Monodromy and irreducibility of type $A_1$ automorphic Galois representations

TL;DR

This work addresses the problem of when automorphic Galois representations attached to regular algebraic polarized cuspidal representations have monodromy data that are independent of the chosen prime $\lambda$, and when the representations themselves are irreducible. It combines potential automorphy, $\,\lambda$-independence results for monodromy groups, and rigidity theorems (notably Larsen–Pink) to show that, under a single $\lambda_0$ with $\rho_{\pi,\lambda_0}$ irreducible and $\mathbf{G}_{\lambda_0}$ of type $A_1$, the complexified monodromy $\mathbf{G}_{\lambda,\mathbb{C}}$ is independent of $\lambda$ and that all $\rho_{\pi,\lambda}$ are irreducible (with residual irreducibility for almost all $\lambda$). The paper further extends the conclusions to the case $K=\mathbb{Q}$ or odd $n$ without polarization, and, in the $K=\mathbb{Q}$ case, shows the system can be realized from two-dimensional modular representations up to twist. The approach hinges on constructing compatible three-dimensional pieces via potential automorphy and leveraging formal bi-character stability to deduce Lie-type rigidity, culminating in a classification that connects automorphic and Galois-theoretic data in a unified framework.

Abstract

Let $K$ be a totally real field and $π$ be a regular algebraic polarized cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb A_K)$. Let $\{ρ_{π,λ}:\mathrm{Gal}_K\to\mathrm{GL}_n(\overline E_λ)\}_λ$ be the compatible system of Galois representations attached to $π$ and denote by $\mathbf G_λ$ the algebraic monodromy group of $ρ_{π,λ}$. Suppose there exists $λ_0$ such that (a) $ρ_{π,λ_0}$ is irreducible; (b) $\mathbf G_{λ_0}$ is connected and of type $A_1$; and (c) the tautological representation of $\mathbf G_{λ_0}$ is of a certain type. We prove that $\bullet$ $\mathbf G_{λ,\mathbb C}\subset\mathrm{GL}_{n, \mathbb C}$ is independent of $λ$; $\bullet$ $ρ_{π,λ}$ is irreducible for all $λ$, and residually irreducible for almost all $λ$. Moreover, if $K=\mathbb Q$ or $n$ is odd, we prove that the same conclusions hold without the assumption that $π$ is polarized. We also prove that if $K=\mathbb Q$, then the compatible system $\{ρ_{π,λ}\}_λ$ is constructed from certain two-dimensional modular compatible systems up to twist.

Theorems & Definitions (41)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5