automorphic forms
Marie-France Vign´ eras
Abstract. The purpose of this paper is to present and to extend the method of [9] in order to obtain congruences between algebraic automorphic forms [7]. The congruences are “easy” to obtain; this is “soft” mathematics as Steve Rallis likes to say. However they are quite important; via the “hard” functori- alities of Langlands, they allow one to get congruences between automorphic representations ofGL(n) or Galois representations, as for instance in[5],[6], or in the proof of the Langlands correspondence modulo`[9]. It is a pleasure to dedicate this work to Steve Rallis, congruences are related to his work via the theory ofL-functions where his contribution is so fundamental.
Letpbe a prime number,F a number field, andGa reductive connected group over F such that {1} is an arithmetic subgroup. We denote by G0 = ResF /QGthe scalar restriction ofGto Q [8]and by GA=G0A=G∞×G∞ the adele group ofGor ofG0.
Theorem 0.1. We consider
- V, V0 two absolutely irreducible algebraic representations ofG0,
- v a finite place of F andKv an open compact subgroup of G(Fv)of pro- order prime top,
-Kv,pan open compact subgroup of the componentG∞,v,pofG∞outside of v, p, andWp a complex irreducible smooth representation ofKp=Kv×Kv,p.
Then we have the following property:
Any irreducible automorphic representation πA of GA of algebraic type V which contains (Kp, Wp)is rational over a number field, its componentπ∞,p outside of∞, p is p-integral, and is congruent modulo pto an irreducible au- tomorphic representation ofGA of algebraic typeV0 which contains(Kp, Wp) and has finite slope atp.
The term finite slope or congruent modulo p used in the theorem are defined as follows. An irreducible automorphic representation πA of GA has a finite slope at p, if the p-component πp of πA contains the trivial representation of a pro-p-Iwahori subgroupIwpofGp=G0(Qp).
Two irreducible automorphic representations πA and πA0 ofGA are called congruent modulopif there exists an isomorphismipfromCto an algebraic closureQp ofQp, a finite extensionE/Qp contained in Qpand a finite set S of places ofF containing the places above∞and p, such that
- the componentsπS andπ0,S ofπAand πA0 outside ofS are unramified, - the values of the charactersipλS, ipλ0,S :HS →C→Qp of the commu- tative Satake-Hecke algebra HS given by πS and π0,S belong to the ring of integers OE of E and are congruentipλS ≡ipλ0,S modulo the maximal ideal ofOE.
The theorem has a converse:
LetWpbe an irreducible complex representation of the pro-p-Iwahori group Iwp. Any irreducible automorphic representation of GA of algebraic type V0, of finite slope at p, which contains (Kp, Wp), is congruent modulo p to an irreducible automorphic representation of GA of algebraic type V, which contains(Iwp×Kp, Wp×Wp).
When p /=2, we obtain a slightly stronger form of the theorem and of its converse in§20.
One gets also congruences modulo pn, n >1, between automorphic forms instead of representations, in§17.
The method of the proof is a combination of arguments given in[9]only for automorphic forms trivial at∞, and in the excellent introduction to algebraic automorphic forms[7].
The beautiful recent results of Buzzard, Chenevier, Emerton and Kisin on p-adic interpolation of automorphic forms make this paper look out of date, but their results do not imply our theorem. The interest of Barry Mazur when I sent him these congruences in July 2002, motivated me to write their proof.
It is a pleasure to thank him and Steve Rallis for many friendly discussions.
Proof of the theorem (see §19)
1 We consider a number field F and a reductive connected groupGover F such thatG(F)is discrete in the groupG∞ of finite adeles.
Equivalent properties are given in[7]. One of them is that the arithmetic subgroups are finite. This condition is very restrictive. There are only three possibilities:
-F =Qand the split centreS ofGis a maximalR-split torus ofG, -F is an imaginary quadratic field andGis aF-split torus,
-F is totally real and the infinite partG∞ ofGis compact (in particular the split centre ofGis trivial).
