Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T17:15:23.151Z Has data issue: false hasContentIssue false
  • English
  • Français

The sign of Galois representations attached to automorphic forms for unitary groups

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  27 July 2011

Joël Bellaïche  and
Gaëtan Chenevier
Joël Bellaïche
Affiliation:
Brandeis University, 415 South Street, Waltham, MA 02454-9110, USA (email: jbellaic@brandeis.edu)
Gaëtan Chenevier
Affiliation:
C.N.R.S., Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau Cedex, France (email: chenevier@math.polytechnique.fr)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let K be a CM number field and GK its absolute Galois group. A representation of GK is said to be polarized if it is isomorphic to the contragredient of its outer complex conjugate, up to a twist by a power of the cyclotomic character. Absolutely irreducible polarized representations of GK have a sign ±1, generalizing the fact that a self-dual absolutely irreducible representation is either symplectic or orthogonal. If Π is a regular algebraic, polarized, cuspidal automorphic representation of GLn(𝔸K), and if ρ is a p-adic Galois representation attached to Π, then ρ is polarized and we show that all of its polarized irreducible constituents have sign +1 . In particular, we determine the orthogonal/symplectic alternative for the Galois representations associated to the regular algebraic, essentially self-dual, cuspidal automorphic representations of GLn (𝔸F) when F is a totally real number field.

Keywords

Galois representationautomorphic formunitary groupsignsymplecticorthogonaleigenvarietyendoscopyp-adic

MSC classification

Secondary: 11F80: Galois representations 11F55: Other groups and their modular and automorphic forms (several variables) 11F85: $p$-adic theory, local fields
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 5 , September 2011 , pp. 1337 - 1352
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[BC09]Bellaïche, J. and Chenevier, G., Families of Galois representations and Selmer groups, Astérisque, vol. 324 (Société Mathématique de France, Paris, 2009).Google Scholar
[BR92]Blasius, D. and Rogawski, J., Tate class and arithmetic quotient of two-ball, in The zeta function of Picard modular surfaces, eds Langlands, R. P. and Ramakrishnan, D. (Université de Montréal, Centre de Recherches Mathématiques, Montreal, QC, 1992), 421–443.Google Scholar
[Car94]Carayol, H., Formes modulaires et représentations galoisiennes à valeurs dans un anneau local complet, in p-adic monodromy and the Birch and Swinnerton-Dyer conjecture, Contemporary Mathematics, vol. 165 (American Mathematical Society, Providence, RI, 1994), 213–237.Google Scholar
[Che]Chenevier, G., Une application des variétés de Hecke des groupes unitaires, in[GRFAbook, II].Google Scholar
[Che04]Chenevier, G., Familles p-adiques de formes automorphes pour GL(n), J. Reine Angew. Math. 570 (2004), 143–217.Google Scholar
[CC09]Chenevier, G. and Clozel, L., Corps de nombres peu ramifiés et formes automorphes autoduales, J. Amer. Math. Soc. 2 (2009), 467–519, 22.Google Scholar
[CH]Chenevier, G. and Harris, M., Construction of automorphic Galois representations, book 2 of [GRFAbook].Google Scholar
[CR10]Chenevier, G. and Renard, D., On the vanishing of some non semisimple orbital integrals, Expo. Math. 28 (2010), 276–289.Google Scholar
[CHL]Clozel, L., Harris, M. and Labesse, J.-P., Endoscopic transfer, in [GRFAbook, ch. I.5.2].Google Scholar
[CHT08]Clozel, L., Harris, M. and Taylor, R., Automorphy for some l-adic lifts of automorphic mod l representations, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1–181.Google Scholar
[Eme06]Emerton, M., On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms, Invent. Math. 164 (2006), 1–84.Google Scholar
[GRFAbook]GRFA seminar of Paris 7 University, the two volumes of the book project, http://fa.institut.math.jussieu.fr/node/29.Google Scholar
[Gro]Gross, B., Odd Galois representations, Preprint. Available at www.math.harvard.edu/∼gross.Google Scholar
[HT01]Harris, M. and Taylor, R., The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151 (Princeton University Press, Princeton, NJ, 2001).Google Scholar
[Lab]Labesse, J.-P., Changement de base CM et séries discrètes, in [GRFAbook, ch. I.5.1].Google Scholar
[Shi11]Shin, S.-W., Galois representations arising from some compact Shimura varieties, Ann. of Math. (2) 173 (2011), 1645–1741.CrossRefGoogle Scholar
[Tho84]Thompson, J. G., Some finite groups which appear as Gal(L/K) where K⊂ℚ(μ n), in Group theory, Beijing 1984, Lecture Notes in Mathematics, vol. 1185 (Springer, New York, 1984).Google Scholar