On the $p$-adic pro-\'etale cohomology of Drinfeld symmetric spaces

TL;DR

This paper computes the $p$-adic pro-étale cohomology of Drinfeld symmetric spaces using $p$-adic Hodge theory, providing an alternative proof of a known theorem and describing cohomology via differential forms with new condensed mathematics tools.

Contribution

It offers a new proof of a key theorem on $p$-adic cohomology of Drinfeld spaces and describes the cohomology of the de Rham period sheaf using differential forms, incorporating condensed mathematics.

Findings

Computed the geometric $p$-adic pro-étale cohomology of Drinfeld symmetric spaces.

Described the cohomology of the positive de Rham period sheaf via differential forms.

Introduced condensed mathematics techniques to $p$-adic Hodge theory.

Abstract

Via the relative fundamental exact sequence of $p$-adic Hodge theory, we determine the geometric $p$-adic pro-\'etale cohomology of the Drinfeld symmetric spaces defined over a $p$-adic field, thus giving an alternative proof of a theorem of Colmez-Dospinescu-Niziol. Along the way, we describe, in terms of differential forms, the geometric pro-\'etale cohomology of the positive de Rham period sheaf on any connected, paracompact, smooth rigid-analytic variety over a $p$-adic field, and we do it with coefficients. A key new ingredient is the condensed mathematics recently developed by Clausen-Scholze.