Cohomologie de syst\`emes locaux $p$-adiques sur les rev\^etements du demi-plan de Drinfeld

TL;DR

This paper extends the understanding of p-adic Galois representations in the étale cohomology of Drinfeld's half-plane coverings, revealing new non-crystalline representations through the study of symmetric power local systems.

Contribution

It generalizes previous results to arbitrary weights by analyzing symmetric power local systems, introducing potentially semistable non-crystalline representations, and developing a new computational approach.

Findings

Identification of new potentially semistable non-crystalline representations

Extension of cohomological results to arbitrary weights

Development of a recipe using Hyodo-Kato and de Rham cohomology

Abstract

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.