Cohomologie des courbes analytiques $p$-adiques

TL;DR

This paper investigates the cohomology of one-dimensional $p$-adic analytic curves, introducing a modified geometric framework to compute and understand their cohomological properties, especially for curves like the Drinfeld half-plane.

Contribution

It develops a new approach using adoc geometry to compute cohomologies of affinoids and curves, providing explicit descriptions of $p$-adic pro-étale cohomology in terms of de Rham and Hyodo-Kato complexes.

Findings

Cohomology of affinoids in dimension 1 is better-behaved than previously thought.

Explicit computations for curves like the Drinfeld half-plane.

Description of $p$-adic pro-étale cohomology in terms of known complexes.

Abstract

Cohomology of affinoids does not behave well; often, this can be remedied by making affinoids overconvergent. In this paper, we focus on dimension 1 and compute, using analogs of pants decompositions of Riemann surfaces, various cohomologies of affinoids. To give a meaning to these decompositions we modify slightly the notion of $p$-adic formal scheme, which gives rise to the adoc (an interpolation between adic and ad hoc) geometry. It turns out that cohomology of affinoids (in dimension 1) is not that pathological. From this we deduce a computation of cohomologies of curves without boundary (like the Drinfeld half-plane and its coverings). In particular, we obtain a description of their $p$-adic pro-\'etale cohomology in terms of de the Rham complex and the Hyodo-Kato cohomology, the later having properties similar to the ones of $\ell$-adic pro-\'etale cohomology, for $\ell\neq p$.