Monodromy and irreducibility of type $A_1$ automorphic Galois representations

TL;DR

This paper proves the independence and irreducibility of Galois representations attached to automorphic forms under certain conditions, and relates these systems to modular forms when the base field is rational.

Contribution

It establishes the independence of algebraic monodromy groups and irreducibility of Galois representations for automorphic forms of type A1, extending results without polarization assumptions in specific cases.

Findings

Monodromy groups are independent of the prime $\\lambda$.

Galois representations are irreducible for all $\\lambda$.

Compatible systems originate from two-dimensional modular systems when $K=\mathbb{Q}$.

Abstract

Let $K$ be a totally real field and $\pi$ be a regular algebraic polarized cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb A_K)$. Let $\{\rho_{\pi,\lambda}:\mathrm{Gal}_K\to\mathrm{GL}_n(\overline E_\lambda)\}_\lambda$ be the compatible system of Galois representations attached to $\pi$ and denote by $\mathbf G_\lambda$ the algebraic monodromy group of $\rho_{\pi,\lambda}$. Suppose there exists $\lambda_0$ such that (a) $\rho_{\pi,\lambda_0}$ is irreducible; (b) $\mathbf G_{\lambda_0}$ is connected and of type $A_1$; and (c) the tautological representation of $\mathbf G_{\lambda_0}$ is of a certain type. We prove that $\bullet$ $\mathbf G_{\lambda,\mathbb C}\subset\mathrm{GL}_{n, \mathbb C}$ is independent of $\lambda$; $\bullet$ $\rho_{\pi,\lambda}$ is irreducible for all $\lambda$, and residually irreducible for almost all $\lambda$. Moreover, if $K=\mathbb Q$ or $n$ is odd, we prove that the same conclusions hold without the assumption that $\pi$ is polarized. We also prove that if $K=\mathbb Q$, then the compatible system $\{\rho_{\pi,\lambda}\}_\lambda$ is constructed from certain two-dimensional modular compatible systems up to twist.