The group G∞ has a unique maximal compact subgroup K∞. The con- nected componentsGo∞andK∞o are related to the split and anisotropic centres S andC ofG0 as follows. We haveK∞o = (C(G0, G0))∞andGo∞=S∞o ×K∞o . We denote byS1the finite algebraicQ-group ofx∈Swithx2= 1. The groups G∞, K∞ are equal to G∞ =S∞K∞ =S1,∞Go∞ and K∞ =S1,∞K∞o. They have the same group of componentsπG,o=G∞/Go∞=K∞/K∞o , isomorphic to the 2-groupS1,∞/(S1,∞∩Go∞). We haveS1,∞=S1(Q).
2 The representations are always left representations (unless right is spec- ified).
An irreducible automorphic representation ofGAof dimension 1 is a Hecke character, i.e. a continuous morphism G(F)\GA → C∗ (for any reductive connected group over F). We denote by Ω the set of Hecke characters ω of SA=S(Q)×S∞o ×Q
pS(Zp).
An automorphic form onGAis a finite sumf =P
ω∈Ωfωof functionsfω: G(F)\GA→C which are smooth (continuous andK∞-finite) onG∞, locally constant onG∞, andfω(sg) =ω(s)fω(g)for alls∈SA, g∈GA. The action of GA by right translations on the spaceSC=⊕ω∈ΩSC,ω of automorphic forms is semi-simple. The irreducible constituents are the irreducible automorphic representations ofGA.
A non-degenerate bilinear symmetric form < fω,1, fω,2 > on SC,ω is the integral of fω,1fω,2η−1 on the compact space G(F)SA\GA, for an invariant Haar measure, where η is the positive Hecke character ofGA equal toωω on SA.
3 Let (R, H, G, M, X) be any commutative ring R, a subgroup H of a group G, an R-module M with an action of H, and a space X with a right action of G. We denote by C(X, R) the R-module of functions f : X →R with the natural action of G and by ˜M the R-module of linear forms on M with the contragredient action of H, such that<m, m >˜ =< hm, hm >˜ for allh∈H, m∈M,m˜ ∈M˜. Then
HomRH(M, C(X, R))
and theR-moduleAH(X, M) of functions onX of type (H, M) φ:X →M ,˜ φ(xh) =h−1φ(x) for allh∈H,
are isomorphic by the map associating to aRH-morphismm→fmthe func- tion φsuch that < φ(x), m >= fm(x). The relationsfhm(x) = fm(xh) and
< φ(xh), m >=< φ(x), hm >=< h−1φ(x), m >are equivalent.
When the groups and spaces have a topology, the representations and the functionsφare supposed continuous.
4 We decompose SC according to the set ˆGo∞ of isomorphism classes of irreducible representations ofGo∞,
SC' ⊕πo
∞∈Gˆo∞π∞o ⊗ρπo∞.
For each(πo∞, V∞)∈Gˆo∞, the complex representationρπ∞o of the locally profi- nite groupGo∞\GA is smooth, semisimple and admissible.
The semisimplicity results from §2 because G∞/Go∞ is finite, the admis- sibility results from the finiteness ofG(F)G∞\GA/K for any open compact subgroupK ofG∞, for any reductive connected group overF [2]5.1.
The model AGo∞(G(F)\GA, V∞) of ρπ∞o 'HomCGo∞(V∞,SC) obtained by applying§3, is the complex space of functions
φ:G(F)\GA→V˜∞, φ(gg∞) =g−1∞φ(g) for allg∞∈Go∞, locally constant onG∞, withGo∞\GA acting by right translations.
We can extend the representationπ∞o ofGo∞toG∞becauseπG,ois a finite 2-group. We choose an extensionπ∞. Then we set
φ(g) =g∞−1ψ(g),
where g∞ is the infinite component of g ∈ GA. Now ψ is invariant by right translation byGo∞ and we have, forγ∈G(F),
ψ(γg) =γg∞φ(γg) =γψ(g).
The natural action of Go∞\GA by right translations on the space A(V∞) of locally constant functions
ψ:Go∞\GA→V˜∞, ψ(γg) =γψ(g) for all γ∈G(F), is a new model of ρπo
∞, depending on the choice of π∞. We could replace Go∞\GAbyπG,o×G∞ of course.
5 When π∞o is trivial, we deduce that ρπo
∞ is isomorphic to the natural representation ofGo∞\GA on the space of locally constant complex functions on G(F)Go∞\GA. The functions with values in Z form a Go∞\GA-stable Z- lattice. An irreducible automorphic representation of GA trivial on Go∞ is defined over the ring of integers of a number field [9].
