Lifting $N$-dimensional Galois representations to characteristic zero

Abstract: Let $F$ be a number field, let $N\geq 3$ be an integer, and let $k$ be a finite field of characteristic $\ell$. We show that if $\rb:G_F\longrightarrow GL_N(k)$ is a continuous representation with image of $\rb$ containing $SL_N(k)$ then, under moderate conditions at primes dividing $\ell\infty$, there is a continuous representation $\rho:G_F\longrightarrow GL_N(W(k))$ unramified outside finitely many primes with $\rb\sim\rho\mod{\ell}$.