [D-S] P. Deligne and J.P. Serre, Formes modulaires de poids 1, Ann. Sci. Ecole Norm. Sup. 4 (1974), 507-530. [G-J-1] S. Gelbart and H. Jacquet, Forms on GL(2) from the analytic point of view, these PROCEEDINGS, part 1, pp. 213-251. , A relation between automorphic forms on GL(2) and GL(3), Proc. Nat. Acad. [G-J-2]
Sci. U. S. A. (1976). [J-L] H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Math., vol. 114, Springer, Berlin, 1970. [J-PS-S] H. Jacquet, I. I. Piatetskii-Shapiro and J. Shalika, Construction of cusp forms on GL(3), Lecture Notes, Maryland. [J-S] H. Jacquet and J. Shalika, Comparaison des formes automorphes du groupe lineaire, C. R. Acad. Sci. Paris 284 (1977), 741-744. [K] R. Kottwitz, Orbital integrals and base change, these PROCEEDINGS, part 2, pp.111-113.
[L] R. P. Langlands. Base change for GL(2), Institute for Advanced Study, Princeton, N. J., 1975 (preprint). [Sa] H. Saito, Automorphic forms and algebraic extensions of number fields, Lectures in Math.,
no.8, Kinokuniya Book Store Co. Ltd., Tokyo, Japan, 1975. [S-1] T. Shintani, On liftings of holomorphic automorphic forms, U.S.-Japan Seminar on Number Theory (Ann. Arbor, 1975). [S-2] -, Two remarks on irreducible characters of finite general linear groups. J. Math. Soc. Japan 28 (1976), 396-414. [S-3] -, On liftings of holomorphic cusp forms, these PROCEEDINGS, part 2, pp. 97-110. [T] J. Tate, Number theoretic background, these PROCEEDINGS, part 2, pp. 3-26. [Tu] J. Tunnell, On the local Langlands conjecture for GL(2), Invent. Math. 46 (1978), 179-198. UNIVERSITE PARIS VII
Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 135-138
REPORT ON THE LOCAL LANGLANDS CONJECTURE FOR GL(2) J. TUNNELL Let o(GL(2)/K) be the set of isomorphism classes of two-dimensional F-semisimple representations of the Weil-Deligne group WW of a nonarchimedean local field K. The purpose of this report is to discuss the following conjecture of Langlands relating o(GL(2)/K) and the set 1I(GL(2, K)) of isomorphism classes of irreducible admissible representations of GL(2, K). Conjecture. For each representation of in O(GL(2)/K) there exists a representation
7c = ir(i) in II(GL(2, K)) such that the determinant of 6 and the central quasicharacter of 7r are equal and such that L(7r Ox Z) = L(6 0 x) and
s(7r O x) = s(6 (D x)
for all quasi-characters x of K*. The map 6 I--. 7r(a) is a bijection of O(GL(2)/K) with II(GL(2, K)). The existence statement in the conjecture was formulated in [4], where it was shown that for a given representation or there is at most one representation 7r satisfying the desired conditions. The injectivity statement is equivalent to saying that a two-dimensional F-semisimple representation of WW is determined by its twisted L- and c-factors and determinant. It is straightforward to show directly that reducible two-dimensional representations are in fact determined by the twisted Lfactors alone. The known proofs of the injectivity statement for irreducible representations use admissible representation techniques. As discussed in [1, 3.2.3] it follows from the work of Jacquet and Langlands that rr(oi) exists when 6 is reducible, and that this establishes a bijection of the set of isomorphism classes of completely reducible (repectively reducible indecomposable) two-dimensional F-semisimple representations of WW with the set of isomorphism classes of principal series (respectively special) representations of GL(2, K). Let oirr(GL(2)/K) consist of isomorphism classes of irreducible two-dimensional representations of the Weil group WK, and let 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 L-factors are all equal to 1. To prove the conjecture it is enough to verify that rr(o-) exists for all of in Oirr(GL(2)/K) and to show that this establishes a bijection of Oirr(GL(2)/K) and AMS (MOS) subject classifications (1970). Primary 12B30, 12B15; Secondary 10D99. © 1979, American Mathematical Society 135
136
J. TUNNELL
IIIusP(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 1 which do not contain the cube roots of unity. References for the proof are given in the following survey.
1. Existence of r(o). If u is a representation induced from a one-dimensional representation of an index two subgroup of WK there is an explicit construction (due to Weil) of an irreducible admissible representation 7t(o) of GL(2, K) with the desired properties [4, 4.7]. When Khas odd residue characteristic all representations in q;II(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 Weil group which are not induced from a proper subgroup [11, paragraph 29]. There is no explicit construction of ir(o') known for such representations. The approach of Jacquet and Langlands to the existence of z(a) in this case is to imbed the local problem in a global one as follows. Let F be a global field and let p be an irreducible continuous two-dimensional complex representation of the Weil group WF. For each place v of F let pv be the restriction of
p to the Weil group of the local field F. THEOREM 1 [4, 12.2]. If the global L -Junctions L(s, p (D x) are holomorphic and bounded in vertical strips as functions of the complex variable s for all quasi-characters
x of AF/F* then 2c(pv) 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 composition 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 (Weil), when p is of A4 type (Langlands-Jacquet-Gelbart), and when F = Q, p is of S4 type and the image of complex conjugation has determinant -1 (Langlands-Serre-Deligne). THEOREM 2 [10, THEOREM A]. Let K be a nonarchimedean local field which contains
the cube roots of unity if it is a proper extension of Q2. Then ir(6) exists for all o in O(GL(2)/K).
This theorem is proved by constructing a global field F, a place v of F such that K, and a representation p of WF satisfying the hypotheses of Theorem 1 such o. The nonarchimedean local fields of even residue characteristic which that pv do not contain the cube roots of unity are precisely those for which there exist representations of the Weil group of S4 type [11, paragraph 25]. The fields excluded in Theorem 2 are those for which there are two-dimensional representations of the Weil group which are not restrictions of global representations known to satisfy the hypotheses of Theorem 1. Deligne has indicated that if 1-adic representations can be associated to certain automorphic representations related to Shimura varieties 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 Fv
THE LOCAL LANGLANDS CONJECTURE FOR GL(2)
137
at the infinite places. Theorem 2 will then be true for K a completion of Fat 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 ir(o) for o in O;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)). Suppose that a is induced from a 1-dimensional representation A of the Weil group WE of a quadratic extension E of K. Denote the quasi-character of E* corresponding to d by the same symbol. The restriction of the character function of 'r(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 7r(o) determines A up to Gal(E/K) conjugation, and hence determines the induced representation o%
3. Positive characteristic. The discussion of §1 shows that ir(a) exists for all representations of the Weil group of a local field of positive characteristic. The bijectivity assertion of the conjecture seems to have first been proved by Deligne [2] as a consequence of Drinfeld's results relating automorphic representations of GL(2) over global function fields to 1-adic representations. 4. A method is presented in [10] that gives alternate proofs of Langlands' Conjecture for the cases discussed in §§2 and 3 and proves the conjectured bijection for all fields in Theorem 2. PROPOSITION [10, 2.2]. Let of and Q2 be two-dimensional representations of the Weil
group of a local field, each of which is the restriction of a global representation satisfying the hypotheses of Theorem 1. If n(o1) x ,r(o'2) then o1 z 0'2.
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 O;rr(GL(2)/K) and K)) into finite sets 0z and IIJ respectively (indexed by a common set A) such that (1) If o' e Ox and Ir(o-) exists, then ir(u) e III. (2) Card(OA) = Card(IIA) for all 2 e A.
The partition elements are determined by conditions on the Artin conductor and determinant (resp. conductor and central quasi-character) of elements in O;rr(GL(2)/K) (resp. (GL(2, K))). Since the conductor of a representation is determined by the twisted e-factors, the first statement follows from the definitions.
138
J. TUNNELL
The computation of the cardinality of the sets 01 is done by constructing all irreducible two-dimensional representations of WK as in [11] and counting those with a given Artin conductor and determinant. The sets 17x are studied by utilizing the correspondence between square-integrable representations of GL(2, K) and admissible
representations of the group of invertible elements in the quaternion division algebra over K. Theorem 2 together with the injectivity proposition and the counting results of Theorem 3 show that 0A and 1 A correspond bijectively by means of u H 7r(a) 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 Q2 which is purely local. They calculate the necessary e-factors for irreducible two-dimensional representations of WQ, 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 a-factors and matching with the factors of representations of WK to be done in general. REFERENCES
1. P. Deligne, Formes modulaires et representations de GL(2), in Modular functions of one variable, Springer Lecture Notes, No. 349, 1973, pp. 55-106. 2. , Review of "Elliptic A-modules" by V. Drinfeld, Math. Reviews 52,1976), #5580. 3. I. M. Gelfand, M. I. Graev and I. I. Piatetskii-Shapiro, Representation theory and automorphic functions, Saunders, 1969. 4. H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Springer Lecture Notes, No. 114, 1970. 5. P. Kutzko, On the supercuspidal representations of GL,.1,11, Amer. J. Math. 100 (1978), 43-60. 6. A. Nobs, Supercuspidal representations of 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 local field, Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 1231-1237. 8.
, The Plancherel formula for SL(2) over a local field, Proc. Nat. Acad. Sci. U.S.A.
63 (1969), 661-667. 9. A. J. Silberger, PGL(2) over the p-adics, Springer Lecture Notes, No. 166, 1970. 10. J. Tunnell, On the local Langlands Conjecture for GL(2), Thesis, Harvard University (1977). 11. A. Weil, Exercises dyadiques, Invent. Math. 27 (1974), 1-22. PRINCETON UNIVERSITY
IV ARITHMETICAL ALGEBRAIC GEOMETRY AND AUTOMORPIDC L-FUNCI'IONS
Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 141-163
THE HASSE-WEIL c-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 c 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 complex structure; if 1' is an arithmetic subgroup of G small enough to contain no torsion then 1'\X will be a nonsingular algebraic variety. In many cases one can show that it has a model defined over a rather explicit number field, and under
certain further assumptions-as Deligne and Langlands may explain-one can choose a canonical model defined over an abelian extension of a special number field E determined, roughly speaking, by G and the complex structure on X. One might expect that the Hasse-Weil c-function of a canonical model is a product of L-functions of the sort Langlands associates to automorphic representations 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 1'=1'o(N), then 1'\X has a model over Q and its Hasse-Weil c-function is, as far as all but a finite number of factors in its Euler product are concerned, a product of L-functions defined in this case by Hecke. This result was extended by others, notably Shimura, Kuga, Ihara, Deligne, and Langlands, to include: (a) other 1' in this G, (b) other G, (c) c-functions associated to nontrivial locally constant sheaves, and finally (d) factors of the c-function corresponding to primes where the variety behaves badly. With one exception-some unpublished work of Shimura-all this work is concerned with X of dimension one. (Refer-for a sampling-to [11], [28], [14], [6], and [19].) As a consequence of these results one had a generalization of Ramanujan's con-
jecture, applying the result of Deligne on the roots of the c-function of varieties over finite fields. A further consequence was a functional equation for and analytic continuation of the Hasse-Weil c-function concerned, which turned out-excepting again Shimura's example-to be a c-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 > 1, and what Milne and I are going to discuss in our lectures is the simplest case he deals with. AMS (MOS) subject classifications (1970). Primary 1OD20; Secondary 14G25, 22E55. © 1979, American Mathematical Society 141
142
W. CASSELMAN
To be more precise, let: F = a totally real field of degree, say, n; OF = integers in F; B = a quaternion algebra over F;
oB = a maximal order in B. We recall that this means simply that B is an algebra of dimension four over F such that B Q F = M2(P), where P 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 OF F is isomorphic either to M2(Fv) or to this unique division algebra, depending on whether or not By has 0-divisors. The algebra B is said to be split at v in the first case, ramified in the second. It is in fact split at all but a finite number of valuations; quadratic reciprocity says this number is even, and another classical result says that for each even set of valuations there is a unique quaternion algebra ramified at exactly those valuations. Thus B itself is a division algebra if and only if it is ramified somewhere. If v is nonarchimedean, the closure of oB in B is a maximal compact subring of By unique if B is ramified at v, otherwise only unique up to conjugacy by an element of By . Let G be the algebraic group over Z defined by the multiplicative group of oB. Thus for any ring R, G(R) = (oB (& R)x. In particular, G(Q) = Bx (canonically)
and G(R) = GL2(R)I x (Hx)J (noncanonically), where I is the set of real valuations of F where B is split, J those where it is ramified. For every rational finite prime p over which B does not ramify, G(Zp) _ 11 GL2(oF, n),
G(Qp) = II GL2(Fv),
where the product is over all primes p of F dividing p. Of course the simplest case is B = M2(F), but that is unfortunately the case we will not allow-i.e., from now on we assume B to be a division algebra. Furthermore, we will assume B totally indefinite at the real primes of F-i.e., that J = 0. Langlands himself does not make these assumptions, but acknowledges gaps in the argument unless they hold.
Let d be a finite set of rational primes containing those over which either F or B ramifies, and let Kf be a compact open subgroup of G(Zf) of the form Ka ]j pea G(Zp), where Ka is a compact open subgroup of f[ pEa G(Zp). Let Z be the center of G, and ZK = (Z(Af) (1 Kf) Z(R). Consider Cx as embedded in GL2(R):
a + b -1 '-' (b
a)
and let KR be the image of (Cx)I in G(R). (This is the connected component of the KR used before.) Then G(R)/KR is a product of n copies of C - R. Let K be KR Kf. The set KS(C) = G(Q)\G(A)/K is (as will be explained later) the union of a finite number of compact complex analytic spaces of dimension n; it will be nonsingular if (as we assume from now on) Kf is small enough. Milne (in his lectures at this Institute) will show that this space is the set of Cvalued points on a certain moduli scheme KS which is defined, smooth, and proper over Spec Z[1/d], where d is the product of primes in d. He will also discuss the
THE HASSE-WEIL c-FUNCTION
143
structure of its points over finite fields. What I will do is to identify its Hasse-Weil
c-function as one of Langlands'; the basic idea in doing this will be to use the Selberg trace formula to calculate p-factors of Langlands' L-function, and Milne's results and an unorthodox application of the trace formula to calculate p-factors of Hasse-Weil-both for p not dividing d. (This will be a lot of trouble, and I should explain at the outset that, although when KS has dimension one it is possible to use a congruence relation to obtain the final result, this will not suffice in general.) One may then apply Deligne's results on the roots of the Frobenius to get a generalization of Ramanujan-Petersson; however, results about a functional equation for Hasse-Weil are incomplete. In everything both Milne and I do we are essentially reporting on Langlands' work. The result we discuss is only a special case of a more complicated result of his involving subgroups of Bx. We avoid, in treating this special case, problems of what he calls "L-indistinguishability," which perhaps he himself will talk about. His result is more general in other ways, too; I include remarks on this in the last
section, where I have collected together a number of substantial parenthetical remarks. My main reference is the summary [21]; complete statement and proofs are in
[22]. Also relevant are Langlands' talk in the Hilbert problems Symposium [20] and his talks at this Institute. 1. Cohomology of KS(C) and representations. 1.1. Let v be the reduced norm: G - G.. Since B is a division algebra, the coset space G(Q)\G'(A) is compact, where G1(A) is the kernel of the modulus homomorphism I v I : x --p 1 v(x) 1. Since the image of GOO-(R), the connected compact of G(R),
under lvi is all of RPOS, the set G(Q)\G(A)/Gco""(R) is compact as well. Since Kf is open in G(Af), the set G(Q)\G(A)/GCO""(R)Kf is finite. Therefore there exists a finite set . ' of elements of G(A) such that G(A) = UG(Q)xGc-(R)Kf (x (= X). (Strong
approximation gives one a better parametrization of ", but we won't need that; see [7].)
I.I.I. LEMMA. For any x e G(A), the space G(Q)\G(Q)xGC011(R)Kf1Kf as a GB0""(R)-space is isomorphic to 1'x\GC0=(R), where f'x is the image in GC0-(R) of G(Q) n Gconn(R) xKfx 1.
This is because GC0""(R) certainly acts transitively on this space and rx is the isotropy subgroup of the coset G(Q)xKf. Let be the upper half-plane in C. As a consequence of the above: 1.1.2. PROPOSITION. The Gc0m(R)-space G(Q)\G(A)/Kf is isomorphic to a disjoint union of spaces rx\Gco""(R) (x e "). The variety KS(C) is the disjoint union of the
The same argument as that used to prove Lemma 2.1 of [19] may be applied to show that if Kf is only small enough, each F x acts freely and KS(C) is nonsingular. We assume this from now on.
1.2. Let sago be the space of automorphic forms on G(Q)Z(R)\G(A). It is a direct sum (D 7r of irreducible, admissible, unitary representations of G(A) (an abuse of language since not G(R), but only its Lie algebra g, acts). If g is the Lie algebra of
144
W. CASSELMAN
G(R) = G(R)/Z(R), then in fact the representation on moo factors through q. Let f be the Lie algebra of KR = KR/KR n ZR. 1.2.1. PROPOSITION. The de Rham cohomology H*(KS(C), C) is naturally isomorphic to OH*(q, f, 7r_) D 7 Kf.
The cohomology is the relative Lie algebra cohomology. The sum is over all constituents z of moo (with multiplicity if necessary), which one factors as i _ ire 0 z f where 'r_ is an irreducible admissible representation of G(R) (more abuse of language as before) and z f one of G(A f). PROOF SKETCH. Let to f be the subspace of d0 of functions fixed by elements of
Kf. Since q commutes with Kf, this is a representation of q which is clearly iso-
morphic to Q z- Qx
f f. Thus the proposition amounts to the claim that
H*(KS(C), C) is isomorphic to H*(q, i, ,moo f). Now from 1.1.2 it follows that ago f is the direct sum of subspaces sago x (x e X), where each ,moo is the space .d(r \Gc0-(R)) of KR-finite, Z(q)-finite functions on r \Gc°"" (R) (Px is the image of T. in G). Thus the proof of 1.2.1 reduces to: 1.2.2. LEMMA. There exists a natural isomorphism:
H*(r,,\G/K, C) = H*(9, , (rx\G)) This sort of thing holds in fact for any semisimple Lie group G, cocompact F c G and maximal compact K. is that of the de Rham complex, whose To see it: the cohomology of X nth term is the space of C" m-forms on X. Now the projection r,,\G -> X is a principal bundle, and the bundle of m-forms is associated to it and the K-space Am(q/t)A. Therefore C°° m-forms correspond to certain C°° functions from r, \G to Am(q/j)A,
which by means of an obvious duality may be thought of as k -linear maps from Am(gl k) to C°°(Px\G). In short, the de Rham complex on X may be identified with a complex whose mth term is Homt(Am(qj), C°°(I'x\G)). But this is the mth term of
the complex by which H*(q, i, C`°(P \G)) is calculated, and it turns out (by an explicit calculation) that the differentials are the same. Therefore 1.2.2 is true with sl(r.,\G) replaced by C°°(r\G). To get the final step, roughly, one recalls that ac-
cording to Hodge theory cohomology classes may be represented uniquely by harmonic classes, and observes that these lift in the above process to elements of .d. (See Chapter IV of [3] for details on this and other points.)
1.3. Although the number of representations occurring in the sum in 1.2.1 is infinite, all but a finite number of terms vanish. To be more precise, we must say more about the relative Lie algebra cohomology of admissible representations. First of all, since G(R) = PGL2(R)f (notation as in the introduction), each 7r_ factors as where each 7r,,,, is an admissible representation of PGL2(R). One has an easy Kiinneth formula: H*(9, i, Ox
Qx H*012,
2r00J
so that the problem for G(R) is reduced to one for PGL2(R). (Note that the Lie algebra of PGL2 is the same as that of SL2.) For this: let C be the Casimir element in U(312), and recall that it lies in the center of U012) and is centralized by the maxi-
145
THE HASSE-WEIL C-FUNCTION
mal compact 0(2), hence acts as a scalar on any irreducible admissible representation of PGL2(R).
1.3.1. LEMMA. Let (2c, V) be an irreducible, admissible, unitary representation of PGL2(R). Then
if v (C) 0 0,
H*0l2, X02, V) = 0 HomIDOZ(A*0I2*2), V)
if Z (C) = 0.
The idea of the proof here is that if 7r is unitary one can put a natural inner product on the complex Hom,LZ(A*0120102), V), such that ir(C) = dd* + d*d where d*
is the adjoint of d. (This is an observation good for all semisimple groups due to Kuga.) The lemma follows immediately. For PGL2(R), there are only three irreducible admissible representations 'r with
z(C) = 0: (a) the trivial representation C; (b) the character sgn(det(g)) ; (c) a single discrete series representation z0, which may be identified with the quotient of
the 0(2)-finite functions on PI(R) by the constant functions. Lemma 1.5 and an easy calculation concerning the restriction of these to 0(2) give : 1.3.2. COROLLARY. If it is an irreducible admissible unitary representation of PGL2(R) other than one of these then H*(7r) = H* 0I2, X02, 'r) = 0. For these: (a) when r is C or sgn (det), m = 0, Hm(2c) = C, 0,
m= 1,
C,
m=2;
(b) when' = 7ro,
Hm(7r)0,
C+C, 0,
m=0, m=1, m=2.
The assumption of unitarity is not in fact necessary (see [3]).
Hence if the irreducible admissible representation ire. of the original G(R) is cohomologically nontrivial, it must be of the form where each z, is one of the above three representations, and its cohomology may be calculated accordingly. If r is an irreducible admissible representation of PGL2(R), set m(lr) = 0
_ -1
if ir is cohomologically trivial, if ir is either C or sgn(det), if it is ir0.
If z_ = Qx 7r, is an irreducible admissible representation of G(R) define m('r 0) to be I If it = it Ox z f is an irreducible admissible representation of G(A) where n is trivial on Z(R), then define m(2r, K) = m(7r_) dim ff. Thus the size of m(2r, K) is just dim2rff and its sign reflects the parity of its contribution to cohomology. As a final remark let me point out that by applying the strong approximation theorem one can show that if r = (D z, (v over valuations of F) is an irreducible admissible automorphic representation of G(A) and irv is one-dimensional at a
146
W. CASSELMAN
place where B is split, then 7rv is one-dimensional everywhere. In particular, if one factor of x- is one-dimensional so are all. As a consequence of this one recovers a result of Matsushima-Shimura [23] which says that the interesting cohomology of KS(C) occurs in the middle dimension. 2. The main theorem and some consequences. 2.1. I recall the L-group attached to G. First of all, if G is considered in the most
straightforward way as a group over F -i.e., so that for any extension F' of F, G(F') = (B OF F')x -then its L-group is just the direct product of LGO = GL2(C) and Gal (F/F). This is because it is an inner twisting of GL2(F). Since G as a group over Q is obtained from this one by restriction of scalars, its L-group LGO is the one in some sense induced from this one: it is the semidirect product of Gal(Q/Q) by LG° = GL2(C)I, where I is the set of embeddings of F into Q and Gal acts on LGO by permutation of factors. (Since Q may be identified with a subfield of C, this I is essentially the same as before.) Without any serious loss for our purposes we may (and will) replace Gal (Q/Q) by Gal (Fn(,rm/F), where Fnorm is the smallest extension
of F normal over Q. It may be helpful if I point out that it is unramified over Q whenever F is. I recall also that if p is a prime of Q unramified in F and 0 is a Frobenius in Gal(Fnorm/F) over p, then the local L-group LGQ, = LGp may be identified with the subgroup of LGQ whose image in Gal(Fnorm/F) is the cyclic subgroup generated by 0. Thus one has an exact sequence 1 --+ GL2(C)I -+ LGp --> <0> -+ 1. (Refer to [2] for everything about L-groups.) In order to define L-functions associated to automorphic representations, one must also introduce finite-dimensional representations of LG. There is only one (for each G) that we will be concerned with, and it is defined as follows: the space of this representation p is a tensor product of copies of C2, one for each element of I; the group LGO = GL2(C)I acts through the standard representation on each factor, and Gal(Q/Q) (or Gal(Fnorm/Q), it makes no difference) acts by permuting the factors. This does indeed define a representation of LG, since it is a semidirect product of these two groups. The dimension of this representation is 2n. When F = Q, for example, LG = GL2(C) and p is just the standard representation itself. This particular choice of p may seem arbitrary, but in fact it was motivated originally (in Langlands' formulation) by general considerations about Shimura varieties (one should refer to the Introduction of [22] for a discussion of this point). 2.2. If X is any smooth, proper scheme over Z[1/d] for some d >_ 1 and p is a prime not dividing d, then the zeta-function of X over Fp is defined to be the function Zp(s, X), rational in p-, such that (at least formally) log Zp(s, X) _
mpms
As a consequence of the etale theory, this agrees with the definition in terms of 1-adic cohomology:
Zp(s, X) _
2-dimX
TI
det(I -
0.p s)H1(X,Q,)
:=o
where 0 is the geometric Frobenius. At least as far as factors other than those dividing d are concerned, its Hasse-Weil zeta-function Z(s, X) is the product of these Zp(s, X).
147
THE HASSE-WELL C-FUNCTION
The main theorem of Langlands is: 2.2.1. THEOREM. Up to prime factors in d, the Hasse- Weil zeta-function of KS agrees with jj L(s - n/2, i, p)m(n.K) where the product is over all 7r occurring as constituents of do.
Of course one should consider the v with multiplicities if necessary. In fact, however, because of the theorem in §16 of [16] relating automorphic forms on G with those for GL2, together with the result for GL2 in §9 of [16], each r occurs exactly once. In this product, m(7r, K) = 0 for all but a finite number of 7r, since after all the cohomology of KS(C) is finite. What is actually to be proven is a purely local result: for p 0 d, the p-factors of Z(s, KS) and jj L(s - n/2, 'r, p) coincide. Now each is = (2 2rv with m(7r, K) # 0 has the property that p for p 0 d is unramified, hence corresponds to an element
in the local L-group LGp. The p-factor of L (s - n/2, ir, p) is then det(I p(g(1rp))/ps-n/2)-1. Upon expansion, the theorem reduces to a formal
g('rp)
equation (p1, 0 J):
E
1ms
M=1 MP
LI m(i, K)
(#KS(Fpm)) =
in do
Epmn/2
l
trace
p(g(pjp)m).
mPms
m=1
This in turn amounts to an equation of coefficients (for p 0 d, m > 1): (2.1)
#KS(Fpm) _
n in do
m(ir, K)pmn1'2 trace p(g(lrp)m).
This is the form in which the theorem is actually proven. In these lectures I will give a complete proof only in the simplest case, when Fis split overp. 2.3. Before beginning the proof, we give an example and some consequences. The case n = 2 is the first interesting one, in the sense that, as already mentioned, the case n = 1 is an old result and can be (and has been) done more elementarily by means of a congruence relation. So, suppose for a while that F is quadratic over Q, and consider the possible p-factors occurring in L(s, ir, p). There are two cases: (1) when p = p1P2 splits in F and (2) when p = p remains prime. We look at (1) first. In this case G(Qp) GL2(F,,) x GL2(F12) and an unramified representation of this must be of the form r1 Qx r2, where each zi is an unramified representation of GL2(Fp), hence corresponds to a pair of unramified characters (a1, (3i) of F. More precisely, ni may be the whole principal series representation parametrized by the pair when it is irreducible, or the associated one-dimen-
sional character of GL2(FF.) when it is not. (By strong approximation, this last happens only when the global representation at hand is also one-dimensional.) In either case, observe that the local L-group LGp is (because p splits) simply the direct product GL2(C) x GL2(C) and that the corresponding element g(irp) of LGp is ((at(P)
\\
ll (a 2(P) 11 R2(P)// (31(P)7 \
The representation p is simply C2 O C2, so the p-factor of L(s - 1, r, p) is
(1 (2.2)
a1(P)az(P)P1-s)-1(1
-
a1(P)/32(P)P1-s)-1(1
-
01(P)a2(P)P1-s)-1
(1 - (31(P)(32(P)P s)-1.
W. CASSELMAN
148
Next case (2). Here 7rp is a single unramified representation of GL2(F), say corresponding to the characters (a, ) of Fp /op . On the other hand, the local L-group is still a semidirect product of GL2(C) x GL2(C) by the Galois group of order 2, since the Frobenius 0 generates Gal. I leave it as an exercise to verify that the element g(rp) of LGp is 01)'
(3(p))/0.
\a(0
\(0
If el, e2 are the standard basis elements of C2 then g(7cp) acts as follows on the space
of p (recall that 0 interchanges factors) : el Ox el i-' a(p)es Ox el,
el 0 e2 - a(p)e2 OO el,
e2 O el
e2 O e2 - 0(p)e2 O e2
/3(p)ei O e2,
Therefore the p-factor of L(s - 1, z, p) is (2.3)
(1 -
a(p)ps-s)-i(1
-
(3(p)pl-s)-i(1
-
a(p)0(p)p2-s)-i.
For n > 2, the analogous formulae can be rather complicated. Incidentally, the unpublished result of Shimura we have referred to before is precisely that the pfactors are of the form (2.2) and (2.3) for certain quadratic fields F. For F = Q (n = 1), one consequence of the expression of the Hasse-Weil zetafunction as an L-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 zetafunction of KS over F itself, at least for certain small degrees. This depends on a generalization of an idea of Rankin due to Jacquet and Shimura (see, for example, [15] and [29]).
3. The analytical trace formula. In this section I will develop a formula for the right side of equation (2.1), in the next one for the left, and in §5 the two will be compared. Beginning in §4 I will assume (although Langlands certainly does not) that p is split in F. This makes things much simpler. The argument will still involve several important ideas, but will avoid what is at once the most complicated and intriguing point of all, the use of paths in a certain Bruhat-Tits building. I hope that what remains is still of some interest. Langlands will presumably say something on this matter in his own talks (see [17]). 3.1. The right side of equation (2.1), (3.1)
E m(Tr, K) pmn/2 trace p(g(zp)m), ir in s.4'0
may be expressed as the trace of a certain operator R f, for some f e C-(G(A)), acting on sago. Recall that moo is a discrete direct sum of irreducible, admissible, uni-
tary representations 7r = ZR Q irp O ,cp of G(A), and for any f e Cc(G(A)/ZK) one has therefore trace Rf trace 2r(f), which is equal in turn to (3.2)
Z trace irR(fR) trace ip(fp) trace zp(fp) Jr in do
149
THE HASSE-WEIL C-FUNCTION
if f factors as f,Q fP fp. Recall that m(rc, K) = m(lrR) dim(1cp)KP. A comparison of
(3.1) with (3.2) suggests choosing the three factors off so that trace n (fR) = m(7r)
(3.3)(a)
for any irreducible admissible representation v of G(R)/Z(R); (3.3)(b)
trace 2c(fp) = dim 1rKP
for any irreducible admissible representation (3.3)(c)
of G(Af);
trace 7C(f pmt) = pmn12 trace p(g(9C)m)
=0
z unramified, otherwise.
This is just what will be done. The choice of fP is simple:
fP = (meas Kp)-I char KP.
That of fpm) is not so explicit but just as simple: according to the Satake isomorphism [4] there exists a unique fp(m) e(G(Qp), G(Zp)) satisfying (3.3)(c). But the matter offR is more complicated. 3.2. If the representation z of G(R) factors as (3z, according to G(R) = GL2(R)I
then m(r) = jj m('r ). If f = f l ff accordingly then trace z(f) = fl trace ir(f). Thus the problem of findingff reduces to the problem of finding f e CC°(PGL2(R)) such that
trace 2r(f) = m(2r)
(3.4)
for any irreducible admissible representation 'r of PGL2(R). One can do this in several ways-for example directly from the Paley-Wiener theorem as in [10], and also from considerations of orbital integrals (see [26]). But Serge Lang has pointed out an argument which is in some sense more elementary than either of these and which I present.below. From this point I follow Lang's book [18, Chapter VI, §7] rather closely. The connected component of PGL2(R) is PSL2(R). Now the maximal compact of SL2(R) corresponding to the one of GL2 (R) at hand is SO(2); let e: SO(2) -. C" be the fundamental character
e0 (cos-sin 0) = cos 0 + ``sin 0
cos 0
-1
sin 9.
The single discrete series representation 7co of PGL2(R) which is cohomologically nontrivial has the property that its restriction to SO(2) is a direct sum Qen (Ini > 2, n even). No other discrete series representation has e2 in its SO(2)-decomposition. I recall that the Hecke algebra of all compactly supported bi-SO(2)-finite distributions acts on the space of any admissible representation of SL2(R). If D satisfies
Lk1RkZD = em(ki)s n(k2)D for all k1, k2 e K, then for any such representation (7r, V) the operator ir(D) takes all of V into the em-eigenspace and annihilates all but the en-eigenspace. In the special case m = n and D amounts simply to integration against an on K, 2r(D) is the identity on the en-eigenspace. Therefore, by choosing f e C°°(SL2(R)) such that
150
W. CASSELMAN
f(klgk2) = E n(k1)s n(k2)f(g)
and approximating this D closely enough-i.e., choosing f positive, normalized, and with support close enough to K-one may assume that 7r(f) is also the identity on this eigenspace.
Apply this to the case m = n = 2, v = 7ro to obtain fl e C?°(SL2(R)) such that a-2(k1)S-2(k2)f(9) for all k1, k2 e SO(2), g e SL2(R); (b) 2ro(fl) is (a) f1(klgk2) = the identity on the e2-eigenspace and 0 on any other SO(2)-eigenspace. Because of the remark above about other discrete series representations of PGL2(R) and e2, one even has
trace ir(fi) = 1, (3.5)
7r = pro,
ir is any discrete series representation of
0,
PGL2(R) other than the iro. At this point recall what Lang calls the Harish transform of an f e C,°(SL2(R)) :
H f(a) = Ix - x 1I f
A\Gf (g lag)
dg
for a = (x
x 1),
x 0 ± 1.
Here A is the group of all diagonal matrices. This function Hf turns out to lie in CC-(A) and even in C,°(A)w, where W is the Weyl group. In fact, f -+ Hf induces an isomorphism of the space of functions f e Cc°(SL2(R)) bi-invariant under SO(2) with CCO°(A)W [18, V, §2]. Thus, one may choose f2 E CC°(SL2(R)SO(2)) with Hf1
Hf2. Set f = fi - f2, so Hf = 0. Since a discrete series representation possesses no vectors :0 fixed by SO(2), equation (3.5) holds for f instead of fl. Furthermore, if x is any character of A and 1rx is the corresponding principal series representation of SL2(R) then (3.7)
trace 2tx(f) = S
AH
f(a)x(a) da = 0
[18, VII, §3]. Since any principal series representation of PGL2(R) restricts to some Z. on PSL2(R), the same holds for all principal series representations of PGL2(R).
Finally, if 7r is any irreducible finite-dimensional representation of PGL2(R) then because its character and that of some unique discrete series representation add up to a principal series representation, trace
-1 =0
'r = C or sgn(det), other finite-dimensional 7r.
Define now fR on PGL2(R) to be -f on PSL2(R) and 0 on the other connected component. Equations (3.5), (3.7), and (3.8) yield (3.4) for f = fR. 3.3. To recapitulate: the function f = fR fP fpm) e C°°(G(A)/ZK) has the property that trace RfJ, o = E m(ir, K)pmn/2 trace p(g(7rp)m). n in ,ado
In §5 the trace will also be expressed by means of the Selberg trace formula. I will recall it here in a rather general form for purposes of contrast a little later: Let
THE HASSE-WEIL C-FUNCTION
151
W be any locally compact group, r a discrete subgroup with P\W compact. The conjugacy class of any r E F will be closed in c (a nice exercise) and so for any F n C'°(c) the orbital integral S,,r\, F(g 1rg) dg is defined and finite, where 9r is is compact also. Under the centralizer of r in W. If rr = qr (1 F, then clearly r7.\',.
some mild assumption the operator RF is of trace class on LZ(F\c) and
trace RF = E meas(F7\1cr) J (r}
F(g 1rg) dg 97r\W
where the sum is over all conjugacy classes in F. This will be applied in §5 to the case
9 = G (A)/ZK, F = G(Q)/G(Q) n ZK, F is the f just constructed. 4. The algebraic trace. In this and the next section I assume, as mentioned earlier, that p splits completely in F.
4.1. There exists a formula for
remarkably similar to that which the
Selberg trace formula yields for the right-hand side of equation (2.1).
The set KS(Fp) is partitioned naturally by the equivalence relation of isogenyi.e., two points lie in the same class when the abelian varieties-plus-structure that they parametrize are isogenous. The isogeny classes are of two types: (1) Those associated to systems (F',{qi}), where F' is a totally imaginary quadratic extension of F which splits at some prime over p and the qi are certain ideals of F'-if pl, , p, are the primes of F which split in F' then qi is one of the two prime factors of pi. Two systems (F/, {q1,i}) and (FF, {q2,i}) determine the same isogeny class if and only if Fi = FF and {q1,i}, {q2,1} are either equal or conjugate. (2) A single isogeny class associated to the quaternion division algebra D/F which is ramified precisely (a) where B/F is, (b) at all primes of F over p, and (c) at all real valuations of F. This latter case may be thought of as an amalgamation of all the quadratic imaginary F'/F which do not split over p. If Y = (F', {qi}) or D, the class of points in KS(Fp) associated to it will be denoted KS(Y). It is stable under the Frobenius; one may describe rather explicitly its structure together with this action. (For the parametrization of isogeny classes as well as the structure of each class, refer to Milne's lectures.) To each Y is associated: (1) A certain algebraic group H = Hy defined over Q. If Y = (F', {qi}) then H is (so to speak) the multiplicative group of F', while if Y = D it is the multiplicative group of D. (2) A certain algebraic group Gy = G defined over Qp. First of all, for each prime p of F over p one has Go = the multiplicative group of Fp = (Fp )Z
if Y = (F', {qi}) and p splits in F', = the multiplicative group of Dp otherwisei.e., if Y = (F', {qi}) and p does not split in F' or if Y = D. The group G is f16, (3) A coset space X = fl Xp, where X, = Gp(Fp)/Gv(ev). On each Xp define the transformation 00 = multiplication by a uniformizing elements of q if Y = (F', {gi}), p splits in F', and q is the qi over p, = multiplication by a uniformizing element of the prime ideal of D. otherwise.
152
W. CASSELMAN
Expressing Gp as (F1)2 )2 in the first case, with the first factor corresponding to p, cVu is
a little more explicitly multiplication by (p, 1). Let X = XY = fl XP' 0 = cY = 11 00.
Let R(Q) = H(Q)/H(Q) n ZK, G(Af) (an abuse of language) = G(A f) x G(Qp)/ZK n Af In every case, observe: 4.1.1. LEMMA. (a) The algebraic group Z is canonically embedded in H;
(b) There exists a canonical class of equivalent embeddings of H(Q) into G(A f) x G(Qp), rendering H(Q) as a discrete subgroup of G(Af).
Only the second needs explaining. There clearly exist natural embeddings of H(Q) into G(Af) and into G(Qp), all equivalent up to inner automorphism. The group H(R)/Z(R) is compact in all cases (no accident) so that since H(Q) is discrete
in H(A) = H(Af) x H(R) the group H(Q) is discrete in H(Af)/ZK n Af, hence in G(A f).
Langlands' main result on the structure of KS(Y) is that there is a canonical class of bijections between its points and the double coset space
H(Q)\G(Af) x G(Qp)/Kp x G(Zp) = II(Q)\G(Af)/KP x G(Zp)
H(Q)\G(A/) x X/Kp.
The Frobenius on KS(Y) corresponds to 0 acting on this coset space through its action on X. 4.2. The number of points of KS(Y) rational over Fp is of course the number of fixed points of the mth power of the Frobenius. To express this conveniently requires a digression of some generality. Suppose ' to be any locally profinite group, r a discrete subgroup. Then ' and hence also C°°(T) acts on the space C°°(F\5) through the right regular representation. Explicitly, for every F e C°°('), f e C°°(F\c),
RFf(x) = f F(g)f(xg) dg = f F(x ly).f(y) dy KF(x, y)f(y) dy
ErF(x 1 ry). The operator RF (or, sometimes, F itself) will be said to possess a formal trace if KF(x, x) has compact support on r\9, and this
where KF(x, y)
trace is then defined to be f r,, KF(x, x) dx. One can presumably relate this to other notions of trace, but all that is important here is that the number of fixed points of a power of the Frobenius can be expressed as such a trace and that one has a formula for it.
For each r e r, let 9r be the centralizer of r in 9 and F1 = F n 9r. If F lies in C°°(c), then its orbital integral f `$r\, F(g 1rg) dg is defined in many different
circumstances, but the most elementary hypothesis that guarantees this is that ; this in turn is assured by the assumption F(g-1 rg) has compact support on that the conjugacy class of r is closed in 9.
4.2.1. PROPOSITION. Let F be in C°°(c). Suppose that the r c Fsuch that F(g lrg) 0 0 for some g e lie in only a finite number of conjugacy classes in r, and that for
153
THE HASSE-WEIL t-FUNCTION
each such r the space
is compact and the conjugacy class of r in g is closed.
Then F has a formal trace which is given by the formula
meas(I'r\gr)
F(g-lrg) dg,
J
where the sum is over all conjugacy classes on P.
PROOF. Choose a compact open subgroup go s W small enough so that I (1 go = {I } and F is bi-invariant under go. Thus r acts freely on ig/go. Let -T be a set of representatives, so that c is the disjoint union of the Tx go (x e T). By one assumption on F, for each x e ' the sum ErF(xlrx) may be restricted .
to a finite number of conjugacy classes in F, hence may be written as E (r) Er,\r F(x 1b-1rox) where the first sum is over representatives of conjugacy classes and only a finite number need be taken into account. Because 1'r\'7. is com-
pact and the conjugacy class of r is closed in ', the function F(g 1rg) lies in C,-(P7.\1); this implies that the above sum is nontrivial for only a finite subset of X. Thus F does possess a formal trace, which is given by (meas
'o)
E E Er7\rF(x15-lrbx)
xe,T (r)
where in fact each sum may be restricted to a finite but arbitrarily large subset. Rearranging this it therefore becomes
F(xlrx) ds = E ir} IrT\g
meas(Pr\gr) {r)
J g7r\v
F(xlrx) dx.
4.3. Now take I' to be II(Q), 9 to be G(Af)-both corresponding to some isogeny class Y. Define FP = (meas KP)-1 char KP, Fpm)
FP
= (meas
6,(o,))-1 char(onm
G(op)),
)=flFm)
F(m) =FP FP
).
4.3.1. LEMMA. The hypotheses of 4.2.1 are satisfied for this choice of F, c, F = F(m) with m>_ 1.
When Y = D, this is clearly so since F\' is actually compact. Thus let Y = (F', {q;}). The only serious hypothesis to verify is the finiteness of the number of conjugacy classes with elements r such that F(glrg) 0 0 for some g e 9. Now in this case H(Q) = (F')" and G(Qp) = 116,(F,) where Gp is (a) (F¢)" or (b) Dp .
In either case F$)(g 1rg) = F(r), and FP )(r) 0 0 if and only if (a) at each Pi where F' splits r has order m at q1 and is a unit at q1; (b) where F' does not split r is a generator of pD. Furthermore, in order that FP(g 1rg) # 0 for some g e G(A) it is necessary and sufficient that r be a unit at all primes not dividing p. Thus if F(m)(glrg) 0 0 for some g, the order of r is specified at all primes of F'. Any two such r differ by a unit. But the units of F', modulo ZK, are a finite set. The function F(m) is important because of:
154
W. CASSELMAN
4.3.2. PROPOSITION. The formal trace of F(m) is the number of fixed points of the mth power of the Frobenius on KS(Y).
I leave this as an easy exercise.
5. The comparison. 5.1. From §3 one sees that the right-hand side of equation (2.1) is
meas(Pr\c,)
(5.1)
J
F(g lrg) dg
where r = G(Q)/G(Q) fl ZK, 1' = G(A)I ZK, F = fR - ft f. Let (p: G(A) --). W be the canonical projection. It is easy to see that gp(Gr(A))\'r is always compact, so that one may replace 9r by rp(G,(A)) in the above formula. But
then one may note that (p(G,(A))\' = G,(A)\G(A). Using the factorization of G(A) each term may be written as the product of (5.2) (a)
meas (Gr(Q)ZK\GT(A)),
(5.2) (b)
f
(5.2) (c)
fcr(Qy)\G(sy)f p)(g 1rpg) dg,
(5.2) (d)
SG7(R)\G(R) fR(g 1rRg) dg.
Gr(Aj)\G(Apf) fp(g lrpg) dg,
From §4 one sees that #KS(Fpm) may be expressed as (5.3)
F(m)(glrg) dg
2] E meas (Tr\'r) Y (T)
19?,\W
where :
Y ranges over all systems (F', {q;}), identifying a system and its conjugate, and the single D; r ranges over all conjugacy classes of Hy(Q); 9 = GY(A!) = G(Af) X GY(Qp); F(m) = FP . F(pm)
as at the end of §4. Just as above, each term here factors as the product of (5.4)(a)
meas(Hr((?)(ZK fl A f)\Gr(Af)Gr(Qp)),
(5.4)(b)
SGr(A)\GD(Af) Fp(g lrpg) dg,
(5.4)(c)
f J c,(eD>\c(eD>
Fpm)(g-lrpg) dg.
The proof of the main theorem reduces to a comparison of measures and orbital integrals. 5.2. Even the comparison of measures is not trivial. For the moment let k be an
arbitrary local field, 0 an additive character of k. If H is any finite-dimensional semisimple algebra over k, Tr its reduced trace, then 0 o Tr is an additive character of H. There exists on H a unique measure dx self-dual under the Fourier transform
THE HASSE-WEIL a-FUNCTION
155
determined by this character, and this in turn gives rise to an invariant measure dxx = dx/IxIH on Hx. If now k is any global field, = jj 0, a character of Ak/k, and H a semisimple algebra over k, one obtains thus measures on all the local groups Hv as well as on Hx(Ak), which is called the Tamagawa measure determined by 0 (§15 of [16]). Apply this construction in turn to the field F itself, quadratic extensions of F, and quaternion algebras over F (including M2(F)). I will always assume such measures chosen from now on. One classical result is that the global measure of Hx(A)lAF H"(F) is independent of the quaternion algebra H (§16 of [16], or the lectures [12] of Gelbart and Jacquet). I will also need a comparison of local measures: 5.2.1. LEMMA. Let k be any nonarchimedean local field, H the unique quaternion division algebra over k. Then
meas off = (q - 1)-1 meas GL2(ok). PROOF. Changing 0 only multiplies both measures by the same constant; choose to be of conductor ok. {Let f be the characteristic function of M2(ok). Then
J (x) =$Mz(k)f(y)b(Tr(xy)) dy =measM2(ok) 50,
x 0 WOO, _ (meas M2(or)l) f(x). Therefore meas M2(oe) = 1. The group GL2(or) is open in M2(or) and since x e M2(oe) lies in GL2(o,) if and only if x mod pE is nonsingular,
meas GL2(of) _ # GL2(FQ)/# M2(FQ)
= (1 - 1/q)2 (1 + 1/q). Now let f be the characteristic function of oH. Then
AX) =
SG(Tr(xy)) dy SDy
= meas off
1,
xepH 1 ,
0,
x011tr,
or
f = (meas DH)(characteristic function of pH l).
The Fourier transform of this in turn is f = (meas oH)(meas(pf. )) f.
Since
meas(pH1) = q2 . meas OH, meas off = q-1. Reasoning as above,
meas off = (1 -1/q)(1 + 1/q)(1 /q). The lemma follows. One can similarly compare measures on GL2(R) and Hx, but that will prove to be unnecessary. 5.2. Again for a while let k be any local field of characteristic 0, 9 either GL2(k), the multiplicative group Hx(k) of the quaternion division algebra over k, PGL2(k),
156
W. CASSELMAN
or H"lk'<. For any x e 9 define D(x) = Ia - (312/lap1 of the eigenvalues of x (over an algebraic closure of k) are a, 3. If T is any torus of 9, then for every regular t e T and f E C°°(c) define OT(f, t) = fT\T f(g ltg) dg.
The function F(t) = OT(f t) satisfies (5.5) (a)
F is smooth on the regular elements of T,
(5.5) (b)
Dl i2 F is locally bounded,
(5.5) (c)
Fhas compact support on T.
If it is an irreducible admissible representation of ' then the character ch,r of r also satisfies (5.5) (a)-(b). For any f e C'°(9),
f(g) dg =
2E fT)
t)
dt,
where the sum is over all conjugacy classes of tori in ', and if ir is an irreducible admissible representation then (5.6)
trace r (f) = 2 E f TD(t) ch,(t)OT(f t) dt.
(For all this see §7 of [16], and for more about orbital integrals see [12] and [26].)
5.2.1. LEMMA. Suppose that 9 = H", F a conjugation-invariant function on 'leg whose restriction to any torus T satisfies (5.5) (a)-(c). If
2 E f T D(t) cho(t)F(t) dt = 0 for all irreducible admissible representations of ,- then F - 0. Fix a torus T. The set of elements of c conjugate to regular elements of T is open. If f e C0°(TLeg), there exists h e C.°(greg) with support in this set and such that OT(h) = f Then
h(g)F(g) dg = J9
D(t)OT(h, t)F(t) dt JT
= fD(t) f(t)F(t) dt. But the characters of irreducible representations are complete in the space of central functions on 9, so that f = E fo cho (if k is nonarchimedean one can reduce this question to one about finite groups) and hence by hypothesis the above integral is 0. In other words, the integral of D F against any f e Q°(T7eg) is 0, which implies that F itself is 0. One may (and I will) identify conjugacy classes in Hx with elliptic and central conjugacy classes in GL2(k). There exists (by §15 of [16]; see also [12]) a bijective correspondence a H ir between (a) irreducible, smooth (hence finite-dimensional) representations a of HX and (b) irreducible, admissible representations it of GL2(k), space-integrable modulo Z, such that
157
THE HASSE-WEIL C-FUNCTION
cho(t) = - ch,(t)
(5.7)
for all regular elliptic elements t. Representations of Hxlkx correspond to representations of PGL2(k). When k = R, the trivial representation of Hx corresponds to the single discrete series representation ro of PGL2. For nonarchimedean k, the one-dimensional representations of Hx correspond to the Steinberg (or special) representations and all others correspond to absolutely cuspidal representations of GL2. 5.2.2. PROPOSITION. Let fR E C,°(PGL2(R)) be as in §3. Then for semisimple x E
PGL2(R) = 9, (g 1xg) dg = 0,
x hyperbolic,
= (meas Cx/Rx)-1,
x elliptic,
= -(meas Hx/Rx)-l,
x = 1.
PROOF. The case of hyperbolic x fell out in the very construction of fR. Recall that fR has the property
z = 1 or sgn(det),
trace 2r(fR) = 1,
r=Zo, = 0,
otherwise.
Consider now the constant function hR on Hx/Rx with the value (meas Hx/Rx)-1. It satisfies
trace o'(hR) = 1,
= 0,
6 = 1, otherwise.
Thus by (5.7) and remarks just afterwards trace n (fR) = - trace a(hR) whenever r and o- correspond to one another. This and equation (5.6) now yield JCx/Rx
D(t) ch,r(t) dt
fR(g 1tg) dg
fC"\PGL2(R)
_
-`
J cxiRx
D(t)cho(t) dt
fCx\Hx
hR(g-ltg) dt
whenever ir and u correspond. By (5.7) and 5.2.1 (or in this case just Fourier inversion on Cx/Rx),
JCx\PGI,20 f (g-ltg) dg =
hR(g ltg) dg = (mess
Cx/Rx)-1
fCx \Hx
for all t# ±lECx/Rx. Since the orbital integrals of fR vanish on hyperbolic elements, fR and - hR corres-
pond to one another in the sense of [121 (the section on "matching orbital integrals"). Hence
fR(l) = -hR(l) = - (meas Hx/Rx)-1. 5.3. Now let k be nonarchimedean and local, and for the moment let G = GL2(k),
158
W. CASSELMAN
K = GL2(ok). For each m >-- 0 let f (m) be the unique function in the Hecke algebra
,*'(G, K) defined according to the Satake isomorphism by trace 2r(f(m)) = qm/2(am + 3m) whenever 7c = 2r(a, 3) is the unramified principal series represen-
tation associated to the characters x - (aord(x), 3ord(x)) of V. One can exhibit f (m) rather explicitly. For every i, j e Z recall the Hecke operator T(pi, pi) = (meas K)-1 char K(7i 12 )K \\T(pm) = F, T(pi, pt) (i, j > 0, i + j = m). There are the more or less classical Hecke operators and satisfy the
where rJ is a generator of p. For m >-- 0 recall also
relation T(p)T(pm) = T(pm+l) + qT(p, p) T(pn-1) which may be written formally as
(1 - T(p)X + qT(p,
(5.7)
p)X2)-1
T(pm)Xm.
5.3.1. LEMMA (IHARA). One has
f(0) = 2 . I, f(l) = T(p), .f cm) = T(pm) - qT(p,
p)T(pm-2)
(m > 2).
PROOF. The case m = 0 is clear. And it is well known that trace z(T(p)) _ q1 /2(a + ) if ir = ir(a, (3), which gives the case m = 1. Formally, then T(p) t--> q1 /2(a + i3) and, also elementary, T(p, p) <-->
(I - ql/2aX)-1 + (1 - ql/2
Now
qm/2(am + 3m)Xm,._, E fcm)Xm
(3X)-1 = 0
but this expression also equals (I - q1/2aX) + (1 -q 112/3X) 2 - T(p)X (I - q1/2aX) (I - q112RX) `-' 1 - T(p)X + gT(p,p)X2* The proposition follows from this and (5.2). 5.3.2. COROLLARY. If
x=
(V1 I
7)1
then
f cm)(x) = - (meas K)-1(q - 1), = 0,
m = 21, otherwise.
Continue to let H be the quaternion division algebra over k. From each m >-- 0 let h(m)
= (meas
oH)-1 char(pH - pmH+l).
5.3.3. PROPOSITION. Let x be a semisimple element of G, m > 1. Then: (a) If x is hyperbolic,
J Gx\Gf
(m)(g lxg) dg = 1
(b) If x is elliptic
if x =
7)m
= 0 otherwise.
1) or (1
\ 12m
I mod (K (l A),
159
THE HASSE-WEIL C-FUNCTION
L\G f(m)(g lxg) dg =
fHY HX h(m)(g l xg) dg
(c) If x is central, f (m) (x) = -h(m)(x).
PROOF. Recall (say from [4]) the relationship between the Satake isomorphism and orbital integrals: for f e .*'(G, K)
f(a) = D(a)1 /2 f
A\G
trace rx(f) = f Af (a)x(a) da. J
fig -lag) dg,
By definition of f (m), then,
a=
f(m) (a) =qm/2,
1) or (1 otherwise.
=0
(yl'n
m),
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 rr ranges over all essentially square-integrable irreducible representations of GL2:
f
trace 9L(f (m)) _
+
2
D(a)ch,,(a) A
da
f f (m)(glag) dg A\G
E I T D(t)ch, (t) dt f T\Gf (m)(g ltg) dg
where now T ranges over all anisotropic tori. Similarly consider (5.6) as r ranges over all irreducible admissible representations of H": trace u(h(m)) =
2 (Z f T
D(t)cho(t) dt f
h(m)(gltg) dg. T \G
Applying 5.2.1 and equation (5.7), it thus suffices to prove that
- trace z(f (m)) + 2
f
A
D(a)ch,,(a) daJ
A\Gf
(m)(glag) dg = trace o(h(m))
whenever a and z correspond. For this: (1) no iv occurring here possesses K-fixed vectors 0 0, so trace ir(f (m)) = 0 always. (2) If iv is absolutely cuspidal then ch,(a) = 0 unless a e A(ok) (first observed in Lemma 6.4 of [19]; see also [8] and [5]). According to case (a) already done, OA(f) = 0 on A(o), so the whole left-hand side vanishes. But in this case a is not one-dimensional, hence not trivial on 011H, so that the right-hand side vanishes also. (3) Suppose iv is special, corresponding to the character u = x o NM of H'<. Thus c(h(m)) = x(ylm) on the one hand and on the other 2
f
A
D(a)ch"(a) da f Avcf (m)(glag) dg =
also, by case (a) again.
5.4. Return to earlier notation. It will now turn out that there is a bijective correspondence between the nonzero terms of the two sums (5.1) and (5.3), and that corresponding terms agree. This will conclude our proof.
160
W. CASSELMAN
According to 5.2.2 the orbital integral of fR in (5.2) is 0 unless rR 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 (5.8) (a)
meas(G(Q)ZK\G(A)),
(5.8) (b)
fpm)(rp) _ (- 1)n (meas D>(Zp))-1,
(5.8) (c)
.fp(rp),
(5.8) (d)
fR(rR) _ (- 1)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 (5.9) (a) (5.9) (b) (5.9) (c)
meas (Dx(Q)(ZK fl Af)\Dx(Af)), F(-)(7-p) = (meas D'(Zp))-1, FP(rP).
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 (1 Af)\D>(Af)) meas(Z(R)\Dx(R)).
Now suppose r e G(Q) such that rR 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 = Y. is one of the (F', {q;}), and the q1 must be specified. Now by the definition of f pm) as the product of the f,(-) and 5.3.3, if the term for r in (5.2) (c) does not vanish then rp is conjugate in GL2(FF) to 1
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 Y,. = (F', {qr}). In either case the element r clearly gives rise to an element of Hy (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 Hecke operators and the Frobenius. Although such a relation is likely to hold very generally, it does not yield a formula for the c-function. I want to explain this a bit more carefully. First of all, the integral Hecke operators always define algebraic correspondences
on the scheme KS. When F = Q, the congruence relation says that modulo p T(p) _ 0 + 0* T(p, p) where 0 is the Frobenius and V* 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 0 is a root of the poly-
nomial X2 - 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 isomorphism it corresponds to a polynomial whose coefficients are Hecke operators.
THE HASSE-WEIL C-FUNCTION
161
These will in fact be integral, and it looks not very difficult to show that 0 is a root. In more concrete terms, this will imply immediately that the roots of the Frobenius
(acting on l-adic cohomology) lie among the roots of this polynomial, or that factors of Zp(X, KS) lie among the factors of ll,insl,, det (I - (g(,p)))r"1, 1P). Now when F = Q, one obtains by a further relation which I have always found a little mysterious that in fact the roots coincide, but it is this second step that does not seem to generalize (I refer to 7.10(2) of [28] ; see also Piatetskii-Shapiro's argument on pp. 333-336 of [24] for a representation-theoretic analogue). Langlands has remarked that there is in some sense a good reason for this difficulty, inasmuch as
when problems of L-indistiguishability arise one may get in some sense only partial coincidence of the roots of the Frobenius with those of the Hecke polynomial. Incidentally, in his proof of the main theorem in the cases already mentioned, Shimura also used the Selberg trace formula.
I might also mention that it is apparently Ihara who first applied the trace formula to matters of this kind-in [14] he showed, modulo some technical problems, that the Hasse-Weil c-function associated to certain sheaves on SL2(Z)\.W is a product of Hecke L-functions. Both he and Shimura used what one might call a global formulation. The first application involving representation theory and local orbital integrals is Langlands' Antwerp talk [19] ; this earlier work seems to have played a role in Langlands' own further development as well as in Drinfeld's (note the remark in [9] on this point). 6.2. It is very little extra work to extend the proof of the main theorem to allow for nontrivial sheaves as well as a refinement which accounts for a factorization of the c-function according to the direct sum decomposition of .sago. Let me first sketch how to deal with sheaves. Suppose : G -> GLn to be a rational representation of G (i.e., rational over Q) trivial on NF/Q, the kernel of the norm map from F" to Q" (considered as algebraic groups) which is canonically embedded as scalars in the centre of G. Then one may associate to a locally constant Q-sheaf EE on KS(C) and 1-adic sheaves EF(Qi) on KS (as in a special case in [19]), and consider for each "good" p the p-factor of a c-function 2n
ep(S, KS,
fJ det(I - (OIH=(KS, Ee(Q,)))P 5)(-1)'. i=o
By the Lefschetz trace formula this satisfies
log Cp = Z M=1
Impms
Z trace a ,(O-).
XEKS(Fp*)
In both these formulas 0 is the Frobenius, and for convenience I have written i; x(Om) as the action of Om on the stalk of E£ at the fixed point x of Om.
On the other hand one can define numbers m(zR, ) analogous to the numbers 0 and 2rR)
m(i R) in 1 satisfying, for example, m(2rR, ) = 0 unless set m(lr) = m(rrR)m(lrP). The main theorem extends to:
Lp(S, KS, S) =
7 11
L(s - n/2, 9G p, P)m(Ir.f)
lr in .ato
The proof is almost exactly the same as for e trivial; one uses an extended version of
162
W. CASSELMAN
the trace formula for the trace of Rf, f e C°°(c), on the representation of c which is L2-induced from e :
trace Rf =
meas(Pr\Wr) Tr e(r) f
f(g-1 rg) dg.
Vr\T
And one also uses a result of Langlands analogous to the one used above which gives not only the structure of KS(Fp) but of E1 on KS(FP) (the natural guess is correct). Now the second point. As Langlands shows, essentially, in [19], corresponding to the decomposition Q = @z one has a decomposition H*(KS(C), = (DH,*.
The only rr which occur are those with m(i) 0 0, and for a 7r = rrR Qx z f which does occur the representation z f may be defined over Q. Corresponding in turn to this decomposition is one of H*(KS, Qt), and hence even of H*(KS X FP, Qt). Another extension of the main theorem is that for good p 2n
Lp(s, r, p) = f j det(I - (0I Hn(Q))ps) (-1)
To prove this: (1) one observes that the subspace H*(Q) is determined in H*(KS,Q) by the effect of operators in Xe(G(A f), KP) and (2) applies the reasoning given above, but allowingfP to be an arbitrary element of this Hecke algebra. It is this result which Piatetskii-Shapiro is concerned with proving (by means of a congruence relation) in [24], with F = Q. 6.3. It has not escaped me that the result in §5 on the measures on GL2 and H" says, essentially, that the measures given by Jacquet-Langlands are multiples, by the same constant, of what Serre calls in [25] the Euler-Poincare measures on each group. (This is not quite true since strictly speaking the E-P measures on these are zero: but there is a clear relationship between E-P measures on SL2 and NH.)
Does this observation generalize? For real groups, I would expect a factor card(WG(c)/WK),to play a role, as it does in Harder's work [13]. Indeed, I suspect that Harder's paper does essentially relate inner twisting of measures to Serre's measures in the case when G is the inner twist of an anisotropic group. If G is not such an inner twist then Serre's measure is identically 0; how to describe the relationship of measures then? As for comparison of orbital integrals, Drinfeld has a nice result for GL, and division algebras of degree n (see Kazhdan's talk at this Institute). Diana Shelstad had proven in her thesis [27] many pretty results on real groups. REFERENCES
1. T. Asai, On certain Dirichlet series associated with Hilbert modular forms and Rankin's method, Math. Ann. 226 (1977), 81-94. 2. A. Borel, Automorphic L -Junctions, these PROCEEDINGS, part 2, pp. 27-61. 3. A. Borel and N. Wallach, Cohomology of discrete subgroups of semisimple groups, Notes, Inst. Advanced Study, Princeton, N. J., 1977. 4. P. Cartier, Lectures on representations of p-adic groups: A survey, these PROCEEDINGS, part 1, pp. 111-155. 5. W. Casselman, Characters and Jacquet modules, Math. Ann. 230 (1977), 101-105. 6. P. Deligne, Formes modulaires et representations 1-adiques, Sem. Bourbaki 1968/69, no. 355,
Lecture Notes in Math., vol. 179, Springer, New York.
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163
7. P. Deligne, Travaux de Shimura, Sem. Bourbaki 1970/71, no. 389, Lecture Notes in Math., vol. 244, Springer, New York. 8.
, Le support du caractere dune representation supercuspidale, C. R. Acad. Sci. Paris
Ser. A-B 283 (1976), A155-A157.
9. V. G. Drinfeld, Proof of the global Langlands conjecture for GL(2) over a function field, Functional Anal. Appl. 61 (3) (1977), 74-75. (Russian) 10. M. Duflo and J-P. Labesse, Sur la formule des traces de Selberg, Ann. Sci. Ecole Norm. Sup. 4 (1971), 193-284. 11. M. Eichler, Introduction to the theory of algebraic numbers and functions, Academic Press, New York, 1966. 12. S. Gelbart and H. Jacquet, Forms of GL(2) from the analytic point of view, these PROCEEDINGS,
part 1, pp. 213-251. 13. G. Harder, A Gauss-Bonnet formula for discrete arithmetically defined groups, Ann. Sci. Ecole Norm. Sup. 4 (1971), 409-455. 14. Y. Ihara, Hecke polynomials as congruence cfunctions in elliptic modular case, Ann. of Math. (2) 85 (1967), 267-295. 15. H. Jacquet, Automorphic forms for GL(2) II, Lecture Notes in Math., vol. 278, Springer, New York. 16. H. Jacquet and R. P. Langlands, Automorphic forms for GL(2), Lecture Notes in Math., vol. 114, Springer, New York, 1973. 17. R. Kottwitz, Combinatorics and Shimura varieties mod p, these PROCEEDINGS, part 2, pp. 185-192.
18. S. Lang, SL2(R), Addison-Wesley, 1975. 19. R. P. Langlands, Modular forms and 1-adic representations, Lecture Notes in Math., vol. 349, Springer, New York, 1973. 20. , Some contemporary problems with origins in the Jugendtraum, Proc. Sympos. Pure Math., vol. 28, Amer. Math. Soc., Providence, R.I., 1976, pp. 401-418. 21. , Shimura varieties and the Selberg trace formula, Canad. J. Math. (to appear) 22. , On the zeta-functions of some simple Shimura varieties (to appear). 23. Y. Matsushima and G. Shimura, On the cohomology groups attached to certain vector-valued differential forms on the product of upper half-planes, Ann. of Math. (2) 78 (1963), 417-449. 24. I. I. Piatetskii-Shapiro, Zeta functions of modular curves, Lecture Notes in Math., vol. 349, Springer, New York, 1973.
25. J-P. Serre, Cohomologie des groupes discrets, Prospects in Mathematics, Ann. of Math. Studies, no. 70, Princeton Univ. Press, Princeton, N. J., 1971. 26. D. Shelstad, Orbital integrals for GL2(R), these PROCEEDINGS, part 1, pp. 107-110. , Some character relations for real reductive algebraic groups, Thesis, Yale Univ., 1974. 28. G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton, 1971. 29. , On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. 31 (1975), 79-98. 27.
UNIVERSITY OF BRITISH COLUMBIA
Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 165-184
POINTS ON SHIMURA VARIETIES mod p J. S. MILNE There is associated to a reductive group G over Q with some additional structure
a Shimura variety S, defined over C. In most cases it is known that S, has a canonical model SE defined over a specific number field E. For almost all finite primes v of E it is possible to reduce SE modulo the prime and obtain a nonsingular
variety S over a finite field F9,. As is explained in [3], in order to identify the Hasse-Weil zeta-function of SE or, more generally, of a locally constant sheaf on SE it is necessary to have a description of (Sv(Fq,), Frob) where Sv(Fq) is the set of points of S with coordinates in the algebraic closure Fgv of Fg, and Frob is the Frobenius map Sv(Fgv) - Sv(Fgv) which takes a point with coordinates (a1,.., a,n) to (aq,, , aqg). To be useful, the description should be directly in terms of the group G. Recently [13] Langlands has conjectured such a description of (Sv(FFq), Frob) for any Shimura variety S and any sufficiently good prime v. In [12] he has given a
fairly detailed outline of a proof of the conjecture for those Shimura varieties which can be realized as coarse moduli schemes for problems involving only abelian varieties, (weak) polarizations, endomorphisms, and points of finite order. (So G(Q) is of the form AutB(Hl(A, Q), 0) where B is a semisimple Q-algebra containing an order which acts on the abelian variety A, 0 is a Riemann form for A whose Rosati involution on End(A) D Q stabilizes B, and AutB refers to Blinear automorphisms g of H1(A, Q) such that ci(gu, gv) = ci(u, u(g)v) with ,u(g) lying in some fixed algebra F contained in the centre of B and fixed by the Rosati involution; there is also a Hasse principle assumption.) Earlier [8, Conjecture 1] Ihara had made a similar conjecture when S is a Shimura curve and had proved it when G = GL2 [9, Chapter 5]. When G = Bx, B a quaternion division algebra over Q, Morita [15] proved Ihara's conjecture for all primes p of E (= Q) not dividing the discriminant of B. Both he and Shimura have obtained partial results for more general quaternion algebras (unpublished). More recently Ihara has proved his conjecture for all Shimura curves and sufficiently good primes (announcement in [11]). While Ihara bases his proof on the EichlerShimura congruence relations, Morita's method, as described in [10], appears to be quite similar to that of Langlands. In order to give some idea of the techniques Langlands uses in his proof I shall describe it in the case that G is the multiplicative group of a totally indefinite quaAMS (MOS) subject classifications (1970). Primary 14K10; Secondary 14G10, 14L05. © 1979, American Mathematical Society 165
166
J. S. MILNE
ternion algebra over a totally real number field. In § 1 it is shown that S, parametrizes, in a natural way, a family of abelian varieties with additional structure. The following section describes how Artin's representability criteria may be used to prove the existence of a variety SQ over Q which is a canonical model for Sc and which, when reduced mod p, parametrizes a family of abelian varieties (with additional structure) in characteristic p. Thus the problem of describing (Sp(Fp), Frob) becomes one of describing this family. In §5 the Tate-Honda classification of isogeny classes of abelian varieties over finite fields is used to determine the isogeny classes in the family, and in §6 the individual isogeny classes are described.
Since this requires the use of p-divisible groups and their Dieudonne modules, these are reviewed in §3. Notation. F is a totally real number field of degree d over Q, B is a quaternion division algebra over F which is split everywhere at infinity, b H b* is a positive F-involution on B, and OB is a maximal order in B. G is the group scheme over Z such that G(R) = (O°BPP (D R)x for all rings R, where OAPP is the opposite algebra to OB.
A is the ring of adeles for Q; A= R x A1 = R x A f x Q p; A f= Zf Q Q, Zf = proj lim Z/mZ; Zf = Zf x Zp. K is a (sufficiently small) open subgroup of G(Z f). J is a product of rational prime
numbers such that if p ,' d then p is unramified in F, B is split at all primes of F dividing p, and K = KPG(Zp) where KP = K n G(A f). SC = KSc is the Shimura variety over C defined by G, K, and the map h: Cx G(R) defined in §1; thus its points in C are Se(C) = G(Q)\G(A)/K K where K00 is the centralizer of h in G(R).
If V = V(Z) is a Z-module then V(R) = V Oz R for any ring R. 1. Sc as a moduli scheme. Recall that an abelian variety over a field k is an algebraic 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 + 27b2 0 0.
It is impractical to describe abelian varieties of dimension greater than one by equations, but fortunately over C there is a classical description in terms of lattices in complex vector spaces. Let V be a lattice in Cg, i.e., V is the subgroup generated by an R-basis and so V ©Z R Cg. Then Cr!V is a compact complex-analytic manifold which becomes a commutative Lie group under addition. When g = 1 the Weierstrass p-function corresponding to V, and its derivative, define an embedding
Z - (p(z), p'(z), 0: C/ V c PP of C/Vas an algebraic subset of the projective plane. Thus Cl 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 Cg, Cg/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 Cg such that, for v e V, 0(z + v) = 0(z) exp (2iri(L(z, v) + J(v))) where L(z, v) is a C-linear function of z and J(v) depends only on v. One shows that L(z, v) is additive in v, and so extends to a function L: Cg x Cg -+ C which is C-linear in the first variable and R-linear in the second. Set E(z, w) = L(z, w) - L(w, z). Then (a) E is R-valued, R-bilinear, and alternating; (b) E takes integer values on V x V; (c) the form (z, w) H 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 Cr! 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 fundamental 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 Cg!V is compact the algebraic structure on Cr! V (but not the projective embedding) defined by a Riemann form is independent of the form. Thus, given a lattice in Cg 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 (according 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(C). The lattice V can be described as the kernel of the exponential exp : Lie(A) -+ A(C),
or as the fundamental group of A(C) which, being commutative, is equal to HI(A, Z). We shall always regard the isomorphism W/V x. A(C) as arising from the exact sequence,
0-HI(A,Z)->tAe+A(C)-i0. Since HI(A, Z) is a lattice in tA; we have HI(A, R) = HI(A, Z) px R tA. Thus A is determined by HI(A, Z) and the complex vector space structure on HI(A, R).
A complex structure on a real vector space V(R) defines a homomorphism h: Cx --> AutR(V(R)), h(z) = (v f- zv), and the complex structure is determined by
h. Thus an abelian variety A is uniquely determined by the pair (HI(A, Z), h) where h: Cx --* Aut(HI(A, R)) is defined by the complex structure on HI(A, R) = tA. Moreover every pair (V(Z), h) for which there exists a Riemann form arises from an abelian variety. Let V(Z) = HI(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
168
J. S. MILNE
finite order on A may be identified with V(Q)/V(Z) c V(R)/V(Z). For any integer
m > 0, the group Am(C) of points of order m is equal to m-1V(Z)/V(Z) V(Z/mZ) x (Z/mZ)21im(A) We define TfA to be proj limmAm(C) = V(Zf) and, for any prime 1, T1A to be proj limmAi,,,(C) = V(Z1); thus TfA = 11T1A and T1A N Z12dim(A)
A homomorphism A ---> A' of abelian varieties induces a C-linear map tA --> tA' such that H1(A, Z) is mapped into H1(A', Z). Conversely, if A and A' correspond respectively to (V, h) and (V', h') then a map of Z-modules a:V(Z) - V'(Z) extending to a C-linear map a O 1: V(R) -* V'(R) (i.e., such that a O 1 ° h(z) = h'(z) ° a O 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 End°(A) for End(A) Qx z Q. Since End°(A) has a faithful representation on H1(A, Q), it is a finite-dimensional Qalgebra; it is also semisimple, and its possible dimensions and structures are well understood. To define a homomorphism is OB -> End(A) when A corresponds to (V, h)
is the same as to define an action of OB on V such that h maps Cx into AutoBoR(V(R)). When such an i is given we say that OB acts on A provided i(l) = 1. Such an i induces an injection is B <., End°(A). A nondegenerate Riemann form E for A defines an involution a a' of End°(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 0 where Tr denotes the reduced trace from End°(A) to Q. Suppose OB acts on A. We say that two Riemann forms E and E' on A are F-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 F-equivalence class A of nondegenerate Riemann forms. Since F is the centre of
B, the Rosati involutions defined by any two elements of such a A induce the same map on i(B). We shall be interested in triples (A, i, A) such that E(i(b)u, v) _ E(u, i(b*)v) for u, v e V(R), b e B, E e A, i.e., we require that the Rosati involutions defined by A stabilize i(B) and induce the given involution b H b* on B. We next review some notations concerning B. The main involution b --. b' of B is so defined that under any R-isomorphism B OF R M2(R), if b corresponds to
M = (a d) then b' corresponds to M' = (d - a) ; thus b + be = TrB,F(b) and MI _
NmB,F(b). The Skolem-Noether theorem shows that there exists a t E B such that b* = t-lb't = tb't-1 for all b e B; automatically t2 e F and the positivity of b F-, b* implies that t2 < 0, i.e., t2 has negative image under all embeddings F C+ R. We fix an isomorphism B px Q R-. M2(R) x x M2(R) such that if b H (M1, then b* H (MMr, , Mnr) where M;r is the transpose of M.. Since
Mtr = t maps to an element (c1(° -o),
so that c > 0.
,
(0
-10)_M`(0 -10)
-o)) with each c; E R, and t may be chosen
The next lemma implies that if OB acts on a complex manifold C2dl V then there is
a Riemann form E for V whose corresponding Rosati involution induces b H b* on B and any two such forms are F-equivalent, i.e., that there is a unique weak
POINTS ON SHIMURA VARIETIES mod p
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polarization which is compatible with the OB-action and the given involution. LEMMA 1.1. Let V = V(Z) be a free Z-module of rank 4d on which OB acts. There is a nondegenerate alternating form 0 on V(Q) such that: (a) ci(u, v) e Z if u, v E V(Z); (b) cb(ut, u) < 0 for all u zA 0, u e V(R); (c) 0(bu, v) = cb(u, b* v) for all b e B and u, v e V(Q); (d) for any B-automorphism a of V(Q) there exists a p(a) e Fx such that cb(au, av) = cb(u, ,u(a)v) for all u, v e V(Q). Moreover' if 0' is a second nondegenerate alternating form on V(Q) satisfying (c) then there exists a c e Fx such that 0(u, cv) _ cb'(u, v) for all u, v c V.
PROOF. V(Q) has dimension one over B and so, after choosing an appropriate basis vector, we may identify V(Q) with B and V(Z) with a left ideal in OB. Define cb(u, v) = TrB/Q(uvct) = TrBIQ(uty*). Then ci(u, v) = Tr (utv*) _
Tr(vt*u*) = Tr(v(-t)u*) = -ci(v, u), and so 0 is alternating. (a) is obvious, and c,(ut, u) = TrB,Q(ut2u*) = TrF,Q(t2TrB,F(uu*)) < 0 for u # 0, which proves (b) and that 0 is nondegenerate. For (c) we note that ci(bu, v) = Tr(utv*b) = 'fr(ut(b*v)*) = cb(u, b*v). Finally, any B-automorphism a of V(Q) = B is multiplication on the right by an element b e Bx. Thus cb(au, av) = Tr (ubb'v't) _ cb(u, u(a)v) with ,u(a) = NmB/F(b). For the last part, consider the Q-linear map v H 0'(1, v): B -> Q. Since TrBIQ: B x B -+ Q is nondegenerate, there is a unique b e B such that 0'(1, v) = Tr (btv*) for all v c B. Then cb'(u, v) = 0' (1, u*v) = Tr (btv*u) = Tr(ubtv*) = Tr(ubv't). We also have 0'(1, v) _ -cb'(v, 1) = - Tr (vbt) = - Tr (t,b,v,) = Tr (b'v,t) = Tr (b,tv*). Thus b = b', which implies that it is in F, and we may take c = b. For the remainder of this section V(Z) will be OB regarded as an OB-module and 0 will be as in the lemma. For any ring R we may identify G(R) with AutoB®R(V(R)) since any O® x R-endomorphism of V(R) = OB Qx R is right multiplication by an element of OB Qx R. Define h to be the homomorphism Cx --* G(R) = AutB®R(V(R)) such that h(i) is right multiplication on V(R) = B O R M2(R) x ... x M2(R) by ((°10), , (°_10)). Thus K_ _ {(M1, , Md)} with Mi of the -=
form(ab Q), a, b e R. The form E is a Riemann form for (V(Z), h), e.g., cb(iu, iv) = c (uh(i), vh(i)) = cb(u, NmBIF(h(i))v) = 0(u, v) and 0(iu, u) ¢(ut/ - (- t2)1 /2, u) > 0 for u 0. Thus (V(Z), h) defines an abelian variety A. The action of OB on V(Z) induces a map i : OB - End(A) and the Rosati involution defined by the weak polarization A containing 0 induces b H b* on B. Recall that K is an open subgroup of G(Zf). Two isomorphisms 01, 02: TfA V(Zf) are K-equivalent if there is a k e K such that 01 = k02. For example, if K = K,n = Ker(G(Zf) -+ G(Z/mZ)) then to give a K-equivalence class of isomorphisms Tf A --j V(Zf) is the same as to give an isomorphism A,,(C) x. V(Z/mZ), i.e., a level m structure. THEOREM 1.2. There is a one-one correspondence between the set of points Sc(C) =
G(Q)\G(A)/K0K and the set of isomorphism classes of triples (A, i, ) where A is an abelian variety of dimension 2d, i defines an action of OB on A, and 0 is a Kequivalence class of OB-isomorphisms TfA
V(Zf).
