Cohomologie de systèmes locaux $p$-adiques sur les revêtements du demi-plan de Drinfeld
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866910581754167296 |
|---|---|
| author | Vanhaecke, Arnaud |
| author_facet | Vanhaecke, Arnaud |
| contents | 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 étale 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 étale 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 proétale 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_10048 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cohomologie de systèmes locaux $p$-adiques sur les revêtements du demi-plan de Drinfeld Vanhaecke, Arnaud Number Theory 11E95, 11F80, 11F85, 14F30, 22E35 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 étale 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 étale 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 proétale 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. |
| title | Cohomologie de systèmes locaux $p$-adiques sur les revêtements du demi-plan de Drinfeld |
| topic | Number Theory 11E95, 11F80, 11F85, 14F30, 22E35 |
| url | https://arxiv.org/abs/2405.10048 |