Hodge filtration and crystalline representations of $\mathrm{GL}_n$
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915673111789568 |
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| author | Breuil, Christophe Ding, Yiwen |
| author_facet | Breuil, Christophe Ding, Yiwen |
| contents | Let $p$ be a prime number, $n$ an integer $\geq 2$ and $ρ$ an $n$-dimensional automorphic $p$-adic Galois representation (for a compact unitary group) such that $r:=ρ\vert_{\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)}$ is crystalline. Under a mild assumption on the Frobenius eigenvalues of $D:=D_{\mathrm{cris}}(r)$ and under the usual Taylor-Wiles conditions, we show that the locally analytic representation of $\mathrm{GL}_n(\mathbb{Q}_p)$ associated to $ρ$ in the corresponding Hecke eigenspace of the completed $H^0$ contains an explicit finite length subrepresentation which determines and only depends on $r$. This generalizes previous results of the second author which assumed that the Hodge filtration on $D$ was as generic as possible. Our approach provides a much more explicit link to this Hodge filtration (in all cases), which allows to study the internal structure of this finite length locally analytic subrepresentation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12153 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hodge filtration and crystalline representations of $\mathrm{GL}_n$ Breuil, Christophe Ding, Yiwen Number Theory Representation Theory Let $p$ be a prime number, $n$ an integer $\geq 2$ and $ρ$ an $n$-dimensional automorphic $p$-adic Galois representation (for a compact unitary group) such that $r:=ρ\vert_{\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)}$ is crystalline. Under a mild assumption on the Frobenius eigenvalues of $D:=D_{\mathrm{cris}}(r)$ and under the usual Taylor-Wiles conditions, we show that the locally analytic representation of $\mathrm{GL}_n(\mathbb{Q}_p)$ associated to $ρ$ in the corresponding Hecke eigenspace of the completed $H^0$ contains an explicit finite length subrepresentation which determines and only depends on $r$. This generalizes previous results of the second author which assumed that the Hodge filtration on $D$ was as generic as possible. Our approach provides a much more explicit link to this Hodge filtration (in all cases), which allows to study the internal structure of this finite length locally analytic subrepresentation. |
| title | Hodge filtration and crystalline representations of $\mathrm{GL}_n$ |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2512.12153 |