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REMARK 1.3. (a) We say that two triples (A, i, ) and (A', i', ') are isomorphic if there exists an isomorphism a: A -> A' such that a a i(b) = i'(b) a a for all b E OB
and 0' o(Tfa)eq for all 0'e0'. (b) Normally when considering families of abelian varieties parametrized by Shimura varieties it is necessary to work with quadruples (A, i, A, ) with A a (weak) polarization. This is not necessary in our case because, as we observed above, A
always exists uniquely. PROOF OF .1.2. We first show how to associate to any g e G(A) a triple (Ag1 ig,
If g = 1 we take (A, i, ) with (A, i) as defined before and
the class of the identity
map TfA = V(Zf) 'd. V(Zf). We write a general g as g = g0gf, g- e G (R), gf e G(Af), and use g_ and g f to modify respectively the complex structure on V(R) and the lattice. Define hg: Cx -> G(R) by the formula hg(z) = g0h(z)g-1 and define gV(Z) to be the lattice g fV(Zf) n V(Q), the intersection taking place inside V(Af). Then Ag is to be the abelian variety defined by the pair (gV, hg). Since OB still acts on gV(Z) we have an obvious map ig : OB y End(A). We define (bg : TfAg = g1V(Zf) V(Zf) to be multiplication by g f'.
If g is replaced by gk0. with k- e K00 then hg is unchanged since K00 is the centralizer of h in G(R). If g is replaced by gk f with k f e K then hg and gV(Z) are unchanged while (bg is replaced by kfl (5g, which is K-equivalent to 0g. If g is replaced by qg with q e G(Q) then q-1: V(R) --> V(R) defines an isomorphism (Aqg, ) (Ag, ). Thus (Ag, ) depends only on the double coset of g. Conversely, an isomorphism a: (Ag, . ) -> (Ag., ) is induced by an isomorphism V(R) => V(R) which sends gV(Z) isomorphically onto g'V(Z). In particular a defines a B-isomorphism q: V(Q) -> V(Q). Thus q e G(Q) and so, after replacing g' by q-1g' and a by q-1a, we may assume that the map V(Q) -> V(Q) corresponding to a is the identity. Thus g0h(z)g--1 = g-h(z)g-1 for all z, and so g_-1 g- e K . Moreover, gV(Z) = g'V(Z) impliesg f1 g; e G(Zf), and g71: g V(Zf) -> V(Zf) being K-equivalent to gf1: g'V(Zf) -> V(Zf) implies that gf1 gf e K. Finally we have to show that every (A, i, 0) arises from some g. Since B is a division algebra there is a B-isomorphism H1(A, Q) V(Q) which we may use to identify H1 (A, Q) with V(Q). Then H1(A, Z) is a lattice in V(Q) and so is of the form g fV(Z) for some gf e G(A f). The isomorphism V(R) x to induces a complex structure on V(R), and we let h': Cx -> AutR(V(R)) be the corresponding map. Since B acts C-linearly on tA, h maps into AutBoR(V(R)) = G(R). Obviously there exists a g00 e G(R) such that h'(z) = g0h(z)g--1. Any 0 E is of the form v i-> gi 1g f1 v: g fV(Zf) -> V(Zf) for some g1 e G(Zf). It is now clear that (A, ) z (Ag ...) with g = g0gfg1. REMARK 1.4. (a) A map a: A -> A' of abelian varieties is an isogeny if it is surjective and has finite kernel; when OB acts on A and A', a is called an OB-isogeny if it commutes with the two actions. Clearly any isogeny (over C) induces an isomorphism on the tangent spaces and so Ag is isogenous to Ag, only if g00 and
g' define the same double coset in G(Q)\G(R)/K0.. On the other hand, the set EndB(A)x\G(Af)/K classifies the triples (A, i, 0) for which there is an OB-isogeny A -* A1. For example, if g = g f then, after replacing g f by an integral multiple, we may assume that gV(Z) c V(Z). The identity map V(R) -> V(R) now defines an isogeny Ag -* Al with kernel V(Z)/gV(Z) (cf. §6 below).
(b) In the case that F = Q, the theorem may be strengthened. Consider the
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projection V(R) x (G(A)/K K) -> G(A)/K K. We give G(A)/K K its usual complex structure and the copy of V(R) over gK,K the complex structure defined by hg. Inside each Vg we have a lattice gV(Z), and these vary continuously with g. Thus we may divide out and obtain a map of complex manifolds -+ G(A)/K K such that the fibre over gK,K is the abelian variety Ag. We may now let G(Q) act on
both manifolds and divide out again to obtain an analytic family .,C/ - Sc of abelian varieties. Each fibre Ag has the structure defined by (ig, ag), and these vary continuously. In fact cV -* Sc is an algebraic family, i.e., d is an algebraic variety and the map is algebraic. If F Q the above construction fails because units of F may act on (Ag, ig, 0g) and so the action of G(Q) on is not free. However we may "rigidify" the situation as follows: consider quadruples (A, i, , e) where A, i, and are as before and e is an injection from the unique weak polarization A to F" such that e(cb') = c.-(O) if 0'(u, v) = cb(u, cv). The isomorphism classes of quadruples are classified by F" x Sc(C) which is a disjoint union of copies of Sc(C), one for each element of F", on which Fx acts by permuting the copies. F" x Sc may be regarded as a scheme over C which is an infinite disjoint union of varieties and the previous process gives an algebraic family of d -> F" x Sc of abelian varieties with structure.
References. The most elegant elementary and nonelementary treatments of abelian varieties over C are to be found respectively in [20] and [17, Chapter I]. Families of abelian varieties parametrized by Shimura varieties were extensively studied by Shimura in the 1960's (see his Annals papers of that period). They are also discussed briefly in [4].
2. S as a scheme over Z [4-1]. We shall see shortly that Sc has a model S. over Q, i.e., that there is a scheme SQ over Q whose defining equations, when considered over C, give Sc. There is no reason to believe that S. will be unique but Shimura has given conditions which will be satisfied by at most one model; such a model (when it exists) is said to be canonical. For example, let F' be a quadratic totally imaginary extension of F which splits B and let A0 be the abelian variety of dimension d defined by the lattice OF' c F' Q R. Then OF. acts on A0 and A0 is said to have complex multiplication by F'. Let A = A0 x A0. If we embed F' in B and choose a basis {e1, e2} for B over F' with e1, e2 e OB, then we get a map B C+ M2(F')
c M2(End°(Ao)) = End°(A) sending OB into End(A). Also we get a map TfA =
(OF' (D OF') x0 Zf O® x Zf = V(Zf) (in the notation of §1). The triple (A, i, ) defines a point of Sc, and hence a point of SQ with complex coordinates. For SQ to be canonical these coordinates must be algebraic over Q and generate a certain explicitly described class field. For the reasons explained in the introduction we would like to have a scheme S defined by equations in Z[d-1] which, when regarded over Q, is the canonical model SQ of Sc, and which is such that it is possible to describe explicitly (S(Fp), Frob) for any p f J. Such an S will define a functor R F-> S(R) which associates to any ring R in which d is invertible the set of points of S with coordinates in R. (More generally, it associates to any scheme T over spec Z[4-1] the set S(T) of maps T -* S.) Since the functor determines the scheme uniquely this suggests that in constructing S we should write down a functor So such that, in particular, .(C)
= Sc(C) = G(Q)\G(A)/K,,K and try to prove that it is the points functor of a
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scheme. After §1 it is natural to define .7(R) to consist of isomorphism classes of triples (A, i, ) where each of the three objects is the analogue over R of the corresponding object over C. Thus A is a projective abelian scheme of dimension 2d over R. Intuitively, A can be thought of as an algebraic family of abelian varieties, each of which is defined over a residue field of R. More precisely it is a projective smooth group scheme over spec R with geometrically connected fibres. As before i is to be a homomorphism OB c. End(A) such that i(l) is the identity map. We assume that A has a polarization whose Rosati involution induces b " b'° on B. Two problems arise in defining which may be best understood if we write 0: T fA -+ V(Zf) as a product 1101: ]-[ T1A -> II V(Z1) of maps. Firstly, if p is not invertible in
R there will never exist an isomorphism 0p: TpA V(Zp); thus we take !5 to be a map defined only over R[p I]. Secondly, unless R is an algebraically closed field it is unrealistic to expect there to be an isomorphism 01: T1A -* V(Z1) for any 1, for this would imply that all coordinates of all /-power torsion points of A are in R. Instead we assume that K K. = Ker(G(Zf) -+ G(Z/mZ)) some m, and consider isomorphisms 0: A,, -Z V(Z/mZ) defined on some etale covering of R, two such isomorphisms 01 and 02 being K-equivalent if 01 = k¢2, k e K, locally on spec(R), and we take 0 to be a K-equivalence class in this new sense. It is necessary to put one extra
condition on the triple (A, i, 0): if the R-algebra R' is such that OB O R' M2(OF (0 R') then the two submodules of tA/R corresponding to the idempotents (0 10) and ($ °) should be free OF OO R'-modules of rank 1 locally on spec(R'). (This condition holds automatically if R = C; for examples where it fails in an analogous situation in characteristic p, see [18, 1.29].) Having defined our functor Y we now have to see whether it is the points functor of a scheme. Generally speaking this is a very delicate question but M. Artin has given an often-manageable set of criteria for a functor to be the points functor of an algebraic space. An algebraic space is a slightly more general object than a scheme, but for our purposes it is just as good; it makes good sense to speak of its points with coordinates in a ring, and the proper and smooth base change theorems in etale cohomology, which are the theorems which allow us to compute Hasse-Weil zeta-functions by reducing modulo a prime, hold for algebraic spaces. (In fact, the algebraic spaces we get are almost certainly schemes, and this surely could be proved by using Mumford's methods [16] instead of Artin's.)
Consider first the case that F = Q. Then Artin's criteria may be checked and show that there is an algebraic space S, proper and smooth over Z [J-1], such that S(R) _ .9(R) for any ring R in which J is invertible. In particular S(C) = .7(C) = S,(C) and S(Pp) _ .(Fp) for any p not dividing J. The algebraic family sad -* S, constructed in 1.4(b) is an element of ,7(Sc) = S(SA), and so gives a map S, -* S. This induces a map Sc -+ S x spec C which is an isomorphism. Moreover it is known that SQ is the canonical model. When F 0 Q, then a slightly weaker result holds, but one which is just as useful to us. Since there are nontrival automorphisms of (A, i, ) there can be no algebraic space S with S(R) = .7(R) for all R. However, there does exist an algebraic space S, proper and smooth over Z [J-1], and a functorial map ,7(R) S(R) which is an isomorphism whenever R is an algebraically closed field. Thus S(C) = 9(C) = Sc(C) as before, and SQ is the canonical model of Sc. To prove these facts one may
"rigidify" the moduli problem as in the second paragraph of 1.4(b), make the
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constructions as in the case F = Q, and then form quotients under the left action by F", or else work directly with stacks. Note that in either case, S (Pp) = Y(Pp) has a description in terms of abelian varieties with additional structure. References. [1] contains a short introduction to Artin's techniques for representing functors by algebraic spaces and [2] a more complete one. In [5] and [18] these techniques are applied to a situation which is very similar to ours. (In fact, it is almost identical; see §7 of the Introduction to [5].) The basic definitions concerning abelian schemes can be found in [16]. 3. Finite group schemes, p-divisible groups, and Dieudonne modules. In the remain-
ing sections we shall need to consider the finite subgroup schemes of an abelian variety and so, in this section, we review some of their properties. We fix a perfect field k of characteristic p 0 0. Let R be a finite k-algebra (so R is finite-dimensional as a vector space over k) and let N = spec R. For any k-algebra R', a point of N in R' is simply a map of k-algebras R -> R'; thus N(R') = Homk_alg(R, R'). If every N(R') is given the structure of a commutative group in such a way that the maps N(R') -+ N(R") induced by maps R' -+ R" are homomorphisms, then we call N, together with the family of group structures, a finite group scheme over k. As for affine algebraic groups, giving the family of group structures corresponds to giving a comultiplication map R m, R x®k R. EXAMPLE 3.1. (a) Any (commutative) finite group M can be regarded in an obvious way as an algebraic group over k and hence as a finite group scheme. Indeed, let R be a product of copies of k, one for each element of M, and let N = spec R. Then N, as a set, is equal to M. The group law on M induces a comultiplication on R which, in turn, induces compatible group structures on N(R') for all R'. If R' has
no idempotents other than 0 and 1, then N(R') = M. (b) pp" = spec k[T]/(TP" - 1). Then pp,(R') = {C e R'ICP" = 1 } is a group under multiplication for any R', and these group structures make pp, into a finite group scheme. Note that p ,(R) = {I } if R has no nilpotents and, in particular, if R is an integral domain. (c) ap = spec k[T]/(TP). Then ap(R') = {a e R'Iap = 0}. As (a + b)P = aP + by in any k-algebra, ap(R') is a group under addition, and these group laws make ap into a finite group scheme. Again ap(R') has only one element if R' has no nilpotents.
(d) Z/pZ = spec k[T]/(TP - T). If R' has no idempotents other than 0 and 1 (e.g., R' an integral domain) then (Z/pZ) (R') = Fp, the prime subfield of R', which
is a group under addition. This example is a special case of (a), because k[T]/(TP - T) = k[T]/T(T - 1) ... (T - (p - 1)) s: k x x k (p copies). The rank or order of a finite group scheme N = spec R is the dimension of R as a vector space over k. For example the order of the group scheme defined by M in 3.1(a) is the order of M, while the orders of pp,, ap, and Z/pZ are p^, p and p respectively.
A homomorphism from one finite group scheme NI = spec RI to a second N2 =
spec R2 is a k-algebra homomorphism R2 -> RI such that the induced maps NI(R')
N2(R') are all homomorphisms of commutative groups.
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From now on we consider only finite group schemes of p-power order. The essential facts are the following.
Facts. 3.2.(a) They form an abelian category. Thus we may form kernels, quotients, etc. exactly as if we were working with a category of modules. (b) When k is algebraically closed the only simple objects are pp, ap, Z/pZ.
This means that any finite group scheme of p-power order has a composition series whose quotients are pp, ap, or Z/pZ. There can be no homomorphism from one simple object to another of a different type. (c) The category is self-dual, i.e., there is a contravariant functor N-F-. N (= Cartier dual of N) which is an equivalence of the category with itself. More precisely, for each N there is a pairing N x N Gm (= GL1) such that, for any k-algebra R, the pairing induces isomorphisms N(R) x, HomR(N, Gm), N(R) Z HomR(N, Gm). For example, (Z/pZ)A = It. and the pairing (Z/pZ)(R) x pp(R) --> Gm(R) is (n, C) Cn; Gap = ap and the pairing ap(R) x ap(R) -> + (ab)p-1/(p - 1)!. Gm(R) is (a, b) --> exp(ab) = I + ab + (d) Hom (Z/pZ, Z/pZ) = Z/pZ, Hom (pup, pup) = Z/pZ, Hom (ap, ap) = k. The statement for Z/pZ is obvious, and that for dip follows by Cartier duality. The map ap -> ap corresponding to c e k is (T' -> cT): k[T]/(TP) -> k[T]/(TP) on the algebra of ap and (a >- ca): ap(R') --> ap(R') on its points. (e) If 0 -> N' -+ N -- N" -> 0 is exact then order(N) = order(N')order(N").
Let A be an abelian variety over k. For each n, Ap Lf Ker(pn: A -> A) is a finite group scheme of order (pn)2dim(A) i.e., the order is the same as when p 0 characteristic(k). The system Ap c- Apz c-> is called the p-divisible (or BarsottiTate) group A(p) of A. More generally, a p-divisible group of height h is a system of finite group schemes and maps N = (N1 1, N2 t?, N3 ...) such that N,, has
Nn). For example order pnh and in_1 identifies Nn_1 with the kernel of (N,, Q p/Zp = (Z/pZ - Z/p2Z -> ) and fep- = (1up --> jupz -+ tlp3 --> ) are p-divisible groups of height one. A(p) is of height 2 dim(A). A homomorphism 0: N -+ N' of p-divisible groups is a family of maps fin: N,, -* N,, commuting with the maps i,, and i,,.
Exercise 3.3. (k algebraically closed.) For any abelian variety A there are maps fin: A1, --> Ap such that Ker(5n) = Ker(g5n+1) for all sufficiently large n. Deduce
that A has
pdimA points of order p, and that when equality holds A(p) _
(Qp/Zp)dim(A) x (p,,)dim(A). (Such abelian variety is said to be ordinary.)
Let W = Wk be the ring of Witt vectors over k; it is a complete discrete valuation ring of characteristic zero whose maximal ideal is generated by p and which has residue field k. There is a unique automorphism a a(p) of W which induces the pth power map on k. If k Pp then W is the completion of the ring of integers in the maximal unramified extension Qpn of Qp and a >-> a(p) is induced by the usual Frobenius automorphism of Qpn over Qp. Let W[F, V] be the ring of noncommuta-
tive polynomials over W in which the relations FV = p = VF and Fa = a(p)F, aV = Va(p), hold for all a e W. There is a contravariant functor, N u-> DN = Dieudonne module of N, associating to each p-power order finite group scheme a W[F, V]-module which is of finite length as a W-module; D defines an antiequivalence of categories. The length of DN as a W-module is equal to the order of N. Thus manipulations with finite group schemes correspond exactly to manipulations with modules over the noncommutative ring W[F, V]. Examples:
D(pp) = W/pW = k; Facts as 0, V acts as 1;
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D(ap)=k;F=O, V=O;
D(Z/pZ)=k;F=1,V=0. If N is unipotent and pN = 0, then DN = Lie(N); the bracket operation on Lie(N)
is zero but it has the structure of a p-Lie-algebra and F acts as the "p-power" operation and V acts as zero. More generally, if N is unipotent and killed by p's, then DN = Hom(N, where WI = Ga = the additive group and W = the Witt vectors of length n regarded as an algebraic group. There are canonical, nondegenerate, W-bilinear pairings <, >: DN x D1V -> W ® Q,,/Z,, such that = <m, Vn> c p), .
The notion of Dieudonne module can be extended to p-divisible groups. On applying D to N = (NI --> N2 -+ ) we obtain a sequence of modules and maps (DNI <- DN2 <- ), and we define DN = proj lim DN,,. This is a W[F, V]-module which is free of finite rank equal to height(N) as a W-module. In classifying p-divisible groups one begins by considering them up to isogeny: N and N' are isogenous if there is a surjective homomorphism N -+ N' with finite kernel or, equivalently, if there exists an injective homomorphism DN' --> DN whose cokernel has finite length over W. If we write W' = W[1/p] = W ®zp Qp, W'[F, F-I] = W'[F, V] = W[F, V] ®Zp Qp, and D'N for DN ®Zp Qp regarded as a W'[F']-module, then we see that N and N' are isogenous if and only if D'N D'N'. Let .mil be the category of W'[F]-modules whose objects occur as D'N for some p-divisible group N. When k is algebraically closed one knows that .fl has exactly one simple object Dz = W'[F]/(Fr - ps) for each rational number A, 0 < A < 1, 2 = s/r, (r, s) = 1. DA has dimension r over W', End(DA) is the unique division algebra over Qp of degree r2, and any D e & can be written uniquely as a finite direct sum D = (D21)ml ... +Q 01)mt with distinct A. Then A1, , At are the slopes of D and m1r1i where d1 = sllr1i is the multiplicity of A1. We sometimes write (Dsir)m
as Dsmirm. Thus Ds1r may now be a multiple of a simple module; it has slope sir with multiplicity r and has dimension r over W'. When k is algebraically closed and N is a p-divisible group over k, the slopes of D'N are called the slopes of N. Clearly N is uniquely determined up to isogeny by its slopes and their multiplicities. For example, all p-divisible groups of height one are isogenous (in fact, isomorphic) to pp- or Qp/Zp because D'(pp-) = DI and
D'(Qp/Zp) = DO are the only Dz of dimension one over W'. There is only one simple DA of dimension two over W'; it is DI i2 = D'(A(p)) where A is a supersingular elliptic curve (cf. §5).
Let k have algebraic closure k : k. Any p-divisible group N over k defines a p-divisible group Nk over k and it is known that DNk DN ®Wk Wk. If k = Fq with q = pa then Fa: DN -+ DN is W-linear and so its characteristic polynomial det(T - FaIDN) _ lit(N)(T - a1) is defined. The set of slopes of D'Nk is {ordq(al), ordg(a2), } where ordq is the valuation of the algebraic closure of WW such that ordq(q) = 1. If A is an abelian variety, we write DA for DA(p). When A is defined over Fq the Frobenius endomorphism n : (a,, a2, ) H (alq, a2, ) of A induces Fa on DA. The characteristic polynomial PA(T) of is A --+ A in the sense of [17, X19] is det(T - FaJDA). Thus the slopes of D'Ak can be read off from PA(T).
We can also define profinite group schemes TfA = proj limA and T1A = proj limA1,,. If 1
char(k) and k is algebraically closed then TIA can be regarded
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(as before) as a free Z1-module of rank 2 dim(A). We write TfA = TfA x TpA. Finally we note that to classify p-divisible groups up to isomorphism, it is necessary to classify the (F, V)-stable lattices in the objects of X. References. The best introduction to the subject matter of this section is [6].
4. S(Fp) as a family of abelian varieties. Fix a prime p not dividing J. From §2 we know that points of S(Pp) correspond to isomorphism classes of triples (A, i, i) where A is an abelian variety of dimension 2d over Pp, i is an action of OB on A, and 0 is a KP-equivalence class of isomorphisms 0: TfA -+ V(Z f) where TfA = proj limpx (Recall that Op: TpA -> V(Zp) is defined only over the ground ring with p inverted, and Fp[p 1] is the zero ring.) The OB Q Fp module to satisfies the following condition: (4.1)
the subspaces corresponding to the idempotents (o $) and ($ °) in OB p Pp x M2(Fp) are free OFD Fp modules of rank 1.
If A is defined by equations 2a(;) TU>, let A(p) be the abelian variety over Fp defined by the equations X'ap1)Tz. There is a Frobenius map F = FA: A --+ A(P) which takes a point with coordinates (t1, , t;;) to (tp1, , ti). The map FA is a purely inseparable isogeny of degree p2d, which means that AF, the kernel of F, is a finite group scheme of order p2d with only one point in any field (so that only ap and pup occur in any composition series for it). As groups, D(A) = D(A(p>), but the
identity map DA -+ DAW is (p)-linear, i.e., am F- a(p)m for a e W, m e DA. The composite DA 'd, DA(P) DF,3 DA is a multiplication on the left by F C= W[F, V] (see [6, p. 63]). Since FA is zero on tA, to tAF, x DAF x (DAF)* = dual(Coker(DA F, DA)). Thus (4.1) may be checked on DA/F(DA) instead of tA. If P e S(Pp) corresponds to (A, i, ) then, intuitively, we may think of the coordinates (a1, , a,) of P as being the coefficients of the equations defining A. Thus
Frob(P) corresponds to (A(P), i(P), 0cp>) where i(P) and (P) are such that FA defines a map of triples (A, i, ) -+ (A (P), icp>,
(p>).
Finally we observe that there are "Hecke operators" acting. Let g e G(A f) and suppose that K' is an open subgroup of G(Zf) such that g-1K'g c K; then x F-+ xg: G(A) --+ G(A) induces a map G(Q)\G(A)/KKK' -+ G(Q)\G(A)/K,K which arises from a map of varieties (g): x'SC -+ KSe. If P corresponds to (A', i', ') then -(g)P corresponds to (A, i, 0) if there is an OB-isogeny a: A -+ A' such that TfA
-* TJA'
V (Zf) -+ V (Zf) commutes with m some positive integer. When we pass to Sp, only G(A f) continues to act: if g e G(A f) and P e K,S(Pp) and (g)P e KS(Fp) correspond respectively to
(A', i', ') and (A, i, ) then there is an isogeny a: A -+ A' whose kernel has order prime top and a commutative diagram Tfa
TfA-*TfA' V(Z) f- VV P
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177
with m a positive integer prime to p. This definition is compatible with that over C in the sense that both mappings (g) come by base change from a mapping 9(g): K'S x spec Z(p) --> KS x spec Z(p) where Z(p) = {m/n a Qj(n, p) = 1}. (More concretely, this means that if (A, i, ) in characteristic zero specializes to (A, t, Vii) in characteristic p then (g)(A, i, ) specializes to (g)(A, t,
5. The isogeny classes. Fix a prime p not dividing d and consider pairs (A, i) where A is an abelian variety of dimension 2d over Fp and i is a homomorphism B c End°(A) such that i(1) = 1. We write A - A' if A and A' are isogenous, and (A, i) '' (A', i') if the pairs are B-isogenous in an obvious sense.. denotes the set of all B-isogeny classes and (A, i) Qx Q the class containing (A, i). It will turn out (last paragraph below) that the map (A, i, )'-+ (A, i) px Q: S(Pp) --> . rp is surjective and so, to describe S(Fp), it suffices to describe 0-p and the fibres of the map. The first is done in this section and the second in the next. Note that Frob (and (g)) preserves the fibres. We first remark that, as in characteristic zero, there is a unique weak polarization on A inducing the given involution on B, and that it gives an F-equivalence class of pairings A x A -> Gn, for all n (cf. [17, §23]). In turn these pairings give an equivalence class of skew-symmetric pairings Sb1: TA x TIA --> T1Gm Z1 with nonzero discriminant for each 10 p, and a similar pairing Op: DA x DA -f W ; this last pairing satisfies the conditions cip(Fm, n) = Op(m, Vn)(P), cip(Vm, n)(P) Op(m, Fn). All pairings satisfy cb(bm, n) = cb(m, b*n), b e B.
The description of . rp will be based on the following classification of isogeny classes over a finite field. (Recall that an abelian variety over a field k is simple if
it contains no nonzero, proper abelian subvariety defined over k and that any abelian variety is isogenous to a product of simple abelian varieties. If A is defined (a4, az ).) over Fq then 'r = IA is the Frobenius endomorphism (a,, a2, THEOREM 5.1. (a) Let A be a simple abelian variety over Fq and let E = End°(A). Then E is a division algebra with centre Q[z], it is an algebraic integer with absolute value qI /2 under any embedding Q[n ] c-, C, and for any prime v of Q[Z] the invariant of E at v is given by
inv,(E) = 2 =0 ordv(q) [Q[n ],, : Qp]
if v is real, i f vll, 1
p,
if v1 p.
Moreover 2 dim(A) = [Q[ir] : Q] [E: Q[2r]] 1,12 and e = [E: Q[2c]] I i2 is the least The characteristic polynomial PA(T) of r: common denominator of the A -f A is m(T)e where m(T) is the minimal polynomial of ir over Q. (b) The simple abelian varieties A and A' over Fq are isogenous if and only if there is an isomorphism Q[lrA]
Q[lA'] such that 7rA F--' ZA'
(c) Every algebraic integer 'r which has absolute value q' /2 under any embedding C arises as the Frobenius endomorphism of a simple abelian variety A,, over Q[x] Fq.
(d) For any abelian varieties A and B over F. and any prime l (including I = p) the canonical map
178
J. S. MILNE
Hom(A, B) O Z1 -> Hom(A(l), B(l)) = Hom(T1A, TIB)
is an isomorphism. (If 10 p then A(1) and B(1) 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], [22], [23].
For example, if in (c) we take 7r = pa, q = pza then we obtain an elliptic curve Ay, such that End°(A1) 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[2r] has a real prime then either A is a supersingular elliptic curve or becomes isogenous to a product of two such curves over Fq2.
From now on we, let p factor as (p) _ pl p,,, in OF, where the pi are distinct prime ideals, and we let d; be the residue class degree of pi over p; thus d = Edi. PROPOSITION 5.2. Let (A, i) be as above. The centralizer of B in End°(A) is either:
(a) a quaternion algebra B' over F which splits except at the infinite primes, the primes where B is not split, and the pi for which di is odd, and there does not split; or (b) a totally imaginary quadratic field extension F' of F which splits B. In the first case A - Aod where A0 is a supersingular elliptic curve and in the second A - Ao where A0 is an abelian variety such that F' c End°(Ao). PROOF. Suppose A ' Ar x A1i r 1, where A0 is a supersingular elliptic curve and Hom(A0, A1) = 0. Then End°(A) x Mr(E) x End°(A1), where E = End°(Ao), and B embeds into Mr(E). Consider F c- Mr(E); we must have dj2r, but d = 2r is impossible because F does not split E [19, Theorem 10], and so r = d or 2d. The Skolem-Noether theorem shows that, when composed with an inner automorphism, the map F -> Mr(E) factors through Mr(Q). Thus the centralizer C(F) of F in Mr(E) is isomorphic to Mr/d(F) O E = Mrld(E O F). Let C be the centralizer of B in Mr(E). Then B OF C -- C(F) because C(F) and B are central simple alge-
bras over F [19, §8]. It follows that either rld = 1, C = F, and B = E Qx F, or rld = 2 and C is a quaternion algebra over F such that, in the Brauer group of F, [B] + [C] = [E OO 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 z e End°(A), and the assumption implies that there is no homomorphism Q[2r] -+ R. Consider
B -B [2r]-B OF C - E F- F [7r]- C Q -Q 17r)
where E is End°(A) and C is the centralizer of B in E. Clearly, F[7r] = F would contradict our assumption. On the other hand we must have [F[7r] : F] < 2 and F[7r] = C for otherwise E would contain a commutative subring of dimension
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179
> 4d = 2 dim(A) over Q, which is impossible by 5.1(a). Let F' = C = F[ir]; 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 10 p, (T,A) ©Z, Q, is free of rank 2 over FI = F' O Q,, from which it follows that B Q F, -_ M2(F,), and we are assuming that B splits at any infinite prime or prime dividing p. Let e be an
idempotent 0 0, 1, in (B px 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 End°(Ao). REMARK 5.3. In case (b) of 5.2, A0 is isogenous to a power of a simple abelian variety, A0 - Ai, because the centre of E = End°(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 End°(A) is a central simple algebra over the field Q[9r]. Let (A, i) and (A', i') be such that there exists an isogeny a: A -+ A'. The SkolemNoether theorem shows that the map B =, End°(A) End°(A'), where a*(r) = ara 1, differs from V: B -* End°(A') by an inner automorphism (r I-' 13x(3-1) of End°(A'). Thus f3a is a B-isogeny A -> A', and we have shown that A ' A' implies(A, i) '' (A', i'). We now consider in more detail the situation in 5.2(b). Let p1, , p,, 0 t < m, be the primes of F dividing p which split in F' and write pi = qiq' for i t. Since OF n End(Ao) and Zp both act on A0(p), their tensor product does, and the splitting Fpl x F xp Qp x Fpm induces an isogeny A0(p) - Ao(p1) x x Ao(pm) and an isomorphism D'A0 . D'Ao(p1) x x D'A0(pm). Clearly A0(pi) has height 2di and so D'(Ao(pi)) has dimension 2di over W'. Since 0(am, n) = O (m, an) for a e F the decomposition of D'A0 is orthogonal for 0p, and Op restricts to a nondegenerate form on each D'(A0(pi)). This implies that the set of slopes {.11, A2,
} of
D'(A0(pi)) is invariant under A ,-. 1 - A.
Fix an i < t. As F. . F. x F,, acts on D'AQ(pi), A0(pi) splits further: Ao(pi) ,., Ao(g1) x Ao(gi), D'Ao(pi) z D'(Ao(gi)) x D'(A0(q )). Since Fq, c End°(A(gi)) has
degree d1 = height(Ao(g1)) over Qp, A(qi) is isogenous to a power of a simple p-divisible group : we may write D'A(gi) = Dki/di, 0 < k, < di. Correspondingly, D'A(gi) = Dk;/d; with k. + k= = di. Fix an i > t. The [F,,: Qp] = 2di = height A0(pi) and so, as above, A0(pi) is isogenous to a power of a simple p-divisible group and we may write DA0(pi) _ Ds/r. Since sir = 1 - s/r we must have s/r = di/2di. We write ki = di/2. Note that for some i, 1 < i < m, we must have k, 0 di/2 for otherwise all slopes of A(p) would equal 1. Then (see §3 and 5.1(a)) 12r1g1121 = 1 for all primes v of Q[2r] and so some power of it would equal one. On replacing F9 by a larger finite field we would have i = q1 '2, 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 pi splits in F'. THEOREM 5.4..O'p contains one element for each pair (F', (kt)IS,Sm) where F' is a totally imaginary quadratic extension of F which splits B and is such that at least one
pi splits in it; if pi splits in F' then ki is an integer with 0 < ki 5 d1 and otherwise k, = di/2; for at least one i, ki 0 di - ki. When pi splits in F' we regard ki and k= as being associated to qi and q;, and we do not distinguish between two pairs (F', (ki))
and (F', (ki)) 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 pi 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) px Q corresponding to (F', (k1, , km)). As before we let pi, 1 < i <_ t, be the primes dividing p which split in F'. Consider the ideal in OF,, a = q(f/d1)k1 gl(f/d1)k1 ... gt+ldt+1)kt+1 ...
qmf/dm)km
where f = 2d1 dm. For some h, ak is principal, say ah = (2c). If we write a i-, a for the nontrivial F-automorphism of F' then 'roc e F and (q f
gif ... gm)k nI OF =
p1 k
... l,f,=k = (pfh).
Thus zz = upfk with u a unit in OF. If u is a square in F then we may replace 7r by 2r/uli2 and obtain an equation srz = q with q = pfk. If u is not a square then we replace is by Z2/u and obtain a similar equation with q = p2fk. Note that the condi-
tion k, 0 k,. for some i implies that i 0 F and hence that F' = F[n]. Under any embedding F' c+ C, F maps into R. Thus complex conjugation on C induces a'--f a
on F'. In particular t is the complex conjugate of the complex number r and so zr = q implies that JzJ = q"2. Let A,, be the abelian variety corresponding, as in 5.1(c) to ir, and let E _ End°(A,,). For any prime v of Q[z] 0
if vIp,
= (ki/d1)[Q[ir]v: Qp]
if vIp and g1Iv,
= (kr/di) [Q[2r] : Q p]
if v J p and Q v. v.
Let e be the denominator of invv(E) (when it is expressed in its lowest terms) and
let e be the least common multiple of the e,,. Then 2 dim(A,,) = re where r = [Q[n]: Q]. Clearly Q[2c]U] for any qlv, vIp, which implies (by class field theory) that F' splits E and (trivially) that e divides [F': Q[Z]] = 2d/r. As [M2d/ re(E) : Q[ir]] = (2d/re)2e2 = [F': Q[ir]]2, F' embeds into M2di.e(E) [19, Theorem 10]. Let A0 = A2d1re, The characteristic polynomial PA,(T) of 7r on A,, is c,,(T)e where c,,(T) is the minimal polynomial of 7r e Q[2r] over Q (5.1(a)). Thus PAO(T) is c,,(T)2d/r which equals the characteristic polynomial of r e F' over Q.
Corresponding to the splitting Fp = F41 x Fqi x , we have A0(p) ^' A0(gl) x A0(gi) x and PAO(T) = P1(T)Pi(T) . . where Pi(T) (resp. Pt (T)) is the characteristic polynomial of the image ici of r in F. (resp. 7r; of r in Fq-) over Qp. Thus (see §3)A(gi) has slopes equal to ord9(2ci) = (fh/di)ki/fh = kidi and A(q;) has slopes equal to kildi. Thus A = A0 x A0, regarded as an abelian variety over Pp, and the map i induced by B M2(F') represent an isogeny class corresponding to (F', (kl, ... , km)) . (A 2d, i), where A0 is a supersingular elliptic curve, represents the supersingular class.
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Obviously if (A, i)
'
181
(A', i') then both represent the supersingular class or
correspond to the same pair (F', (k1, , km)). It remains to show that if (A, i) and (A', i') both correspond to (F', (k1, ..., km)) then (A, i) - (A', i'). By considering A and A' to be defined over some finite subfield of Fp we get elements 2r = 7CA e F' and n' = 7CA. e F'. The assumption implies that ordq(z) = ordq(z') for all qip, q a prime of F', and 5.1(a) then shows that I ir17C'I, = 1 for all primes of F'. Thus z and 7r' 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 completes the proof. The proof that any class in .gyp is represented by an element of S(Fp) requires the following lemma. LEMMA 5.5. Let T c TfA be such that TfA/T is finite; then there exists an isogeny a : A' --+ A such that T fa maps TfA' isomorphically onto T.
PROOF. The finiteness of TfAIT means that, for all n » 0, the cokernel N of TInT -> TfA/nTfA is independent of n. Thus there is a map An = TfA/nTfA 0D N. Define A' to be the cokernel of a '- (q(a), a) : An -4 N x A, and a : A' -* A to be (b, a) H na; then (T fa)(TfA') = T. Let (A, i) represent a class in .gyp and let 0' = B fl End(A); it is an order in B. Regard (A, i) as being defined over a large finite field Fq; then End(A) Ox Z!
EndFq(TIA) and 0' O ZI = (B O Zi) n Endpq(TIA). For almost all 1, 0' O ZI will equal O® O ZI and we take TI = TIA; for the remaining 1 we may choose a TI of finite index in TA which is stable under OB, i.e., such that EndFq(TI) n (B 0 ZI) = OB Ox Z1. Note that D(TplpTp) = M is a W[F, V]-module of finite length over W. We may choose Tp such that M/FM satisfies (4.1). Let A' correspond to T = fj ITI as in the lemma. Then A' together with the obvious i and some ¢ lies in S(Fp) and represents (A, i) O Q.
6. An isogeny class. It remains to describe the set Z = Z(A, i, OA) of elements
(A', i', ') of S(Fp) such that (A', i') is isogenous to a given pair (A, i). An OB-isogeny A' " A determines an injective map T fa : TfA' -> TfA whose image A satisfies the following conditions: (a) A is OB-stable. (b) TfA/A is a finite group scheme. (More precisely, Coker(A/nA -+ TfA/nTfA)
= Coker(A', -
Ker(A' ", A) for n » 0.)
(c) D(A/pA)/FD(A/pA) satisfies (4.1). (For A/pA = Ap and so D(A/pA)/FD(A/pA) = DA/F(DA).)
Consider all subobjects A of TfA satisfying (a), (b), (c). Any such A may be written A = AP x Ap with AP a Zf-lattice in TfA (in the usual sense of modules over Z f) and Ap c TTA. We let Y be the set of pairs (A, 5) with A as above and V(Z f). Since (A, ) is determined 0 a K-equivalence class of isomorphisms AP by a pair ((AP, ), Ap), we may write Y = Yp x Yp. By 5.5, every (A, ) e Y arises from a triple (A', i', ') e S(FF) equipped with Z c S(FF). an isogeny a; A' -+ A. Thus we have a surjective map Y and we define Y Ox Q to be For n a positive integer we set n(A, 0) = (nA, the set of pairs (y, n) with y e Y and n e Zoo, where (y, n) and (y', n') are identi-
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fled if n'y = ny'. Then we write Y ® Q = (YP (9 Q) x (Y, O Q) where Yp O Q may be identified with the set of OB-stable lattices A in (TfA) px Q equipped with a K-equivalence class of isomorphisms A --, TfA. There is an action of H(Q) = Endo8(A)x on Y Q Q: for a E H(Q) we choose a positive integer m such that ma is an isogeny of A and define a(A, ¢, n) = (T f(ma), oT f(ma)-I, mn). LEMMA 6.1. The map Y -+ Z described above induces a bijection H(Q)\Y O Q Z.
PROOF. (A, 0, n) and (A', 0', n') map to the same element of S(Fp) if and only if there exist OB-isogenies
A
A' a,
and an OB-isomorphism 00: TfA' - V(Zf) such that n((Tfa)TfA', oo(T'fa)) = (A, ¢) and n'((Tfa')TfA', 00(Tfa')) = (A', a'). Then a'a 1 makes sense as an element of End°(A) and a'a-1(A, , n) = (A', ', n'). LEMMA 6.2. The map G(Af) -+ Yp Ox Q, g H (g(TfA), OAg1), induces a bijection G(A)/K --+ Yp O Q. PROOF. Obvious.
LEMMA 6.3. There is a one-one correspondence between Y® x Q and the set X of W[F, V]-submodules M of D'A which are free of rank 4d over W, OB-stable, and such that M/FM satisfies (4.1). PROOF. p: A --+ A induces maps i,,: A/pnA y Alp1,+IA which define a p-divisible
The exact sequence 0 -+ A -* TpA -+ N -+ 0 (N finite) group A(p) = (A/p-A, gives rise to 0 -+ N -* A(p) -+ A(p) - 0. On applying D we get
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, 0) x H(Q)\G(A f) x X /Kp. Frob acts by sending M E X to FM; the Hecke operator
(g), g e G(A f), "acts" by
multiplication on the right on G(A f).
PROOF. This simply summarizes the above.
It remains to give a more explicit description of X. Note that, corresponding x D'A(pm), we have X Xl x x X,,,. to the splitting D'A D'A(pl) x It is convenient to write G1(Zp) = Auto5(A(pt)) and 6,(Q,,) = End,B(A(pI))x = Endo8(D'A(}it))x. In the simplest cases G1(Qp) acts transitively on the lattices M c D'A(pt) which belong to Xt, and in this case X= : Gt(Qp)/G1(Zp). (To say that
Gt(Qp) acts transitively means that each A'(pt) is isomorphic to A(pt) and not merely isogenous; cf. [6, p. 93].)
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183
EXAMPLES 6.5. (a) F = Q, F' is a quadratic extension of Q, (p) = qq' in F', and (A, i, 0) is in the isogeny class corresponding to (F', (0)).
Then A(p) (Qp/Zp)2 x (pp_)2 with OB 0 Zp = M2(Zp) acting in the obvious way on each factor. Thus G(Zp) = {(o as°) la, b e Zx} and G(Qp) _ {(g °) la, b e Qp }. In this case X = G(Qp)/G(Zp). Frob acts as (I p).
(b) As above, except (A, i, ) corresponds to (F', (1)). Then A(p) (pp_)2 x (Qp/Zp)2 (i.e., in the splitting Fp = Fq x F,,, F, now corresponds to the pp_ factor). G(Zp), G(Qp) and X are as before but Frob acts as ($ °)-
(c) F = Q, (A, i, 0) is in the supersingular class. Then D'A(p) -_ DI /2 x DI /2 and End(DI"2) = Bp, the unique division quaternion algebra over Qp. B acts through the embedding
B p Qp
M2(Qp) `ar non M2(Bp).
Thus G(Qp), the centralizer of B Qx Qp in M2(Bp), is (B;)x. Moreover G(Zp) may be
taken to be Ox where 0 is the maximal order in B. In this case X A, 6(Qp)1QZp). Frob acts as multiplication by w, a generator of the maximal ideal of O. (d) F arbitrary, p splits completely in F, (p) = pI pd, (A, i, 0) corresponds to (F', (k1, ..., kd)). Then X XI x . . . x Xd where Xi is as in case (a) if pi splits in F' and k, = 0, as in case (b) if pi splits and ki = 1, and as in case (c) otherwise. (e) The general case. For a statement of the result, see [14]. (This case is treated in detail in: J. Milne, Etude d'une classe d'isogenie, Seminaire sur les groupes reductifs et les formes automorphes, Universite Paris VII (1977-1978).) Added in proof (November 1978). The outline of a proof in [12] of the conjecture for those Shimura varieties which are moduli varieties is less complete than appeared at the time of the conference. The above proof (completed in the report
referred to in 6.5(e)) for the case of the multiplicative group of a quaternion algebra differs a little from the outline in that it depends more heavily on the Honda-Tate classification of isogeny classes of abelian varieties over finite fields. The complete seminar referred to in 6.5(e), which redoes in greater detail much of the material in this article and [3], will be published in the series Publications Mathematiques de l'Universite Paris 7. BIBLIOGRAPY
1. M. Artin, The implicit function theorem in algebraic geometry, Colloq. in Algebraic Geometry, Bombay, 1969, pp. 13-34. 2. , Theoremes de representabilite pour les espaces algebriques, Presses de l'Universite de
Montreal, Montreal, 1973. 3. W. Casselman, The Hasse- Weil cfunction of some moduli varieties of dimension greater than one, these PROCEEDINGS, part 2, pp. 141-163.
4. P. Deligne, Travaux de Shimura, Seminaire Bourbaki, 1970/71, no. 389. 5. P. Deligne and M. Rapoport, Les schemas de modules de courbes elliptiques, Antwerp II, Lecture Notes in Math., vol. 349, Springer, New York, pp. 143-316. 6. M. Demazure, Lectures on p-divisible groups, Lecture Notes in Math., vol. 302, Springer, New York, 1972. 7. T. Honda, Isogeny classes of abelian varieties over finite fields, J. Math. Soc. Japan 20 (1968), 83-95.
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8. Y. Ihara, The congruence monodromy problems, J. Math. Soc. Japan 20 (1968), 107-121. 9. , On congruence monodromy problems, Lecture Notes, vols. 1, 2, Univ. Tokyo, 1968, 1969. 10.
, Non-abelian class fields over function fields in special cases, Actes Congr. Internat. Math., vol. 1, Nice, 1970, pp. 381-389. II. , Some fundamental groups in the arithmetic of algebraic curves over finite fields, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 3281-3284. 12. R. Langlands, Letter to Rapoport (Dated June 12, 1974-Sept. 2, 1974). 13. , Some contemporary problems with origins in the Jugendtraum, Proc. Sympos. Pure Math., vol. 28, Amer. Math. Soc., Providence, R. I., 1976, pp. 401-418. 14. , Shimura varieties and the Selberg trace formula, Canad. J. Math. 29 (1977), 12921299.
15. Y. Morita, Ihara's conjectures and moduli space of abelian varieties, Thesis, Univ. Tokyo (1970).
16. D. Mumford, Geometric invariant theory, Springer, 1965. 17. , Abelian varieties, Oxford, 1970. 18. M. Rapoport, Compactifications de 1'espace de modules de Hilbert-Blumenthal, Compositio Math. (to appear). 19. J. P. Serre, Expose 7, Seminaire Cartan 1950/51. 20. H. Swinnerton-Dyer, Analytic theory of abelian varieties, L.M.S. lecture notes 14 (1974). 21. J. Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134144. 22.
, Classes d'isogenie des varietes abeliennes sur un corps fini, Seminaire Bourbaki,
1968-1969, 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., 1971, pp. 53-64. UNIVERSITY OF MICHIGAN
Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 185-192
COMBINATORICS AND SHIMURA VARIETIES mod p (BASED ON LECTURES BY LANGLANDS)
R. E. KOTTWITZ One of the major problems in the study of Shimura varieties is that of expressing
their Hasse-Weil zeta-functions as products of L-functions associated to automorphic 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 ResF/Q(B*) norm, ResF,Q(F*) of some connected Q-subgroup of ResF/Q(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 = ResF,Q(B*). In this special case, the zeta-functions of the corresponding Shimura varieties can be expressed as products of automorphic L-functions associated to the group G itself (in general, groups besides G are needed as well). This case has already been discussed in [2] and the purpose of the present discussion is to provide further details on the combinatorial 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 ResF/Q(B*) mentioned previously, but even so it is interesting to carry out the exercise in our situation (with further simplifying 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(Af) (where Af 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 zetafunction Z(s, SK) is a product over the places v of Q of local factors SK).
Let F' be a finite extension field of F which is Galois over Q. For o, e Gal(F'/Q)/Gal(F'/F), let Ho = GL2(C) and let Vo be the standard representation of HQ on C2. Then LG° = r[ H, has a natural representation r on V = & Vo, which may be extended to a representation of the L-group LG = Gal(F'/Q) K LG° by putting r(z)(®x ovo) = Qx owo for v e Gal(F'/Q) where-wa = yr-,Q. The result of [2] is that Z(s, SK) = jj,r L(s - d12, 2r, r)m cn,K) up to a finite number
of local factors, where d = [F: Q], 7r runs over the representations of G(A)/Z(R) (where Z is the center of G) which occur in L2(G(Q)Z(R)\G(A)), and m(ir, K) is an integer which is associated to is and K in a way that is described in [2]. The product over 'r is actually finite since m(2r, K) turns out to be 0 for all but a finite number of v. The precise result is AMS (MOS) subject classifications (1970). Primary 05C05, 14G10, 22E50, 22E55. © 1979, American Mathematical Society 185
R. E. KOTTWITZ
186 (1)
Zp(S, SK) = 11 L(s - d/2, 9rp, r)m(a,K) s
where 9rp is the local factor of rr at a finite rational primep which satisfies: (A) F is unramified at p; (B) B splits at every prime of F above p;
(C) K = KpKP where Kp is a maximal compact subgroup of G(Qp) and KP is a compact open subgroup of G(A f), AI being the ring of adeles of Q with component
0 at p and oo. For the rest of 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(rr, K) = 0 unless zp is unramified. Take the logarithm of each side of (1), and use the power
series expansion of log(1 - X). Comparing the coefficients of the powers of pin in the two sides, we see that (1) is equivalent to (2)
Card SK(FP,) = Y, m(z, K)pid, 2 tr(r(g,)') 7r
for all positive integers j, where rr runs over all the representations of G(A)/Z(R) which are unramified at p and which occur in L2(G(Q)Z(R)\G(A)), and where gyp is an element of the semisimple conjugacy class of LG associated to irp. For every place v of Q choose a Haar measure dgv on G(Q,) such that the measure of G(ZZ) is 1 for almost all finite places v. Use dgv to define the action of
on representations of G(Q,). Choose a Haar measure dz, on Z(R), and use the quotient measure dg_/dz0. to define the action of L'(G(R)/Z(R)) on representations of G(R)/Z(R). LetfP be the characteristic function of the subset KP of G(A f) divided by the measure of KP (with respect to 11,,p, - dgv). It is pointed out in [2] that there
exists f0. e C0°(G(R)/Z(R)) such that m(rr, K) = m(rr)trrrp(fPf) where V =
Qx $pr and m(2r) is the multiplicity of rr in L2(G(Q)Z(R)\G(A)) (it is known that this multiplicity is 1, but we will not need to use this fact). By the theory of Hecke algebras, there exists a unique function fpi) in the Hecke algebra ye(G(Qp), Kp) such that
tr zp(fp')) = p;di2 tr(r(g,p)i) if it is unramified,
=0
if 7rp is ramified.
So the right side of (2) becomes E,, m (rr) tr z(fpi?fPf ), 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 jj vdgv on G(A) by some Haar measure dgr on Gr(A) (where as usual Gr denotes the centralizer of r in G), and let mr = meas(Gr(Q)Z(R)\Gr(A)) where the measure used is the quotient of dgr by the measure on Gr(Q)Z(R) Gr(Q) x
Z(R) which is the product of the Haar measure on Gr(Q) which gives points measure 1 with dz-. Applying the trace formula, we find that (2) is equivalent to (3)
Card SK(FP;) _
r
mr
f
J G,(A)\G(A)
f(J)ffpf (g-1 rg) dg
where r runs over a set of representatives of the conjugacy classes in G(Q). It turns out that the integral f G,(R) \G (R) f (g 1rg) dg. is 0 unless r is central or elliptic at oo, and so the right side of (3) is
187
COMBINATORICS
Ear r
(4)
JGT(AI)\G(AI)
fnf(i)(g 1 rg) dgf
where ar = mrf Gr(R)\G(R) Mg I rg) dg. and r runs over a set of representatives for the conjugacy classes in G(Q) which are elliptic or central at 00. To go further we must know more about Card SK(Fp). 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 0 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 integers m', m" such that m" > m' >_ 0 -and m' + m" = d, there is an associated 0-stable subset Yp of SK(Fp), and SK(Fp) is the disjoint union of the sets Yp and another 0-stable subset Yo. 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 IP(Q)\(G(Af) x XP)/Kp for some set Xp on which IP(Qp) and 0 act, the two actions commuting with each other. An embedding L -+ B gives us a map I, --> G, and Ip(Q) acts on G(Af) via Ip(Q) -> IP(A f) --> G(A); also Ip(Q) acts on Xp via IP(Q) - IP(Qp), and KP acts on the product G(Af) x X. by acting on G(Af) alone.
For any positive integer.j and for x e XP, let Tx = {g c IP(Qp): oix = gx}, and let di be the characteristic function of Tx. Choose a Haar measure ,u on IP(Q p), and for r e Ip(Q) let (5)
ID
(r)
xExpm dIP(Qp) p(Ix)
x(r)
where I. is the stabilizer of x in IP(Qp). Choose a Haar measure v on Gr(Af). Let V' = H v#p, - dgv where dgv is the Haar measure on 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 Y. to Card(SK(Fp;)) is
$Q(P)G(P)fp(g I rg) dv where br = meas(IP(Q)\GT(Af) x IP(Qp)). The measure used is the quotient of v x p by the measure on IP(Q) which gives every point measure 1. We see that (6)
rEIP(Q) brPP`''(r)
(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 fp') can be written in a natural way as a sum of functions, and it is this sum which corresponds to the sum over m', m".
The function fpi) in the Hecke algebra = . °(G(Qp), Kp) is characterized by the equation tr 2rp(fpf)) = pfdi2 tr(r(g,p)i) for all unramified representations rcp of G(Qp). Since p remains prime in F, Fp is a field (of degree d over Qp) and G(Qp) x 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 --> f\ from .e to the algebra of polynomials in a, b, a 1, b-I which are symmetric in a, b (a and b are two indeterminates). Regarding GL2(Fp) as the Qp points of ReSF/Q GL2, we get an iso-
morphism f f from Y to the algebra of functions on LG° x {0} which are
188
R. E. KOTTWITZ
obtained by restricting linear combinations of characters of finite dimensional complex analytic representations of the complex Lie group LG. The relation between f v and f - is 1
{
//
bl/ x ... x (ad 0d) x 0) = J v((al
.. ad bl
'0 bd))
We want tofind f, i) E W such that (fpi))-(x) = pid/2tr(r(xi)) forallx E LG° x {0}. Let l be the greatest common divisor of j and d. Then for x=(Oa,1b ) x ... x (ad 1
b) d) x
we have
tr r(xi) = tr r((a, 'O ai bl
0
b) x
_ ((al ... ad)i/l + (b1 ./.
ai+1
(a2
0 b2
b
) x ... X 0'
bd)i/l)'.
It follows that (fpi))v((O 6)) = p,d/2 (ail' + bi/l)1. For k' e Z with 0 < k' < 1/2, let fi, k, be the function in the Hecke algebra such that J vk' = pid/2 (h) (aik'lt bik"/l + aik"lt bik'lt)
where k" = I - V. If l is even, then fork' = k" = 1/2, let fi, k, be the function in the Hecke algebra such that f Uk' = pid /2( k,)aik'llbik'll By the binomial theorem we have fp') = E kl[ 21 f, k'.
Now consider a triple p = (L, m', m"), and let 7 E L - F. Recall that m" is determined by m' (since m' + m" = d). Choose an embedding L --> B. Then 7 gives us an element of IP(Q) and of G(Q). We will show that the contribution of 7 to (6) is 0 unless m' is divisible by d/l, in which case it is equal to the contribution of k' _ Im'ld and 7 to the following rewritten version of (4): [1/21
E
k=oar
1G,(Af)\G(AfP)
f p(g-1 l 7 g)
dv' (' dgp dv J Gr(Bp)\G(Bp) A k'(g-' 7 g) dg,
where dg, is the Haar measure on G7(Qp) corresponding to the Haar measure u on IP(Qp) under the isomorphism IP(Qp) -Z Gr(Qp) induced by the embedding L -> B. It can easily be shown that a.. = br with the choice of measures that we have made, so it is enough to show the following: THEOREM. With notations as above, tpU)(7) = 0 unless d/1 divides m', in which case it is f GT(Qp)\G(Qp) fi, k' (g-1 7g) (dgp/dg,.) where k' = lm'/d.
We begin the proof of this theorem by evaluating SGr(ep)\G(ep) fi,k' (9-' 7g) (dgp/dg,.). This is easy since L splits at p and the hyperbolic orbital integrals of any
f e .$P can be read off from f v. Let a, 0 E Fp be the two eigenvalues of 7 and assume without loss of generality that the valuation v(a) of a is less than or equal to the valuation v(/3) of /3. Then the integral of fi, k' over the orbit of 7 is equal to 0 unless v(a) = jk'll and v((3) = jk"/I, in which case it is IaN/(a - N)2 IF pid/21( r)(meas(Io)-1) P
COMBINATORICS
189
where Io is the subgroup of Ip(Qp) corresponding to 0* x OF*.p under IP(Qp)
Fp x Fp The final result is that the integral is equal to 0 unless v(a) = jk'11 and .
v(/3) = jk"/1, in which case it is (k,)p1k'd11 u(Io)-1.
We must now evaluate cpi(r). This is the combinatorial exercise referred to in the beginning of this paper. First we will describe XP and the actions of IP(Qp) and 0 on X. Let Qpn be a maximal unramified extension of Qp containing Fp, let (9pn be its valuation ring, let 0 be the set of (9p"-lattices in the two dimensional vector space Qpn Q+ Qpn over Qpn, and let .4 be the set of classes of lattices in Qpn p+ Qpn (two lattices M1, M2 are said to be in the same class if there exists a nonzero element c e Qpn such that M1 = cM2). For any lattice M, let M denote the class of M. The set R is the set of vertices of the Bruhat-Tits building of SL2(Qpn). This building is a tree. The groups GL2(Qp°) and Gal (QpnlQp) act on 9' and .4. Let
B1Omp and let o be the Frobenius element of Gal (Qpn/Qp). Let Xp be the set of sequences {Mi}iEZ of elements M1 of R'satisfying: (A) M; Mi_1 pMi for all i c Z; (B) B6dMi = Mi-d for all i e Z.
The action of 0 on X. is given by 1: {M1} -, {M;} where M, = Mi-1. Identify IP(Qp) with A(Fp), where A = {(o 0)). The action of an element r e IP(Qp) on Xp is
given by r{Mi} = {rMi}. Recall that we are trying to calculate cpi(r) = Jx,u(Ix)-1 ox(r), where the sum is taken over x E XP mod IP(Qp). Let Il be the subgroup of IP(Qp) = A(Fp) consisting of all matrices of the form 0
(c0
where n e Z and c e Fp . As before let 10 = A((9F). Then IP(Q p) = Io x I,. Furthermore u(Io) = ,u(Ix) [Io: Ix} _ ,u(II) [IP(Qp): I1Ix], so cp'(r) is equal to ,u(Io)-1Card S(r, j) where S(7-,j j) = {x E Xp mod Il : O ix = rx}. Let W be the apartment in 2 corresponding to the split torus A. Choose a point
po in W, and let Il = {x e R: po is the point of W nearest x}. Let -&' be a set of representatives for the lattice classes in .it, or in other words, let .A1" be a subset of 9' such that no two elements of mil' are in the same class and such that {M: M e .ill'} = M. Then the set of {M1} E Xp which satisfy (C) MO E .ill'
is a set of representatives of Xp mod I,. The condition 0i{M1} = r{Mi} just says
(D) rMi = M=-;. We can summarize this discussion by saying that S(7-,j j) is the set of all sequences of
elements of 3' satisfying (A)-(D). Any sequence {M1} in S(r, j) determines a sequence {x1} = {Mil of elements of 9 satisfying: (A') xi is a neighbor of xi-1; (B') BUdX1 = x i-d;
(C')xoeJI; (D') rxi = xi-1. Let S'(r, j) be the set of sequences {x1} satisfying (A')-(D'). If the valuation of
190
R. E. KOTTWITZ
det(r) is not equal to j, then conditions (A) and (D) are incompatible, and S(7', 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 = 1. Conditions (B') and (D') together are equivalent to the two conditions:
(E') o.dj/l xi = B-i" rail x:.,
(F') rrBs6sdxi = x i-r We will now show that S'(r, j) is empty unless B-iitrdi' a lo. Suppose S'(7, j) is nonempty, and let {x1} e S'(r, j). Then po is the point of 21 closest to xo, so rpo is the point of v2t closest to rxo, where z = 6 d f i' B-J i' rd 1'. But zxo = xo by (E'), and ,r% = s21 since ci = fit, B W = s?t and 7-2t = W. So z fixes po. Since U -dill fixes po, we conclude that B-iiird/i fixes po. But B-ii'rd/i is diagonal and its determinant
is a unit (since we are assuming that v(det r) = j), so that the fact it fixes po implies it belongs to I. So we have shown that S'(r, j) is empty unless B-,i'rdi' e lo. Let a, e Fp be the diagonal entries of r, so that r = (g go). Then B-j I rd it e Io if and only if v(a) = jm'ld and v(/i) = jm'ld. In particular we see that cpi(r) = 0 for all r unless jm'ld and jm"/d are integers. Recalling that 1 = (j, d) and that m' + m" = d, we see that jm'ld, jm"ld e Z if and only if d/1 divides m', and that if d11 does
divide m', then cpi(r) is still zero unless v(a) = jm'ld = jk'11 and v(13) = jm"ld = jk"11. Comparing this with the orbital integrals we have computed, we find that all we have left to show is the following lemma. LEMMA. Suppose B-i'ird/I e Io. Then Card S'(r, j) = ('k,)p;k'da where k' = lm'/d.
To prove this we first look more closely at conditions (E') and (F'). Let 4" = . : v(x) = x}, where z is the automorphism of 4 defined previously. Then (E') simply says that xi e R= for all i e Z. Let Qp be the completion of Qp . There exists a unique continuous action of GL2(Qp) on .4 which extends the action of {x e
GL2(Qp°) on .4 (M is given the discrete topology). Each element of GL2(Qp) acts by a simplicial automorphism of the tree .4. Using the fact that B-i ltrd it e Io, it is easy
to show that there exists a diagonal matrix o with diagonal entries in Qp- {0} such that 6°1d1' 6-1 = B-ii'rd/'. A simple calculation shows that .41' = 5'q°d'/1 where .g°dj1' denotes the set of fixed points of adi '1 in R. The subtree 31rd'" of .q can be canonically identified with the Bruhat-Tits building of SL2 over the fixed field of Qp" under Udi". So °d'/` is a tree in which every vertex has exactly pdJ'i + 1 neighbors. Since gT is simply the translate of god'" by 6, it too is a tree in which every vertex has exactly pdi /i + 1 neighbors. A further consequence of B-i llyd i' e Io is that 3" contains N. The reason for this is that 6diii fixes .t pointwise, as does every element of Io. In (F) the automorphism rrBs,sd of 9 appears. We will need to know the effect
of this automorphism on W. The automorphism 6sd acts trivially on W. The automorphism B acts on slt by translation by m" - m'. The automorphism rd acts on 2t in the same way that Bi does, since B-ii'rai' e Io, and therefore r acts on 9t by translation by (m" - m')j/d. So rrBs3.sd acts on W by translation by 1 - 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 2t be a
191
COMBINATORICS
subtree of Tin which every vertex has 2 neighbors. Let po e W. Let I and k be nonnegative integers such that 1 - 2k is positive. Let e be an automorphism of T which acts on It by translation by 1 - 2k. Let N(l, k) be the set of sequences of length 1 + 1 of points x0, , x1 in T satisfying: (i) x1 is a neighbor of xs+1 for i = 0, ,1 - 1; (ii) po is the point of W closest to x0;
(iii) x1 = x0. Then N(l, k) = (k)qk.
Let us agree to call a sequence of points x0, 1 joining x0 to x1. Then
x1 satisfying (i) a path of length
N(l, k) = E Card {paths of length l joining x0 to exo}, x0E ((
where mil 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 x0 and ex0 is equal to 1 - 2k + 2n where n is the distance from x0 topo. s xo
xo
n 11
po
11
1-2k
po
So N(l, k) = Yn_0 Card {x0 e &: distance from x0 to po is n} M(1, k - n). The number of points x0 at distance n from po such that po is the point of 21 closest to x0
is (1 - q-1)qn if n > 0 and is 1 if n = 0. Therefore
N(1, k) = M(l, k) + (1 - q-1) En=1 q'M(l, k - n). The function M(l, k) on the set of pairs of integers (1, k) such that l > 2k > 0 is characterized by the three properties : (i) M(l, 0) = I for 1 >_ 0;
(ii) M(l, k) _ (q + 1)M(l - 1, k - 1) for l = 2k > 0; (iii) M(l, k) = M(1 - 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 I from x toy is the same as to give a neighbor x' of x and a path of length 1 - 1 from x' to y. For 1, k e Z, let m(l, k) be the coefficient of X1 Y" in the formal power series ex-
pansion of f(X, Y) _ (1 - qY) (1 - Y)-1 (I - X - qXY)-1. We will show that M(l, k) = m(l, k) for 1 >_ 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 - qY) (1 - Y)-1. Comparing coefficients of X1Yk on the two sides
we see that m(1, k) - m(l - 1, k) - qm(l - 1, k - 1) = 0 if 1 > 0, which shows
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R. E. KOTTWITZ
that m satisfies (iii). Setting Y = 0 in the definition off(X, Y) we get E o m(1, 0)X1 _ (1 - X)-I = 1 + X + X2 + , which shows that m satisfies (i).
We still have to show that m satisfies (ii). Let g(X, Y) = (1 - X -
qXY)-1 and
let n(l, k) be the coefficient of X1 Yk in the formal power series expansion of g(X, Y). We have
(1 - X -
(X + qXY)' _
qXY)-1 =
1=0
(k)X1-k(gXY)k
1=0 k=0
[Z
ZO(k)gkX 1 Yk,
and therefore n(l, k) = (k)qk. In particular for k > 0 we have
n(2k - 1, k) = (2k k
1)qk
=
(k
-
1)qk-1 q
= gn(2k - 1, k - 1).
This shows that the coefficient of X2k-1 yk in (1 - qY)g(X, Y) is zero. But (1 - qY) g(X, Y) is equal to (1 - Y) f (X, Y), so the coefficient of X2k-1 Yk in (1 - Y) f(X, Y)
is zero, which means that (7)
m(2k - 1, k) = m(2k - 1, k - 1) for all k > 0.
We have already seen that if ! > 0, then m(l, k) = m(1 - 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(1, k), we see that
N(l, k) = m(1, k) + (1 - q-I) E gnm(l, k - n) n=1
since m agrees with M whenever M is defined and m(!, k - n) is 0 for n > k. We may use the equation above to extend the definition of N(l, k) to all integers 1 and k. With this convention we find that EN(l, k)X'Yk is equal to Y' m(!, k)X1Yk + (1 - q-1) E qn E m(l, k - n)X1Yk n=1
1, k
1'k
=.f(X, Y) + (I - q-1) E gnyn 2] m(l, k n=1
= f(X, Y) [1 + (1
n)X1yk-n
1, k
-q-1) 2] gnYn] n=1
= f(X, Y)(1 - Y)(1 -
qY)-1 = g(X,Y)
We have already computed the formal power series expansion of g(X, Y). Looking back at the answer, we find that N(!, k) = (k)qk. This proves the lemma. REFERENCES
1. R. P. Langlands, On the zeta-functions of some simple Shimura varieties, Notes, Institute for Advanced Study, Princeton, N. J., 1977.
2. -, Shimura varieties and the Selberg trace formula, Canad J. Math. (to appear). UNIVERSITY OF WASHINGTON, SEATTLE
Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 193-203
NOTES ON L-INDISTINGUISHABILITY (BASED ON A LECTURE OF R. P. LANGLANDS)*
D. SHELSTAD 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 formulations (cf. [7]) which suggest possible generalizations. We assume that F is a field of characteristic zero. In the case that F is local there are only finitely many
irreducible admissible representations of G(F) which are "L-indistinguishable" from a given representation. We structure this set, an L-packet, by considering not the characters of the members but rather sufficiently many linear combinations of these characters. In the case that F is global we consider certain L-packets of representations of G(A) and describe the multiplicity in the space of cusp forms of a representation 7r = &7r, 'rv in terms of the position of the local representations 7rv in their respective L-packets.
1. AC(T), T(T) and 8(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 AC(T) _ {g e G(F) : ad g 1/Tis defined over F} and Z (T) = T(F)\W(T)/G(F). If g e AC(T) then a -> gQ = a(g)g-1 is a continuous 1-cocycle of 03 = Gal(F/F) in T(P). The map g --> (6 --> gv) induces an injection of Z(T)into a subgroup 6(T) of H1(03, T(F)) defined as follows.
Let 1
be the preimage of T in the simply-connected covering group G5C of the
derived group of G. Then 8(T) is the image of the natural homomorphism of H1((53, TSe(F)) into H1((53, T(F)). If Hl(t 3, G,c(F)) = 1 and so, in particular, if F is
local and nonarchimedean then `,(T) coincides with 8(T). L-indistinguishability appears when G contains a torus T such that ¶(T) is nontrivial.
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 (l3 by JK/F =
Gal(K/F) in the definitions of the last section. An application of Tate-Nakayama duality then allows us to identify 8(T) with the quotient of P e X*(T.) : EoEdxiF6d = 0} by
up,
S E X*(Tsc): A l
po,
, e X*(T)
QE@X/F
AMS (MOS) subject classifications (1970). Primary 22E50, 22E55.
*This work has been partially supported by the National Science Foundation under Grant MCS76-08218. © 1979, American Mathematical Society 193
194
D. SHELSTAD
) denoting Hom(GLI, ). A quasi-character x on X*(TS°) trivial on this latter module defines, by restriction, a character on s(T). In [7] there is attached to each triple (G, T, x) a quasi-split group over F (there denoted H). We will pursue this just in the case that G is an inner form of SL2. X*(
3. Groups attached to an inner form of SL2 (local case). Suppose that G is an inner form of SL2 and that Fis local; we will continue with this assumption until §14. We have then two groups to consider: SL2 and the group of elements of norm one in a quaternion algebra over F. We may take PGL2(C) as LG° (notation as. in [1]) and the diagonal subgroup as distinguished maximal torus LT °.
Fix a maximal torus T in G, defined over F. A quasi-character x from the last section is just a BK/F-invariant quasi-character on X*(T). We fix an isomorphism between X*(T) and X*(LT°) = Hom(LT°, C") as follows. If G is SL2 and T the
diagonal subgroup we use the map defined by the pairing between X*(T) and X*(LT°); 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*(LT°) we transfer K to a quasicharacter on X*(LT°); using the canonical isomorphism betweenHom(X*(LT°), C") and LT° we then regard s as an element of LT°. At the same time we transfer the action of 03KIF on X*(T) to X*(LT°) and LT°, writing UT for the new action of E03K/F Here are the possibilities. If Tis split then (53K/F acts trivially and K is an arbitrary element of LT°. If T is anisotropic, suppose that T is defined by the quadratic extension E of F. We shall assume that K is some fixed large but finite Galois extension of F containing, in particular, E; OK/F acts on T through 3E/F. Let a° be the nontrivial element of 03EIF and av be a coroot for Tin G. Then o °av = - av so that generates X*(T) there are then just two possibilities for K. (K(av))2 = 1. Since
The nontrivial K defines the element (-o °)* of LT°; here, and throughout these notes, we use (a d)* to denote the image in PGL2(C) of the matrix (a d) in GL2(C). The action of UT on LT° is described as follows: if O 'C= OK/F maps to the trivial element in (53E/F under C3K/F -' 3E/F then 0'T acts trivially and if U maps to 6° then UT acts by CO
b)*
- a)*
As an element of LT°, T is UT-invariant, U E 03K/F. We define LH° to be the connected component of the identity in the centralizer of x in LG°. Whatever T, if x is trivial then LH° = LG° and if x is nontrivial then LH° = LT°. Let a e 03K/F. Then since both LH° and LT° are invariant under 6T we may multiply UT by an inner automorphism of LH° to obtain an automorphism 6H stabilizing LT° and fixing each root, if any, of LT° in LH°. The collection {6H, U E WK/F} defines a semidirect
product LH = LH° x WK/F where WK/F, the Weil group of K/F, acts through 3K/F. In duality, we obtain a quasi-split group Hover F. Specifically: PROPOSITION. (a) If x is trivial (whatever T) then LH = LG = LG° x WK/F and
H = SL2. (b) If T is split and r nontrivial then LH = LT° x WK/F and H = T.
195
L-INDISTINGUISHAnILITY
(c) If T is anisotropic and c nontrivial then LH = LT° x WK/F where w e WK/F acts trivially on LT° if w maps to 1 under WK/F -' OK/F -' '3E/F, and w acts by (o n)* -' (o a)* if w maps to o-° (E, 6° as before); and H = T.
To indicate that H is defined by (T, ,) we write H = H(T, T). Note that the choice (of diagonalization) made in defining the isomorphism between X*(T) and X*(LT°) does not affect H(T, s). 4. Embedding L-groups. For later use we specify an embedding CH of LH 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 LT° in LG° by: (0
\0
b)* x w
b)*
xw
if w e WK/F maps to 1 under WK/F -' 03K/F -' 03E/F, and
a0 (0
b)* x
(a
w
0
1 (0 0)* x w
b)* \1
if w maps to 6°. 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 LH embeds in LG; [7] indicates how to accommodate this (see Lemma 1 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, K).
We choose Haar measures dt and dg on T(F) and G(F) respectively. If f e C,°(G(F)), b e ¶(T), and r e T(F) is regular in G then we set
01(r, .f) = f h-1T(F)h\G(F) f(g 1h-lrhg)
dg (dt)h
where It e W(T) represents d and (dt)h is the measure on h-1T(F)h obtained from dt by means of ad h. For S e 5(T) - Z (T) we set 06( J) - 0. Recall that W(T), Z(T), &(T) were defined in §1. In the case that x is trivial (whatever T) we define OT/x(
,f) _ OT/1( ,f) = e(G) E 01( $ee(T)
,f)
where e(G) = 1 if G = SL2 and s(G) _ -1 if G is anisotropic. We call 0T/1( a "stable" orbital integral.
)
If T is split and x nontrivial then we define
OT/K(r,f) = Ti I- T21 IT1r21 1'2
E K(,3)0a(r, f)
de&(T)
where 7-1, T2 are the eigenvalues of 7%
Suppose that T is anisotropic and that K is nontrivial. Then our normalization depends on two choices. First choose a nontrivial additive character OF on F. If E is the quadratic extension of F attached to T then we define A(E/F, OF) as in [5]. Also choose a regular element r° of T(F). Let K' denote the quadratic character of Fx attached to E. Then we define
196
D. SHELSTAD
0T/K(r,J) = A(E/F, Y'F)K' (T 1
- r2)
Ti
I1 7211/2
a
E(T)
where the order on the eigenvalues Ti' 7-2 of r and r°, TO of r° is prescribed by fixing a diagonalization of T. A different choice of OF or T° causes at most a sign change in the normalizing factor.
6. Transferring orbital integrals. The integrals OT/,r( , f) can be transferred to stable orbital integrals on H = H(T, s) in the following sense: LEMMA. If f E2 C°°(G(F)) then there {exists existsfH e C0°(H(F)) such that OT / 1(r, fH) = cT /" (r, .f )
and (i) if G is anisotropic ands trivial (so that H is SL2) then
OT'/1( J H) __ 0 for T' split; (ii) if G is SL2 ands trivial (so that H is SL2 again) then OT /1(
, fH)
OT'/1(
,
f)
,for all T', provided that the Haar measures dt' are chosen consistently.
This is proved (in [4]) by a case-by-case argument. Note that if H is T then OT/1( J H) is just fH itself. For general (real) groups a formalism for transferring the OT1'( , f) to His developed 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 -' OT/1(7., f), where r is a regular semisimple element of G(F) and T denotes the torus containing T. The transfer of the oT /x( , f) to H establishes a correspondence (f, fH) between C,°(G(F)) and C°°(H(F)). Dual to this correspondence there is a well-defined map from the space of stable distributions on H(F) to the space of invariant distributions on G(F); that is, if OH is a stable distribution on H(F) then we may define a (con-
jugation-) invariant distribution 0 on G(F) by the formula e(f) = OH(fH), f e CC°(G(F)). This map, OH --> 0, will be central to our study of L-indistinguishability. We denote by x.n the character of (the infinitesimal equivalence class of) an irreducible admissible representation of G(F); we regard xn 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 SL2, 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 W(T), T c G.
L-INDISTINGUISHABILITY
197
8. Local L-packets and some stable characters. We assume that the Langlands correspondence has been proved for G; in fact enough has been proved in each of the cases being considered. In the following definitions we also allow G to be a torus. We denote by 0(G) the set of equivalence classes of admissible homomorphisms of WF into LG° i WW (notation and definitions as in [11). To each {(p} e 0(G) there is attached a finite collection 1Ic.} of irreducible admissible representations of G(F), an L-packet. Two irreducible admissible representations of G(F) are said to be L-indistinguishable if they belong to the same L-packet. In the case that G is an inner from of SL2 there is a simpler definition: i1 and 2r2 are L-indistinguishable if and only if there exists g E 6(F) such that 7U2 is equivalent to 7cI ° adg. We set x{,P} = Ef(, Z,,. Clearly: PROPOSITION. xc,} is stable.
Since the Langlands correspondence has been proved for any (connected reductive) real group [6] we can define characters x{,} in that case. In general x{,,} need not be stable; however if the constituents of 1I{,P} are tempered (cf. [3]) then x{,} is stable [9].
