Solid locally analytic representations
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866918444083970048 |
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| author | Jacinto, Joaquín Rodrigues Camargo, Juan Esteban Rodríguez |
| author_facet | Jacinto, Joaquín Rodrigues Camargo, Juan Esteban Rodríguez |
| contents | We develop the $p$-adic representation theory of $p$-adic Lie groups on solid vector spaces over a complete non-archimedean extension of $\mathbb{Q}_p$. More precisely, we define and study categories of solid, solid locally analytic and solid smooth representations. We show that the category of solid locally analytic representations of a compact $p$-adic Lie group is equivalent to that of quasi-coherent modules over its algebra of locally analytic distributions, generalizing a classical result of Schneider and Teitelbaum. For arbitrary $G$, we prove an equivalence between solid locally analytic representations and quasi-coherent sheaves over certain locally analytic classifying stack over $G$. We also extend our previous cohomological comparison results from the case of a compact group defined over $\mathbb{Q}_p$ to the case of an arbitrary group, generalizing results of Lazard and Casselman-Wigner. Finally, we study an application to the locally analytic $p$-adic Langlands correspondence for $\mathrm{GL}_1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_03162 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Solid locally analytic representations Jacinto, Joaquín Rodrigues Camargo, Juan Esteban Rodríguez Representation Theory Number Theory We develop the $p$-adic representation theory of $p$-adic Lie groups on solid vector spaces over a complete non-archimedean extension of $\mathbb{Q}_p$. More precisely, we define and study categories of solid, solid locally analytic and solid smooth representations. We show that the category of solid locally analytic representations of a compact $p$-adic Lie group is equivalent to that of quasi-coherent modules over its algebra of locally analytic distributions, generalizing a classical result of Schneider and Teitelbaum. For arbitrary $G$, we prove an equivalence between solid locally analytic representations and quasi-coherent sheaves over certain locally analytic classifying stack over $G$. We also extend our previous cohomological comparison results from the case of a compact group defined over $\mathbb{Q}_p$ to the case of an arbitrary group, generalizing results of Lazard and Casselman-Wigner. Finally, we study an application to the locally analytic $p$-adic Langlands correspondence for $\mathrm{GL}_1$. |
| title | Solid locally analytic representations |
| topic | Representation Theory Number Theory |
| url | https://arxiv.org/abs/2305.03162 |