On the images of certain $G_2$-valued automorphic Galois representations

Abstract

In this paper we study the images of certain families $\{{\rho }_{\pi ,\ell }{\}}_{\ell }$ of $G_2$-valued Galois representations of $\mbox{Gal}(\overline{F}/F)$ associated to $L$-algebraic regular, self-dual, cuspidal automorphic representations $\pi$ of $\mbox{GL}_7(\mathbb{A}_F)$, where $F$ is a totally real field. In particular, we prove that, under certain automorphic conditions, the images of the residual representations ${\overline{\rho }}_{\pi ,\ell }$ are as large as possible for infinitely many primes $\ell$. Moreover, we apply our result to some examples constructed by Chenevier, Renard and Taïbi.

In this new version, we extend the results of the previous version to totally real fields. Comments are welcome