From §12 on we will not need to distinguish in notation between an admissible homomorphism (p: WW -* LG° x WW and its equivalence class; we will then denote both by (p and write II,, x9, etc.
9. Character identities (introduction). We return to the map of stable distributions
on H(F) to invariant distributions on G(F). If {(PHI e 0(H) then, as we have observed, x{,H} is stable. Its image in invariant distributions on G(F) is represented by some function x on the regular semisimple elements of G(F). This function x is computed by the Weyl integration formula. We write: XIPH} x Z. Recall the embedding CH of LH in LG (§4). We could have used WW in place of WK/F in defining LG, LH, and CH. With these modifications, suppose that (PH is an admissible homomorphism of WW into LH. Then (p = CH ° (pH maps WW to LG. Suppose that (p is admissible; recall that (local) admissibility imposes the condition that the image of cp lie only in parabolic subgroups of LG which are "relevant to G" (cf. [1]). Then we say that {cp} e 0(G) factors through {(pH} E 0(H). Suppose that {cp} factors through {(pH}. Then linear combinations of the characters of the representations in II{,} make natural candidates for x (~ x{PH}).
10. "S°\S". We introduce a useful group. It is easier to work with a homomorphism (p: W. --> LG rather than an equivalence class {(p}. We exclude the case of sp(2) [1] and corresponding special representation of G(F), and consider just an
admissible homomorphism (p: WK/F -- LG where K, LG (and LH, cH) are as earlier. We define SP to be the centralizer in LG° of the image of (p; SS will be the con-
nected component of the identity in S,. If (p' is equivalent to (p then there exists g E LG° such that S,. = gSPg 1 and SO, = gS;g 1. Suppose that (p = CH ° (pH where (pH is an admissible homomorphism of WK/F into LH and H = H(T, x). We regards as an element of LT°. Recall that LH° is the connected component of the identity in the centralizer of x in LG°. By definition (§3), 6H fixes x, u e CASK/F. It follows then that s lies in the center of the image of LHin LG. Therefore a centralizes the image of cPH(WK/F) in LG; that is, x centralizes
198
D. SHELSTAD
cp(WK/F). We have then that s e So,. We define sg,(s), or just s(s) when (p is understood, to be the coset of s in S°\Sq,. Recall that this quotient appeared in [3]. Suppose that cp' is equivalent to cp and that cp' = tH ° pH where co : WK/F -+ LH
and H = H(T', s'). Then x' r= 5,,. Write cp' as ad g. cp, g e LG°. Then g ls'g e S. We define s,,(s') to be the coset of g 1E'g in SO\Sp. We will see that S,\S9, is abelian. Therefore s,,(K') is independent of the choice for g. We continue with the same cp, cp'. The conjugation ad g induces an isomorphism between SO\S9, and SO.\Sc, which carries s(p(s) to s,,,(K). Because both groups are abelian this isomorphism is independent of the choice of g in the equivalence cp' = ad g ° cp. We may therefore regard SS\S, and the elements sp(K) as attached to {cp}.
11. Calculations. We compute explicitly S, \S, and the elements s(K) = sw(s). We need take just one homomorphism cp: WK/F __+ LG from each equivalence class. If we write cp(w) as cpl(w) x w, w e WK/F, then the homomorphism cp1: WK/F -+ LG° = PGL2(C) lifts to a two dimensional representation 01 of WK/F [7, Lemma 3]. (i) Suppose that 01 is reducible.
Then 01 factors through z: WK/F --> Fx and is defined by a pair (, v) of quasicharacters on Fx. We take 01(w) =
(9(14w))
0
W e WK/F
v( (x ))/
If (,u/v)2 0 1 then S. = S° = LT° and if u = v then Sq' = Sq' = LG°. However, if ,u/v has order two then S. = LTO >
Hwhen H = H(, 1); s(1) is trivial also. Suppose that ,u/v has order two. In (ii) we will show that a homomorphism equivalent to cp factors through LT, where T is a torus defined by the quadratic extension E of F attached to u/v, and that the associated s(K) = s,(K) is the coset of (oi i o) *-
(ii) Suppose that b1 factors through a representation 2 = Ind(WE/F, Ex, 0) of WE/F where E c K is a quadratic extension of F and 0 is a quasi-character on Ex. Then (p, factors through cp2i the projective representation defined by 2. Let 6° be the nontrivial element of (SSE/F and define B(x) = 0(x°°), x e Ex. We realize WE/F explicitly as {x x p; x e Ex, p e (1, 6 }}, with multiplication rule (x x p)(x' x p') = x(x')P ap, p' X pp,
where ap, P. = 1 unless p = p' = 6° and a,,Q° is some (chosen) element a of Fx NmE/FE>. Then we may assume that
02(x x 1) =
00
( 0
0)
x c Ex,
0(x) '
and 0
P2(1 x 6 ° ) = (z
zl
0/
L-INDISTINGUISHABILITY
199
where z is a chosen square root of 0(a). Thus B(x)
02(X x 1) =
x e Ex,
0
0(,X))*
and
(P20 x 6°) _
1
0)*
Let T be a torus attached to E. Then rp factors through LT; indeed we chose the embedding CT of LT in LG (§4) so as to insure this. Recall that T = H(T, t) where a, as element of LT°, is (-o °)* ; s(x) is the coset of (-o °)* in S°/S,. To compute S p, suppose first that (0/0)2 0 1. Then
S=
{1' (
0
1)*}
so that S,' \Sv = Z/(2); we have already recovered the nontrivial element of S°\S, as an s(ee).
Suppose now that 0/0 has order two. Then Sc, coincides with' 1(WK/F); that is, Sc
- {1' \
1)*' (1
0
0)'
1
(-Ol
0
)*}.
Therefore S,\Sc = Z/(2) Q+ Z/(2). We have recovered only (-o °)* (and 1) as an s(rc). To recover the remaining elements of S,'\Sc, we recall homomorphisms rp', rp": WK/F + LG which are equivalent to rp and factor through other LH.
Let E0 be the quadratic extension of E defined by 0/0. Then Fx/NmE,IFEE = Z/(2) p+ Z/(2) so that there are two distinct quadratic extensions E' and E" of F which are distinct from E and contained in E0. We pick a quasi-character 0' on (E')x such that O'°NmEOIE. = 0°NmEO,E and 0'/0' is the quadratic character attached
to E0/E' (0'(x) = 0'(x°'), 1 0 6' e CSSEVF); we pick 0" similarly. Then 02' = Ind(WE'IF, (E')x, 0') and " = (E")x, 0") define representations Bp, (Pi of WK/F, each equivalent to 01, and hence homomorphisms (P', rp° : WK/F LG, each equivalent to rp. We define (pi, rpz and rp", rp2 as we did cpl, cp2; we realize rp2 and (p2 as we did rp2. Then
coincide and equal
As with rp earlier, rp' (... rp") factors through L(T') (... L(T")), where T' (...T") is
some torus attached to E' (... E"). Suppose T' = H(T', rc') and T" = H(T", K"). Then, writing p = ad g' ° (p' = ad g" ° rp", g', g" E LG°, we have 1
S(Ic)
g' (0 1)* g -1
=
and
s(rc") =
g"
(-0 1
0)* g"-i
200
D. SHELSTAD
An elementary argument shows thatnone of the conjugations ad g', ad g", ad(g')-'g" may fix (-o °)* and so we conclude that
{40, AX")} CO1
0)*
10
0)* J
Thus we have recovered each element of S°\S, as an "s(ic)". We have one further case to consider: 0 = 0. We pick a quasi-character u on F" such that 0 = u ° NmE,F. Then rp is equivalent to the homomorphism of type (i) above attached to the pair (u, Cu) where C is the quadratic character of F" defined
by the extension EIF. We now call that homomorphsim (p'. We had that S,,\S,, has two elements and that (° o)* is a representative for the nontrivial coset. We now
recover this coset as an s,(s). Let T be a torus attached to E. Then (P factors through LT and the associated s is (-o °)*. If cp' = ad g ° (p, g e LG°, then 0
g\
0
1/* g 1
(10
0)*
(mod LT°),
and we are done. (iii) Suppose that 01 is of tetrahedral or octahedral type. Then S. = SS = 1. This is a straightforward exercise. In summary: PROPOSITION. (1) SO\S,, is one of 1, Z/(2) or Z/(2) Q+ Z/(2).
(2) For each element s of
there is a pair (T, ic) such that s = s(ic).
12. Character identities (continued). From now on we will not distinguish in notation between an admissible homomorphism ro: WK/F --.> LG and its equivalence
class; that is, we denote both by cp ... as we have noted, S, \S, can be regarded as attached to the equivalence class. Suppose that (p factors through cpf, where H = H(T, K). Let s = s(ic). Then, in the notation of §§8, 9: LEMMA. There exist integers <s, 7r), r e III, such that
XH - E,rEnp <S,Z>xn. The proof [4] is again a case-by-case argument. Here is a summary of the explicit results. (A) G = SL2. We consider (p as in (i), (ii), (iii) of the last section. (i) If (u/v)2 0 1 or if ,u = v then 14 contains one element which we denote by z;
S°\S,p = 1 and <1,'> = 1. If p/v has order two then III has two elements which we denote by n1 and ir2. Recall that SS\S, = Z/(2); <s,'1) and <s, 22> are the two characters in s. These characters depend on the choices we made in normalizing orbital integrals (§5); that is, for some choices < , 21> is the trivial character and for others < , 1r2> is the trivial one. (ii) We have already considered the case 0 = 0. If 0/0 is not of order two then II4 has two elements and SS\S. = Z/(2); the result is as in (i). Suppose that 0/0 has order two. Then we had that SS\Sp is Z/(2) Q+ Z(2); III has four elements, too. Suppose I lw = {'r i = 1 , , 4). Then we may (and do) normalize orbital integrals so that the < , z1> are the four characters on SO,\S,.
201
L-INDISTINGUISHABILITY
(iii) Here 14, has one element, say ,r; Sg'\S,, = I and <1, 1c) = 1. (B) G anisotropic. We have only to consider rp as in (ii) with 0 0 0, and (iii). Suppose F = R. Then Sy,\Sp = Z/(2) since we exclude (iii) also. However II, has only one element, say 7r; <s, ir> is a character in s, trivial or nontrivial according to our choice of normalizing factors for orbital integrals. Suppose that F is nonarchimedean. Then ff. has the same number of elements as the corresponding L-packet in SL2, and the result is as there, except when cp is of type (ii) with 0/0 of order two. Then the corresponding packet in SL2 has four elements; II,, has only one element, say ir. We must set <1, z> = 2 and <s, z> = 0, s
1; in particular, < , z> is not a character. As for generalizing these identities we will report some progress for real groups
in a forthcoming paper; in the case that x is trivial, so that H is a quasi-split inner form of G, an appropriate identity is known provided that the constituents of II4, are tempered [9]. 13. Structure of local L-packets. From the results of the last section we conclude: PROPOSITION. If G is the quasi-split form (SL2) then the pairing < II4 (noncanonically) as the dual group of S,'\Sp.
,
> identifies
If G is anisotropic we remark only the existence of the functions < , iv>, 7r e III. With some qualifications, we can expect an analogue of the proposition for a general quasi-split real group (cf. [3]).
14. A global application. Suppose now that F is a global field and that G is SL2. There are analogues for the groups H of the local case; again H may be a maximal torus in G, defined over F, or G itself. We assume that K is some large but finite Galois extension of F and consider
just homomorphisms cp: WK/F -> LG defined by representations of the form Ind(WK/F, WK/E, 0), where E is a quadratic extension of F contained in K and 0 is a grossencharacter for E not factoring through NmE/F. If Tis a torus attached to E then cp factors through LT (as earlier, provided that LT is correctly embedded in LG). We define III to be the set of representations 7r = Qx v7rv of G(A) which are irreducible, admissible (as in [1]) and such that 1r, e 14, for each place v. The set II4, is an L-packet in the following sense. We define two irreducible admissible representations 7r1 = Qx,1r, and ire = Qx, it of G(A) to be L-indistinguishable if for all places v, irv and iv are L-indis-
tinguishable and for almost all v, iv and irv are equivalent. An L-packet is an equivalence class for this relation. We have singled out the packets ff. for the following reason. As is proved in [4], two representations from such a packet may appear with different multiplicities in
the space of cusp forms for G. One may appear, with multiplicity one, and the other not appear. For the remaining L-packets the members do appear with equal multiplicity (conjectured to be either zero or one). In [4] there is a formula for the multiplicity with which a representation from II,, appears in cusp forms; it is this we wish to discuss. As in the local case, we define SP to be the centralizer of cp(WKIF) in LG° and S.' to be the connected component of the identity in S.. As in (ii) of § 11 we have that either S,\Sp = Z/(2) or S,'\Sp = Z/(2) Q+ Z/(2). For each place v, Sp embeds
202
D. SHELSTAD
naturally in SS, and SS in Sp,. There is then a natural map of SS\S, into SSv\S,,v. In
notation we will not distinguish between an element of S,' \S, and its image in S,\S'Pv.
Recall that the local characters < , zv> depend on the choices we made when normalizing orbital integrals (§§5, 12). We chose a nontrivial additive character OF, on F and regular element r° = rv of T(FF) for each torus T anisotropic over F. Instead, we now choose a nontrivial additive character 0 on F\A and for each torus T anisotropic over F a regular element r° in T(F). At each place v where T does not split we use cb and r° to specify OT,,,( , ), x nontrivial. Fix s e S, \S, and it a 11.. Then we have <s, acv> = 1 for almost all v. Therefore we may define <s, 2r> = 14<s, 7rv>. Then [4] : PROPOSITION. (i) <S, lr> is independent of the choices made for 0 and r° (ii) < , > induces a (canonical) surjection of Ifp onto the dual group of SP'\Sco.
Also: THEOREM. The multiplicity with which 'r e 11, appears in the space of cusp forms is : ° 1S
[Srp\
,p]
S
<s, Z> .
SES;\S'
That is, those r for which < , z> is the trivial character appear with multiplicity one and those z for which < , 'r> 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 ResE,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]: d,, °
[Sw Sw
E 1.7s,
<s, 21)
where d. is defined as follows. If cp, cp': WF -> LG are admissible homomorphisms (cf. [1, §16]) call (p and (p' locally equivalent everywhere if cpv is equivalent to (P, for
each place v. Call cp and (p' weakly globally equivalent if cp' is equivalent to cusp where co is a continuous 1-cocycle of WF with values in the center of LG° such that the restriction of cu to each local group is trivial. Then d, is the number of weak global equivalence classes in the everywhere local equivalence class of cp. In [8] L-indistinguishability plays a role in relating the zeta-functions of certain Shimura varieties to automorphic L-functions. Briefly, in the case discussed in [2], where G is the full multiplicative group of some quaternion algebra over a totally real field and there is no L-indistinguishability, functions L(s, Ir, p) appear (7r 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 multiplicative group, we must consider at least some of the L-packets 11. where different multiplicities in cusp (...automorphic) forms occur. Then cp: WF -* LG factors through CT: LT -+ LG, with T a nonsplit torus of G. If cp = CT ° (PT, ZT e HIT and PT = p cT then L(s, z, p) = L(s, 2VT, PT). There is a natural decomposition PT = pT p+ pT, where pT, p2T may be reducible, and it is the functions L(s, irT, p`T) which are
relevant.
L-INDISTINGUISHABILITY
203
REFERENCES
1. A. Borel, Automorphic Lfunctions, these PROCEEDINGS, part 2, pp. 27-61. 2. W. Casselman, The Hasse-Weil 1function of some moduli varieties of dimension greater than one, these PROCEEDINGS, part 2, pp. 141-163.
3. A. W. Knapp and G. Zuckerman, Normalizing factors, tempered representations and L-groups, these PROCEEDINGS, part 1, pp. 93-105.
4. J-P. Labesse and R. P. Langlands, L-indistinguishability for SL(2) (to appear). 5. R. P. Langlands, On Artin's L -Junctions, Rice Univ. Studies, vol. 56, no. 2, pp. 23-28, 1970. 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 of some simple Shimura varieties (to appear). 9. D. Shelstad, Characters and inner forms of a quasi-split group over R, Compositio. Math. (to appear). 10. , Orbital integrals and a family of groups attached to a real reductive group, Ann. Sci. Ecole Norm. Sup. (to appear). COLUMBIA UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 205-246
AUTOMORPHIC REPR + SENTATIONS, SHIMURA VARIETIES, AND MOTIVES. EIN MARCHEN R. P. LANGLANDS 1. Introduction. It had been my intention to survey the problems posed by the study of zeta-functions of Shimura varieties. But I was too sanguine. This would be a mammoth task, and limitations of time and energy have considerably reduced the compass of this report. I consider only two problems, one on the conjugation of
Shimura varieties, and one in the domain of continuous cohomology. At first glance, it appears incongruous to couple them, for one is arithmetic, and the other representation-theoretic, but they both arise in the study of the zeta-function at the infinite places.
The problem of conjugation is formulated in the sixth section as a conjecture, which was arrived at only after a long sequence of revisions. My earlier attempts were all submitted to Rapoport for approval, and found lacking. They were too imprecise, 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 importance for the study of motives of CM-type. It is defined in the fifth section, where its hypothetical properties are rehearsed. With the introduction of motives and the Taniyama group, the report takes on a tone it was not originally intended to have. No longer is it simply a matter of formulating 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 relation between automorphic representations and motives. The relation is usually AMS (MOS) subject classifications (1970). Primary 14G10, 14F99. © 1979, American Mathematical Society
205
206
R. P. LANGLANDS
couched in terms of L-functions, and no one has suggested a direct connection. It may be provided by the principle of functoriality and the formalism of tannakian categories. This possibility is discussed in the highly speculative second section. However there is a small class of automorphic representations which are certainly not amenable to this formalism. I have called them anomalous, and in order to make their significance clearer I have discussed an example of Kurokawa at length in the third section. Although the anomalous representations form only a small part of the collection of automorphic representations, they are frequently encountered, 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 conform to the facts, and is prepared to abandon it when that is called for. The views of this report are in any case not peculiarly mine. I have simply fused my own observations and reflections with ideas of others and with commonly accepted tenets. I have also wanted to draw attention to the specific problems, on which expert advice would be of great help, and I hope that the report is sufficiently loosely written that someone familiar with continuous cohomology but not with arithmetic can turn to the seventh section, overlooking the first few pages, and see what the study of Shimura varieties needs from that theory, or that someone familiar with Shimura varieties but not automorphic representations or motives will be able to find the definition of the Serre group in the fourth section and the Taniyama group in the fifth, and then turn to read the sixth section. Finally a word about what is not discussed in this report. The investigations of Kazhdan [26] and Shih [46] on conjugation of Shimura varieties are not reviewed; their bearing on the problem formulated here is not yet clear. L-indistinguishability is not discussed. Little is known, and that is described in another lecture [44]. Problems caused by noncompactness are ignored. There is a tremendous amount of material on compactification and on the cohomology of noncompact quotients, but no one has yet tried to bring it to bear on the study of the zeta-functions. The
omission I regret most is that of a discussion of the reduction of the Shimura varieties modulo a prime [36]. Here there is a great deal to be said, especially about the structure of the set of geometric points in the algebraic closure of a finite field, starting with the work of Ihara on curves. I hope to report on this topic on another occasion.
2. Automorphic representations. Our present knowledge does not justify an attempt to fix a language in which the relations of automorphic representations with motives are to be expressed, Nonetheless that of tannakian categories [40] appears promising and it might be worthwhile to take a few pages to draw attention to the
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
207
problems to be solved before it can be applied in the study of automorphic representations. 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 r has a central character co and z -> Ico(z)I 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 i is said to be cuspidal if for any K-finite vectors u and v,
in the space of r and its dual, v 1(g) <7c (g)u, v> is square-integrable on the quotient Z(F)\GL(n, F). To construct an arbitrary irreducible admissible representation one starts from a partition {ni, , n,} of n and cuspidal representations zI, , z, of GL(ni, F). If wi is the central character of zi, there is a real number si such that Icoi(z)I = Izlsi if z lies in the centre of GL(ni, F), a group isomor-
phic to Fl. Changing the order of the partition, one supposes that sI > > s,. The partition defines a standard parabolic subgroup P of GL(n) and o = Qx 1ci a representation of M(F), because the Levi factor M of P is isomorphic to GL(n1) x ... x GL(n,). The representation c yields in the usual way an induced representation Io of G(F). IQ may not be irreducible, but it has a unique irreducible quotient, which we denote nl J+ F+ z,. Every representation is of this form and iI 0 0 Zr ^' pi f . ff acs if and only if r = s and after renumbering 'r1 _- r . Thus every
representation can be represented uniquely as a formal sum, in the sense of this notation, of cuspidal representations. The representation 'I HI 0 Zr is said to be tempered if all the si 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 Z _ ZI 0+ 7r Z' = zi 0+ f acs, with ni and n; cuspidal, we set
Hom(ic, ic') =
E[ C. (i. i I,ri -,r1l
The composition is obvious. The tempered representations form a subcategory H°(F). If F is a global field then z is said to be cuspidal if i is a constituent of the representation of GL(n, AF) on the space of measurable cusp forms (P satisfying (a) iof(zg) = w(z)co(g), Z E Z(4F), (b) JZ(AF)G(F)\G(AF) L
If nI,
, n, is a partition of n and 7CI,
, z, cuspidal representations of GL(ni, AF)
>_ s, and then construct the we may again change the order so that sI >_ induced representation I,. Every automorphic representation is a constituent of some I. For an adequate classification, one needs more. The following statement may eventually result from the investigations of Jacquet, Shalika, and PiatetskiiShapiro, but has not yet been proved in general. A. If r is a constituent of I. and of IQ, then the partitions {n1, , n,} and ns} have the same number of elements, and, after a renumbering, ni = n;
and 'ci - ri.
R. P. LANGLANDS
208
If iri = Qx 7ri(v) then one constituent of I. is the representation it = 0+ z,(v). This representation will be with local components 7r(v) = 1r1(v) ff
denoted 'r = r1 (+ J .. 0+ ir 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 i(F).
We agree to call r = z1 PB
T+ z, tempered if each of the cuspidal re-
presentations ici has a unitary central character, that is, if each of the si is 0. However 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 HowePiatetskii-Shapiro and Kurokawa allow us to cling. B. If it = &z(v) is a cuspidal representation with unitary central character then each of the factors n(v) is tempered.
The tempered representations form a subcategory H°(F) of 17(F).
It is clear that we have tried to define the categories H(F) and 17°(F) in such a way that if v is a place of F there are functors 17(F) --> 17(F,) and 17°(F) taking it to its factor ir(v). However without a natural definition of the arrows, there is no unique way to define Hom (zr, z') - Hom (vr(v), z'(v)). If J. is not irreducible it will have other constituents in addition to i1 + 41 x,. 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 7r is any automorphic representation there is a unique isobaric representation 7r' such that z(v) - 7r'(v) for almost all v. If F is a global field the purpose of the tannakian formalism would be to provide us with a reductive group over C whose n-dimensional representations, or rather their equivalence classes, are to correspond bijectively to the isobaric automorphic representations of GL(n, AF). This hypothetical group will have to be very large, a projective limit of finite-dimensional groups. We denote it by Gf(F). The category Rep(Gf(F)) of the finite-dimensional representations over C of
the algebraic group Gff(F) would certainly be abelian, but in addition it is a category in which tensor products can be defined. Moreover there is a functor to the category of finite-dimensional vector spaces over C. If (cp, X), consisting of the space X and the representation cp of Gff (F) on it, belongs to Rep (Gf(F)) one simply ignores cp. The tensor product satisfies certain conditions of associativity, commutativity, 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 Gu(F) we will have to associate to each pair consisting of a cuspidal representation z of GL(n, A) and a cuspidal representation z' of GL(n', A) an isobaric representation it Xx 'i' of GL(nn', A). In general, if g = z1 EB ors we would + 7r, and z' = zi 0+ set
z3x z' _L+ i, i
ix
;).
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
209
In addition, we will have to associate to each isobaric representation 7r of GL(n, A), n = 1, 2, , a complex vector space X(7r) of dimension n, together with isomorphisms
X(z M 0 " ' X(n) (D X(r')
X(r X ir') ^' X(ir) Ox X(ir'). There are a large number of conditions to be satisfied, among them one which is perhaps worth mentioning explicitly. Suppose 2r and 2r' are cuspidal and r [5 7c' = Ji=1n1 with n= cuspidal. Then the set {2ct, ,7c,} contains the trivial representation of GL(l, A) if and only if z' is the contragredient of z, when it contains this representation exactly once. At the moment I have no idea how to define the spaces X(ic); indeed, no solid reason for believing that the functor ' -> X(z) exists. Even though the attempt to introduce the groups Gff(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 r and z' are tempered then i Xx ir' is also, and thus be able to introduce a group GQ°(F) classifying the tempered automorphic representations. If QF is the group of multiplicative type whose module of rational characters is the module of positive characters of the topological group F"\IF then GH(F) will be a direct product GII°(F) X QF.
One will also wish to introduce, by a similar process, groups GH(F) and Ga°(F), attached to a local field F and classifying the irreducible, admissible representations
of GL(n, F), n = 1, 2, , and the tempered representations of GL(n, F). The formalism is clearly intended to be such that if F, is a completion of F there are homomorphisms Gn(F0)
II °(F)
GQ(F) and Gp°(F.) -> Gffo(F) dual to 11(F) -+ 1I(F0) and
11 °(F0).
If F is a local field the conjectured classification of the representations of GL(n, F) [3], verified when F is archimedean, provides a concrete description of the category 17(F) with its product X. If F is archimedean and WF is the Weil group of F then 17(F) is equivalent to the category of continuous semisimple representations a of WF on complex vector spaces X. The tensor product is the usual one (o-, X) O (a', X') _ (o (D o', X 0 X') and the fibre functor is (6, X) -> X. Thus Gff (F) is a kind of algebraic hull of the topological group WF. In particular there is a homomorphism WF --> Gff (F)(C) whose image is Zariski-dense. The subcategory corresponding to H°(F) is obtained by taking only those (o, X) for which the image of 6(WF) is relatively compact. If F is nonarchimedean one should take not the Weil group but a direct product WW = SL(2, C) x WF. Conjecturally at least, o is to be replaced by a continuous, semisimple representation of WW whose restriction to SL(2, C) is complex analytic. To obtain a category
equivalent to H°(F) one should take only those 6 for which a(WF) is relatively compact. Observe that in order to obtain a semisimple category we have replaced the group WDF employed by Borel and Tate [47] by the group WW. If w
I w I is the
usual positive character of the Weil group, there is an obvious homomorphism of WDF into WW which takes w e WF S WDF to
/lall/2 0
0
IwI-h/2
/
x w.
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R. P. LANGLANDS
Notice that according to these classifications there are homomorphisms of algebraic groups GH(F) --> Gal(F/F) and
GH.(F) -> Gal(F/F),
the group on the right being a projective limit of finite groups. The principle of functoriality cannot be valid unless there are similar homomorphisms when F is a global field. Let QF, Q +F, QF, be the groups of multiplicative type whose modules of rational
characters are, respectively, the module of all characters of F", or F"\IF if F is
global, of all positive characters, or of all unitary characters. Then OF = S1F x OF and there will be homomorphisms
Gd(F) ' QF, G/1°(F) Q F. If F is nonarchimedean and local we may also define Q,,,,, Q ;,, and Q°,,, by replacing the modules of characters of various types by the modules of unramified characters of the same type. The groups Q,,,,, Q ;,, and Q . contain a distinguished point over C, the Frobenius 0, which is simply the image of a uniformizing parameter in F". In any case the formalism will certainly allow us to introduce for any representation a of the algebraic group GH(F) over C an L-function L(s, a) and if a corresponds to the representation' of GL(n, AF) then L(s, a) = L(s, ir). But the reasons for wishing to introduce the groups GH(F) and GH.(F) and the associated formalism are not simply, or even primarily, aesthetic. There are problems which will be difficult to formulate exactly without them. Suppose, for example, that r = Ox vr,, is an isobaric representation of GL(n, A) and each of the factors zv is tempered. For almost all v, zv is unramified and associated to a conjugacy class {g,,} =
in GL(n, C). Since pr,, is supposed tempered this class
meets the unitary group U(n) and I may, as I prefer, regard it as a conjugacy class in U(n). The general analytic analogue of the Tchebotarev theorem or the Sato-
Tate conjecture would be a theorem or conjecture describing the asymptotic distribution of the classes
Suppose the formalism existed and it were associated to a representation a of GH°(F). The image o,(Gdo(F) (C)) would be a reductive subgroup H(C) of GL(n, C) with a maximal compact subgroup KH. For almost all v, a, would factor through
would be defined. Since its conjugacy class in H(C) would meet KH, it would define a conjugacy class in KH, which we denote Gno(F,) -> Q'un and
s {gv} and the asymptotic distribution of the classes Of course can be inferred from that of the classes There is a natural probability measure on the space of conjugacy classes in KH. If X is a set of conjugacy classes
and if the set Y = U,x x is measurable in KH, one takes meas X = meas Y. It is natural to suppose that it is this measure which defines the asymptotic distribution To verify the supposition, it will be necessary to establish of the classes that if p is any representation of GQ°(F) over C then the order of the pole of L(s, p) at s = 1 is equal to the multiplicity with which the trivial representation of GR0(F) occurs in p. If the existence of Gno(F) were established, it would be easy enough to deduce this from the recent results of Jacquet and Shalika [25].
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
211
Within this formalism, the principle of functoriality asserts that if F is a local field and G a reductive group over F then any L-packet of representations of G(F) is associated to a homomorphism rp: GQ(F) ._., LG of algebraic groups over C for which Gg(F) _ LG
Gal(F/F) is commutative. If Gf(F) is replaced by Ggo(F), the L-packet should be tempered. If F is global, some caution will have to be exercised. If the L-packet II consists of
z = (3z, for which the ir, are always tempered, it should correspond to a cp : GH.(F) --> LG. Otherwise this may not be so, for a reason which will perhaps be clearer after an example of Kurokawa [29] is discussed in the next section. If there is a representation
%: LG -> GL(n) x Gal(F/F) and if the image cb*(II) of II given by the principle of functoriality is not isobaric then II can be associated to no (p. One may nonetheless hope to prove, both locally and globally, that to each q : Gf(F) --p LG is associated an L-packet, provided rp commutes with the homomorphisms to the Galois group. If G is not quasi-split the local behaviour of c with respect to parabolic subgroups will also have to be taken into account [3]. For archimedean fields one recovers the usual classification. The few examples studied [44] suggest that questions about the multiplicity with which the elements of II occur in the space of automorphic forms will have to be answered in terms of cp.
The principal reason for wishing to define the group Gff (F) is that it provides the only way visible at present to express completely the relation between automorphic forms and the conjectural theory of motives [40]. The category of motives over F, a local or a global field, is Q-linear and tannakian, but it does not always possess a fibre functor over Q and seldom a single naturally defined one. Thus tan-
nakian duality associates to it not a group over Q but a group-like object, a "gerbe" in the rustic terminology which has become so popular in recent years. Over C this object becomes a group GMot(F), and the relations between motives and automorphic representations will probably be adequately expressed by the existence of a homomorphism PF: G1(F) -' GMot(F) defined over C. The field F can be local or global. The local and global homomorphisms are to be compatible with each other and with the formation of L-functions. Both the image and the kernel of pF will probably be rather large when F is a number field (cf. C.6.2 of [43]) but rather small when Fis local.
3. Anomalous representations. Since the anomalous representations cause some difficulty in the study of the zeta-functions of Shimura varieties, it will be useful to acquire some feeling for them before going on. The brief remarks of the previous
section suggest that an automorphic representation z of the reductive group G, or rather the L-packet II containing it, should be called anomalous if for some homomorphism 0: LG -* GL(n, C) x Gal(F/F) the principle of functoriality takes
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R. P. LANGLANDS
r or II to an anomalous representation of GL(n, A). It may be that the counterexamples to the Ramanujan conjecture of Howe-Piatetskii-Shapiro [19] are anomalous in this sense. Since their paper is not available to me as I write, I have to test this suggestion on another example, discovered by Kurokawa and quite explicit. The group G is to be the projective symplectic group in four variables over Q. The L-group is then the direct product of LG°, the symplectic group in four variables and the galois group Gal(Q/Q). Since all the groups with which we shall deal in this section will be split, we may ignore the factor Gal (Q/Q). Let G, be the product of PGL(2) over Q with itself, so that LG' = SL(2, C) x SL(2, C) and let G2 be GL(4) over Q. Define cpl: LGI --> LG to be the homomorphism
(r2 \Ti
oi/
x
62)
and let cp2 be the standard imbedding of LG° in LGz which is GL(4, C).
In order to analyze Kurokawa's example one must formulate his statements representation-theoretically. I state the facts necessary to this purpose, but have to ask that the reader understand the discrete series sufficiently well to verify them for himself. There is nothing to prove. It is simply a question of writing down explicitly for the special case of concern to us here some of the results of [17] and and [18], and some definitions from [31].
An automorphic representation ir = Qx lrv of G(A) is associated to a holomorphic form of weight k in the classical sense of [39] if and only if 7r_ is a member of the holomorphic discrete series and lies in an L-packet Ho(,), where the restriction of cb(k) to C" s Wc/R has the form zPz
z-+
z-& 2
z Pz2
with A = (2k - 3)/2, ,u = Z. Since G is the projective group, k must be even. Notice that zzz x is to be calculated as z2A(zz)-2 = f-22 (zz)k. We will eventually
take k = 10. On the other hand an automorphic representation r = (D z, of PGL(2, A) is associated to a holomorphic form of weight 2k - 2 if and only if n,. belongs to the discrete series and to an L-packet Il'ck), where o(k) : z
(
with A as above. In particular (p, takes the L-packet AP ck) O'I (2), consisting in fact of a single representation, to ffock>.
We want to apply the principle of functoriality to rp, and a very special auto-
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
213
morphic representation z of G1(A). z must be a tensor product 7C' Q Z" of two representations of PGL(2, A). 7r. and z" will both be members of the discrete series, the first in "p(10: and the second in IIp(2). In fact z' will be the automorphic representation associated to the cusp form of weight 18, but it" will be anomalous or, to be more exact, its pull-back k" to GL(2, A) will be anomalous. To construct it we begin with the partition {1, 11 of 2 and the two characters
7: x
Ixl1/2,
v: X -p
IXI-1/2
of GL(1, A), and then construct the induced representation of GL(2, A) as in the preceding section. Any constituent k" of the induced representation factors through a representation f" of PGL(2, A). We so choose k" that 7r' e 119 (2: while 3t = 7)p ff] vp for all p. The representation k" is clearly anomalous. The representation z is unramified. Thus the automorphic representation 2c° of G(A) lies in the L-packet cpl.({2c}) defined by z, cpl and the principle of functoriality if np is unramified for all p and 2r e cp1.({7rp}), and z' is in 14(10). We take z° to be the representation defined by the cusp form X1o of weight 10 [39]. Then i°° lies in 14(10:. According to the definitions [3] the relation zr e cp1.({7rp}) is a statement about eigenvalues of Hecke operators. These statements have been verified for small primes by Kurokawa [29]. The necessary equalities are too complicated to be merely coincidences, and we may assume with some confidence that z° e cpl.({z}).
In any case the representation z° certainly is a counterexample to Ramanujan's conjecture. If cp = cp2 ° (P1 then the principle of functoriality yields the same L-packet when applied to IV and cp as it does when applied to z° and (P2- Since G2 is GL(4) the
L-packet consists of a single representation. It is easily seen to be anomalous and to be equivalent almost everywhere to the isobaric representation Z' 0+ 7) ff] V. Thus zr° itself is anomalous in the sense described at the beginning of the section. 4. Shimura varieties. In this section we review the definition of Shimura varieties,
taken for now over C, and their relations with motives. The point of view is Deligne's and most of what follows has been taken from his papers [9], [10], or learned in conversation with him. Of course the book of Saavedra Rivano [40] has again
been a basic reference; many of the facts and definitions below will be found in it. Recall that the data needed to define a Shimura variety are a connected reductive
group G over Q and a homomorphism h:.--+ G defined over R. The symbol. is used to denote the group Resc,RGL(1). Thus we have a canonical isomorphism GL(l) x GL(1) . over C and we may speak of the restriction of h to the first or the second factor. It is not h which matters but the set
51 = {Adg°hjgEG(R)} and it will be best simply to let h denote an arbitrary element of 8.5. Recall that the pair (G, h) is subject to three conditions [10, X1.5]:
(a) If w is the diagonal map GL(1) -> GL(1) x GL(1) then the homomorphism h ° w is central. (b) The Lie algebra 0 of G(C) is a direct sum 13 + t + p and if (z1, z2) r= AC) then
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R. P. LANGLANDS
ad h(z)(X) = zi 1z2X,
X E p,
= X,
X E t,
= Z1Z21X,
X r=
1.
In fact the summands p, t, and p vary with h, and when it is useful to make the dependence on h explicit we write Ph, th, and ph.
(c) The adjoint action of h(i, - i) on the adjoint group is a Cartan involution. Since G(R) acts on the real manifold sp by conjugation every element of 03 defines a complex vector field on .Y). Let Xh be the value of the vector field associated to X
at h E 5i. The complex structure on . is so defined that the holomorphic tangent space at h is {XhI X E p} and the antiholomorphic tangent space is {Xhl X E p}. If A f is the ring of adeles over Q whose component at infinity is 0 and K is an open compact subgroup of G(Af) then G(Af)/K is discrete and XK = x G(A f)/K is a complex manifold on which G(Q) acts to the left. If K is sufficiently small then any r E G(Q) with a fixed point lies in z(Q) n K and thus fixes the whole manifold. We shall always assume that Kis sufficiently small and then
ShK(C) = G(Q)\. x G(A f)/K is a complex manifold, proved by Baily-Borel to be the set of complex points on an algebraic variety ShK = ShK(G, h) = ShK(G, 55) over C. Deligne anticipates that ShK will often be a moduli space for a family of motives
over C. This is sometimes so, the motives then being those attached to abelian varieties, but can certainly not yet be proved in general. Nonetheless there is a good deal to be learned from a rehearsal of the considerations that suggest such an interpretation of ShK. In essence one observes that ShK(C) is the parameter space for a family of polarized Hodge structures; the difficulty is to show that these Hodge structures all arise from motives. A real Hodge structure V is a finite-dimensional vector space VR over R together with a decomposition of its complexification Vc = Op,gEZVp,q, satisfying Vq,P = Vp,q. The collection of real Hodge structures forms a tannakian category over R, whose associated group is R. Indeed to a real Hodge structure V, one associates the
representation a of, defined by (4.1)
The relations Vq,P =
a(Z1, Z2)v = Z1 PZ2 qv,
v E Vp' q.
imply that, o- is defined over R. Conversely each represen-
tation of .? that is defined over R yields a Hodge structure, the elements of Vp,q being defined by (4.1). The real Hodge structure V is said to be of weight n if Vp,q =
0 whenever p + q : n. Certainly any Hodge structure is a direct sum V = Q+ nVn with Vn of weight n. We are however interested in the category of polarized rational Hodge structures. A rational Hodge structure V is formed by a finite-dimensional vector space V. over Q, a direct sum decomposition VQ = Q VQ, and real Hodge structures of
weight n on V; = VQ pQ R. There is a distinguished object of weight -2, the Tate object Q(1), in the category of rational Hodge structures. The underlying rational vector space is Q(1), = 2xiQ s C and, by definition, Q(1) 1, 1 = Q(1)c
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
215
It seems to be customary to identify the underlying vector space of Q(n) = Q(1)0" with (2ici)"Q and Q(n)R with (2 ri),,R s C. The factors 2vi have been chosen for reasons which need not concern us. It is no trouble to carry them along. If V is a rational Hodge structure and u the associated representation of R let C be o(- i, i) acting on VR. If V is of weight n, a polarization of V is a bilinear form
P: V x V -> Q (- n) satisfying: (a) For all u and v in Vc and all r e R(C)
P(o(r)u, o(r)v) = o(r)P(u, v). Thus the form is compatible with the Hodge structures.
(b) P(v, u) = (-1)"P(u, v). (c) The real-valued form (22ci)^P(u, Cv) on VR is symmetric and positive-definite.
A rational Hodge structure is said to be polarizable if each of its homogeneous components admits a polarization, a polarization of the full structure being defined
by polarizations of the homogeneous components. The category .Yee-9(Q) of polarizable Hodge structures is tannakian, with a natural fibre functor wHod: V -> VQ and an associated group GHod, reductive but overwhelmingly large. It does have factor groups of manageable size. If V is a polarizable rational Hodge structure, one may take the tannakian category generated by V and Q(1) and the repeated formation of duals, sums, tensor products, and subobjects. The associated group is called the Mumford-Tate group of V and denoted by .,&9-(V). It is finite-dimensional and reductive, and there is a surjection GHod ---> (V) defined over Q. If a is the representation of R attached to V then &g -(V) is simply the smallest subgroup of the group of automorphisms of the rational vector space underlying V which contains o(m) and is defined over Q [37]. It is consequently connected. The polarizable rational Hodge structures for which -&g -(V) is abelian play a particularly important role in the study of Shimura varieties. They are said to be of CM type. The second description of the groups .£ (V) shows that the category of such Hodge structures is closed under sums and tensor products and thus is a tannakian category. The associated group has been studied at length in [41] and is often called the Serre group. At the risk of making a comparison with [41] difficult, for Serre himself employs a different notation, we shall denote the group by Y. It is not difficult to describe 9. Let Q be the algebraic closure of Q in C and let c e Gal(Q/Q) be complex conjugation. To construct X*(.So), the module of rational characters of 9, we start with- the module M of locally constant integral-valued functions on Gal(Q/Q). The Galois group acts by right translation and
X*(Y) = p e MI (a - 1)(c + 1)2 = (c + 1)(u - 1)2 = 0
Vo e Gal(Q/Q)}.
In particular if A e X*(Y),
(6 - 1)2(1) + (a - 1)2(() = 0 because the left side is (c + 1)(0- - 1)2(1). The lattice of rational characters of 9?
is canonically isomorphic to Z Q Z and the homomorphism ho: R - . dual to the homomorphism X*(9) X*(M) which sends 2 to (2(1), 2(c)) is defined over R. The composite h o w: GL(l) -> .9' is dual to the homomorphism X*(,V) --> Z taking A to 2(1) + 2(c) and is defined over Q.
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R. P. LANGLANDS
To verify that the group .1' just defined in terms of its module of characters is the Serre group defined in terms of Hodge structures is easy enough. The existence of the two homomorphisms ho and ho a w implies that every representation of 9 defined over Q defines a rational Hodge structure. It is enough to show that these
are polarizable when the representation is irreducible. To obtain the irreducible representations, one takes a2 e X*(.) and defines the field F by Gal(Q/F) _ {a e Gal(Q/Q) I A = 2}.
The underlying space of the representation is the vector space over Q defined by
F, and the representation r = r2 is that defined symbolically by rA(s): x c F A(s)x. The weight of the associated Hodge structure is - (A(1) + A(c)) = n. There is
an a e F such that ((a) _ (- 1)na and (-1)aW ina is totally positive. A possible polarization is P(u, v) = (27ri)-n TrF/Quac(v).
Conversely suppose one has a rational Hodge structure V whose Mumford-Tate group (V) is abelian. Since there is a homomorphism 9 -+ fg (V), the coweight GL(l) -> W of the group q defined by z -> (z, 1) also defines a coweight
VV of fg (V). The lattice Y. of coweights of ff (V) is a Gal(Q/Q) module generated by vv. We define an injective homomorphism of its lattice of rational characters V* into X*(.9') by sending v e Y* to the element A given by A(U) = <6v, vV>,
u e Gal(Q/Q).
The dual homomorphism Y --> .A'.i (V) is surjective and V is defined by a rational representation of Y. We return to the Shimura varieties ShK(C) and show how one attaches families of rational Hodge structures to ShK(C). Let Go be the largest quotient of G such that
(a - 1)(c + 1)v' = (c + 1)(6 - 1)v' = 0 for every coweight of the centre of Go. Let be a rational representation of G on V which factors through Go. To each x = (h, g) in XK we associate a triple (Vx, V, (px). Vx is a rational Hodge structure whose underlying space is VQ = VQ, the Hodge structure being defined by the representation a h of R. The third term cpx is an isomorphism Vx f -+ VAf and is given by v -> g(g)-1 v. It is only defined up to composition with an element of e(K). The homogeneous components of VQ are independent of x and may be written V. It follows from Lemma 2.8 of [11] that if x e XK there is at least one collection Px of bilinear forms Pn : VQ x VQ -> Q(- n)Q which is invariant under the derived
group of G and is a polarization of Vx. Let $x denote the collection of all such polarizations. If g e G(Q), Px e ,1x, and x' = rx then the collection Px' given by Pn': (u, v) ---> Pn(e(7-1)u, e(r 1)v) lies in $.7x'.
We must also verify that the family { Vx} over XK is a family of rational Hodge structures in the sense of [9]. Otherwise it could not possibly be attached to a family of motives. There are two points to be verified. Let Vp q be the subspace of VV of
type p, q and set Vp = (Dp,ZpVV,q.. The space Vp c Vc must be shown to vary holomorphically with x. In other words if v(x) is any local section of Vp and Y any antiholomorphic vector field then Yv(x) also takes values in V. The condition of transversality must also be established, to the effect that for the same v(x) and any holomorphic vector field Y the values of Yv(x) lie in VPL1.
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
217
There is certainly no harm in supposing that v(x) takes values in Vp q. Then one has to show that if x° = (h°, g') is fixed and Y is the vector field defined by Xe 03 then Yv(x°) lies in Vp when X e ph° and in V._1 when X E ph°. Let Kh° be the stabilizer of h° in G(R). We represent XK as the quotient G(A)lKh° K and lift v(x) to a function on G(A) which we write as (g, gf) -* (g)u(g, gf),
g e G(R), gf e G(A f).
The function u takes values in the constant space Vp q. Moreover
Yv(x°) = e(X)u(1, gf) + Xu(1, gf). The second term lies in V p ° q for all X E ( l 3 . If X e p then e(X)Vpx q c V 1, q_1 and if X e p then (X)Vpx q C Vp°1,q+1.
If r e G(Q) and x' = rx then v -> e(r)v provides an isomorphism between (Vx $x, px) and (Vx', $x', Thus, if s e SK(C), any two elements of {(Vx, $,,, lpx) I x - s} are canonically isomorphic, and we may take (VS, $5, (PI) to be any one of them, and redefine our family as a family of rational Hodge structures, with supplementary data, over the base SK(C). The locally constant sheaf
FA(Q) of rational vector spaces underlying this family is the quotient of VQ x XK by
the action r: (v, x) , (e(T)v, rx) of G(Q). For this quotient to be well defined, the group K must be sufficiently small, for G(Q) n KhK is then contained in the kernel of for all h (cf. II.A.2 of [41]). It is here that the condition that i; factors through Go intervenes.
When dealing with motives, one does not need to introduce the polarizations explicitly as part of the moduli problem, but in order to introduce a Hodge structure on the cohomology groups of the sheaves FA(Q) one must verify that on each connected component of SK(C) a locally constant section s -- Ps can be defined. Let G°(R) be the connected component of G(R) and choose x° = (h, gf) in XK. If XK = {(Ad g° h, gfk) I g e G°(R), k e K}
then the image of XX in SK(C) is open. If x and x' lie in XX and x' = rx then r lies in
(4.2)
G(Q) n G°(R)KhgfKg f 1.
All we need do is find a collection P = {Pn} such that P e S 3x for all x e XX and Pn(e(r)u, e(r)v) = Pn(u, v) if r lies in the group (4.2). Choose any P in $T:x°. Then general P e $x for all x e XX. There are certainly homomorphisms An of G into linear group of Vn such that the
Pn((r)u, (r)v) = Pn(A (r)u, v). The eigenvalues of A(r) are positive if r lies in G°(R)Kh. Moreover A is trivial on the
derived group of G and factors through Go. It therefore follows from the results of II.A.2 of [41] that each tin is the identity on all of (4.2). Although we have defined the families (Vs, $s, (ps) for any G, it is clear that SK(C) is not going to appear as a moduli space unless G is equal to Go, and so for the rest
of this section we assume this. The moduli problem is best formulated completely in the language of tannakian categories. We can drop the polarizations and retain only the pairs (Vs, cps), or (Vx, cpx), but we now have to emphasize that (Vx, cpx) is
218
R. P. LANGLANDS
defined for every (finite-dimensional) representation e of G over Q, and so we write V(e)) for the representation and (Vx(e), cpx(E)) for the pair (Vx, (p-). On the category R&9(G) of finite-dimensional representations of G we have the natural fibre functor (ORep(G) : V(g)Q and 7jx: (E, V(e)) -> Vx(e) is a Q-functor from M'.9(G) to -Y0_9(Q) which satisfies cuHod ° ylx = WRep(G) Since
Aff_ (Q) and W&9(GHod) are the same categories, iyx defines a homomorphism [40, 11.3.3.1] cpx: GHod -+ G and x may be defined by V(e)) - (e ° cpx, V(e)). When we emphasize rJx, (px appears as an isomorphism of two fibre functors (px: 0) Hod ° Vx --- O)Rep(C)
However, when we emphasize cpx, as we shall, then these two fibre functors are the same, for they are both obtained from 0Hod ° lix = wRep(G) by tensoring with Af, and tpx may be interpreted as an isomorphism of UOR p(G). Such an isomorphism is given by a g-1 E G(A f) [40, 11.3]. This is the g appearing in x = (h, g). Only the coset gK s G(A f) is well defined. We have arrived, by a rather circuitous route, at the conclusion that XK parametrizes pairs (cp, g), cp being a homomorphism from GHod to G defined over Q, and g
in G(Af) 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 cp with the canonical homomorphism Jl --* GHod lies in 5). If r c G(Q) the pairs (cp, g) and (ad r ° cp, rg) will be called equivalent. The variety ShK(C) parametrizes equivalence classes of these pairs.
One of the important tannakian categories is the category 'ffO (k) of motives over a field k. It cannot be constructed at present unless one assumes certain conjectures 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 p-functor h811: M0 (C) --> .&2(Q). .ill0 (C) together with the fibre functor 0Mot(c) of rational cohomology also defines a group GMot(c) over Q and hBH is dual to a homomorphism hB*H: GHod -' GMot(c) defined over Q.
Implicit in Deligne's construction is the hope that any homomorphism cp': GHod
-> G satisfying H is a composite cp' = cp o hBH. According to the Hodge conjecture, cp would be uniquely determined [40, VI.4.5] and ShK(C) would appear as the moduli space for pairs (cp, g), with g as before, but where cp is now a homomorphism from GMot(c) to G defined over Q and satisfying: H'. The composition of cp with the canonical homomorphism R -* GMot(c) lies in S,5.
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 r is an automorphism of C. Then r-1 defines a O-functor 72(r): M09 -(C) -> -&09-(C). Let GMot(c) be the group defined by M(99(C) and the fibre functor 0Mot(c) ° li(z) The dual of V(r) is then an isomorphism over Q: cc(z) : GMot (c) - GMot(C) -
The homomorphism cp has a dual, a OO -functor ij: Rfg(G)
M(99 (C) and the
fibre functor coMot(c) °)7(z) °)7 defines a group G' c° over Q. The OO -functor li then defines a dual
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
coy: GMot(c) -- GT,w,
219
and co' = or ° cp(r): GMot(c) -+ G=,w,
Moreover the two fibre functors WMot(c) and w4ot(C) o V(z) are canonically isomorphic. As a consequence, there is a canonical isomorphism G(Af) -> Gz'c(Af). Let g' be the image of g.
The pair ((p', g') seems once again to define a solution to our moduli problem. the composition of The difficulty is that G (p' with the canonical homomorphism may not lie in 5i. One of the purposes of the next two sections is to discover what G=,w is likely to be.
5. The Taniyama group. There is one type of Shimura variety which is very easy to study, that obtained when G is a torus T. Then the set reduces to a single point {h}. For each open compact subgroup U of T(Af) the manifold ShK(C) consists of a finite set and Shu = Shu(T, h) is zero-dimensional. In general a special will be a pair (T, h) with T s G and h c= ,). If U = K n T(Af) then point of (G, ShK(C) is a subset of ShK(C), the points of which have traditionally been referred to as special points, and I shall continue this usage. But it is best to give priority to the pair (T, h) rather than to the points of ShK(C) it defines. There are a number of unsolved problems about Shimura varieties and their special points that I want to describe in the next section. To formulate them some Galois cocycles have to be defined. Deligne has shown me that my original construction gave, in particular, a specific extension of Gal(Q/Q) by the Serre group, .So, an extension I venture to call the Taniyama group and denote by . Since the cocycles needed are, as Rapoport observed, often easily defined in terms of , I begin by constructing it. will also be defined over Q. The group .9' is an algebraic group over Q, and Thus we will have an exact sequence 1 -* .9 -+
(5.1)
- Gal(Q/Q) -> 1.
Recall that X*(.) is a module of functions on Gal(Q/Q), and that the Galois action on X*(,9') giving the structure of Y as a group over Q is defined by right translation. We are still free to use left translation to define an algebraic action of Gal(Q/Q) on 9, and it is this action which is implicit in (5.1). The extension will not split over Q but it will be provided with canonical splittings over each 1-adic field Q1, Gal(Q/Q) -+ (Q1), which will fit together to give Gal(Q/Q) -* .i (A f). Rather than attempting to work directly with 5°, I choose a finite Galois extension L of Q, let .2L be the quotient group of . whose lattice of rational characters consists of all functions in X*(..P) invariant under G(Q/L), and define extensions (5.2)
1 -> 9OL ->
L - * Gal(Lab/Q) -> 1,
afterwards lifting to Gal(Q/Q), and then passing to the limit. To motivate the construction we suppose that the extension is defined and that there is a section a -> a(r) of L - Gal(Lab/Q) with a(v) e L(L). Let a(v1)a(T2) _ dy1,'12a(2IZ2) with d
(5.3)
1, z2
E ,99L(L), and with zl(dz2, z3)dy1, T2T3 =
d 1, z2dz1T2i
13.
Observe that the elements of the Galois group play two different roles. They are first of all elements of a quotient group of L, and secondly they are automorphisms of Lab and thus act on L(Lab), since L is defined over Q. In the first role
220
R. P. LANGLANDS
they will be denoted by r, perhaps with a subscript added, and in the second by p
or u. We have p(a(r)) = cp(v)a(z) with cp(r) E AL(L). Certainly
(5.4)
cpa(a(r)) = P(ca(z))cp(z).
In addition (5.5)
d,1,1'2cp(ri)zi(cp(rz)) = p(d1,,)cp(zir2)
Conversely if we have collections {cp(r)} and {d,1,,} satisfying (5.3), (5.4), and (5.5), we can construct g -L over Q, together with the section a. Any splitting Gal(Lab/Q) --v (A f) will be of the form c -> b(z)a(z) with b(z) E 9 L(Af(L)). In order that it be a splitting, we must have (5.6)
b(r1)r1(b(r2))d1,z2 = b(zlr2)
If the b(r)a(z) are to lie in .9L(A1) we must have (5.7)
p(b(v))cp(r) = b(r).
Again any collection {b(z)} satisfying (5.6) and (5.7) defines a splitting, and it is our task to construct {b(r)}, {cp(z)}, and {dr1,T2}.
The group ° is a quotient of Gxod and thus is provided with a canonical homomorphism h: R -* Y. Over C the group q is canonically isomorphic to GL(l) x GL(1). Restricting h to the first factor we obtain a coweight p of 6, the canonical coweight. If 2 E X*(6) then = A(1). Since 6°L is a quotient of ,", p also defines a coweight of S°L, which for convenience will also be denoted by u. If v is any coweight of Y L and x any invertible element of L or of A f(L) then xv will be the element of ."L(L), or 9L(Af(L)), satisfying A(xv) = x for all A E X*(YL). Recall that we have two actions of Gal(L/Q), or Gal(LablQ), on X*($ L) or on
X*(6L) = Hom(X*(6 L), Z),
the lattice of coweights. That defined by right translation we write v - ow, and that defined by left translation we write v --> vv. Thus <2, or p> _ A(o' 1) while = A(r), because an inverse intervenes in the action by left translation, and pz 1 = zp. Moreover ai(xl,) = a(x)ai, while z(xP) = xPr. These aspects of the notation have to be emphasized because at some points our convention of distinguishing between p, a on the one hand, and r on the other, fails us. For the study of Shimura varieties, it is best to take Q to be the algebraic closure
of Q in C, and we shall do this. Thus we provide ourselves with an extension of the infinite valuation on Q to Q. There is a property of the Weil groups that will play a prominent role in our discussion. Let v be a valuation of Q and hence of Q, and let Qv and Qv be the completions of Q and Q with respect to v. Eventually v will be defined by the inclusion Q s C, and Q will be R and Qa will be C. In any case, the data provide us with imbeddings (5.8)
(5.9)
Fv y CF,
Gal(Fv/Q,,)
Gal(F/Q),
if F is any finite Galois extension of Q in Q. The local and global Weil groups WFVQ, and WF/Q are defined as extensions
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
1 -- F -* WFWQv ->
221
1
and
1 -+ CF ---> WF/Q -> Gal(F/Q) -, 1.
We may imbed the arrows (5.8) and (5.9) in a commutative diagram 1 -> F, --, WFv/Q,, - Gal(Fa/Q,,) -> I I IF
I I 1 -+ CF - WF/Q --> Gal(F/Q) -* I
Moreover we may so choose the central arrows that they are compatible with field extensions and upon passage to the limit yield I: WQa -> WQ. It is the image of WQv in WQ that will be fixed, and IF may be changed to w -+ xIF(w)z 1 where x e CF and xu(x)-1 e Fv for all u e Ga1(F JQa). Now let v be the valuation given by Q s C. Let FF = The natural map FF -> CF is an imbedding, and we sometimes regard FF as a subgroup of CF. If we take an element r of Gal(Q/Q), lift to WQ, and then project to WL/Q we obtain an element w = w(v) of WL/Q which is well defined modulo the connected component,
and in particular modulo the closure of L. We choose a set of representatives wa, a e Gal(L/Q), for the cosets of CL in WL/Q in such a way that the following conditions are satisfied : (a) w1 = 1. (b) If u e Gal(Lv/Q,,) then wa e (c) If p e Gal(L/Q) and or e then wPwa = aP,awpa with ap,a e L-1To arrange the final condition we may choose a collection f of representatives 97 for the cosets Gal(L/Q)/Gal(Lv/Qy) and set w.na = wnwa if o e Gal(Lv/Q,). We suppose that f contains 1. With this choice we also have: (d) If f ap,a} is the cocycle defined by wPwa = ap,aw,, then a,,,P = I for r e f and or e Gal(Lv/Qv) If w e WL/Q, let wow = ca(w)wa, ca(w) e CL. If w = w(z) we set
bo(r) =
n
aEGal (L/Q)
c (w)a,`
It lies in CL O X*(.L), but is not well defined, because w is not. However we can show that it is well defined if taken modulo L-1 ® X*(YL), and that, in addition, it behaves properly under extensions of the field L. The ambiguity in w now has no effect, for we are only free to replace w by uw where u = and u lies in the image of L-U, where U is a subgroup of the group of units of L defined by a strong congruence condition. But [41, II.A.2],
H
a(v)au = 1
a=Ga1(Q/Q)
for all v E U. Consequently 77-77
01
(f6(u)a`)(llCo(w)a/) a
is congruent modulo L' 0 X*(9L) to ]Z aca(w)ap. Suppose the representatives wa are replaced by ewa, and, hence, ap,a by
R. P. LANGLANDS
222
a,,,, = epp(ea)e-, ap, c.
If or e Gal(Lv/Qv) then ea a Lv and p(ea) e L. Since ap,a and ap, a must then both be
in L-', we infer that ep =-apa (mod L.1) when a e Gal(Lv/Qv). Moreover ca(w) is replaced by ca(w) = ea=e, lc(w) and bo(r) by bo(a) = {lleo1,e111`}bo(r) or
The factor may be written
fl
fjG (S-'-1)11 -= fl
,ref aeGal(Lv/Qv)
a
e'1,1(1-`1)11.
Since
E
aeGal
(1 + 6)(1
(Lv'Q, )
0,
this change has no effect on bo(a). In this argument we have denoted the image in
Gal(L/Q) of r e Gal(Q/Q) by the same symbol, a practice we shall continue to indulge in. If we modify I then w is replaced by xwx 1 with x c= CL and xa(x)-1 E Lv for all U e Gal(L,/Q). Then ca(xwx1) = U(x)Qr(x1)ca(w) and 11U(x)-0'P0'a(x)0'P = a
RU(x)v(r
1
1)p
o
Since or(x) = x (mod L-') if a E Gal(Lv/Qv), the same argument as before shows that bo(a) is unchanged.
Finally suppose that L s L'. Then bo(a) e CL ® X*(9"') and bo(a) e CL p X*(.SOL) are both defined, and we must verify that bo(a) is taken to bo(r) by the canonical mapping of the first group to the second. Either Lv = R or Lv = C, and the two cases must be treated separately. Suppose first that Lv = C and hence that Gal(L'/L) () Gal(L/Qv) = {I }. Since both bo(r) and bo(a) are independent of the choices of coset representatives, we may choose those which make it easiest to verify that bo(z-) is the image of ba(r). Let e be a set of representatives for the cosets Gal(L'/L)\Gal(L'/Q)/Gal(L,',/Qv) containing 1. Every element 6 of Gal(L'/Q) may be written uniquely as a product o, = Cijp, C e Gal(L'/L), 97 e e, p E Gal(Lv/Q1). We may suppose that wa = ww,,,wp with wj = 1 and wP E WL;,Q,. Let w' be w(r) with respect to L', and w be w(r) with respect to L. Then under the canonical map r: WL,/Q -- WL/Q the element w' maps to w. If a E Gal(L/Q) it lifts to a unique element of Gal(L'/Q) of the form jp. We suppose that wa is the image of wwP. Thus if w,wpw' = d,),p(W )w,.wp,
with d,,,p(W) E WL;,L then ca(w) _ z(d,,p(w')). On the other hand if U1 = Cep lies in
Gal(L'/Q) then ca,(w')wg.
Consequently, by the very definition of Tic, CA(W) = 11 cat(W'). ol-a
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It follows immediately that bo(i) is the image of bo(r).
If Lv = R then X*(.7L) ^-, Z and the Galois group acts trivially. Suppose we replace wQ by eQwQ with eQ e CL. Then cQ(w) is replaced by eQe- . Since ]l(eQe1 )Q'` = ]J(eQeQT )'`
= 1,
this has no effect on bo(a), and when defining bo(r) we need not suppose that the collection {w;} is subject to the constraints (a), (b), and (c). If we want to define bo(a), we may still need to be careful about the choice of the coset representatives wo, o a Gal(L'/Q). However, since we are only interested in the image of bo(a) in .5PL, we may again ignore (a), (b), and (c). We choose a set e of representatives for the cosets Gal(L'/L)\Gal(L'/Q), write a = pi), p e Gal(L'/L), 72 e e, and take wo = w,w.,. If or e Gal(L/Q) is the image of 7), we take wQ = lr(wn). The argument can now proceed as before. Let b(r) be a lift of bo(a) to IL 0 X*(9L) _ 'L(A(L)) and let b(r) be the projection of b(r) on VL(A f(L)). The element b(a) is well defined modulo boL(L) and, as we shall see, this bit of ambiguity will cause us no difficulty. But we have to fix one choice. The first point to verify is that dt1, T2 =
b(v1)r1(b(rz))b(r1rz)-1
lies in ,SoL(L). When verifying this, we may choose the liftings b(r1), b(r2), and b(r1r2) in any way we like. We choose liftings cQ(w1) and cQ(w2) of cQ(w1) and cQ(w2) to IL
and take
b(rl) = Hl
(wl)QP,
b(r2) = llea(w2)QP-
Q
Q
Since cC(wlw2) = cQ(w1)cQT1(w2), we may take CQ(wlw2) to be CQ(wt)CaT1(w2). Because
ri 1(b(r2)) = l lMw2)aTl a = fkT1(wz)Q'`, a a
the element dT1, 12 will then be 1.
Finally we have to establish that the elements c,,(r) defined by equation (5.7) lie in .9'L(L), for equations (5.4) and (5.5) will then follow immediately. It will suffice to show that for any w e WL/Q and any p e Gal(L/Q) (5.10)
Ip(ca(w))-PQ"}
a {ucQ(w)QP} {I Q
lies in L_x Qx X*(7L). Suppose w = w1w2 and wl projects to r1 e Gal(L/Q). Then cQ(w) = cQ(w1)C,,,1(w2) and (5.1/0) is equal to 77 PQ's}r1 {l lca(w2)Q1`p(co(w2))-PQP}
1! Q
Q
Consequently we need only verify that (5.10) lies in Lx O X*(.92L) for w in CL and for w = w;. If w lies in CL then cQ(w) = a(w) and jj Qa(w)Qi, = jj Qpu(w)Pay. The expression (5.10) is therefore equal to 1. If w = wT then cQ(w) = a,,T and (5.11)
Haaip(aa,T)-pa' = llapa0'
a
a
However it follows from condition (c) that ap,Q - a,,,, (mod L-). Since (I + 6)
(1 - Z1)/ z = 0, the right side of (5.11) lies in Lx Q X*(6L).
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R. P. LANGLANDS
Since b(z), although defined for r E Gal(Q/Q), depends only on the image of r in Gal (Labl Q), the groups g-L and 9` are now completely defined. The ambiguity in the b(r) is easily seen to correspond to the ambiguity in the choice of the section a(r). If E s Q is any finite extension of Q, we let 9'E be the inverse image of Gal(Q/E) in . If E s Lab we may also introduce L,. The group L has been introduced by
Serre [41], who uses it to formulate some ideas of Taniyama. He makes its arithmetic significance quite clear, but his definition is sufficiently different from that given here that an explanation of the reasons for their equivalence is in order. If x is an idele of L then O(x) = 1 I GaI(L/Q) o(x)ai, is an element of ,'L(A(L)). By 11.4 of [41] there is an open subgroup U of the group of ideles IL such that O(x) = 1 if x e U n Lx. The standard map of IL onto Gal(Lab/L) restricts to U and if r = r(x) is the image of x we may take w = w(r) to be the image of x in CL. Then O(x) is a lifting of bo(r) to .9L(A(L)). However O(x) depends only on r, and thus we may take b(r) = O(x). Then d1,2 = 1 and cp(r) = 1 if z, rl, z2 lie in the image Gal(Lab/F) of U. Here F is the finite extension of L defined by U. The elements cp(r) are in fact 1 for all v e Gal(Lab/L). If we choose a set of representatives e for Gal(Lab/F)\Gal(Lab/L) and set b(ra) = r E Gal(Lab/F), ij e e, then, in general, d,,.TZ will depend only on the images of rl, r2 in Gal(F/L), and L may be ob tained by pulling back an extension 1 -> YL _
u -->
Gal(F/L) -> I
to Gal(Lab/L). Here v is the quotient of "L by the normal subgroup {a(z)f r E Gal (Lab/F)). It is the extension u that Serre defines directly. He denotes it by the symbol The map 0 defines a homomorphism of Lx/Lx n U Lx U/LX into .9'L(L) and to verify that u is the group studied by Serre, we have only to verify that it can be imbedded in a commutative diagram 1 -> Lx U/LX -. IL/Lx - IL/Lx U -> 1 1
A(L) -* 57-U(L) - Gal(F/L) -* 1
The right-hand arrow is x - r(x)-1. We do not have to pass to the quotient but may define the homomorphism (5.12)
IL/Lx
L(L)
directly. The central arrow is then obtained by composing with the projection -L(L) -->
U(L). The homomorphism (5.12) is
x . { f or(x)"l b(r) 1a(z) a
1
J
if r is the image of x in Gal(Lab/L). The composition of our splitting Gal(Lab/L) L(Q1) with . L(Q1) -> u(Q1) is either Serre's el or its inverse, presumably its inverse, for we are so arranging matters that the eigenvalues of the Frobenius elements acting on the cohomology of algebraic varieties are greater than or equal to 1. The cocycle p --> c,,(r) certainly becomes trivial at every finite place, but is not necessarily trivial at the infinite place, v. Indeed under the isomorphism
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
225
H1 (Gal(Lv/Qv), .L(Lv)) ^' H -I (Gal (Lv/Q,,), X*(.L))
given by the Tate-Nakayama duality it corresponds to the element of the group on the right represented by (I - v1),u. If, as has been our custom, we denote the image of r in Gal(L/Q) again by r we may suppose that w(r) = wT, for the class of {cp(a)} depends only on this image. According to the discussion of formula (5.11) 11 a
aPaa(z-1-1)k
= 11
1 (ap,IcaP,V)P7(i
1)k = eP(r)
77E f
lies in L-1 Q X*(.L) = .9'L(L,). By definition, the classes of {c,(z)} and {ep(r)} are inverse to one another in H'(Gal(L/Q), YL(A(L))). Thus all we need do is calculate the projection of ep(v) on .9L(Lv) for p e Gal(Lv/Qv). Observe first of all that if we agree to choose coset representatives satisfying(d), then ap,,/c ap,1 = p(a ) Again if pi, = 1lpl then ap,7,, = a,11p1, r = ?1(ap1, )a7)1, plc
a-nt,1p1
= 7]1(ap1,,).
Since apl, e Lv , the term on the right has a projection on Lv different from 1 only if 7)1 = 1. Then 7] too equals 1, and so the projection of ep(z) on YL(Lv) is
a'''' =
Gal
If
a')',
in conformity with our assertion. Suppose T is a torus over Q, provided with a coweight p such that (5.13)
(1+()(r-1),u=(a-1)(1+(),u=0
for all z e Gal(Q/Q). Then there exists a unique homomorphism 0: 9 -> T such that the composition of 0 with the canonical coweight of 9 is p. We can transport the cocycles p -> cp(z) from . ' to T, obtaining cocycles {cp(z, ,u)} as well as b(r, u) E T(A f(L)). However if T and h: .. -* T define a Shimura variety and u is the restriction of h to the first factor of l the condition (5.13) will not necessarily be satisfied.
Nonetheless, we may repeat the previous construction and define b(z, u) and {cp(,r, ,u)} for all z such that (1 + c)(v 1 - 1)p = 0. This generalization is necessary for the treatment of those Shimura varieties Sh(G, h) for which G is not equal to Go. It should perhaps be observed that b(v, u) is not insensitive to the ambiguity in the choice of w = w(r), although {cp(r, ,u)} is. However, if Z is the centre of G the ambiguity all lies in Z(A f) n K, and may be ignored. If E s C the motives over E whose associated Hodge structure is of CM type are themselves said to be of CM type. They form a tannakian category CM(E) with a natural fibre functor coCM(E), given by rational cohomology. Since one expects that the natural functor CM(Q) -> CM(C) is an equivalence, we may as well sup-
pose that E s Q. According to the hopes expressed at the end of the previous section there should be an equivalence 72: CM(Q) ---> R,99(Y) and an isomorphism (1)Rep(9,) ° r) -' wcM(e), which would enable us to identify .9' with the group GCM(Q), defined by the category CM(Q) and the functor WCM(Q). If E s Q, one hopes that in the same way it will be possible to identify E with GcM(E). There are properties which this identification should have, and it is necessary to describe them explicitly. First of all, if F s E the diagram
226
R. P. LANGLANDS
Y- E-F I
I
I
GCM(Q) -3 GCM(E) - GCM(F)
should be commutative. The other properties are more complicated to describe. Suppose r e Gal(Q/Q) and r takes E to E'. Its inverse then naturally defines a x®-functor 7)(r) from CM(E') to CM(E). Let GcM(E.) be the group defined by the category CM(E') and the functor WCM(E) ° V(r). The dual of V(r) is then an isomorphism rp(r): GcM(E) -' GcM(E'). In
terms of representations, zj(r) associates to every representation (a', GCM(E') a representation
of GCM(E) Since WCM(E') and (OcM(E)
of become
isomorphic over Q there is a family of homomorphisms, one for each ', ci(e'): compatible with sums and tensor products. If ci'(e) is another possible family then there is a t e E,(Q), such that Finally since the two functors cocM(E,) and mcM(E) ° 7J(r) are canonically isomorphic, arising
as they do from the 1-adic cohomology, there is a canonical family of isomorphisms `VAf(S ) V(S')Af - V/(OAf
On the other hand, suppose a(r) e (Q) maps to r. Let o'(a(r)) = ca(r)a(r). We may use a(r) to associate to every representation (g', V(')) of E, a representation of E. The space V(g) is obtained by twisting V(e') by the cocycle {'(ca(r) 1)}. The representation e is t -> e'(a(r)ta(r)-1). This functor (a', V(e'))V(g)) is to be (isomorphic to) that obtained from i(r) by identifying E and is to be the GcM(E) and E. and GcM(E'). Moreover one possible choice for isomorphism cbo(e') : V(e')Q -implicit in the definition of V(e). If b(r)a(r) is the image of r under the canonical splitting Gal(Q/Q) - (Af), then is to be cio(e') °
If r e Gal(QIE) then E' = E and 77(r) : CM(E') --> CM(E) is the identity functor. from Rep(TE,) = Rep(TE) to Rep(TE) must be canonically isomorphic to the identity. A final property, which seems to be independent of the preceding ones, is that this isomorphism should be given by ' -> and e'(a(r)) : V(e') -> V(e). Observe that a(r) is now in E and these transformations are defined over Q. I assume that all this is so, just to see where it leads, and especially to see what it suggests about the groups Gr,`c introduced in the previous section. But there are some lemmas to be verified first. I conclude the present section by describing a
Thus the functor (g', V(g')) ->
property of the Taniyama group whose significance was pointed out to me by Casselman. It will be needed to show that the zeta-functions of motives, and especially abelian varieties, of CM type can be expressed as products of the Lfunctions associated to representations of the Weil group. The point is that there is a natural homomorphism rp of the Weil group WQ of Q into -(C) and thus for any finite extension F of Q a homomorphism !pF : WF --> .r F(C). To define it we work at a finite level, defining WL/Q -> `L(C), and afterwards passing to the limit. Fix for now a set of coset representatives w, which satisfies (a), (b), and (c). If w r= WL,Q we define bo(w) = 11 aEGal (L/Q) ca(w)s)`. If r is the image of w in Gal(Lab/Q)
then bo(w) = bo(r) (mod L_x Qx X *(.9L)), and we may lift bo(w) to b(w) in 5L(A(L))
in such a way that the projection of b(w) in SoL(Af(L)) is b(r). However a simple
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
227
calculation shows that if Tl, z2 are the images of w1, w2 then e AL(L). Since its projection on SL(A f(L)) is equal to dZ,, z2, it is itself equal to dzj, Z2 ,VL(C), we may define cp by cp: w -> If b(w) in
b(w1)z1(b(w2))b(w1w2)-1
If the coset representatives wa are changed then (P is replaced by cp' _ ad a o cp, a e SoL(C), but this is of no importance.
If GWF is the group over C defined by the tannakian category of conti:iuous, finite-dimensional, complex, semisimple representations of WF then cPF is the com-
posite of the imbedding WF -> GWF(C) and an algebraic homomorphism OF: GWF -> cF. Moreover if the principle of functoriality is valid, there is a surjection GJ(F) -* GWF and we can expect to have a diagram PF
Gil (F)
' Gmot(F)
I
I
G
WF
F
whose two composite arrows differ by ad s, s e Y (C) g
F(C).
6. Conjugation of Shimura varieties. The principal purpose of this section is to formulate a conjecture about the conjugation of Shimura varieties, a conjecture whose first justification is that it is a simple statement which implies what we need for the study of the zeta-functions at archimedean places and is compatible with all that we know. Some lemmas are necessary before it can be stated, and we shall see that these lemmas together with the hypothetical properties of the Taniyama group suggest an answer to the question that arose at the end of the fourth section. This answer in its turn throws new light on the conjecture, so that we can weave a consistent pattern of hypotheses, and our task will be ultimately to show that it has some real validity. We need a construction, which we make in sufficient generality that it applies to all Shimura varieties and not just those associated to motives. Suppose the pair (G, 55) defines a Shimura variety, T and T are two Cartan subgroups of G defined over Q, and h: 2 -> T, h -> R --). T both lie ins,-). Let fc and u be the coweights of T and T obtained by restricting h and h to the first factor of R, and choose a finite Galois extension L which splits T and T. Let z e Gal((2/Q); then the coweights
(1 + ()(v 1 -1),u and (1 + c)(z 1 -1),u are both central and they are equal. Choose w = w(z) as before, and set T7
b0(v, u) = I I c (w)a1,, a
/
b0(z, p) = IflI c (w)°i` or
Let b(r, u) and b(r, ,u) be liftings to T(A(L)) and T(A(L)) and let B(v) = B(z, u, p) be the projection of b(z, p)-1 b(r, ,u) on G(Af(L)). Although we may not be able to define the cocycle fcp(z, u)}, we can define {ce(z, ,uad)} if pad is the composition of u with the projection to the adjoint group. It may be as well to check that B(r) is indeed independent of the choice of w and of the coset representatives wa, provided the usual conditions (a), (b), and (c) are satisfied. If wa is replaced by eawa then, apart from a factor in L_' Qx XX(T), bo(z, a) 1) Since these is multiplied by fl,7,s e (1+r)(_ ' 1)u, and bo(z, ,u) by TI,ef two terms are central and equal by assumption, the change has no effect on B(z). If w is replaced by xwx 1 with xo-(x1) e Lv for a e Gal(L IQ,), then bo(z, a) is modified
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R. P. LANGLANDS
by the product of an element in L_ xp X*(T) and fl x7(1+t) (r-'-1) z. Since bo(r, ,u) undergoes a similar modification, B(r) is not affected. One can also show easily that B(r) is not changed when L is enlarged; the argument is once again basically the same as that used to treat b(z). B(r) does depend on the choice of b(r, p) and b(r, p). These choices made, we will use them consistently to define cl,(r, ,uad), Co(r, Pad), and, when
(1 + c)(v 1 -1)a = (1 + 6)(z--1 -1),u = 0,
c,(r, ,u), c,(r, ,u). Thus Cp(v, uad) is to be the projection of p(b(r, ,u) in Gad(Af(L)). With these conventions, the ambiguity in B(r) will cause no harm in the construction to be given next. Let Gr.P and Gr. i be the groups obtained from G by twisting with the cocycles {Cp(r, ,uad)-1} and {Cp(r, ,uad)-1}. We are going to verify the following: FIRST LEMMA OF COMPARISON. (i) If CP = CP(r, Pad) then
rP = B(r)ad cP I(p(B(r)-1)) lies in Gr',,(L). (ii) The cocycle fri) in Gr,11(L) bounds.
