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 $4$).