The pro-étale cohomology of Drinfeld's upper half space

Abstract

We determine the geometric pro-étale cohomology of Drinfeld's upper half space over a $p$-adic field $K$. The strategy is different from the one given by [P. Colmez et al., Invent. Math. 219, No. 3, 873–985 (2020; Zbl 1475.11110)] and is based on the approach developed in [S. Orlik, Invent. Math. 172, No. 3, 585–656 (2008; Zbl 1136.22009)] describing global sections of equivariant vector bundles.