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

TL;DR

This paper studies compatible systems of Galois representations, establishing their algebraic monodromy groups, descent properties, and invariance of abelian quotients, with applications to residual irreducibility.

Contribution

It introduces a method to attach algebraic monodromy groups to compatible systems and proves invariance of their abelian quotients, extending residual irreducibility results.

Findings

Construction of algebraic monodromy groups for compatible systems

Descent of compatible systems to strongly E'-rational systems

Invariance of maximal potentially abelian quotients across the system

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.