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

Abstract

Let $\{\rho_\lambda:G_K\rightarrow GL_n(\overline E_\lambda)\}$ 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 $\{\rho_{\lambda'}: G_K\rightarrow GL_n(E'_{\lambda'})\}$ 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.