Cohomologie de systèmes locaux $p$-adiques sur les revêtements du demi-plan de Drinfeld

Abstract

Colmez, Dospinescu and Niziol have shown that the only $p$-adic representations of $\mathrm{Gal}({\bar{\mathbb{Q}}}_{\mathrm{p}}/{\mathbb{Q}}_{\mathrm{p}})$ 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.

In French