On the images of the Galois representations attached to certain RAESDC automorphic representations of $\mbox{GL}_n(\mathbb{A}_{\mathbb{Q}})$

Adrian Zenteno

Published 2017-06-13Version 1

In this paper we prove the existence of infinitely many compatible systems $\{ \rho_\ell \}_\ell$ of $n$-dimensional Galois representations associated to regular algebraic, essentially self-dual, cuspidal automorphic representations of $\mbox{GL}_n(\mathbb{A}_{\mathbb{Q}})$ ($n$ even) such that the image of $\overline{\rho}_{\ell}^{proj}$ (the projectivization of the reduction of $\rho_\ell$) is an almost simple group for almost all primes $\ell$. Moreover, applying this result to some low-dimensional cases, we prove that the symplectic groups: $\mbox{PSp}_n(\mathbb{F}_{\ell^s})$ and $\mbox{PGSp}_n(\mathbb{F}_{\ell^s})$, for $6\leq n \leq 12$, and the orthogonal groups: $\mbox{P}\Omega^+_n(\mathbb{F}_{\ell^s})$, $\mbox{PSO}^+_n(\mathbb{F}_{\ell^s})$, $\mbox{PO}^+_n(\mathbb{F}_{\ell^s})$ and $\mbox{PGO}^+_n(\mathbb{F}_{\ell^s})$, occurs as Galois groups over $\mathbb{Q}$ for infinitely many primes $\ell$ and infinitely many positive integers $s$.