(iii) If (1 + c)(z 1 -1),u = (I + 0(r -I -1),u = 0 then rP = cp(r, W, cp(r, ,u).
The third assertion is clear; it is the other two with which we must deal. The element rP can be obtained by projecting (6.1)
b(r, p)-1 p(b(z, ,a))p(b(r, 0-1)b(r, u) on G(Af(L)). This makes it perfectly clear that rP is not affected if w = w(-r}is-replaced by xw(r) with x e CL. Thus we may assume that w = wr, where z is here also used to denote the image of r in Gal(L/Q). Then we have to show that b(r, ,u)-1 p(b(r, p)) = b(r, ,u)-1 p(b(r, ,u))
(mod G(L,,) G(L)).
It follows easily from (5.11) that if aP,7 is a lift of ap,7 to IL, then both sides are congruent to f aPna+C> (1-r-1),U = H P.7 7)
ap7)(l+c) (1-r->) p.
P.7 7)
The desired equality follows. To prove the second assertion we shall apply Hasse's principle, but for this we need a group G whose derived group Gder is simply connected. Let Gs, be the simply connected covering of Gder. The Cartan subgroup T defines Tder and TSe. We have an imbedding X*(T5C) -+ X,(T) and Gder = G57 if and only if the quotient is torsionfree. If we can construct a diagram of Gal(L/Q)-modules
0-*X*(T.) - Q
P-0
0_X*(T5J_X*(T)_M_0 in which P is torsion-free and P -+ M is a surjection whose kernel is a free
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
229
Gal(L/Q)-module, then we can use it to define a central extension G' of G with XX(T') = Q. We will have Gael = GSe = G,,, and G'(R) -+ G(R) will be surjective. To construct the diagram, we choose an exact sequence of Gal(L/Q)-modules
0-N-+ P-aM--a0, with P torsion-free and N free. Then Q is the set of all (x, p) in XX(T) Q+ P for which x and p have the same image in M. We lift u to u' = (u, v) with v e P. If u = ad g o u and g is the image ofg', we set
= ad g' o ,u'. If the assertion is valid for u', ,u', and G', it is valid for ,u, u, and G. Consequently we may suppose that Gderis simply-connected. There are two types of ambiguity in B(r). It can be changed to tB(r) with t c T(L).
Then {rp} is replaced by {(trp ad cp1(p(t)-1)}, and its cohomology class is not affected. We can also change B(r) to B(v)t with t e T(L). Then {cp} is replaced by {cp} with cp = p(tad )Cptad, tad being the image of tin Gad, and {rp) is replaced by {7-P}, with
rp = rp ad ep
1(p(t-1))t.
If rp = 6 ad CP1(p(5-1)) then cp(p(5t)-I).
rp = (St)ad-1 Consequently the ambiguity in B(r) has no effect on the assertion (ii). Rather than prove (ii) directly for a given choice of the pair 'u, u, we want to prove it for a succession of pairs. For this one should first check that the validity of (ii) defines an equivalence relation. If u and u are interchanged then {y,} is replaced by {rP 1}, and if rp = S ad CP1(p(S-1)) then rp 1 = 6-1 ad cp 1(p(S))
To show the transitivity, we introduce a new notation, denoting rp by rp(u, ,u), and cp by cp(u). Suppose rp(/12, /13) = t ad rp(,ul, /12) = sad CP
CP1(/L3)(p(t-1))
1(/2)(p(S-I)).
Observing that rp(/12, 93) ad Cp 1(P3)(p(S 1))rp(,U2, /13)-1 = ad
Cp'(u2)(p(S-I))
and that rp(u1, /13) = 7'p(/'I, ,u2)rp(f-2, ,u3), one deduces with little effort that
rp(uI, /G3) = St ad
Cp(a3)(p(st)-I).
Transitivity established, we return to the original notation. If u is conjugate to p under G(Q), say ad x o u then rp = x ad cP1(x1) = x ad cP1(p(x1)),
and certainly bounds. In general u and p are not conjugate under G(Q), but they are conjugate under G(R). Since G(Q) is dense in G(R) we may take advantage of the transitivity and assume that they are conjugate under Gd,,(R). It is now that the assumption that Gder is simply connected intervenes. If we are
R. P. LANGLANDS
230
careful in our choice of b(z, u) and b(v, ,u), defining them by liftings of co(w) to 7L,
then B(z) and the rP will lie in G. Moreover the cocycle {7-P} in Gde (L) certainly bounds at every finite place. Since we are applying Hasse's principle, we need only verify that it bounds at infinity as well. One begins with a calculation similar to the one made while studying the cocycle {cp(z)}. If p e Gal(L/Q), set
eP(r' ) =
P9(T-1-1)k
rE9
aP9,1
and define ep(r, u) in a similar fashion. The projection of {rp} on GD,P(L_) is cohomologous to ep(z, ,u)ep(r, ,u)-1. If p E Gal(Lv/Q,) the projection of ep(r, u) on is
fP('r )
_
aP0(= 1-1)a
All
Thus all we need do is show that fp(r, p)f'(z,,u)-1 bounds in G=.11(L ). Recall that the cocycle defining Gz'P(L,) is hp = P
aPa(`-1),u-d. P,a
For brevity, we write { fl a aP Q} = {ap(v)}. If x E G(R) we write x(u) = ad x o ,u. If
x (u) _ i then xff(r, p)ff(z, u)-1 ad hp(p(x 1)) = ap((xi 1 x-1 -r 1)u) because p(x) = x and fP(r, u) -l ad hp(x1) = x1 fP(r, ,u)-1 If w lies in the normalizer of Tder in (6.2)
then
wad hp(p(w 1)) = wp(w1)ap((w-1)(i 1-1),u).
However {wp(w1)} is a cohomology class in TdeL(Lv) or Tdel(Lv), the two groups being equal, and it is shown in [45] that it is equivalent to {ap((w-1),u)}. Thus the is {a1((w-1)z 1du)}. Since we may take w so that its action class of (6.2) in on the weights of T is the same as that of xv 1x1r, the verification is complete. There is another fact that we should verify while this proof is fresh in our minds. Suppose there is an w in the Weyl group such that (6.3)
w,u = 'r-1 P.
Then (1 + c)(v 1 -1),u = 0 and {cP 1(r)} = {cP 1(r, ,u)} is defined. It bounds in G(L). Once again it is enough to verify this when Gder is simply connected, although this time the modifications made to arrive at a simply-connected derived group would
be different. It is no longer important that P -+ M be surjective, or that its kernel be free, but there must be a v e P which maps to p and is fixed by Gal(Q/E). The set of all r which satisfy (6.3) for at least one co form a group. Let E = E(G, h) = E(G, .Y1) be its fixed field. Then E s L. In order to apply Hasse's principle we
have to arrange that {cp(r)} lies in Gder(L) and that it obviously bounds in Gder(Af(L)). Since cp(r) only depends on the image of z in Gal(L/Q), we may calculate it when w = wT. If C5 is a set of coset representatives for Gal(L/Q)/Gal(L/E), then
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
231
[
bo(a) _ veG H oeGa1(L/E) a°" °o, We write
v(ao,)a,,,,, av,a and vo,a = v,u + v(6 - 1),u. Now for o, e Gal(L/E),
v(o-1),u is a weight of the derived group and rlvTlo al'Q°T')/I may be lifted to Tde1(A(L)). On the other hand ]-l v flo av ax av o = 1. If a = fl o ao,x then a lies in CE and lifts to a' in 'E. Moreover H,, v(a)yu lifts to ]-[ v(a')v" which lies in T(A). If to IL then we so choose cp(a) that a lifting of (6.4)
pv(a 1)µl
Cp(a) = 1 11
v,o
J
Clva?x 1)µ 1
v,o
J
modulo T(L-). It remains to verify that the {cp 1(r)} so defined bounds at the infinite place. The projection of the right side of (6.4) on CL © X*(Tde7) is
n
v, or
apvo(x-'-1)µ
p' vo
=
1[ oeGal (L/Q)
aPO(r i-1)µ. p' or
Thus, restricting {cP'(a)} to and projecting on we obtain a class cohomologous to {ap((i '-1)u)} _ {ap((co-1),u)}. We have observed already that it is shown in [45] that the right side bounds in Gde1(Lv). Although there is one more consequence to be derived from (6.3), there are some things that must first be said about the general (T, h), or (T, ,u).
We drop the assumption (6.3) for a while, and return to it later. We have associated to the pair (T, p) and v a twisted form Gx'i of G. The twisting of Tin G is trivial, and Tx = T is a Cartan subgroup of GV,p. Let ux be r 1p. There is a unique homomorphism hr: R -+ T= whose restriction on the first factor is ux and which is defined over R. The pair (GV,K, hx) defines a Shimura variety.
The roots {r} of T in G are the same as the roots of Tx in Gro. However the classification into compact and noncompact differs for the two pairs. The root r of Tin G is compact or noncompact according as (- l) is 1 or -1. In order for (Gx,u, hr) to define a Shimura variety, r must be compact or noncompact as a root (-1) is 1 or - 1. However the ideas used in the proof of of Tx according as Lemma A.8 of [34] show that the type of r is changed on passing from T, G to Tx, Gx,p if and only if
-1.
We are now almost ready to formulate a conjecture about the conjugation of Shimura varieties. After discussing the conjecture and its consequences, we shall show how it can be heuristically justified in terms of the Taniyama group and motives.
Recall that ShK(C) = G(Q)\5-5 x G(A f)/K. Thus if g e G(A f) and K1 = g'Kg, then right multiplication by g defines a morphism
gr(g) : ShK(C) - ShK,(C). It is algebraic and is called a Hecke correspondence. If v is an automorphism of C, we set ShK(G, h) = Shk = ShK px z i C, the Shimura varieties being at the moment only defined over C. Then ax(g): ShK -> ShK,. If G = T is a torus and K = U then Shu is 0-dimensional. Let a also denote the element of Gal(Q/Q) defined by a. If we set Ux = U then
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232
ShU(C) = T(Af)l U= Tr(Af)/Ur = ShU,. This gives us an isomorphism ShU = ShU(T, h) --> Shuz = Shu,(Tr, hr). In addition there is the natural map z from complex points of ShU(T, h) to complex points of Shu(T, h). Define cpr = cpr(U, T, h) by the commutativity of the diagram Shu(T, h) ShU(T, h)
Shuz(Tr, hr) In general Gr,' and G are different. But Gr,,u is defined by the cocycle {cp 1(z, /Cad)} and in Gad (A f(L)) cP 1(Z, Pad) = b(r, ,uad)-1 p(b(r, Pad))Thus
g - gr = ad b(v, Pad)-1(g) defines an isomorphism of G(Af) with Gr,fL(Af).
Conjecture. There is a family of biregular maps cpr = cpr(K, G, h), K s G(Af) defined over C, taking Shk (G, h) to ShK,(Gr,", hr), and rendering the following diagrams commutative: Shu(T, h)
.,
ShK(G, h)
y
Ir ShK(G, h)
c..,
ShK,(Gr'u,hr)
1 (a)
Sh&(T, h) wTI
Shuz(Tr, hr)
iw
Here U = T(A f) n K and the cpr in the left column is cpr(U, T, h).
Shk(G, H)
=cg'
Shk1(G, h)
(b)
ShK,((;r,r,, hr)
9(91)
ShKI(Gr,1,, hr)
The conjecture in this form refers to a specific T and a specific h factoring through T. If we choose another pair (T, h) we obtain another group Gr, P and another collection {or = r(K; G, h)}. The conjecture is inadequate as it stands, and must be supplemented by a statement relating (p, and Apr. Observe that there can be at most one family {cpr} satisfying the conditions (a) and (b). We have already associated to the two pairs (T, h) and (T, h) a cocycle {Tp} which bounds in Gr,u(L). Let rp = u adc,1(p(u 1)). Then g -> ugu 1 defines an isomorphism of G",,4(Q)with Gr, P(Q) or of GI(A1) with Gr, K(A f). Set uKr = uKru 1. SECOND LEMMA OF COMPARISON. The homomorphism ad u o hr is conjugate under Gr', P(R) to hr.
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
233
If this statement is true when u is replaced by a v in G(Lv) = G(C) which also trivializes {Tp} restricted to Gal(Lr/Q,) then it is true for u. An examination of the proof of the second property of {7-P} shows that we can take v to lie in z 1wT(L,),
x and w being as in that proof. Since x lw(z lll) = v lp, we have ad v o hr = hr. We infer that ad u carries .»r to r and hence defines a bijection Gr,'(Q)\s-y1 x Gr.P(A f)/Kr -, GT, k(Q)\.T x Gr,'KAf)/UKr
and an isomorphism 0: ShK4Gr,i,, hr)
Sha,,(GD.', hr).
Since u and B(r) both trivialize {7-p} in Gr,P(Af(L)) there is a y in Gr'p(Af) with u = B(z)y. Let rp be the composite 0 a ar(y). Supplement to the conjecture. The diagrams Shk(G, h)
ShK,(Gr,", hr)
w
Sh (GI, 11, hr) K'
are commutative.
The conjecture as it stands certainly implies that the conjugate of a Shimura variety is again a Shimura variety. Together with its supplement, it implies the usual form of Shimura's conjecture [10]. To verify this one applies the Weil criterion [491 for descent of the field of definition. For this we need families of isomorphisms fp : Shq(G, h) ---> ShK(G, h) defined for automorphisms p of C over E(G, h) and satisfying fop = fp fg.
Choose a Cartan subgroup T and an h which factors through it. We know that when z fixes E(G, h) the cocycle {cP 1(a, ,u)} is defined and bounds in G(L). Let co 1(T, u) = vp(v 1). Then g -+ vgv 1 is an isomorphism of G(Q) with Gr,u(Q). Methods which we have already used show easily that The composite ad v o h is conjugate under GD,i(R) to hr.
Consequently ad v defines an isomorphism obtain from G(Q)\
x G(Af)/K
-+
r and then, as before, we
Gr,KIQ)` r x Gr,p(Af)lvK
an isomorphism SK(G, h) -, SVK(GT,P, hr).
On the other hand zv = b(z,,u)-1 with z c= GD,P(A f). We define fT by the commutativity of Shk(G, h)
f=
lprl
SK(G, h) I
hr) c=>
' ShK(Gr,, hr)
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234
I omit the calculations, lengthy but routine, by which it is deduced from the conjecture and its supplement that f, does not depend on the choice of T and h and that the cocycle condition fop = f fg is satisfied. Up to now the SK have been taken as varieties over C, but by the criterion for descent we may now define them over E(G, h) in such a way that the fr are simply the identity maps. It has to be verified that the models thus obtained are canonical, but the construction is clearly such that only the case that G is a torus T need be considered. Let a be the transfer of w = w,r to CE and a' a lifting of a to 'E. The proof that in this case cp(z, p) is trivial shows in fact that we may take it to be 1 and b = b(r, u) to be r 17Ga1(L/Q)/Ga1(L/E) v(a )"u. We take v to be I and z to be b-1 = b(r,,u)-1. The composition Shu(T, h)
r'
Shu(T, h)
$(b)
''Shu(T, h)
is then the identity for all a fixing E(T, h), and this is just the condition that Sh(T, h) be the canonical model. Suppose E(G, h) s R. If we take the canonical model for ShK then the complex conjugation defines an involution 0 of the complex manifold ShK(C). It is necessary to have a concrete description of this involution in terms of the representation ShK(C) = G(Q)\5) x G(A f)/K,
and one purpose of the conjecture is to provide it. Choose some special point (T, h). If E(G, h) s R then we may define b(c, u) and {cp(c, ,u)}. However the condition (c) on the coset representatives wo used to define b(c, u) implies that b(t, u) = 1. We may also take cp(c, u) to be 1, and then v may be taken to be 1 as well. It follows that h and h' are conjugate in G(R). Since Kh, _ Kh, 7): ad g h' > ad g h is a well-defined map of to itself. Consequence of the conjecture. The involution 0 may be realized concretely as the mapping (h, g) -> (vj(h), g) of G(Q)\S-.) x G(Af)IK to itself.
Since we are comparing two continuous mappings which commute with the a(g), g e G(Af), it is enough to see that they coincide on the point in Su(T', h') represented by (h', 1). Since f is the identity and v = z = 1, (p, takes this point to the point in ShK,(G', k, h') = ShK(G, h) represented again by (h', 1). It follows immediately from condition (a) of the conjecture that c applied to the point represented by (h', 1) is (h, 1). For each T and p let v(u) be the fibre functor from Af1(G) to Jl&i (GT,/') which with _ ', and with V(s) being the space obtained takes (a', V(e')) to from V(e') by changing the Galois action to o : x -+ '(co(z, lc) 1)6(x). If (T, h) is another special point, and if v(u, ,u) is defined by the diagram
Reg(G)
"(K P) v(µ)
..&1(G1, u)
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
235
then, according to the first lemma of comparison, the two fibre functors and WRep(G1'P)°v(P, ,u) are isomorphic.
On the other hand, we associated, at the end of the fourth section, to each pair (rp', g) a group G- P and a pair(cp', g'). The groups G and Gr,`p are associated to the
same tannakian category, and there is thus an equivalence of categories v((p): Aeg(G) --> .q .(Gr,,P), determined up to isomorphism. If tP factors through the Serre group, then it can be factored through a Cartan subgroup T of G and defines a coweight ft of T. The hypothetical properties of the Taniyama group imply that G1', may be taken to be Gr' P with v(cp) being v(, i). Thus we have a diagram
Afg(Gr,p) v(u, gyp)
Reg(G)
Jlfg(Gr, P)
which is commutative up to isomorphism of functors. Moreover the two fibre functors WRep(G=,u) and wRep(GT,v) ° v(a, (p) are isomorphic. If we choose an isomor-
phism between them, we obtain [40, 11.3.31 an isomorphism over Q, Gr''p -> Gr,.'. Composing with (p' we obtain a homomorphism (p": GM°t(c) ---> Gr,l,. Since there are so many special points, it is not unreasonable to hope that v(c, (p) exists for all rp, and that WRep(G'',+') and v(,a, (p) are always isomorphic. The second lemma of comparison in conjunction with the hypothetical properties
of the Taniyama group implies that the composition of (p" with the canonical homomorphism -+ GM°t (c) lies in pr when 'p factors through the Serre group, and once again we may surmise or hope that this will be so in general.
If (pr is the biregular map appearing in the conjecture then the composition (pr o r 1 defines a map from the set of complex points on ShK(G, h) to the set of complex points on ShK,(Gr,p, hr). The idea is that (pr ° r-1 will take (cp, g) to a pair ((p", g"),
by a process which can be defined within the moduli problem. We have just seen how to obtain rp", at least at the hypothetical level at which we are working. To obtain g" we observe that we have two homomorphisms (6.5)
(0Rep(G"W) ° v(LL,'p) - WRep(Gr. )
One is obtained from the chosen isomorphism over Q by extending scalars to A f. The other is obtained by a lengthy composition. The g from which we start provides an isomorphismwMot(c) °)7 ' wR p(G) The canonical isomorphism (,t)Mot(c) ° V(z)
->WJ°t(c) can be composed with V. Finally the definitions provide an isomorphism ° ij(v) ° 7j Putting these all together, we obtain an isomorphism from WRep(G:.P) ° V(p) to WMot(c)
(6.6)
wRep(GT'w) ° 40 -' wRep(G)'
However we also have an isomorphism wRep(G)
WRep(GT'N) 0 v(p)'
It is given by the isomorphism x -> e'(b(a, u)-1)x of V(g') with V(e). Composing
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R. P. LANGLANDS
(6.6) and (6.7) we obtain a second isomorphism between the two fibre functors figuring in (6.5). According to general principles, it can be obtained by composing the first with an element (g")-1 in G=,,u(Af) [40, §11]. If one can establish, in some way or another, that the map cr: (tp, g) --> (cp", g")
is really defined, then to prove the conjecture and its supplement one will only need to verify that the composite 0 .,r is complex analytic. However our purpose here has been to see how the wheels mesh, not to find the mainspring.
7. Continuous cohomology. If G = Go then, according to the principles of the fourth section, we should be able to attach to each point of ShK(C) an equivalence class of pairs ((p, g). Here rp is a homomorphism from GMot(c) to G defined over Q and if yl is the associated O-functor g6'9(G)-+ X(96 7-(C), then g defines an isomorphism 0)Mot(C) °
(URep(G).
In general we have mappings ShK(G, h) --> ShK0(Go, ho), and by pulling back we can associate to each point of ShK(C) a pair (tp, g) where g is again in G(Af), but cp now takes GMot(c) to Go.
If (i;, V(i;)) is a representation of Go over Q or, what is the same, a representation of G factoring through Go, then to each point s of ShK(C) we may associate the motive M(s, ) defined by ° 0, together with the isomorphism 0°t (C) (M(s, ))
V() a f
defined by g. The variety ShK(G, h) should be defined over E = E(G, h) and so should this family of motives. Suppose now that the variety ShK(G, h) is proper. In the best of all possible worlds, one might be able to form the cohomology groups M', 0 < i <_ 2 dim ShK, of the family M(., ), which would again be motives, now over E, and thus correspond to representations 6' of GMot (E) If the formalism of the second section were established, one could compose u' with PF to obtain representations p® 6 Qx PF of Gff (E). Then the basic problem would simply be to describe the image p=(Gf(E)).
But we do not have all this formalism, and one of the principal reasons for studying Shimura varieties is the hope that by grappling with the specific arithmetic problems they pose we will obtain an insight that will help us with its construction. Informed by the general principles and hypotheses we are attempting to establish,
we can try to formulate questions that are, at least in part, tractable and which if answered will confirm or, if the answer is other than expected, perhaps refute these principles.
In the present context we can first observe that even if the Mt remained undefined, the zeta-function Z(z, MZ) can be defined directly in terms of the data at our disposal. It is a product over the places of E, j-[ Mi). At a nonarchimedean place it can be defined by the l-adic representation of the Galois group on the ith cohomology group of the 1-adic sheaf FE(Ql) associated to , as in the papers [30] and [34]. Since our principal concern now is with the factors for the archimedean places, we need not enter into details. The field E is contained in C, and the archimedean places are obtained by applying automorphisms of C, or of Q, to E. We first define the factor Mi) for the place v given by E s C. We have seen that we can associate to a locally con-
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
237
stant sheaf FA(Q) over ShK(C). Moreover, we have an analytic family of polarized Hodge structures on FE(C) = FF(Q) O C. By a construction of Deligne [12], [50] this defines a Hodge structure on the cohomology groups H= = Hi(ShK(C), FF(C)). In accordance with the ideas of Serre [42] the factor Z (z, M1) will be defined as
L(s, pi) where pi is a representation of the Weil group Wc,E, on H. The Hodge structure on H= defines a representation ofR(R). Since Cx = R(R), this can be used to define p= on Cx s Wc/E,. If E,, is equal to C, this defines p1 completely. If E, = R then to define pi completely we also have to define p1(w), if w is the element of the Weil group which projects to c and has square - 1. Since ShK is defined over E, c also defines an involution on ShK(C) which we denoted by 0. What we need is a map of order two
0: 0*F _ F,
F = FE(C),
such that on each fibre 6*FP, a = FB,> _, Fa' P S
The associated map e* on cohomology takes HP to Hq,P and we set pi(w) equal to (- 1)PC*.
To define 0 we have to assume that the consequence of the conjecture which was described in the previous section is valid. It can be proved directly in several cases. Then 0 can be obtained by taking the map (h, g) --* (77(h), g) and passing to the quotient. Given (h, g), the fibres at the image of (h, g) and (as(h), g) may both be identified with V(i;)c and 0 is simply the identity map.
If we replace the imbedding E s Cby v-1: E --> C, a being an automorphism, then the complex manifold ShK(C) is replaced by ShK(C), which we may identitfy by means of rpT with ShK,(C), the manifold associated to ShK,(GT,.u, hi). Thus the factor of the zeta-function defined by the place associated to Z--': E = E(G, h) - C can be calculated by replacing G by GT.,,, h by hT, and E by %--'(E) = E(GT,u, h) with the place defined by its inclusion in C. The space V(i;) has also to be twisted by the cocycle {i;(cQ(z,
The function Z(z, M=) defined, the immediate problem is to show that it can be expressed as a product of L-functions associated to automorphic representations (7.1)
Z(z, M`) = fl L(z - a,, z1, r j).
Here a; e C is a translation, 7c; is an automorphic representation of some group H., and r; is a representation of the L-group LH;. The first step is to decide which H;, which 7r;, and which r; intervene in the product.
The first step is to use the theory of continuous cohomology to compute the cohomology groups of the sheaves Fe(C) together with their Hodge structure, and thus to compute Z,(z, M% v being again defined by E c C. Using this together with an analysis of the L-packets of automorphic representations of G [44], one searches for an identity (7.1) which is at least valid when both sides are replaced by their factors at v. An example is discussed in detail in [34]. The identity found, it must be verified for the local factors at the other places V. If v' is an archimedean
place, then the theory of continuous cohomology will allow us to compute Z,,.(z,M1) in terms of the automorphic representations of a Gr,p, a group which
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R. P. LANGLANDS
differs from G by an inner twisting. To make the comparison it will be necessary to have established the principle of functoriality for the pair G and G=,.-, and to have understood in detail how it manifests itself. This is bound up with the study of Lpackets and is primarily an analytic problem, for we expect that the trace formula will give us a good purchase on it [23].
At the finite places the identity (7.1) is difficult to treat as it stands, and for reasons familiar from topology one replaces the left side by (7.2)
Z(z) = r[ Z(z, M=)c 1>`
modifying the right side accordingly. The right side is then analyzed by the trace formula, at least if there is no ramification or, at worst, a mild sort [8], [14], [30]. I do not see at the moment any general way of dealing with a truly nonabelian situation, although a rather curious method has been discovered by Deligne for treating the group GL(2) [13]. If there is no ramification, the factor of (7.2) at a finite place can be analyzed by the fixed point formulae of /-adic cohomology. Apart from combinatorial difficulties [28] the critical factor is to have a reasonably explicit description of the set of geometric points on ShK(G, h) in x., the algebraic closure of the residue field at a prime p of E, together with the action of the Frobenius on it [32]. This is an idea first applied by Ihara [20], who has since intensively studied the structure of this set for Shimura curves [21], [22].
Not much has been done when there is ramification. The first thing is to analyze in reasonably simple cases the manner in which the variety reduces badly. Some interesting discoveries have been made for curves [8], [14], [15], but higher dimensional varieties behave in a more complicated manner. However tools are available for studying their reduction, and it is time to begin. None of these steps will be easy to carry out. The study of L-packets is in an embryonic stage, and even the combinatorial problems will demand considerable ingenuity in their solution [27]. There is still a great deal to be learned from the study of specific examples.
The theory of continuous cohomology is itself in its infancy, and my purpose in
this section is to draw attention to some problems which arise in the study of Shimura varieties and which the mature theory should resolve. I begin by introducing a representation r of the L-group LG which will play a fundamental role in the discussion. The group LG is a semidirect product LG° x Gal(Q/Q). If T is a Cartan subgroup of G over Q then there is an isomorphism of X*(T), the lattice of coweights of T, with X*(LT°), the lattice of weights of LT°, defined up to an element of the Weyl group. In particular if (T, h) is a special point then ,u defines an orbit 6 in X*(LT°), and 0 is independent of(T, h). Let r° be the representation of LG° whose set of extreme weights is 6. The group Gal((?/Q) acts
on the weights of LT° and preserves the set of dominant weights. The group Gal(Q/E) fixes the set 8. Theus Gal(Q/E) fixes the dominant element uv in 6, and we may extend r° to LG° x Gal(Q/E) in such a way that Gal(Q/E) acts trivially on the weight space of ,u". The extended representation will also be called r°, and we define r to be the induced representation
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
239
r = Ind(LG, LG° x Gal(Q/E), r°). Led d be the dimension of ShK(G, h). One expects that the function (7.2) will be equal to a product of functions
L(z - d/2, ic, p).
(7.3)
Here ir is an automorphic representation of one of the groups H attached to G in [33]. There is, in general, an imbedding rp: LH c. LG and p is a subrepresentation
ofr°T. If w is any place of Q, let rw be the restriction of r to the L-group LGw, which equals LG° x Gal(Qw/Qw). Implicit in this notation is an imbedding Q s Qw. Then the double cosets in Gal(Q/E)\Gal(Q/Q)1Gal(QwlQw) parametrize the places of E dividing w, and rw = @111w rv, with
rv = Ind(LGw, LG° x Gal(Qw/Ev), rv)
and rv(v) = r°(a,Z6v 1) if av is some element in the coset defining v. The representation p will also be a direct sum p = O+v,wpv, and the function (7.3) will be a product ]-[ w II viw L(z - d/2, 7tw, pv). The factor corresponding to the place v is L(z -d/2, 2rw, pv) Since we shall only be interested in the place v given by E s C, we shall write r and p instead of rv and pv. Moreover we shall write an automorphic representation of G(A) (or of H(A)) as ir O icf, ' being a representation of G(R) and Z f of G(Af). Any irreducible representation z of G(R) lies in some L-packet H, where cp is a homomorphism from WWR to LG. If a is the character of WCIR obtained by composing Wc,R ---> R with the absolute value, let 01(2r) = a"2 Ox (r ° cp). Then
L(s - d/2, ir, r) = L(s, 01(v)) On the other hand, suppose ir ®7r f is an automorphic representation of G(A). Let it act on the subspace U p Uf of the space of automorphic forms. If h e 5=) then, according to the principles of continuous cohomology [4], its contribution to the cohomology of ShK(C) with values in FC(C) in dimension i is (7.4)
HomKh(A'g/t Ox V, U) Ox UK .
Here t is the Lie algebra of Kh and V the dual of V(e). The space UK is the space of vectors in Uf fixed by K. The action of z e Cx = R(R) defining the Hodge structure sends (p Qx u to (p' Qx u with
co'(X O v) = cc(rj(z)X O (h(z 1))v)
Here X - * yi(z)X is the action which multiplies the exterior product of p holomorphic and q antiholomorphic vectors by z p z Q. Thus 72(z) = ad h(z1) ° 72(z) and (P'(X(Dv) = z(h(z1))(P(V(z)X O v).
If E s R we may extend this action of Cx to an action of WWR = Let n e G(R) be such that ad n ° h = h'. Then the element of w which projects to c and has square -1 sends rp O u to cp' Q u with (p'(X ©v) = 2c(n)rp(ad n-1(X) Ox (n 1)v).
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R. P. LANGLANDS
In either case the representation of Wc,E, on (7.4) factors as cb (7r) px 1, where cii(7r) acts on HomKh(A1C33/f ® P7, U). Let 02(7c) be the element in the representation ring of Wc/R defined by 0200 = (D(-1)= Ind(Wc/R. Wc,E,, O`(Z))Let m(7r f, K) be the dimension of UK. . If for all 7r we had (7.5)
cbz(v) = m(')01(2r)
we could expect a relation (7.6)
Z(z) = 11 L(z - d/2, 7r, r)m(ir-)m(irJ.K) Z
Here, as a single exception, we have taken 7t to be a representation of G(A), its component at infinity being denoted 7rc, and r to be the representation of LG. However (7.5) is not always valid, and it is the true form of the relation between 01(7r) and 02(7r) that we must discover, for it is the clue to the correct expression of(7.2) as a
product of L-functions associated to automorphic representations. Let 1I(g) = {7r1, , Zr) be the set of discrete series representations with the same central and infinitesimal characters as t. Then 11(x) = H is an L-packet fl, and the representations 01(7r1) are all equal. We denote them by cb1(II). The continuous cohomology of the representations 7r, is completely understood [4], and it is a simple exercise to prove the following lemma. LEMMA. @i=1 ybz(') = (- 1)101(I) Thus, in this case, the relation (7.5) fails when the L-packet has more than one element. In order to correct (7.6) one has to replace r by a subrepresentation. However r is in general irreducible as a representation of LG, and so we have to introduce the groups LH of [33], and begin the study of L-indistinguishability. If we accept L-indistinguishability, but expect no other difficulties with the correction of (7.6), then we have to be prepared to prove that every irreducible component of 02(7r) is a component of 01(7r). But we will again be deceived. There is another difficulty. It appears already in the simplest of the examples considered by Casselman [6] and Milne [36], although they had no occasion to draw attention to it. Suppose G is the group associated to a quaternion algebra over Q which is split at infinity but not atp. Let z f = 7rp OO 7rp, and suppose 7r ®x z f is trivial on the centre, 7rp is onedimensional and trivial on the maximal compact subgroup Kp of G(Qp), and 7r is either one-dimensional or the first element of the discrete series. is taken to be trivial. If K = KPKp then L(s - 1, 7z (9 7rf, r) should appear in the zeta-function Z(s, ShK) with the exponent ±m(7rp, KP). Here m(7rp, KP) is the multiplicity with which the trivial representation of KP occurs in 7r f, and the sign is positive if 7r is
one-dimensional and negative if it is the first element of the discrete series. As Casselman and Milne show in their lectures, this is so locally almost everywhere. One can probably show without great difficulty that the local statement is correct at p as well when 7t belongs to the discrete series, for 7r then contributes to the cohomology in dimension one and 7rp = 7r(Op) where vp is a special representation of the thickened Weil group. In particular
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
L(z - Z, 7rp, r) _
241
1 I
- e/p=
,
Iel =I.
However if 7c is one-dimensional then r contributes to the cohomology in dimensions zero and two and the corresponding local contribution to the zeta-function should be 1
m(,rP,KP)
1
(1 - e/pz)(l - ep/pz) I The factor inside the brackets is not L(z - 2, 9rp, r).
The difficulty is resolved if we realize that when zc, and hence ir O 7rf, is one-dimensional we should not be using L(z - 2, ir px 7r f, r) at all but rather L(z - 21, 7r' O -c"f, r) where 7r' O lr f is the one-dimensional representation of G'(A) = GL(2, A) defined by the same character of the idele-class group as 7r O rf.
Since G'(Q,) - G(Q,) and Tc - 7rv for almost all places v, the error of using Ir px Z f instead of ir' xp zf is not detected when one only considers the local zeta-
function almost everywhere. The significance of the considerations of the second and third sections begins to
appear. The representation r Q 2r f and the representation z" O 7r f of G'(A) associated to it by the principle of functoriality are anomalous, because 7r v is onedimensional for almost all places v while zp is infinite-dimensional. The isobaric
representation equivalent to pr" O of almost everywhere is r' Q irf. It was implicit in the discussion of the second section that anomalous representations would have nothing to do with motives, and so it should come as no surprise now that we must discard 7r p z f and replace it by z' Qx 2c f. In this example it itself was not changed forG(R) - G'(R) and ir '- 2r'. However in general we must expect that itself will have to be modified. Thus the proper factor will not be L(z - d/2, r O z f, r) but L(z - d/2, 'r' (D 'r'f, r), where 7r' O 7r f is an automorphic representation of a group G' obtained from G by an inner twisting. Again ir, will have to be equivalent to prv almost everywhere. Since at the moment we are primarily interested in the infinite place. we simply
ask whether it is possible to find a candidate for 7r' or, rather, for an L-packet {ir'} = 17'. There are apparently two conditions to be satisfied, the first arising from the compatibility of functional equations. (a) Let z E II. and let {2r'} = H. For any additive character 0 of R and any representation 6 of LG = LG', E,\Z,a o (P, ) = E(z, 6o (P, 0)
L(l - z, u o () L(z, 7 o (P)
is equal to E'\Z, a o 7r', cb) = a (Z, a o (P', )
L(1 - z, a ow') L(z, 6 0 (P')
(b) It is possible to find a summand (bo(7r') of cbl('r') which is such that 02(tc) _ acbo(2r'), a E Z.
These conditions are only tentative, and may have to be modified in the course of time, but they will serve for the explanation of our problem.
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R. P. LANGLANDS
The first condition involves only cp and cp' and we begin by constructing some pairs that satisfy it. Fix an element w of Wc/R that projects to c and satisfies w2 = -1. We may suppose that (p(w) = a x c with a in the normalizer of LT° in LG°. Then (p(w) also normalizes LT°. Let (p(c) denote the transformation of X*(LT°) or of X*(LT°) defined by cp(w). We may also suppose that (p takes Cx to LT° and that (p(z) = z'1 zo(c)e with A E X*(LT°) Qx C and A - (p(c)A e X*(LT°).
The representation (p' will be defined in a similar way. Thus cp'(w) = a' x c with a' in the normalizer of LT°, and (p'(z) = ze 2P'(". Notice that A is to be the same for cp' as for cp. However a, which is given, is replaced by a', which we must now define.
We suppose that (p(c) sends every root to its negative, and choose 2 in X*(LT°) such that Av(a) = e2nr
1+aA-
(1-a)(1+(p'(c))
2
8
A+q. 2
Observe that the action of (p'(c) is the same as that of c. We are assuming that (p is a given, well-defined homomorphism, and hence [31]
that .1 + (p(c)A =
A
-
((c)!1
-q
(mod X*(LT °)).
In order to show that (p' is also well defined we must verify that .1' + (p'(c)A' =
A
-2
(P(c)/1
(mod X*(LT°)).
We begin with the equations a(p'(c) = (p'(c)a = (p(c) and all + (p(c)) = 1 +(p(c), remarking also that the square of both (p(c) and (p'(c) is the identity. We infer that the left side of (7.7) is equal to (1 + (p'(c))(l + (p(c))A 2
-
(1 - a)(1 + (p'(c)) 4
A + q,
because (1 + (p'(c))q/2 = q. The sum is in turn congruent to
1 - (c) A - (1 - a)(1+ cp'(c)) A=1
(c)
A
2
Consequently the homomorphsim (p' can be constructed whenever (p is defined and (p(c) sends every root to its negative. The following lemma is valid in this generality. LEMMA. For any representation a of the Weil form of LG and any nontrivial character 0 of R, E'(Z, o o
(p,
') = &'(v, 6 o (P', `Y).
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
243
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 (p', defined by an a' in LT °, and, reversing the process, passed to A and a. More generally if LM and LM' are two parabolic subgroups of LG, and (p: Wc/R -+ LM has an image which lies in no proper parabolic subgroup of LM, then we can use the process to pass to a (p" whose image lies in the minimal parabolic of LM, and thus of LG or LM', and afterwards reverse it to pass from (p" to a (p': WCIR -> 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 7c in some 1I,, one can choose one of the (p' just described so that the condition (b) is
satisfied for the elements ic' of ff.,. 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 e Qx iris zero there is no difficulty satisfying
(b). We take (p' _ (p and a = 0. The simplest nontrivial example is obtained by taking trivial and iv trivial. Let iv e II,,. Then (p(z) = zQZp(,)a, with q equal again to one-half the sum of the positive roots. If LM is the parabolic subgroup of LG corresponding to the minimal parabolic of G over R, then the image of (p lies in LM and (p(c) takes every root of LT° in LM to its negative. Define (p' as above, with a' E LT °. G' can be taken to be the quasi-split form of G over R. The continuous cohomology of iv 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 ic' E III, the representation cbi(ic'), which depends in reality only on (p', is equivalent to 02(Z)-
If G is not quasi-split over R then (p' is different from (p. If G is not quasi-split over R then it is certainly not quasi-split over Q, and the trivial representation of G(A) is anomalous. Once again we see that the passage from iv to ic' is the local expression of the passage from an anomalous representation to one which is not anomalous. Other interesting examples are the representations iv = Ji,j of PSU(n, 1) discussed in Chapter XI of the notes of Borel-Wallach [4]. Take e trivial. In this case 02(i0 is (- 1)i+j times a representation induced from C> , the representation of C" used having the weights (7.8)
Z i fj, Z ---i-1 Z j-1 ... Z (n-j) Z (n-i)
Here 0 5 i + j < n - 1 and 0 S 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 LG°is SL(n + 1, C) and LT° may be taken to be the group of diagonal
matrices. The representation r° is the standard representation of SL(n + 1, C). It is easy enough to deduce from [4] that if 7c e II4 then (p(Z) = ZAZ,)A1
with A being equal to (n/2 - i, n/2, n/2 - 1,
Z E Cx
, n/2 - i + 1, n/2 - i - 1,
,
R. P. LANGLANDS
244
- n/2 + j + 1, - n/2 + j - 1, , - n/2, - n/2 + j). The numbers occurring here are n/2, n/2 - 1, , - n/2, but the order is somewhat unusual. The transformation (p(c) is given by
(XI, ..., Xn+1) - (- xn+1, - X2, ... - Xn, - Xl).
We are of course using the obvious representation of the elements of X,(LT°) Q C as sequences of n + 1 complex numbers whose sum is 0. Suppose, to be definite, that i < j. We will choose (p' to be such that the transformation (p'(e) takes (x1, Xn+1) to (- xn+1, - Xn,"', - Xn-i+1, - Xn-j, - Xi+2, - Xn-j+l, .. , - Xn-i, - Xi+l, . ', - x1). The indices within the gaps decrease or
,
increase regularly by one. If i' e II., then the representation 0l(2r') is induced from a representation of Cx with weights
ziz j, 1 , z 1
1
, ... zi+lz j+l
zi-lzj-1 ... z (n-j-D z (n-i-D z(n-j+1)z (j-1), ... z(n-i)z-i z(n-i+l)z (n-i+ll ..., znz n z(n-f)z (n-i) Happily the set (7.8) is a subset of (7.9) and the condition (b) is satisfied. It should be observed that the representation cio(z') that is chosen to satisfy (b) will have to be, except for some degenerate values of i and j, a proper subrepresen-
tation of cil('r'). This phenomenon will, I hope, be taken into account by L-indistinguishability. For example if e is the element of LT° with diagonal entries t
1, -1,
.1
-1,
-1, 1
then s commutes with (p'(Wc,R) and cbo(2c') may be taken to be the restriction of cbi(z') to the + 1 eigenspace of r(e). REFERENCES
1. J. Arthur, Eisenstein series and the trace formula, these PROCEEDINGS, part 1, pp. 253-274. 2. W. Baily and A. Borel, Compactifzcation of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (1966), 442-528. 3. A. Borel, Automorphic Lfunctions, these PROCEEDINGS, part 2, pp. 27-61. 4. A. Borel and N. Wallach, Seminar on the cohomology of discrete subgroups of semi-simple groups, Institute for Advanced Study (1976/77). 5. P. Cartier, Representations of p-adic groups: A survey, these PROCEEDINGS, part 1, pp. 111-155. 6. W. Casselman, The Hasse- Weil function of some moduli varieties of dimension greater than one, these PROCEEDINGS, part 2, pp. 141-163. , GL(n), Algebraic Number Fields (L-Functions and Galois Properties), A. Frohlich, 7.
ed., Academic Press, New York, 1977. 8. I. V. Cherednik, Algebraic curves uniformized by discrete arithmetical subgroups, Uspehi Mat.
Nauk 30 (1975), 183-184. (Russian)
AUTOMORPHIC REPRESENTATIONS, SHIMURA VARIETIES, AND MOTIVES
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9. P. Deligne, Travaux de Griffiths, Sem. Bourbaki (1970). , Travaux de Shimura, Sem. Bourbaki (1971). 10. , La conjecture de Weil pour les surfaces K3, Invent. Math. 15 (1972), 206-226. 11. , Theorie de Hodge. III, Inst. Hautes Etudes Sci. Pub]. Math. 44 (1974), 5-78. 12. , Letter to Piatetski-Shapiro. 13.
14. P. Deligne and M. Rapoport, Les schemas de modules de courbes elliptiques, Modular Functions of One Variable. II, Lecture Notes in Math., vol. 349, Springer, New York, 1972, pp. 143-316.
15. V. G. Drinfeld, Coverings of p-adic symmetric spaces, Functional Anal. Appl. 10 (1976), 29-40. (Russian) 16. J. Giraud, Cohomologie non-abelienne, Springer-Verlag, Berlin and New York, 1971. 17. Harish-Chandra, Representations of semi-simple Lie groups. VI, Amer. Math. J. 78 (1956), 564-628. 18. , Discrete series for semi-simple Lie groups. I, Acta Math. 113 (1965), 241-318; II, 116 (1966), 1-111. 19. R. Howe and I. Piatetski-Shapiro, A counterexample to the "generalized Ramanujan coniecture" for (quasi-) split groups, these PROCEEDINGS, part 1, pp. 315-322. 20. Y. Ihara, Hecke polynomials as congruence cfunctions in elliptic modular case, Ann. of Math. (2) 85 (1967), 267-295. 21. , On congruence monodromy problems, vols. 1, 2, Lecture notes from Tokyo Univ., 1968, 1969. 22. , Some fundamental groups in the arithmetic of algebraic curves over finite fields, Proc. Nat. Acad. Sci. 72 (1975).
23. S. Gelbart and H. Jacquet, Forms of GL(2) from the analytic point of view, these PROCEEDINGS, part 1, pp. 213-251. 24. H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Math., vol. 114, Springer, New York, 1970.
25. H. Jacquet and J. Shalika, A non-vanishing theorem for zeta-functions of GL(n), Invent. Math. 38 (1976).
26. D. Kazhdan, On arithmetic varieties, Lie Groups and Their Representations, Halsted, New York, 1975. 27. R. Kottwitz, Thesis, Harvard Univ., 1977. , Combinatorics and Shimura varieties mod p, these PROCEEDINGS, part 2, pp. 185-192. 28. 29. N. Kurokawa, Examples of eigenvalues of Hecke operators in Siegel modular forms, 1977 (preprint). 30. R. P. Langlands, Modular forms and l-adic representations, Modular Forms of One Variable. II, Lecture Notes in Math., vol. 349, Springer, New York, 1973, pp. 361-500. , On the classification of irreducible representations of real algebraic groups, Institute 31. for Advanced Study, Princeton, N. J., 1973. , Some contemporary problems with origins in the Jugendtraum, Proc. Sympos. Pure 32. Math., vol. 28, Amer. Math. Soc., Providence, R. I., 1976, pp. 401-418. , Stable conjugacy: definitions and lemmas, Canad. J. Math. (to appear). 33. , On the zeta functions of some simple Shimura varieties, Canad. J. Math. (to appear). 34. , On the notion of automorphic representation, these PROCEEDINGS, part 1, pp. 203-207. 35. 36. J. Milne, Points on Shimura varieties modp, these PROCEEDINGS, part 2, pp. 165-184. 37. D. Mumford, Families of abelian varieties, Proc. Sympos. Pure Math., vol. 9., Amer. Math. Soc., Providence, R. I., 1966, pp. 347-351. 38. I. Piatetski-Shapiro, Multiplicity one theorems, these PROCEEDINGS, part 1, pp. 209-212. 39. H. Resnikoff and R. L. Saldana, Some properties of Fourier coefficients of Eisenstein series of degree two, Crelle 265 (1974), 90-109. 40. N. Saavedra Rivano, Categories tannakiennes, Lecture Notes in Math., vol. 265, Springer,
New York, 1972. 41. J.-P. Serre, Abelian 1-adic representations and elliptic curves, Benjamin, New York, 1968. , Facteurs locaux des fonctions zeta des varietes algebriques (definitions et conjectures), 42. Sem. Delange-Pisot-Poitou, 1970. , Representations 1-adiques, Algebraic Number Theory, Tokyo, 1977. 43.
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44. D. Shelstad, Notes on L-indistinguishability, these PROCEEDINGS, part 2, pp. 193-203. 45.
, Orbital integrals and a family of groups attached to a real reductive group, 1977
(preprint). 46. K.-Y. Shih, Conjugations of arithmetic automorphic function fields, 1977 (preprint). 47. J. Tate, Number theoretic background, these PROCEEDINGS, part 2, pp. 3-26. 48. N. Wallach, Representations of reductive Lie groups, these PROCEEDINGS, part 1, pp. 71-86. 49. A. Weil, The field of definition of a variety, Amer. J. Math. 78 (1956), 509-524. 50. S. Zucker, Hodge theory with degenerating coefficients. I, 1977 (preprint). THE INSTITUTE FOR ADVANCED STUDY
Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 247-290
VARIETES DE SHIMURA: INTERPRETATION MODULAIRE, ET TECHNIQUES DE CONSTRUCTION DE MODELES CANONIQUES PIERRE DELIGNE SOMMAIRE
Introduction ........................................................247
Rappels, terminologie et notations ...................................250 1. Domaines hermitiens symetriques ....................................251 1.1 Espaces de modules de structures de Hodge ........................251 1.2 Classification ..................................................257
1.3 Plongements symplectiques ......................................259
2. Varietes de Shimura ................................................262
2.0 Preliminaires ..................................................262 2.1 Varietes de Shimura ............................................265 2.2 Modeles canoniques ............................................268
2.3 Construction de modeles canoniques ..............................270 2.4 Loi de reciprocite: preliminaires ..................................274
............................280 ......................284
2.5 Application: une extension canonique 2.6 La loi de reciprocite des modeles canoniques
2.7 Reduction au groupe derive, et theoreme d'existence ................285 Bibliographie ........................................................289
Introduction. Cet article fait suite a [5], dont nous utiliserons les resultats essen-
tiels (ceux des paragraphes 4 et 5). Dans une premiere partie, nous tentons de motiver les axiomes imposes aux systemes (G, X) (2.1.1) a partir desquels sont definies les varietes de Shimura. On montre que, grosso modo, ils correspondent aux espaces de modules de structures de Hodge X+ du type suivant. (a) X+ est une composante connexe de 1'espace de toutes les structures de Hodge
sur un espace vectoriel fixe V relativement auxquelles certains tenseurs t1 .. t sont de type (0, 0). Le groupe algebrique G est le sous-groupe de GL(V) qui fixe les ti et X est l'orbite G(R) X+ de X+ sous G(R). 4NIS (MOS) subject classifications (1970). Primary 10D20, 14G99; Secondary 20G15, 20G30. © 1979, American Mathematical Society
247
248
PIERRE DELIGNE
(b) La famille de structure de Hodge sur V parametree par X+ verifie certaines conditions, verifiees par les families de structures de Hodge qui apparaissent naturellement en geometrie algebrique: pour une structure complexe convenable (et uniquement determinee) sur X+, c'est une variation de structure de Hodge polarisable. L'espace X+ est automatiquement un domaine hermitien symetrique (= espace
hermitien symetrique a courbure < 0). Les domaines hermitiens symetriques peuvent tous etre ainsi decrits comme des espaces de modules de structures de Hodge (1.1.17), et je crois cette description tres utile. Par exemple: le plongement d'un domaine hermitien symetrique D dans sondual b (une variete de drapeaux) correspond a 1'application (une structure de Hodge),-, (la filtration de Hodge correspondante). Les descriptions comme "domaine de Siegel de 3e espace" s'interpretent en disant que, sous certaines hypotheses, si on superimpose a une structure
de Hodge une filtration par le poids, on obtient une structure de Hodge mixte, d'oi une application de D dans un espace de modules de structures de Hodge mixtes (cf. les constructions de [1, III, 4.1]). Ce dernier point ne sera ni mentionne, ni utilise dans l'article. Ce point de vue, et la description de certaines varietes de Shimura comme es-
paces de modules de varietes abeliennes, sont lies par le dictionnaire: it revient au meme (equivalence de categorie A H HI(A, Z)) de se donner une variete abelienne ou une structure de Hodge polarisable de type {(-1, 0), (0, -1)} (il s'agit ici de Z-structures de Hodge sans torsion; par passage au dual (A H HI(A, Z)), on peut remplacer {(-1, 0), (0, -1)} par {(1, 0), (0, 1)}). Polariser la variete abelienne revient a polariser son HI. Avec parametres, de meme, it revient au meme de se donner un schema abelien polarise sur une variete complexe lisse S, ou une variation de structures de Hodge polarisee de type {(- 1, 0), (0, -1)} sur 1'espace analytique San. Une famille analytique de varietes abeliennes, parametree par San 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 it 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 classifions un certain type de plongements de domaines hermitiens symetriques dans le demi-espace de Siegel. Les resultats sont paralleles a ceux de Satake [11]. Une application de l'astuce unitaire de Weyl, pour laquelle nous renvoyons a [7], ra-
mene la classification a la connaissance d'un fragment de la table, donnee par exemple dans Bourbaki [4], dormant 1'expression des poids fondamentaux comme combinaison lineaire de racines simples. Le lecteur desireux d'en savoir plus sur les variations de structure de Hodge, et
la facon dont elles apparaissent en geometrie algebrique, pourra consulter [6] (dont nous ne suivons pas les conventions de signes); certains faits, enonces daps [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
Mc(G, X) est un modele de MC(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) Ox E(c.x) C Mc(G, X), cette faton de definir Mc(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). Its jouent un role technique dans la construction de modeles canoniques. Les differences apparentes 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 l'introduction de [5]. Pour une breve description, sur des exemples, de comment on passe du langage adelique a un langage plus classique, je renvoie a [5, 1.6-1.11, 3.14-3.16, 4.11-4.16]. Dans [5], nous avions systematise les methodes introduites par Shimura pour construire des modeles canoniques. Dans la 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 1'ensemble des composantes connexes geometriques d'un modele faiblement canonique (suppose exister) de Mc(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 universel de son groupe derive Gder et M une classe de conjugaison, definie sur un corps de nombres E, de morphismes de G,, dans G. On construit un morphisme qyr 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)
NFiEG(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 deduit du theoreme d'approximation forte que co(G(A)/pG(A)G(Q)) = 2'o(G(A)/G(Q)),
et qM fournit une action sur 'ro(G(A)/G(Q)) de zoC(E), le groupe de Galois rendu abelien Gal(Q/E)ab d'apres la theorie du corps de classes global.
La seconde idee nouvelle-en fait un retour au point de vue de Shimura-est l'observation suivante: les resultats de 2.6 permettent de reconstruire un modele faiblement canonique ME(G, X) de MC(G, X) a partir de sa composante neutre M4(G, X+) (une composante connexe geometrique, dependant du choix d'une composante connexe X+ de X), munie de l'action (semi-lineaire) du sous-groupe H de G(Af) x Gal(Q/E) qui la stabilise. Soient Z le centre de G, Gad le groupe adjoint, Gad(R)+ la composante neutre topologique de Gad(R), et Gad(Q)+ = Gad(Q) n Gad(R)+
L'adherence Z(Q)- de Z(Q) dans G(Af) agissant trivialement sur MC(G, X), l'ac-
250
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tion de H sur MQ(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 Gad(Q)+ pour la topologie des images des sous-groupes de congruence de A isomorphisme unique pres, cette extension ne depend que de Gaa Gder et de la Gder(Q).
projection X+ad de X+ dans Gad (2.5). On la note gE(Gad Gaer X+ad). La composante neutre M$(G, X+) est la limite projective des quotients de X+ad par les sous-groupes
arithmetiques de Gad(Q)+, images de sous-groupes de congruence de Gder(Q). On verifie enfin que les conditions que doivent verifier son modele MQ(G, X+) sur Q, et l'action de SE(Gad Gder X+ad), pour correspondre a un modele faiblement canonique, ne dependent que du groupe adjoint Gaa, de X+ad, du revetement Gder de Gad et de 1'extension finie E (contenue dans C) de E(Gad X+ad). Ces conditions definissent les modeles faiblement canoniques (resp. canoniques, pour E = E(Gaa X+ad)) connexes (2.7.10).
Le probleme de 1'existence d'un modele canonique ne depend donc, en gros, que du groupe derive. Cette reduction au groupe derive est une version beaucoup plus commode de la maladroite methode de modification centrale de h dans [5, 5.11].
En 2.3, nous construisons une provision de modeles canoniques a l'aide de plongements symplectiques, en invoquant [5, 4.21 et 5.7]. Les resultats de 1.3 nous permettent d'obtenir les plongements symplectiques desires avec tres peu de calculs. En 2.7, nous expliquons la reduction an groupe derive esquissee ci-dessus et nous deduisons de 2.3 un critere d'existence de modeles canoniques qui couvre tous les cas connus (Shimura, Miyake et Shih). Dans Particle, nous utilisons l'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). Il eut ete plus naturel de transposer les demonstrations, et de transposer de meme la fonctorialite [5, 5.4], et le passage a un sous-
groupe [5, 5.7] (evitant par la la sybilline proposition [5, 1.15]). Le manque de temps, et la lassitude, m'en ont empeche. J'ai recemment demontre qu'on pouvait donner un sens purement algebrique a la 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 la le critere d'existence 2.3.1 de modeles canoniques, et 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, 1.15]) et fournit des renseignements partiels sur la 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 la nullite de Hl(K, G) pour G semi-simple simplement connexe sur un corps local non archimedien. Des indications bibliographiques sur ces theoremes sont donnxes dans [5, (0.1) a (0.4)]. Signalons en outre Particle de G. Prasad (Strong approximation for semi-simple groups over function fields, Ann. of Math. (2) 105 (1977), 553-572) prouvant le theoreme d'approximation forte sur un corps global quelconque. Soit G un groupe semisimple simplement connexe, de centre Z, sur un corps global K. Nous n'utiliserons
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le principe de Hasse pour HI(K, G) que pour les classes dans l'image de H'(K; Z). En particulier, les facteurs E8 nous indifferent. 0.2. Groupe reductif signifiera toujours groupe reductif connexe. Un revetement d'un groupe reductif est un revetement connexe. Groupe adjoint signifiera groupe reductif adjoint. Si G est un groupe reductif, nous noterons Gad son groupe adjoint, Gder son groupe derive, et p: G --> Gder le revetement universel de 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 la composante neutre du centre Z de G). L'exposant + designera une composante connexe topologique (par exemple: G(R)+ est la composante neutre topologique du groupe des points reels d'un groupe G). On notera aussi G(Q)+ la trace de G(R)+ sur G(Q). Pour G reductif reel, nous noterons par un indice + l'image inverse de Gad(R)+ dans G(R). Meme notation + pour la trace sur un groupe de points rationnels. Pour Xun espace topologique, noun notons iro(X)1'ensemble de ses composantes connexes, muni de la topologie quotient de celle de X. Dans l'article, l'espace 'ro(X) sera toujours discret, ou compact totalement discontinu. 0.4. Un domaine hermitien symetrique est un espace hermitien symetrique a courbure < 0 (i.e. sans facteur euclidien on compact). 0.5. Sauf mention expresse du contraire, un espace vectoriel est suppose de dimension 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 Z = proj lim Z/nZ = flp Z p, Af = Q O Z = ]1 p Q p (produit restreint) et on note A = R x Af l'anneau des adeles de Q. On designera parfois encore par A 1'anneau des adeles d'un corps global quelconque. 0.7. G(K), G OF K, GK : pour G un schema sur F (par exemple un groupe algebrique sur F) et K une F-algebre, on designe par G(K)1'ensemble des points de G a valeur dans K, et par GK ou par G OF K le schema sur K deduit de G par extension des scalaires. 0.8. Nous normaliserons l'isomorphisme de reciprocite de la theorie du corps de classes global (= choisirons lui ou son inverse) Gder.
1roAE/E*
Gal(Q/E)ab
de telle sorte que la 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 la justification be. cit. 3.6, 8.12). 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 = (DVpq du complexifie de V, telle que Vpq soit le complexe conjugue de Vqp. Definissons une action It de C* sur VV par la formule (1.1.1.1)
h(z)v = z pz-qv pour v r= Vpq.
Les h(z) commutent a la conjugaison complexe de Vc, donc 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, considers comme groupe algebrique reel (autrement dit, S = RCIRGm (restriction des scalaires a la 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 algsbrique reel S). A l'inclusion R* c C* correspond une inclusion de groupes algebriques reels G. c S. Nous noterons Wh (ou simplement w) la restriction de h-1 a Gm, et 1'appellerons le poids w : Gm --> GL(V). On dit que Vest homogene de poids n si Vpq = 0 pour p + q 0 n, i.e. w(2) est l'homothetie de rapport 2,,. Nous noterons fUh (ou simplement u) l'action de Gm sur Vc definie par ,u(z)v = z pv pour v e Vpq. C'est un compose Gm -> Sc -->h GL(Vc). La filtration de Hodge Fh (ou simplement F) est definie par FP = Q+rZp Vrs. On
dit que Vest de type e' c Z x Z Si Vpq = 0 pour (p, q)
'.
Plus generalement, si A est un sous-anneau de R tel que A Qx Q soit un corps (en pratique, A = Z, Q ou R), une A-structure de Hodge est A-module de type fini V, muni d'une structure de Hodge sur V (DA R. EXEMPLE 1.1.2. L'exemple fondamental est celui ou V = Hn(X, R), pour X une
variete kahlerienne compacte, et ou 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 structure de Hodge facteur direct, ou passage an dual. Ainsi, le dual Hn(X, R) de Hn(X, R) est muni d'une structure de Hodge de poids - n. L'homologie, ou la cohomologie, entisres fournissent des Z-structures de Hodge. EXEMPLE 1.1.3. Les structures de Hodge de type {(- 1, 0), (0, -1)} sont celles pour lesquelles faction h de C* = S(R) est induite par une structure complexe sur V; pour V de ce type, la 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/P de son algebre de Lie L par un reseau F'. On a ]' D R -Z L, d'ou une structure complexe sur r OO R. Considsrons celle-ci comme une structure de Hodge (1.1.3). Via 1'isomorphisme P = H1(A, Z), c'est celle de 1.1.2. EXEMPLE 1.1.5. La structure de Hodge de Tate Z(1) est la Z-structure de Hodge de type (- 1, - 1) de reseau entier 27riZ c C. L'exponentielle identifie C* a C/Z(1), d'ou un isomorphisme Z(1) = H1(C*). La structure de Hodge Z(n) = dln Z(1)®11 (n r= Z) est la Z-structure de Hodge de type(-n, -n) de reseau entier (2iri)nZ. On note ...(n) le produit tensoriel de ... par Z(n) (ttivist a la Tate). 1.1.6. REMARQUE. La rsgle h(z)v = z-pz-qv pour v E Vpq est celle que j'utilise dans (Les constantes des equations fonctionnelles desfonctions L, Anvers II, Lecture Notes in Math., vol. 349, pp. 501-597) et l'inverse de celle de [6]. Elle est justifise d'une part par 1'exemple 1.1.4 ci-dessus, d'autre part par le desir que C* agisse sur R(1) par multiplication par la norme (cf. loc. cit., fin de 8.12). 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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253
(b) en chaque point de S, une structure de Hodge sur la fibre de V en s, variant continument avec s. On exige que la filtration de Hodge varie holomorphiquemefit avec s, et verifie 1'axiome dit de transversalite: la derivee d'une section de FP est dans FP-1. II sera souvent donne un systeme local VZ de Z-modules de type fini, tel que V = V® O R. On parlera alors de variation de Z-structure de Hodge. De meme pour Z remplace par un anneau A comme en 1.1.1. REMARQUE 1.1.8. Regardons S comme Rant une variete reelle, dont 1'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 [To,-1, To,-I] c TO,-1. De filtration de Hodge du complexifie du fibre tangent: meme, l'axiome des variations de structure de Hodge exprime que la derivation (fibre tangent) OR (sections C`° de V) --> (sections C°° de V) (ou plutot le complexifie
de cette application) est compatible aux filtrations de Hodge : aDFP c Fp pour D dans T°.-1 (holomorphie) et BDFP 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 1'espace des parametres. L'exemple fondamental est 1.1.2, avec parametres : si f : X - S est un morphisme propre et lisse, de fibres XS kahleriennes, les Hn(X5, Z) forment un systeme local sur S, et la filtration de Hodge sur le complexifie Hn(X5, C) varie holomorphiquement avec s et verifie l'axiome de transversalite.
1.1.10. Une polarisation d'une structure de Hodge reelle, de poids n, V est un morphisme T : V Qx V --* R(- n) tel que la forme (29ri)n V(x, h(i)y) soit symetrique et definie positive. De meme pour les Z-structures de Hodge, en remplacant R(-n)
par Z(- n), .... Puisque ?F(h(i )x, y) = ?1(x, h(- i)y) (h(i) est trivial sur R(- n)), et que h( - i)y = (-1)nh(i)y, la condition de symetrie revient a : ??t symetrique pour n pair, alterne pour n impair. Les structures de Hodge qui apparaissent en geometrie algebrique sont des Zstructures de Hodge homogenes et polarisables. Exemple fondamental : les theoremes de positivite de Hodge assurent que Hn(X, Z), pour X une variete projective et lisse, est polarisable (noter que h(i) est l'operation notee C par Weil dans son livre sur les varietes kahleriennes). 1.1.11. Soient des espaces vectoriels reels (Vi)i,1 et une famille de tenseurs (s1)1 1 dans des produits tensoriels de puissances tensorielles des Vi et de leurs duaux. On s'interesse aux familles de structures de Hodge sur les Vi, pour lesquelles les s; sont
de type (0, 0). Pour interpreter cette condition "type (0, 0)" dans des cas particuliers, noter que f : V -* West un morphisme si et seulement si, en tant qu'element de Hom(V, W) = V* p W, i1 est de type (0, 0). Soit G le sous-groupe algebrique de ]T GL(Vi) qui fixe les s;. D'apres 1.1.1, une famille de structures de Hodge sur les Vi s'identifie a un morphisme h : S fl GL(Vi). Pour que les s; soient de type (0, 0), it faut et it suffit que h se factorise par G : it s'agit de considerer les morphismes algebriques h : S -> G. On pent regarder G, plutot que le systeme des Vi et s;, comme l'objet primordial: si G est un groupe algebrique lineaire reel, it revient au meme de se donner h : S --
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G, ou de se dormer sur chaque representation V de G une structure de Hodge, fonctorielle pour les G-morphismes et compatible aux produits tensoriels (cf. Saavedra, [10, VI, §2]). Les morphismes wh et ,ah de 1.1.1 deviennent des morphismes de Gdans G, et de G,n dans Gc respectivement.
1.1.12. La construction 1.1.11 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 1'espace des morphismes (= homomorphismes de groupes algebriques sur R) de S dans G. Soit G1 le plus petit sous-groupe algebrique de G par lequel se factorisent les h e X : X est encore une composante connexe de 1'espace des morphismes de S dans G1. Puisque S est de type multiplicatif, deux elements quelconques de X sont conjugues : 1'espace X est une classe de Gi(R)+-conjugaison de morphismes de S dans G. C'est aussi une classe de G(R)+-conjugaison, et G1 est un sous-groupe invariant de la composante neutre de G. 1.1.13. Vu 1.1.9 et 1.1.10, nous ne considerons que les Xtels que, pour une famille fidele V1 de representations de G, on ait (a) Pour tout i, la graduation par le poids de Vt (de complexifiee la graduation de Vic par les VC = Q fi+q=n V?l) est independante de h E X. Conditions equiva lentes: h(R*) est central dans G(R)°; ]a representation adjointe est de poids 0. (/3) 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 Vest la composante homogene d'un poids n d'un V it existe 1 : V OO V R( - n) qui, pour tout h E X, soit une polarisation de V. PROPOSITION 1.1.14. Supposons verifie (a) ci-dessus.
(i) Il existe une et une seule structure complexe sur X telle que les filtration de Hodge des V. varient holomorphiquement avec h e X. (ii) La condition 1.1.13(/3) est verifiee si et seulement si la representation adjointe
est de type {(-1, 1), (0, 0), (1, -1)}. (iii) La condition 1.1.13(r) est ver fee si et seulement si G1 (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 la somme des V1. C'est une representation fidele de G. Une structure de Hodge est determinee par la 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 Vp9 = FP fl Fe. L'application (p de X dans la grassmannienne de Vc : h H la filtra-
tion de Hodge correspondante, est done injective. Nous allons verifier qu'elle identifie X a une sous-variete complexe de cette grassmannienne; ceci prouvera (i) : la structure complexe sur X induite de celle de la grassmannienne est la seule pour laquelle rp soit holomorphe.
Soient L 1'algebre de Lie de G et p : L -+ End(V) son action sur V. L'action p est un morphisme de G-modules, injectif par hypothese. Pour tout h e X, c'est aussi un morphisme de structures de Hodge. L'espace tangent a X en h est le quotient de L par 1'algebre de Lie du stabilisateur de h-a savoir le sous-espace L00 de L pour la structure de Hodge de L definie par h. L'espace tangent a la grassmannienne en cp(h) est End(Vc)/F°(End(Vc)). Enfin, dcp est le compose
VARIETES DE SHIMURA
L/L00
P
255
End(V)/End(V)0° ji
Ii
Lc/F°Lc---End(Vc)/F°End(VV)
Puisque p est un morphisme injectif de structures de Hodge, dlp est injectif; son image est celle de Lc/F°Lc, un sous-espace complexe, d'oii ]'assertion. (ii) L'axiome de transversalite signifie que l'image de dcp est dans F-1End(Vc)/F°End(Vc), i.e. que Lc = F-'Lc. Pour prouver (iii), nous ferons usage de [7, 2.8], rappele ci-dessous. Rappelons qu'une involution de Cartan d'un groupe algebrique lineaire reel (non necessairement connexe) G est une involution o de G telle que la forme reelle G° de G (de conjugaison complexe g -+ 6(g)) soit compacte: G°(R) est compact et rencontre toutes les composantes connexes de Ga(C) = G(C). Pour C G G(R) de carre central, une C -polarisation d'une representation V de G est une forme bilineaire ? G-invariante, telle que T (x, Cy) soit symetrique et defini > 0. Pour tout g e G(R), on a alors tr(x, gCg ly) = I(g lx, Cg ly) : la notion de C-polarisation ne depend que de la class de G(R)-conjugaison de C. Rappel. 1.1.15 [7, 2.8]. Soient G algebrique 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 reelle de G est C-polarisable; (iii) G admet une representation reelle C polarisable fidele. On notera que la condition 1.1.15(i) entraine que G° est reductif--pour avoir une forme compacte. Elle ne depend que de la classe de conjugaison de C. Prouvons 1.1.14(iii). Soit G2 le plus petit sous-groupe algebrique de G par lequel
se factorisent les restrictions des h e X a U1 c C*. Pour qu'une forme bilineaire T: V Qx V --> R(- n) verifie 1.1.13(r), it faut et it suffit que (27Gi)nW: V p V --+ R soit invariant par les h(U')-donc par G2-(ceci exprime que ?' est un morphisme), et une h(i)-polarisation. D'apres 1.1.15, 1.1.13(r) equivaut a: int h(i) est une involution de Cartan de G2. On en deduit d'abord que G1 est reductif: G2 1'est, pour avoir une forme compacte, et G1 est un quotient du produit G,,, x G2. Puisque G2 est engendre par des sous-groupes compacts, son centre connexe est compact: it est isogene au quotient de G2 par son groupe derive. L'involution 0 = int h(i) est donc une involution de Cartan de G2 si et seulement si c'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 1.1.16. Les conditions 1.1.13(a), ((3), (r) ne dependent pas de lafamille
fidele choisie de representations V,.
COROLLAIRE 1.1.17. Les espaces X de 1.1.13 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 1.1.13. (2) Que G est adjoint: par(t), G est reductif, et son quotient par un sous-groupe
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central fini est le produit d'un tore T par son groupe adjoint Gad. L'espace X s'identifie encore a une composante connexe de 1'espace des morphismes de dans Gad : 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. Soit dons G un groupe simple adjoint, et X une G(R)+-classe de conjugaison de morphismes non triviaux h : SIG,, -> 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; it existe done sur X une structure riemannienne G(R)+-equivariante. D'apres (ii), h(i) agit sur 1'espace tangent Lie(G)/Lie(G)00 de X en h par -1: l'espace X est riemannien symetrique. On verifie enfin qu'il est hermitien symetrique pour la structure complexe 1.1.14(i). It 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 la multiplication par u (Jul = 1) sur 1'espace tangent TX a Xen x se prolonge en un automorphisme mx(u) de X. Soient A le groupe des automorphismes de X, et h(z) = m(z/z) pour z E C*. Le centralisateur Ax de x commute a h, et la condition de 1.1.14(ii) est done verifiee: Lie(Ax) est de type (0, 0), et Tx = Lie(A)/Lie(Ax) de
type {(- 1, 1), (1, -1)}. Enfin, on sait que A est la composante neutre de G(R), pour G adjoint, et que 1'espace riemannien symetrique X est a courbure < 0 si et settlement si la symetrie h(i) fournit une involution de Cartan de G (voir Helgason [8]).
1.1.18. Indiquons deux variantes de 1.1.15 (cf. [7, 2.11]).
(a) On donne un groupe algebrique reel reductif (0.2) G, et une classe de G(R)conjugaison de morphismes h : S - G. On suppose que wh-note w-est central, done independant de h (condition 1.1.13(a)), et que int h(i) est une involution de Cartan de G/w(Gm).
Puisque G est reductif, w(Gm) admet un supplement G2 : un sous-groupe invariant connexe tel que G soit quotient de w(Gm) x G2 par un sous-groupe central fini. 11 est unique: engendre par le groupe derive et le plus grand sous-tore compact du centre. 11 contient les h(Ul) (h e X), et int h(i) en est une involution de Cartan. Si Vest une representation de G, sa restriction a G2 admet done une h(i)-polarisation 0. Si V est de poids n, w(Gm) agit par similitudes, done G de meme : pour une representation convenable de G sur R, 0 est covariant. Pour cette representation, R est de type (n, n); ceci permet de faire agir G sur R(n), de facon compatible a sa structure de Hodge, et de voir Xb = (2 vi)-nO comme une forme de polarisation Ginvariante: V © V -+ R( - n). (b) Supposons que G se deduise par extension des scalaires a R de GQ sur Q, et que w soit defini sur Q. Le groupe G2 est alors defini sur Q, car c'est l'unique supplement de w(Gm), et tout caractere de G/G2 est defini sur Q, car ce groupe est trivial ou isomorphe, sur Q, a Gm. Si une representation (rationnelle) V de GQ est de poids n, les formes bilineaires G-invariantes V O V -> Q(- n) forment un espace vectoriel F sur Q. L'ensemble de celles qui sont de polarisation (rel. h E X) est la trace sur F d'un ouvert de FR, et cet ouvert est non vide d'apres (a). Il existe done des formes de polarisation ': V 0 V -+ Q(n) G-invariantes.
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257
On prendra garde que les formes en (a) et (b) ne sont pas toujours de polarization pour tout h' E X: si h' = int(g)(h), la formule ?lr(x, h'(i)y) = g (g lx, h(i)g-ly) montre que la forme (22ri)nPI(x, h'(i)y) est symetrique et definie-mais definie positive ou negative selon l'action de g sur R( - n). 1.2. Classification.
Dans la suite de ce paragraphe, nous utilisons la relation 1.1.17 entre domaines hermitiens symetriques et espaces de modules de structures de Hodge, pour reformuler certains resultats de [1] et [81, et donner quelques complements. 1.2.1. Considerons les systemes (G, X) formes d'un groupe algebrique reel simple adjoint G, et d'une classe de G(R)-conjugaison X de morphismes de groupes algebriques reels h : S -> G, verifiant (les notations sont celles de 1.1.1, 1.1.11).
(i) La representation adjointe Lie(G) est de type {(- 1, 1), (0, 0), (1, -1)} (en particulier, h est trivial sur Gm c S); (ii) int h(i) est une involution de Cartan; (iii) h est non trivial ou-ce qui revient an meme (cf. 1.1.17)-G est non compact. D'apres 1.1.17, les composantes connexes des espaces X ainsi obtenus sont les domaines hermitiens symetriques irreductibles. L'hypothese (ii) assure que les involutions de Cartan de G sont des automorphismes interieurs, donc que G est une forme interieure de sa forme compacte (cf. 1.2.3). En particulier, G, etant simple, est absolument simple.
La classe de G(C)-conjugaison de uh: G. -> Gc ne depend pas du choix de h e X. Nous ]a noterons M. PROPOSITION 1.2.2. Soit Gc un groupe algebrique complexe simple adjoint. A chaque systeme (G, X) forme dune forme reelle G de G, et de X verifiant 1.2.1(i), (ii), (iii) associons Mx. On obtient ainsi une bijection entre classes de Gc(C)-conjugaison de systemes (G, X), et classes de Gc(C)-conjugaison de morphismes nontriviaux ,u: Gm -- Gc verifiant la condition suivante. (*) Dans la representation ad u de Gm sur Lie(Gc), seuls apparaissent les caracteres z, 1 et z-1.
Pour verifier 1.2.2, nous utiliserons la dualite entre domaines hermitiens symetriques, et espaces hermitiens symetriques compacts: 1.2.3. Soient G une forme reelle de Gc, X une classe de G(R)-conjugaison de morphismes de S/Gm dans G, et h e X. La forme reelle G correspond a une conjugaison complexe or sur Gc ; definissons G* comme la forme reelle de conjugaison complexe int(h(i)) 6 :
G*(R) = {g a (C)Ig = int(h(i))a(g)}.
Le morphisme h est encore defini sur R, de S/Gm dans G* : on a h(C*/R*) c G*(R); definissons X* comme la classe de G*(R)-conjugaison de h. La construction (G, X) --+ (G*, X*) est une involution sur 1'ensemble des classes de Gc(C)-conjugaison des systemes (G, X) formes d'une forme reelle G de Ge, et d'une classe de G(R)conjugaison de morphismes non triviaux de S/Gm dans G. Elle echange les (G, X) comme en 1.2.2, et les (G, X) tels que G soit compact et que Xverifie 1.2.1(i). On sait que les formes reelles compactes de Ge sont toutes conjuguees entre elles.
Puisque si g e G. normalise une forme reelle G, on a g e G(R) (ceci parce que G est adjoint), la dualite ramene 1.2.2 a 1'enonce suivant:
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LEMME 1.2.4. Soit G une forme compacte de Gc. La construction h -> ,uh induit une bijection entre (a) classes de G(R)-conjugaison de morphismes h : S/Gm G, verdant 1.2.1(i), et
(b) classes de Gc(C)-conjugaison de morphismes ,u: Gm -+ G, verifiant 1.2.2(*).
Soient T un tore maximal de G, et Tc son complexifie. On verifie d'abord que l'application. h - ph: Hom(S/Gm, T) -> Hom(Gm, Tc) est bijective. Si W est le groupe de Weil de T, on sait que
Hom(Ul, T)/W -, Hom(Ul, G)/G(R) et
Hom(Gm, Tc)/W
Hom(Gm, Gc)/Gc(C).
