Galois representations modulo $p$ that do not lift modulo $p^2$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917805001015296 |
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| author | Merkurjev, Alexander Scavia, Federico |
| author_facet | Merkurjev, Alexander Scavia, Federico |
| contents | For every finite group $H$ and every finite $H$-module $A$, we determine the subgroup of negligible classes in $H^2(H,A)$, in the sense of Serre, over fields with enough roots of unity. As a consequence, we show that for every odd prime $p$, every integer $n\geq 3$, and every field $F$ containing a primitive $p$-th root of unity, there exists a continuous $n$-dimensional mod $p$ representation of the absolute Galois group of $F(x_1,\dots,x_p)$ which does not lift modulo $p^2$. This answers a question of Khare and Serre, and disproves a conjecture of Florence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_12560 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Galois representations modulo $p$ that do not lift modulo $p^2$ Merkurjev, Alexander Scavia, Federico Number Theory 12G05 (Primary) 11F80, 12F12, 20J06 (Secondary) For every finite group $H$ and every finite $H$-module $A$, we determine the subgroup of negligible classes in $H^2(H,A)$, in the sense of Serre, over fields with enough roots of unity. As a consequence, we show that for every odd prime $p$, every integer $n\geq 3$, and every field $F$ containing a primitive $p$-th root of unity, there exists a continuous $n$-dimensional mod $p$ representation of the absolute Galois group of $F(x_1,\dots,x_p)$ which does not lift modulo $p^2$. This answers a question of Khare and Serre, and disproves a conjecture of Florence. |
| title | Galois representations modulo $p$ that do not lift modulo $p^2$ |
| topic | Number Theory 12G05 (Primary) 11F80, 12F12, 20J06 (Secondary) |
| url | https://arxiv.org/abs/2410.12560 |