Irreducibility of automorphic Galois representations of $GL(n)$, $n$ at most $5$

[Irréductibilité des représentations galoisiennes associées à certaines représentations automorphes de $GL(n)$ pour $n$ inférieur ou égal à $5$]

Calegari, Frank ; Gee, Toby

Annales de l'Institut Fourier, Tome 63 (2013), p. 1881-1912 / Harvested from Numdam

Résumé

Nous prouvons l’irréductibilité pour $n$ inférieur ou égal à $5$ des représentations galoisiennes $l$-adiques associées aux représentations automorphes cuspidales algébriques et régulières de $\textrm{GL}_n$ sur un corps totalement réel qui sont autoduales à torsion près. Nous prouvons également l’irréductibilité des représentations galoisiennes modulo $l$ pour presque tout $l$, et nous montrons l’indépendance en $l$ de l’algèbre de Lie de la clôture Zariskienne de la représentation $l$-adique.

Abstract

Let $\pi $ be a regular, algebraic, essentially self-dual cuspidal automorphic representation of $\textrm{GL}_n({\mathbb{A}}_F)$, where $F$ is a totally real field and $n$ is at most $5$. We show that for all primes $l$, the $l$-adic Galois representations associated to $\pi $ are irreducible, and for all but finitely many primes $l$, the mod $l$ Galois representations associated to $\pi $ are also irreducible. We also show that the Lie algebras of the Zariski closures of the $l$-adic representations are independent of $l$.

Détail

Publié le : 2013-01-01

MR 3186511 | Zbl 1286.11084

DOI : https://doi.org/10.5802/aif.2817
Classification:  11F80,  11R39

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