On Galois representations with large image

Abstract

For every prime number $p\geq 3$ and every integer $m\geq 1$, we prove the existence of a continuous Galois representation $\rho: G_\mathbb{Q} \rightarrow Gl_m(\mathbb{Z}_p)$ which has open image and is unramified outside $\{p,\infty\}$ (resp. outside $\{2,p,\infty\}$) when $p\equiv 3$ mod $4$ (resp. $p \equiv 1$ mod $4$).