6 We consider - a number fieldL,
- a placeλ∞∈Hom(L,C),
- a finite dimensionalL-vector spaceV with an absolutely irreducible alge- braic action ofG0.
In §4, we fixπ∞, resp. πo∞, to be the representation of G∞, resp. Go∞, equal to the restriction of the algebraic action ofG0(C) on V∞=V ⊗L,λ∞C.
The representation π∞⊗ε of G∞, twist of π∞ by a character ε of the component groupπG,o, is called ofalgebraic type(V, ε)orV. An irreducible automorphic representation ofGAof infinite componentπ∞⊗εis also called of algebraic type(V, ε)or V.
The irreducible complex representations ofG∞of algebraic type are charac- terized by the property that their restriction to the componentGo∞is algebraic.
Any continuous irreducible complex representation toK∞o is algebraic, and any continuous irreducible complex representation of Go∞ is the twist by a character of an algebraic representation becauseGo∞=So∞×K∞o (see§1).
TheL-vector spaceA(V) is aGo∞\GA-stableL-structure of the semi-simple admissible representationA(V∞). The finite part of an irreducible automor- phic representation ofGAof algebraic type isdefined over a number field.
7 In order to obtain an integral structure whenV is not trivial (§5), we will workp-adically. We choose
- an isomorphismip:C→Qp.
The place λ∞ and ip give ap-adic place λp =ipλ∞∈Hom(L,Qp) which allows to identify the complex spaceV∞defined in§6 with theQp-vector space Vp=V ⊗L,λpQp.
The interest is that there is a natural action of the p-adic group Gp = G0(Qp)⊂G0(Qp) onVp, such that the actions ofG(F) onVpas a subgroup of Gp and onV∞ as a subgroup ofG∞coincide via the isomorphism V∞ →Vp. One identifies AGo∞(G(F)\GA, V∞) and AGo∞(G(F)\GA, Vp) and the trick is to set
f(g) =g−1p ψ(g) = (g∞, gp−1)φ(g)
where φ∈ AGo∞(G(F)\GA, V∞) and (g∞, gp) is the (∞, p)-component ofg ∈ GA. There exists an open compact subgroup K =Kp×Kp of G∞ =Gp× G∞,p(depending onφ) such thatφ, ψ are right invariant byK. The relation ψ(γgk) =γψ(g) for (γ, k)∈G(F)×K is equivalent to
f(γgk) = (γgpkp)−1ψ(γgk) = (γgpkp)−1γψ(g) =k−1p f(g).
These are the functions studied in§3 when
(R, H, G, M, X) = (Qp, K, G∞, Vp, G(F)Go∞\GA).
the action ofKonVp is the inflation of the action ofKp (trivial onKp).
8 TheQp-vector spaceAK(G(F)Go∞\GA, Vp)ofp-adic automorphic forms onGA of locally algebraic type(K, Vp)is
{f :G(F)Go∞\GA→V˜p, f(gk) =k−1p f(g) for allk∈K}, withGo∞\GpAacting by right translations.
The inductive limit
A(G(F)Go∞\GA, Vp) = limKAK(G(F)Go∞\GA, Vp)
over all open compact subgroups K = Kp×Kp of G∞, is the space of p- adic automorphic forms onGAof locally algebraic typeVp. The complex and p-adic spaces
AGo∞(G(F)\GA, V∞), A(G(F)Go∞\GA, Vp)
are isomorphic viaip. The isomorphism isπG,o×G∞,p-equivariant only. The action of Gp is locally algebraic on A(G(F)Go∞\GA, Vp) and is smooth on AGo∞(G(F)\GA, V∞).
9 We denote byE=Lλp, OE the ring of integers,pE an uniformizer,kE the residual field. We choose:
- an open compact subgroupKpo ofGp, - anOE-structureL(Vp)ofVp stable byKpo.
The linearOE-dual ˜L(Vp) =L( ˜Vp) is anOE-structure of the contragredient V˜p. In§8, we can suppose that the groupsKp are contained inKpo.
