$p$-adic Hodge parameters in the crystabelline representations of $\mathrm{GL}_n$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916791171678208 |
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| author | Ding, Yiwen |
| author_facet | Ding, Yiwen |
| contents | Let $K$ be a finite extension of $\mathbb{Q}_p$, and $ρ$ be an $n$-dimensional (non-critical generic) crystabelline representation of the absolute Galois group of $K$ of regular Hodge-Tate weights. We associate to $ρ$ an explicit locally $\mathbb{Q}_p$-analytic representation $π_1(ρ)$ of $\mathrm{GL}_n(K)$, which encodes some $p$-adic Hodge parameters of $ρ$. When $K=\mathbb{Q}_p$, it encodes the full information hence reciprocally determines $ρ$. When $ρ$ is associated to $p$-adic automorphic representations, we show under mild hypotheses that $π_1(ρ)$ is a subrepresentation of the $\mathrm{GL}_n(K)$-representation globally associated to $ρ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_21237 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $p$-adic Hodge parameters in the crystabelline representations of $\mathrm{GL}_n$ Ding, Yiwen Number Theory Representation Theory Let $K$ be a finite extension of $\mathbb{Q}_p$, and $ρ$ be an $n$-dimensional (non-critical generic) crystabelline representation of the absolute Galois group of $K$ of regular Hodge-Tate weights. We associate to $ρ$ an explicit locally $\mathbb{Q}_p$-analytic representation $π_1(ρ)$ of $\mathrm{GL}_n(K)$, which encodes some $p$-adic Hodge parameters of $ρ$. When $K=\mathbb{Q}_p$, it encodes the full information hence reciprocally determines $ρ$. When $ρ$ is associated to $p$-adic automorphic representations, we show under mild hypotheses that $π_1(ρ)$ is a subrepresentation of the $\mathrm{GL}_n(K)$-representation globally associated to $ρ$. |
| title | $p$-adic Hodge parameters in the crystabelline representations of $\mathrm{GL}_n$ |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2407.21237 |