Solid locally analytic representations
| Autor: | Jacinto, Joaquín Rodrigues, Camargo, Juan Esteban Rodríguez |
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| Rok vydání: | 2023 |
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| Druh dokumentu: | Working Paper |
| Popis: | We develop the $p$-adic representation theory of $p$-adic Lie groups on solid vector spaces over a complete non-archimedean extension of $\mathbf{Q}_p$. 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 of $G$. We also extend our previous cohomological comparison results from the case of a compact group defined over $\mathbf{Q}_p$ 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 $\mathrm{GL}_1$. Comment: Comments welcome! |
| Databáze: | arXiv |
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