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arXiv · 2501.18906

The lifting problem for Galois representations

Alexander Merkurjev·Federico Scavia
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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 $k$.

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Alexander Merkurjev, Federico Scavia. 2025-01-31. The lifting problem for Galois representations. https://arxiv.org/abs/2501.18906

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