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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2501.00259 |
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| _version_ | 1866915125134360576 |
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| author | Aoki, Kensuke |
| author_facet | Aoki, Kensuke |
| contents | For a finite extension $K/\mathbb{Q}_p$ and a split reductive group $G$ over $\mathcal{O}_K$, let $\overlineρ \colon \mathrm{Gal}_K \to G(\overline{\mathbb{F}}_p)$ be a continuous quasi-semisimple mod $p$ $G$-valued representation of the absolute Galois group $\mathrm{Gal}_K$. Let $\overlineρ^{\mathrm{ab}}$ be the abelianization of $\overlineρ$ and fix a crystalline lift $ψ$ of $\overlineρ^{\mathrm{ab}}$. We show the existence of a crystalline lift $ρ$ of $\overlineρ$ with regular Hodge-Tate weights such that the abelianization of $ρ$ coincides with $ψ$. We also show analogous results in the case that $G$ is a quasi-split tame group and $\overlineρ \colon \mathrm{Gal}_K \to {^L}G(\overline{\mathbb{F}}_p)$ is a semisimple mod $p$ $L$-parameter. These theorems are generalizations of those of Lin and Böckle-Iyengar-Paškūnas. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_00259 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Crystalline lifts of semisimple $G$-valued Galois representations with fixed determinant Aoki, Kensuke Number Theory For a finite extension $K/\mathbb{Q}_p$ and a split reductive group $G$ over $\mathcal{O}_K$, let $\overlineρ \colon \mathrm{Gal}_K \to G(\overline{\mathbb{F}}_p)$ be a continuous quasi-semisimple mod $p$ $G$-valued representation of the absolute Galois group $\mathrm{Gal}_K$. Let $\overlineρ^{\mathrm{ab}}$ be the abelianization of $\overlineρ$ and fix a crystalline lift $ψ$ of $\overlineρ^{\mathrm{ab}}$. We show the existence of a crystalline lift $ρ$ of $\overlineρ$ with regular Hodge-Tate weights such that the abelianization of $ρ$ coincides with $ψ$. We also show analogous results in the case that $G$ is a quasi-split tame group and $\overlineρ \colon \mathrm{Gal}_K \to {^L}G(\overline{\mathbb{F}}_p)$ is a semisimple mod $p$ $L$-parameter. These theorems are generalizations of those of Lin and Böckle-Iyengar-Paškūnas. |
| title | Crystalline lifts of semisimple $G$-valued Galois representations with fixed determinant |
| topic | Number Theory |
| url | https://arxiv.org/abs/2501.00259 |