Cohomologie de syst\`emes locaux $p$-adiques sur les rev\^etements du demi-plan de Drinfeld
| Autor: | Vanhaecke, Arnaud |
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| Jazyk: | francouzština |
| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Colmez, Dospinescu and Niziol have shown that the only $p$-adic representations of $\rm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)$ appearing in the $p$-adic \'etale cohomology of the coverings of Drinfeld's half-plane are the $2$-dimensional cuspidal representations (i.e. potentially semi-stable, whose associated Weil-Deligne representation is irreducible) with Hodge-Tate weights $0$ and $1$ and their multiplicities are given by the $p$-adic Langlands correspondence. We generalise this result to arbitrary weights, by considering the $p$-adic \'etale cohomology with coefficients in the symmetric powers of the universal local system on Drinfeld's tower. A novelty is the appearance of potentially semistable $2$-dimensional non-cristabelian representations, with expected multiplicity. The key point is that the local systems we consider turn out to be particularly simple: they are "isotrivial opers" on a curve. We develop a recipe to compute the pro\'etale cohomology of such a local system using the Hyodo-Kato cohomology of the curve and the de Rham complex of the flat filtered bundle associated to the local system. Comment: In French |
| Databáze: | arXiv |
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