TheOE-moduleA(G(F)Go∞\GpA,L(Vp)) is free, Go∞\GA-stable, and gen- erates theQp-vector spaceA(G(F)Go∞\GA, Vp). The finite part outside ofpof an irreducible automorphic representation ofGA of algebraic type isdefined over the ring of integers of a finite extension of Qp.
We could not do this in the complex realization of§4, because the repre- sentationV∞ofG∞, when it is not trivial, has no integral structure stable by G(F), by density.
10 We return to the generalities of §3. We suppose that G is a locally profinite group acting continuously on a locally profinite space X, and that H =K is an open subgroup of Gacting smoothly on an R-module M. We
consider the compact induction from the R-representations of K to the R- representations ofG
indGKM ={f :G→M, f(kg) =kf(g), (k, g)∈K×G, f with compact support},
the restriction resGK in the other direction, and the HeckeR-algebra H(G, K, M) = EndRGindGKM.
These functors respect smoothness. In the abelian category of smooth R- representations, the induction from K to G is left adjoint to the restriction fromGtoK [10],
HomRG(indGKM, V)'HomRK(M,resGKV).
Hence HomRK(M,resGK·) simply denoted by HomRK(M,·) is a functor from the smoothR-representations ofKto the rightH(G, K, M)-modules.
The sumVsm of the smooth subrepresentations of anR-representationV of G is a smooth subrepresentation ofV and for any smooth representation M ofK
HomRK(M, V) = HomRK(M, Vsm)
because K is open. One then deduces that AK(X, M) is naturally a right H(G, K, M)-module.
This remains true when the action of K on M is locally algebraic. The proof follows the same pattern.
11 We can add to the machine:
- a finite dimensional L-vector space W with an absolutely irreducible smooth action of an open compact subgroupK ofG∞.
The functorV →HomRK(W, V) (see§10) gives a bijection from the isomor- phism classes of the irreducible smooth representations ofG∞ which contain (K, W) onto the isomorphism classes of the simple right modules of the Hecke algebraH(G∞, K, W) (consider the idempotent of the global Hecke algebra of G∞associated to (K, W)).
The representation ρπo
∞ of πG,o×G∞ is admissible and semisimple. The isomorphism class of the subrepresentation ofρπo
∞ generated by the irreducible subrepresentations which contain (K, W), is identified with (see§3,§4):
The right H(G∞, K, W)-moduleAGo∞×K(G(F)\GA, V∞⊗L,λ∞ W)of au- tomorphic forms onG∞ of algebraic typeV and smooth type(K, W) .
This semi-simple module is isomorphic to
A(V∞⊗L,λ∞W) ={ψ:Go∞\GA→V˜∞⊗L,λ∞W ,˜
ψ(γgk) = (γ, k−1)ψ(g), for all (γ, k)∈G(F)×K}.
The finite set G(F)Go∞\GA/K has a system of representatives g1, . . . , gn
inG∞ becauseG(F) is dense inG∞(for any reductive connected group) [2].
The values ψ(gi) are invariant by the finite arithmetic group Γi = G(F)∩ (Go∞×giKgi−1). As a complex vector space,
A(V∞⊗L,λ∞W)' ⊕ni=1( ˜V∞⊗L,λ∞W˜)Γi, ψ7→(ψ(gi))1≤i≤n. It is easy to chooseK such thatG(F)∩Kis trivial, or even such that all the finite groups Γi are trivial, noting that ifK=Q
vKv is factorizable, then the order of Γi divides the pro-order ofKv for any finite placev ofF.
12 One deduces from §11 a formula for the finite dimension of the space of automorphic representations with given algebraic and smooth types. In particular:
For any algebraic absolutely irreducible representationV ofG0, there exists an irreducible automorphic representation of algebraic typeV. IfKis an open compact subgroupK of G∞ such that K∩G(F) ={1}, for any irreducible complex representationW ofK, there exists an irreducible automorphic rep- resentation of algebraic typeV which contains(K, W).
13 There exists a finite set S of places of F which contains the infinite places, such thatK =KS×KS (see§11) where KS ⊂GS and outside of S, KS is a good maximal open compact subgroup ofGS, andWS is trivial.
