Monodromy of four dimensional irreducible compatible systems of Q

TL;DR

This paper investigates the monodromy groups of four-dimensional compatible systems of Galois representations over totally real fields, establishing conditions under which these systems are symplectic and potentially automorphic, with significant residual image properties.

Contribution

It proves that for certain four-dimensional systems over eland, the monodromy groups are symplectic and potentially automorphic, extending understanding of Galois representations and their automorphic connections.

Findings

ne-dimensional systems are symplectic for almost all mbda.

Residual images contain sp_4(\u0010F_l) subgroups for almost all mbda.

Systems are potentially automorphic under specified conditions.

Abstract

Let $F$ be a totally real field and $n\leq 4$ a natural number. We study the monodromy groups of any $n$-dimensional strictly compatible system $\{\rho_\lambda\}_\lambda$ of $\lambda$-adic representations of $F$ with distinct Hodge-Tate numbers such that $\rho_{\lambda_0}$ is irreducible for some $\lambda_0$. When $F=\mathbb{Q}$, $n=4$, and $\rho_{\lambda_0}$ is fully symplectic, the following assertions are obtained. (i) The representation $\rho_\lambda$ is fully symplectic for almost all $\lambda$. (ii) If in addition the similitude character $\mu_{\lambda_0}$ of $\rho_{\lambda_0}$ is odd, then the system $\{\rho_\lambda\}_\lambda$ is potentially automorphic and the residual image $\bar\rho_\lambda(\text{Gal}_\mathbb{Q})$ has a subgroup conjugate to $\text{Sp}_4(\mathbb{F}_\ell)$ for almost all $\lambda$.