L'application h ->,uh induit donc une bijection
Hom(S/Gm, G)/G(R) - Hom(Gm, G)/Gc(C), et, pour que h verifie 1.2.1(i), it faut et it suffit queph verifie 1.2.2(*). 1.2.5. Soit G un groupe algebrique complexe simple adjoint. Nous allons enume-
rer les classes de conjugaison de morphismes non triviaux u: Gm -+ G verifiant 1.2.2(*), en terme du diagramme de Dynkin D de G. Rappelons que ce dernier est canoniquement attache a G-en particulier, les automorphismes de G agissent sur D-On pent identifier ses sommets aux classes de conjugaison de sous-groupes paraboliques maximaux. Soient T un tore maximal, X(T) = Hom(T, Gm), Y(T) = Hom(Gm, T) (le dual de X(T) pour l'accouplement X(T) x Y(T) ->0 Hom(Gm, Gm) = Z), R c X(T) l'ensemble des racines, B un systeme de racines simples, ao l'oppose de la plus grande racine et B+ = B U {ao}. Les sommets de D sont parametres par B, et ceux du diagramme de Dynkin etendu D+ par B+. Une classe de conjugaison de morphismes de G. dans G a un unique representant ,u e Y(T) dans la chambre fondamentale > 0 pour a e B. Il est uniquement determine par les entiers positifs (a a B) et, G etant adjoint, ceux-ci peuvent etre prescrits arbitrairement. La condition 1.2.2(*), pour y non trivial, se recrit (*)'
Ecrivons la plus grande racine comme combinaison lineaire de racines simple L B+n(a)a = 0, avec n(ao) = 1, et appelons speciaux les sommets de D+ tels que, pour la racine correspondante a E B+, on ait n(a) = 1. On sait que le quotient du groupe des copoids par celui des coracines agit sur D+, et de facon simplement transitive sur 1'ensemble des sommets speciaux. Les sommets speciaux sont done les conjugues sous Aut(D+) du sommet correspondant a ao, et leur nombre est I'indice de connexion I zl(G) I de G (cf. Bourbaki [4, VI, 2 ex 2 et 5a)]).
La condition (*)' se recrit (*)" Pour une racine simple a e B correspondant a un sommet special de D, on a = 1. Pour les autres racines simples, = 0. 1.2.6. An total, les classes de GG(C)-conjugaison de systemes (G, X) comme en 1.2.2 sont parametrees par les sommets speciaux du diagramme de Dynkin D de Gc. En particulier, pour G une forme reelle donnee de Gc, X est determine par le
VARIETES DE SHIMURA
259
sommet special s(X) correspondant (G(R) c Gc(C) est en effet son propre normalisateur). Le sommet correspondant a X-1 = {h-11h c X} est le transforme de s(X) par l'involution d'opposition. Dans 1.2.3, G et G* sont des formes interieures l'un de l'autre. S'il existe X verifiant 1.2.1(i), (ii), (iii), G est done une forme interieure de sa forme compacte. En d'autres termes, la conjugaison complexe agit sur le diagramme de Dynkin de Gc par l'involution d'opposition. PROPOSITION 1.2.7. Soit G un groupe algebrique reel simple adjoint, et supposons qu'il existe des morphismes h : C*/R* -* G verdant 1.2.1(i), (ii), (iii). L'ensemble de ces morphismes a alors deux composantes connexes, echangees par It H h-1. Chacune a pour stabilisateur la composante neutre G(R)+ de G.
L'hypothese (ii) assure que le centralisateur K de h(i) est un sous-groupe compact maximal de G(R). En particulier, ico(K) -z ,c0G(R). Il a meme algebre de Lie que le centralisateur de h. Ce dernier est un groupe algebrique connexe-en tant que
centralisateur d'un tore-et compact-en tant que sous-groupe du centralisateur de h(i). Il est donc topologiquement connexe et Centr(h) = K+ = K n G(R)+. Le centre de K+ est de dimension 1: le complexifie de K+ est le centralisateur de,uh donc,
d'apres (*)", un sous-groupe de Levi d'un sous-groupe parabolique maximal. On peut aussi le deduire de ce que la representation de K+ sur Lie(G)/Lie(K+) est irreductible (cf. [8, preuve de V, 1.11). Le morphisme h est donc un isomorphisme de S/Gavec le centre connexe de K+, et, K+ determine h au signe pres. A fortiori, h(i) determine h au signe pres. Des lors (a) L'application h H h(i) est 2 : 1. (b) Elle envoie isomorphiquement l'orbite G(R)+/K+ de h sous G(R)+ sur 1'ensemble G(R)/K de toutes les involutions de Cartan dans G(R). La proposition en resulte. COROLLAIRE 1.2.8. Soit (G, X) comme en 1.2.1, et s le sommet correspondant du diagramme de Dynkin de Gc. (i) Si s nest pasfixe par 1'involution d'opposition, G(R) et X sont connexes. (ii) Si s estfixe par l'involution d'opposition, G(R) et X ont deux composantes connexes; les composantes de X sont echangees par h F-. h-1, et par les g a G(R) - G(R)+.
Signalons que le cas (i) est encore caracterise par les conditions equivalentes (i') le systeme de racines relatif de G est de type C (plutot que BC); (i") Xest un domaine tube. 1.3. Plongements symplectiques. 1.3.1. Soit V un espace vectoriel reel, muni d'une forme alternee non degeneree F. Le demi-espace de Siegel S+ correspondent admet la description suivante: c'est
1'espace des structures complexes h sur V, telles que V soit de type (1,1) (pour l'identification (1.1.3) entre structures complexes et structures de Hodge de type {(-1, 0), (0, -1)}) et que la forme '(x, h(i)x) soit symetrique et definie positive. Si on remplace "defini positif" par "defini", le double demi-espace de Siegel S± obtenu est une classe de conjugaison de morphismes h: S -* CSp(V) (CSp = similitudes symplectiques; dans [5], ce groupe est note Gp). 1.3.2. Soient G un groupe algebrique reel adjoint (0.2) et X une classe de conjugaison de morphismes h: S - G. On suppose verifiees les conditions (i), (ii) de 1.2.1, et on remplace (iii) par
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(iii') G est sans facteur compact. Le systeme (G, X) est donc un produit de systemes (Ge, XX) comme en 1.2.1, et X, correspond a un sommet special du diagramme de Dynkin de G, (1.2.6). Considerons les diagrammes
(G, X) F- (G1, XI) -> (CSp(V), S+), ofi G est le groupe adjoint du groupe reductif GI, et ou XI est une classe de GI(R)-conjugaison de morphismes de S dans G1. On dispose d'une section G -+ G1, de sorte que Vest une representation de G. Notre but est la determination 1.3.8 des representations complexes irreductibles nontriviales W de G, qui est essentiellement equivalent a figurent dans la complexifiee d'une representation ainsi obtenue. Ce probleme celui resolu par Satake dans [11]. LEMME 1.3.3. Il suffit qu'existe (GI, X1) -> (G, X), comme ci-dessus, et une representation lineaire (V, p) de type {(- 1, 0), (0, -1)} de G1, telle que W figure dans V,
Remplagant GI par le sous-groupe engendre par le groupe derive Gj et par l'image de h, on se ramene a supposer que int h(i) est une involution de Cartan de GI/w(Gm). Il existe alors sur V une forme de polarisation ? (1.18(a)), telle que p soit un morphisme de (GI, XI) dans (CSp(V), S+).
1.3.4. Considerons le systeme projectif (H,,),,N suivant: N est ordonne par divisibilite, H = Gm, et le morphisme de transition de Hd dans H. est x H xd (lim proj H est le revetement universel-au sens algebrique-de Gm). Un morphisme fractionnaire de Gm dans un groupe H est un element de lim inj Hom(HH, H). De meme pour le groupe S. Pour p : Gm - H fractionnaire, defini par ,un : H. = Gm -j
H, et V une representation lineaire de H, V est somme des sous-espaces V. (a a (1/n)Z) tels que, via un, Gm agisse sur Va par multiplication par xaa. Les a tell que Va 0 0 sont les poids de ,u dans V. De meme, un morphisme fractionnaire h : S --> H determine une decomposition de Hodge fractionnaire Vr,s de V (r, s e Q). LEMME 1.3.5. Pour h e X, soit ph le relevement fractionnaire de ,uh d Gc. Les representations W de 1.3.2 sont celles telles que Ph n'ait que deux poids a et a + 1.
La condition est necessaire: Relevant h en h1 e X1, on a Phl = Ph v, avec v central. Sur V, ph, a les poids 0 et 1. Si -a est l'unique poids de v sur W irreductible dans Vc, les seuls poids de Ph sur W sont a et a + 1. Pour W non trivial, l'action de Gm via uh (n assez divisible) est non triviale (car G. est simple), donc non centrale,
et les deux poids a et a + 1 apparaissent. La condition est suffisante: Prenons pour V 1'espace vectoriel reel sous-jacent
a W, et pour groupe GI le groupe engendre par l'image de G, et par le groupe des homotheties. Pour h e X, de relevement fractionnaire h a G, soit h1(z) = h(z)z azl-a. Si W. et Wa+1 sont les sous-espaces de poids a et a + 1 de W, h agit sur W. (resp. Wa+I) par (z/z)a (resp. (z/z)I+a), et h1 par z (resp. z) : h1 est un vrai mor-
phisme de S dans G1, de projection h dans G, et V est de type {(-1, 0) (0, -1)} rel. h1. Il ne reste qu'a appliquer 1.3.3. 1.3.6. Traduisons la condition 1.3.5 en terme de racines. Soient Tun tore maximal de G, T son image inverse dans G, B un systeme de racines simples de T, et p e Y(T) le representant dans la chambre fondamentale de la classe de conjugaison de ,uh (h a X). Si a est le poids dominant de W, le plus petit poids est - z (a), pour r
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VARIETES DE SHIMURA
l'involution d'opposition. 11 s'agit d'exprimer que <,u,,3> ne prend que deux valeurs a et a + 1, pour /3 un poids de W. Ces poids etant tons de la forme (a + une combinaison Z-lineaire de racine), et les , pour r une racine, etant entiers, la con-
dition s'exprime par = - 1, soit = 1.
(1.3.6.1)
Determinons les solutions de (1.3.6.1). Pour tout poids dominant a, <,u, a + T(a)>
est un entier, car a + T(a) est combinaison Z-lineaire de racines. Si a 0 0, it est > 0, sans quoi u annulerait tons les poids de la representation correspondante. Un poids dominant a verifiant (1.3.6.1) ne peut donc etre somme de deux poids: LEMME 1.3.7. Seuls les poids fondamentaux peuvent verifier 1.3.6.1.
1.3.8. D'apres 1.3.7, les representations W cherchees se factorisent par un facteur simple G, de G, et leur poids dominant est un poids fondamental; it correspond a un sommet du diagramme de Dynkin D, de G8,. La condition necessaire et suffisante (1.3.6.1) ne depend que de la projection de ,u dans G8c; celle-ci correspond a un sommet special s de D, (1.2.6), et s a racine simple as. Le nombre <,u, w>, pour w un poids, est le coefficient de as dans 1'expression de w comme combinaison Qlineaire de racines simples. Pour cv fondamental, ces coefficients sont donnes dans les tables de Bourbaki [4]. Its sont donnes par la table suivante, oix sont enumeres les diagrammes de Dynkin munis d'un sommet special (entoure). Chaque sommet correspond a un poids fondamental co, et on 1'a afecte du nombre . Les sommets correspondant aux poids qui verifient (1.3.6.1) sont soulignes. TABLE 1.3.9. Dans la table, ... indique une progression arithmetique. Ap+g_1 (sommet special en pierce position)
ql(p + q) ... pgl(p + q)
...
pl(p + q)
.- ...... .................... -.i............................... i
1
B,
C,
DR
1
.. .
1 /2
. ................................................... .
1
. ...................................................
.. .
1 /2
(k+2>5)
. ...................................................
Dk+ 2
1
E
2/3
2
4/3
5/3
4/3
5/2
4/2
6
3/2
E7
1/2
o- ..................................................
1
2
3 1
3/2
-d
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REMARQUES 1.3.10. (i) Pour G simple exceptionnel, aucune representation W ne verifie 1.3.2. (ii) Pour G simple classique, sauf le cas Df (1 >_ 5), les representations W de 1.3.2
forment un systeme fidele de representations de G. Pour DH, on obtient seulement une representation fidele d'un revetement double de G (a savoir, la composante connexe algebrique du groupe des automorphismes d'un espace vectoriel sur H
muni d'une forme antihermitienne non degeneree-une forme interieure de SO(2n)).
2. Varietes de Shimura. 2.0. Preliminaires.
2.0.1. Soient G un groupe, run sous-groupe et tp: r --* d un morphisme. Supposons donnxe une action r de d sur G, qui stabilise r, et telle que (a) r(rp(7-)) est l'automorphisme interieur intr de G; (b) tp est compatible aux actions de d, sur r par r, et sur lui-meme par automorphismes interieurs: cp(r(o)(r)) = inta(rp(r)). Formons le produit semi-direct G x J. Les conditions (a), (b) reviennent a dire que 1'ensemble des r (p(?-)-l est un sous-groupe distingue, et on definit G*rd comme le quotient de G x d par ce sous-groupe. On notera que les hypotheses entrainent que Z = Ker((p) est central dans G, et que Im(cp) est un sous-groupe distingue de 4. Les lignes du diagramme
o -z ---> r --' 4 -> a/r ---> o (2.0.1.1)
r G*r4-ad/r-'0
0->Z-'G II
II
sont exactes, d'oii un isomorphisme (2.0.1.2)
r\G
-
d\G *r d
et, mice en evidence, une action a droite de G *r4 sur r\G. Pour cette action, G agit par translations a droite, et d par 1'action a droite r-1. Si G est un groupe topologique, que d est discret, et que 1'action r est continue, la groupe G *r d, muni de la topologie quotient de celle de G x d, est un groupe topologique, G/Ker(cp) en est un sous-groupe ouvert, et l'application (2.0.1.2) est un homeomorphisme. La construction 2.0.1 garde un sens dans la categorie des groupes algebriques sur un corps. Si G est un groupe reductif sur k, on a un isomorphisme canonique G = G *zcc> Z(G) (pour l'action triviale de Z(G) sur G). 2.0.2. Soient G un groupe algebrique sur un corps k, et Gad le quotient de G par son centre Z. L'action par automorphismes interieurs de G sur lui-meme (x, y) -'
xyx 1: G x G -' G est invariante par Z x {e} agissant par translation, donc se factorise par une action de Gad sur G. Prendre garde que l'action de r E Gad(k) sur G(k) n'est pas necessairement un automorphisme interieur de G(k) (la projection de G(k) dans Gad(k) nest pas toujours surjective). Un exemple typique est l'action de PGL(n, k) sur SL(n, k).
De meme, l'application "commutateur" (x, y) = xyxly 1: G x G -' G est invariante par Z x Z agissant par translation, et se factorise par une application "commutateur" ( , ) : Gad x Gad -+ G.
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VARIETES DE SHIMURA
Tout ceci, et le fait que ces "commutateurs" et "automorphismes interieurs" verifient les identites usuelles se voit au mieux par descente, i.e. en interpretant G comme un faisceau en groupes sur un site convenable, et Gad comme le quotient de ce faisceau en groupes par son centre. En caracteristique 0, si on s'interesse seulement aux points de G sur des extensions de k, it suffit d'utiliser la descente galoisienne-cf. 2.4.1, 2.4.2. Variante. Pour G reductif sur k, les groupes G et G ont meme groupe adjoint, et les constructions precedentes pour G et G sont compatibles. En particulier, l'application commutateur ( , ) : G x G -> G a une factorisation canonique (
,
):GxG
Gad x Gad
-_, G
G.
On en deduit que le quotient de G(k) par le sous-groupe distingue pG(k) est abelien. 2.0.3. Soient k un corps global de caracteristique 0, A 1'anneau de ses adeles, G
un groupe semi-simple sur k et N = Ker(p: G -> G). Soient S un ensemble fini de places de k, AS 1'anneau des S-adeles (produit restreint etendu aux v S) et posons f's = pG(As) n G(k) (intersection dans G(AS)). C'est le groupe des elements de G(k) qui, en toute place v S, peuvent se relever dans G(kv) (se rappeler que p: G(A) -> G(A) est propre). La suite exacte longue de cohomologie identifie G(k)lpG(k) a un sous-groupe de Hl(Gal(klk), N(k)), et 1's1pC(k) aux elements localement nul, en les places v S,
de ce sous-groupe. En particulier, 1'slpC(k) est contenu dans le sous-groupe H1(Gal(k/k), N(k)) des classes dont la restriction a tout sous-groupe monogene est triviale (argument et notations de [12]). Si Im Gal(k/k) est l'image de Galois dans Aut N(k), on a H1(Gal(klk), N(k)) = H1(Im Gal(klk), N(k)) (loc. cit.); en particulier, Ps/pG(k) est fini. PROPOSITION 2.0.4. (i) Ts ne depend que de l'ensemble des places v e S of le groupe de decomposition Dv c Im Gal(k/k) est non cyclique. En particulier, it ne change pas si on ajoute a S les places a 1'infmi.
(ii) Ps/pG(k) s'identifie au sous-groupe du groupe fini H1(Im Gal(k/k), N(k)) forme des classes de restriction nulle a tout sous-groupe de decomposition D, v S. En particulier, pour S grand, on a rs/pG(k) = H1(Im Gal(k/k), N(k)).
La restriction d'un element de H1(Im Gal(k/k), N(k)) a un groupe de decomposition cyclique est automatiquement nulle, d'oii (i). Pour (ii), on peut supposer que S contient les places a l'infini. Le principe de Hasse pour G (pour des classes venant du centre) assure alors que tous les elements du groupe (ii) sont effectivement realises comme classe d'obstruction. COROLLAIRE 2.0.5. Tout sous-groupe de S-congruence assez petit de G(k) est dans
rs. Si U est un sous-groupe de S-congruence, U/Un pG(k) est fini: l'obstruction a relever dans G(k) meurt dans une extension galoisienne de degre et de ramification bornees, donc est dans H'(Gal, N(k)) pour Gal un quotient fini de Gal(k/k). Des
conditions de S-congruence permettent alors de passer de ce HI a Ps/pG(k), cf. [12].
REMARQUE 2.0.6. On notera par ailleurs que si 6 verifie le theoreme d'approxi-
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mation forte rel. S, tout sous-groupe de S-congruence U c rs de G(k) s'envoie sur rs/pG(k). COROLLAIRE 2.0.7. Pour toute place archimedienne v, un sous-groupe de S-congruence assez petit U de G(k) est daps la composante connexe topologique G(kv)+ de G(kv).
Puisque G(kv) est connexe, on a G(kv)+ = pG(kv) et U (-- rs = (2.0.4 et 2.0.5).
c G(kv)+
COROLLAIRE 2.0.8. Le sous-groupe G(k)pG(As) de G(AS) est ferme, topologiquement isomorphe a pG(As)*rs G(k) (i.e. pG(As) en est un sous-groupe ouvert).
C'est un sous-groupe parce que, vu 2.0.2, pG(As) est distingue dans G(As), avec
un quotient commutatif. Soit T S assez grand pour que G(k) soit dense dans G(AT) (approximation forte). Notons kT_s le produit des kv pour v e T - S. Pour K un sous-groupe compact ouvert de G(AT), on a
G(k)pG(As) = G(k)p(G(k) G(kT-s) x p-1K) c G(k)(pG(kT-s) x K).
D'apres 2.0.5, pour K assez petit, on a dans G(AT): G(k) n K c TT, d'oir dans G(As) : G(k) n (pG(kT-S) x K) c r s c pG(As). L'intersection de G(k)pG(As) avec le sous-groupe ouvert pG(kT-s) x K est done contenue dans pG(As), et le corollaire en resulte. COROLLAIRE 2.0.9. Si G verifie le theoreme d'approximation forte rel. S,1'adherence
de G(k) dans G(As) est
2.0.10. Soit T un tore sur k et S un ensemble fini de places contenant les places
archimediennes. Soit U c T(k) le groupe de S-unites. D'apres un theoreme de Chevaelly, tout sous-groupe d'indice fini de U est un sous-groupe de congruence (voir [12] pour une demonstration elegante). 11 en resulte que si T' -+ Test une isogenie, l'image d'un sous-groupe de congruence pour T' est un sous-groupe de congruence pour T. 2.0.11. Soient G reductif sur k, p: G - Gder le revetement universel de son groupe derive, et ZO ]a composante neutre de son centre Z. Voici quelques corollaires de 2.0.10 (on suppose que 1'ensemble fini S de places contient les places archimediennes).
COROLLAIRE 2.0.12. Pour U d'indice flni dans le groupe des S-unites de Z(k) it existe un sous-groupe compact ouvert K de G(As) tel que
G(k) n (K Gder(As)) c Gder (k)
U.
Appliquons 2.0.10 a l'isogenie ZO -> G/Gder: pour K petit, un element r de G(k) dans K Gder(As) a dans (G/Gder)(k) une image petite, pour la topologie des sousgroupes de S-congruence, done peut se relever un petit element z de Z(k), et r = (7'z-') z. COROLLAIRE 2.0.13. Le produit d'un sous-groupe de congruence de Gder et dun sous-groupe d'indice fini du groupe des S-unites de Z°(k) est un sous-groupe de Scongruence de G(k).
VARIETES DE SHIMURA
265
COROLLAIRE 2.0.14. Tout sous-groupe de S-congruence assez petit de G(k) est contenu dans la composante neutre topologique G(R)+ de G(R). Appliquer 2.0.13, 2.0.7 a Gder, et 2.0.10 a Zo. 2.0.15. On sait que GdeC(k)pG(A) est ouvert dans G(k)pG(A) (car image inverse de {e} c le sous-groupe discret G/Gder(k) de (G/Gder)(A)). D'apres 2.0.8, G(k)pG(A) est donc un sous-groupe ferme de G(A). On pose (2.0.15.1)
z(G) = G(A)/G(k)pG(A).
L'existence de commutateurs 2.0.2 montre que I'action de Gad(k) sur zc(G), deduite de 1'action 2.0.2 de Gad sur G, est triviale. 2.1. Varietes de Shimura. 2.1.1. Soient Gun groupe reductif, defini sur Q, et X une classe de G(R)-conjugaison de morphismes de groupes algebriques reels de S dans G. On suppose verifies les axiomes suivant (les notations sont celles de 1.1.1 et 1.1.11): (2.1.1.1) Pour h e X, Lie(GR) est de type {(-1, 1), (0, 0), (1, -1)}.
(2.1.1.2) L'involution int h(i) est une involution de Cartan du groupe adjoint
G. Gad
(2.1.1.3) Le groupe adjoint n'admet pas de facteur G' defini sur Q sur lequel la projection de h soit triviale. L'axiome 2.1.1 assure que le morphisme wh (h e X) est a valeurs dans le centre de G, donc est independant de h. On le note wX, ou simplement w. Quelques simplifications apparaissent lorsqu'on suppose que: (2.1.1.4) Le morphisme w : Gm -+ GR est defini sur Q. (2.1.1.5) int h(i) est une involution de Cartan du groupe (G/w(Gm))R. D'apres 1.1.14(i), X admet une unique structure complexe telle que, pour toute representation V de GR, la filtration de Hodge Fh de V varie holomorphiquement
avec h. Pour cette structure complexe, les composantes connexes de X sont des domaines hermitiens symetriques. La preuve de 1.1.17 montre aussi que si l'on decompose Gad en facteurs simples, h se projette trivialement sur les facteurs compacts, et que chaque composante connexe de X est le produit d'espaces hermitiens symetriques correspondant aux facteurs non compacts. L'axiome 2.1.1.3 pent encore s'exprimer en disant que Gad (resp. G, cela revient au meme) n'a pas de facteur G' (defini sur Q) tel que G'(R) soit compact, et le theoreme d'approximation forte assure que G(Q) est dense dans G(Af). 2.1.2. Les varietes de Shimura KMC(G, X)-ou simplement KMc-sont les quotients KMc(G, X) = G(Q)\X x (G(Af)/K) pour K un sous-groupe compact ouvert de G(Af). D'apres 1.2.7, et avec les notations de 0.3, faction de G(R) sur X fait de zco(X) un espace principal homogene sous G(R)/G(R)+. Puisque G(Q) est dense dans G(R) (theoreme d'approximation reel), on a G(Q)/G(Q)+=- G(R)/G(R)+, et, si X+ est une composante connexe de X, on a KMc(G, X) = G(Q)+\X+ x (G(Af)/K)
Ce quotient est la somme disjointe, indexee par l'ensemble fini G(Q)+\G(Af)/K de doubles classes, des quotients Pg\X+ du domaine hermitien symetrique X+ par les images l g c Gad(R)+ des sous-groupes r, = gKg I (1 G(Q)+ de G(Q)+. Les r, sont des groupes arithmetiques, d'oii une structure d'espace analytique sur f'g\X+.
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L'article [2] fournit une structure naturelle de variete algebrique quasi-projective sur ces quotients, donc sur KMC(G, X). Si r, est sans torsion (tel est le cas pour K assez petit); it resulte de [3] que cette structure est unique. Plus precisement, pour tout schema reduit Z, un morphisme analytique de Z dans l g\X+ est automatiquement algebrique. 2.1.3. On a
Zo KMC = G(Q)\2ro(X) x (G(Af)/K) = G(Q)\G(A)/G(R)+ x K = G(Q)+\G(Af)/K.
Puisque G(Af)JK est discret, on peut remplacer G(Q)+ par son adherence dans G(Af). La connexite de G(R) assure que pG(Q) c G(Q)+. Par le theoreme d'approximation forte pour G, pC(Q) est dense dans pC(Af), et G(Q)+ pG(Af). Des lors, (2.1.3.1)
Zo KMC = G(Q)+pC(Af)\G(Af )/K,
= G(Af)/pG(Af)
G(Q)+ K = G(Af G
puisque pG(Af) est un sous-groupe distingue, avec un quotient abelien. De meme, posant 7coic(G) = zco'r(G)/zroG(R)+, on a
(2.1.3.2)
G(R)+ x K = i(G)/G(R)+ x K = zoir(G)/K.
Zo KMc = G(A)/pG(A)
G(Q)
En particulier,'o KMc 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 KMC forment un systeme pro-
jectif. Il est muni d'une action a droite de G(Af): un systeme d'isomorphismes g-'Kg M. Il est commode de considerer plutot le schema Mc(G, X) g: KMC -ou simplement Mme--limite projective des KMc. 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 it redonne les KMc: KMC = McIK.
Nous nous proposons de determiner Mc et sa decomposition en composantes connexes.
DEFINITION 2.1.5. Fixons une composante connexe X+ de X. La composante neutre M8 de Mc est la composante connexe qui contient 1'image de X+ x {e} c X x G(Af). DEFINITION 2.1.6. Soient Go un groupe adjoint sur Q, sans facteur Go defini sur Q tel que Go'(R) soit compact et G1 un revetement de Go. La topologie z(GI) sur G0(Q) est celle admettant pour systeme fondamental de voisinages de l'origine les images des sous-groupes de congruence de G1(Q).
Nous noterons ^ (rel. GI), ou simplement n, la completion pour cette topologie.
Soit p : Go -> G1 I'application naturelle, notons - l'adherence dans G1(Af), et posons F= pGo(A) (1 G,(Q). Puisque G0(R) est connexe, r c GI(Q)+. On a (2.0.9, 2.0.14) (2.1.6.1) (2.1.6.2)
Go(Q)A(rel. G1) = G1(Q)- *G,(Q) Go(Q) = pGo(Af) *r Go(Q), Go(Q)+A(rel. G1) = G1(Q)+ *G,(Q)+ Go(Q)+ = pGo(Af) *r Go(Q)+.
PROPOSITION 2.1.7. La composante neutre M$ est la limite projective des quotients
F\X+, pour r un sous-groupe arithmetique de Gad(Q)+, ouvert pour la topologie z(Gder)
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D'apres 2.1.2, c'est la limite des r\X0, pour r ]'image d'un sous-groupe de congruence de G(Q)+. Le Corollaire 2.0.13 permet de remplacer G par 2.1.8. La projection de G dans Gad induit un isomorphisme de X+ avec une classe de G(R)+-conjugaison de morphismes de S dans Gan et, d'apres 2.1.7, M$(G, X) ne depend que de Gad, Gder 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 GI un revetement de G. Les varietes connexes de Shimura (rel. G, G1, X+) sont les quotients 1'\X+, pour run sous-groupe arithmetique de G(Q)+, ouvert pour ]a topologie r(G1). On note M8(G, G1, X+) Gder.
leur limite projective, pour r de plus en plus petit. On notera que 1'action par transport de structure de G(Q)+ sur MO(G, G1, X+) se prolonge par continuite en une action du complete G(Q)+A (rel. G1). Avec les notations de 2.1.7, et l'identification ci-dessus de X+ avec une classe de G(R)+-conjugaison de morphismes de S dans Ga , on a M°(G, X) = MO(Gad Gder X+)
2.1.9. Soient Z le centre de G, et Z(Q)- l'adherence de Z(Q) dans Z(Af). D'apres Chevalley (2.0.10), c'est le complete de Z(Q) pour la topologie des sous-groupes d'indice fini du groupe des unites; it recoit isomorphiquement l'adherence de Z(Q) dans r0Z(R) x Z(Af). Pour K c G(Af) compact ouvert, on a Z(Q) K = Z(Q)- K (dans Z(Af)), et KMC = G(Q)\X x (G(Af )/K) =
G(Q) X Z(Q)
\X x (G(Af)/Z(Q) K)
Z(Q) x (G(Af)/Z(Q) K).
L'action de G(Q)/Z(Q) sur X x (G(Af)/Z(Q)-) est propre. Ceci permet le passage a la limite sur K: PROPOSITION 2.1.10. On a
Mc(G, X)
Z(Q)
\X x (G(Af)IZ(Q)-)
COROLLAIRE 2.1.11. Si les conditions 2.1.4 et 2.1.5 sont verifiees, on a Mc(G, X) G(Q)\X x G(Af).
Dans ce cas, Z(Q) est discret-dans Z(Af) et Z(Q)- = Z(Q). COROLLAIRE 2.1.12. L'action a droite de G(Af) se factorise par G(Af)/Z(Q)-.
2.1.13. Soient Gad(R)1 1'image de G(R) dans Gad(R), et Gad(Q)1 = Gad(Q) n Gad(R)1. L'action 2.0.2 de Gad sur G induit une action (a gauche) de Gad(Q)1 sur le systeme des KMC
int(r): KMC
rKr-IMC,
et a la limite sur Mc. Pour r e Gad(Q)+, cette action stabilise la composante neutre (donc toutes les composantes, cf. ci-apres) et y induit l'action 2.1.6. Convertissons cette action en une action a droite, notee T. Si r est l'image de
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6 e G(Q), l'action r coincide avec l'action de 0, vu comme element de G(Af) : pour u e Mc image de (x, g) e X x G(Af), u r est image de
(r-I(x), intr 1(g)) = (0-I(x), 8-1g0) - (x, g8) mod G(Q) a gauche. Au total, nous obtenons ainsi une action a droite sur Mc du groupe (2.1.13.1)
Z(Q)) *G(Q)/Z(Q) Gad(Q)1 - Z(Q)) * G(Q)+/Z(Q) Gad(Q)+.
PROPOSITION 2.1.14. L'action a droite de G(Af) sur tcoMc fait de iroMc un espace principal homogene sous son quotient abelien G(Af)/G(Q)+ = tcotc(G).
Cela resulte aussitot par passage a la limite des formules 2.1.3. 2.1.15. Puisque Gad(Q) agit trivialement sur 2r(G) (2.0.15), et que Gad(Q)+ stabilise au moins une composante connexe (2.1.13), le groupe Gad(Q)+ les stabilise toutes. Pour faction 2.1.13 du groupe (2.1.13.1) sur Mc, le stabilisateur de chaque composante connexe est done (2.1.15.1)
Z(Q)+ *G(Q)+/Z(Q)
Gad(Q)+ =
2.0.13
Gad(Q)+A(rel. Gder).
Resume 2.1.16. Le groupe G(Af)/Z(Q)- *G(Q)/Z(Q) Gad(Q)1 agit a droite sur Mc. L'ensemble profini iroMc est un espace principal homogene sous faction du quotient abelien G(Af/G(Q)+ = toir(G) de ce groupe par l'adherence de Gad(Q)+. Cette adherence est le complete de Gad(Q)+ pour la topologie des images des sous-groupes de congruence de Gder(Q) L'action de ce complete sur la composante neutre, une fois convertie en une action a gauche, est faction 2.1.8. 2.2. Modeles canoniques. 2.2.1. Soient Get X comme en 2.1.1. Pour h e X, le morphisme Ph (1.1.1, com-
plete par 1.1.11) est un morphisme sur C de groupes algebriques definis sur Q: Ph: G. -* G. Le corps dual (= reflex field) E(G, X) c C de (G, X) est le corps de definition de sa classe de conjugaison. Si X+ est une composante connexe de X, on le notera parfois E(G, X+).
Soient (G', X') et (G", X") comme en 2.1.1. Si un morphisme f : G' --> G" envoie X' dans X", on a E(G', X') = E(G", X"). 2.2.2. Soient T un tore, E un corps de nombres, et p un morphisme, defini sur E, de G,,, dans TE. Le groupe E*, vu comme groupe algebrique sur Q, est la restriction des scalaires a la Weil RE/Q(G,"). Appliquant.RE/Q 'a ,u, on obtient RE/Q(,u): E* ---> RE/QTE
On dispose aussi du morphisme norme NE/Q : RE/QTE -+ T (sur les points ra-
tionnels, c'est la norme, T(E) -* T(Q)). D'oii par composition un morphisme NE/QO RE/Q(a) : E* - T. Nous le noterons simplement NRE(u), ou meme NR(u). Si E' est une extension de E, p est encore defini sur E', et (2.2.2.1)
NRE'(u) = NRE(,u) o NEVE.
2.2.3. Soient en particulier Tun tore, h : S - TR et X = {h}. Si E c C contient E(T, X), le morphisme Ih est defini sur E, d'ou un morphisme NR(,uh) : E* -* T. Passant aux points adeliques modulo les points rationnels, on en deduit un homomorphisme du groupe des classes d'ideles C(E) de E dans T(Q)/T(A), et, par passage aux ensembles de composantes connexes, un morphisme
VARIETES DE SHIMURA
269
'roNR(uh) :7roC(E) - ro(T(Q)/T(A)) La theorie du corps de classe global identifie iroC(E) an groupe de Galois rendu abelien de E. Le groupe 2co(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 2c0T(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(Af)/K. Leur limite projective T(Af)/T(Q)-, calculee en 2.1.10 est le quotient de 2co(T(Q)\T(A)) par 2coT(R).
Nous appellerons morphisme de reciprocite le morphisme rE(T, X): Gal(Q/E)ab -> T(Af)/T(Q)- inverse du compose de l'isomorphisme de la theorie du corps de classe global (0.8), de 2rONR(uh) et de la projection de 2coT(A)/T(Q) sur T(Af)/T(Q)-. Il definit une action rE de Gal(Q/E)ab sur les KMC(T, X) : 6 H la translation a droite
par rE(T, X) (6). Le cas universel (en E) est celui oix E = E(T, X): it resulte de (2.2.2.1) que ]'action rE de Ga1((2/E) est la restriction a Gal(Q/E) c Ga1(Q/E(T, X)) de rE(T,X) 2.2.4. Soient G et X comme en 2.1.1. Un point h e X est dit special, on de type CM, si h: S --> G(R) se factorise par un tore T c G defini sur Q, On notera que si Test 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 donc compacte. Le corps E(T, {h}) ne depend que de h. C'est le corps dual E(h) de h. Nous transporterons cette terminologie aux points de KMC(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(Af), la classe de G(Q)-conjugaison de It ne depend que de x. Nous dirons que x est special si h Pest, que E(h) est le corps dual E(x) de x, et que la classe de G(Q)-conjugaison de h est le type de x. Sur ]'ensemble des points speciaux de KMC(G, X) (resp. de MC(G, X)) de type
donne, correspondant a un corps dual E, nous allons definir une action r de Gal(Q/E). Soient donc 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, 6 e Gal(Q/E), et r(6) un representant dans T(Af) de rE(T, {h})(6) e T(Af)/T(Q)-. On pose r(6)x = classe de (h, r(6)g). Le lecteur verifiera que cette classe ne depend que de x et de 6. L'action ainsi definie commute a ]'action a droite de G(Af) sur MC(G, X). 2.2.5. Un modele canonique M(G, X) de MC(G, X) est une forme sur E(G, X) de MC(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 r donne, correspondant a un corps
dual E(z), le groupe de Galois Gal((?/E(a)) 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 OO E(C,x) C- MC(G, X). Soit E c C un corps de nombres qui contient E(G, X). Un modele faiblement canonique de MC(G, X) sur E est une forme sur E de MC(G, X), muni de ]'action a droite de G(Af), qui verifie (a) et
(b*) meme condition que (b), avec Gal(Q/E(T)) remplace par Gal((?/E(,r)) n Gal(Q/E). 2.2.6. Dans [5, 5.4, 5.5], nous inspirant de methodes de Shimura, nous avons
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montre que Mc(G, X) admet au plus un modele faiblement canonique sur E (pour E(G, X) (-- E c C), et que, lorsqu'il existe, it 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, 4.21, 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 T, et S± le double demi-espace de Siegel correspondant (cf. 1.3.1). S'il existe un plongement G y 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 X) et (V, p) une
representation fidele de type {(-1, 0), (0, - l)} de G. Si int h(i) est une involution de Cartan de GR/w(G,n), it existe une forme alternee ?II sur V, telle que p induise (G, X) L+ (CSp(V), S).
Par hypothese, la representation fidele V est homogene de poids - 1. Le poids w est done defini sur Q, et on prend pour tll une forme de polarisation comme en 1.1.18(b).
COROLLAIRE 2.3.3. Soient (G, X) comme en 2.1.1, w = wh (h e X), et (V, p) une representation fidele de type {(-1, 0), (0, -1)} de G. Si le centre ZO de G se deploye sur un corps de type CM, it existe un sous-groupe G2 de G, de meme groupe derive et par lequel sefactorise X, et uneforme alternee ?II sur V, telle que p induise (G2, X) (CSp(V), S+)).
L'hypothese sur ZO revient a dire que le plus grand sous-tore compact de ZR est defini sur Q. On prend G2 engendre par le groupe derive, ce tore, et l'image de w, et on applique 2.3.2. 2.3.4. Soit (G, X) comme en 2.1.1, avec G Q-simple adjoint. L'axiome (2.1.1.2) assure que GR estforme interieure de saforme compacte. Exploitons ce fait. (a) Les composantes simples de GR sont absolument simples. Si on ecrit G comme obtenu par restriction des scalaires a la Weil: G = RF,QGS avec Gs absolument
simple sur F, cela signifie que F est totalement reel. Posons les notations: I = 1'ensemble des plongements reels de F, et, pour v e I, G = GSQF,,, R, D = diagramme de Dynkin de GUc. On a GR = [T GU, Gc = [jGUc, et le diagramme de Dynkin D de Gc est la somme disjointe des D.U. Le groupe de Galois Gal(Q/Q) agit sur D et I, de facon compatible a la projection de D sur I. (b) La conjugaison complexe agit sur D par l'involution d'opposition. Celle-ci est centrale dans Aut(D). Des lors, Gal((?/Q) agit sur D via une action fidele de Gal(KDIQ), avec KD totalement reel si l'involution d'opposition est triviale, quadratique totalement imaginaire sur un corps totalement reel sinon. 2.3.5. On a X = fl XU, pour XU une classe de GU(R)-conjugaison de morphismes
de S dans G. Pour G compact, X est trivial. Pour G noncompact, XU est decrit par un sommet sU du diagramme de Dynkin DU de GUc (1.2.6).
Quelques notations: I. = 1'ensemble des v e I tell que GU soit compact, I- Ic, DU (resp. la reunion des DU pour v e IU (resp. v e GU (resp. le produit des GU pour v e IU (resp. v e IU); de meme pour les revetements universels; enfin, f(X) = 1'ensemble des sU pour v E La definition 2.2.1 donne:
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VARIETES DE SHIMURA
PROPOSITION 2.3.6. Le corps dual de (G, X) est le sous-corps de KD fixe par le sous-groupe de Gal(KD/Q) qui stabilise .(X).
2.3.7. Supposons qu'il existe un diagramme (2.3.7.1)
(G, X)
(G1, X1) c --' (CSp(V), Si).
Le revetement universel G de G se releve dans G1, ce qui permet de restreindre la 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
(G., X) .- (Ker(G1R - G,)°, X1)
(CSp(V), S±).
On trouve que les composantes irreductibles non triviales de la representation Vc
de Gc se factorisent par l'un des
(v e IJ, et que leur poids dominant est
fondamental, de l'un des types permis par la Table 1.3.9. L'ensemble des poids dominants des composantes irreductibles de la representation Vc de cc est stable par Gal(Q/Q). Puisque Gal(Q/Q) agit transitivement sur I, et que 0 0, on trouve que (a) Toute composante irreductible W de Vc est de la forme (9,1T W,,, avec WV une representation fondamentale de (v e T (-- I), correspondant a un sommet v(v) de D. Nous noterons.( V) 1'ensemble des z(T) c D pour W c Vc irreductible.
(b) Si S e 5P( V), S n D, est vide ou reduit a un seul point ss E DV (v e dans la Table 1.3.9 pour (D,,, ss est l'un des sommets soulignes.
et,
(c) . est stable par Gal(Q/Q). On a . {01. Si un ensemble de parties Y de D verifie (b) et (c), nous noterons G(,)c le quotient de Gc qui agit fidelement dans la representation correspondante de Gc. La condition (c) assure qu'il est defini sur Q. Le cas le plus interessant est celui oil (d) 5o est forme de parties a un element. Si . verifie (b), (c), 1'ensemble Y' des {s} pour s e S E . verifie (b), (c), (d), et G(9°') domine G(So).
Dans la table ci-dessous-deduite de 1.3.9-nous donnons La liste des cas oil it existe Y 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 5o verifiant (b), (c), (d) maximal, et le groupe G(.9') correspondant (il domine tous les 6(.9'), pour 9 verifiant (b), (c)). TABLE 2.3.8.
Types A, B, C. Le seul . verifiant (b), (c), (d) est 1'ensemble des {s}, pour s une
extremite-correspondant a une racine courte pour les types B, C-d'un diagramme D,, (v c= I). Le revetement G(am) est le revetement universel.
Type D1 (1 >_ 5). Pour qu'il existe . verifiant (b), (c), it faut et it suffit que les
(GV, X) (v e Ij soient ou bien tous de type DR, ou bien tous de type D. Distinguons ces cas Sous-cas DR. Le Y verifiant (b), (c), (d) maximal est 1'ensemble des {s} pour s a 1'extremite "droite" d'un DV. Le revetement 6(.9') est le revetement universel.
Sous-cas Dg. L'unique Y verifiant (b), (c), (d) est 1'ensemble des {s} pour s
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1'extremite "gauche" d'un D. Le revetement 6(,9') de G est de la forme RF/QG*, pour G* le revetement double de G forme de SO(21), cf. 1.3.10. Type D4. Remplacons .9', verifiant (d), par S = {s I {s} e ,9'}. La condition (b)
sur ,9' devient: S est contenu dans 1'ensemble E des extremites de D, et s n I(X) = 0. Pour la definition de ?(X) voir 2.3.5. La partie de E stable par GaI(Q/Q) maximale pour cette propriete est le complement de Gal(Q/Q) 2'(X). Elle rencontre chaque Dv en 0, 1 ou 2 points. Dans le premier cas, it n'existe pas .9' verifiant (b), (c). Dans le second (resp. 3e), elle (resp. son complement) est l'image d'une section r de X -* I, invariante par Galois ; z(I) est disjoint de (resp. contient).'(X). Appelant r(v) le sommet "gauche" de Dv, on retrouve la situation de D1 (1 < 5) : Sous-cas DR. II existe une section r de X -> I avec r(I) D ?'(X). Cette section est alors unique, et la situation est la meme qu'en DR, 1 5. Sous-cas Dn. On donne une section r de X -> I avec r(I) n 0. Si on n'est pas dans le cas D4, cette section est unique, et l'unique .' verifiant (b), (c), (d) est 1'ensemble des {s} pour s 1'extremite "gauche" d'un D. Le revetement 6(.9') de G est de la forme RF/QGS, pour GS un revetement double de GS qui se decrit en terme de r. Pour la suite de cc travail, it nous sera commode de redefinir le cas Dg comme excluant D. Avec cette terminologie, it existe .9' verifiant (b), (c) si et seulement si (G, X) est de l'un des types A, B, C, DR, D' et, sauf pour le type Dh', it existe .' verifiant (b), (c), (d) tel que 6(.9') soft le revetement universel de G. 2.3.9. Nous aurons a considerer des extensions quadratiques totalement imaginaires K de F, munies d'un ensemble T de plongements complexes: un au-dessus de chaque plongement reel v e I,. Un tel T definit une structure de Hodge hT : S --+ KR sur K(considers comme un espace vectoriel rationnel, et sur lequel K* agit par multiplication): si J est 1'ensemble des plongements complexes de K, on a K O C = CJ, et on definit hT en exigeant que le facteur d'indice u e J soit de type
i(X)
(-1, 0) pour a e T, (0, - 1) pour 6 e T, et (0, 0) si a est au-dessus de
Le
resultat principal de ce numero est la PROPOSITION 2.3.10. Soit (G, X) comme en 2.1.1, avec G Q-simple adjoint, et de Pun des types A, B, C, DR, DA. Pour toute extension quadratique totalement imaginaire K de F, munie de T comme en 2.3.9, it existe un diagramme
(G, X) i- (G1, X1) I--' (CSp(V), S±) pour lequel (i) E(G1, XI) est le compose de E(G, X) et de E(K*, hT).
(ii) Le groupe derive G1' 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} I 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(Ks, C), pour Ks un produit convenable d'extension de Q, isomorphes a des sous-corps de KD puisque Gal(Q/KD) agit trivialement sur D, donc sur S. En particulier, Ks est un produit de corps totalement reels ou CM. A la projection S -+ I correspond un morphisme
F
Ks.
Pour s e S, soit V(s) la representation complexe de Gc de poids dominant le
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poids fondamental correspondant a s. La classe d'isomorphie de la representation QQ V(s) est definie sur Q. Ceci ne suffit pas ace qu'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 donc V une representation de G sur Q, avec Vc - (D V(s)n, pour n convenable. Nous noterons VS l'unique facteur de Vc isomorphe a V(s)n. Ces facteurs sont permutes par Gal(Q/Q) de facon com-
patible a faction de Gal(Q/Q) sur S, et la decomposition Vc - D VS correspond donc a une structure de Ks-module sur V: sur V5, KS agit par multiplication par l'homomorphisme correspondant de KS dans C. Notons G' le quotient de G' qui agit fidslement sur V. C'est le revetement de G considers en (ii). Soit h e X, et relevons h en un morphisme fractionnaire (1.3.4) de S dans GR. On en deduit une structure de Hodge fractionnaire sur V, de poids 0. Soit s e S, et v son image dans I. Le type de la decomposition de VS est donne par la Table 1.3.9:
(a) si v e I,, V, est de type (0, 0);
(b) si v e In,, V. est de type {(r, -r), (r - 1, 1 - r)} oil r est donne par 1.3.9: c'est le nombre qui affecte le sommet s de D,,, muni du sommet special qui definit Xv.
On definit une structure de Hodge h2 de V, en gardant VS de type (0, 0) pour v e I,,, et, pour v e In,, en renommant la partie de type (r, -r) (resp. (r - 1, 1 - r)) de V. comme etant de type (0, - 1) (resp. (- 1, 0)). Si G2 est le sous-groupe algebrique de GL(V) engendre par G' et K,*, la structure de Hodge h2 est un morphisme S -+ G2R. Notant X2 sa classe de G2(R)-conjugaison, on dispose de (G2, X2) -> (G, X), et E(G2, X2) = E(G, X).
Munissons K OF V de la structure de Hodge h produit tensoriel de celle de V et de celle de K (2.3.9). On a (K OF V) 0 R = OO F,v R) OR (V OF,, R). Cette decomposition est compatible a la structure de Hodge, et sur le facteur cor-
respondant a v e I, (resp, v e Is), la structure de Hodge est le produit tensoriel d'une structure de type {(- 1, 0), (0, -1)} sur K OF,v R (resp. V OF,, R) par une de type {(0, 0)} sur V OF,, R (resp. K OF,, R). Au total, h3 est de type {(- 1, 0), (0, -1)}. Si G3 est le sous-groupe algebrique de GL(K OF V) engendre par K* et G2, la structure de Hodge h3 est un morphisme S -+ G3R
Si X3 est la classe de conjugaison de h3, on dispose de (G3, X3) -* (G, X). Le groupe derive de G3 est G', et E(G3, X3) est le compose de E(G2, X2) = E(G, X) et de E(K*, hT). Pour obtenir (GI, XI) cherche, it ne reste plus qu'a appliquer 2.3.3 a (G3, X3) et a sa representation linsaire fidele K OO F V.
REMARQUE 2.3.11. La construction donnee se generalise pour fournir un diagramme (2.3.7.1) ou .(V) est n'importe quel ensemble de parties de D verifiant 2.3.7(b), (c). En gros : (a) si Y vsrifie 2.3.7(b), (c), on definit K5, par Hom(K5,,, C) _ <9', on construit
une representation V de G telle que .(V) =9, et ]a decomposition Vc = QQ Vs (S e 9) fournit sur V une structure de K,-module; (b) la structure de Hodge fractionnaire de Vs est de type (0, 0) pour S au-dessus
de Ic, de type {(r, -r), (r - 1, 1 - r)} avec r decrit-comme ci-dessus-par le point de S au-dessus de In, sinon;
(c) on convertit {(r, - r), (r - 1, 1 - r)} en {(0, - 1), (- 1, 0)} comme plus haut;
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(d) pour convertir le (0, 0) en {(- 1, 0), (0, -1)}, on tensorise V, sur K,, avec K,, de type CM muni d'une structure de Hodge convenable h. Par cette methode, on obtient pour (GI, X1) un groupe derive G(Y), et un corps dual compose de E(G, X) et E(KK*, h). Noter que, meme pour . verifiant (b), (c), (d), la conversion indiquee de (0, 0) est plus generale que celle de 2.3.10. REMARQUE 2.3.12. Pour les types A, avec 2,'(X) fixe par l'involution d'opposition, B, C et DR, le corps dual E(G, X) est le sous-corps (totalement reel) de KD fixe par est le sous-groupe de Gal(KD/Q) qui stabilise I,. Si I, = 0, c'est Q. Si I, (resp. reduit a un element v, c'est v(F). Les E(K, hT) sont des extensions de E(G, X). sont de REMARQUE 2.3.13. Pour ces types, et DR, les V, de 2.3.10, pour v-e type {(- Z, 2), (Z, -21)j. Ceci permet, dans 2.3.10, de remplacer G2 par le sousgroupe de GL(V) engendre par F* et G'. Si I, = 0, on peut meme le remplacer par et le critere 2.3.2 s'applique direcle sous-groupe de GL(V) engendre par Q* et tement ace groupe, d'ob (GI, X1) avec E(G1, X1) = E(G, X) (= Q hors le cas Dr). 2.4. Lois de reciprocite: preliminaires. Les constructions de ce numero nous permettrons, au numero 2.6, de calculer la
loi de reciprocite des modeles canoniques, i.e. l'action du groupe de Galois sur l'ensemble des composantes connexes geometriques.
Bien qu'elles s'expriment mieux dans le langage de la descente fppf, nous les avons exprimees dans celui de la descente galoisienne, le croyant plus familier aux non-geometres. Ceci expose a quelques redites et inconsequences, et introduit des hypotheses parasites de separabilite ou de caracteristique 0.
Soit G un groupe reductif sur un corps global k. Avec la notation de 2.0.15, notre but est de construire des morphismes canoniques des deux types suivant. a. Pour k' une extension finie (qu'on supposera separable) de k, et G' deduit de G par extension des scalaires a k', un morphisme norme (2.4.0.1)
Nk,,k: 7r(G') - 7r(G).
b. Pour Tun tore, et M une classe de conjugaison, definie sur k, de morphisms de T dans G, un morphisme (2.4.0.2)
qM : 7r(T)
7r(G).
Si m e M(k), qM sera le morphisme q,, induit par m; la difficulte est de montrer que ce morphisme ne depend pas du choix de m, et de construire qM meme si M n'a pas de representant defini sur k. Les proprietes de fonctorialite de ces morphismes seront evidentes sur leur definition.
2.4.1. Nous utiliserons systematiquement le langage des torseurs (que je prefere a celui des cocycles), et celui de la descente galoisienne, sous la forme que lui a donnee Grothendieck (cf. SGA 1, ou SGA 41/2[Arcata]).
Descente galoisienne: Soit K une extension separable finie d'un corps k. Pour construire un objet X sur k (par exemple un torseur), it suffit de construire (a) pour toute extension separable k' de k, telle qu'il existe un morphisme de k-algebre de K dans k', un objet Xk, sur k'; (b) pour k" une extension de k', un isomorphisme Xk°,k': Xk, O k" -z Xk,,; et de verifier (c) une compatibilite Zk"'k' En pratique, cela signifie que pour construire X, on peut.supposer 1'existence d'objets auxiliaires qui n'existent que sur une extension separable K de k-a charge
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de montrer le X construit ne depend pas-a isomorphisme unique pres-du choix d'un tel objet auxiliaire. REMARQUE 1. La descente galoisienne est un cas particulier de la localisation en
topologie etale; une construction comme en (a), (b), (c) ci-dessus sera souvent introduite par l'adverbe "localement". EXEMPLE 2.4.2. Expliquons le relevement canonique utilise en 2.0.2 de l'applica-
tion commutateur. L'usage de la descente galoisienne-plutot que fppf-nous oblige a supposer que la projection de G sur Gad est lisse, et a ne considerer que ( , ) : Gad(k) x Gad(k) -+ G(k), plutot que le morphisme Gad x Gad -+ G. Si x1, x2 e Gad(k), on peut, localement, ecrire xi = p(ki)zi avec z1 dans le centre de G. L'ulement zi est unique, a multiplication par un element du centre de G pres. Le commutateur de xl et x2 ne depend pas de l'arbitraire dans le choix des ti, et on pose (xl, x2) = 919291-192-1-
2.4.3. Pour G un groupe algebrique sur un corps k, un G-torseur est un schema P sur k, muni d'une action a droite de G qui en fasse un espace principal homogene.
Le G-torseur trivial Gd est G muni de l'action de G par translations a droite. On identifie les points x e P(k) aux trivialisations de P (isomorphismes ip: Gd -2. P) par p(g) = xg. Si f : G1 -+ G2 est un morphisme, et P un G1-torseur, it existe un G2-torseurf(P) muni de f : P -a f(P) verifiant f(pg) = f(p)f(g), et it est unique a isomorphisme unique pres. Nous nous interesserons a la categorie [G1 --+ G2] des G1-torseurs P munis d'une trivialisation def(P). Pour morphismes, on prend les isomorphismes de G1-torseurs, compatibles a la G2-trivialisation. On note H°(G1 - G2) le groupe des automorphismes de (Gld, e) (c'est Ker(G1(k) -+ G2(k))) et H1(G1 --+ G2) 1'ensemble (pointe par (Gld, e)) des classes d'isomorphie d'objets. Chaque x e G2(k) definit un objet [x] de [G1 -+ G2] : le G1-torseur trivial Gld, muni de la trivialisation x de f(Gid) = G2d. Quand cela ne prete pas a confusion, nous le noterons simplement x. L'ensemble des morphismes de [x] dans [y] s'identifie a {g e G1(k) If(g)x = y} : a g, associer u --+ gu : Gld -a Gld. Un objet est de
la forme [x] si et seulement si, en tant que G1-torseur, it est trivial-d'oil une suite exacte
1 - a H°(G1 --a G2) (2.4.3.1)
G1(k) -+ G2(k)
--+ H1(G1 -a G2) -a H'(G1) -+ H1(G2)
(ceci ne decrit pas l'image inverse de p e H1(G1) ; pour la decrire, it faut proceder par torsion, comme dans [13]). 2.4.4. Si f est un epimorphisme, de noyau K, it revient au meme de se dormer le G1-torseur P G2-trivialise par x e f(P)(k), ou le K-torseur f-1(x) c P: le foncteur naturel [K -+ {e}] -a [G1 -a G2] est une equivalence. Plus generalement, si g: G2 -+ H induit un epimorphisme de G1 sur H, et que Ki = Ker(Gi -+ H), le foncteur naturel est une equivalence [K1 -+ K2] --+ [G1 --+ G21-
2.4.5. Si G est commutatif, la somme s: G x G -a G est un morphisme, et on definit la somme de deux G-torseurs par P + Q = s(P x Q). Si G1 et G2 sont commutatifs, on additionne de meme les objets de [G1 -a G2], qui devient une categorie ,de Picard (strictement commutative) (SGA 4, XVIII, 1.4).
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Tout ce qui precede vaut pour des faisceaux en groupes sur un topos quelconque. 2.4.6. Si k' est une extension finie de k (le cas oft k'lk est separable nous suffit) et G' un groupe algebrique sur k', le foncteur de restriction des scalaires a la Weil Rk'ik est une equivalence de la categorie des G'-torseurs avec celle des Rk./kG'torseurs. Ceci correspond au lemme de Shapiro H1(k', G') = H'(k, Rk-IkG). Si G'
se deduit par extension des scalaires de G-commutatif-sur k, on dispose d'un morphisme trace Rk'/kG' -> G-d'o%1 un foncteur trace Trk,,k des G'-torseurs dans les G-torseurs. Plus generalement, pour G1 -+ G2 un morphisme de groupes commutatifs on trouve un foncteur additif (2.4.6.1)
Trk.ik:
[Gi -f G'] -' [G1
G2].
De tels foncteurs sont decrits avec une grande generalite dans [SGA 4, XVII, 6.3].
Pour k'lk separable, on peut donner une definition simple par descente: localement, k' est somme [k': k] copies de k, [Gi --> Gz] s'identifie a la categorie des [k' : k]uPies d'objets de [G1 -a G2], et Trk,,k a la somme.
Quand les groupes sont notes multiplicativement, on parlera plutot de foncteur norme Nk'/k
2.4.7. Soient G un groupe reductif (0.2) sur k, et p
G le revetement uni-
versel de son groupe derive. Le cas particulier de 2.4.3 qui nous importe est jG -- G]. Soient Gad le groupe adjoint de G, Z le centre de G, et Z celui de G. Le morphisme G -+ Gad est un epimorphisme, d'ou une equivalence (2.4.4)
[2 ---a Z] --a [G - G].
(2.4.7.1)
Puisque Z et 2 sont commutatifs, [Z -a Z] est une categorie de Picard strictement commutative (2.4.5). Utilisant l'equivalence (2.4.7.1), on fait de [G --a G] aussi une telle categorie. Nous allons calculer [x1] + [x2] dans [G -* G], et les donnees d'associativite et de commutativite. On suppose p: G --> Gder separable pour pouvoir
proceder par descente galoisienne. Ecrivant (localement) xj = p(gl)z1, avec z1 central, on a des isomorphismes g; : [z;] -a [x1] et 9192: [z1z2] -' [xlx2], d'oi1 un isomorphisme (2.4.7.2)
[x1] + [x2]
g1+g2
[z1] + [z2] _ [z1z2]
g'gz
[x1x
on change de decomposition: x= = p(g;)zi, avec g, = g;u1 (u1 e 2), le diagramme [Z1] + [Z2] _ [Z1Z2]
[x1l + [x2]
est commutatif. L'isomorphisme (2.4.7.2). (2.4.7.3)
[x1] + [x2] = [x1x21
est done independant des choix faits. Le lecteur verifiera facilement que, via cet
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isomorphisme, la donnee d'associativite est deduite de l'associativite du produit, et que la donnee de commutativite est (x1, x2): [x2x1] -+ [x1x2]. II verifiera aussi que si y, = p(g,)x1(i = 1, 2), la somme des g.: [x,] - [yi] est g1 + g2 = g1 intx,(g2): [xlx2] -> [Y1Y2],
oiI int designe l'action de G sur G (definie par transport de structure, ou via l'action 2.0.2 de Gad = Gad),
La categorie [G --> G] etant de Picard, 1'ensemble H1(G - G) des classes d'iso-
morphie d'objets est un groupe abelien. Les formules ci-dessus montrent que l'injection G(k)/p(C(k)) -+ H1(C -* G) est un homomorphisme. D'apres (2.4.3.1), c'est un isomorphisme si HI(C) = 0. Soient k' une extension finie de k, et notons par un ' 1'extension des scalaires a V. L'equivalence (2.4.7.1) permet de deduire de 2.4.6.1 un foncteur trace (que nous baptiserons norme) (2.4.7.4)
Nk',k: [C' --> G'] --. [C -->G].
PROPOSITION 2.4.8. Si k est un corps local ou global, le morphisme deduit de (2.4.7.4) par passage a 1'ensemble des classes d'isomorphie d'objets induit un morphisme de G(k')/pC"(k') dans G(k)/pC(k): G(k')/pC(k')
G(k)/pC(k)
(2.4.8.1)
H1(G'
,H1(C_ a G).
G')
Si H1(C) = 0, la fleche verticale droite est un isomorphisme, et l'assertion est evidente. Cette nullite vaut pour k local non archimedien. Pour k local archimedien,
le seul cas interessant est k = R, k' = C, et le diagramme commutatif Z(C)
H1(Gc-Gc)
INCli? 1
Z(R)
HI(C ---> G)
montre que (2.4.8.1) est encore defini-a valeurs dans l'image de Z(R) (et meme de sa composante neutre). Pour k global, et x E G(k')/p(C(k')), l'image de Nk,,k(x) e H1(G -a G) dans HI(C) est donc localement nulle. D'apres le principe de Hasse, elle est nulle. Nous n'utilisons ici le principe de Hasse que pour les classes de cohomologie dans l'image de H1(Z), de sorte que les facteurs E8 ne creent aucun trouble. L'image Nk.,k(x) est donc dans G(k)/pG(k), comme promis. 2.4.9. Pour k local non archimedien d'anneau des entiers V, k' non ramifie sur k, d'anneau des entiers V', et G reductif sur V, le morphisme 2.4.8 induit un morphisme de G(V')/pC(V): on le voit en repetant les arguments qui precedent sur V, la descente galoisienne etant remplacee par la localisation etale (ici, formellement identique a une descente galoisienne sur le corps residuel). On peut donc adeliser 2.4.8: pour k global, le produit restreint des morphismes 2.4.8, pour les completes de k, est un morphisme Nk,ik: G(A')/pC(A')
G(A)/pG(A).
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278
Divisant par le morphisme trace global, on obtient enfin le morphisme (2.4.0.1) Nk,ik : 2r (G') -> ir(G).
De meme que la construction du morphisme (2.4.0.1) repose sur celle du foncteur (2.4.7.1), celle de (2.4.0.2) reposera sur la
Construction 2.4.10. Soient G reductif connexe sur k, p: G -> G le revetement universe[ du groupe derive, T un tore sur k, et M une classe de conjugaison, definie sur k, de morphisme de T dans G. On definira unfoncteur additif
qM : [{e} ---> T] -> [G -, G]. Nous donnerons de la construction deux variantes. lere methode. Localement, it existe m dans M. Posons X(m) = Z m(T) c G et Y(m) = p 1X(m) = 2 (p-1 m(T))°. Les groupes X(m) et Y(m), extensions de tores par un sous-groupe central de type multiplicatif, sont commutatifs. Its donnent lieu a un diagramme
2_Y(M) I
I
Z
'(m) -T
Si g dans G conjugue m en m', it conjugue (1)m en (1)m. (on le fait agir sur G par faction de Gad = Gad). De plus, l'isomorphisme int(g) de (1)m avec (1)m, ne depend
pas du choix de g: si g centralise m, it centralise Z, m(T), Z, ainsi que (p 1(T))° (un tore isogene a un sous-tore de T), donc X(m) et Y(m). Deux m dans M etant localement conjugues, ceci permet d'identifier entre eux les diagrammes (1)m, et d'en deduire un diagramme unique
2 (1)
1
-Y 1
Z
X
T
defini sur k. On definit qx comme etant le foncteur compose
qM : [{e} --> T]
[Y
X]
2.4.4
[Z -
> Z]
[G
G].
2eme methode. On suppose p : G - Gder separable, pour proceder par descente galoisienne. Les objets de [{e} -> T] n'ayant pas d'automorphismes, un foncteur T] dans [G -> G] est simplement une loi qui a t e T(k) assigne un Gde [{e} torseur G-trivialise qM([t]). Procedons par descente galoisienne. Localement it existe m e M(k). Soit qm le foncteur [t] [m(t)]. Nous allons definir un systeme transitif d'isomorphismes entre les qm. Ceci fait, nous pourrons definir qx comme etant l'un quelconque des qm. q,,', on choisit g tel que m' = gmg 1(nouvelle applicaPour definir cm , m : q, tion de la methode de descente) et on pose cm. m([t]): [m(t)] -> [m'(t)] est (g, m(t)) e G(k). On a bien m'(t) _ (p((g, m(t)))m(t), et it reste a verifier que (g, m(t)) est independant de g. L'espace C des g qui conjuguent m en m' est connexe et reduit, en tant que torseur sous le centralisateur du tore m(T). La fonction (g, m(t)) de C dans 6 a une
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projection p((m, m(t))) = m'(t)m(t)-1 dans G constante. La fibre de p etant discrete, elle est constante. La construction est recapitulee dans le diagramme commutatif qM([t])
(qm([t]) _ [m(t)], m' =gmg1)
(2.4.10.1)
qm([t])
(g.m(t))
qm'([t])
Construisons la donnee d'additivite de qM. C'est la donnee, pour chaque t1, t2 E T(k), d'un isomorphisme gM([tlt2]) -+ gM([t1]) + gM([t2]), ces isomorphismes etant compatibles aux donnees d'associativite et de commutativite pour la somme dans
[G -+ G]. On les definit par descente: pour m E M, et m' = gmg 1, le diagramme suivant est commutatif [m(tlt2)]
2.4.7
[m(t1)] + [m(t2)] (g. m(t t ))+
(g.m(ittz))
gM([t1t2])
gM([tID + q ([t2])
(9,m(t2))
I
1
[m'(t1t2)]
2.4.7_
[m'(tI)] + [m'(t2)]
(fleches obliques 2.4.10.1), et definit l'isomorphisme cherche independamment de m. Sa commutativite exprime une certaine identite, dans G, entre commutateurs (2.4.2), et la projection de cette identite dans G resulte de ce que les fleches ecrites ont un sens. Pour la prouver, on note que localement (descente) c'est la projection
d'une identite analogue pour G x Z°-de projection vraie dans G x Z0, done vraie dans G. La compatibilite a l'associativite et a la commutativite se voit par descente, en fixant m; on utilise que le commutateur 2.4.2 est trivial sur m(T). 2.4.11. Complements. (i) La construction 2.4.10 est compatible a ]'extension des scalaires: notant par un ' l'extension des scalaires de k a une extension k' de k, on definit de facon evidente un isomorphisme de foncteurs additifs rendant commutatif le diagramme [{e} - T]
'm
I
[{e} -T]
[G - G]
I
9M
[G'- G']
(ii) On peut repeter la construction 2.4.10 sur une base quelconque; la descente galoisienne est a remplacer par la localisation etale. (iii) La construction 2.4.10 est compatible aux foncteurs normes. Pour definir l'isomorphisme de foncteurs additifs rendant commutatif le diagramme
[{e} - T]
'm
, [G' -p G] }2.4.7.4
[{e} --> T]
G]
(notations de (i), avec k'lk fini separable), et verifier ses proprietes, le plus simple
280
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est de proceder par descente : localement, k' devient une somme kl de copies de k, [G' -- G] devient [G' --> G]', la norme (trace) devient la somme, et tout est trivial.
2.4.12. Pour k local non archimedien, HI(G) = 0 et, passant aux ensembles de classes d'isomorphie d'objets, on deduit de 2.4.10 un morphisme qM: T(k) G(k)/pG(k). Pour G reductif sur l'anneau des entiers V de k, Tun tore sur Vet M sur V, it induit un morphisme de T(V) dans G(V)/pG(V) (2.4.11(ii)).
Pour k archimedien, on peut rencontrer une obstruction dans HI(G), mais elle disparait pour x dans la composante neutre (topologique) T(k)+ de T(k): par 2.4.11(ii), elle depend continument de x, et est nulle pour x = e. On a donc encore un morphisme T(k)+ G(k)/pG(k). Pour k global, prenant le produit restreint de ces morphismes pour les completes
de k, on trouve (2.4.12.1)
qM: T(A)+ - G(A)/pG(A)
Si T(k)+ = {x e T(k) I pour v reel, x est dans comme en 2.4.8, fournit (2.4.12.2)
le principe de Hasse, utilise
qM: T(k)+ --> G(k)/pG(k)
Puisque T(A)+/T(k)+-Z T(A)/T(k) (theoreme d'approximation reel pour les tores),
on obtient finalement par passage au quotient le morphisme (2.4.0.2) promis: qM : 7r(T) --> z(G).
2.5. Application: une extension canonique. 2.5.1. Soit G un groupe reductif sur Q. On suppose que G n'a pas de facteur G' (defini sur Q) tel que G'(R) soit compact. Le theoreme d'approximation forte assure des lors que G(Q) est dense dans G(Af). Le cas qui nous importe est celui d'un groupe comme en 2.1.1. Reprenons le calcul de 2.1.3. Pour K compact ouvert dans G(Af), (2.5.1.1)
zo(G(Q)\G(A)/K) = G(Q)\iro(G(R)) x (G(Af )/K) = G(Q)\G(A)/G(R)+ x K
= G(Q)+\G(Af)/K
et on peut remplacer G(Q) (re'sp. G(Q)+) par son adherence dans G(Af). Celle-ci contient pG(Af), un sous-groupe distingue a quotient abelien de G(Af), et, avec les notations de 2.0.15, (2.5.1.2)
2r0(G(Q)\G(A)/K) = 7r(G)/G(R)+ x K = 2ron(G)/K = G(Af)/G(Q)+ K.
Passant a la limite sur K, on en deduit que iro(G(Q)\G(A)) = irox(G) = G(Af)/G(Q)+Soit toic(G) le quotient G(Af)/G(Q)+ de toic(G) par 2roG(R)+ (0.3). La suite (2.5.1.3)
0 - G(Q)+/Z(Q)- -p G(Af)/Z(Q)- ---a zoz(G) -a 0
est exacte. L'action de Gad sur G induit une action de Gad(Q) sur cette suite exacte. L'existence du commutateur (2.0.2) montre que 1'action de Gad(A) sur G(A)/p6(A)
est triviale-a fortiori celle de Gad(Q) sur zo'r(G). L'application du sous-groupe G(Q)+/Z(Q) de G(Q)+/Z(Q)- dans Gad(Q)+ verifie les conditions de 2.0.1. Le groupe G(Q)+IZ(Q)- * Gad(Q)+ n'est autre que le complete de Gad(Q)+ pour la G(Q)+ Z(Q)
VARIETES DE SHIMURA
281
topologie r(Gde1) (2.1.6). Appliquant de meme la construction *Gad(Q)+ au terme *Gad( central de 2.5.1.3, on obtient finalement une extension
(2.5.1.4) 0--aGad(Q)+A(rel. Gde,)-a G(Af) *G(Q)+/Z(Q) Gad(Q)+_ror(G)--a0. Z(Q)
2.5.2. Du fait que Gad 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 adjoint 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 Had son image dans Gad. Tel etant le cas dans les applications, on suppose Had(R)
compact-donc connexe puisque Had est connexe. Cette hypothese assure que Had(Q) est discret dans Had(Af), et que H(Q) c G(Q)+. Posons Z' = Z n H. Le diagramme commutatif
G(Af) -a G(Af)/G(Q)+ = zoir(G) I
T
H(Af) --a H(Af)JH(Q)- = zoz(H) fournit un morphisme de suites exactes G(Af) *G(e)+Iz(e)
Z(Q)(2.5.2.1)
I
I
0- Had(Q)
Z (Q)
T
*x(e)iz'(e) Had(Q)_z0z(H)-->0
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 -a GE. Par passage aux ico, le morphisme compose de (2.4.0.1) et (2.4.0.2)
NEIQ qM : i (T) - z(GE) - ir(G) fournit un morphisme lro2r(T) -a 2co2r(G) -a zo'r(G). Nous noterons (G, M) son inverse, et ffE(G, M) 1'extension image inverse par r(G, M) de ]'extension (2.5.1.4):
Gder)- eg (G, M)
2rolr(T)--a0
0-'Gad(Q)+A(rel. Gder)_ G(Af) *G(Q)+iz(Q) Gad(Q)+_zor(G)_0 Z(Q)
Soient u : H --a G un morphisme comme en 2.5.2(a) (Had = Gad), 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 SE(H, N) par Ker(Gad(Q)+n(rel. Gder) -a Gad(Q)+n(rel. Hder)) avec &E(G, M).
Pour H --+ G un tore comme en 2.5.2(b), muni de m : T -- HE dans M, on trouve un morphisme d'extensions
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0 --> Gad(Q)+^(rel. Gder) ---> &E(G,M) - 7roir(T) --> 0 ...
0 ---' Had(Q)
--. 7rolr(T) -s 0
2.5.4. Soient E, G, T, M comme ci-dessus, avec G adjoint. Considerons les systemes (G1, MI, u) formes d'une extension centrale u: GI --> G de G (Gad = G), definie sur Q, et d'une classe de conjugaison de morphismes M1 de T dans G1, definie sur E, qui releve M. Sur Q, pour ml dans MI d'image in dans M, le centralisateur de ml 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 ml contient le centre de G1. On a donc MI =. M. LEMME 2.5.5. Il existe des systemes (GI, MI, u) pour lesquels Gder est un revetement
arbitrairement prescrit de G.
Il suffit de montrer qu'on peut obtenir le revetement universel G. Sur Q, pour tout m dans M, la composante neutre de l'image inverse par m du revetement a de G est un revetement z: T -> T de T. 11 ne depend pas de m, donc 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(7r), on deduit de M x Id: T - CE X T un relevement Mi : T -> (GE X T)/Ker(7r). Ce relevement est a valeurs dans un groupe GE, defini sur E, de groupe adjoint GE. Il reste a remplacer GE par un groupe defini sur Q. Ecrivons GE = GE*2EZE. L'idee est de remplacer ZE par le coproduit, sur 2E, de ses conjugues: si on pose Z = REIQ(ZE)l Ker(TrEie: REIQ(ZE) ' 2) et G1 = G *2 Z, on a GE c G1E, et Ml fournit le relevement voulu. Construction 2.5.6. A isomorphisme unique pres, 1'extension &'(G1, M1) ne depend que de Met Glen
Soient deux systemes (G', Ml') et (G', Mi), de meme groupe derive. Considerons la composante neutre G1 du produit fibre de G' et G' sur G, et al classe M1 = Mi x m M. Le diagramme d'extensions &'(G', Mi) `
.0'(GI, MI)
'(Gi, 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, definie sur E, de morphismes de T dans G. L'extension 19E (G, G', M) de 7co7r(T) par le complete G(Q)+^ (rel. G') est 1'extension °E(G1, MI), pour un quelconque systeme (G1, M1) comme en 2.5.4, tel que Gder = G'.
Pour F une extension de E, la classe M fournit, par extension des scalaires de E a F, une classe de conjugaison MF de morphismes de TF dans GF; 1'extension SF(G, G', M) correspondante est image inverse de 6'E(G, G', M) par la norme NF/E: 7ro7C(TF) -* ZOZ(T) :
0 -'
G') --' &F(G, G', MF) - ' zoz(TF) 1
>0
NFIE
0 --> G(Q)+A(rel. G') --> 6E(G, (", M) -----> 7co7c(T) --' 0
283
VARIETES DE SHIMURA
Les extensions IffE(G, G', M) se deduisent toute de -ffE(G, G, M) par passage au quotient: remplacer G(Q)+n (rel. G) par son quotient G(Q)+n (rel. 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, MI) --> (G, M) comme ci-dessus, soient Hl la composante neutre de l'image inverse de H dans GI, et ml 1'element de Ml audessus de m (2.5.4.) Prenons l'image inverse par r(HI, {mI}) du morphisme d'extensions (2.5.2.1): G(Q)+n (rel.
G') - SE(G, G', M) ...
H(Q)
zoir(T) icoir(T)
On voit comme en 2.5.6 que, a isomorphisme unique pres, ce diagramme ne depend pas du choix de (GI, MI). Comme en 2.5.7, ce diagramme, rel. un revetement G' de G, se deduit du meme diagramme, rel. G, par passage au quotient. On a aussi la meme fonctorialite en E qu'en 2.5.7. En particulier, pour m dans M, defini sur une extension Fde E, qui se factorise par H on trouve un morphisme d'extensions
0 --> G(Q)+"(rel. G') ---> IffE(G,G',M)
'r0ir(T) ---> 0 INFIE
(2.5.8.1)
0 -H(Q)
...
zoz(TF) ----> 0
Nous utiliserons ce diagramme de la facon suivante: Si IffE(G, G', M) agit sur un
ensemble V, et qu'un point x e V est fixe sous H(Q) c G(Q), it a un sens de demander qu'il soit fixe "par 1coz(TF)" i.e. par le sous-groupe image de 1'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 = la classe de conjugaison de uh, pour h e X. Elle est definie sur E. Le groupe r(T) est le groupe des classes d'ideles de E, et la theorie du corps de classe global identifie lro2r(T) a Gal(Q/E)ab. Si G' est un revetement de G, l'image inverse par le morphisme Gal(Q/E) -* Gal(Q/E)ab de 1'extension &E(G, G', M) est une extension (2.5.9.1)
0 --> Gad(Q)+A(rel. G')
gE(G, G', X)
Gal(Q/E) -+ 0.
Le cas universe] est celui oii E=E(G, X), et ou G'= 6: d'apres 2.5.7, cE(G, G', M)
est l'image inverse de Gal(Q/E) c Gal(Q/E(G, X)) dans f'E(c,x)(G, G', X), et SE(G, G', X) est un quotient de 'E(G, G, X). 2.5.10. Soit h E X+ un point special: h se factorise par H c G, un tore Mini sur Q. Puisque int h(i) est une involution de Cartan, H(R) est compact. On peut donc appliquer 2.5.8 a H et a ,uh (defini sur 1'extension E(H, h) de E(G, X)). Par image inverse, on deduit de (2.5.8.1) un morphisme d'extensions
G(Q)+' (rel. G') - SE(G, G', X) , Gal(Q/E) (2.5.10.1) T
H(Q)
T
...
