Note on a certain category of mod $p$ representations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911441560272896 |
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| author | Sorgdrager, Reinier |
| author_facet | Sorgdrager, Reinier |
| contents | Let $p>3$ be a prime number, $f\geq1$ an integer. We consider a certain full subcategory $\mathcal C$ of the category of smooth admissible mod $p$ representations of either $\text{GL}_2\mathbf Q_{p^f}$ or of the group of units of the quaternion algebra over $\mathbf Q_{p^f}$. This category was introduced in the context of the mod $p$ Langlands program by Breuil-Herzig-Hu-Morra-Schraen in the $\text{GL}_2$-case and by Hu-Wang in the quaternion case. We prove that whether a smooth admissible mod $p$ representation $π$ (with central character) belongs to $\mathcal C$ is completely determined by the restriction of $π$ to an arbitrarily small open subgroup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_17773 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Note on a certain category of mod $p$ representations Sorgdrager, Reinier Representation Theory Number Theory Let $p>3$ be a prime number, $f\geq1$ an integer. We consider a certain full subcategory $\mathcal C$ of the category of smooth admissible mod $p$ representations of either $\text{GL}_2\mathbf Q_{p^f}$ or of the group of units of the quaternion algebra over $\mathbf Q_{p^f}$. This category was introduced in the context of the mod $p$ Langlands program by Breuil-Herzig-Hu-Morra-Schraen in the $\text{GL}_2$-case and by Hu-Wang in the quaternion case. We prove that whether a smooth admissible mod $p$ representation $π$ (with central character) belongs to $\mathcal C$ is completely determined by the restriction of $π$ to an arbitrarily small open subgroup. |
| title | Note on a certain category of mod $p$ representations |
| topic | Representation Theory Number Theory |
| url | https://arxiv.org/abs/2503.17773 |