Monodromy of subrepresentations and irreducibility of low degree automorphic Galois representations

TL;DR

This paper investigates the monodromy and irreducibility properties of low-degree automorphic Galois representations, establishing residual irreducibility and irreducibility results for compatible systems over certain number fields.

Contribution

It proves residual irreducibility of subrepresentations for almost all primes and establishes irreducibility of Galois representations attached to automorphic forms under specified conditions.

Findings

Residual irreducibility of subrepresentations for almost all primes.

Irreducibility of Galois representations attached to automorphic forms when $K$ is totally real or CM.

Residual irreducibility when $K=\mathbb{Q}$.

Abstract

Let $X$ be a smooth, separated, geometrically connected scheme defined over a number field $K$ and $\{\rho_\lambda\}_\lambda$ a system of n-dimensional semisimple $\lambda$-adic representations of the \'etale fundamental group of $X$ such that for each closed point $x$ of $X$, the specialization $\{\rho_{\lambda,x}\}_\lambda$ is a compatible system of Galois representations under mild local conditions. For almost all $\lambda$, we prove that any type A irreducible subrepresentation of $\rho_\lambda\otimes \bar{\mathbb{Q}}_\ell$ is residually irreducible. When $K$ is totally real or CM, $n\leq 6$, and $\{\rho_\lambda\}_\lambda$ is the compatible system of Galois representations of $K$ attached to a regular algebraic, polarized, cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_K)$, for almost all $\lambda$ we prove that $\rho_\lambda\otimes\bar{\mathbb{Q}}_\ell$ is (i) irreducible and (ii) residually irreducible if in addition $K=\mathbb{Q}$.