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

TL;DR

This paper investigates the images of certain G_2-valued automorphic Galois representations, proving that under specific conditions, their residual images are maximally large for infinitely many primes, with applications to explicit examples.

Contribution

It establishes the largeness of residual images of G_2-valued Galois representations associated to automorphic forms, extending understanding of their image structure under automorphic conditions.

Findings

Residual images are as large as possible for infinitely many primes.

Results apply to specific examples by Chenevier, Renard, and Ta"ibi.

Provides new insights into the image structure of G_2-valued 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 Ta\"ibi.