On coefficients, potentially abelian quotients, and residual irreducibility of compatible systems

Abstract

Let $\{{\rho }_{\lambda }:G_K\to G{L}_n({\overline{E}}_{\lambda })\}$ be a semisimple E-rational compatible system of a number field K. In a first step, building upon the theory of pseudocharacters [Ro96],[Ch14], we attach to each ${\rho }_{\lambda }$ an algebraic monodromy group $G_{\lambda }$ defined over $E_{\lambda }$ and also prove that the compatible system can be descended to a strongly E'-rational compatible system $\{{\rho }_{{\lambda }^{\prime }}:G_K\to G{L}_n(E_{{\lambda }^{\prime }}^{\prime })\}$ for some finite extension E'/E. Secondly, we demonstrate that the maximal potentially abelian quotient of $G_{\lambda }$ is independent of $\lambda$ in a strong sense. Finally, as an application, we generalize a result of Patrikis--Snowden--Wiles on residual irreducibility of compatible systems.