Toward the $p$-adic Hodge parameters in the potentially crystalline representations of $\mathrm{GL}_n$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914347504107520 |
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| author | He, Yiqin |
| author_facet | He, Yiqin |
| contents | Let $p$ be a prime number, $n$ an integer $\geq 2$, and $L$ a finite extension of $\mathrm{Q}_p$. Let $ρ_L$ be an $n$-dimensional (non-critical but not necessary generic) potentially crystalline $p$-adic Galois representation of the absolute Galois groups of $L$ of regular Hodge-Tate weights. By generalizing the previous results and strategy for the crystabelline case of Ding and the recent work of Breuil-Ding, we construct an explicit locally analytic representation $π_{1}(ρ_L)$, and describe explicitly the information of Hodge filtration of $ρ_L$ it determines. When $ρ_L$ comes from a patched $p$-adic automorphic representation, we show that $π_{1}(ρ_L)$ is a subrepresentation of the $\mathrm{GL}_n(L)$-representation globally associated to $ρ_L$, under some mild hypothesis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21063 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Toward the $p$-adic Hodge parameters in the potentially crystalline representations of $\mathrm{GL}_n$ He, Yiqin Number Theory Let $p$ be a prime number, $n$ an integer $\geq 2$, and $L$ a finite extension of $\mathrm{Q}_p$. Let $ρ_L$ be an $n$-dimensional (non-critical but not necessary generic) potentially crystalline $p$-adic Galois representation of the absolute Galois groups of $L$ of regular Hodge-Tate weights. By generalizing the previous results and strategy for the crystabelline case of Ding and the recent work of Breuil-Ding, we construct an explicit locally analytic representation $π_{1}(ρ_L)$, and describe explicitly the information of Hodge filtration of $ρ_L$ it determines. When $ρ_L$ comes from a patched $p$-adic automorphic representation, we show that $π_{1}(ρ_L)$ is a subrepresentation of the $\mathrm{GL}_n(L)$-representation globally associated to $ρ_L$, under some mild hypothesis. |
| title | Toward the $p$-adic Hodge parameters in the potentially crystalline representations of $\mathrm{GL}_n$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2602.21063 |