Galois representations with large image in the global Langlands correspondence

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 $\{{\rho }_{\pi ,\mathrm{\ell }}{\}}_{\mathrm{\ell }}$ of $n$-dimensional $\mathrm{\ell }$-adic Galois representations of $\text{Gal}(\bar{F}\mathrm{/}F)$. In general, it is expected that the image of the residual Galois representation ${\bar{\rho }}_{\pi ,\mathrm{\ell }}$ of ${\rho }_{\pi ,\mathrm{\ell }}$ should be as large as possible for almost all primes $\mathrm{\ell }$, 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 $\{{\rho }_{\pi ,\mathrm{\ell }}{\}}_{\mathrm{\ell }}$ 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 ${\bar{\rho }}_{\pi ,\mathrm{\ell }}$ are as large as possible for infinitely many primes $\mathrm{\ell }$. In fact, we prove the large image conjecture (i.e., large image for almost all primes $\mathrm{\ell }$) when $F=\mathbb{Q}$ and $n=5$.