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

Adrian Zenteno

Published 2020-08-02Version 1

In this paper we study the images of certain families $\{\rho_{\pi,\lambda} \}_\lambda$ of $G_2$-valued Galois representations associated to algebraic regular, self-dual, cuspidal automorphic representations $\pi$ of $\mbox{GL}_7(\mathbb{A}_\mathbb{Q})$. In particular, we prove that under certain conditions in the weights of $\pi$ the images of the residual representations $\overline{\rho}_{\pi,\lambda}$ are as large as possible for infinitely many primes $\lambda$ and provide some examples where such conditions are satisfied.