The Hecke Z-algebra HS = H(GS, KS) is commutative and acts semi- simply on A(V∞⊗L,λ∞ W) with eigenvaluesλπS where λπS is the character of HS acting on the one dimensional space of KS-invariants of πS, for all irreducible automorphic representationsπA=πS⊗πS ofGAof algebraic type V which contain (K, W).
14 We suppose in §11 that K = Kp×Kp with Kp ⊂ Kpo (see §9) and W =Wp⊗LWphas a factorization compatible with the factorization ofG∞= Gp×G∞,p. The groupKacts onVp⊗L,λpW by the diagonal locally algebraic irreducible action of Kp and by the smooth action of Kp natural on W and trivial onVp. We choose
-a K-stableOL-structure L(W) =L(Wp)⊗OLL(Wp)ofW over the ring of integersOL ofL(see§11).
The linearOL-dual ˜L(W) =L( ˜W) is anOL-structure of the contragredient W˜. As in§8,
15 The Qp-space of p-adic automorphic forms of locally algebraic type (K, Vp⊗L,λpW)is
AK(G(F)Go∞\GA, Vp⊗L,λpW) ={f :G(F)Go∞\GA→V˜p⊗LW ,˜ f(gk) =k−1f(g) for allk∈K}.
This is a rightπG,o× H(G∞, K, Vp⊗L,λpW)-module as in§10, but the proof of the theorem uses only its πG,o× H(G∞,p, Kp, Wp)-module structure. The natural isomorphism
AGo∞×K(G(F)\GA, V∞⊗L,λ∞W)→ AK(G(F)Go∞\GA, Vp⊗L,λpW) isπG,o× H(G∞,p, Kp, Wp)-equivariant.
16 The integral structures chosen in§9 and in§14, give anOE-structure L=L(Vp)⊗OLL(W) of the locally algebraic representation ofKonVp⊗LW. As in§11, theOE-moduleAK(G(F)Go∞\GA,L) is isomorphic to
⊕ni=1L˜Γi,
isOE-free, stable byπG,o×H(G∞,p, Kp,L), and generates theQp-vector space AK(G(F)Go∞\GA, Vp⊗L,λpW).
17 We suppose thatthe orders of the finite arithmetic groupsΓi in
§11, are prime to p. The congruences between automorphic forms rely on the fact that the naturalπG,o× H(G∞,p, Kp,L)-equivariant map between
AK(G(F)Go∞\GA,L)/ pnEAK(G(F)Go∞\GA,L) and
AK(G(F)Go∞\GA,L/pnEL) is an isomorphism for any integern≥1.
The congruences between automorphic representations when n = 1, rely on the Deligne-Serre lemma [4]. Let X be any commutative subalgebra of πG,o× H(G∞,p, Kp,L). For any eigenvalueµofX on the reduction
AK(G(F)Go∞\GA,L/pEL)
there exists a finite extensionE0/E and an eigenvalueλof X acting on AK(G(F)Go∞\GA,L ⊗OEOE0) =AK(G(F)Go∞\GA,L)⊗OEOE0, congruent modulopto µ, i.e. λ≡µ modpE0.
18 The pro-p-radicalIwp of an Iwahori subgroupIw ofGp, called apro- p-Iwahori subgroup of Gp, should be considered as the pro-p-Sylow ofGp
because it is a pro-p-group, uniquely defined modulo conjugation, which con- tains the pro-p-radical of any parahoric subgroup of Gp containing Iw. As Iwp is a pro-p-group, it satisfies the following properties:
1) An irreducible locally algebraic representation of a pro-p-group over a field of characteristicpis trivial.
2) For any locally algebraicE-representationW of a pro-p-groupIp, there exists a finite extensionE0/E and an OE0-structure ofW ⊗EE0 with trivial reduction modulopE0 (for the action ofIp)[3].
19 We are now ready to prove the theorem and its converse. Let πA be an irreducible automorphic representation ofGA of algebraic type V and containing (Kp=Kv×Kp,v, Wp), letV0be as in the theorem, and letWp be an irreducible smooth complex representation ofIwp.
In§11, we takeK=Iwp×Kp. The orders of the arithmetic groups Γi in
§11 are trivial because the pro-orders of Kv and ofIwp are relatively prime.