T
Gal(Q/E E(H, h))
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PIERRE DELIGNE
2.6. La loi de reciprocite des modeles canoniques.
2.6.1. Soient (G, X) comme en 2.1.1 et E c C un corps de nombres qui contient
E(G, X). Supposons que Mc(G, X) admette un modele faiblement canonique ME(G, X) sur E. Le groupe de Galois Ga1(Q/E) agit alors sur 1'ensemble profini iro(Mc(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, l'action (a droite) de G(Af) fait de 'roMc(G, X) un espace principal homogene sous le quo-
tient abelien z0'G = G(Af)/G(Q)+. L'action de Galois est donc definie par un homomorphisme rG,X de Gal(Q/E) dans zoz(G), dit de reciprocite. Convention de signe: faction (a gauche) de or coincide avec l'action (a droite) de rG,X(o). Ce
morphisme se factorise par le groupe de Galois rendu abelien, identifie par la theorie du corps de classe global a 7ro2E(GmE), d'ob (2.6.1.1)
rG,X: Z01i(GmE)
roz(G)
2.6.2. Soit M la classe de conjugaison de ,uh, pour h E X. Puisque E D E(G, X),
elle est definite sur E. Composant les morphismes 2.4.0, on obtient NE/QqM: 1(GmE) --. A(GE) --> Z(G)-
Par passage an z0, on en deduit (2.6.2.1)
IoNE/QgM : r0'(GmE) , 7roir(G) --> ron(G).
THEOREME 2.6.3. Le morphisme (2.6.1.1), donnent faction de Gal(Q/E) sur 1'ensemble des composantes connexes geometriques dun modele,faiblement canonique ME(G, X) de Mc(G, X) sur E, est 1'inverse du morphisme it NE/QqM de 2.6.2.
L'idee de la demonstration est que, pour chaque type z de points speciaux (2.2.4), on connait faction d'un sous-groupe d'indice fini Gal, de Gal(Q/E) sur les points speciaux de ce type (par definition des modeles faiblement canoniques)donc sur 1'ensemble des composantes connexes puisque 1'application qui a chaque point associe sa composante connexe est compatible a faction de Galois. Que l'action de Gal, obtenue soit la restriction a Gal, de l'action definie par l'inverse de lro(NE/QqM) est verifie en 2.6.4 ci-dessous, et it reste a verifier que les Gal, engendrent Gal(Q/E).
Un type z de points speciaux est defini par h e X se factorisant par un tore c : T -> G defini sur Q. Le sous-groupe Gal, correspondant est Gal(Q/E) (1 Ga1(Q/E(T, h)) = Gal(Q/E E(T, h)). D'apres [5, 5.1], pour toute extension finie Fde E (G, X), it existe (T, h) tel que 1'extension E(T, h) de E (G, X) soit lineairement disjointe de F. Ceci est plus qu'assez pour assurer que les Gal, engendrent Gal(Q/E). 2.6.4. Soient T et h comme ci-dessus, et ,u = uh. Le morphisme ,u: Gm -+ Test defini sur E(T, h) et le morphisme voNR(uh) de 2.2.3 se deduit, par application du foncteur pro, de NE(T,h),Q o qp: i(GmE(T,h)) -' r(T). On en deduit que l'action de
Gal(Q/E) n Gal((?/E(T, h)) sur les points speciaux de type z est compatible a l'action de Gal(Q/E(T, h))ab = 1ro'Z(GmE(T,h)) sur7ro(Mc(G, X)) deduite, par applica-
tion du foncteur z0, de l'inverse de
G o NE(T,h)IQ o ql,: z(GmE(T,h)) - 7r(T) - r(G) De la fonctorialite de Net de q, it resulte que ce compose est NE(T,
h)/Q o qM :
VARIETES DE SHIMURA
" h1(GmE(T,h))
285
7r(TE(T,h)) - z(T) z(GE(T,h)) --' 7r(G)
egal a NE(G,X)/Q ° qM ° NE(T,h)/E(G,X) 7r(GmE(T,h))
7r(GmE(G,X))
9M
7r(GE(T,h)) - g(G) z(GE(G,X)) - 7r(G)
Puisque la norme NE(T,h)/E(Q,x) correspond, via la theorie du corps de classe a l'inclusion de Gal(Q/E(T, h)) dans Gal(Q/E(G, X)), on a bien 1'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 1' un groupe localement compact totalement discontinu. Nous nous interesserons a des systemes projectifs, munis d'une action a gauche de r, du type suivant. (a) Un systeme projectif, indexe par les sous-groupes compacts ouverts K de 1', de schemas SK. (b) Une action p de ]'sur ce systeme (definie par des isomorphismes pK(g) : SK- . SgKg-1).
(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 K/L, et on suppose que (K/L)\SL =.SK.
Un tel systeme est determine par sa limite projective S = lim proj SK, munie de l'action de 1': on a SK = K\S. Nous appelerons S un schema muni dune action a gauche continue de I. On definit de meme la continuite d'une action a droite par la condition S = lim proj S/K. 2.7.2. Soit 7r un ensemble profini, muni d'une action continue de I'. On suppose que 1'action est transitive, et que les orbites d'un sous-groupe compact ouvert sont ouvertes: pour e e 7r, de stabilisateur d, la bijection P14 -> 7r est un homeomorphisme.
Si r agit continument sur un schema S, muni d'une application continue equivariante dans 7r, la fibre SQ est munie d'une action continue de d: pour K compact ouvert dans r, Kn 4\SQ est la fibre en l'image de e de K\S -* K\7r, et Se est ]a limite de ces quotients. LEMME 2.7.3. Lefoncteur S -> S. est une equivalence de la categorie des schemas S, munis d'une action continue de r et d'une application continue equivariante dans 7r, avec la categorie des schemas munis d'une action continue de J.
Le foncteur inverse est le foncteur d'induction de d a r : formellement, indr(T)
est le quotient de r x T par d agissant par 6(7-, t) = (7-5-1, 8t); ceci a un sens parce que faction de d sur Test propre; pour K compact ouvert dans r, on a
286
PIERRE DELIGNE
K\Indr(T) = K\F x T divise par 4 H
rEK\a=K\n
(rKr 1 f1 4)\T.
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 continument sur Spec(F). Plus generalement, si X est un schema sur E, il agit continument (par transport de structure) sur XF = X X Spec(E) Spec(F). On a (descente galoisienne) LEMME 2.7.5. Lefoncteur X -+ XF est une equivalence de la categorie des schemas sur E avec la categorie des schemas sur F, munis dune action continue de Gal(F/E) compatible a faction de ce groupe de Galois sur F.
2.7.6. Soient E Q un corps de nombres, r un groupe localement compact totalement discontinu, r un ensemble profini, muni d'une action de I' comme en 2.7.2, sauf qu'on prend ici une action a droite, et e e ir. On se donne aussi une action a gauche de Gal(Q/E), commutant a I'action de I'. Soit Fe le stabilisateur de e. Si on convertit I'action a droite de r 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 S de Gal(Q/E) par Fe.
0 -> Fe 1
e 1
-
Gal((?/E) -- f 0 11
0 -+ F--- ,I' x Gal(Q/E)-Gal(Q/E)->0 2.7.7. Lorsque I'action de I fait de r un espace principal homogene sous un quotient abelien ir(F) de r, faction de Galois est definie par un morphisme r : Gal(Q/E) - ir(F), tel que o x = x r(oo), .9 ne depend pas de e: Fe est le noyau de la projection de Fsur ir(F), et 1'extension & est l'image inverse, par r, de 1'extension Fde 2r(F) par Fe. 2.7.8. Considerons les schemas S sur E, munis d'une action a droite continue de
F et d'une application Gal(Q/E) et F-equivariante de Sii dans'. On note Se la fibre en e. C'est un schema sur Q, muni d'une action continue (a gauche) de 1'exten-
sion C et I'action de 9' sur SQ 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. Lefoncteur S -f S, est une equivalence de categories.
Le cas qui nous interesse est celui ob les S/K sont de type fini sur E, pour K compact ouvert dans r, et oi11'application de Sd sur 'r identifie r a zo(SQ). Ces conditions correspondent a: les K\Se, 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 M°(G, G', X+) sur E consiste en (a) un modele M° de M°(G, G', X+) sur Q, i.e. un schema MQ sur Q, muni d'un isomorphisme du schema sur C qui s'en deduit par extension des scalaires avec M°(G, G', X+);
287
VARIETES DE SHIMURA
(b) une action continue de SE(G, G', X+) (2.5.9.1) sur le schema MQ°, compatible a I'action du quotient Gal(Q/E) de 19E sur Q, et telle que I'action du sous-groupe G(Q)+A (rel. G') (une action Q-1ineaire cette fois) fournisse par extension des scalaires a C I'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 M°(G, G', X+) defini par h-fixe par H(Q)-soit defini sur Q et (en tant que point ferme de MQ) fixe par l'image de 1'extension en deuxieme ligne de (2.5.10.1) (rel. ,uh).
Lorsque E = E(G, X), on parle de modele canonique (connexe). 2.7.11. Les proprietes de fonctorialite suivantes sont immediates. (a) Soient des systemes (G, G,, X+) comme en 2.7.10, en nombre fini, et E c Q un corps de nombres contenant les E(G,, X+'). Si les M3 sont des modeles faiblement canoniques, sur E, des MS(G,, G,, X+), ), leur produit est un modele faiblement canonique de M22(HG,, fl G,, fl X+) sur E. (b) Soient (G, G', X+) comme en 2.7.10, et G" un revetement de G, quotient de G'. Si MQ est un modele faiblement canonique, sur E, de MAO G, G', X+), son quotient par Ker(G(Q)+ ' (rel. G') -> G(Q)+n (rel. G")) est un modele faiblement canonique, sur E, de M$(G, G", X+). 2.7.12. Soient Gun 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 Gad sur G induit donc une action, par transport de structure, de Gad(Q)+ sur ME(G, X). Convertissons cette action en une action a droite. Combinee a I'action de G(Af), elle fournit une action a droite de G(Af)
Z(Q)-
*G(Q)/z(Q)
Gad(Q)+ - G(Af)
Z(Q)-
Gad(Q)+.
*G(Q)+iz(Q)
Apres extension des scalaires a C, c'est I'action (2.1.13). Soit it 1'ensemble profini 'ro(Mii(G, X)) = 2r°(Mc(G, X)), et e e r la composante neutre (2.1.7) rel. X+. Le foncteur 2.7.9 transforme ME(G, X), muni de la projec-
tion naturelle de MQ(G, X) dans z, en un schema M°(G, X) sur Q, muni d'une action continue de 1'extension (2.5.9.1).
PROPOSITION 2.7.13. L'equivalence de categories 2.7.9 fait se correspondre les modeles faiblement canoniques de M(G, X) sur E et les modeles faiblement canoniques de M°(Gaa Gder X+) sur E.
Dans la definition 2.2.5 des modeles faiblement canoniques, nous avons impose l'action d'un sous-groupe Gal(Q/E(z)) n Gal((?/E) de Gal((?/E) sur 1'ensemble des points speciaux de type v. Ceux-ci forment une seule orbite sous G(Af), et faction prescrite commute a I'action de G(Af). Dans la definition 2.2.5, on peut donc se contenter d'exiger que pour un point special de type v ses conjugues par Galois soient comme prescrit. En particulier, it 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 l'image de (h, e) E X x G(Af) dans Mc(G, X). On retrouve ainsi la variante [5, 3.13] de la definition: Mc(H{h}) a trivialement un modele cano-
288
PIERRE DELIGNE
nique (c'est un ensemble profini, et on prend le modele sur E (H,{h}) pour lequel
l'action de Galois sur ces points est l'action prescrite), et on impose au morphisme naturel MC(H, {h}) - MC(G, X) d'etre defini sur E E(H, {h}). Nous laissons au lecteur le soin de verifier que 1'equivalence de categorie 2.7.9 transforme cette condition de fonctorialite en celle qui definit les modeles faiblement 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 puh, pour h e X+: une classe de conjugaison de morphismes de Gm dans G. Si G1 est un groupe reductif de groupe adjoint G, un relevement Xl de X+ en une classe de GI(R)+-conjugaison de morphismes de S dans GR definit un relevement M(Xj) de M: la classe de conjugaison de ,ah, pour h e Xl (cf. 2.5.4). LEMME 2.7.15. La construction Xl -> M(XX) met en bijection les relevements de X+ et ceux de M.
La construction h -p /h est une bijection de 1'ensemble des morphismes h de S dans un groupe reel G avec 1'ensemble des morphismes ,u de G. dans Gc qui commutent a leur complexe conjugue: on a h(z) _ ,u(z),u(z). Via ce dictionnaire, le probleme devient de verifier que si ,ul : Gm -* Gc commute a ,ul. Cela resulte de la rigidite des tores: le morphisme int jil(z)(/21) coincide avec ,uI pour z = 1, et releve /21 pour toute valeur de z. 11 est donc constamment egal a pl. Ce dictionnaire permet de traduire 2.5.5 en le LEMME 2.7.16. Soient G, G' et X+ comme en 2.7.10. Il existe un groupe reductif G1,
le groupe adjoint G et de groupe derive G', et une classe de G1(R)° conjugaison X, de morphismes de S dans G qui releve X+ et telle que E(G, X+) = E(G1, Xl ). 2.7.17. Ce lemme, et 1'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 equivalence 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, it Taut et it suffit que M°(Gaa, Gael X+) en admette un. En particulier, 1'existence d'un tel modele ne depend que de (Gad, Gaer 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) M°(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, it existe une extension finie F de E dans Q, lineairement disjointe de F, et un modele faiblement canonique de M°(G, G', X+) sur F'. Alors, it existe un modele faiblement canonique de M°(G, G', X+) sur E. Le corollaire 2.7.19 et 2.3.1, 2.3.10 fournissent de nombreux modeles canoniques. THEOREME 2.7.20. Soient G un groupe Q-simple adjoint, G' un revetement de G,
VARIETES DE SHIMURA
289
et X+ une G(R)+-classe de conjugaison de morphismes de Sdans GR, verifiant (2.1.1.1), (2.1.1.2), (2.1.1.3). Dans les cas suivants, M°(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 DR et G' est le revetement universel de G. (c) (G, X) est de type Dr', et G' est !e revetement 2.3.8 de G.
Appliquant 2.7.11, 2.7.18, on en deduit le COROLLAIRE 2.7.21. Soient G un groupe reductif, X une G(R)-classe de conjugaison de morphisme de S dans GR, verifiant les conditions de 2.1.1, et X+ une composante connexe de X. Pour que M(G, X) admette un modele canonique, il suffit que, (Gad, X+) soit unproduit de systeme (G,, X,) du type considers en 2.7.20, et que le revetementGder de Gad soit un quotient du produit des revetements des G, consideres en 2.7.20. BIBLIOGRAPHIE
Pour la bibliographie des articles de Shimura consacres a ]a construction de modeles canoniques, je renvoie a [5].
1. A. Ash, D. Mumford, M. Rapoport and Y. Tai, Smooth compactifications of locally symmetric varieties, Math. Sci. Press, 1975. 2. W. Baily and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (1966), 442-528. 3. A. Bore], Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, J. Differential Geometry 6 (1972), 543-560. 4. N. Bourbaki, Groupes et algebres de Lie, Chapitres IV-VI, Hermann, 1968. 5. P. Deligne, Travaux de Shimura, Sem. Bourbaki Fevrier 71, Expose 389, Lecture Notes in Math., vol. 244, Springer-Verlag, Berlin, 1971. , Travaux de Griffiths, Sem. Bourbaki Mai 70, Expose 376, Lecture Notes in Math., 6. vol. 180, Springer-Verlag, Berlin, 1971. , La conjecture de Weil pour les surfaces K3, Invent. Math. 15 (1972), 206-226. 7. 8. S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, 1962. 9. D. Mumford, Families of abelian varieties, Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math., vol. 9, Amer. Math. Soc., Providence, R.I., 1966, pp. 347-351. 10. N. Saavedra, Categories tannakiennes, Lecture Notes in Math., vol. 265, Springer-Verlag, Berlin, 1972. 11. I. Satake, Holomorphic imbedding of symmetric domains into a Siegel space, Am. J. Math. 87 (1965), 425-461. 12. J. P. Serre, Sur les groupes de congruence des varietes abeliennes, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 3-20. , Cohomologie galoisienne, Lecture Notes in Math., vol. 5, Springer-Verlag, Berlin, 13. 1965.
SGA Sem. de Geometrie Algebrique du Bois-Marie. SGAI, SGA4 t.3 et SGA4'"z sont parus aux Lecture Notes in Math. nos. 224, 305, 569, Springer-Verlag, Berlin. INSTITUT HAUTES ETUDES SCIENTIFIQUES, BURES-SUR-YVETTE
Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 291-311
CONGRUENCE RELATIONS AND SHIMURA CURVES YASUTAKA IHARA Introduction. Nonabelian reciprocity for the one-dimensional function field K over the finite field Fq has been investigated in two directions. One is the recent beautiful work of Drinfeld (realizing a part of Langlands' philosophy), which associates to each automorphic representation of GLZ(KA) a system of 1-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 [5b] as explicit conjectures (cf. also [5a, c]). Its aim is a sort of arithmetic uniformization theory for algebraic curves X over F9 by means of discrete sub-
groups 1' of PSL2(R) x PGL2(k), where k is a p-adic field with N(p) = q. For example, if XN is the canonical modular curve of level N A 0 (mod p) over Fp2, p a prime, then the space 3(XN) of all ordinary closed scheme-theoretic points of XN can be expressed as the quotient rN\.r, where TN is the modular group of level N over Z[I/p] and W is the space of all imaginary quadratic subfields M of M2(Q)
with (M/p) = 1. This simultaneous uniformization $(XN) = rN\.' is important, because this naturally describes all the Frobenius elements in the covering system {XNIXI}.1 These were proved in [5b, Chapter 5 of Vols. 1, 2] by using several exquisite results of Deuring on the reduction of elliptic curves parametrized by points of W. A simple characterization of the system {XNIXI} was proved in [6]. The present work started with an observation that these can be proved with only the Kronecker congruence relation T(p) O Fp = ]I U I17 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 f', and that of X with some additional structures, and the clarification of this equivalence seems significant. The main purpose of this paper is to outline our theory which gives a systematic study of abstract congruence relations (or equivalently, CR-systems, §1). It mainly states that whenever there is an unramified symmetric CR-system ' over o (Definitions 1.1.1, 1.5.1), there is a satisfactory arithmetic uniformization theory for an algebraic curve X over F9 by means of a discrete subgroup r of PSL2(R) x Aut( ). AMS (MOS) subject classifications (1970). Primary 14G15, 12A90; Secondary 14H30, 10D10. 'A brief review of this uniformization is given in §7.1. © 1979, American Mathematical Society
291
YASUTAKAIHARA
292
Here, o is the ring of integers of a p-adic field k (Np = q), and is the tree associated with PGL2(k) (Main Theorems 1-111). This will then be applied to the case where X is a smooth reduction of a Shimura curve and P 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, including 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; p: the maximal ideal of o; k: the quotient field of o; Spec o = {rJ, s} (ij: the generic point, s: the closed point). If Z is an o-scheme, Z., = Z (D,, k denotes its fiber over 7) (the general fiber) and ZS = Z , FQ denotes its fiber over s (the special fiber). For any field F, F denotes its algebraic closure. If Z is a proper smooth irreducible algebraic curve over F, we write Z = Z OF 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 Ukd over k. 1. Congruence relations. 1.1. Let X be a proper smooth irreducible algebraic curve over F9. We do not assume that X is absolutely irreducible. Let II (resp. t11) be the graphs on X x Fq X
of the qth power morphisms X -> X (resp. X +- X). Consider II, III and II U t]I as closed reduced subschemes of X X Fq X. DEFINITION 1.1.1. A triple (XI, X2; T) of two-dimensional integral o-schemes is called a congruence relation w.r.t. (X, o), if (i) XI, X2 are proper smooth over o, and have X as the special fiber; (ii) T is a closed subscheme of S = XI x o X2, flat over o, such that T x s
(XXFgX)=IIUIII.
Let ,u: Xo T be the normalization of the two-dimensional integral scheme T, and put tpt = pri o It (i = 1, 2), where pr, are the projections of T to X,. The system (1.1.2)
.
' = {XI
Xa
! X2}
thus obtained will be called a CR-system w.r.t. (X, o). Since T is the image of X0 in XI x,, X2, the association (XI, X2 ; T) --> X is invertible. Among the two equivalent
notions, congruence relations and CR-systems, we shall use the latter more frequently. 1.2. Let F9, (c > 1) be the exact constant field of X. Then it is easy to see that the
exact constant rings of Xl (i = 0, 1, 2) are o,. Moreover, since T is irreducible, and since the connected components of XI X,, X2 and of X x Fq X correspond bijectively, 17 and t]I must lie on the same component of X X Fq X. This shows that
either c = 1 or c = 2.
DEFINITION 1.2.1..x"' belongs to Case 1 if c_= 1, and Case 2 if c = 2.
We denote by g the genus of X = X 0 FV Fq.
SHIMURA CURVES
293
1.3. It follows easily from the definition that Xj,, = X; po k (i = 0, 1, 2) are proper smooth irreducible algebraic curves over k. Put X=v = X,q pk k, and let gi denote the genus of X=,7 (i = 0, 1, 2). Then g1 = g2 = g. Put cp=,, = (p, ®, k (i = 1, 2). Then (pi, are finite k-morphisms of degree q + 1. Therefore, the Hurwitz formula gives (1.3.1)
go - 1 = (q + 1)(g - 1) + i o,
where 8 is the degree (over k) of the differental divisor ("Zif f erente") of (p=q.
1.4. When x runs over all the Fq2-rational points of X, y = (x, xq) runs over all the geometric points belonging to the intersection II fl tff on T5. Consider y as a point of the two-dimensional scheme T. Then the local ring OT, , may or may not be normal, depending on x. DEFINITION 1.4.1. An Fqz-rational point x of X is called a special point if y = (x, xq) is a normal point of T. The special points of X will be denoted by x1, ..., 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 p. These two components meet at above those y = (x, xq) for which x is special. Therefore, the coincidence of the Euler-Poincare characteristics of Xo, and of Xos gives (1.4.2)
go - 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 - 1)(g - 1) + 1 8.
By the Zariski connectedness theorem [4, III, 4.3.1], His always positive. 1.5. DEFINITION 1.5.1. (i) . ' is called unramified, if rpl7q and (P2p are unramified. (ii) . ' is called symmetric, if X2 = XI and IT = T, where IT is the transpose
of T; or more precisely, if there exists a pair (el, S2) of mutually inverse o-isomorphisms .91 : XI -=a X2, (ff2 : X2 = X1 which lift the identity map of X and for which (8'1 x d"2)(T) = IT. (It follows easily that such a symmetry (&1, &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 = Z., the ring of p-adic integers, we can prove, as an application of our previous work [10], the following rigidity of unramified congruence relations. THEOREM 1.6.1. Let X, and a set S of Fp2-rational points of X be given. Then there exists at most unique unramified CR-system . ' with respect to (X, Zp) which has C5 as the set of special points. Moreover, when it exists, it is symmetric.
Thus, when o = Zp, X: unramified implies T: symmetric. As for the existence,
the question is more difficult. We already know a necessary condition IS1 = (q - 1)(g - 1) for the existence of X. But this is not sufficient. The question is essentially connected with a certain differential on a formal lifting of X (see [10], [7]).
I hope to discuss the further developments of [10] in the near future.
1.7. The most well-known congruence relation is the Kronecker congruence
YASUTAKAIHARA
294
relation (X1, X2; T), where XI = X2 is the projective j-line over Zp and T is the closed subscheme of XI x zp X2 defined by the modular equation Op(j, j') = 0 of order p. This is symmetric, but not unramified. As is well known, the ordinary (resp. special) points are the singular (resp. supersingular) j-invariants of elliptic curves over Pp, with the exception of the cusp which is also ordinary. For other examples, see §6 and [9a]. 2. The first Galois theory. Let ' = {X1 wI F- X0 - wz X2} be any CR-system w.r.t. (X, o). The purpose of this section is to establish a fundamental Galois-theoretic property of the system X7) _ {Xl,,
X°ij -n?i X2J
over k. This will be basic for our further studies. 2.1. Let K1 denote the function field of X; (i = 0, 1, 2). Then each K1 is an algebraic function field of one variable with exact constant field k, (cf. §1.2). The mor-
phisms (pi (i = 1, 2) induce the inclusions K. y K0, and we have K° = KIK2,
[K°:K;]=q+1(i=1,2).
DEFINITION 2.1.1. L is the smallest Galois extension of K° such that L/K1, L/K2
are both Galois extensions. Call V, the Galois group of LIK, (i = 0, 1, 2). Note that V° = VI n V2. DEFINITION 2.1.2. Gp is the subgroup of Aut(L/k) generated by VI and V2.
The Krull topologies of V, will be extended to the topology of G+ defined by the characterization " V, are open". 2.2. Suppose for a moment that . ' is the Kronecker CR-system (see §1.7). Then (in the usual sense of notations) K1 = Qp(j(z)), K2 = Qp(j(pv)), and L is the field generated over Qp by all functions of the form j(pna + b) (n e Z, b e Z[1/p]). This shows that
Vi = PGL2(Zp),
V2 =
0 -01\
-
1
Vl
p
011,
and Gp = PGLZ (QP), where (in general) PGLZ (k) = {g e GL2(k); ordp det(g) = 0 (mod 2)}/k". So, in this case, we already know the structure of Gp relative to V1, V2. In fact, we
know the structure of the related double coset ring and, as its consequence, we know such a property of Gp that Gp is the free product of V1 and V2 with amalgamated subgroup V° [5b, Vol. 1, Chapter 2, §28], [16]. We can show that these are general phenomena attached to the congruence relations. 2.3. To be precise, return to an arbitrary CR-system . ' and consider the disjoint union (V1\Gp) Li (V2\Gp) of two left coset spaces as a point-set. Call it °. Two points V1g, V2g' (belonging to different coset spaces) are called "mates" if VIg n V2g' # 0. Since (V1: V°) = q + 1 (i = 1, 2), each point has exactly q + 1 mates. (Gp ; V1, V2) obtained from this point-set ° by conConsider the diagram necting each pair of mates by a segment. Then Gp acts on 9- by the right multiplications, and the action is effective, due to the minimality of L. ' be any CR-system and put = (Gp ; V1, V2). Then (i) is connected and acyclic; in other words, for any points A, B e °, there exists a
THEOREM 2.3.1. Let
29S
SHIMURA CURVES
unique sequence of points A == A0, Ah · · ·, A1 = B of §"Osuch that A;, A;+l are mates for 0 ~ i ~ I - I and A;-1 :;. Ai+ 1 for I ~ i ~ I - I. (We shall write I == I(A, B) and call it the length between A and B.) (ii) If A, B, A', B' e §"O are such that I(A,B)- I(A', B'), and A, A' belong to the same coset space, V1\Gt or Va\Gt, then there exists g e Gt such that A' = A', B' = B•. In particular, 9"0 is an infinite set; hence (Gt: V0) == oo. Moreover, by (ii), we obtain (V0 : {I})== co, i.e., [L: K0] ==co. This implies in particular that K1 n Kz = ke- As a formal consequence of (i), we obtain CoROLLAllY
2.3.2. Gt is the free product of V1 and V2 with amalgamated subgroup
Vo.
2.4. The picture of 9" looks like (for q == 2):
I
-0
(2.4.1)
I
"•
0-
/
I
0
-· "·/
1
I
It is the same as the tree associated with the group PGI.t(k). REMARK 2.4.2. The groups Gt and PGlJ(k) act effectively on the "same" diagram !7, and they have the same domains of transitivity "o" and "•". But an example shows that, in general (at least if we do not impose a condition such as the unramifiedness on ~). Gt can be essentially bigger than PGL{(k). 2.S. The proof of Theorem 2.3.1 is based only on the following fact. "There exist discrete valuations v1 of Kh v2 of K2, and wh w2 of K0, satisfying the following two conditions: (a) for each i == 1, 2, wh w 2 are all the distinct extensions of v; to K0 ; (b) for each i == I, 2, K; is w,-adically dense in K0 ." The local rings 8x;,x1,, Bx.u. Bx.•u define the valuations v;, W~t K'2 having these properties. 2.6. Let k L be the algebraic closure of kin L. Then Gt acts on k L, and the fixed field is ke. Therefore, (k L)/ke is a Galois extension, and if Gt denotes the L, then Aut((k L)/ke) ~ Gt/Gt canonically. kernel of the action of Gt on k The following statement is exactly what we need for our present purpose about the knowledge of the "size" of k L.
n
n
n
n
n
n
PR.OPOSITION 2.6.1. For any g, g' e Gt and I in Kf · Kf; in other words,
~
i, j
~
2, k, is algebraically closed
o: == l n (g'-1 v~"~'>1. o:. This can be proved easily by using the basic properties of the valuations v;, w; given above. 2.7. When !£is symmetric, there is an isomorphism K1 ...::::. K2 defined by the equality X2 = Xh and since 1T == T, this isomorphism extends uniquely to an involution of K0 denoted by c. In Case 2, c induces an involution of k 2 over k. By the definition of L, e can be extended to an automorphism of L, again denoted
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YASUTAKA IHARA
by e. Then Vo c is well defined, c-1Voc = V0, c2 e V0, and c-1VIc = V2, c-1V2c = V1. DEFINITION 2.7.1. When ' is symmetric, Gp is the group generated by Gn and c. Note that G. is generated by VI and c, and that (Gp: Gp) = 2. We shall identify (VI\Gp) Li (V2\G,) with VI\Gp, by VIg H V1g, V2g F-, Vlcg (g e G,+). Thus, V1g and VIg' are mates if and only ifg'g 1 e V1tVI. This group Gn acts on effectively, and
Theorem 2.3.1(ii) can be rewritten as: (ii*) If A, B, A', B' e 0 are such that l(A, B) = 1(A', B'), then there exists g e- G¢
such that A' = Ag, B' = Bg. As a corollary of Theorem 2.3.1(i) we obtain: COROLLARY 2.7.2. Gp is the free product of VI and Vo U Voc with amalgamated subgroup V0.
When . ' is the Kronecker CR-system, we have (for a suitable extension c):
G, = PGL2(Qp),
2.8. Now return to an arbitrary CR-system X. When it is unramified, K0/KI, K0/K2 are unramified extensions of algebraic function fields, and this implies that L/K0 is also unramified. DEFINITION 2.8.1..x"' is called almost unramified if almost all prime divisors of K0/k are unramified in L. By the above remark, X: unramified implies X: almost unramified. I do not know
whether the almost-unramifiedness also implies the symmetricity when 0 = Z. This definition of almost-unramifiedness singles out the class of CR-systems that are related to automorphic functions. There are examples of ' that are not almost unramified. 2.9. The arithmetic fundamental group T. Let ' be almost unramified and symmetric. Then to each ' and an embedding e : k c C (C: the complex number field) we can associate a discrete subgroup 1' of (2.9.1)
PSL2(R) x G.,
determined up to conjugacy, in the following manner. Consider the set I of all those places c of L into C U (oo) such that (i) sic extends s, and (ii) the valuation ring of c is either L itself or is a discrete valuation ring. Let Gp act on I as ec --> (ec e -1, g e G, a e L). Then 2' carries a natural G, ggc, where invariant complex structure (cf. [5b, Vol. 1, Chapter 2] or [9b]). Moreover, each connected component -yo of Iis isomorphic to the complex upper half-plane 8,5, and Gp acts transitively on the set of all connected components of 2. Choose any connected component X'0 of T. and put (2.9.2)
1' = {r e Gv r'Xo = Xo}.
Then 1' acts effectively on 10 = S,-), and hence T can be considered as a subgroup of
Aut(.h) = PSL2(R). On the other hand, T is naturally a subgroup of G,. By these two embeddings, 1' will henceforth be considered as a subgroup of PSL2(R) x G. Up to conjugacy in PSL2(R) x G,, I' does not depend on the choices of To and of the above two isomorphisms. (On the other hand, P depends essentially on s.)
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DEFINITION 2.9.3. 1' is called the arithmetic fundamental group belonging to X and e. Put
r+ = r n (PSL2(R) x Gp ),
(2.9.4)
4i = r n (PSL2(R) x Vi)
(i = 1, 0, 2).
Then the projections of 4i to PSL2(R) are fuchsian groups of the first kind, and when . ' is unramified, they are nothing but the universal covering groups of Xi.,,®x k,C.2 Therefore, 1'+ is discrete in the product group (2.9.1), and (in view of (2.9.6) below) moreover it is torsion-free when . ' 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 ' is unramified. The projection of r+ in PSL2(R) is dense, and that in Gp is dense in Gp G.
Now for r itself: In Case 2, c acts as an involution of k2/kl; hence [7 = I'+. In Case 1, the symmetry c of Ko can be extended to an element of r; call it c again. Then Joe is well defined, and we have (2.9.5)
e-14oc = Jo,
e-Idle = 42,
e2 E 4o;
e-1421
= 41.
Therefore, r is generated by r+ and c. Since Gp = Vor+, and Gp = Vof' (Case 1), we obtain immediately from (2.3.2), (2.7.2) the following COROLLARY 2.9.6. (i) r+ is the free product of 41 and 42 with amalgamated subgroup 4o. (ii) In Case 2, we have I' = I'+, and in Case 1, r is the free product of 41 and 4o U Joe with amalgamated subgroup 4o.
When X is the Kronecker CR-system, we have
4I = PSL2(Z),
f+
42
=
(0 0 -1o)
0 -11
41p
0
,
= PSL2(Z[1/p]),
and
F = {r e GL2(Z [1/p]); det r E pz}/±pz, up to conjugacy in PSL2(R) x PGL2(Qp). 3. The canonical liftings. Let . ' = {XI w'+- Xo -->R X2} be any CR-system w.r.t. (X, o). We shall study in detail the reduction mod p of some special kind of places of Llk. Roughly speaking, we consider those e whose stabilizer in Gn 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 Gp -orbits Gn of those and that of all ordinary closed points (scheme-the-
oretic points) of X Q Fq2. This is achieved by constructing a canonical lifting x --> i of each ordinary geometric point x of X to a geometric point i of Xi,, (i = 1, 2). When " is unramified and symmetric, our main results are reformulated .
in terms of T(§3.14, Main Theorem I, and §3.15). 'When c = 2, ®x is w.r.t. the embedding k c, C determined by Eo.
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YASUTAKA IHARA
3.1. Reduction of Gp -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 e -+ where (a E L, g e Aut(L/k)). For each E Pl(L/k), denote by i its restriction to K1 (i = 1, 2) considered as a point of Xi,,, and by his the unique specialization of i i on X. Since T ©o F9 = II U 1ff, we have b2s = e&'. Since Gp is generated by VI and V2, the repeated use of this fact shows that for any g e Gp and 1 i, j 2, and ,s are conjugate over F9, and that when i = j they are conjugate over F92.
When X is symmetric, and ;s are conjugate over F9 for any g e G. We shall pay attention to the following mappings induced by his (i = 1, 2): Pl(L/k) H {Points of X},
(3.1.1)i
Gp \Pl(L/k) H {F9z-conjugacy classes of points of X},
(3.1.2)i
and when X is symmetric, Ge\Pl(Llk) -+ {FQ conjugacy classes of points of X}.
(3.1.3)symm
As for (3.1.2)i (i = 1, 2) they are the transforms of each other by the involution of F92/F9.
3.2. For each e E Pl(L/k), define its (transcendental) decomposition group D£ and the inertia group I+ by (3.2.
{g e Gp ; g - g} _ {g e G, ; 9 = 0£}, I+ _ {g E Gp ; g = {g e DF ; g acts trivially on 0£/me},
D£
1)
where - is the equivalence of places, 0£ is the valuation ring of e, and me is its maximal ideal. When . ' is symmetric, we define De and Ie in the same manner, i.e., just by dropping the symbol + in (3.2.1). Obviously, I+ (resp. IC) is normal in Dt (resp. De). For each g c G,+, and g e G, when . ' is symmetric, define its degree Deg(g) by Deg(g) = Min 1(A, Ag).
(3.2.2)
Then Deg(g) is a nonnegative integer, and it is even if and only if g E G,+. Moreover, Deg(g) = 0 if and only if g is G+-conjugate to some element of V1 or V2.
Put (3.2.3)
.
IF = {g e IF ; Deg(g) = 0).
3.3. We consider the following condition [A] for e Pl(L/k): [A] IOforms a subgroup of I£ with infinite index. If [A] is satisfied for e, then also for gie (g e G,+, resp. Gn). When . " is unramified,
then L/Kg (g e Gp) are unramified, so that the usual inertia groups It n g-1 vg are trivial. Therefore, IF = {1}. Therefore, the condition [A] is equivalent with [A] (X: unramified). Ie is an infinite group.
Return to the general case of X, and define Pl(Llk; [A]) as the set of all
E
PI(L/k) satisfying [A]. It is stable under G, (resp. Gp). For each e PI(Llk; [A]), define
(resp. DegO when X: symmetric) as the minimum value of Deg(g) where depends only on g runs over all elements of IF - 10 (resp. II - IF). Then
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SHIMURA CURVES
Gn
,
and when
'
and Deg() depend only on
is symmetric, both
Gp . 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 a --> j, induces a bijection between the set of all G+-orbits in Pl(L/k, [A]) and that of all Fqz-conjugacy classes of ordinary points (see § 1.4) of X; i. e.,
red;: Gn \Pl(L/k, [A]) x. {ordinary closed points of X& Fqz}. 9
Moreover, we have 2 Deg(ejs/Fqz). (ii) When " is symmetric, red; (i = 1, 2) induce one and the same bijection between the set of all G, -orbits in Pl(L/k; [A]) and that of all FQ conjugacy classes of ordinary points of X; i.e.,
red= : Gn\Pl(L/k; [A]) - {ordinary closed points of X}.
Moreover, we have DegO = (iii) For any e Pl(L/k; [A]), IF is a normal subgroup of I+ such that IW/IO Z, and D+/I+ is canonically isomorphic to the full Galois group of OF/mE over k2 n When is symmetric, I0 is also normal in IF, I /I° Z, and DEIIe is canonically isomorphic to the full Galois group of OF/n
over k.
The proof will be outlined in the rest of §3. The main point is the construction of the inverse maps the liftings of j, to such that satisfy [A] (see §§3.103.11). But before this, we shall give some preparatory materials. 3.5. Rivers on the tree . To look more closely at the G+-orbits Gn such that (Gp ; V1, V2) is very ei5 are ordinary, the following notion of "river" on _ helpful. DEFINITION 3.5.1. A river on .i is defined when each segment has a direction in such a manner that for each A e 0 and its mates B0, B1i , Bq, precisely one of AB; (0 _< i < q), say ABo, is directed outward as ABo and the rest are all directed inward; AB,,..., ABq.
-0
0
A river on - is determined uniquely by an arbitrary infinite flow going downstream
YASUTAKAIHARA
300
on . It is clear that two such flows on determine the same river if and only if they meet somewhere in their downstreams. 3.6. Let e Pl(L/k). Since his (i = 1, 2) are mutually conjugate over FQ, 1, is
ordinary if and only if 2s is so. Call
ordinary when j, are so. Each ordinary
element e E Pl(L/k) determines a river on -, called Riv(g), in the following manner.
Let A = V1g1 and B = V2g2 be mates, so that Vig1 n V2g2 : 0. Take g e V1g1 n V2g2. Let C be the restriction of g to Ko considered as a point of So., and Cs be its unique specialization on X. Since g is ordinary, g is also ordinary. Therefore, Cs does not belong to II n tH (the intersection taken on Xos). Therefore, C, lies on just one of 17 or W. If it is I7, give the direction AB, and if it is 111, give it the other way. It is easy to see that this defines a river p = Riv(e) on . Now let [A0 -, Al --> ] be any infinite flow in this river, and take g E DF (resp. De when X: symmetric). Then g leaves Riv(e) invariant, so that the g-transform of the above flow is another flow [Ag -> Af --> . . .] of Riv(e). Therefore, they must meet
somewhere in their downstream. Therefore, Ag = A;+3 holds for some 3 E Z if i is sufficiently large, and 8 is independent of the choice of [A0 --> Al -> ]. The mapping g --> 3 = 8(g) defines a homomorphism b: D£ -> Z (resp. De -* Z), and it is easy to see that (3.6.1)
Deg(g) = Io(g)I,
In particular, IF° is the kernel of 51
g E DF (resp. DF).
(resp. SJI,, since 10 c I+ ); hence 10 is always a
normal subgroup of IF (resp. II) and we have II /I°
(0) or Z (resp.
(0)
or Z) whenever is ordinary.
3.7. The mapping X. Let i = 1, 2. A point i of Xi,, is called ordinary if its specialization his E X is so. Let (X.,7)ord denote the set of all ordinary points of Xi,,. The mapping x of Xlnd LJ XZrd into itself is defined as follows. Let e1 r. lord (resp. (resp. 2 E Xznd), and let C (resp. C') be the unique point of TO, such that 01Q 02(C') = 2) and that its specialization Cs (resp. Cs) on Xos lies on 17(resp. III). Here,
Bpi = (Pi pk k. Then x is defined by x(ei) = 02G), X(S2) = 0i(C') Thus, x maps Xlnd into XZrd, and vice versa. Generically, x is a q-to-one mapping. (An important p-adic analytic property, the rigidity of x, has been investigated by Tate, Deligne and Dwork (cf. [3]) in the case where X is the Kronecker CR-system.) 3.8. Illustration of X. Take any e r= Pl(L/k) which is ordinary, and put (3.8.1)
(Gp g)i = {(gS)1; g r= Gp.}
(i = 1, 2).
Then the disjoint union (Gp g)1 LJ (Gn g)2, which is a subset of points of 11, LJ X2ij, can be naturally identified with (3.8.2)
(Vi\Gu /IF) LJ (V2\Ga
Moreover, the action of x on this set corresponds to the arrows defined from Riv(e)
on 9-; "(Gp )1 LJ (Gp )2 with the action of x" = { with Riv(C-)}/It . For example, if IF = { 1 } and IF - Z, then this quotient looks like (for q = 2).
SHIMURA CURVES
(3.8.3)
.
301
.. .
.
The length of the central cycle is equal to 3.9. The schemes T?1(p'). For each i, j (1 < i, j < 2) and 1 ? 0 with
i-j-1
(3.9.1)
(mod 2),
a closed subscheme Ti1(p') of Xi ®, X; is defined in the following way. Let A1, A2 be the points of 0 corresponding to VI, V2i respectively. Fix i, j and I satisfying (3.9.1), and let B be any point of 0 such that l(A1, B) = 1. Then l(A1, B) is even, so that B = Ag with some g e G,+. Note that the double coset V1gVi is independent of
the choice of B. Consider the ring homomorphism Ki Ox k K; -- Ki Kg c L defined by E Auz O vz -+ E 1u2vAg. Then the kernel defines a closed integral subscheme of Xi x o X1 depending only on i, j,1. Call it T11(p'). It is easy to check that T1i(p') _ tTi1(p'). When . ' is symmetric, Ti1(p') is symmetric and depends only on 1.
PROPOSITION 3.9.2. Let Ti1(p')S be the special fiber of Ti1(p'). Then Ti1(p')3 is a closed subscheme of X x Fq X determined by the following two properties: (i) it is locally defined by a single equation, (ii) its irreducible components and their multiplicities are given by the following formula:
Ti1(p'Js = (1' + 'H') + E qk-1(q - 1) (,U1-2k + t]J1-2k) +
s(l)q(1-2) 12 (q - 1) 4
1=k<1/2
where ifr is the graph of the qrth power morphism of X, tllr is its transposed graph, 4 is the diagonal of X x F, X, and s(l) = 1 (resp. 0) according to l: even (resp. l: odd).
3.10. The canonical liftings. Let i = 1 or 2, and I > 1. For each ordinary point x of X with degree l over Fq2, we are going to define its canonical lifting i e Xi,7.
Define j = 1 or 2 by the congruence i - j = I (mod 2), and look at Proposition 3.9.2. Observe that the points of X with degree I over Fqz are in a one-to-one correspondence x - (x, x9) with those geometric points of H' n t17' not lying on any other irreducible components of T=,(p')5 than I!' or II!'. PROPOSITION 3.10.1. If x is ordinary, (x, xa) is not normal on the two-dimensional scheme Ti1(p').
This can be proved easily by using a suitable morphism from the normalization
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YASUTAKA IHARA
of T11(p') onto X0. The following lemma, which is crucial to the canonical liftings, is an elementary exercise in two-dimensional local rings: LEMMA 3.10.2. Let R be any complete discrete valuation ring, and let k (resp. ,r) denote its quotient field (resp. residue field). Put Spec R = {72, s} (s: the closed point).
Let Z be a two-dimensional integral scheme having a structure of a proper and flat R-scheme, and let z be a x-rational ordinary double point of Z, = Z On s which is not normal on Z. Then there is a unique point C on Z. which is not normal and which has z as its specialization. Moreover, C is a k-rational ordinary double point of Zq.
Since z is reduced on Z, (being an ordinary double point) and since Z is flat over R, the two-dimensional local ring Oz, z is a Cohen-Macaulay ring. This gives the existence of C, by the Serre's criterion for normality [4, IV, 5.8.6]. The uniqueness and the last assertion follow from the isomorphism ©z, z ^ R [[X, Y]]/XY for the completion of Oz, _.
Now apply Lemma 3.10.2 for R = o2t, Z = T.,(p') po 02, and z = (x, x9) to obtain the following THEOREM 3.10.3. Let i, j, land x be as at the beginning of §3.10. Then there exists a of T,;(p').v (g, E X,q, ; e 1;.,1) which is not normal unique k21-rational point is an ordinary double point of T ,t(p')n Qx k and which lifts (x, xe'). Moreover, k21
DEFINITION 3.10.4. g, is the canonical lifting of x on
3.11. The following basic properties of the canonical liftings follow from the uniqueness of (g,, .). First, since T;,(p') _ 'T,;(p'), the definition and the uniqueness of (.,, I;;) tell us immediately that is the canonical lifting of x9' on More important properties will be given in the following THEOREM 3.11.1. Let x be an ordinary point of X. For each i = 1, 2, let , be the canonical lifting of x on X,.. Put d = Deg(x/F9). Then (i) , is kd-rational, and is of degree d over k; (ii) ;q' is the canonical lifting of x9 on X,.; (iii) x(eI) (resp. x(e2)) which is the canonical lifting of xq on X2, (resp. XI,); (iv) , is the unique point of (v) when ' is symmetric, we have eI = e2 specializes to x and satisfies In the case where . ' is the Kronecker CR-system, the canonical lifting is the same as the Deuring's lifting of (the j-invariant of) a nonsupersingular elliptic curve over Fp to (the j-invariant of) such an elliptic curve over Qp that has the same endomorphism ring as the former. The above characterization (iv) follows the Dwork's characterization of the Deuring lifting given in [3]. 3.12. Fix i (1 <_ i < 2), and consider the canonical lifting x --> , of the ordinary points of X to the points of X. By Theorem 3.11.1 it induces the lifting of each ordinary closed point (x, xq2, , xq2!-2) (1 = Deg(x/F92)) of X Qx Fq Fq2 to a closed ... =q]2r-2) 2e]2, of X,n 0, k2. Moreover, since aq]2 = x2( ,), the point to the elements of Pl(Llk) belong to one and the same extensions of IF contains an element g with 5(g) = 21. From G+-orbit G . Since x21(e,) 21, and that DE /It this we can show easily that Gp E Pl(Llk; [A]), induces the full automorphism group of OE/mE over k2 fl (Oe/rn ). So, as for
Theorem 3.4.1, it remains to prove that "these Gp a exhaust G+\Pl(L/k; [A])".
SHIMURA CURVES
303
This can be checked in the following way. First, count the degree of the intersection product T;i(p21)n do on X;n x k X;,1 by using Proposition 3.9.2. Then count the number of those distinct points of T=t(p2!), d, that either belong to the G,+-transforms of canonical liftings, or do not correspond to an element of Pl(Llk; [A]), by using Riv(e). We can check that the latter number reaches the former, and this gives rise to that "Gn exhaust Gn \Pl(Llk; [A])". The details [9b] will be published shortly. 3.13. One can deduce some more conclusions from Theorem 3.11.1. For example, for each e P1(L/k; [A]), the residue field 0e/mf is abelian over (OE/me) (1 k2 (and
also over k if X: symmetric), and its norm group can be determined explicitly. Roughly speaking (i.e., disregarding ramifications), this is determined by the slope of T='10'21), at (fit, _). At this stage, we use the work [12] of Lubin-Tate. 3.14. The First Main Theorem. Now let .` ' be an unramified symmetric CR-system
w.r.t. (X, o). Fix an embedding e: k ca C, and let I' be the arithmetic fundamental group belonging to X and e (§2.9). In §§3.14-15, we shall give some reformulations of Theorem 3.4.1 and Theorem 3.11.1 in terms of P. and conAs in §2.9, take a connected component 2'o of 2', identify _Y0 with sider I' = {r e Gp; rEo = _Y'o} as a group of transformations of 8.5. For each z e 55, let Pz denote its stabilizer in r, and put (3.14.1)
Obviously, .
.Ye={'re S -) ;
I
= co}.
is a P-stable subset of .Y>. The points of .
will be called P-points on
MAIN THEOREM I. Let J3(X) denote the set of all ordinary closed points of X. Then the reduction mod p induces a bijection (3.14.2)
ir: P\.ye z
(X).
This is a direct corollary of Theorem 3.4.1(ii), because there is a canonical bijection between 1'\. and Gp\Pl(Llk; [A]), defined as follows. First, an element of 2' is algebraic over k if it has a nontrivial stabilizer in G. Therefore, YP is embedded in Pl(L/k). Moreover, for any z e . n Pl(L/k), Pz is nothing but the inertia group Iz defined in §3.2. Since the condition [A] (for X: unramified) is equivalent with JI,j = oo (§3.3), this implies that n P](L/k; [A]) _ ,V. Since G, acts transitively on the set of connected components of 2', this induces the above bijection. 3.15. Although the Main Theorem I has a simple form, it does not contain our best knowledge on the reductions and the liftings of geometric points, which will now be described in terms of 1' as follows. Let d, (i = 1, 2) be the subgroups of P corresponding to V1 (see §2.9). Then d;\. i can be identified with the set of points of Xze = XXv Qx k, C, where Qx is w.r.t. the embedding k, 4 C defined by the restriction of 2 to k,. Let x be a geometric point of X over FQ with degree d, and P = (x, x9, , x9°-') be the closed point of X containing x. Let .YPP denote the set of all
those points z e Ye such that the I'--orbit containing v corresponds to P by it ((3.14.2)).
Case 1. In this case, P contains an involution c satisfying (2.9.5), so that there is no distinction between the two choices of i. The quotient J1\..P c Xlc is illustrated by the diagram:
YASUTAKAIHARA
304
4
0
0.4-0 1/ (q = 2, d = 4)
YIN
o
(3.15.1)
t where the arrow is the x-mapping which is q-to-1, and the central cycle is of length d which consists of the canonical liftings of xq" (0 5 v < d) on X1.
Case 2. In this case, the simultaneous o2-structure for X gives mutually conjugate structures for X1 and X2. Therefore, X1c and X2c are generally nonisomorphic. The disjoint union (41\-?IfP) U (42\. 'P) is illustrated by the diagram (3.8.3), where o e41\. P, e42\.rr, the arrow is the x-mapping, and the central cycle is of length d (which is even) which consists of the canonical liftings of xqv on X,c (i is distinguished by the parity of v). Finally, in each case, P. is free cyclic and the homomorphism 5:Dz -* Z of §3.6 induces an isomorphism P. = d Z. Therefore, if rT is a generator of fz, the degree d of P over Fq is equal to Deg (rz), where r= is considered as an element of G. Moreover, we can distinguish the two generators of P. by the signs of 6(r,). This sign has also the following interpretation. Let d/dz -+ 2 d/dz be the linear transform of the tangent space of .S=) at z induced by rz. Then 2 belongs to k" (via the embedding e: k -+ C), and ord.2 has the same sign as 6(7-,).
4. The second Galois theory. The subject of second Galois theory is a system f = (fl, fo, f2) of three finite etale morphisms f= : X * -* X. connecting two CRsystems . ' = {X1 In, Xo -*wz X2} and ,1"* = {Xl P1 Xo --+,P2' Xz } in a compatible way. The purpose of §4 is to present our main results on the two categorical
equivalences induced by f --> f O C and f - f Q F. The former is highly nontrivial.
4.1. In general, let
Uo -02 U2} be a system formed of three
_ { U1 01
schemes U, (i = 0, 1, 2) and two morphisms cb;: Uo -- U. (i = 1, 2). Let 0&*
{U*
Gi
4 U*p ----
U* 1 2
}
be another such system. We say that a pair (Qi*, f) is a finite etale covering of 0ll, if f is a triple (fi, fo,f2) of finite etale morphisms ft: U* --+ U. (i = 0, 1, 2) satisfyU=* x u; Uo (canonically; i = 1, 2). ing f o 0* _ O= o fo Q = 1, 2) and Uo Now let . ' = {X1' - Xo ->P2 X2} be a CR-system w.r.t. (X, o), and X* _ {Xl '1,- Xo ->91a X2 } be another CR-system w.r.t. (X*, o), with the common base ring o. Suppose that (*,/) is a finite etale covering of .9i' and that the constituents
f (i = 0, 1, 2) of f = (fl, fo, f2) are o-morphisms. Then we shall call (X*, f ) a finite etale CR-covering of X. In this case, it follows from the definitions that the
two finite etale morphisms f.,: X* --> X obtained from J. (i = 1, 2) by the base change Qx o Fq must coincide. This morphism f,.s will be denoted by f. It also follows
easily that S* = f -'(ft where CS (resp. V) is the set of special points w.r.t.X (resp.
'*). When X is unramified, X* is also unramified, because rp*q are the base
305
SHIMURA CURVES
changes of pi.. When ' is symmetric, we can prove easily by using [4, IV 18.3.4] that X* is also symmetric. 4.2. Now let -T be any unramified symmetric CR-system w.r.t. (X, o), and fix an embedding e: k c+ C. Let F be the arithmetic fundamental group belonging to X, e. Then our main result on the second Galois theory reads as follows. MAIN THEOREM II. The following three categories (i), (ii), (iii), are canonically equivalent:
(i) Finite etale CR-coverings (X*, f) of X. (ii) Subgroups T* of F with finite indices. (iii) Finite etale coveringsf: X* -+ X, with X*: connected, such that all points of X* lying above the special points xI, , xH of X are Fqz-rational points of X*.
The equivalence functors are as follows; (i) -+ (ii) is the functor of taking the arithmetic fundamental group, and (i) -3 (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. In §5, ' is an unramified symmetric CR-system w.r.t. (X, o), s is a fixed embedding k cc C, and F is the ,
arithmetic fundamental group belonging to 21', e.
5.1. Let f'* be a subgroup of r with finite index, (.1'*, f) be the corresponding finite etale CR-covering of .1", and (X*, f) be the corresponding finite etale covering of X(§4). Let. (resp. ye*) be the set of all F-points (resp. F*-points) on S5 (§3.14).
Then Y* _ .°, because (r,: F ') < oo for any z e 5-. Let $(X*) denote the set of all ordinary closed points of X* w.r.t. X*. As we noted in §4, 3(X*) is the inverse image of $3(X) w.r.t. f. By Main Theorem I for .1'*, we have a canonical bijection (5.1.1)
ir.: r*\,YP z 3(X*)
defined by the reduction mod V. When X*/X is a Galois covering, and P* is a closed point of X*, the Frobenius
automorphism ((X*/X)/P*) is by definition the element a of the Galois group Gal(X*/X) of X*/X 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 u of the Galois group on points are from the left, and that on functions are from the right connected by fa(g) = f(o ), the geometric and the arithmetic Frobeniuses do not have inverse expressions here.) MAIN THEOREM 111. (I) The diagram
(5.1.2)
canon.
F\.°
_r
f
$(X)
is commutative; (ii) when F* is a normal subgroup of F, the natural action of 1'/F* on F*\.r and the action of Gal(X*/X) on l(X*) correspond with each other through ir. and through the canonical isomorphism F/F* = Gal(X*/X) of Main Theroem II;
YASUTAKAIHARA
306
(iii) moreover, when f'* is normal in ]', the Frobenius automorphism of P* _ ir. (F* r) (z e .e) over X is given by r* rT, where rT is the generator of 1'z such that a(rT) < 0 (§3.15); (5.1.3)
1'* \ P/X ) =
rT.
T
Thus, as a universal expression of the Frobenius automorphism, we may write in a more suggestive way as
( lX) = r=.
(5.1.4)
\ z
6. The case of Shimura curves. The above results'apply to each Shimura curve for almost all p by Theorem 6.3.3 which is the "p-canonical version" of the Shimura congruence relation. 6.1. Let F be a totally real algebraic number field of finite degree m, and B be a quaternion algebra over F which is unramified at one infinite place 001 of F and ramified at all other infinite places 002, , oo,,, of F. As usual, the subscripts I, 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 bR e BR, N(bR) is positive at 002, , co,,. Let (BA)+ denote the group of all bA e BA with N(bR): totally positive, and put (B,')+ = (BA)+ n B,. Now let 2 be the function field over F constructed in Shimura [15] for B/F (the case "n = r = 1"). It is an infinitely generated 1-dimensional extension of F, and is obtained as the total field of arithmetic automorphic functions on B. In [15] Shimura established an isomorphism (6.1.1)
Aut(f3/F) = (BA)+/(Bs)+ F',
where - denotes the topological closure. Recall that the algebraic closure of F in 2 is the maximum abelian extension Fab of F, and that the diagram
Aut(2/F)
(B A)+
(6.1.2)
I
N-1
FA
P C
Aut(Fab/F)
is commutative, where t is the homomorphism defined by (6.1.1), c is the homomorphism defined by the class field theory, p is the restriction homomorphism, and N-1(bA) = N(bA)-1. For each open compact subgroup U of Aut(.V/F), let 2u (resp. (Fab)u) be its fixed
field in 2 (resp. Fab). Then 2u is an algebraic function field of one variable over (Fab)u Let S(U) be the proper smooth curve with function field 21. 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 Fp as a (central) subgroup of (B')-l- and let P be the closure of t(Fp ). Then P is a central compact subgroup of Aut($2/F). Let k(l) (resp. k(°°)) be the decomposition field (resp. the inertia field) of p in FablF, and k(d' (d >_ 1) be the unique subextension of k(°) lk(1) with degree d. Let £P be the fixed field of Pin 2. Then 2P n k(°°) = k(2).
307
SHIMURA CURVES
Put
BB =,$-,H'p B,,
(6.2.1)
F,x = II' F, x, t#oo, p
oo is the abbreviation for I 0 oo 1, , co), and for each subgroup H of (where I BA (resp. F;1) denote by H- the topological closure in BB (resp. F,) of the projection of H in BB (resp. in FA x). Now look at the canonical isomorphism (6.2.2)
Aut(2/F)/P = (Ba)+/(Be)+
. F¢ Fx
(B1 /Fn) x (Bil(Fx)").
Then the groups of (6.2.2) act on 2P in the natural manner. Put (6.2.3)
BAx 11) = {bA E BA x; N(bA) E ((F ")+)r},
where (Fx)+ is the group of all totally positive elements of F. The isomorphism (6.2.2) induces (6.2.4)
Aut(2/ka')/P = (B." /Fu) x (BAx(l)/(Fx)_).
DEFINITION 6.2.5. For each open compact subgroup UA of B;(1)/(Fx)", the fixed field of UA in WP, with the natural action of Bp/Fp, will be called the Bp/Fp field associated with UA, and denoted by L = L(UA).
Thus, L(UA) is a one-dimensional infinitely generated extension over V on which Bp /Fp acts effectively. DEFINITION 6.2.6. For each UA, the group I' = P(UA) is the subgroup of Bx/F> formed of all r E BxlFx satisfying proo,(r) E (B,,)+/Rx,
pr4(r) e
UA,
where pr-1(resp. pro) are the projections of B x/Fx into B_1/Rx, Bn /(Fx)-, respectively.
By the Eichler-Kneser approximation theorem, r(UA) = F(U;,) implies UA = U, .
Now fix an isomorphism Bp _- M2(Fp) (over Fp), which induces B,'/Fl PGL2(Fp). Let V1, V2 be the open compact subgroups of Bp /Fp corresponding to PGL2(op), wp 1 PGL2(op)wp respectively, where op is the ring of integers of Fp and wp = (° o) (pr: a prime element of op). Put Vo = V1 n V2. Let UA be an open
compact subgroup of B;')l(Fx)-, put L = L(U4), and let Ki (i = 0, 1, 2) be the fixed fields of V2 in L (i.e., the fixed fields of V, x UA in 2P). Then each Ki is an algebraic function field of one variable over 01, and Ko = K1K2. Moreover, w, induces an involution c of Ko which is trivial on k(' and inverts K1 and K2. As for L, it is the smallest extension of Ko such that L/K1 (i = 1, 2) are both Galois extensions. Now let Xi.i (i = 0, 1, 2) be the proper smooth curves over k(1) with function fields K1, and (pi,,: X0 - X10 (i = 1, 2) be the natural morphisms. Then Xln = X2.. (via c) and tT,i = T. for the image T,, of X0,, in X1, x k(1) X2,,. The system, (6.2.7)
{XJ'
n,
2n
Xon
X20}
thus defined does not depend on the choice of the isomorphism Bp ^ M2(Fp). We shall call (6.2.7) the k11)-system associated with (p, UA).
Each extension .
of p in VD defines a -adic embedding of k(l) into F. Let
'( , UA),, denote the system of curves over Fp obtained from (6.2.7) by the base
change OF, with respect to this embedding of VD.
308
YASUTAKA IHARA
Question 6.2.8. Does there exist a symmetric CR-system '(p, U4) over o, which has X (p, U4),, as its general fiber? Shimura's work provides an affirmative answer to this question for "almost all p" in the sense described below, and it has been supplemented by Y. Morita to some results for "individual" (p, U4). But as far as the author knows, this question has
not yet been solved for all cases. Note here that by our Main Theorem II (§4) and the last statement of §4.1, when (6.2.8) is valid for (p, U4) and when (pin (i = 1, 2) are unramified, then this is also true for (p, Ui) for all open subgroups U,* of U4.
6.3. Let p be as in §6.2. An open compact subgroup U of Aut (2/F) is said to be coprime with p if U contains the image t(V,) of a maximal compact subgroup Vp of B'. In this case, V, is uniquely determined, and is called the p-component of U. Obviously, U is coprime with almost all p. Let U be the coprime with p. Then (Fab)11 (-- V°°). Put (6.3.1)
U (P' = (U (1 Aut(2/k (-1)) P,
and S(UIP') be the proper smooth curve whose function field is the fixed field of U(P' in 2. Then the exact constant field of S(U(P)) is k(2), and (6.3.2)
S(U(P)) Ox V°°> ~ S(U) (D VW)
where the tensor products are over the exact constant fields. This curve S(U(P)), obtained from S(U) by dividing S(U) OO k(°°) by the action of P, will be called the p-canonical model of S(U). If we consider U(P) as a subgroup of the group (6.2.4), then U(P) decomposes as U(P) = Vp x U4, where Vp is the image in B1 1F,1 of the p-component Vp of U, and U is an open compact subgroup of Bn (l>. We shall call U,1 the A-component of U. By definitions, S(U(P)) is k(1)-isomorphic with X1 (i = 1, 2) where {Xlv v1,)-- XOV -->" X2J is the V11-system associated with (p, U4). Now, from the Shimura's congruence relation [15, (I) Theorem 2.23], we obtain just by passing to the p-canonical models, the following THEOREM 6.3.3 (SHIMURA). Let U be any open compact subgroup of Aut(2/F) which is an image of an open compact subgroup of (BA)+. For each finite place p of F at which B is unramified and such that U is coprime with p, let U4 denote the Acomponent of U. Then Question 6.2.8 has an affirmative answer (for all extensions p of p) for almost all p.
In his Master's thesis [13], Y. Morita proved among others the following: THEOREM 6.3.4 (MORITA). When F = Q, Question 6.2.8 has an affirmative answer for all p at which B is unramified and with which U is coprime.
6.4. Let (p, U4) be such that Question 6.2.8 has an affirmative answer. We shall
see how the objects L, G, F,, etc. are given explicitly for ' _ ,(p, U4). Note first that the base ring o = o. is the ring of integers of F. (i) The field L and the group Gp (§2). The field L for this case is given by L(U,1) PGL2(FF). Ok« FF, and the group Gp is nothing but Bp /Fp (ii) The ramifications and (AF, F) (§§2, 3). It is clear that X is almost unramified.
Moreover, if B qo M2(Q) and U4 is sufficiently small, then , ' is unramified. Let e: Fp c-> C be an extension of ooI. Then the group r and its two embeddings are
SHIMURA CURVES
309
given by I' = 1'(U,,), pr-,, prp, respectively. To give ," explicitly, let J4(B) denote the set of all such quadratic extensions M of F contained in B that M is totally imaginary and that ((M/F)/p) = 1. Put (Bx)+ = Bx (1 (BA)+. Then (Bx)+ acts on 5 through the localization at ool, and for each M e J4(B) there is a unique common fixed point zM of Mx on 5i. Now YP is given by (6.4.1)
. Ile = {VM; Me J,(B)}.
Since ZM 0 VM, for M 0 M', ° can be identified with J4(B). The corresponding action of P on J4(B) is then given by M -* rMr-1. (iii) The second Galois theory. The open subgroups U,* of UA correspond bijectively with the congruence subgroups T* of 1'(U,,). Our theory Q§2-5) is valid for any subgroups of 1'(U,,) with finite indices. We do not know whether all such subgroups r* are congruence subgroups, except for the case of B = M2(Q) where it was proved to be valid by Mennicke and Serre. (iv) The reciprocity. For each M e J4(B), the group 1'T for z = VM is given by
r (1 (Mx/Fx). Let P = PIP2 be the decomposition of p in M. Then the degree Deg(z) is the smallest positive integer d such that pd is principal; pd = (a). When ' is unramified, 1'T is free cyclic and is generated by the class of a (mod Fx). The .
two generators of fT are distinguished by the choice of pl. Choose pl to be the restric-
tion of [i to M and let r,, be the generator of I',. represented by a (where pd = (a)). Then this is the generator describing the reciprocity of Main Theorem III. Thus, we have described explicitly the objects of Main Theorems I, II, III for the case of Shimura curves. A numerical example is given in [9a].
7. Comments and remarks. 7.1. Brief review of the elliptic modular case (cf. [5b, Chapter 5 in Vols. 1, 2]; [6]). Let X = Spec Z[j ] be the affine j-line over Z, and put Xc = X Q C. Let be
the complex upper half-plane, J = PSL2(Z), and identify J\5l with XX via the modular j-function.3 Let 97t be the set of all imaginary quadratic subfields of M2(Q).
For each Me 9N, denote by rm the common fixed point of elements of Mx in Si, and put jM = j(rM) which is a point of Xc. Then jM is determined by the J-conjugacy class of Min 9]2, and is an algebraic integer.
Fix a prime number p, and its extension p in the algebraic closure Q. Put X = X Qx Fp. A geometric point of X will be called special (resp. ordinary), if it is the j-invariant of a supersingular (resp. singular) elliptic curve in the Deuring's sense [1]. As is well known, the special points are Fpz-rational. For each M e 99R, denote by jM the geometric point of X obtained by the reduction mod b of jM. Then: (I) 2 3 M -f jM e X OFp is surjective. (II) If (M/p) 0 1, then jM is special. (III) If (M/p) = 1, then jM is ordinary; moreover, when this is so, jM' (M' E 1) is F, ,-conjugate with jM if and only if M' and M are I'-conjugate in M2(Q), where (7.1.1)
1' = IT E GL2(Z[1/p]); det (r) e pz}/±pz.
'The constant multiple of j is normalized in such a way that j(,/-1) = 12'.
YASUTAKAIHARA
310
Therefore, if Y denotes the set of all ZM with M E 9R and (M/p) = 1, then the reduction mod p induces a bijection (7.1.2)
ir: r\.ye x {Fp-conjugacy classes of ordinary geometric points of X}.
Moreover, for each Fp conjugacy class P = (j, jP, , jp°-') of ordinary geometric points of X, the collection d\.*P of all points jM such that jM e P can be illustrated by the diagram (3.15.1), where the arrow is the unique p-adic lifting of the pth power map satisfying E T(p) p Q, T(p) being the Hecke correspondence
c X x z X defined by the double coset J(" ,)J. The mapping x is generically p-to-1, the central cycle is of length d, and the distance k of fM from the central cycle is the p-exponent of the conductor of M2(Z) n M. Each ordinary geometric point j e X has a unique lifting in the central cycle, called the canonical lifting (or Deuring representative) of j. These are obtained by the modular reconsideration of the Deuring's results on the reduction of elliptic curves [2]. (IV) Subgroups 1'* of Pwith finite indices are categorically equivalent with those finite coverings X* -> X over Fp2 satisfying (a) the Igusa's ramification properties,
and (b) all points of X* lying above the special points of X are Fp2-rational (cf. [6])
(V) The reciprocity in this system of coverings of X is described as follows. Let P be as above, and a = rm E YP. Let PT be the stabilizer of v in I. Then, modulo torsion (which occurs when j = 0 or 123), PT is isomorphic to Z and has a generator r= represented (modulo pz) by an element of M" which generates a positive power of p e M. The I'-conjugacy class {rz}r is well-defined by P (modulo torsion), and {rz}r- is the Frobenius conjugacy class for P in this system of coverings. Here, r is the projective limit of all finite factor groups of r [5b, Vol. 2, Chapter 5]. By (IV), (V), the basic bijection (7.1.2) holds also between r* and X*. 7.2. Remarks on its relations with other works. The chief datum defining a Shimura variety (of GL(2)-type) is a quaternion algebra B over a totally real number field F. The dimension of the variety is the number of distinct infinite places of F at which B is split. In the Shimura theory, the totally indefinite case and the onedimensional case are the two extremes. They coincide only when F = Q. (A) Morita's contribution on the Shimura curves over Fp for the case F = Q is briefly mentioned in [5c].
(B) Langlands' conjecture on the explicit description of the points of Shimura varieties over finite fields, together with that of the Frobenius action, is (essentially) a higher dimensional generalization of our conjectures and results given in [5b]. Langlands discussed extensively, obtaining definitive results, the case where B is totally indefinite [11]. When F = Q, his results essentially coincide with the Morita's solution of our conjecture.4 When F 0 Q, the object varieties are different. (C) With Drinfeld. Let X be a Shimura curve over F9, P be its arithmetic funda-
mental group, and consider the system of coverings X* -> X corresponding to congruence subgroups P c T. Then the Galois group of this system {X*/X) is an adelic compact group (without infinite and li factors), and T is the "maximal global
subgroup" of this adelic Galois group. Recall (Main Theorem III) that the Frobeniuses in {X*/X} are represented by the global element rz e T. 'There is however a difference about the aspect of supersingular moduli; mainly because of the (current) difference in the standpoint.
SHIMURA CURVES
311
Now let F = Q. Then this adelic presentation of the Galois group gives a system of 1-adic representations (not always in GL2(Qt) but in GL2(Q1) when 1 JD(B)). It is very plausible that there exists an automorphic representation of GL2(KA) (K = FQ(X)) which corresponds with this system via the Drinfeld theorem.
On the other hand, if F # Q, then the reduced norm of 7, over F is not usually a power of p; hence it cannot correspond with an automorphic representation of GL2 (perhaps possible for GL2m (m = [F: Q])). At any rate, this relation would not reduce one theory to another. REFERENCES
1. M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkdrper, Abh. Math. Sem. Univ. Hamburg 14 (1941), 197-272. , (a) Teilbarkeitseigenschaften der singuldren Moduln der elliptischen Funktionen and die 2. Diskriminante der Klassengleichung, Comment. Math. Hely. 19 (1947), 74-82. (b) Die Struktur der elliptischen Funktionenkdrper and die Klassenkdrper der imagindren quadratische Zahlkdrper, Math. Ann. 124 (1952), 393-426. 3. B. Dwork, p-adic cycles, Inst. Hautes Etudes Sci. Publ. Math. 37 (1969), 27-116. 4. A. Grothendieck, Elements de geometrie algebrique (EGA). I - IV, Inst. Hautes Etudes Sci. Publ. Math (1960-1967). 5. Y. Ihara, (a) The congruence monodromy problems, J. Math. Soc. Japan 20-1, 2 (1968), 107121.
(b) On congruence monodromy problems, Lecture Notes, Univ. Tokyo, vols.1, 2, 1968, 1969. (c) Non-abelian classfields over function fields in special cases, Actes du Congres Internat. Math., Nice, 1970, Tome 1, pp. 381-389. , On modular curves over finite fields, Discrete Subgroups of Lie Groups, Proc. Internat. 6. Colloq., Bombay, Oxford Univ. Press, 1973, pp. 161-202. , On the differentials associated to congruence relations and the Schwarzian equations 7. defining uniformizations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 21 (1974), 309-332. 8. Y. Ihara and H. Miki, Criteria related to potential unramifiedness and reduction of unramified coverings of curves, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), 237-254. 9. Y. Ihara, (a) Some fundamental groups in the arithmetic of algebraic curves over finite fields, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 3281-3284.
(b) Congruence relations and Shimura curves. II (to appear in J. Fac. Sci. Univ. Tokyo Sect. IA Math. 25-3). , On the Frobenius correspondences of algebraic curves, Algebraic Number Theory, 10. Papers contributed for the Kyoto Internat. Symposium, 1976; Japan Soc. Promotion of Sci., Tokyo, pp. 67-98. 11. R. P. Langlands, (a) Shimura varieties and the Selberg trace formula (to appear). (b) On the zeta functions of some simple Shimura varieties (to appear).
12. J. Lubin and J. Tate, Formal complex multiplication in local fields, Ann. of Math. (2) 81 (1965), 380-387.
13. Y. Morita, Ihara's conjectures and moduli space of abelian varieties, Master's thesis, Univ. Tokyo (1970). 14. G. Shimura, Construction of classfields and zeta functions of algebraic curves, Ann. of Math. (2) 85 (1967), 58-159. , On canonical models of arithmetic quotients of bounded symmetric domains. I, II, 15. Ann. of Math. (2) 91 (1970), 144-222; 92 (1970), 528-549. 16. J.-P. Serre, Arbres, amalgames, SL,, Asterisque 46, Soc. Math. France, 1977. UNIVERSITY OF TOKYO
Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 313-346
VALEURS DE FONCTIONS L ET PERIODES D'INTEGRALES P. DELIGNE Dans cet article, j'enonce une conjecture (1.8, 2.8) reliant les valeurs de certaines fonctions Len certains points entiers a des periodes d'integrales.
Les fonctions L considerees sont celles des motifs-un mot auquel on n'attachera pas un sens precis. Ceci inclut notamment les fonctions L d'Artin, les fonctions L attachees a des caracteres de Hecke algebriques (= Grossencharakter de type A0), et celles attachees aux formes modulaires holomorphes sur le demi-plan de
Poincare, supposees primitives (= new forms; on considere toutes les fonctions L, Lk, attachees aux puissances symetriques, Symk, de la representation l-adique correspondante). Cet article doit le jour a D. Zagier: pour son insistance a demander une conjecture, et pour la confirmation experimentale qu'il en a donnee, sitot emise, pour les fonctions L3 et L4 attachees a 4 = Ez(n)gn (voir [18]). C'est cette confirmation qui m'a donne la confiance necessaire pour verifier que la conjecture etait compatible aux resultats de Shimura [13] sur les valeurs de fonctions L de caracteres de Hecke algebriques. 0. Motifs. Le lecteur est invite a ne consulter ce paragraphe qu'au fur et a mesure des besoins. On y rappelle une partie du formalisme, du a Grothendieck, des motifs. Pour les demonstrations, je renvoie a [8]. 0.1. La definition de Grothendieck des motifs sur un corps k a la forme suivante.
(a) Soit r(k) la categorie des varietes projectives et lisses sur k. On construit une categoric additive X"(k), pour laquelles les groupes Hom(M, N) sont des espaces vectoriels sur Q, munie de
a. Un produit tensoriel p, associatif, commutatif et distributif par rapport a l'addition des objects ("associatif" et "commutatif" n'est pas une propriete du foncteur px , mais une donnee, soumise a certaines compatibilites-cf. Saavedra [19]);
/3. Un foncteur contravariant H*, de 'Y,'(k) dans G&'(k), bijectif sur les objets et
transformant sommes disjointes en sommes, et produits en produits tensoriels (donnee d'un isomorphisme de foncteurs H*(X x Y) = H*(X) Qx H*(Y), compatible a l'associativite et a la commutativite). Il s'agit, pour 1'essentiel, de definir Hom(H*(X), H*(Y)). Pour la definition, utiAMS (MOS) subject classifications (1970). Primary 10D25, 14K10. © 1979, American Mathematical Society
313
314
P. DELIGNE
lisee par Grothendieck, de ce groupe comme un groupe de classes de correspondances entre X et Y, voir 0.6.
(b) Rappelons (SGA 4 IV 7.5) qu'une categorie additive est karoubienne si tout projecteur (= endomorphisme idempotent) est defini par une decomposition en somme directe, et que chaque categorie additive a une enveloppe karoubienne, obtenue en lui adjoignant formellement les images des projecteurs. La categoric 'eff(k) des motifs effectifs sur k est l'enveloppe karoubienne de . #'(k). (c) On definit le motif de Tate Z(-1) comme un facteur direct H2(PI) convenable de H*(Pl). On verifie que la symetrie (deduite de la commutativite de (D): Z(-1)
Q Z(- 1) -> Z(-1) O Z(-1) est l'identite, et que le foncteur M -* M 0 Z(-1) est pleinement fidele.
(d) La categorie il(k) des motifs sur k se deduit de &eff(k) en rendant inversible le foncteur M --> M p Z(- 1). Notons (n) l'itere (- n)ieme de 1'auto-equivalence M ---* M Qx Z(- 1). La categorie il(k) admet .Oeff(k) comme soul-categorie pleine, et tout object de X(k) est de la forme M(n) pour M dans .il'eff(k) et n un entier. Par definition, si F est un foncteur additif de '(k) dans une categorie karoubienne .c/, it se prolonge a .illeff(k). Si .c/ est munie d'une auto-equivalence A
A(- 1), et le prolongement de F d'un isomorphisme de foncteurs F(M(-1)) _ F(M)(-1), it se prolonge a .il(k). EXEMPLE 0.1.1. Si k' est une extension de k, et F le foncteur H*(X) -> H*(X (&k k') de ff'(k) dans -&(k'), on obtient le foncteur extension des scalaires de G'(k) dans X (k'). Si k' est une extension finie de k, on dispose du foncteur de restriction des scalaires a la Grothendieck JL k',k: ,(k') -* *-(k): (X -* Spec(k')) - (X -* Spec(k') -> Spec(k)).
