Conditional base change for Unitary groups en collaboration avec Michael Harris

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Niveau: Supérieur, Doctorat, Bac+8
Conditional base change for unitary groups Michael Harris ? Jean-Pierre Labesse ?? Introduction It has been known for many years that the stabilization of the Arthur-Selberg trace formula would, or perhaps we should write “will,” have important consequences for the Langlands functoriality program as well as for the study of the Galois representations on the -adic cohomology of Shimura varieties. At present, full stabilization is still only known for SL(2) and U(3) and their inner forms [LL,R]. The automorphic and arithmetic consequences of stabilization for U(3) form the subject of the influential volume [LR]. Under somewhat restrictive hypotheses, one can sometimes derive the expected corollaries of the stable trace formula. Examples of such “pseudo-stabilization” include Kottwitz' analysis in [K2] of the zeta functions of certain “simple” Shimura varieties attached to twisted forms of unitary groups over totally real fields, and the proof in [L1] of stable cyclic base change of automorphic representations which are locally Steinberg at at least two places. These conditional results have been used successfully to provide non-trivial examples of compatible systems of -adic representations attached to certain classes of automorphic representations of GL(n) [C3], and of non-trivial classes of cohomology of S-arithmetic groups [BLS, L1].

  • let

  • group

  • hermitian forms

  • now let

  • tate-nakayama isomorphism

  • jacquet-langlands transfer

  • galois cohomology

  • conditions when

  • trivial examples


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Conditional base change for unitary groups
Michael HarrisJean-Pierre Labesse∗∗
Introduction
It has been known for many years that the stabilization of the Arthur-Selberg trace formula would, or perhaps we should write “will,” have important consequences for the Langlands functoriality program as well as for the study of the Galois representations on the` present,-adic cohomology of Shimura varieties. At full stabilization is still only known forSL(2) andUinner forms [LL,R]. The automorphic and(3) and their arithmetic consequences of stabilization forU(3) form the subject of the influential volume [LR].
Under somewhat restrictive hypotheses, one can sometimes derive the expected corollaries of the stable trace formula. Examples of such “pseudo-stabilization” include Kottwitz’ analysis in [K2] of the zeta functions of certain “simple” Shimura varieties attached to twisted forms of unitary groups over totally real fields, and the proof in [L1] of stable cyclic base change of automorphic representations which are locally Steinberg at at least two places. These conditional results have been used successfully to provide non-trivial examples of compatible systems of`-adic representations attached to certain classes of automorphic representations of GL(n) [C3], and of non-trivial classes of cohomology ofS results Conditional-arithmetic groups [BLS, L1]. also suffice for important local applications, such as the local Langlands conjecture forGL(n) [HT,He].
The present article develops a technique for obtaining conditional base change and functorial transfer. LetUnbe a unitary group over a number fieldFattached to a quadratic extensionE/F technique. The applies to quadratic base change fromUntoGL(n)E, and to transfer between inner forms of unitary groups. Roughly speaking, ifπis an automorphic representation ofUwhich is locally supercuspidal at two places of Fsplit inEexpected consequences of the stable trace formula hold for, then the π; in particularπadmits a base change to a cuspidal automorphic representation ofGL(n)E(Theorem 2.2.2). Slightly more general results are available whenFis totally real andEis totally imaginary, and whenπis of cohomological type. Automorphic descent fromGL(n)EtoUncan be proved under analogous hypotheses (Theorem 2.4.1, Theorem 3.1.2). Finally, we prove transfer between distinct inner forms of unitary groups (Jacquet-Langlands transfer) under quite general local hypotheses (Theorem 2.1.2 and, in a more precise form, Theorem 3.1.6 and Proposition 3.1.7). As in [L1], all results are obtained from the simple version of the Arthur-Selberg trace formula, in which non-elliptic and non-cuspidal terms are absent.
As a principal application, we obtain results similar to those of [C3] and [HT] for the cohomology of Shimura varieties attached to unitary groups of hermitian forms, or rather for the part of the cohomology satisfying the supercuspidality hypotheses (Theorem 3.1.4). An initial motivation for this project was the construction of non-trivial examples of families of nearly equivalent cohomological automorphic forms on unitary groups of hermitian forms, to which the analysis of special values ofL-functions and periods in [H1,H2] could be applied (Theorem 3.1.6 and Proposition 3.1.7). It is important to be able to work with unitary groups of hermitian forms, rather than the unitary groups of division algebras with involutions of the second kind, because the theta correspondence used in [H2] applies only to the former (cf. also work in progress of Harris with Li and Skinner on the Iwasawa main conjecture forp-adicL-functions).
