Monodromy of four dimensional irreducible compatible systems of Q

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 $\lambda$.