Residual irreducibility of compatible systems

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.

11 pages