The lifting problem for Galois representations

TL;DR

This paper completely characterizes when Galois representations in any dimension and characteristic can be lifted to Witt vector rings, solving a fundamental problem in number theory and algebraic geometry.

Contribution

It provides a comprehensive solution to the lifting problem for Galois representations across all dimensions and characteristics, identifying all pairs (n, k) where lifting is possible.

Findings

Determined all pairs (n, k) with lifting property.

Established criteria for lifting Galois representations.

Unified understanding of lifting in all dimensions and characteristics.

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 $\Gamma_F\to \mathrm{GL}_n(k)$ lifts to $\mathrm{GL}_n(W_2(k))$, where $\Gamma_F$ is the absolute Galois group of $F$ and $W_2(k)$ is the ring of $p$-typical length $2$ Witt vectors of $k$.