On the de Rham flip-flopping in dual towers

TL;DR

This paper establishes a comparison between de Rham and Hyodo-Kato cohomologies for dual towers of rigid analytic spaces, including local Shimura varieties, using pro-étale cohomology and descent techniques.

Contribution

It proves a flip-flopping comparison theorem for de Rham and Hyodo-Kato cohomologies in dual towers, extending to local Shimura varieties and finite level Drinfeld spaces.

Findings

Cohomologies are expressed as pro-étale cohomology of period sheaves.

De Rham and Hyodo-Kato cohomologies are shown to be admissible representations.

Comparison theorems hold for dual towers including local Shimura varieties.

Abstract

We prove a version of de Rham and Hyodo-Kato flip-flopping for dual towers of rigid analytic spaces including those coming from dual basic local Shimura varieties. The main tool are comparison theorems expressing the two cohomologies as pro-\'etale cohomology of corresponding relative period sheaves that, by definition, satisfy pro-\'etale descent. As an application, we show that de Rham and Hyodo-Kato cohomologies of finite level coverings of the Drinfeld space of any dimension $d$ over $K$ are admissible as representations of $\mathbb{GL}_{d+1}(K)$.