Galois representations with large image in the global Langlands correspondence

Abstract

The global Langlands conjecture for ${\text{GL}}_n$ over a number field $F$ predicts a correspondence between certain algebraic automorphic representations $\pi$ of ${\text{GL}}_n({\mathbb{A}}_F)$ and certain families $\{{\rho }_{\pi ,\ell }{\}}_{\ell }$ of $n$-dimensional $\ell$-adic Galois representations of $\text{Gal}(\overline{F}/F)$. In general, it is expected that the image of the residual Galois representation ${\overline{\rho }}_{\pi ,\ell }$ of ${\rho }_{\pi ,\ell }$ should be as large as possible for almost all primes $\ell$, unless there is an automorphic reason for the image to be small. In this paper, we study the images of certain compatible systems of Galois representations $\{{\rho }_{\pi ,\ell }{\}}_{\ell }$ associated to regular algebraic, polarizable, cuspidal automorphic representations $\pi$ of ${\text{GL}}_n({\mathbb{A}}_F)$ by using only standard techniques and currently available tools (e.g., Fontaine-Laffaille theory, Serre's modularity conjecture, classification of the maximal subgroups of Lie type groups, and known results about irreducibility of automorphic Galois representations and Langlands functoriality). In particular, when $F$ is a totally real field and $n$ is an odd prime number $\le 293$, 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 $\ell$. In fact, we prove the large image conjecture (i.e., large image for almost all primes $\ell$) when $F=\mathbb{Q}$ and $n=5$.

arXiv commentThis article supersedes arXiv:2008.00556. Accepted in the Fields Institute Monographs