ON GALOIS REPRESENTATIONS WITH LARGE IMAGE

. For every prime number p ? 3 and every integer m ? 1, we prove the existence of a continuous Galois representation & rho; : GQ N GlmpZpq which has open image and is unramified outside {p, oo} if p = 3 mod 4 and is unramified outside {2, p, oo} if p = 1 mod 4. We also revisit the question of the lifting of residual Galois representations in terms of embedding problems; that allows us to produce Galois representations with open image in the group of upper triangular matrices with diagonal entries equal to 1, unramified outside {p, oo}, for m "small" comparing to p.