The pro-\'etale cohomology of Drinfeld's upper half space

TL;DR

This paper computes the geometric pro-étale cohomology of Drinfeld's upper half space over a p-adic field using a novel approach that differs from previous methods, leveraging equivariant vector bundles.

Contribution

It introduces a new method for calculating pro-étale cohomology of Drinfeld's upper half space, expanding the toolkit beyond existing approaches.

Findings

Determined the geometric pro-étale cohomology of Drinfeld's upper half space.

Developed a new approach based on equivariant vector bundles.

Provided results that differ from prior methods by Colmez, Dospinescu, and Niziol.

Abstract

We determine the geometric pro-\'etale cohomology of Drinfeld's upper half space ${\mathcal X}$ over a p-adic field. The strategy is different from the one given by Colmez, Dospinescu and Niziol. It uses the approach developed in a former paper of the author describing global sections of equivariant vector bundles on ${\mathcal X}$.