On le prolonge en F : H*(X) F-. H*(JLk-,kX), d'oI un foncteur de restriction des scalaires Rk-,k : X(k') --> .1l(k). EXEMPLE 0.1.2. Soit Y une "theorie de cohomologie", a valeurs dans une cate-
gorie karoubienne a, fonctorielle pour les morphismes dans Ol' (k). Le foncteur .*' se prolonge a .&eff(k). Si d est muni d'un produit tensoriel, que IP verifie une formule de Kiinneth et que la tensorisation avec .yP(Z(-1)) est une auto-equivalence de d, it se prolonge a .bil(k). Ce prolongement est le foncteur "realisation d'un motif dans la theorie .*". .
Nous noterons (n) l'auto-equivalence de .,q/ itere (- n)ieme de la tensorisation avec (Z( - 1)). Pour la determination de .°(Z( - 1)) dans diverses theories , voir 3.1. 0.2. Nous utiliserons les realisations suivantes: 0.2.1. Realisation de Betti HB. Correspondant a k = C, sal = espaces vectoriels
que Q, .1° = cohomologie rationnelle: X'- H*(X(C), Q); 0.2.2. Realisation de de Rham HDR. Correspondant a k de caracteristique 0, .4 = espaces vectoriels sur k, W = cohomologie de de Rham : Xi- H*(X, Q X*); 0.2.3. Realisation 1-adique H1. Correspondant a k algebriquement clos, de carac-
teristique 0 1, d = espaces vectoriels sur Qt, .P = cohomologie 1-adique: X H*(X, Q1) Et leurs variantes: 0.2.4. Realisation de Hodge. k est ici une cloture algebrique de R, et ,2/ la categorie des espaces vectoriels sur Q, de complexifie V 0 k muni d'une bigraduation
315
VALEURS DE FONCTIONS L
V Q k = QQ Vp,q telle que VqP soit le complexe conjugue de Vpq. Pour theorie de cohomologie, on prend le foncteur X u- H*(X(k), Q) muni de la bigraduation de son complexifie H*(X(k), k) fournie par la theorie de Hodge. La realisation de Betti est sous-jacente a celle de Hodge. 0.2.5. Pour k quelconque, et o, un plongement complexe de k, on note H0(M) la
realisation de Betti du motif sur C deduit de M par 1'extension des scalaires o: k -+ C. Notons c la conjugaison complexe. On obtient par transport de structure un isomorphisme F,: HQ(M) -, H,(M), et F, © c envoie HQq sur Hf,. Pour o reel, F_ est une involution de H,(M), dont le complexifie echange Haq(M) et Hop(M).
La o-realisation de Hodge est H,(M), muni de sa bigraduation de Hodge et, si o est reel, de l'involution F . Pour k = Q, on k = R et o le plongement identique,
on remplace 1'indice o par B. Pour k = R, et M = H*(X), 1'involution F, de HB(M) = H*(X(C), Q) est l'involution induite par la conjugaison complexe F, : X(C) --+ X(C).
0.2.6. Realisation de de Rham. La cohomologie de de Rham est munie d'une filtration naturelle, la filtration de Hodge, aboutissement de la suite spectrale d'hypercohomologie Epq = Hq(X, Sip)
Hggq (X).
De la, une filtration F sur HDR(M). 0.2.7. Realisation 1-adique. Si X st une variete sur un corps k, de cloture
algebrique k, le groupe de Galois Gal(k/k) agit par transport de structure sur H*(Xk,QI). Si M est un motif sur k, et qu'on definit H1(M) comme la realisation 1-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 1-adiques en une cohomologie adelique H*(X, AJ) = (IIH*(X, Z1)) Oz Q On pent 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 pent remplacer
la categorie c par celle, c*, des objets gradues de .Q/: utiliser la graduation naturelle de H*. Pour que la formule de Kiinneth fournisse un isomorphisme de (M) Qx , (N), compatible a 1'associativite et a la foncteurs °(M Qx N) commutativite, it faut prendre pour donnee de commutativite dans d* celle donnee par la regle de Koszul. 0.3. Pour k = C, le theoreme de comparaison entre cohomologie classique et
cohomologie etale fournit un isomorphisme H*(X(C), Q) O Ql -, H*(X, Q). Si X est defini sur R, cet isomorphisme transforme F, (0.2.5) en 1'action (0.2.7) de la conjugaison complexe. Pour un motif, on a de meme HB(M) Qx Q1 . H,(M). 0.4. Pour k = C, le complexe de de Rham holomorphe sur Xan est une resolution du faisceau constant C. Par GAGA, on a donc un isomorphisme
H*(X(C), Q) O C = H*(X(C), C)
jP*(Xan Q*an)
H*(X, Q*).
Pour un motif, on obtient un isomorphisme (compatible aux filtrations de Hodges 0.2.4 : FP = @p,Zp Vp'q' et 0.2.6) HB(M) O 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) ® C
HDR(M) Ok.O C.
Notons le cas particulier k = Q, ou on trouve deux Q-structures naturelles sur la realisation de M en cohomologie complexe : l'une, HA(M), liee a la description de la cohomologie en terme de cycles, l'autre, HDR(M), He 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'1k (M') (0.5.1)
HQ(M) = O HH(M'), r
ou la somme est etendue a 1'ensemble J(o) des plongement complexes de k' qui prolongent u. Cet isomorphisme est compatible aux bigraduations de Hodge, et a
F.
(0.5.2)
HDR(M) = HDR(M')
(restriction des scalaires de k' a k).
Cet isomorphisme est compatible a la filtration de Hodge. (0.5.3) (0.5.4)
H1(M) = Ind H1(M') (representation induite de Gal(k/k') a Gal(k/k)). Via l'isomorphisme k' x®k Q C = G (a), et l'isomorphisme HDR(M') Ox k,u
C = (D,-J(,) HDR(M') Qk,TC qui s'en deduit, (0.4.1) pour M et o, est la somme des isomorphismes (0.4.1) pour M' et a (r E AO: HDR(M) O k,, C
HA(M) Ox C
(0.5.2)
(0.5.1)
11
OHT(M') O C T
(041) . .
11
OO HDR(M') Ok'.z C. T
0.6. Pour X projectif et lisse sur k, notons Zd(X) l'espace vectoriel sur Q de base 1'ensemble des sous-schemas fermes irreductibles de codimension d de X, et ZR(X) son quotient par une relation d'equivalence R. Pour k de caracteristique 0, une des definitions de Grothendieck des motifs s'obtient en faisant (pour X et Y connexes) Hom(H*(Y), H*(X)) = ZRm(1')(X X Y), R etant 1'equivalence cohomologique.
En ce qui concerne la caracteristique p, signalons seulement deux dificultes: on ignore si l'equivalence cohomologique-en cohomologie 1-adique (10 p)est independante de 1, et la definition de la classe d'un cycle pose des problemes en cohomologie cristalline. 0.7. Soit X une variete projective et lisse complexe. Nous appellerons cycle de Hodge de codimension d sur X un element de H2d(X(C), Q) de type (d, d)-soit, ce qui revient au meme, un element de H2d(X(C), Q)(d) de type (0, 0). Soient k un corps algebriquement clos et X une variete projective et lisse sur k. Posons
VALEURS DE FONCTIONS L
317
H2d(X, k x Af)(d) = HDf(X)(d) x Hld(X, Af)(d). Pour k = C, on appelle encore cycle de Hodge l'image dans
HZd(X, k x Af)(d) = H2d(X(C), Q)(d) O (k x Af)
d'un cycle de Hodge. Pour k algebriquement clos, admettant des plongements complexes, un cycle de Hodge absolu de codimension d sur X est un element de H2d(X, k x Af)(d), tel que, pour tout plongement complexe u de k, son image dans H2d(X (9k C, C x Af)(d) soft de Hodge. On verifie que PROPOSITION 0.8. (i) L'espace vectoriel sur Q Zha(X) des cycles de Hodge absolu
est invariant par extension des scalaires de k a un corps algebriquement clos k' (admettant encore un plongement complexe). (ii) Pour k algebriquement clos de caracteristique 0, et X defini sur un sous-corps algebriquement clos ko de k, admettant un plongement complexe: X = Xo Qx ko k, on pose
Zha(X) = Zha(Xo) cz H21(X0, ko x Af)(d) c H2d(X, k x Af)(d). D'apres (i), cette definition ne depend pas des choix de X0 et ko. (iii) Pour X defini sur un sous-corps k0 de k: X = X0 Qx ko k, be groupe Aut(k/k0) agissant sur H2d(X, k x Af)(d) stabilise Zha(X). II agit sur Zha(X) a travers un groupe fini, correspondant a une extension finie kp de k0. On pose Zha(X0) = Zha(X) Aut(k/ko)
0.9. Une notion utile de motif s'obtient en faisant (pour X et Y connexes)
Hom(H*(Y), H*(X)) = Zham(Y)(X x Y).
Les composantes de Kiinneth de la diagonale de X x X sont absolument de Hodge. Ceci permet de decomposer H*(X) en une Somme de motifs Hi(X), de munir la categorie des motifs d'une graduation (avec H=(X) de poids i) et de modifier la contrainte de commutativite pour O comme en [19, VI, 4.2.1.4]. On verifie que, ceci fait, la Qx -categorie des motifs sur k est tannakienne, isomorphe a la categorie des representations d'un groupe proalgebrique reductif. Pour k = C, la conjecture suivante, plus faible que celle de Hodge, equivaut a dire que le foncteur "realisation de Hodge" est une equivalence de la categorie des motifs 0.9 avec la categorie des structures de Hodge facteur direct de la cohomologie de variete algebriques (ou deduites par twist a la Tate de tels facteurs directs). Espoir 0.10. Tout cycle de Hodge 1'est absolument.
Si X est une variete abelienne a multiplication complexe par un corps quadratique imaginaire K, avec Lie(X) libre sur k Qx K, les methodes de B. Gross [7] prouvent que certains cycles de Hodge non triviaux sont absolument de Hodge. Ajoute sur epreuves: partant de ce cas particulier, j'ai pu verifier 0.10 pour les varietes abeliennes. J'ai verifie aussi que la xp-categorie de motifs (0.9) engendree par les H^(X), pour X une variete abelienne, contient les motifs H*(X), pour X une surface K3 ou une hypersurface de Fermat. 0.11. Le principal defaut de la definition 0.9 des motifs est qu'elle ne se prete pas a la reduction mod p. On ignore si un motif 0.9 sur un corps de nombres F fournit un systeme compatible de representations 1-adiques de Gal(F/F).
318
P. DELIGNE
0.12. Nous utiliserons le mot "motif" de facon libre, sans nous preoccuper de faire rentrer les motifs consideres dans le cadre de Grothendieck. L'essentiel pour nous sera de disposer de realisations.r(M), pour les theories' considerees en 0.2, et d'avoir pour ces groupes le meme formalisme que pour les.o*(X). 1. Enonce de la conjecture (cas rationnel). 1.1. Soit M un motif sur Q. Nous admettrons que les realisations 1-adiques HI(M)
de M forment un systeme strictement compatible de representations 1-adiques, au sens de Serre [11, I.11]. A savoir : it existe un ensemble fini S de nombres premiers, tel que chaque H1(M) soit non ramifie en dehors de S U {1}, et que, notant par Fp e Gal(Q/ Q) un element de Frobenius geometrique en p (l'inverse d'une substitution de Frobenius (pp), le polynome det(1 - Fpt, HI(M)) e Q,[t] (p 0 S U {l}) soit a coefficients rationnels, et independant de 1. Notons Zp(M, t) son inverse, et posons Lp(M, S) = Zp(M, P -S). La serie de Dirichlet a coefficients rationnels donnee par le produit eulerien Ls(M, s) = 11 pas L p(M, s) converge pour R s assez grand. Pour s quelconque, on definit Ls(M, s) par un prolongement analytique (qu'on espere exister). Notre but est d'enoncer une conjecture donnant la valeur de Ls(M, s) en certains points entiers, au produit pres par un nombre rationnel. Puisque, pour s entier, p-5 est rationnel, le choix de S est sans importance-a ceci pres qu'aggrandir S peut introduire des zeros mal venus. 1.2. Pour ecrire proprement 1'equation fonctionnelle conjecturale des fonctions L, it y a lieu de completer le produit eulerien Ls(M, s) par des facteurs locaux en p e S, et a l'infini. La definition des facteurs locaux en p e S requiert une hypothese additionnelle, que nous supposerons verifiee: (1.2.1) Soientp un nombre premier, Dp c Gal(Q/Q) un groupe de decomposition en p, Ip c Dp son sous-groupe d'inertie et Fp e Dp un Frobenius geometrique. Le polynome det(1 - Fpt, H1(M)IP) e Qt [t] (1 = p) est a coefficients rationnels, et est independant de 1. Posons Zp(M, t) = det (1 - Fpt, H1(M)IP)-1 E Q(t), et Lp(M, s) = Zp(M, p -s). On definit (1.2.2)
L(M, s) = H Lp(M, s). P
Le facteur a l'infini L00(M, s) (essentiellement un produit de fonctions F) depend
de la realisation de Hodge de M-en fait seulement de la classe d'isomorphie de 1'espace vectoriel complexe HB(M) (D C, muni de sa decomposition de Hodge et de l'involution F . Sa definition est rappelee en 5.2.
Posant A(M, s) = L,,(M, s)L(M, s), l'equation fonctionnelle conjecturale des fonctions L s'ecrit (1.2.3)
A(M, s) = e(M, s)A(11I, 1-s)
ou 1lI est le dual de M (de realisations les duales des realisations de M) et ou e(M, s),
comme fonction de s, est le produit d'une constante par une exponentielle. Sa definition est rappelee en 5.2. Elle depend d'une hypothese additionnelle. DEFINITION 1.3. Un entier n est critique pour M si ni L,(M, s), ni LOO(M, 1-s) n'ont de pole en s = n. Notre but est de conjecturer la valeur de L(M, n) pour n critique, a multiplica-
319
VALEURS DE FONCTIONS L
tion par nombre rationnel pres. On a L(M(n), s) = L(M, n + s) (3.1.2), et de meme pour L,0. Ceci nous permet de ne considerer que les nombres L(M) = dfn L(M, 0). Nous dirons que M est critique si 0 est critique pour M. On verifie que pour que M soit critique, it faut et it suffit que les nombres de Hodge hp9 = dfn dim Hp9(M) de M, pour p q, ne soient non nuls que pour (p, q) dans la partie hachuree du diagramme ci-dessous, et que F, agisse sur HP,P, par l'identite si p < 0, par - I si
p>0.
P
Supposons M homogene de poids w : hpq = 0 pour p + q # w, et posons .q(M) = - w/2. 11 resulte de la conjecture de Weil que, pour S assez grand, la serie de Dirich-
let Ls(M, s) converge absolument pour M(M) + s(s) > 1. Pour R(M) + a(s) _ 1, on est au bord du demi-plan de convergence, et on conjecture (a) que Ls(M, s) ne s'annule pas, (b) que Ls(M, s) est holomorphe, sauf si M est de poids pair - 2n, et contient Z(n) en facteur: on s'attend alors a un pole en s = 1 - n (une valeur non critique). L'analogie avec le cas des corps de fonctions, et les cas connus, menent a croire que les facteurs locaux Lp(M, s) (p quelconque, y compris oo) n'ont de pole que pour M(M) + s(s) < 0. Si tel est le cas, (a), (b) et 1'equation fonctionnelle conjecturale (1.2.3) impliquent que L(M) # 0, oo pour M critique et M(M) = Z. Pour R(M) = 2L (M) s'annulle parfois. Notre conjecture (1.8) est alors vide. PROPOSITION 1.4. Soit M un motif sur R. Via l'isomorphisme (0.4.1) HB(M) OO C
HDR(M) OR C,
HDR(M) s'ident f e au sous-espace de HB (M) Qx Cfixe par c -+ F,,c.
Prenons M = H*(X). La conjugaison complexe sur HDR(M)B = H*(Xc, Q*) se deduit par fonctorialite de l'automorphisme antilineaire F,, du schema X. Remontant la fleche composee (0.4) qui definit I, on l'identifie a l'involution deduite de l'automorphisme (F,,, F-*) de (X(C), Q*a"), puis au compose de FF: H*(X(C), C) -* H* (X(C), C) et de la conjugaison complexe sur les coefficients. Ceci verifie 1.4.
320
P. DELIGNE
1.5. Pour Mun motif sur R, nous noterons HB(M) (resp. HB(M)) le sous-espace
de HB(M) fixe par F (resp. ou F = -1). Posons d(M) = dim HB(M) et d±(M) _ dim HB(M). Pour M un motif sur k, et u un plongement complexe de k qui se factorise par R, on note HQ (M), do (M) et d(M) ces objets pour le motif sur R deduit de M par o-. Pour k = Q, on omet la mention de or. Le corollaire suivant resulte aussitot de 1.4, et de ce que tant F que la conjugaison complexe echangent Hpe et Hap.
COROLLAIRE 1.6. Soit M un motif sur R. Pour la structure reelle HDR(M) de HB(M) Qx C, les sous-espaces Hp4 sont definis sur R; le sous-espace HB(M) Q R est reel, et HB(M) Qx R est purement imaginaire.
1.7. Dans la fin de ce paragraphe, nous ne considererons que des motifs sur Q; sauf mention expresse du contraire, nous les supposons homogenes. Si leur poids w est pair, nous supposerons aussi que F agit sur HPP(M) (w = 2p) comme un scalaire: soit + 1, soit -1. Cette hypothese est verifise pour M critique. Puisque F echange HP et HqP, elle assure que les dimenions d+(M) et d-(M) sont egales l'une a Ep>a hp9, I'autre a EpZ9 hpq. En particulier, ces dimensions sont egales a
celles de sous-espaces F+ et F- figurant dans la filtration de Hodge. Posant HDR(M) = HDR(M)/F+, on a encore dim HDR(M) = d±(M).
Il resulte de ce que F echange Hpq et HqP que les applications composees (1.7.1)
I±: HB(M)c -* HB(M)c
HDR(M)c - ' HDR(M)c
sont des isomorphismes. On pose (1.7.2)
c+(M) = det(II),
(1.7.3)
a(M) = det(I),
le determinant etant calculs dans des bases rationnelles de HB et HoR (resp. HB et HDR). La definition de o(M) ne requiert pas les hypotheses faites sur M. D'apres 1.6, I+ est reel, i.e. induit I+: HB(M)R --> HDR(M)R, tandis que Lest purement'imaginaire. Les nombres c+(M), id-(M) c -(M) et id- (M) a(M) sont donc reels non nuls. A multiplication par un nombre rationnel pres, it ne dependent que de M. Les periodes de M sont classiquement les <w, c> pour cu E HDR(M) et c e HB(M)v. Par exemple, si X est une variete algebrique sur Q, cu une n-forme sur X, dsfinie sur Q et que c est un n-cycle sur X(C), = f, co est une periode de Hn(X). Exprimons c±(M) en terme de periodes. Le dual de HDR(M) est le sous-espace F± de HDR(MV), oii My est le motif dual de M. Si la base choisie de HDR(M) a pour duale la base (co) de F±(HDR(MV)), et que (c) est la base choisie de HB(M), la matrice de I± est , 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 la conjecture (cas general).
2.1. Jusqu'ici, nous n'avons considers que des fonctions L donnses par des series de Dirichlet a coefficients rationnels. Pour faire mieux, it nous faudra considerer des motifs a coefficient dans des corps de nombres.
Voici deux facons, 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 categoric des motifs sur k.
Il s'agit d'une construction valable pour toute categorie additive karoubienne (0.1(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 structure de E-module: E --> End(M). B. La categorie G1k,E, des motifs sur k, a coefficient dans E, est 1'enveloppe
karoubienne (cf. 0.1(b)) de la categorie d'objets les motifs sur k-considers Iffk,E, un motif M se notera ME-et de morphismes donnes par comme objet Hom(XE, YE) = Hom(X, Y) O E. Passage de B a A. Pour X un motif, et V un espace vectoriel sur Q de dimension finie, on note X O V le motif, isomorphe a une somme de dim(V) copies de X, caracterise par Hom(Y, X Qx V) = Hom(Y, X) px V (ou par Hom(X O V, Y) = Hom(V, Hom(X, Y))). On passe de B a A en associant a ME le motif M O E, muni de sa structure de E-module naturelle. Passage de A a B. Si M est muni d'une structure de E-module, on recupere sur
ME deux structures de E-module: celle deduite de celle de M, et celle qu'a tout objet de 1(k,E. L'objet de .ALk,E correspondant a M est le plus grand facteur direct de ME sur lequel ces structures coincident. En detail: l'algebre E Qx E est un
produit de corps, parmi lesquels une copie de E dans laquelle x 0 1 et 1 O x se projettent tous deux comme x. L'idempotent correspondant e agit sur ME (qui est un E px E-module) et son image est l'object de JI'k,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 OE YE = (X Q Y)E et (XE)v = (XV)E. Darts le langage A, X OE Y est le plus grand facteur direct de X Qx Y sur lequel coincident les deux structures de E-module
de X O Y, et X est le dual usuel de Xv, muni de la structure de E-module transposee. Si on applique ces remarques a la categorie des espaces vectoriels sur Q, plutot qu'a celle des motifs, on retrouve l'isomorphisme du F-dual d'un espace vectoriel sur E avec son Q-dual, donne par co H la forme TrE,Q(((u, v>).
Nous avons defini en 0.1.1 des foncteurs de restriction et d'extension du corps k des scalaires. Its 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 H X (DE F; dans le langage B, c'est XE F- XF.
Restriction des coefficients. Dans le langage A, on restreint a E la structure de F-module. Le lecteur prendra soin de ne pas confondre les roles de k et E. Un exemple type
qu'on peut retenir est celui du HI des varietes abeliennes sur k, a multiplication complexe par un ordre de E. En terme de la variete abelienne-prise a isogenie pres-les foncteurs ci-dessus deviennent: extension du cops de base, restriction des scalaires a la Weil, construction OE F, qui multiplie ]a dimension par [F: E], restriction a E de la structure de F-module.
322
P. DELIGNE
2.2. Soit M un motif sur Q, a coefficient dans un corps de nombres E. Pour chaque nombre premier 1, la realisation l-adique H1(M) de M est un module sur le complete 1-adique Et de E. Ce complete est le produit des completes EA, pour 2 ideal premier au-dessus de 1, d'oix une decomposition de H1(M) en un produit de Ez-modules H2(M).
On conjecture pour les HA(M) une compatibilite analogue a 1.2.1; si elle est verifiee, on peut pour chaque plongement complexe 6 de E definir une serie de Dirichlet a coefficients dans 6E, convergente pour 9?s assez grand:
L(6, M, s) = 11 Lp(6, M, s),
oii
p
(2.2.1)
Lp(6, M, s) = 6Zp(M, p -s) avec Zp E E(t) cz EA(t) donne par Zp(M, t) = det(1 - Fit, HA(M)'P)-I pour Alp.
Pour 6 variable, ces series de Dirichlet se deduisent les unes des autres par conjugaison des coefficients. Nous regarderons le systeme des L(6, M, s) comme une fonction L*(M, s) a valeurs dans la C-algebre E Qx C, identifiee a CHom(E' C) par (2.2.2)
EDCi
CHom(E,C) : e x0 z
- (z, c(e))o
Cette fonction peut aussi titre definie directement par un produit eulerien. On espere comme en 1.1 qu'elle admet un prolongement analytique en s.
11 y a lieu de completer le produit eulerien L(6, M, s) par un facteur a l'infini L_(6, M, s) dependant de la realisation de Hodge de M. Posant 11(6, M, s) _ L_(6, M, s)L(6, M, s), 1'equation fonctionnelle conjecturale des fonctions L s'ecrit (2.2.3)
11(6, M, s) = &(6, M, s)11(6, k, 1 - s),
oiu e(6, M, s), comme fonction de s, est le produit d'une constante par une exponentielle. Les definitions de L00 et a sont rappelees en 5.2. Comme ci-dessus, on re-
gardera le systeme des A et celui des e, pour 6 variable, comme des fonctions 11* et a* a valeurs dans E Q C. Il resulte de 2.5 ci-dessous que L,(6, M, s) est independant de 6, et que la fonction Lco, pour le motif RE,Q M deduit de M par restriction du corps des coefficients (2.1) est la puissance [E: Q]ieme de L_(6, M, s). Ceci justifie la PROPOSITION-DEFINITION 2.3. Soit M un motif sur Q d coefficient dans E. Un entier
n est critique pour M si les conditions equivalentes suivantes sont verifiees (i) 1'entier n est critique pour RE,Q M;
(ii) ni L,(6, M, s), ni L_(6, M(1), 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 la valeur de L*(M) =
dfn
L*(M, 0), pour M critique,
a multiplication par un element de E pres. En d'autres termes, it s'agit de conjecturer simultanement les valeurs des L(6, M) = dfn L(6, M, 0), a multiplication pres par un systeme de nombres 6(e), e e E. 2.4. La realisation HB(M) de M en cohomologie rationnelle est munie d'une structure de E-espace vectoriel. Sa dimension est le rang sur E de M. L'involution
F est E-lineaire; les parties + et - sont donc des E-sous-espaces vectoriels. On note d+(M) et d-(M) leur dimension.
VALEURS DE FONCTIONS L
323
Le complexifie de HB(M) est un E ® C-module libre. Identifiant E ® C a CHom(E,C) (2.2.2), on en deduit une decomposition
HB(M) ® C = OO HB(a, M), a
avec
HB(6, M) _ (HB (M) ® C) OO EOC, n C,
soit HB(o , M) = HB(M) (&E, o C.
Les HB,q(M) de la decomposition de Hodge etant stables par E, chaque HB(o, M) herite d'une decomposition de Hodge HB(c, M) _ +QHB' q(6, M). L'involution F permute Hpq et HqP, en particulier stabilise HPP, qu'elle decoupe en des parties
+ et-. On note hpq(o, M) la dimension de HBq(u, M), et hpp±(G, M) celle de HBp±(6, M). La proposition suivante permet d'omettre u de la notation. PROPOSITION 2.5. Les nombres hpq(6, M) et W+(6, M) sont independant de u.
On peut supposer, et on suppose, que M est homogene. Dans ce cas, Hpq(M) s'identifie au complexifie du E-espace vectoriel Gr PF(HDR(M)): c'est un E ® Cmodule libre, et la premiere assertion en resulte. Pour la seconde, on observe que hpp±(6, M) est 1'exces de d±(M) sur Ep>q hpq(oc, M).
2.6. Soit M un motif sur Q a coefficient dans E. Dans la fin de ce paragraphe, sauf mention expresse du contraire, nous supposons que RE/Q M verifie les hypotheses de 1.7. Les espaces F± et H,R sont cette fois des espaces vectoriels sur E. Les isomorphismes (0.4.1) et (1.7.1)
I: HB (M) ® C
( 2 . 6 . 1) (2 . 6 . 2)
1!: HB (M) ® C
H DR(M) ® C , HDR(M) 0 C ,
sont des isomorphismes de E ® C-modules entre complexifies d'espaces vectoriels sur E. On pose c±(M) = det (1) G (E (D C)*,
o(M) = det (I) E (E (D C)*,
le determinant etant calcule dans des bases E-rationnelles de HB(M) et HDR(M) (resp. HB(M) et HDR(M)). La definition de 6(M) ne requiert pas les hypotheses faites sur M. A multiplication par un element de E* pres, ces nombres ne dependent que de M. 11 resulte a nouveau de 1.6 que c+(M), id (M) c -(M) et id-(M) o(M) sont dans (E ® R)*. Conjecture 2.7. Posons M(M) Zw. Si M est critique, (i) L(ar, M, s) n'a jamais de pole en s = 0, et ne peut s'annuler en s = 0 que pour
.q(M) = 2. (ii) La multiplicite du zero de L(o,, M, s) en s = 0 est independante de u.
Pour (i), je renvoie a la discussion a la fin de 1.3. Que (ii) soit raisonnable m'a ete suggere par B. Gross. Conjecture 2.8. Pour M critique et L(ai, M) 0 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 sont couvertes par notre conjecture, 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(o-, M, s) sont des produits euleriens, indexes par les places finies de k. Il y a lieu de les completer par les facteurs a l'infini Lv(a, M, s), indexes par les places a l'infini, dont la definition est rappelee en 5.2. Us dependent en general de a. Seul Lo(o,, Rk,Q M, s), produit sur v a l'infini des L (u, M, s), est independant de a. REMARQUE 2.10. Soient F une extension de E, c le morphisme structural de E
dans F et cc son complexifie: E O C c> F O C. On a L*(M OE F, s) = ccL*(M, s) et c+(M OE F) = cc c+(M). La conjecture est donc compatible a 1'extension du corps des coefficients. Pour F une extension galoisienne de E, de groupe de Galois G, le Theoreme 90 de Hilbert H'(G, F*) = 0 assure que ((F O C)*/F*)G = (E (D C)*/E* : la conjecture est invariante par extension du corps des coefficients. REMARQUE 2.11.
Si E est une extension de F, on a L*(RE/FM, S) _
NEIF L*(M, s). Les periodes c± verifiant la meme identite, la conjecture est compatible a la restriction des coefficients. REMARQUE 2.12. Soit D une algebre a division de rang d2 sur E. Un motif M sur
Q, muni d'une structure de D-module, et de rang n sur D, definit une serie de Dirichlet a coefficients dans E dont les facteurs euleriens sont presque tons de degre nd: pour A une place finie de E, HA(M) est un module libre sur le complete DA = D (DE EA, et on reprend la definition (2.2.1) en posant
Zp(M, t) = det red (1 - Fpt,
HA(M)IP)-1
(si DA est une algebre de matrices sur EA, et e un idempotent indecomposable, le determinant reduit d'un endomorphisme A d'un DA-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 fonctions L elles aussi sous le chapeau 2.8: choisissant une extension c: E - F de E qui neutralise D, et un idempotent indecomposable e de D (DE F, on a ccL*(M, s) _ L*(e(M OE F), s). On peut aussi definir c+(M) directement dans ce cadre. C'est ce qui est explique en 2.13 ci-dessous. La fin de ce paragraphe est inutile pour la suite.
2.13. Soit D une algebre simple sur un corps E (voire une algebre d'Azumaya sur
un anneau...). Pour le formalisme tensoriel, it est commode de regarder les Dmodules comme de "faux E-espaces vectoriels" : (a) Se dormer un espace vectoriel V sur E revient a se donner, pour toute exten-
sion etale F de E, un F-espace vectoriel VF, et des isomorphismes compatibles VG = VF OO F G pour G une extension de F. On prend VF = V OE F. Par descente, it suffit de ne se dormer les VF que pour F assez grand.
(b) Soit W un D-module. Pour toute extension etale F de E, et tout F-isomorphisme D O F - EndF(L), avec L libre, posons WF,L = HomD®F (L, W (D F) (les produits tensoriels sont sur E). On a un isomorphsime de D O F-module WOE F = LOF WF,L
VALEURS DE FONCTIONS L
325
Si F est assez grand pour neutraliser D, L est unique a isomorphisme non unique pres; la non-unicite est due aux homotheties, qui agissent trivialement sur End(L).
C'est pourquoi la donnee des WF,L n'est pas du type (a); c'est un "faux espace vectoriel sur E". Soient (We) une famille de D-modules, et T une operation tensorielle. Si les homotheties de L agissent trivialement sur T(WF,L), le F-espace vectoriel T(WF,L) est independant du choix de L et on obtient un systeme du type (a), d'oil un espace vectoriel T(Wa) sur E. EXAMPLE. Si W' et W" sont deux D-modules de rang n [D: E]1/2 sur E, on pent prendre T = Hom(AnWF,L, A WF L); on obtient un espace vectoriel 8(W', W")
de rang 1 sur E, et tout homomorphisme f: W' -+ W" a un determinant reduit det red(f) e 5(W', W"). Pour definir c+(M), on applique cette construction aux D-modules HB (M) et a HDR(M). Posons 6 = d(HB(M), HDR(M)). Le determinant reduit de l'isomorphisme de D (D C-modules I+: HAM), --> HDR(M)c est dans le E p C -module libre de rang 1 5 Qx C. On pose det red(I+) = c+(M) e pour e une base de 6. 3. Exemple: la fonction C.
3.1. Pour comprendre les diverses realisations du motif de Tate Z(1), le plus simple est de l'ecrire Z(1) = Hi(Gm). Le groupe multiplicatif n'etant pas une variete projective, ceci ne rentre pas dans le cadre de Grothendieck, qui demande qu'on definisse plutot Z(l) comme le dual du facteur direct H2(Pl) de H*(PI). La realisation en Z,-cohomologie de Z(1) est le module de Tate Tj(Gm) de Gm
Z,(1) = proj lim En realisation de Hodge, on a HB(Z(1)) = H1(C*) isomorphe a Z (a Q plutot, en homologie rationnelle). Ce groupe est purement de type (-1, - 1) et F_ = -1. En realisation de de Rham, HDR(Z(1)) est le dual de HI (Gm), isomorphe a Q, de generateur la classe de dz/z. L'unique periode de H1(Gm) est (3.1.1)
dz = 27ri. z
Sur Z1(1), le Frobenius arithmetique (pp (p 0 1) agit par multiplication par p. Le Frobenius geometrique agit donc par multiplication par p-1; ceci justifie l'identite citee en 1.3 (3.1.2)
L(M(n), s) = L(M, n + s).
Puisque F00 agit sur (- 1)n sur HB(Z(n)), HB(Z(n)) est nul pour a = - (-1)n, et ce(Z(n)) = 1. Pour a = (- 1)n, d'apres (3.1.1), ce(Z(n)) _ 3(Z(n)) _ (2zi)n: (
3'1 3)
pour a = (_ 1)n, ce(Z(n)) _ (2iri)n 5(Z(n)) = (2ni)n pour a = - (- 1)n. ce(Z(n)) = 1,
3.2. La fonction c(s) est la fonction L attachee an 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 = 1, 0 n'est pas un zero trivial et it serait raisonnable de definir les entiers critiques comme incluant 0. La equation (3.1.3) et les valeurs
326
P. DELIGNE
connues de C(n) = L(Z(n)) pour n critique verifient 1.8: t(n) est rationnel pour n impair <_ 0, et un multiple rationnel de (2iri)n pour n pair >_ 0. 4. Compatibilite a la conjecture de Birch et Swinnerton-Dyer.
4.1. Soient A une variete abelienne sur Q, et d sa dimension. La conjecture de Birch et Swinnerton-Dyer [15] affirme notamment: (a) L(HI(A), 1) est non nul si et seulement si A(Q) est fini.
(b) Soit co un generateur de H°(A, Qd). Alors, L(HI(A), 1) est le produit de un nombre rationnel. Le motif HI(A)(1) est isomorphe au dual H1(A) de HI(A): ceci traduit 1'existence
JAW IC 01
d'une polarization, autodualite de HI(A) a valueurs dans Z(- 1). D'apres 1.7, c+(HI(A)(1)) se calcule donc comme suit: si wI, ,wd est une base de H°(A, QI) _ F+HDR(A), et cI, , Cd une base de HI(A(C), Q)+, on a
(4.1.1)
c+(HI(A)(1)) = det<w;, e.>.
Designant encore par e; un cycle representatif, on a <w;, e,> = f1i w;. Prenons pour base (e;) une base sur Z de HI(A(R)°, Z) c HI(A(C), Z). Le produit de Pontryagin des e; est represents par le d-cycle A(R)° dans A(C), pour une orientation convenable, et, si (o est le produit exterieur des w=, le determinant (4.1.1) est l'integrale JA(R)0 w. On a
IwI = [A(R): A(R)°] SA(R)
fA (R) °
w
et 1.8 pour HI(A)(1) equivaut donc a 4.1(b) ci-dessus.
4.2. La conjecture de Birch et Swinnerton-Dyer donne la valeur exacte de L(H'(A), 1); la description du facteur rationnel en 4.1(b) a partir du motif HI(A)(1) a la forme suivante: (a) La donnee de M = HI(A)(1) equivaut a celle de A a isogenie pres. II faut commencer par choisir A. Cela revient a choisir un reseau entier dans HB(M), dont les /-adifies soient stables sous 1'action de Gal(Q/Q). (b) On choisit alors co, par example comme produit exterieur des elements d'une base d'un reseau entier dans HDR(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 6gaux a 1, et la formule du produit assure que c+(M) Ij p cp(M) est independant de w.
(c) Un autre nombre rationnel, h(M), est defini en terme d'invariants cohomologiques de A ; le nombre rationnel cherche est h(M) - f p cp(M)-I. L'invariance de la conjecture par isogenie est par ailleurs un theoreme non trivial.
4.3. Pour generaliser 4.1(a) a un motif de poids -1 M quelconque, it faudrait disposer d'un analogue de A(Q). Le groupe A(Q) peut s'interpreter comme le groupe des extensions de Z(0) par HI(A), dans la categorie des 1-motifs [6, §10] sur Q. Ceci suggsre de considerer le groupe des extensions de Z(0) 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 H2d-1(X)(d): on dispose d'une suite exacte de cohomologie 0
H2d-1(X)(d)
.
H2d-I(X
- Y)(d) a HI (X)(d) -' H2d(X)(d),
Y definit une classe de cohomologie cl(Y) e HY(X)(d), d'image nulle dans H2d(X)(d), et on prend a-1cl(Y). Cette construction correspond a celle qui a un diviseur de degre 0 sur une courbe associe un point de sajacobienne.
5. Compatibilite it l'equation fonctionnelle. Soit E un corps de nombres. Dans (E (9 C)*, nous noterons - la relation d'equivalence definie par le sous-groupe E*. PROPOSITION 5.1. Soit M un motif sur Q, a coefficient dans E. On suppose verjees les hypotheses de 1.7. On a alors
c+(M) -
(22ri)-d- (M)
. 5(M) . c+(M(1))
Pour L un module libre de rang n sur un anneau commutatif A (A sera E, on E xp C), posons det L = n L. On etend par localisation cette definition au cas oil L est seulement projectif de type fini (cette generalisation n'est pas indispensable a la preuve de 5.1). On a des isomorphismes canoniques
det(L") =
(5.1.1)
det(L)-1
et, pour tout facteur direct P et L, (5.1.2)
det(L) = det(P) det(L/P)
(on a designe par un produit tensoriel, et par -1 un dual). 11 y a ici des problemes
de signe, qu'on resoud an mieux en considerant det(L) comme un module gradue inversible, place en degre le rang de L. Nos resultats finaux etant modulo -j, 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 a(X, Y). On deduit de (5.1.1) et (5.1.2) des isomorphismes (5.1.3)
o(X, Y) = 8(YV1 XV)
et, pour F c X et G c Y, (5.1.4)
5(X,Y) = 8(F, Y/G) 8(G, X/F)-1.
LEMME 5.1.5. Via 1'isomorphisme (5.1.3), on a det(u) = det(tu). LEMME 5.1.6. Soient F et G des facteurs directs de X et Y. On suppose que 1'isomor-
phisme f: X -+ Y induit des isomorphismes fF: F - Y/G et fG1: G -> X/F. Via l'isomorphisme (5.1.4), on a det(f) = det(fF) det(f G1)-1. La verification de ces lemmes est laissee an lecteur. Pour 5.1.6, notons seulement
que, pour G1 = f-1(G) et F1 = f(F), on a X = F (1 G', Y = G Q+ F1, et que f echange F et F', Get G1. Appliquons le Lemme 5.1.6 aux complexifies de H+(M) r- HB(M) et de
F- c HDR(M). On trouve que, via l'isomorphisme (5.1.4), le determinant de I : HB(M) O C - HDR(M) O C est le produit du determinant de I+: HB(M) p C
328
P. DELIGNE
(HDR(M)/F-) D C et de l'inverse du determinant du morphisme induit par l'inverse de 7:
J : F- O C --> (HB(M) / HB (M)) O C.
Le morphisme J- est le transpose du morphisme I- pour le motif k dual de M. Appliquant 5.1.5, et prenant des bases sur E des espaces 5(X, Y) en jeu, on obtient finalement que (5.1.7)
5(M) - c+(M) C -(k)-I.
On en deduit 5.1 en appliquant au motif M la formule suivante, consequence de 3.1: (5.1.8)
c±(M) =
c±(-I)°(M(n))
Notons aussi, pour usage ulterieur, la formule analogue (5.1.9)
3(M) _
S(M(n)),
5.2. Rappelons la forme exacte de l'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 plongement complexe a (cf. 2.9). La forme generale d'abord: (a) Pour chaque place v de k, on definit un facteur local L,(a, M, s). Pour v fini, la definition de Lv(u, M, s) = oaZ, (M, Nv s) (Z,(M, t) e E(t)) depend d'une hypothese
de compatibilite analogue a (1.2.1). On a Z,(M, t) = det(1 - Ft,
HA(M)r")-I.
Pour v infini, induit par un plongement complexe v, on obtient L,(a, M, s) en decomposant HH(M) Qx E,. C en somme directe de sous-espaces minimaux stables par les projecteurs qui donnent la decomposition de Hodge, et par F00 pour v reel, en associant a chacun le "facteur 1"' de la Table (5.3), et en prenant leur produit. Pour v complexe, les sous-espaces minimaux sont de dimension un, d'un type (p, q). Pour v reel, it sont soit de dimension 2, de type {(p, q), (q, p)}, p : q, soit de dimension 1, de type (p, p), avec F, = ± 1. On note A(a, M, s) le produit des L,(a, M, s). (b) Soient 0' un caractere non trivial du groupe A Q k/k des classes d'adeles de k, et ?If ses composantes. Soient aussi, pour chaque place v de k, une mesure de Haar dxv sur k,,. On suppose que, pour presque tout v, dx donne aux entiers de k la masse 1, et que le produit Qx dx des dxv est la mesure de Tamagawa, dormant la masse 1 au groupe des classes d'adeles. On definit des constantes locales ev(o-, M, s, ',,, dxc), presque toutes egales a I et toutes, comme fonction de s, produit d'une constante par une exponentielle. Soit e(6, M, s) leur produit (independant de et des dxv). (c) L'equation fonctionnelle conjecturale est
A(a, M, s) = e(6, M, s)A(a,
s).
Pour definir les ev, une hypothese additionnelle de compatibilite entre les HA(M) est requise. Elle permet d'associer a v, a, M une classe d'isomorphie de representations complexes du groupe de Weil W(klk,,) pour v infini, du groupe de Weil epaissi ' W(kv/kv) pour v fini, et on prend le a de [4, 8.12.4], avec t = p -s. Pour v infini, ceci revient a decomposer HH(M) (DE,,, C comme en (a), a associer
329
VALEURS DE FONCTIONS L
a chaque sous-espace de la decomposition un facteur rv, et a prendre leur produit. La table des e;,, pour un choix particulier de ?'v et de dx,,, est donnee en 5.3. Pour v fini, on commence par restreindre les representations HA(M) A un groupe de
decomposition Gal(kvlkv) c Gal(k/k), puis au groupe de Weil W(kv/kv). Appliquant [4, 8.3, 8.4], on deduit de HA(M) une classe d'isomorphie pA de representations de ' W(kvlkv) sur E. 11 est loisible ici, et utile, de la remplacer par sa F-semi-simpli-
fiee [4, 8.6]. On demande que ces representations, pour A variable, soient compatibles, i.e., que si on etend les scalaires de EA a C, par a: EA --+ C prolongeant a, la classe d'isomorphie de la representation obtenue soit independante de A et de o. 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
e((p, N), s, jr, dx) = s(p 0 cos,
dx)
det(- FNv S, VP(')/Ker(N)P(I)).
REMARQUE 5.2.1. La fonction de s
e((p, N), s, 1, dx)L((p, N)", 1 - s)L((p, N),
s)-I
est la meme pour (p, N) et pour (p, 0). Ceci permet d'enoncer l'equation fonctionnelle conjecturale des fonctions L en supposant seulement une compatibilite entre les semi-simplifiees des restrictions des HA(M) a un groupe de decomposition. 5.3. Dans la table ci-dessous, on donne les facteurs locaux, et les constantes, associees aux divers types de sous-espaces minimaux de la realisation de Hodge. Pour les constantes, on a suppose ?If et la mesure dxv choisis comme suit: exp(27rix), mesure dx, v reel: v complexe: lv(z) = exp(27ri Trc/R(z)), mesure I dz A dz 1, soit pour z = x + iy, exp(47rix) et 2dxdy. On utilise les notations PR(s) = 7r-s/2 1'(s/2), Pa(s) = 2 (27r)-s F(s). place
facteur r constante
type
(pour 0, dx ci-dessus) complexe (p, q) ou (q, p), p < q reelle {(p, q), (q, p)}, p < q
(p, q), F, _
(-1)p+E,e = 0 ou 1
ie-p I'c(s -p) I o(s - p) ie p+I I PR(s+e-p)I it
Le cas qui nous interesse est celui oii k = Q. On peut dans ce cas prendre ?.V (x) = exp(2n ix), ?p(x) = exp(- 27rix) (via l'isomorphisme Qp/Zp = partie pprimaire de Q/Z), dx00 = mesure de Lebesgue dx, dxp = la mesure de Haar sur Qp dormant a Zp la masse 1. PROPOSITION 5.4. Si M est critique de poids w, on a, modulo un nombre rationnel independant de u,
L,(a, M(l))Lj6, M)-1 ,,,
(27r)-d- (M)
. (27r)-wd(M)12
D'apres 2.5, les L_(oo, ) sont independant de a. Ceci nous permet de ne verifier 5.4
que pour o, fixe. La formule est compatible a la substitution M'-+ M(1): M(1) est
330
P. DELIGNE
de poids - 2 - w, son d- est d+(M) et abregeant d(M) et d ±(M) en d et d±, on a d =
d+ + d- et
(_d__
2)+(-d+_ (-2-,d)
0.
Ceci permet de ne verifier 5.4 que pour w > -1. Pours entier, on a, modulo Q*,
PR(s) - (2z)-s72 PR(s) - (27r) (1-s) /2
(5.4.1)
Pa(s) -
(2ir)_,
pour s pair > 0, pour s impair, pour s > 0.
Pour w >-- - 1, la puissance de 2z dans la contribution de chaque sous-espace de HB(M) Qx C, comme en 5.2 (a), est donc donnee par
{(pq), (qp)}, p < q (pp), p pair 0, FF = -1 (pp) , p im p air
> 0,
QM) L0(M(1)) L_(M(1))L.(M)-1 -1 -P -1 -W p
F = -1
2
12p
-1 -P
-1 - 2
-1
-1 -
2p
2
La proposition en resulte aussitot. Posons det M = Ad (M) M (puissance exterieure sur E). PROPOSITION 5.5. e*(M) , e*(det M).
Pour des dxv choisis comme suggere en 5.4, nous prouverons plus precisement des equivalences (5.5.1)
e,*(M, ?fv, dxv) - ev (det M, f, dxv).
Posant iv(o-, M, ;lfv) = ev(u, M, t?fv, dxv) ev 1(0, det M, ?1fv, dxv), ceci equivaut a (5.5.2)
vYlv(o', M, `yv) _ 7)v(2o, M, Vv)
pour tout automorphisme a de C. 11 resulte de [4, 5.4] que, si a e Qv est de valeur absolue 1, et qu'on pose (tlfv
a)(x) = tlfv(ax), on a
(5.5.3)
v(o, M, Tv) = 7)v(u, M, ?v . a)
(pour JaJJv = 1). Pour v un automorphisme de C, et v fini, r f, est de cette forme ?v a avec II a II v =
1;dememe,$f_ = Pour v fini, la definition de ev est purement algebrique, d'oiI v ,,(o, M, 0'v) = 7)v(ro, M, zO'v) et (5.5.2) resulte de (5.5.3).
Pour v = oo, on a encore 0(o-, M, r-) = 77_(6, M, 1r00) = 770(o, M, 0'0); si on prend ?bO comme suggere en 5.3, V_ est une puissance de i independante de e, d'ou 7)-(oo, M, ?If-) = ± 1, independant de a, ce qui verifie (5.5.2). THEOREME 5.6. Modulo la Conjecture 6.6 sur la nature des motifs de rang 1, la Conjecture 2.8 est compatible a 1'equation fonctionnelle conjecturale des fonctions
L:ona
VALEURS DE FONCTIONS L
331
LUM) c+(M) - s*(M)L*(M(1)) c+(M(1)). D'apres 5.1, 5.3 et 5.5, cette formule equivaut a (29ci)-d- (M)
8(M)
(29c)-d- (M)
. (29L)-wd(M)/2 . s*(det M).
Posons D = det M et s = d-(D). On a (3(M) (S(D), d-(M) = e (mod 2), et wd(M) est le poids w(D) de D, de sorte que la formule equivaut encore a s*(D) ,., (29c)w(D)/2
(5.6.1)
.
iE
. (3(D).
Nous prouverons en 6.5 que (5.6.1) vaut pour une classe de motifs de rang 1 qui, conjecturalement (6.6), les englobe tous. 6. Exemple: les fonctions L d'Artin. DEFINITION 6.1. La categorie des motifs d'Artin est 1'enveloppe karoubienne de la duale de la categorie d'objets les varietes sur Q de dimension 0, de morphismes les correspondances definies sur Q. Par definition, chaque variete de dimension 0, X, definit un motif d'Artin H(X), le foncteur H est un foncteur contravariant pleinement fidele,
H: (varietes de dim 0, correspondances) -> (motifs d'Artin) et tout motif d'Artin est facteur direct d'un H(X). 6.2. Explicitons cette definition. Soient Q une cloture algebrique de Q, et G le groupe de Galois Gal(Q/Q). Une variete de dimension 0 est le spectre d'un produit fini A de corps de nombres, et la theorie de Galois (sous la forme que lui a donnee Grothendieck) dit que le foncteur
X = spec(A) - X(Q) = Hom(A, Q) : (categorie des varietes de dimension 0, sur Q, et des morphismes de schemas) -> (categorie des ensembles fins munis d'une action continue de G) est une equivalence de categorie. Le foncteur inverse est I F-' spectre de l'anneau des fonctions G-invariantes de Idans Q. Une correspondance d'une variete de dimension 0, X dans une autre, Y, est une combinaisons lineaire formelle a coefficients dans Q de composantes connexes de
X x Y. L'application E a=Z; H E a.(fonction caracteristique de Z,(Q) c (X x Y) (Q)) identifie correspondances et fonctions G-invariantes, a valeurs rationnelles, sur (X x Y)(Q) = X(Q) x Y(Q), et la composition des correspondances au produit matriciel. Notons Hle foncteur contravariant X H espace vectoriel QX (Q), muni de 1'action
naturelle de G; (correspondance F: X -+ Y) i-' le morphisme F*: QY(Q) , QX (e) de matrice IF. Il est pleinement fidele, et identifie la categorie des motifs d'Artin a celle des representations rationnelles de G. 6.3. Dans ce modele, si Q est la cloture algebrique de Q dans C, le foncteur "realisation de Betti" HB est le foncteur "espace vectoriel sous-jacent". La structure de Hodge est purement de type (0, 0), et l'involution F, est faction de la conjugaison complexe F E G. On a en effet un isomorphisme, fonctoriel pour les correspondances,
332
P. DELIGNE
H*(X(C), Q) = QX(C) = QX(Q) = H(X).
Le foncteur "realisation 1-adique" Ht est le foncteur HI(V) = V O QI On a en effet un isomorphisme, fonctoriel pour les correspondances, H*(X(Q), Q) = QXce> = QXce) O Q1
Calculons de meme la realisation de de Rham. Pour X = Spec(A), on a HDR(X) _ A. Ecrivant que A = HomG(X(Q), Q) = (QX ce> (D Q)c, on obtient que (6.3.1)
HDR(V) = (V O Q)G.
La formule (6.3.1) realise HDR(V) comme un sous-espace de V px Q. Ce sousespace est une Q-structure: on a (V (9 Q)G O Q V O Q. Apres extension des scalaires a Q, HB(V) et HDR(V) sont done canoniquement isomorphes. Etendant les scalaires jusqu'a C, on trouve l'isomorphisme (0.4.1) (le verifier pour V = H(X)). 6.4. Soit E une extension finie de Q. La categorie des motifs d'Artin a coefficient dans E est la categorie deduite de celle des motifs d'Artin comme en 2.1. Nous l'identifierons a la categorie des E-espaces vectoriels de dimension finie, munis d'une action de G. Les motifs d'Artin a coefficient dans E, de rang 1 sur E, correspondent aux caracteres s: G --. E*. Nous allons calculer leurs periodes. Soient done e: G -* E* et f le conducteur de e : s se factorise par un caractere, encore note e, du quotient (Z/fZ)* = Gal(Q(exp(2iri/f))/Q) de G. Notons [s] l'espace vectoriel Es, de dimension 1 sur E, sur lequel G agit par s. La Somme de Gauss
g = E e(u) px exp(2iriu/f) e [e] x0 Q est non nulle, et invariante par G: c'est une base, sur E, de HDR([r]). Le determinant de I: HB([s]) ® C __ HDR([e]) O C, calcule dans les bases 1 et g, vaut g-1. On sait
que, pour tout plongement complexe o de E, on a ag og = f, nombre rationnel independant de a, d'oii (6.4.1)
S([s]) - E --'(u) p exp(-2'riu/f) e (E (D C)*.
PROPOSITION 6.5. Soit D le motif [e](n). C'est un motif sur Q, a coefficients dans
E, de rang 1 et de poids - 2n. Posant s(-1) = (-1)'I, avec ri = 0 ou 1, on a s*(D) ,., (22r)-n i,7-n 5(D).
On sait que la constante de 1'equation fonctionnelle de la fonction L de Dirichlet 1(n) n -s (a plongement complexe de E) est donnee par
L(o, [e], s) = F.
a(6, [a], s) = i'I fs . E o
r(u)-1 exp( -
2iriu/f ).
D'apres (5.1.9), on a par ailleurs 5(D) = (2zri)nS([s]). On conclut en appliquant (6.4.1) et en notant que pours entier (s = n), fn est rationnel independant de Cette proposition verifie (5.6.1) pour les motifs [a](n). Pour achever la preuve de 5.6, it ne reste qu'a enoncer la Conjecture 6.6. Tout motif sur Q, a coefficient dans E et de rang 1 est de laforme [s](n), pour a un caractere de G = Gal (Q/Q) a valeurs dans les racines de l'unite de E et n un entier.
VALEURS DE FONCTIONS L
333
PROPOSITION 6.7. La Conjecture 2.8 est vraie pour lesfonctions L d'Artin.
Pour les motifs d'Artin, on dispose de l'equation fonctionnelle des fonctions L. De plus, le determinant d'un motif d'Artin, tordu a la Tate, est du type predit en 6.6. Les arguments des paragraphes 5, 6 montrent done la compatibilite de 2.8 a 1'equation fonctionnelle, et it suffit de prouver 2.8 pour les motifs V(n), pour V un
motif d'Artin et n un entier < 0. Si V(n) est critique, F. agit alors sur V par multiplication par -(-1)n, et c+ = 1 (cf. 3.1), et it s'agit de prouver que, pour tout plongement a de E dans C et tout automorphisme de C, on a rL(o-, V(n)) _ L(vu, V(n)). On le deduit des resultats de Siegel [14]. Voir [2, 1.2].
7. Fonctions L attachees aux formes modulaires.
7.1. Posons q = ezniz, et soitf = Eangn une forme modulaire holomorphe cuspidale primitive (new form) de poids k >--_ 2, conducteur N et caractere e. La serie de
Dirichlet Eann s admet un developpement en produit eulerien, de facteur local
en p IN egal a (1 - apps +
e(p)pk-1
p2s)-1.
Soit E le sous-corps de C engendre par les an. La forme f doit donner lieu a un motif M(f) de rang 2 a coefficient dans E, de type de Hodge {(k - 1, 0), (0, k - 1)}, de determinant [e 1](1 - k) (notations de 6.4) et de fonction L la serie de Dirichlet
E
ann-s.
Je n'ai pas essaye de definir M(f) comme etant un motif au sens de Grothendieck. Une difficulte est que M(f) apparait de facon naturelle comme facteur direct dans la cohomologie d'une variete non compacte, ou encore comme facteur direct dans la cohomologie d'une courbe modulaire complete, a coefficient dans l'image directe d'un faisceau localement constant (plutot, un systeme local de motifs!) sur la partie a distance finie de cette courbe modulaire. Ceci echappe au formalisme de Grothendieck, mais permet de definir les realisations du motif M(f). 7.2. Supposons tout d'abord que k = 2, et que s est trivial. Soient X le demiplan de Poincare, N le conducteur de f, 1'o(N) le sous-groupe de SL(2, Z) forme
des matrices dont la reduction mod N est de la forme (o *), et posons w f = E a qn dq/q = E angn 2 iridz. La forme c of est une forme differentielle holomorphe sur la courbe completee M(1'o(N)) de M(I'o(N)) = X/1o(N). Elle est vecteur propre des correspondances de Hecke: (7.2.1)
T,*wf = anwf (pour n premier a N)
et est caracterisee a un facteur pres par (7.2.1). Ce fait resulte du theoreme fort de multiplicite 1 et de la theorie des formes primitives; le lecteur peut, s'il le prefere completer (7.2.1) par une condition analogue pour n non premier a N, et faire de meme ci-dessous; l'assertion devient alors elementaire, car la condition (7.2.1) ainsi completee determine (a un facteur pres) le developpement de Taylor de (Of en la pointe ioo. Abregeons M(1'o(N)) en M et M(P'(N)) en M. Ces courbes ont une Q-structure naturelle, pour laquelle les correspondances de Hecke sont defines sur Q. La forme w f est definie sur E, et sa conjuguee par un automorphisme a de C est wo f (appliquer o, aux coefficients).
Definissons le motif M(f) come etant, dans le langage 2.1(B), le sous-motif de H1(M)E noyau des endomorphismes T,* - a,a si on veut ne considerer que des
334
P. DELIGNE
noyaux de projecteurs, remplacer T,* - a par P(T -
pour P un polynSme convenable. On sait que M(f) a les proprietes enoncees en 7.1. Dans chacune des theories cohomologiques W qui nous interessent, l'application (7.2.2)
°IC(M)
IY1(M)
est surjective, et les systemes de valeurs propres des T,* sur le noyau n'apparaissent pas dans .*'1(M). La -yP-realisation de M(f) est donc encore le noyau commun,
dans YI(M) O E, des T,' - an. Le groupe de cohomologie Hcl(M, Q) est muni d'une structure de Hodge mixte, de type {(O, 0), (0, 1), (1, 0)}, dont H'(M, Q) est le quotient de type {(0, 1), (1, 0)}. En particulier, (7.2.2) induit un isomorphisme sur les sous-espaces F1 de la filtration de Hodge; toute forme differentielle holomorphe w sur M definit ainsi une classe c C9 de cohomologie a support propre sur M. Ceci peut se voir directement: si est l'ideal des pointes, H*(M), en cohomologie de de Rham, est 1'hypercohomologie sur M du complexe -> Q1 (analytiquement, ce complexe est une resolution du faisceau constant C sur M, prolonge par 0 sur M), et le H1 recoit H°(M, 01). Supposons pour simplifier que E = Q, et calculons c+(M(f)(1)). Le motif M(f)(1) est le dual de M(f) et c+ est donc une periode de M(f) (1.7) : it faut integrer (of contre une classe d'homologie rationnelle de M, fixe par F_. Relevant M(f) dans .'(M), on voit qu'on peut plutot integrer w f contre une classe d'homologie sans support de M, fixe par F. Toutes les integrales non nulles de ce type seront commensurables.
Pour calculer F_, it est utile d'ecrire M comme quotient de X± = C - R par le sous-groupe de GL(2, Z) forme des matrices ayant pour reduction mod N une matrice (o *). La conjugaison complexe est alors induite par z - z, et l'image dans M(C) de iR+ est un cycle sans support fixe par F_. La formule L(M(f), 1) fowf justifie 1.8 pour M(f)(1):le second membre est,ou bien nul,ou bien - c+(M(f)(1)). REMARQUE 7.3. On a aussi
i
c (M( )(1)) - falb (Of + f-alb
wf
7.4. Pour e (et E) quelconque, it faut remplacer 10(N) par r1(N): le sous-groupe de SL(2, Z) forme des matrices de reduction mod N de la forme (o i). Comme cidessus, pour calculer F_, it est plus commode de travailler avec Xy et le sous-groupe de GL(2, Z) forme des matrices de reduction mod N de la forme (*). Par ailleurs, une dualisation apparait, cachee en 7.2 par la symetrie de la correspondance T. Pour une definition convenable de T., on a (a) M(f) est le noyau commun des T,* - a dans H1(M), M = X/1'1(N). (b) On a tT,*wf = de sorte f, et T,*wf = a, w f (voter la formule a = que c of est dans la realisation de de Rham de M(f) = M(f)v(-1). On trouve que c+(M(f)(1)) e (E (D C)*/E* e C*Hom(E,O)/E* est donne par le systeme de periodes
c+(M(f)(1)) - J0 a'af )a , si ce dernier est non nul. Noter que si dune de ces integrales s'annule, elles s'annulent toutes. Tel est le cas si et seulement si, dans la partie de dhomologie sans support
VALEURS DE FONCTIONS L
335
tendue par le cycle iR+ et ses transformes par les Tn, le systeme de valeurs propres a pour les T n'apparait pas. Ceci justifie 2.8, et, partiellement 2.7, pour M(f)(1). REMARQUE 7.5. Lessystemesdevaleurs propres des Tndans Ker(. "(M) -j 1(lll)) sont lies aux series d'Eisenstein, alors que ceux qui apparaissent dans W'(M) sont lies aux formes paraboliques. C'est pourquoi ces ensembles sont disjoints. Il en re-
sulte que la structure de Hodge mixte de HI(M, Q) est somme de structures de Hodge: 1'extension de H'(M, Q) par Ker(HI(M, Q) ---> H'(M, Q)) splitte. Ceci, generalise au cas de n'importe quel sous-groupe de congruence de SL(2, Z), equivaut au theoreme de Manin [9] selon lequel la difference entre deux pointes est toujours d'ordre fini dans la jacobienne (cf. [6, 10.3.4, 10.3.8 et 10.1.3]). La demonstration donnee ne differe d'ailleurs pas en substance de celle de Manin.
7.6. En poids k quelconque, it faut remplacer la cohomologie de M par la cohomologie de M a coefficient dans un faisceau convenable. Pour decrire ce qui se passe, je remplacerai Met M par M et M,,, relatifs au groupe de congruence I'(n), avec n multiple de N et > 3. On redescend ensuite a M et M en prenant les invariants par un groupe fini convenable. Soient g: E ---> M la courbe elliptique universelle et j l'inclusion de M dans Mn. La cohomologie.a considerer est H1(Mn, j*Symk-2(R1g*Q)). Les realisations de M(f) sont facteur direct de cegroupe, calcule dans la theorie de cohomologie correspondante. On peut comme precedem-
ment les relever dans HI(M,,, Symk-2(Rlg*Q)). Pour calculer c± du dual de M(f). it faut alors integrer la classe dans HI definie par f contre des classes d'homologie sans support de M. a coefficient dans le systeme local dual de Symk-2(R1g*Q). Si on prend le cycle image de iR+, muni de diverses sections du dual (une base), on trouve les integrales d'Eichler J0
f(q) gg (2ziz)l
(0 5 l 5 k - 2)
(periodes paires pour 1 pair, impaires pour l impair)-et on sait comment ecrire L(M(f), n) pour n critique en terme de ces periodes. PROPOSITION 7.7. Soit M un motif de rang 2, a coefficient dans E, de type de Hodge
{(a, b), (b, a)) avec a
b. Alors, d±Symn(M) et c±SymnM sont donnes par les for-
mules suivantes:
(1)sin=21+1:d±=1+1,et C±SymnM = C±(M) (1+1) (1+2) /2 C+(M)1(1+1) /2 o(M)l (1+1) /2;
(2) sin= 21: d+ = 1 + 1, d- = 1, et c+SymnM = (c+(M) c (M))1(1+1) /2 b(M)l(1+1) /2, c SymnM = (c+(M) C (M))1(1+1) /2 5(M)1(1-1) /2.
C'est une question de simple algebre lineaire, que nous traiterons par "analyse dimensionnelle". (a) HB(M) est un espace vectoriel de rang 2 sur E, muni d'une involution F_ de la forme (o _01) dans une base e+, e convenable. Les d±SymnM sont les dimensions des parties + et - de la puissance symetrique nieme, de bases respectives 2, {e+", } et {e+("-') a e+("-3) a 3, } -d'oil les valeurs annoncees. a+("-2)
a
336
P. DELIGNE
(b) Soit co, 72 une base du dual de HDR(M), telle que w annule F+HDR(M). Le
sous-espace F- de HDR(SymnM)V - SymnHDR(M)V admet pour base les wn, wn-d±+1 d±-1 et wn-1 (wn-1 v) A ...
e+n A (e+(n-2) e-2) A ...),
(7.7.1)+
c+SymnM = <wn A
(7.7.1)-
c SymnM = <wn A (w' 1 yi) A ...,(e+(-1) e) A
e3) A ...>.
(c) Posons V = HB(M) O C - HDR(M) O C; c'est un E Q C-module. Les formules ci-dessus montrent que c±SymnM ne depend que du E O C-module V, de sa base e+, e, de (o e V*, et de l'image V de 7) dans V*/<w): le d+-vecteur a gauche du produit scalaire (7.7.1) ne change pas si on remplace 7) par 7) + 2w. Par ailleurs, <w, e+> et <w, e-> sont inversibles, et V est une base de V*/<w>. Le systeme (V, e+, e , w, V) est done decrit a isomorphisme pres par les quantites c+ = <w, e+>, c = <w, e-> et 8 = <w A7), e+ A e >, dans (E Qx C)*. Remplacer e+, e-, w, V par Ae+, ,ue , co, vV/2u remplace c+, c et 3 par Ac+, uc et v6. Pour c+ _
c = 5 = 1, le second membre de (7.7.1)+ est dans Q*. Dans le cas general c±SymnM est done un multiple rationnel du produit de c+(M), c (M) et 8(M)/c+(M)c(M) aux puissances respectives les degres auquels figurent e+, e et 7)
dans (7.7.1). On s'epargne la moitie du calcul en notant que remplacer F par -F echange e+ et e , done c+ et c , respecte 3, et echange les (7.7.1)+ pour n impair, les conserve pour n pair.
n=21+1:d+=d-=1+1, deg 77 dans (7.7.1)± = 0 + 1 +
+ 1 = 1(1 + 1)/2,
deg e+ dans (7.7.1)+ _ (21 + 1) + (21 - 1) + = deg e- dans (7.7.1)-,
+ 1 = (1 + 1)2
deg e dans (7.7.1)+;
n=21:d+=1+l,d-=1, deg V dans (7.7.1)+ = 0 + 1 + deg y2 dans (7.7.1)- = 0 + l +
+ 1=1(1+ 1)/2, + (1 - 1) = 1(1 - 1)/2,
dege-dans(7.7.1)+=21+(21-2)+ +0=1(1+ 1), deg e±dans (7.7. 1)- = (21 - 1) +
=12.
7.8. Cette proposition nous fournit une conjecture pour les valeurs aux entiers critiques de L(SymnM(f), s). Voici les formules: (7.8.1)
L(6, M(f), s) = Ec.an-s = H Lp(a, M(f), s) P
ob pour presque tout p
Lp(M(f), s) = (1 - apps +
e(p)pk-1-2s)-1 =
((1-apps) (1 -app s))-1,
avec e un caractere de Dirichlet. On a A2M(f) = [e 1](1 - k), d'oi (7.8.2)
o(M(f)) -
p exp (-2-riu/F),
si e est de conducteur F, et (7.8.3)
L* (M(f ), m) - (27Ci)m c±(M(f
1)m, pour 1 < m < k - 1.
337
VALEURS DE FONCTIONS L
(7.8.4)
L(6, SymnM(f ), s) = IT Lp(u, SymnM(f ), s) p
ou pour presque tout p Lp(SymnM(f), s)-1 =
(1 - apap `P s); II i=a
conjecturalement, pour m critique, on a L*(SymnM(f), m) .., (29ri)md±SymnM(f) .c SymnM(f),
1)m,
oii c± et d± sont donnes en terme de c±(M) et b(M) (caracterises par (7.8.2), (7.8.3)) par les formules 7.7. Pour 1'evidence numerique en faveur de cette conjecture, voir [18].
8. Caracteres de Hecke algebriques. Le lecteur trouvera dans [3, paragraphe 5], dont nous utiliserons les notations, les definitions essentielles relatives aux caracteres de Hecke algebriques (= grossencharaktere de type A0). Conjecture 8.1. Soient k et E deux extensions finies de Q, et x un caractere de Hecke algebrique de k a valeurs dans E.
(i) Il existe un motif M(x) de rang un sur k, a coefficients dans E, tel que, pour toute place 2 de E, la representation 2-adique HAM(x) soft celle definie par x: le Frobenius geometrique en ,9 premier au conducteur de x et a la caracteristique residuelle 1 de A agit par multiplication par x(9). (ii) Ce motif est caracterise a isomorphisme pres par cette propriete. (iii) Tout motif de rang 1 est de laforme M(x). (iv) Decomposons k Q E en produit de corps: k O E = fl K1, et ecrivons la partie algebrique xaig : k* --> E* de x sous la forme xaig(x) = ]-[ NKiIE(x)ni. La decompo-
sition de k © E induit une decomposition de HDR(M(x)) en les HDR(M(x))i = HDR(M(x)) Qk©E K1; avec cette notation, la filtration de Hodge est la filtration par les (Dn,>p HDR(M(x))1.
L'unicite 8.1(ii) impose aux M(x) le formalisme suivant (8.1.1) (8 1 2)
(8.1.3)
M(x'x") - M(x') o M(x'). Si c : E -* E' est une extension finie de E, M(cx) se deduit de M(x) par extension des coefficients de E a E'. Si k' est une extension finie de k, M(x o Nk,Ik) se deduit de M(x) par extension des scalaires de k a k'.
REMARQUE 8.2. Posons les notations:
c = la conjugaison complexe, Q = la cloture algebrique de Q dans C, S = Hom(k, Q) = Hom(k, C), J = Hom(E, Q) = Hom(E, C). La decomposition de k px E en les Ki correspond a la partition de S x J en les orbites de Gal(Q/Q). Tout homomorphisme algebrique V: k* -* E* s'ecrit sous la forme ]I NK,,E(x)n`. Si l'orbite sous Gal(Q/Q) de (o-, z) e S x Icorrespond a Ki, i.e., si a O z: k 0 E --> C se factorise par Ki, nous noterons n(71; u, z), ou simplement n(u, z), 1'entier ni. La fonction n(u, z) est constante sur les orbites de Gal(Q/Q). Si V est la partie algebrique d'un caractere de Hecke x, on a de plus (8.2.1)
L'entier w = n (7); u, z) + n (iy; ca, r) est independant de 6 et z.
338
P. DELIGNE
C'est le poids de x (et de M(x)). On ecrira parfois n(x; a, r) pour n(Zaig; a, r). Reciproquement, un homorphisme 97 verifiant (8.2.1) est presque de la forme xalg, pour x un caractere de Hecke convenable : (a) une de ses puissances 1'est; (b) it existe une extension finie c: E --> E' de E telle que n) le soit; (c) it existe une extension finie k' de k telle que V o Nk,,k le soit.
La regle 8.1(iv) permet de deduire la bigraduation de Hodge de HM(x) de sa structure de E-module : le facteur direct HoM(x) ©E,. C de HoM(x) Qx C est de type de Hodge (p, q), avecp = n(x; o,, r) et q = w - p = n(x; or, cr). EXEMPLE 8.3. Soit A une variete abelienne sur k, a multiplication complexe par E. On suppose H1(A) de rang I sur E (type CM). Il resulte de la theorie de Shimura et Tanayama que H1(A) verifie la condition de 8.1(i), pour un caractere de Hecke algebrique x de k a valeurs dans E, et que la partie algebrique xalg de x se lit sur le k O E-module Lie(A): Zalg(x) = detE(x O 1, Lie(A))-1 EXEMPLE 8.4. Prenons pour k le corps des racines niemes de l'unite, soit V 1'hyper-
surface de Fermat d'equation projective EmO X," = 0 et soit M le motif "cohomologie primitive de dimension moitie de V". Soit G le quotient de un +1 par son sous-groupe diagonal. Ce groupe agit sur V par (a,) * (X1) _ (a1X,), et sur M par transport de structure. Decomposant M a l'aide de la decomposition de l'algebre de groupe Q[G] en produit de corps, on obtient des motifs verifiant 8.1(i) pour des caracteres de Hecke algebriques convenables : ceux introduits par Weil dans son etude des sommes de Jacobi. EXEMPLE 8.5. Le motif Z(- 1) verifie 8.1(i) pour x = ]a norme. Pour x d'ordre fini, un motif d'Artin convenable verifie 8.1(i). 8.6. Pour chaque a E S, H,(M(Z)) est de rang 1 sur E. Choisissons une base ev de chaque HH(M(Z)). Outre sa structure de E-module, la somme des HQ(M(Z)) O C
a une structure naturelle de k p C = Cs-module -au total, une structure de k Ox E Qx C-module libre de rang 1, pour laquelle e = LeQ est une base. 8.7. La realisation de de Rham HDR(M(x)) est un k OO E-module libre de rang 1.
Choisissons en une base w. La somme des I,: H,(M(x)) Qx C - HDR(M(Z)) pk,v C est l'isomorphisme (0.4) de k Qx E p C-module I: OO Hq(M(Z)) O C
HDR(M(x)) De C.
Soit w une base du k O E-module HDR(M(x)), et posons p'(x) = w/I(e) E (k O E (D C)*. Cette periode depend de x, w et e. Prise modulo (E (& F)* et E*S,
elle ne depend que de )6. Nous noterons p'(x; a, r)-ou simplement p'(o-, r)-la composante d'indice (a, r) de son image par l'isomorphisme E Qx F Qx C -->" CsxJ 8.8. Soit A comme en 8.3 et calculons les periodes p'(6, r), modulo Q*, pour le motif H1(A). On commence par etendre les scalaires de k a Q*, a l'aide de o-. Si
n(6, r) = 1, it existe alors une 1-forme holomorphe w definie sur Q telle que u*w = r(u)o) pour u e E, et, pour Z E H1(A(C)), on a p'(6, r) - f z co. On peut passer de la au cas general a l'aide de la formule p'(a, r) p'(a, cr) - 2n i.
8.9. La conjecture 8.1(ii) affirme en particulier que si deux motifs verifient la condition de 8.1(i), ils ont meme periode p. Pour les motifs 8.4, les periodes s'expri-
VALEURS DE FONCTIONS L
339
ment en terme de valeurs de la fonction I' et, si on travaille mod Q*, 8.1 suggere la conjecture de B. Gross [7] reliant certaine periodes a des produits de valeurs de la fonction I. La comparaison de 8.4 et 8.5 mene a la Conjecture 8.11, 8.13 suivante. Le resultat annonce apres 0.10 permet de la demontrer. 8.10. Soient N un entier, k = Q(exp(2icilN)), 9 un ideal premier de k, premier
a N, k, le corps residuel et q = N. _ Ik,I. On notera t l'inverse de la reduction mod -0, des racines N1eme de I dans k, a celles de k.
Pour a e N-1 Z/Z, a 0 0, considerons la somme de Gauss g(g, a,) _ -1] t(x a(q-1)) (x). La somme est etendue a kj et T: k, --> C* est un caractere additif non trivial. Soit a = En(a)3a dans le groupe abelien libre de base N-IZ/Z - {0}. Si En(a)a
= 0, le produit des g(-0, a, )n(a) est independant de Yf et on pose g(9, a) =
g(g, a, b)n(a). Weil [16] a montre que, comme fonction de 9, g(9, a) est un caractere de Hecke algebrique xa de k a valeurs dans k. Notons le representant entre 0 et 1 de a dans Z/N. Si a = En(a)da verifie (*)
Pout tout u e (Z/N)*, on a En(a) = 0,
it resulte la determination par Weil de la partie algebrique de xa que X. est d'ordre
fini; on note encore xa le caractere de Gal(Q/k) valant xa(g) sur le Frobenius geometrique en 9. Posons 1(a) = P()n(a). Une premiere partie de la conjecture: algebraicite de 1(a) lorsque a verifie (*), a ete prouvee par Koblitz et Ogus: voir l'appendice. On desire avoir, plus precisement: Conjecture 8.11. Si a verifie (*), on a oP(a) = xa(a) P(a) pour tout as e Gal(Q/k). Si a est invariant par un sous-groupe H de (Z/N)*, on peut preciser 8.11 en remplacant Gal(Q/k) par Gal(Q/kH), et en utilisant SGA 41/2 6.5 pour definir un caractere de ce groupe, a valeur dans les racines de l'unite de V. C'est ce que nous faisons ci-dessous.
8.12. Soient Hun sous-groupe de (Z/N)*, 9 un ideal premier de kH, premier a N, i son corps residuel et ?: )e --> C* un caractere additif non trivial. Pour a e N-1 Z/Z - {0}, soient 9a,i les ideaux premiers de kH(exp(2nia)) au-dessus de 9. Si x' est le corps residuel en 9a,i, et que Ix'I = q', on note g(B0,i, a, ?) la somme de Gauss - Et(xa(q -1))tlf(Tr,,I,rx) (somme etendue a x'*). Le produit g(', a, 7Il) = Ili g(ga,i, a, (f) ne depend que de l'orbite de a sous H. Soit a = En(a)oa, invariant par H, et verifiant En(a)a = 0. Pour chaque orbite 0 de H dans N-1 Z/Z - {0}, on note n(0) la valeur constante de n(a) sur 0. On pose
g(g, a) =
[I
g(g, a, ?11)n(a)
a mod H
Comme fonction de 9, g(g, a) est un caractere de Hecke algebrique X. de kH a valeurs dans V. Pour le prouver, on se ramene par additivite a supposer les n(a) > 0. On applique alors [3, 6.5] pour F = kH, F = Q, k = notre k, et I = une Somme
disjointes de copies d'orbites H: n(0) copies de 0; pour i e I, d'image a/N dans N-1Z/Z, on prend pour Ai le compose Z(1)F - (Z/N)(1)F = ,UN(k) -1 k*; le caractere de Hecke obtenu est le produit de X. par le caractere "signature de la
340
P. DELIGNE
representation de permutation de H sur I". Un cas particulier de ce resultat figure deja dans Weil [17]. Le caractere xa de 8.10 est le compose de X. ci-dessus avec la norme Nk/kH Conjecture 8.13. Soient 9 un ideal premier de kH, premier a Net F9 un Frobenius geometrique en J'. Si a invariant sous H verge (*), on a F,,,r(a) = g(.9, a) . 1'(a) En particulier, si le groupe des racines de 1'unite de kH est d'ordre N', on a F(a)N' E kH.
