The lifting problem for Galois representations
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917907383975936 |
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| author | Merkurjev, Alexander Scavia, Federico |
| author_facet | Merkurjev, Alexander Scavia, Federico |
| contents | 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 $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_18906 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The lifting problem for Galois representations Merkurjev, Alexander Scavia, Federico Number Theory 12G05 (Primary) 11F80, 12F12, 20J06 (Secondary) 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 $k$. |
| title | The lifting problem for Galois representations |
| topic | Number Theory 12G05 (Primary) 11F80, 12F12, 20J06 (Secondary) |
| url | https://arxiv.org/abs/2501.18906 |