Galois representations with large image in the global Langlands correspondence

TL;DR

This work investigates the images of residual Galois representations $\overline{\rho}_{\pi,\ell}$ attached to regular algebraic, self-dual, cuspidal automorphic representations $\pi$ of $\mathrm{GL}_n(\mathbb{A}_F)$ within the global Langlands correspondence. It develops and uses only standard tools—Fontaine–Laffaille theory, Serre’s modularity theorem, maximal-subgroup classifications of Lie-type groups, and known automorphy and irreducibility results—to establish large-image outcomes: (i) for totally real $F$ and odd prime $n=p \le 293$, $\overline{\rho}_{\pi,\ell}$ has large image for a density-positive set of primes $\ell$; (ii) a full large-image result for $F=\mathbb{Q}$ and $n=5$; and (iii) complementary $G_2$-valued cases when $n=7$, via Chenevier’s constructions. The paper also provides density-one results under extra local hypotheses when $F=\mathbb{Q}$, using Serre’s modularity and symmetric-power functoriality. Together, these results give concrete evidence for the large-image phenomenon in the global Langlands program and illustrate a robust, tool-driven approach to controlling residual images using only standard ingredients.

Abstract

The global Langlands conjecture for $\text{GL}_n$ over a number field $F$ predicts a correspondence between certain algebraic automorphic representations $π$ of $\text{GL}_n(\mathbb{A}_F)$ and certain families $\{ ρ_{π,\ell} \}_\ell$ 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 $\overlineρ_{π,\ell}$ of $ρ_{π,\ell}$ 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 $\{ρ_{π,\ell} \}_\ell$ associated to regular algebraic, polarizable, cuspidal automorphic representations $π$ 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 $\overlineρ_{π,\ell}$ 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 $n=5$.

Theorems & Definitions (18)

• Theorem 2.1
• Theorem 3.1
• Proposition 3.2
• proof
• Remark 1
• Theorem 3.3
• proof
• Theorem 4.1
• Theorem 4.2
• proof