Residual irreducibility of compatible systems

TL;DR

This paper proves that for a compatible system of absolutely irreducible Galois representations, the residual representations are almost always absolutely irreducible, with key technical results on the image of these representations and their bounded index properties.

Contribution

It establishes the residual irreducibility for almost all primes in compatible systems and introduces a new proof for the bounded index property of the image of Galois representations.

Findings

Residual representations are absolutely irreducible for density 1 set of primes.

The image of the Galois representations is an open subgroup of a hyperspecial maximal compact subgroup.

A new proof is provided for the bounded index property of the image.

Abstract

We show that if $\{\rho_{\ell}\}$ is a compatible system of absolutely irreducible Galois representations of a number field then the residual representation $\overline{\rho}_{\ell}$ is absolutely irreducible for $\ell$ in a density 1 set of primes. The key technical result is the following theorem: the image of $\rho_{\ell}$ is an open subgroup of a hyperspecial maximal compact subgroup of its Zariski closure with bounded index (as $\ell$ varies). This result combines a theorem of Larsen on the semi-simple part of the image with an analogous result for the central torus that was recently proved by Barnet-Lamb, Gee, Geraghty, and Taylor, and for which we give a new proof.