Locally analytic completed cohomology

TL;DR

This paper computes the geometric Sen operator for Shimura varieties and uses it to prove the rational vanishing of completed cohomology, advancing understanding of p-adic Hodge theory.

Contribution

It provides an explicit computation of the geometric Sen operator for arbitrary Shimura varieties and applies this to confirm the Calegari-Emerton conjectures.

Findings

Computed the geometric Sen operator in terms of equivariant vector bundles and the Hodge-Tate period map.

Established the rational vanishing of completed cohomology for Shimura varieties.

Connected the geometric Sen operator to the proof of Calegari-Emerton conjectures.

Abstract

We compute the geometric Sen operator for arbitrary Shimura varieties in terms of equivariant vector bundles of flag varieties and the Hodge-Tate period map. As an application, we obtain the rational vanishing of completed cohomology in the Calegari-Emerton conjectures.