Residual irreducibility of compatible systems
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.