Residual Irreducibility of Compatible Systems.

We show that if {ρℓ} is a compatible system of absolutely irreducible Galois representations of a number field then the residual representation ¯ρℓ is absolutely irreducible for ℓ in a density 1 set of primes. The key technical result is the following theorem: the image of ρℓ is an open subgroup of a hyperspecial maximal compact subgroup of its Zariski closure with bounded index (as ℓ 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. [ABSTRACT FROM AUTHOR]