Available in electronic format
Available in print format		 Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN: 1088-6834(e) ISSN: 0894-0347(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Corps de nombres peu ramifiés et formes automorphes autoduales

Author(s): G. Chenevier; L. Clozel
Journal: J. Amer. Math. Soc. 22 (2009), 467-519.
MSC (2000): Primary 11F70, 11F72, 11F80
Posted: September 17, 2008
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: Let $ S$ be a finite set of primes, $ p$ in $ S$, and $ \mathbb{Q}_S$ a maximal algebraic extension of $ \mathbb{Q}$ unramified outside $ S$ and $ \infty$. Assume that $ \vert S\vert\geq 2$. We show that the natural maps

$\displaystyle \operatorname{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p) \rightarrow \operatorname{Gal}(\mathbb{Q}_S/\mathbb{Q})$

are injective. Much of the paper is devoted to the problem of constructing self-dual automorphic cuspidal representations of $ \operatorname{GL}(2n,\mathbb{A}_{\mathbb{Q}})$ with prescribed properties at all places, which we study via Arthur's twisted trace formula. The techniques we develop also shed some light on the orthogonal/symplectic alternative for self-dual representations of $ \operatorname{GL}(2n)$.

References:

[A1]
J. Arthur, The invariant trace formula I. Local theory, J. Amer. Math. Soc. 1 (1988), 323-383. MR 928262 (89e:22029)

[A2]
J. Arthur, The invariant trace formula II. Global theory, J. Amer. Math. Soc. 1 (1988), 501-554. MR 939691 (89j:22039)

[A3]
J. Arthur, The local behaviour of weighted orbital integrals, Duke Math. J. 56 (1988), 223-293. MR 932848 (89h:22036)

[A4]
J. Arthur, An introduction to the trace formula, in Harmonic Analysis, the trace formula and Shimura varieties, A.M.S., Clay Math. Institute (2005). MR 2192011 (2007d:11058)

[AC]
J. Arthur et L. Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Ann. of Math. Studies 120 (1989). MR 1007299 (90m:22041)

[Be]
J.-N. Bernstein (rédigé par P. Deligne), Le «centre »de Bernstein, in Bernstein, Deligne, Kazhdan, Vignéras, Représentations des groupes réductifs sur un corps local, Hermann, Paris (1984). MR 771671 (86e:22028)

[BLS]
A. Borel, J.-P. Labesse et J. Schwermer, On the cuspidal cohomology of $ S$-arithmetic subgroups of reductive groups over number fields, Compositio Math. 102 (1996), 1-40. MR 1394519 (97j:11026)

[BW]
A. Borel et N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies 94 (1980). MR 554917 (83c:22018)

[Bou]
A. Bouaziz, Sur les caractères des groupes de Lie réductifs non connexes, J. Funct. Analysis 70 (1987), 1-79. MR 870753 (89c:22020)

[BT]
F. Bruhat et J. Tits, Groupes réductifs sur un corps local: I. Données radicielles valuées, Pub. Math. IHES 41 (1972), 5-251. MR 0327923 (48:6265)

[Cas]
W. Casselman, A new nonunitarity argument for $ p$-adic representations, J. Fac. Sci. Univ. Tokyo 28 (1981), 907-928. MR 656064 (84e:22018)

[Ch]
G. Chenevier, On number fields with given ramification, Compositio Math. 143 (2007), 1359-1373. MR 2371372

[Clo1]
L. Clozel, Théorème d'Atiyah-Bott pour les variétés $ \mathfrak{p}$-adiques et caractères des groupes réductifs, Harmonic analysis on Lie groups and symmetric spaces (Kleebach, 1983), Mém. Soc. Math. France 15 (1984), 39-64. MR 789080 (87b:22034)

[Clo2]
L. Clozel, Motifs et formes automorphes : Applications du principe de fonctorialité, in Automorphic forms, Shimura varieties, and $ L$-functions, Clozel, Milne eds. vol. I, Perspectives in Math. 10, Academic Press (1990), 77-159. MR 1044819 (91k:11042)

[CloD]
L. Clozel et P. Delorme, Le théorème de Paley-Wiener invariant pour les groupes de Lie réductifs II, Ann. Sci. École Norm. Sup. 23 (1990), 193-228. MR 1046496 (91g:22013)

[HT]
M. Harris et R. Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Math. Studies 151 (2001). MR 1876802 (2002m:11050)

[He1]
G. Henniart, La conjecture de Langlands locale pour $ \operatorname{GL}(3)$, Mém. Soc. Math. France, no. 11-12 (1984). MR 743063 (86g:11070)

[He2]
G. Henniart, Correspondance de Langlands et fonctions $ L$ des carrés extérieur et symétrique. Prépublications Mathématiques de l'IHES (2003).

