The lifting problem for Galois representations

TL;DR

This work resolves the lifting problem for Galois representations over fields of positive characteristic by identifying exactly when every $n$-dimensional representation of $\Gamma_K$ over $k$ lifts to $W_2(k)$. The authors recast the problem in terms of negligible cohomology classes in $H^2(G,M)$ and apply Merkuljev’s framework to reduce to finite subgroup cohomology, handling odd $p$ and $p=2$ separately with detailed finite-group analyses. They establish a sharp dichotomy: lifting holds for char$(F)=p$, or $n\le2$, or $|k|=2$ with $n\le4$, and fails otherwise; they also classify when Lift$(k,n)$ splits. The results significantly clarify the landscape of liftability, providing explicit non-liftable representations and a complete description of when lifting is possible or splits, with broad implications for deformation theory and modularity approaches in number theory.

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

Theorems & Definitions (61)

• Theorem 1.1
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Theorem 2.5