The lifting problem for Galois representations

Abstract

We solve the lifting problem for Galois representations in every dimension and in every characteristic. That is, we determine all pairs $(n,k)$, where $n$ is a positive integer and $k$ is a field of characteristic $p>0$, such that for every field $F$, every continuous homomorphism $\hat{I}{“}_F\to {\mathrm{GL}}_n(k)$ lifts to ${\mathrm{GL}}_n(W_2(k))$, where $\hat{I}{“}_F$ is the absolute Galois group of $F$ and $W_2(k)$ is the ring of $p$-typical length $2$ Witt vectors of $k$.

38 pages