[Hi]
A. Hitta, On the continuous (co)homology of locally profinite groups and the Künneth theorem, J. Algebra 163 (1994), 481-494. MR 1262715 (94m:22004)

[K1]
R. Kottwitz, Tamagawa numbers, Ann. of Math. (2) 127 (1988), 629-646. MR 942522 (90e:11075)

[K2]
R. Kottwitz, Rational conjugacy classes in reductive groups, Duke Math. J. 49 (1982), 785-806. MR 683003 (84k:20020)

[KR]
R. Kottwitz et J. Rogawski, The distributions in the invariant trace formula are supported on characters, Canad. J. Math. 52 (2000), 804-814. MR 1767403 (2001e:11117)

[KS]
R. Kottwitz et D. Shelstad, Foundations of twisted endoscopy, Astérisque 255 (1999). MR 1687096 (2000k:22024)

[La1]
J.-P. Labesse, Cohomologie, stabilisation et changement de base, Astérisque 257 (1999). MR 1695940 (2001k:22040)

[La2]
J.-P. Labesse, Pseudo-coefficients très cuspidaux et $ K$-théorie, Math. Ann. 291 (1991), 607-616. MR 1135534 (93b:22026)

[Lg]
R. P. Langlands, Automorphic representations, Shimura varieties, and motives. Ein Märchen, in Automorphic Forms, Representations, and L-Functions, Proc. Sympos. Pure Math. 33 vol. II, A.M.S., Providence (1977), 205-246. MR 546619 (83f:12010)

[M]
P. Mezo, Twisted trace Paley-Wiener theorems for special and general linear groups, Compositio Math. 140 (2004), 205-227. MR 2004130 (2004e:22010)

[NSW]
J. Neukirch, A. Schmidt et K. Wingberg, Cohomology of Number Fields, Springer-Verlag, Grundlehren 323 (2000). MR 1737196 (2000j:11168)

[PR]
D. Prasad et D. Ramakrishnan, On self-dual representations of division algebras over local fields. Prépublication.

[Ro]
J. Rogawski, Trace Paley-Wiener theorem in the twisted case, Trans. A.M.S. 309 (1988), 215-229. MR 957068 (89k:22035)

[S1]
J.-P. Serre, Cohomologie des groupes discrets, Prospects in Mathematics, Ann. of Math. Studies 70 (1971), 77-169. MR 0385006 (52:5876)

[S2]
J.-P. Serre, Répartition asymptotique des valeurs propres de l'opérateur de Hecke $ T\sb p$, J. Amer. Math. Soc. 10 (1997), 75-102. MR 1396897 (97h:11048)

[Sh1]
F. Shahidi, A proof of Langlands' conjecture on Plancherel measures; complementary series for $ p$-adic groups, Ann. of Math. (2) 132 (1990), 273-330. MR 1070599 (91m:11095)

[Sh2]
F. Shahidi, Twisted endoscopy and reducibility of induced representations for $ p$-adic groups, Duke Math. J. 66 (1992), 1-41. MR 1159430 (93b:22034)

[She]
D. Shelstad, Characters and inner forms of a quasi-split group over $ \mathbb{R}$, Compositio Math. 39 (1979), 11-45. MR 539000 (80m:22023)

[T]
R. Taylor, Galois representations, Annales de la Faculté des Sciences de Toulouse 13 (2004), 73-119. MR 2060030 (2005a:11071)

[TY]
R. Taylor et T. Yoshida, Compatibility of local and global Langlands correspondences, J. Amer. Math. Soc. 20 (2007), 467-493. MR 2276777 (2007k:11193)

[Ti]
J. Tits, Reductive groups over local fields, in Automorphic forms, representations and $ L$-functions, Proc. Sympos. Pure Math. 33 vol. I, A.M.S., Corvallis (1977), 29-69. MR 546588 (80h:20064)

[V]
D. Vogan, Unitarizability of certain series of representations, Ann. of Math. (2) 120 (1984), 141-187. MR 750719 (86h:22028)

[W1]
J.-L. Waldspurger, Le groupe $ \operatorname{GL}(n)$ tordu, sur un corps $ p$-adique, partie I, Duke Math. J. 137 (2007), 185-234. MR 2309147

[W2]
J.-L. Waldspurger, Le groupe $ \operatorname{GL}(n)$ tordu, sur un corps $ p$-adique, partie II, Duke Math. J. 137 (2007), 235-336. MR 2309148

Similar Articles:

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 11F70, 11F72, 11F80

Retrieve articles in all Journals with MSC (2000): 11F70, 11F72, 11F80

Additional Information:

G. Chenevier
Affiliation: Laboratoire Analyse, Géométrie et Applications, UMR 7539, Institut Galilée, Université Paris 13, 99 Av. J-B. Clément, 93430 Villetaneuse, France

L. Clozel
Affiliation: Centre d'Orsay Mathematique, Université Paris XI, Batiment 425, 91405 Orsay Cedex France

DOI: 10.1090/S0894-0347-08-00617-6
PII: S 0894-0347(08)00617-6
Received by editor(s): 28 juin 2007
Posted: September 17, 2008
Additional Notes: Le deuxième auteur est un membre de l'Institut Universitaire de France
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google