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 $\leq 293$, 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 $n=5$.
Categories: math.NT math.GR