The results of the present paper have already been used by L. Fargues in his thesis, in his realization of the local Langlands and Jacquet-Langlands correspondences on the cohomology of Rapoport-Zink moduli
 in part by the National Science Foundation, through GrantsInstitut de Mathematiques de Jussieu-UMR CNRS 7568. Supported DMS-9203142 and DMS-9423758. ∗∗utitstInseuqitame´htaMedRCNRu-UMssiedeJutitunItsa8dn7S75htaMame´uqitLedeinumUMy,NRRC20S66
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M. Harris J.-P. Labesse
spaces (cf.§ Proposition 3.3.1 answers in part an old question of Rapoport; this should allow3.2). Finally, him to construct non-trivial examples of varieties with ordinary reduction. Recent work of Waldspurger and Arthur has reduced the general stabilization problem to several specific problems in local harmonic analysis, known collectively as the Langlands-Shelstad conjecture or the “funda-mental lemma” for endoscopy, and its variants. At the time of writing, Laumon has announced important progress on the main case of the fundamental lemma for unitary groups, inspired in part by earlier work of Goresky, Kottwitz, and MacPherson. Conditional base change will presumably be unnecessary when this work comes to fruition. We hope that the applications provided here will still be of interest. One of us (M.H.) began work on the present paper in 1993, in the hope of finding non-trivial near-equivalence classes as indicated above. The proofs sketched at that time could not be completed until the techniques of [L1] became available. The present collaboration began in January of 1998, when the two authors were visiting the RIMS and the University of Kyoto. We thank these institutions, and M. Kashiwara and H. Yoshida, for their hospitality. Finally, we thank L. Fargues and M. Rapoport for raising the questions treated in subsections 3.2 and 3.3.
1. Trace formula identities for Unitary groups
1.1 – Basic notation
We denote byθ0the non-trivial automorphism that fixes the canonical splitting inGL(n): θ0(x) =Jtx1J1 and where, as usual,x7→txis the transposition andJis the matrix 001001J=(.1)n.0.0
We observe thatJ2= (1)n+1. LetFzero with some fixed algebraic closurebe a field of characteristic F. LetEbe a quadratic Galois algebra overF(i.e.Eis either a quadratic field extension or the split algebraFF). LetαGal(E/F) denote the non-trivial Galois automorphism. The group overF Gn= ResE/FGL(n). has an automorphismθof order 2 defined byθ(x) =θ0(α(x denote by We)) .Lnthe coset ofθin the semidirect productGno< θ>: Ln=Gnoθand byUn=U(n, E/F), or simply byU, the fixed point subgroup forθinGn. This is the quasisplit unitary group attached toE/F denote by. WeUn,adthe adjoint quotient and bySUnthe special unitary subgroup. We have exact sequences of groups 1U1UnUn,ad1
and
1SUnUnU11. Fichier harris-labesse, compilation le 12-3-2004– 888
Conditional base change for unitary groups
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We shall also use groups of unitary similitudes: letF0be a subfield inFand let RGnResE/F0GnandRUn= ResF /F0Un. = The quasisplitF0-similitude groupGUnis the subgroup ofRGnof pointsxsuch thatν(x) =(x1) belongs toGm=G1, viewed as a group overF0, embedded in the center ofRGn is a short exact. There sequence 1RUnGUnνGm1.
1.2 – Galois cohomology and inner forms
We need an assortment of results in Galois cohomology. We use the notation of [KS] and [L1] for Galois and adelic cohomology. We observe that the special unitary groupSUn, the derived subgroup ofUn, is simply connected. Then the abelianized Galois cohomology in the sense of Borovoi [B] and [L1] is easy to compute forUnandUn,ad. The abelianized cohomology ofUnis the cohomology of its cocenterU1. The abelianized cohomology ofUn,adis, by definition, the hypercohomology of the crossed moduleSUnUn,ad. This complex is quasi-isomorphic to the complex of toriU1nU1and hence, up to a shift by 1, to the diagonalisable groupDn= ker[U1nU1 In] . particular there are isomorphisms Hbai(F, Un)Hi(F, U1) andHbia(F, Un,ad)Hi+1(F, Dn).
1.2.1. Lemma.
(i) We have Ha1b(, Un) =Z/2ZandH2ab(, Un) = 1 where=F(resp.=AF/F) whenE/Fis a quadratic extension of local (resp. Moreover, global) fields. ifFis global, we havekeri(F, U1) = 1fori0. (ii) We have H1ab(,n,ad) =H2(, Dn) =1Z/2Zififnniseenvisodd Uwhere=F(resp.=AF/F) whenE/F is global)is a quadratic extension of local (resp. This fields. also the case ifF=RandE=RR. WhenFis a non archimedean local field andE=FFthen Ha1b(F, Un,ad) =H2(F, Dn) =Z/nZ. WhenF=CthenH2(F, Dn) = 1. (iii) WhenE/Fextension of global fields the mapis a quadratic H2(Fv, Dn)H2(AF/F, Dn) is surjective, unless maybe whenFv=C. Moreoverkeri(F, Dn) = 1fori2. Prooffrom the Tate-Nakayama isomorphism and from the above re- (i) and (ii) follow easily : Assertions marks. We still have to prove (iii). The co-localization map can be computed using Poitou-Tate duality [L2, Corollaire 2.2]; it is known that keri(F, Dn) = 1 fori We are left to prove that3 [L2, Corollaire 2.4]. ker2(F, Dn that Using) = 1 . Dn= ker[U1nU1] Fichier harris-labesse, compilation le 12-3-2004– 888