We choose the number fieldL and the place λ∞ in §6, and the isomorphism ip in§7, such that:
a) The algebraic representationsV, V0 ofG0 and the smooth representation Wp⊗CWp ofK are defined overLembedded inC viaλ∞.
b) The finite dimensionalQp-vector spacesUp =Vp, Vp0, Wp⊗C,ipQphave Iwp-stableOE-structuresL(Up) of trivial reduction modulopE(for the action ofIwp), using§18 2).
c) The finite components prime to pin the finite set of irreducible auto- morphic representations ofGA
- of algebraic typeV and containing (K, Wp⊗CWp), or
- of algebraic typeV0 and containing (K,idp⊗CWp) (trivial onIwp), are defined over OE.
In §11, the L-representation of K is either W0 equal to a Kp-stable L- structure ofWpwithK acting by inflation (trivial onIwp), orW =Wp,L⊗L
W0 for a Kp-stableL-structureWp,LofWp.
We choose aKp-stableOE-structureL(Wp) ofWp⊗C,ipQp. With b), we obtain theOE-structures in§9,§14 and in §16,L for Vp⊗L,λpW andL0 for Vp0⊗L,λpW0. We denote bydthe complex dimension ofWp.
There is aK-isomorphism
L/pEL ' ⊕dL0/pEL0.
The same is true for the contragredients. We deduce as in §17 that the reduction modulopE of
AK(G(F)Go∞\GA,L) and ⊕dAK(G(F)Go∞\GA,L0)
areπG,o× H(G∞,p, Kp,L(Wp))-isomorphic.
We suppose in §13 that S contains p. Then πG,o × HS has the same eigenvalues on the reduction modulopE of
AK(G(F)Go∞\GA,L) and ⊕dAK(G(F)Go∞\GA,L0).
For∗=LorL0, we deduce from c) that for any finite extensionE0/E, the eigenvalues ofHS on
AK(G(F)Go∞\GA,∗) or on AK(G(F)Go∞\GA,∗ ⊗OEOE0)
are the same. Remembering from §1 that πG,ois a finite 2-group, we deduce as in§17:
Whenp /=2, the reduction modulopE of the eigenvalues ofπG,o× HS on AK(G(F)Go∞\GA,L) and onAK(G(F)Go∞\GA,L0)
are the same. Ifp= 2, this remains true withoutπG,o.
20 Using§13, we obtain a slightly stronger version of the theorem and of its converse.
With the notations of the theorem and of its converse, letεbe a character ofπG,o. Ifp /=2, any automorphic irreducible representation ofGAof algebraic type(V, ε), and containing(Iwp×Kp, Wp⊗Wp), is congruent to an automor- phic irreducible representation ofGA of algebraic type(V0, ε)and containing (Iwp×Kp,idp⊗Wp), and conversely.
Ifp= 2, this remains true without the characterε.
References
[1] Nicolas Bourbaki, Alg`ebre I. Hermann (1970).
[2] Armand Borel, Some finiteness properties of adele groups over number fields.
Publ. Math., Inst. Hautes Etud. Sci. 16 (1963) 5-30.
[3] Jean-Francois Dat,ν-tempered representations of p-adic groups. I :`-adic case.
Preprint 2003.
[4] Pierre Deligne et Jean-Pierre Serre, Formes modulaires de poids 1. Ann. Sci.
E.N.S. (iv) 7 (1974), 507-530.
[5] Fred Diamond and Richard Taylor, Lifting modular mod`representations. Duke Math. J., Vol. 74, No 2, (1994), 253-269.
[6] Chandrasekhar Khare, On the local correspondence mod`. J. Number Theory 88 (2001), no. 2, 357–365.
[7] Benedict H. Gross, Algebraic Modular Forms. Israel Journal of Mathematics 113 (1999), 61-93.
[8] T.A. Springer, Linear algebraic groups. Second Edition. Birkh¨auser. PM 9 (1998).
[9] Marie-France Vign´eras, Correspondance de Langlands semi-simple pour GL(n, F) modulo` /=p. Invent. math. 144, 177-223 (2001).
[10] Marie-France Vign´eras, Repr´esentations`-modulaires d’un groupe r´eductif p- adique avec` /=p. Birkha¨user PM 137 (1996).
Marie-France Vign´eras
Institut de Mathematiques de Jussieu 175 rue du Chevaleret
Paris 75013, France [email protected]
