Lifting $N$-dimensional Galois representations to characteristic zero

TL;DR

This paper proves that certain mod $\ell$ Galois representations of number fields can be lifted to characteristic zero representations over Witt vectors, under specific conditions on the image and ramification.

Contribution

It establishes a lifting theorem for $N$-dimensional Galois representations with large image, extending previous results to higher dimensions and more general conditions.

Findings

Existence of characteristic zero lifts under specified conditions

Lifts are unramified outside finitely many primes

Applicable to representations with image containing $SL_N(k)$

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 $\rb\sim\rho\mod{\ell}$.