On peut de 8.13 deduire la variante suivante, apparemment plus generale. On remplace la condition (*) par (*')
En(a) = k est un entier independant de u e (Z/N)*.
Le caractere de Hecke N9-k . g(g, a) est alors d'ordre fini. On l'identifie a un caractere x de Gal(Q/kH), et on espere avoir u((22ri)-k r(a))
= x(6) '
((2'ri)-k 1'(a)).
8.14. Si k est un corps extension quadratique totalement imaginaire d'un corps totalement reel, soit, comme nous dirons, un corps de type CM, Shimura [13] a determine les valeurs critiques des fonctions L des caracteres de Hecke algebriques de k, au produit pres par un nombre algebrique. II les exprime en terme de periodes de varietes abeliennes de type CM, a multiplication complexe par k. Dans la fin du paragraphe, nous montrons que son theoreme est compatible a la Conjecture 2.8, qui les exprime en terme de periodes de motifs sur k, de rang 1.
8.15. Soient x et M(X) comme en 8.1. Excluons le cas 8.5, et supposons que Rk,Q M(x) verifie 1.7. Le corps k est alors totalement imaginaire, et les nombres de Hodge hPP sont nuls : avec les notations de 8.1 et 8.2, aucun n; n'est egal a w/2. Notre premiere tache est de calculer c+Rk,QM(x) en terme des periodes p'(x; a, z). Rappelons que c'est le determinant de l'isomorphisme de E Qx C-module I+: He Rk/QM(x) O C
HDRRk/QM(x) 0 C,
calcule dans des bases definies sur E.
Choisissons les eQ de 8.6 de sorte que FFe0 = e, Les eQ + eCQ forment alors une base de HB Rk,QM(x) c HBRk,Q Mx = OQEsHQM(x). Notons au passage que
d+ = d = Z[k: Q]. Soit S le quotient de S par Gal(C/R). Pour calculer c+ det(I+), nous utiliserons la base (e9 + ecQ) de HB . Elle est indexee par S.
D'apres 8.1(iv), la filtration de Hodge de HDRRkIQM(x) = HDRM(x) se lit sur sa structure de k 0 E-module : si on note (k 0 E)+ le facteur direct de k Qx E produit des Ki tels que n; < w/2, le quotient HDRRk,QM(x) de HDRRk,QM(x) est le facteur direct correspondant: HDRRk,QM(x) = HDRM(x) Ok®o E (k (D E)+.
Soit w comme en 8.7, et utilisons la structure de k Qx E p C = CsxJ-module On a par definition I(eQ) _ de HDRM(X) pour decomposer w: w et donc I+(eo + ecQ) =
E
n (o, z) <w12
p'(6, z)-lwQ,z +
E
n (o, z) <w12
p'(C6, v)-Io)co,z;
VALEURS DE FONCTIONS L
341
c'est la Somme, indexee par v e J, de termes egaux a p'(o,, v)-1wo,r pour n(o, v) < w/2, et a r)-1wca,r pour n(u, r) > w/2. Pour a S, de representant u, posons wb =
E
wo,r +
n (o, r) <w/2
E
n (cc, r) <w12
wco,r
Les I+(ea + e,a) sont des multiples des co, par des elements de E ® C; les wQ forment done une base de HDR. Dans les bases ea + ecv et wQ, la matrice de I+ est
diagonale; son determinant det'(I+) e (E x C)* = C*J a pour coordonnees det'(I+)r =
II
n (a, v) <w/2
p'(u, v)-1.
Soient les applications Es - ES: 1Q --* la + l,a, pour u image de u, ES O C k O E O C, deduit de l'isomorphisme de k Q C avec CS, et la projection de k O E sur (k 0 E)+. Par composition, on obtient un isomorphisme de E Q C-modules: Es
C
(k (D E)+ ®C.
Nous noterons D(x) son determinant, calcule dans des bases definies sur E des deux membres. Identifiant HDR a k Qx E a l'aide de la base w, on voit que c'est le determinant de l'application identique de HDR Qx C, calcule dans la base co,,, a la source, et une base definie sur E, au but. Notant Dc(x) ses composantes dans CJ, on trouve
pour c+ = det'(I+) D(x) la formule suivante. PROPOSITION 8.16. On a C+Rk/QM(x)
fl
n (o, r) <w/2
p'(6, ,r)-1
.
Dc(x))VEJ
REMARQUE 8.17. Supposons k de type CM, extension quadratique de ko totalement reel. Le quotient S de S s'identifie alors a 1'ensemble des plongements complexes de ko, et le diagramme
ES®Ck0QEQC 1
ES®C
k ®E( C
(k( E)+Qx C
est commutatif. L'application composee ko Qx E - (k 0 E)+ est done un isomorphisme, et D(x) est encore le determinant de ES D C ->` ko Qx E p C, deduit par extension des scalaires de Q a E de l'isomorphisme CS -* - ko px C. Ceci fournit pour D(x), bien defini mod E*, un representant dans (Q 0 C)* = C* c (E O C)*, a savoir le determinant de l'inverse de la matrice (o-a), pour a e S et a parcourant une base de ko sur Q. L'isomorphisme CS --+- ko Qx C transforme la forme quadratique xx? en la forme Tr(xy). Ceci permet d'identifier (det(oa))2 au discriminant deko : D(x) - racine carree du discriminant de ko. 8.18. Notons p"(x; o, r) l'image de p'(x; ou, r) dans C*/Q*. Elle ne depend que de x, or et v. Si un homomorphisme algebrique v: k* --> E* verifie (8.2.1), une de ses puissance est la partie algebrique d'un caractere de Hecke: r)N = xalg. De plus, Si xaIg = xa1g, x' et x" ne different que par un caractere d'ordre fini et x'M = x"M pour M convenable. On deduit de (8.1.1) que p"(xM; 6, z) = p"(x; a, z)M, et ceci permet de poser sans ambiguite
P. DELIGNE
342
AV; 0"'r) = p(x; or, ,t)1/N, pour 7N = xeig.
Ces periodes obeissent an formalisme suivant: o, r). (8.18.1)p(v, Vn; a, v) = p(7/; a, (8.18.2) p(72; o-, v) ne change pas quand on remplace E par une extension E' de E, et v par un de ses prolongements a E'. (8.18.3) p(V; a, r) ne change pas quand on remplace k par une extension k' de k, a par un de ses prolongements a k', et x par x o Nk,/k r)p(7In;
(8.18.4) Si a est un automorphisme de k, et 3 un automorphisme de E, on a AV; a, z) = p(j3 a 1, Ca-1 Z./i-1). (8.18.5) Le complexe conjugue de p(7); a, r) estp(7); 6, i).
(8.18.6) Pour k = F = Q, p(Id; Id, Id) = 27ri. Les formules (8.15.1) a (8.15.3) resultent de (8.1.1) a (8.1.3), (8.18.4) et (8.18.5) se voient par transport de structure, et (8.18.6) resulte de (8.5). 8.19. Soit : k* -+ E* un homomorphisme verifiant (8.2.1). On suppose aussi que n(7); a, v) ne vaut jamais w/2, ce qui permet de definir (k px E)+ comme en 8.15. Definissons 7)* : E* -* k* par
7)*(y) = detk(1 0 y, (k (D E)+). Cet homomorphisme verifie encore (8.2.1), et n(r)*; a, v) = 1
si n(ij; a, z) < w/2,
=0
si n(7J ; oa, a) > w/2.
Si c* est la partie algebrique d'un caractere de Hecke x*, M(x*) est le H1 d'une variete abelienne sur E, a multiplication complexe par k, dont 1'algebre de Lie, comme k px E-module, est isomorphe a (k x® E)+. PROPOSITION 8.20. Avec les hypotheses et notations de 8.19, prenons pour E un sous corps de C, et no tons 1 1'inclusion identique de E dans C. On a
H
n (a,1) <w/2
p(7);
1) = 11 p(7)*; 1, o)n(nQ.l), a
Sic : E - E' (-- C est une extension finie de E, le membre de gauche ne change pas quand on remplace 7) par c7) (8.18.2). On a (cj)* _ 7]* o NE-/E et, par (8.18.3), le membre de droite ne change pas non plus. Si c : k --> k' est une extension finie de degre d de k, le membre de gauche est eleve a la puissance d quand on remplace k par k', et 7) par 7) o Nk./k: un plongement complexe de k est induit par d plongements de k', et on applique (8.18.3). De meme pour le membre de droite, par (8.18.2) et 1'egalite (7) o Nk./k)* = c27*.
Ces compatibilites nous ramenent a supposer que E est galoisien et que k est
isomorphe a E. Pour chaque isomorphisme w de k dans E, posons n(c)) = n(7); 1 o cv, cv). Notant additivement le groupe des homomorphismes de k* dans E*, on a 27 = Y, n(w) w. Puisque p(7); 1 o cv, 1) = p(7) o w l; 1, 1) (8.15.4), on a (8.20.1)
II
n (a, 1) <w12
Par ailleurs,
p(r); a, 1) = H
n (w) <w12
p() o a)-1; 1, 1) = p ( E
\n (w) <w/2
27 o cv-1; 1, 1).
VALEURS DE FONCTIONS L
n(w)<w/2
rI ° w-1 =
n(w1)<w/2; o)2
n(w2)w2 o wi 1 =
343
(02 ° wii
n(w2) w2
n(w1)<w/2
_ En((02)w2°71*. W2
Ceci permet de continuer (8.20.1) par fT P(a) o YJ*; 1, l)n(w) = 11 P(1*; 1, 1. ))n (w) = 11 P(7I*; 1, o
w
W
(nouvelle application de 8.15.4), et prouve 8.20.
8.21. Combinant 8.16 et 8.20, on trouve pour la composante d'indice 1 de c+ Rk/Q M(x) E (E (D C)*/E* = C*1/E*, 1'expression suivante, mod Q* Ci Rk/Q M(x) - 11 P(xa g; 1, a)-n(x;''1) 0
Si x (i.e., M(x)) est critique, la Conjecture 2.8 affirme done que L(1 ° x, 0) - II P(xa g; 1, 6)-n(X;°' 1)
(mod Q*).
0
Pour E assez grand, xa g est la partie algebrique d'un caractere de Hecke x*, et les
periodes s'interpretent comme periodes d'integrales abeliennes (8.19), (8.3). L'enonce obtenu est celui que Shimura a prouve pour k de type CM, ou abelien sur un corps de type CM (avec une restriction sur le poids). REMARQUE 8.22. Si 77 : k* -* E* verifie (8.2.1), it existe des sous-corps k' de k et c : E' --> E de E, soit de type CM, soit egaux a Q, et une factorisation 1 = e77 Nk/k.. On a alors p(71; U, z ) = p(77'; U I k', z I k'). Si maintenant k et E sont de type CM (ou Q), et qu'on note encore c leur conjugaison complexe, on a cu = Uc, or = zc, et ijc = ciy, d'ou P(r); U, Z-)- = P(V; CU, cr) = P(71; CC, 'CC) = P(C7C 1; U, Z-) =
U, z)
les periodes, a priori dans C*/Q*, sont reelles, i.e., dans R*/(Q* (1 R*). REMARQUE 8.23. Soient G le groupe de Galois de la reunion des extensions de type CM de Q dans Q c C, et c e G la conjugaison complexe. C'est un element central
de G. Si (p est une fonction localement constante a valeurs entieres sur G, nous poserons (p*(x) = (p(xc). Supposons que (p + co* est constante. Soient G1 un quotient fini de G tel que (p se factorise par une fonction (p1 sur G1, et k le corps correspondant. L'hypothese faite signifie que 1'endomorphisme X(p1(U)U de k* verifie (8.2.1). La periode p(2'(p1(U)u; 1,1) ne depend pas du choix de G1; on pose P((p) _ p(E(p1(U) U; 1,1). La fonctionnelle Pest un homomorphisme dans C*/Q* du groupe des fonctions localement constante a valeurs entieres sur G qui verifient (p + (p* _ constante.
Appendix by N. Koblitz and A. Ogus. Algebraicity of some products of values of
the r function. Let AN = N-1 Z/Z - {0}, and let UN = (Z/NZ)* operate on AN in the obvious way. Iff:AN -> C, define (f>: UN --> C by (u) _ EQEAN denotes the representative of a between 0 and 1.
We first compute H. = {f: AN -> Q: is constant}. EXAMPLE. If n is a divisor of N and a e AN, consider the set Sn,,, = [{a + k/n: k = 0, 1, . . ., n - 11 U { - na}] n AN- It is easy to see that its characteristic function en,Q is in HQ. Note that S1,0 = {a, - a}.
344
P. DELIGNE
PROPOSITION. HQ is generated by {e,,,a : n = 1 or n is prime and na : 0}.
PROOF. The orbits of AN under the action of UN correspond to the divisors d of
N with I < d < N: each orbit can be written uniquely in the form UN(l/d). The stabilizer subgroup Id of 1/d is {u e UN: u =1 mod d} and the orbit of 1/d is canonically isomorphic to Ud. Thus, a function f on AN is determined by the collection of functions fd : Ud -> Q defined by fd(v) = f(v/d).
To prove the proposition, we first complexify, so as to be able to work with characters of UN. LEMMA. If f : AN -> C and if x is a character of UN, then the inner product of and x is given by:
( I x)
E L(O, xd) I Id I
(fd I Xd),
where the sum is taken over those divisors d of N such that x is pulled back from a character xd of Ud. PROOF. We have
( I X) =UEUN E (u) x(u) = E E f (ua) x(u) aEANuEUN
2] E E fd(udv) x(u),
dIN vEUd UEUN
where ud E Ud is the image of u.
Now if we choose a set Ud s UN of coset representatives for UN -> Ud, we can write: fd(udv)x(u) _
E fd(udv)x(u')x(u) = L1
u'EUd uSld
UEUN
u'EUd
fd(udv)x(u') E x(u)' UEld
Of course, this sum is zero unless x is trivial on Id, i.e., unless x is pulled back from a character xd of Ud, in which case it is
E fd(u'v)xd(u')IIdI = E fd(w)xd(w)xd(v)IIdI =
u'EUd
l xd(v)Ild1(fdlxd)
,vEUd
If we substitute this into the above expression for (Ix) and note that IvI/d, where Ivy is the smallest positive representative of v e Ud, we find:
(Ix) _ E E Ivlxd(v)IId1d-1(fdlxd) d
VEUd
- E L(0, xd)IIdk (fdl xd)
as claimed.
d
To prove the proposition, we let d f be the largest divisor of N such that fd j4 0; it suffices to prove that any f in H can be written as g + f', with g a linear combination of the e's and d f. < d f. Let d = d1; we shall show below that fd is a linear combination of functions h1 which factor through Ud -> Ud/{±1}, and functions hp which factor through Ud -> Udlp for some prime divisor p of d, p 0 d. Any function of the first type is invariant under ± 1, and hence the corresponding function on the orbit UN(1/d) is a linear combination of el,a's. A function of the second type is invariant under the kernel K of Ud -> Udl which has order p if p divides d/p and
order p - 1 otherwise. In the first case, K = {l + kd/p: k = 0,...,p - 1}, and in
VALEURS DE FONCTIONS L
345
the second, it is this same set with one element deleted, namely, the value of I + kd/p which is divisible by p. Thus, the set Sp,,,,,d = (Kw)ld, with the addition of one or two elements in the orbit UN(p/d). Hence it is clear that the corresponding function on the orbit UN(1/d) can be written as a linear combination of ep,¢'s and functions supported on the orbit of p/d, and we have obtained our desired decom-
position f = g + f'. It remains for us to prove the claim aboutfd. If d is twice an odd number the map Ud -* Ud12 is an isomorphism, and the claim is trivial. In the other cases, choose a primitive odd character xd of Ud, and let x be the pull-back of xd to UN. Since x is nontrivial and is constant, (< f > I x) = 0. Let us look at the expression for this inner product provided by the lemma; the only nonzero terms come from those fe
such that fe 0 0 and IQs Ker(x). But then Ker(x) contains I¢ and Id, hence also Idle = I,,, where m = g.c.d. (d, e). Since x is primitive, m = d and d divides e, and since fe = 0 fore > d, 0 = ((f > Ix) = -L(0, xd) I Id I (fd l xd) Since xd is odd and primitive, we conclude that (fdl xd) = 0, i.e., fd is orthogonal to every odd primitive char-
acter of Ud. It follows that fd can be written as a linear combination of characters which are even (factor through Ud/{ ± I)) or imprimitive (factor through some Ud, p). When d = p, we should also remark that the constant functions on Up are also invariant under ± 1, and hence are already covered by the first case. THEOREM. Suppose that f : AN --> Q is such that is constant : E,, f(ua) _ k for all u E UN. Then
r(J) =def Z-k III'(
PROOF. The map < >: He -> Q sending any f to E¢ f(a) is evidently linear, and hence r(f + g) = r(f)r(g), for f and g in HQ. By the proposition, any f in HQ is a Q-linear combination of the e's; hence of is a sum of e's for some n. Since f (f )n = r(nf), we are reduced to checking the theorem for the e's described in the proposition. Noting that <e1,¢> = 1 and that <ep,¢> = (p + 1)/2 if pa 0, one has only to appeal to the identities:
1'(x)1'(1 - x) = z(sin
2cx)-1
zk-1 r(x\
k
Pl
pli2 X(27t)(p I) i2 I'(px).
REMARK. Kubert has recently obtained a much more precise result expressed in a
different terminology from which it follows that, if HQ is Z-valued, then 2f is a Z-linear combination of the e's (to appear). BIBLIOGRAPHIE
1. M. V. Borovoi, Sur faction du groupe de Galois sur les classes de cohomologie rationnelles de type (p, p) des varietes abeliennes, Mat. Sb. 94 (1974), 649-652. (Russian) 2. J. Coates and S. Lichtenbaum, On 1-adic zeta functions, Ann. of Math (2) 98 (1973), 498-550.
3. P. Deligne, Applications de la formule des traces aux sommes trigonometriques, dans SGA 4'/Z, 168-232. 4. , Les constantes des equations fonctionnelles des fonctions L, Modular Functions of One
Variable. II, Lecture Notes in Math., vol. 349, Springer-Verlag, New York, 1973, pp. 501-595. 5. , Formes modulaires et representations 1-adiques, Seminaire Bourbaki 355 (Fevrier 1969), Lecture Notes in Math., vol. 179, Springer-Verlag, New York, 139-172.
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P. DELIGNE
6. P. Deligne, Theorie de Hodge. III, Inst. Hautes Etudes Sci. Pub]. Math. 44 (1974), 5-77. 7. B. Gross, On the periods of abelian integrals and a formula of Chowla and Selberg, Invent. Math. 45 (1978), 193-211. 8. Ju. I. Manin, Correspondences, motifs and monoidal transformations. , Points paraboliques et fonction zeta des courbes modulaires, Izv. 36 (1972), 19-66. 9. (Russian) 10. I. I. Piateckii-Shapiro, Relations entre les conjectures de Tate et de Hodge pour les varietes abeliennes, Mat. Sb. 85(4) (1971), 610-620 = Math. USSR-Sb. 14 (1971), 615-625. 11. J. P. Serre, Abelian 1-adic representations and elliptic curves (Mc Gill), Benjamin, New York, 1968. 12.
, Facteurs locaux desfonctions zeta des varietes algebriques (definitions et conjectures), Sem. Delange-Pisot-Poitou 1969/70, expose 19. 13. G. Shimura, On some arithmetic properties of modular forms of one and several variables, Ann. of Math. (2) 102 (1975), 491-515. 14. C. L. Siegel, Berechnung von Zetafunktionen an ganzzahligen Stellen, Nachr. Akad. Wiss. Gottingen Math. Phys. Kl. 11 10 (1969), 87-102. 15. J. Tate, On the conjecture of Birch and Swinnerton-Dyer and a geometric analogue, Seminaire
Bourbaki 306 (1965/66), Benjamin, New York; reproduit dans 10 exposes sur la theorie des schemas, North-Holland, Amsterdam, 1968. 16. A. Weil, Jacobi sums as grossencharaktere, Trans. Amer. Math. Soc. 73 (1952), 487-495. 17. , Sommes de Jacobi et caracteres de Hecke, Nachr. Akad. Wiss. Gottingen Math. Phys. Kl. 11 1 (1974), 1-14. 18. D. Zagier, Modular forms whose Fourier coefficients involve zeta functions of quadratic fields, Modular Forms of One Variable. VI, Lecture Notes in Math., vol. 627, Springer-Verlag, New York, 1977, pp. 105-169. 19. N. Saavedra, Categories tannakiennes, Lecture Notes in Math., vol. 265, Springer-Verlag, New York, 1972. SGA. Seminaire de geometrie algebrique du Bois-Marie. SGA 4. par M. Artin, A. Grothendieck et J. L. Verdier, Theorie des topos et chomologie etale des schemas, Lecture Notes in Math., vols. 259, 270, 305, Springer-Verlag, New York. SGA 4'/Z. par P. Deligne, Cohomologie etale, Lecture Notes in Math., vol. 569, Springer-Verlag, New York, 1977. INSTITUT DES HAUTES ETUDES SCIENTIFIQUES, BURES SUR YVETTE, FRANCE
Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 347-356
AN INTRODUCTION TO DRINFELD'S "SHTUKA" D. A. KAZHDAN We begin with a formulation of the Langlands conjecture for GL(n) over a local field K of positive characteristic p. Fix a prime 10 p and denote by Qt an algebraic closure of Q1.
We introduce some notation. For any topological group G with a compact subgroup U c G we will consider smooth Qt-representations of G, that is, homomorphisms p: G -* Aut V, where V is a Qt-vector space such that the stabilizer of any vector v e V is open. We say that p is admissible if for every open compact subgroup U c G we have dim VU < oo, where VU is the set of U-invariant vectors. DEFINITION. We denote by 0 the set of equivalence classes of admissible irreducible representations of G and by Gn c G (n = 1, 2, ) the subset consisting of classes of n-dimensional representations. REMARK. It is easy to check that all standard results (see [7]) about admissible C-representations are true for our Q,-representations. Now we can formulate the local unramified reciprocity law. Let Kp be a non-
archimedean local field, o = op its ring of integers, 7i: e o a prime element, k = o/(7c) the residue field, q = card k, and v the valuation on K9 such that v(7r) = 1. Recall that an irreducible admissible representation of GL(n, K9) is unramified if the subspace of GL(n, o)-invariant vectors has positive dimension. REMARK. In this case this dimension is one.
The classification of the set Gp,,,,, of equivalence classes of unramified representations of GL(n, K9) is well known. Let B c GL(n, K9) be the subgroup of triangular matrices. To any element x = (x1, , xn) (x1 E Q*) we associate the character xx of B:
xx(b) =
(q1-nx1)" (b11) (q2-nx2)°(b22) ... xn(bnn)
where b11 are the diagonal elements of b. By px we denote the induced representation
indBL(n,xp)xx (which is realized in the space of locally constant functions on GL(n, K9) such that f(bg) = xx(b)f(g) for all b E B, g e GL(n, K9)). The representation px is not necessarily irreducible, but it has a finite Jordan-Holder series which contains exactly one unramified component. We denote it by px. The following statement is well known [1]. THEOREM 1. (a) For every element s of the symmetric group Sn, px (b) The map p: (Q*)n/Sn -+ Gp ,,,, is an isomorphism.
Ps (x)
AMS (MOS) subject classifications (1970). Primary 10H10; Secondary 10D20, 22E55, 20G35. © 1979, American Mathematical Society
347
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D. A. KAZHDAN
REMARK. Our parametrization of Gp,,,,, differs from the usual one by a shift and does not use fractional powers of q. Denote by Kp a separable closure of K, and by kp the corresponding algebraic closure of kp. We denote by WK, the dense subgroup of Gal(K IKp) consisting of elements which induce on k, the map x _ x9" for some n e Z, and we denote by IKb WKp the subgroup fixing kp. Thus WK, is the Weil group of Kp and Ip the inertia group. A representation of WK, is said to be unramified if it is completely reducible and its restriction to IKp is trivial. We can now state and easily prove the local reciprocity law for unramified representations. THEOREM-DEFINITION 2. There is a natural I-I correspondence between Gp, un and the set of equivalence classes of n-dimensional unramified representations of WKp.
REMARK. By an n-dimensional representation we mean a morphism into GL(n, Qt). PROOF-CONSTRUCTION. Let z be an unramified semisimple representation of WKp. Since it is unramified, we may consider it as a -representation of Wxp/IKp Z.
It thus corresponds to an element aZ in GL(n, Qj) (the image of the Frobenius Fr). Since z is semisimple, az is semisimple, and so the conjugacy class of a= is completely determined by the set of eigenvalues x = (x1, , xn) E ((?*)n. We associate to z the representation p = px of GL(n, Kp), and we write p - z.
This construction uses the explicit realization of unramified representations, but there exists a more canonical formulation. It is known (see [1]) that for every natural number m there exists a unique locally constant Q-valued function Xm on GL(n, Kp) with compact support such that (a) xm is bi-GL(n, o)-invariant, (b) for every x = (x1, ..., xn) a (Q* )n: Tr px(Xm) = xi + ... + xn . Now we can reformulate Theorem 2. THEOREM 2': The irreducible unramified representation p of GL(n, Kp) corresponds to the unramified n-dimensional representation z of WK, if and only if Tr p(Xm) Tr v(Frm) for all natural numbers m > 0.
Now consider the global case. Let K be a global field, char K = p.
We may consider K as the field of rational functions on a smooth absolutely irreducible curve X over the finite field k. Let q = card k. We denote by H the set of all points of K. For every p e II we denote by Kp the completion of K at p. We denote by A the adele ring of K and consider the Qt-space L of Q!-valued locally constant functions f on GL(n, K)\GL(n, A) which are cuspidal. That is to say for every K -subspace A = Kn, A o 0, Kn, 1u4(k)\U4(A) f(ux) du = 0 where UA is the subgroup of g e GL(n, Kn) such that gA = A and g acts trivially on A and KnIA. The group GL(n, A) acts by right shifts on L. The support of every element f E L is compact mod A* (the center of GL(n, A)) (see, e.g., Godement-Jacquet, Lecture Notes in Math., vol. 260, Springer, p. 142) and, consequently, the representation T of GL(n, A) on L is smooth.
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We will call an irreducible admissible representation of GL(n, A) a cuspidal automorphic representation if it occurs as a subrepresentation of L. We denote by GL(n, A)^ the set of these representations. Recall from, say, [5] that any irreducible admissible representation^(p, V) of GL(n, A) may be uniquely expressed as a restricted tensor product ©pp(p e 11) of local irreducible admissible representations. If S is a finite subset of 11 such that P. is unramified for p 0 S, we say p is unramified outside S. The next statement (see [10]) is useful for the precise formulation of the Langlands Conjecture. THEOREM 3. Let p, p' be two cuspidal automorphic representations of GL(n, A) such that pp = pp for almost all p e 11. Then p = p'.
Now consider the Galois side. Let K be a separable algebraic closure of K, and let WK be the dense subgroup of Gal(K/K) consisting of elements which induce on k the map x -> x9" for some n e Z. For every p e II we have an imbedding WK, WK defined up to conjugation in WK (see [12]). DEFINITION. Let z7 be a finite dimensional Q1-representation of WK. We say that T is unramified at p if the restriction Tp of r to WK, is unramified. REMARK 1. It is known that for every such z- there exists a finite set S c H such that T is unramified outside S. REMARK 2. If c is unramified at p, then T(Fr p) is well defined up to conjugation. THEOREM 4 (SEE [12]). Let r and r' be two irreducible representations of WK, and suppose that, for almost all p e 11, v(Fr p) is conjugate to T'(Fr p). Then r is equivalent to T'.
We can now formulate the Langlands Conjecture. There exists a one-one correspondence cp : p ,v r between the set GL(n, A), and the set (WK)n which has the following property:
For every cuspidal automorphic p there exists a finite set S c IT such that for all
p e 11- S the representations p and z- are unramified outside S and pp - r, (see Theorem 2).
REMARK 1. Theorems 3 and 4 imply the uniqueness of such a correspondence (if it exists).
REMARK 2. There is another conjecture, in which we ask for a one-one correspondence between cuspidal automorphic representations and systems of 1-adic representations of WK. It follows from results of Deligne [3] and Drinfeld that this is true for n = 2. REMARK. We can deduce from the global conjecture the following local one.
Let M be a local field of positive characteristic. There exists a one-one correspondence between super cuspidal representations pM of GL(n, M) and irreducible n-dimensional representations TM of WM. This correspondence is such that for every global field K, for every point po e II such that M ^ Kpo and for every automorphic representation p = ®pp such that pp0 is super cuspidal, the restriction Tpo of the representation r - p of WK on WK,, WM corresponds to ppo.
We can now ask ourselves a rather general question: how can we arrange a oneone correspondence between representations of two different groups, say G and W?
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One way is for every representation p of G to define a function f'P on some set X and to do the same for every representation r of W. Then we can say that p - r if TP = T, If G = GL(2, A) or GL(3, A), W = WK, X is the set of characters of idele class group K*\A* and 1'P and F. are F-functions, this method works very well (see [7], [9]) and permits us for every system of 1-adic irreducible 2 or 3-dimensional
representations z of WK to find the corresponding representation p of GL(2, A) or GL(3, A). But so far this approach does not give us the possibility of going in the opposite direction and constructing a representation r of WK from a representation
p of GL(n, A). Another more explicit approach to producing a correspondence between representations of two groups G and W was first used by H. Weyl [13] who considered the
case G = GL(n, R), W = Sm = the symmetric group. He considered the representation of G x Won the tensor power L = V©m of V Rm. We can decompose L in a direct sum L = QQ Li of G x W-invariant irreducible subspaces. As is well known, F. is a tensor product L; = pi 00 zi where pi are irreducible representations m) of GL(n, R) and r, are irreducible representations of Sm. He found that (for n in this way we get a one-one correspondence between representations of Sm and part of the representations of GL(n, R). This means that (a) {r,} = Sm, (b) if i, j
are such that pi p; then i = j. Drinfeld applies this method to the case when G = GL(2, A), W = WK. The first idea would be to construct a Q1-representation (T, L) I of GL(2, A) x WK with the following properties. When we consider the decomposition of L = QQ Li into a sum of G x W-invariant irreducible subspaces and write each L, as pi 00 ri, pi e GL(2, A),,, ri e (WK)Z then (a) {pi} = GL(2, A)Q ;
(b) for all i, r, - p,. But, unfortunately, such a representation (T, L) does not exist.
It might be useful to explain why not. To do this we must recall some definitions and results which are well known for representations of finite groups (see [11]) and can be easily restated and reproved for admissible representations. So let G be a group and (p, V) be an irreducible admissible Q1-representation of G. We denote by M(p) c Ql the field of definition of the equivalence class of p. That is, M(p) is the field which corresponds to the subgroup gP c Gal(Q1/Q1) consisting of elements G e Aut(Q,: Q) such that p° - p. DEFINITION. We say that p is unobstructed if it can be realized over M(p). That means that there exist a vector space Vo over M(p) and a morphism po: G -4 Aut Vo such that (po, Vo OO M(P)Q,) is equivalent to (p, V). The following two statements are well known (see [11j):
LEMMA 1. Suppose that (p, V) is a representation of G, H C G is a subgroup and dim VH = 1. Then p is unobstructed.
LEMMA 2. Suppose that L is the space of a smooth Q,-representation of G x W, (p, V) e G, (z, C) e W are such that Homw(C, L) ^ V as a G-module. Suppose that p is unobstructed. Then r is also unobstructed. LEMMA 3. Every irreducible representation of GL(2, A) is unobstructed. 'Because (T, L) is uniquely determined up to isomorphism by its properties, it is natural to expect that it is defined over Q, and not only over Q1.
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PROOF. For every positive ideal a of K we denote by T,, c GL(2, A) the subgroup
of matrices with integer elements which preserves the vector (0, 1) mod a. It is known (see [10]) that there exists an ideal a such that dim Vr° = 1. So our lemma follows from Lemma 1. On the other hand, it is easy to construct a two-dimensional representation a of WK which is obstructed. For example, we can take z to be induced from a one-dimensional representation of a subgroup of index two in WK. In this case we can (see [1]) construct p e GL(2, A), such that p - T. So we see that a representation (T, L) with properties (a) and (b) does not exist. What is to be done? We know that for every irreducible representation z of any
group H the representation z O t (where t = contragredient of a) of H x H is unobstructed. So we can try to realize QQ p Ox z Ox z (p e GL(2, A)2, z - p). This is almost possible. Among other things, we must consider the direct integral instead of the direct sum. To formulate precisely the result of Drinfeld we need some more definitions. DEFINITION. Let (p, V) be a representation of GL(n, A). We say that p is a graded representation if we can write Vas a direct sum V = QtczVt in such a way that for every g e GL(n, A) p(g)Vt = Vt+v(det(g))
REMARK. It is clear that for any graded representation (p, V) the operators p(f), f e C°°(GL(n, A)), are not of trace class and we cannot define the character Try. But sometimes we can define the regularized character def
Trp(f) = Tr P°p(.f) P0,
.f e tp(GL(n, A)),
where P0 is the projection onto V°. It is not hard to check that the definition of Trp is independent of the gradation
V=eVt. Let (p, V) be a graded representation, and let x e A *. We denote by Vx the quotient of V by the subspace generated by {x(det g)v - p(g)v}, v e V, g e GL(n, A). DEFINITION. We say that (p, V) is completely reducible if, for all x e 4*, V. is a direct sum of irreducible representations. The following result is easy. LEMMA. Let (p, V), (p', V') be two graded completely reducible representations such that Trp and Trp, exist and Trp = Trp,. Then p - p'.
Now we can formulate the "constructive" variant of Drinfeld's result. THEOREM 6. There exists a representation (p, V) of GL(2, A) x WK x WK such that (a) the restriction of p to K* c Center GL(2, A) is trivial; (b) for every x e A*/K*, Vx is a direct sum of irreducible representations Vx = @Vx.i,
(c) GL(2, A)' C
(d) for all (x, i), px,i ^ r.,j.
Vx,i = px.i OO Zx.i OTx.l;
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Of course, the proof of this theorem consists of two parts: (A) construction of (p, V) and (B) proof that (p, V) satisfies conditions (a)-(d). We shall only briefly discuss both parts. First, we consider a geometric interpretation of WK x WK. Let A be the field of functions on X x X, A, : A its separable closure, WA c Gal(A,/11) a subgroup consisting of elements which induce the map
x-* x9",neZ,onk. We have the natural map z: WA -+ WK x WK, which is simply the restriction of Aut(A,/A) to K, O K, c A5. It is clear that Im 7r consists of pairs any a e WA
(o', a") E WK x WK such that u'Ik = a"jk. Drinfeld defines an extension (* *)
0- WA --> WA-Z---O
and an epimorphism Tt: WA -* WK X WK. To do this consider the group WA of automorphisms r of the algebraic closure A
of A such that the restriction of r to the perfect closure Ap I`©KP" c A has the form FrrFrz where Fr1, Fr2 are partial Frobeniuses on Kper x0 KpeL. We define W to be WA/{the subgroup generated by the Frobenius}. We have the natural imbedding WA -> WA, and it gives the extension (**). REMARK. Analogously, we may define groups WK and WK, but in this case we will have WK = WK. Our morphism 'r extends to Tt: WA -> WK X WK = WK X WK. THEOREM 7 (DRINFELD). (1) k is an epimorphism.
(2) For every finite group H and every homomorphism cp: WA - H, we can write rp as a composition cp = cp o k, where 0 is a homomorphism 0: WK x WK - H. The proof of this theorem is not difficult, but in order to present it we would have to introduce some new definitions, and this would take too long. So we leave the proof to the reader. Now we can try to imagine the possible construction of the representation (p, V). As we know, we can consider K as a field of functions on a smooth, projective, absolutely irreducible curve X over a finite field k. Let S = X x X. Suppose that we can define a projective system Mi, i e I, of algebraic varieties over the generic point V of S such that: (a) we have the action of GL(2, A) on the projective limit M = proj lim Mi (we consider all Mi as A-varieties), (b) we have liftings of partial Frobeniuses Fri, Fr2 (as endomorphisms of k(V)) to the endomorphisms of Mi, for all i E I, in such a way that Frl o Fr2 lifts to the Frobenius on Mi. Then we can construct a representation (pM, VM) of GL(2, A) x WK x WK. To do this fix any integer r and consider VM = inj lim Hr(Mi, Q1).2 Then the group GL(2, A) x WA acts naturally on VM, and W A preserves the images of Hr(M1, Qt) in VM for every i e I. It follows now from Theorem 7 that the action of GL(2, A) x WA can be factored through GL(2, A) x WK x W. Drinfeld's construction actually follows this scheme. But the varieties Mi are not of finite type. To explain why this is so we have to remind ourselves what we want to realize. We want to obtain a representation (p, V) of GL(2, A) x WK x WK 'By Hr we mean t-adic cohomology with compact support.
DRINFELD'S "SHTUKA"
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such that its restriction to GL(2, A) will be four times the representation in the space L of cuspidal locally constant functions on GL(2, K)\GL(2,.A). We see that in the very definition of L we describe it as a subspace in the bigger space L of all locally constant functions with compact support on GL(2, K)\GL(2, A). So we can expect that the only way to realize the representation (p, V) is to realize it as a subspace of (p, V), and that the restriction of (p, V) to GL(2, A) will be a multiple of L; we can try to realize V as proj lim Hr(M=, Q1) and V as a subspace of V. But the representation L is "big". This means that for a compact open subgroup U c GL(2, A) and a character X E (A*/K*)A, the subspace (Lx)U of U-invariant vectors in L. is infinite dimensional. This means (see below) that dim Hr(M°, Q1) _ 00 3 and M° cannot be a variety of a finite type. After this long explanation of "why the construction cannot be very nice", we describe (but will not present) Drinfeld's construction. For every positive divisor D on X, Drinfeld defines an algebraic space MD over )7, and for every D' = D he defines a morphism MD, -* MD. This space is the union D MD = Un __MD of connected irreducible components, all the spaces M(resp. Mo+1), n e Z, are isomorphic. They have the following structure. (1) Each space
MD is a union MD = U MD' n-K of surfaces of finite type MD, n-K, (2) MD+i, n-K-i
MD, n-x and (3) the difference MD+i, n-K-t - MD. n-K is a union of affine lines. Drinfeld also defines compactifications MD. n-K of each subspace MD, n-K,
and shows that the birational isomorphism MD+i, n-K-i , MD. n-K can be ex. MD,n-K such that (a) the compositended to a morphism I1 n: MD+i,n-K-1 is the identity and tion of ir with the natural imbedding MD,n-K C. MD+i, n-K-1 is (b) the restriction of ' to the boundary f D , n =def MD+i, n-K-i a radical morphism. Define MD = proj limKMD.n-K and M = proj limDMD. Drinfeld shows that the partial Frobeniuses and GL(2, A) act naturally on M. MD+i,n-K-1
Now we can define def
V = H2(M, Q1)
lim H2(MD'n-K, Q1) K, n, D
To define V we consider the subspace Ve c V which is generated by classes in H2(M, Q1) of rational curves on M. 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 MD' n-x so MD, n-K is an example of a surface which has an infinite number of rational curves on it without having a family. It is very possible that for large D, MD' n-K is a surface of generic type.
It can be proved that the restriction to Ve of the canonical cup-product on H2(M, Q1) is nondegenerate. Drinfeld defines V to be the orthogonal complement of V. in V (or you may consider Vas the quotient space V/Ve). Of course, the partial Frobeniuses preserve Ve, 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 3Mo is a connected component of M;.
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D. A. KAZHDAN
can we prove that it gives us the corresponding representation (p, V) which realizes the Reciprocity Law?
The only known way to do this is based on the Trace Formula. We have the following general statement. Let (p, V) be a graded representation of GL(n, A) x WK x WK such that (a) For every Schwartz-Bruhat function on GL(n, A), Tr(f) exists. (b) For any two places p', p" of K, any number m', m" a Z+ and any S-B function f on GL(n, AP', p"), we have the equality
Tr(X°, Q X° (9f) x Fr-' x Frp;')v = Tr(X-, x i
X A, def
where X'(g) = Xp(g 1). Then the representation (p, V) realizes the Reciprocity Law. REMARK. By Frp ' we denote the measure on WK which can be defined in the following way. First of all, consider the local group WK... It contains the inertia group Ip., and WK,,II4, -- Z. We denote by up% the Ip.-invariant measure on WK,' which is concentrated on the preimage of m' and satisfies f WK , PP" = 1. We have an imbedding WK,, c, WK (which is defined up to conjugation), and we denote by Frp' the image of up ' 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 MD are disjoint unions of M. The
same is true for MD and M = UM-. So we can write V = @+Vt, where Vt = H2(M2t U Met+l, Qt). When you have the definition of M, you will see that the condition p(g)Vt = Vt+v(det g), g E GL(2, A), is satisfied.
To find the right side of Tr(Xp' x Xp;' 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 Xp, that is, for very conjugacy class Q c GL(2, Kp) to find the integral f QXp(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, because it might be useful for people who try to prove the Reciprocity Law for GL(n).
THEOREM 9. (1) If Q is a nonsemisimple conjugacy class in GL(n, Kr), then
fnXpdw=0. Let 0 be a semisimple class, r e Q. Then ZG(r) = fl GL(n L.), where L= D K, are finite extensions of Kp, Ent[Lz: Kp] = n. So we can write r = F1 Ti, 7-i L*. (2) If 50X' (w) dw 0 0 then there exists j, such that ntv(NL;/Kp(ri)) = m, and v(NL.:KP (Ti) = 0 for all i = j. To define the J Q we have to fix the Haar measure on 0. Because 0 = Zo(r)\GL(n, Kr), it is sufficient to fix Haar measure dr on GL(r, L) for all r e Z*, L n Kp a finite extension. We do this by the condition meas GL(r, OL) = 1. (3) Let r e 0r be a semisimple element satisfying the conditions in (2). Then J nr
Xp (w) dw = (1
- q1)(l -
(1 -
qe'-1) [li, k],
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DRINFELD'S "SHTUKA"
where 1, is the residue field of L. and qi = card li.
The proof of this theorem is based on the explicit formula for Xp which Drinfeld
found. He proved that (1) Xp (g) 0 0 - g e GL(n, o,) and v(det g) = m. (2) If g satisfies (1), then X, -(g) = (1 - q) (1 - q2) . . . (1 - qr-1), where n - r is the rank of the reduction of g mod r. 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 UDgUD
for g e G, D a divisor of X, UD c GL(n, A) the congruence compact open subgroup of integer adelic matrices equal to Id mod D. To do this, fix some N >- 1 and let NMD = U. MDf (n+N)12], E(n-N+1)/2]. The space NMD is a graded union of algebraic
spaces of finite type. By definition, we have the projection II: M, NMD. Denote by Tg,D C NMD X NMD the image of the map II x ff o g: M --> NMD X NMD. This T'g,D defines a correspondence on NMD. Drinfeld proved that the NMD are quasismooth surfaces (i.e., the etale cohomology of NMD with support in any geometric
point is the same as in the smooth case). So Tg,D defines an operator Ag,D in H2(NMD, Q1).
The variety NMD is a surface over the generic point )7 of X x X. For almost all closed points (p', p") of X x X. it has good reduction R over (p', p"). The cohomology of R is isomorphic to H*(NMD, Qt), and we can consider the operator Cg,D = Frp' Q FrP;' Q Ag,D on H2(R, Q1). The left side of (!) reduces to computing Fr Cg, D1'. To do this, one applies the Lefschetz Trace Formula (see [8]). It tells us that
Z (- 1)i Tr(Cg,DIH`(NMD, Qi)) = E
xedflfg, D
i=1
n(x),
where J c R x R is the diagonal and Tg,D is the reduction of f'g,D.
First look at the left side. Using the existence of the two systems of rational curves on M, one proves that Hl(NMD, Qi) = H3(NMD, Q,) = 0. It is easy to find HO and H4, so it remains to compute the right side. To do this, Drinfeld describes the geometric points of the reduction of M over (p', p") as a Frp. x Frp x GL(2, Ar'" a")-module.4 It gives us the description of d (1 Tg,D. If R were smooth, we could conclude that for large m', m", n(x) = 1. Using the fact that R is quasi-smooth, he proves that this is asymptotically correct (for large m', m"), and comparing with the Selberg Trace side, he deduces (!). REFERENCES
1. P. Cartier, Representations of a p-adic group: A survey, these PROCEEDINGS, part 1, pp. 111155.
2. P. Deligne, Formes modulaires et representations de GL(2), Lecture Notes in Math., vol. 349, Springer-Verlag, Berlin and New York. 3. , Les constants des equations fonctionnelles des functions L, Lecture Notes in Math., vol. 349, Springer-Verlag, Berlin and New York.
4. -, Cohomologie etale, SGA 4Z, Lecture Notes in Math., vol. 569, Springer-Verlag, Berlin and New York. 'A",," are adeles without p' and p" components.
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5. D. Flath, Decomposition of representations into tensor products, these PROCEEDINGS, part 1, pp. 179-183. 6. I. M. Gelfand, M. I. Graev and I. I. Piatetski-Shapiro, Representation theory and automorphic functions, Saunders, 1969. 7. H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Math., vol. 114, Springer-Verlag, Berlin and New York.
8. R. P. Langlands, Modular forms and 1-adic representations, Lecture Notes in Math., vol. 349, Springer-Verlag, Berlin and New York. 9. I. I. Piatetski-Shapiro, Converse theorem for GL(3), Mimeo Notes No. 15, Univ. of Maryland, 1975. 10.
, Multiplicity one theorems, these PROCEEDINGS, part 1, pp. 209-212. 11. J.-P. Serre, Representation lineaires des groupesfinis, Hermann, Paris, 1967. 12. J. Tate, Number theoretic background, these PROCEEDINGS, part 2, pp. 3-26. 13. H. Weyl, The classical groups, their invariants and representations, Princeton Univ. Press, Princeton, 1939. HARVARD UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 357-379
AUTOMORPHIC FORMS ON GL2 OVER FUNCTION FIELDS (AFTER V. G. DRINFELD)
G. HARDER AND D. A. KAZHDAN Introduction. This is a report on unpublished work of V. G. Drinfeld on automorphic forms over function fields. Our aim is to give a description of the so-called scheme of `shtuka', to discuss the computation of the trace of Frobenius on the cohomology of this scheme in terms of the Selberg trace formula and the proof of the Ramanujan conjecture. We will not discuss the much more subtle question concerning the reciprocity law which involves the compactification of the above moduli scheme. We shall not give detailed proofs, but we shall try to give references and hints so
that the reader may fill the gaps. 1. The moduli space of FH-sheaves.
1.0. Let X/Fq be a smooth projective curve. Let X = X X F9 Fq; Drinfeld introduces some objects over X which he calls `shtuka'. These objects are vector bundles
over X together with some additional structure. These additional structure data involve the Frobenius and some kind of `Hecke modifications'. Therefore we suggest to call these objects FH-sheaves. 1.1. Vector bundles. Let X/Fq be our given curve, let K/F9 be its field of meromorphic functions. We denote
X= X XF,Fq, K = The space of adeles of K(resp. KK) is denoted by A (resp. AJ The geometric points of X/Fq are denoted by v, w e X(Fq). For any v e X(Fq) we denote the completion of K, with respect to v by K. and its ring of integers by (9,,. The valuations of the field K/Fq will be denoted by p, . Then KK will be the completion of K at p and (9, its ring on integers. If v e X(Fq) induces p on K we write v -+ p. The points v which lie over a fixed p form an orbit under the action of the
Frobenius on X(Fq). A vector bundle of rank d over X is a locally free sheaf of rank d over the structure sheaf P. We have an alternate way to describe vector bundles. Let us take a vector space V/K together with a basis el, , ed. Let us assume that we have a family M = of (9 lattices M, c V Qx Kv, s.t. d
My = 0 (90ei for almost all v. i=1
AMS (MOS) subject classifications (1970). Primary 10D15, 14F05, 14H10. © 1979, American Mathematical Society
357
358
G. HARDER AND D. A. KAZHDAN
We can associate a locally free sheaf E/X to this family of lattices by defining
T(U,E) = {xE VIxEMvforallve U} where U c Xis an open subset. On the other hand it is very easy to see that every vector bundle can be realized in this form. Let us take for Mo the trivial family Mo,v = (9ve1 +Q . . . O+ (9ved for all v E X(Fq). Then we may use an adele x e GLd(A,) to produce a new family xM0 = {x,Mo,,,}v and it is an easy exercise to check that we have a bijection GLd(K_)\GLd(A_)/.3£'.
-, {set of isomorphism classes of vector bundles of rank d on X } where HvGLd((9v)
1.2. Level structures on vector bundles. Let D be a positive divisor on X/FQ, i.e., a divisor which is rational over Fq. We define as usual (9X (- D) = sheaf of germs of
regular functions f with div(f) > D = sheaf of ideals defining D, and for any vector bundle we define E(- D) = E Qx (9X (- D). A level structure (along D) on
E is an ft isomorphism: E/E(-D) -, ((9X/(9X(-D)d). The isomorphism classes of vector bundles with a level structure along D are given by the elements in
the coset space GLd(K)\GLd(A_)/.'_(D) where %'_(D) is the open congruence subgroup of )C- defined by x = 1 mod D. We are particularly interested in the two cases d = I and d = 2. A vector bundle E/X of rank 2 with level structure along D is called stable if for any line subbundle
L c Ewe have deg L < Z(deg E + deg D).
It follows from Mumford's theory that we have a fine moduli scheme # Spec(Fq) for the functor of stable bundles with level structure along D. (Compare [6].)
If for any v c Z we denote by -&D) the stable vector bundles with deg det E = v then the D(") are smooth quasi-projective schemes over Fq. 1.3. Modifications of vector bundles for d = 2. Let E/X be a vector bundle, let v E X(Fq). We want to define the notion of a modification of E at v. The fiber of our vector bundle Ell at v is a vector space E. = E xp ((9v1. v) = E xp Fq(v) over F. A point all E Pl(Ev)(Fq) defines a nonzero linear form av :Ev-->Fq which is unique up to a scalar. We define E(av) = {sheaf of germs of sections s of E for which a,*(sv) = av(sv) = 0}. This give us a new vector bundle on X and we have an exact sequence
0 --> E(av) c
E -Fq(v) -> 0,
i.e., E/E (a) is a sheaf concentrated on v and the fiber over v is a one dimensional vector space over the residue field P. = Fq(v). We call E(av) a lower modification of E at v (in direction av). We want to give an interpretation of this in terms of lattices. Let us assume that we have a family of lattices M = xM0 with xla GL2(A,). Let us pick an element av e GL2(w) CO v
Ol
/ GL2 (w)
where prv is a uniformizing parameter. Then we put av = ( , 1,
, av,
,
1 ...) and
GL2 OVER FUNCTION FIELDS
359
the bundle defined by M(a) = xav Mo defines obviously a modification of M. If we observe that the set GL2(Ov) (o v 1) GL2(0v)
divided by GL2(0v) on the right is exactly our PI(EE)(Fq) we see that we have an interpretation of the notion of modification in terms of families of lattices.
If w e X(Fq) is another point we define upper modifications at w: For aw e PI(Ew)(Fq) we put E(aw) = E(aw) Ox OX(w). In this case we have E
E(aw) -
Fq(w) and again we have an interpretation in terms of lattices M = xM0 -j 1 ..) and xawMo where Gaw = (... 1 , aw, I
aw e GL2((9w) (o w
01)
GL2((9w)
If v 0 w are two geometric points and av e PI(Ev)(Fq), aw e PI(Ew)(FF) then we define E (av, Cw) = (E(av))(aw). We observe that deg E(av, aw) = deg E. Another important observation is that we have a canonical identification E(av, aw)I X-{v}-{w} = EI X-{v1-(w)
This is clear from the definition of E(av, aw) in terms of locally free sheaves. Therefore we know: If D is a positive divisor on X/Fq and if v, w 0 supp(D) then a level structure 0 on E along D induces a level structure on E(av, aw) which will be denoted by the same letter 0. It is very easy to see that the data of such a double modification E --> E (av, aw) _ E' is equivalent to give a diagram
' locally free of rank 2
(9
where deg .9 = deg E + 1 = deg E' + 1 and where .9/0v(E) is concentrated at w and .9/0w(E') is concentrated at v. We just have to choose = E (a,,) and ov, 0w are the inclusions. 1.4. The FH-sheaves and their moduli space. The Frobenius automorphism x --> xq of Fq induces a map Fr: Spec(Fq) Spec(Fq) and therefore a map Id x Fr : X = X X Fq Spec(Fq) X.
If E/X is a vector bundle we denote by E° the pull-back under the above map, i.e., Ea = (Id x Fr)*E. If E has a level structure 0 along D then we get a level structure
0a: EO'/Ev(- D) - ((Q/(9( - D)2) ° = OX/OX(- D)2
-
since (OX/(9X(- D))2 = (V/2( D))2 ®Fq Fq. A set of data (E, ci, av, aw, 0) is called an FH-sheaf -or a `shtuka'- if we have the following: (1) D is a positive rational divisor and v, w e X(Fq), v 0 w and v, w 0 supp(D). (2) av e PI(Ev)(Fq), aw e PI(Ew)(Fq) (3) 0 is a level structure along D. (4) 0: (El, 01) (E (av, aw), 0) If we fix the two points v, w we call it an FH-sheaf over (v, w).
G. HARDER AND D. A. KAZHDAN
360
Two such FH-sheaves are isomorphic if the bundles are isomorphic and the isomorphism maps the corresponding data into each other. REMARK. If (E, 0, a,,, aw, 0) is an FH-sheaf and if (E, 0) is sufficiently stable for instance, if for any line subbundle L c Ewe have deg L < I (deg E + deg D) - 4 then the additional data a,, aw, v, w and 0 are uniquely determined by the pair (E, 0). This is again a very simple exercise. Let us assume that we have given an FH-sheaf (E, a,,, Law, 0) and a rational positive divisor D on X/F9. We ask for the possible level structures on this FH-sheaf along D. The isomorphism 0 induces an isomorphism OD: Eu/Ec(- D) 4 E/E(- D) and we must have for our level structure map 0: E/E (- D) --> (9X/0X(- D)2 that 0-100 = OD. It follows from Lang's theorem that we can always solve this
equation and the solutions form a principal homogeneous space and the action of GL2((9x/&x(-D)), i.e., all other solutions are of the form g with g e GL2((9x/(9x(- D)) = G (D).
We now want to show that we can construct a coarse moduli space for the functor of FH-sheaves. Let us fix two points v, w e X(F9), different from each other and a D with v, w 0 supp(D). Let us denote for any t e N by G(D,tthe moduli
space of vector bundles E with level structure along D which satisfy deg L S '-z(deg E +deg D) - t for all line subbundles L c E. This is an open subscheme of our &D above. If we look at a modification E (av, au,) of such a bundle it will correspond to a point in old t_,. One checks without too much difficulty that we find a closed subscheme over Fq, . ,,,w c ffD,, x _&D,t-1 whose geometric points are the pairs of bundles (E, E (a,, au,)). If p, q are the valuations of K induced by v and w and if F9d is the union of the residue fields of p and q then we have obviously that .Y,,,u, is defined over F9d.
According to a remark above we see that the directions a,, au, of the modifications are uniquely determined by (x, y) if t >_ 4 (for instance). Moreover some simple infinitesimal arguments will prove that the tangent space at in the point (x, y) projects surjectively to the tangent space of the first component and that the kernel of this projection is of dimension 2. Now let TFr be the graph of the geometric Frobenius (x, xO) c D,t X D,t 1 Then it is clear that M ' " ) _ ,,,w n TFr is a smooth two-dimensional scheme over Fqd and the first components of its geometric points are exactly the bundles with a (uniquely determined) structure as FH-sheaves. On MR, w) we have an action of the finite group G(D). If D' D then we have an obvious mapping of some open part M D; w) c MD: t) to M o, t) (the stability condition changes as D becomes larger) and obviously we get 1
RD O, w) c-, Mo ,,
)
I
M (V W )Iker(G(D')
G(D))
D,t
and therefore we have Mo,tw) MD' w) =
Mo . z)/ker(G(D')
inj lim
G(D)) and we may define
MD r)/ker(G(D') -> G(D))
D'?D;v, wEsupp(D')
for some t >_ 4.
MD(v,tw)
GL2 OVER FUNCTION FIELDS
361
This altogether is a sketch of the proof of the following theorem. THEOREM 1.4.1. (i) The functor of FH-sheaves over (v, w) with level structure along D is representable as an inductive limit {MD.w)}/F9d of schemes. If we fix the degree of the determinant then it will be a limit of quasi projective schemes which are quotients of smooth schemes by the action of finite groups.
(ii) If AD = X X X - J - D x X - X x D we can construct an inductive system MD - AD whose fiber over (v, w) is MD' w)
The assertion (ii) just follows from the fact that we can constructv,w with variable parameters v, w too. REMARK. The formulation is a little bit sloppy since we are not working with the limit itself but rather with the inductive system. Later on we shall also discuss the projective system proj limDMD,w) 1.5. Geometric properties of MD (v, w) and MD.
1.5.1. Let first consider the case d = 1. A line bundle LIX has a structure as an FH-sheaf if Lo-+ L (w - v) or equivalently L_I D L' = C9X (w - v). This tells us that the 1-dimensional FH-sheaves form a principal homogenous space under the action of Picx(F9); this is the group of line bundles on X. Therefore the moduli space for 1-dimensional FH-sheaves over (v, w) is an infinite discrete set. If we introduce a level structure it will become a principal space under the action of PicX,D(Fq) of line bundles on X with a rational level structure along D. If we fix D and vary v, w we find an infinite covering PD - AD whose Galois group of groups of "Decktransformationen" is exactly PicX,D(F9). 1.5.2. If d = 2 and (E, 0) is an FH-sheaf with level structure then we see imme-
diately that (det E, det 0) is a 1-dimensional FH-sheaf with level structure. This way we get a map det : MD -> P.
1.6. Extension of FH-sheaves. Now we shall assume that our two points v, w are not only different from each other but they are also not in the same orbit of the Frobenius map. This is the same as that v and w induce different valuations on K.
Let us assume that we have an FH-sheaf (E, ci, av, aw, 0) and that we have a line subbundle F c E such that 0 : Fa --> Fwhere F is the line subbundle of E (a,), aw) corresponding to F (a line subbundle is given by its generic fiber). One checks very easily that there are two possibilities. (i) av is nonzero on the fiber F Qx F9(v) = F c E,,. Then aw is zero on F. and F = F(w - v), i.e., F itself is a one-dimensional FH-sheaf. In this case we find that the quotient line bundle E/Fis defined over FQ. This follows from 1.5.2. (ii) a,, is zero on Fv and then nonzero on Fw. In this case F = F = F° and the line bundle is defined over F9. In this case E/F is a one-dimensional FH-sheaf. In the first case we call the FH-sheaf a left FH-extension and we shall write L instead of F and 0 -* L -+ E --> H -* 0. In the second case the FH-sheaf is a right extension and we shall write HI instead of F and 0 --> Hl --> E -> LI -f 0. We want to investigate a little bit more closely the conditions when an FH-sheaf is an extension and in how many ways it can be written as an extension.
Let us look at the generic fiber V/K of E. Then 0 induces a linear map 0k: Vor ---> V. Since VQ = Vas an additive group this mapping OK induces a map OK :
362
G. HARDER AND D. A. KAZHDAN
V -> V which is only o= 1-linear, i.e.,
OK(A x) = AOK(x)
Therefore-if we want our (E, 0, av, av 0) to be an extension-this mapping OK must have a one-dimensional invariant subspace W c V. Then we may choose a basis vector el e W and e2 e V/W and we find OK(el) = Ale,, 0K(e2) = 22e2 PROPOSITION 1.6.1. Let p (resp. q) be the valuations induced by v (resp. w) on K. We assume p q. Let A1, A, e KF9n = K and
,u, = NormKv,K (A1) e K,
i = 1, 2.
Then we have up to ordering µ1, ,u2:
(1) ,u1 has a zero at p and is a unit at q. (2) u2 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 +Q 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 (F, ci, av, a,v, 0) in exactly one way as a left and in exactly one way as a right extension :
0->L-->E-+ H->0,
0->H,--*E-->L,->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 H,, Ll how can we describe the FH-extensions 0 -> L -> E -* H --> 0,
0-*H1-+El -->L,-+ 0? We discuss only the first case. Let us fix v, w and a divisor D, still with the standing assumption p 0 q, v, w supp D. Moreover we fix level structures cbL : L/L( - D)
'(xl(x( - D),
OH: H/L( - D)
(9xl ax(- D). The extensions of L by H are classified by the elements in H1(X, Hom(H, L)) and the set
H1(X, Hom(H, L)(-D)) = Hl(!, H-1 (D L(-D)) classifies extensions with a splitting along the divisor D. e Therefore a vector bundle E which corresponds to an element H1(X, H-1 0 L(-D)) is automatically equipped with a level structure along D. We assume that D has large degree compared to deg(H-1 Q L). We ask for those which give rise to a vector bundle with a (then unique) FH-structure. This means that we have to get a
0-* L->E->H--* 0
0-* L(w)->E->H-+0
Z
0-La-->Ec->H->0 Here PL is the composition L° -> OLL(w - v) y L(w). We get a diagram
GL2 OVER FUNCTION FIELDS
HI(X H-I Ox L(-D))
363
, j. m) H1(X, H-1 p L( - D + W))
HI (X H-1 ® L"(-D)) II
HI(X H-1 (D L(-D)) and to get an FH-sheaf we have to look for the classes l*(e)
which satisfy
- (hLla) = 0.
PROPOSITION 1.6.2. (1) This equation defines a smooth one-dimensional group scheme over Fq which is isomorphic to Ga. (2) It can be defined over the field of definition of (L, cbL, OL) and (H, OH). (3) To each pair (L, OL), (H, OH) we find two affine lines GD H, GD > L of right and
left FH-extensions 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 FH-sheaves which we constructed as extensions have a particular kind of level structure in the sense that the level structure is adapted to the extension. The 0: E/E(- D) -* (OX/OX(- D))2 maps the L/L(- D) to the first component in (OX/OX(- D))'. One checks easily that any FH-extension with an adapted level structure can be obtained in the above way as a point on GD H or GH, L 2. On the other hand we have an action of GL2(OX/(9X(- D)) on the scheme of FH-sheaves with level structure along D (1.4). Then we may look at the translates of the affine lines Go H and GD' L and we find some more affine lines on the moduli
scheme. The points on the union of these lines correspond to the FH-extensions 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 FH-sheaf which contains a subbundle F c C with deg F > z deg E + 1 then it is easy to see that 0 has to map Fa to F, i.e., that E is necessarily an FH-extension. If s >-- 2 is an integer and if MD,w)(s) is the subscheme whose points are given by vector bundles E s.t. deg F < z deg E - s for all line subbundles F c E, then Mjj'w>(s - 1) - MD'w)(s) is a union of affine lines of left and right FH-extensions, where the subbundle is running over all FH-sheaves (resp. sheaves defined over FQ) of degree s and rank 1. 5. The affine lines GD H(resp. GDI" Ll) where deg L (resp. deg HI) 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 HI) tends to - oo. 2. The geometric points on the moduli space. In this section we want to discuss the main result which establishes a relation between the number of points on certain open pieces, of our moduli spaces and the traces of Hecke operators on the space of cusp forms.
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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 MD, w) --> Spec(Fgd) where Fqd is the union of the residue fields of p and q. If E(cb, av, aw, 0) is an FH-extension then we have two exact sequences 0 --> L - > E -+ H --* O,
0
E --* L1 --. O,
and we call the above extension strongly decomposed if deg L > deg H or deg H1 >_ deg L1. The nonstrongly decomposed FH-sheaves form an open subscheme UD (, MD(", 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 MD' w)
and this action induces an action of the same group on the cohomology with compact supports H;(UD, Ql) where UD = UD X Fqd Fq and Ql is an algebraic clo-
sure of Q1. Let p c Ql be the group of roots of unity. For any finite character w: PD -> u we denote by H;(UD, (0l)w the corresponding eigenspace in the cohomol-
ogy. We are interested in the trace-i.e., the alternating sum of traces-of the powers Frm of the geometric Frobenius for dim on the cohomology 4
(- 1)= trace FrmlH(Un, Qi)w
trace FrmIH;(UD, Q1). _ i=o
REMARK. The scheme UD -> Spec(Fgd) is not of finite type. We have the determinant map det UD -> PD introduced in 1.5.2. The fibers UD) for E PD(F4) are
quasi-projective schemes and if a r= PD then the tensorisation defines a map (JDE)
UDa2)
If PD, ,, is the kernel of co then
H'(UD, Q1)w c H"(UD,
Q1)PD,- _
Z
FEPD/PD
H (UD), Ql)
The information contained in the above trace is therefore equivalent to some informations on the number of rational points of the UoF). We now want to relate the above trace of Frm to the trace of some Hecke operators acting on a space of automorphic forms. We choose an embedding u 4 C*. Then we can consider the given character (0 also as a character with values in C*. Let us put .)Cp = GL2((9p), ,ir = GL2(01) and X' c ]Ip,Tp q GL2((9p,) to be the full congruence subgroup defined by D. If Z(A) is the centre of GL2(A) we have an identification
Z(A)/Z(K) (Z(A) n )rp x )Co x
PD
and therefore we may view our character w also as a complex valued character on Z(A). We introduce the space
H. = L., disc (GL2(K)\GL2(A)ID x .Cp x X,) 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 of Hecke operators. p and q. We introduce some algebras gyp, w, The elements of p,( are those functions f: GL2(Kp) -> C which are compactly supported modulo the centre, and which satisfy f(xz) = f (x) w (z),
z E Z(Kp), x E GL2(K7),
GL2 OVER FUNCTION FIELDS
365
and which are X , bi-invariant. Each such f defines an operator T1: H.
Tf(h) (y) =
H.
f/Z(K,) h(yx).f(x1) dx, GL2 (K0)
where y e GL2(A) and GL2(Kp) c. GL2(A). In these algebras we have for any n e Z the elements fin?) which are defined as follows: If Vsl, 12 0 0 is a function on GL2(Kp) which satisfies tl 1 1/2
7JsI,s2 W * k"/ = It1Is,1 It2Is,2 t2 I'
k0 E GL2((9,) and has central character w, i.e., co(t) = ItIS1+s2 then
SZ (K0) \GL2 (K0) '7si, sz(Yx)
n)(x 1) dx = (Np)Inl/2(Npns1 + Npnsz) 71s1,sz(y)
where Np is cardinality of the residue field at p. 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 pv( gi(n))7) = (Np)Inl/2(Npns1 + Npns2)y/sl,12
where 7) 0 0 is a Y,-invariant vector and where s1, s2 are the parameters of this representation (compare [1, §3, B]).
We are now able to state the main equality. Let us denote by d0 (resp. dq) the degree of the place p (resp. q). The operator An = T m(')
o T2 (q)dq
: Hw -' H.
is of trace class (compare [1, §2]) and we have
trace FrmIH;(UD, Q1). = trace bmlHW -Eisenstein contribution in the trace formula for the operator 0m. 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 V over X/F9 of degree one. It generates an infinite cyclic group {7)} c PD and PD/{V} = PO,D is finite. Then it is rather easy to see that trace Frml H;(UD, Q1). = trace Fr- (EPD;deg G+ D
H;(UDE ), Q1)WEPO,
c=0,1
We express this trace in terms of the eigenvalues a2,, of the Frobenius acting on the
cohomology H;(Uoe), Q1). Then we obtain E41 Eb= (- 1)t am. for the above trace. Now we look for a similar expression of the trace of On on the spaces H. Here we know that H,, decomposes further,
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G. HARDER AND D. A. KAZHDAN
H. = He,onsp +O He,,l +OHW,p (D 0C'x p
X
where p runs over a finite set of irreducible representations of GL2(A) in the space of cusp forms LW, e Sp (GL2(K)\GL2(A)) and where x : GL2(A) -> C is a function which factors over the determinant x GL2(A) ->det IK -> PD u c C* with (0'2 = (0. We compute the trace of 0m on each of the constituents. The representation p has to be of class one at p and q and therefore ap is obtained by induction from a representation ps1,s2
0
t2 p)
-
It1,CIC1It2,pI12It1,V/t2,pI1/2.
We define ap(p) = q51/d,, ap(p) = qsz/dp, etc. Then
trace O.I H,,,p = (dim H,
p(p)m) (g,(p)m + q(p)m)
This follows from the definition of cm. The contribution of the one-dimensional spaces CX is also easy to compute. This function x is tl,p
x 0
*
t2,p
. x(t1,p t2, p)
It1,pIV
1/2It2,plp 1/2Itl,p/t2,pI1/2
We put 7)p = x(cp)lld,, etc., and get by definition Tom x = ()7
+ 77
qm) (V + ylq qm)
We want to recall that 71,, Ylq are roots of unity.
It can be checked very directly from the trace formula that the Eisenstein contribution to the trace of 0m in the trace formula can be written as s
i=0
e; m = Einsenstein contribution in the Selberg trace formula for the operator 0m,
with sz = ± 1 and where the 7= are algebraic integers of weight 0, 1, 2, i.e., q'', vt = 0,1,2 (compare [8, 3.1]). If we define a(p) = dim H4 p then our main equality reads as follows: 4
b;
E E (- 1)1am, _ 1=o v=1
/
/ /
p
)m)( q(p)m + q(p)m) S
+ +(r1e + Vu 9m)(rI4 + la qm) + E E= rJi' i=o
x
Now we are ready to derive some consequences from this equality. First we recall what is known about the at,,. The a;,,, are algebraic integers and we have Ia,,,,l = g172 where j is an integer less than or equal to i. This number j is the weight of a,,,,. (Compare [9, 14].) On the other hand we have a simple LEMMA 2.2.1. If x1,
and if
xn, xl,
xn', yl,
ym, yl,
ym' are complex numbers
GL2 OVER FUNCTION FIELDS n
m
n'
m'
i=1
i=1
i=1
i=1
367
k = 1, 2,...,
and if xi 0 y; and xz y; for all i, j then n = n', m = m' and up to order we have xi = xi, Yi = Yi. Now we know that ap(p) (p) and q(p) q(p) are roots of unity because w is finite. Moreover it follows from the theory of automorphic forms that q and Ieq(p)I < q. This means that the terms (7)p 7q)mgZm which occur in the contribution from C. x are the only terms of weight 4 on the right-hand side. The number of these terms is equal to 2 IPO.DI This is clear since for each w which is a square we have 2 IP0,D/P0, DI solutions of w'2 = win the group of characters of PD. On the other hand the terms of weight 4 on the left-hand side come from the top cohomology bq
a4 v-
trace FrmIH4(UD), Q1) _ e, deg (e) =0,1
v=1
We shall see in the next section that the UD)/F9 are all nonempty. The number of e is again equal to 2 IPDI. Therefore we find the UD)IF, are irreducible. On the other hand we see that the terms of weight 3 on the left-hand side occur with a minus sign since they have to come from H 3(UD), Ql). But our remark on
the Einsenstein contribution shows that there are no terms with minus sign and weight 3 on the right-hand side. So there cannot be any such term on the left-hand side. This is easily seen to be equivalent to saying that for any smooth compactification UD) c. Ywe have H3(Y, Q1) = H'(Y, Q1) = 0. This absence of terms of weight 3 now implies directly that the terms qm ap(p), etc., have to be of weight 2, i.e., 1, etc. We may summarize: THEOREM 2.2.2. (1) The schemes UD)7Fd are absolutely irreducible.
(2) The cohomology_groups H3(UD), Q1) 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,,,, ,,,p 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 UD/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 Hecke operators in the trace formula. 2.3.1. The set of geometric 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 MD, w) are defined over F9 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 1-1 correspondence to the double cosets (compare 1.2) GL2(K)\GL2(A_)/.%' (D) and this tells us that the set
M(F9) = proj lim
MD(F9)
D, v, w4supp(D)
is a subset M(Fq) c GL2(K)\GL2(A00)/GL2((9v) x GL2((9w). REMARK. For notations compare 1.1. We still consider v, w as valuation of K_, so Kv, Kw are completions of K_ and (9v, C9w are the rings of integers in Kv, Kw.
It is clear that exactly those points x e GL2(A_) give a point in M(FQ) for which we can solve the equation
x° = r lxavaw
(*)
where u is the mapping induced by the action of the Frobenius on the constants in K_, where r E GL2(K) and where (
av E GL2(w) 1 O v
l)1 GL2(w),
1
aw E
(0 " ?)GL2(Ow),
and where av (resp. Caw) is the adele which has v- (resp. w)-component av (resp. aw)
and 1 elsewhere. We observe that r, a, aw are determined by x. If we have two solutions of (*) : xa = r 1 xavaw,
x'a = r'
1 x'av
then x and x' correspond to the same point of M(F9) if and only if x a 1x'kvkw with a e GL2(KK) and kv e GL2((9v), kw E GL2((9w). Then we find
r' = ara°,
av = kvavkv °,
aw = kwawk;".
(Here we take advantage of our assumption that v and w are rational.) The relation
r' = ara a means that r and r' are a-conjugate and we shall say that a;, is aGL2(&v)-conjugate to av and aw is c-GL2(Ow)-conjugate to aw. This gives us: Each geometric point in M(F9) gives rise to the following data: (i) a 6-conjugacy class {r} in GL2(K ), (ii) a u-GL2((9v)-conjugacy class {a,,} in GL2(0v) (w
l) GL2((9v),
(iii) a a-GL2((9w)-conjugacy class {aw} in 7L w
GL2l(9w)' (O
Q
1) GL'21(Ow)-
If these sets of data are given, then we find a point if and only if r is 6-conjugate to av (resp. aw) in GL2(Kv) (resp. GL2(Kw)) and if r is a-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 o-conjugacy classes in GL2(K00). (ii) Given such a class describe the possible classes av, au,.
(iii) Given r, a,, au, describe the set of geometric points which belong to this set of data, i.e., the solutions of x, = with these specific r, a,,, au,. 2.3.2. a-conjugacy. We start to discuss (i). To do this we make use of the SaitoShintani method of norms. (Compare [7].) If r e GL2(KK) then we can find an integer s.t. r E GL2(K If this is the case we define r r° "' r° 1 = ro One can show without difficulties (1) The conjugacy class of To is defined over K.
(2) If we a-conjugate r into ara o = r' with a E GL2(K Fq,) then 1= N (r'), and we may find an a E GL2(KFgn), s.t. we have N (r') = N (ara 0) = To E GL2(K).
The following assertion is slightly less trivial. (3) If and are conjugate in GL2(K,) then r and r' are a-conjugate. To prove (3) one has to use Galois cohomology and class-field theory. We shall give enough hints for the proof of (3) in the following discussion.
We shall always assume that N,,(7-) = 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 a-conjugacy class is central. To is semisimple and no power ro with k 1 is central. In this case we shall say that the u-conjugacy 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 r e and N (r) _ To is a central element, then the image r of r in PGL2(KFq,) defines a 1-cocycle in PGL2(KFgn)). Then it follows from Hilbert's Theorem 90 that r and
r' are 6-conjugate if and only if r and r' define the same cohomology class in H1(K Fqn/K, PGL2(KFO)) and then we find that the central a-conjugacy classes are in 1-1 correspondence with the elements of exponent 2 in the Brauer group of K. We shall not say very much about the case of noncentral or-conjugacy classes. The only thing we want to mention is that if To = N (r) is noncentral, then we have either To e E" y GL2(K) where E/K is a quadratic extension or To e T(K) where T/Kis a split torus. In both cases we must have that r is in the centralizer of To, i.e.,
r C EE or r e T(K ). 2.3.3. We come to the discussion of (ii). Let us assume that we have a E GL2(KF,); we ask:
Can we solve
x, a xv ' = r,
x, e GL2(K ), a,,,= GL2(Ov) (0 ° °)GL2((2 ),
and how many 6-GL2(P )-conjugacy classes for a,, do we get? Let us assume in the first case that
N,,(r)=ro=epzCK"
370
G. HARDER AND D. A. KAZHDAN
where z is a central element, i.e. z e Kx and a is a unit in Kp . Then we may modify/ r into uvruv ° with u eKv s.t. N,,(uv r uv °) = uv uv °" Nn(r) = uv
uv°" e, z
and we can find a u such that uu-°" e = 1.
Then we get xvuavu °xv ° = uru ° and we may change a, -> uavu ° since we do not change the a-GL2((9v)-conjugacy class. This implies that we have Nn(xv av xv °) = xvN(av) xv °" = z e K x.
If we pass to the projective linear group we find tvN(av) xv
1, i.e.,
N(a) = x°v" xv 1.
But if we take n large with respect to the order given by divisibility we find z6v" xv 1 is close to one and therefore we find an element k e PGL2((9v) s.t. kv"kv 1 = zv"zv 1. Changing again av -* kvavkv or we find that within the a-GL2((9v)-conjugacy class of our original av we may assume N,,(av) = zv e K,x. It is easy to see that then 2jn
and Nn(av) =
Zn/2 0 0v7rn/2
Since we know that Hl(KFq"/K, PGL2((9(Fq0)) = 0 it follows that in this case av is a-conjugate to the element
(° We ask for which {r} the condition N,,(r) = To = eo zp is fulfilled. First of all we observe that this is certainly true if {r} is a central class or if {7-1 is nondecomposed and the field generated by To is nonsplit at p. 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 To is noncentral and splits at p. In this case it is clear since r has to be in the centralizer of To that To is conjugate in GL2(KK) to v
(0
1
and then av has to be oa-GL2((9v)-conjugate to 00 (Ov 1),
We shall say that {T} is decomposed at v. Therefore we find that we have only two possible o,-GL2((9v)-conjugacy classes for av namely (0 v 0111'
(10 01J
