On Galois representations with large image

Abstract

For every prime number p ě 3 and every integer m ě 1, we prove the existence of a continuous Galois representation ρ : G Q Ñ Gl m pZ p q which has open image and is unramified outside tp, 8u (resp. outside t2, p, 8u) when p " 3 mod 4 (resp. p " 1 mod 4).