Perfectoid Shimura varieties and the Calegari-Emerton conjectures

TL;DR

This paper advances the understanding of completed cohomology in Shimura varieties by proving new cases of the Calegari-Emerton conjecture, using perfectoid techniques and boundary analysis.

Contribution

It introduces a new perfectoidness result for towers of minimally compactified Shimura varieties, extending Scholze's work and enabling progress on the conjecture.

Findings

Proved new cases of the Calegari-Emerton conjecture.

Established a new perfectoidness result for Shimura varieties.

Developed an inductive analysis of cohomology on the Borel-Serre boundary.

Abstract

We prove many new cases of a conjecture of Calegari-Emerton describing the qualitative properties of completed cohomology. The heart of our argument is a careful inductive analysis of completed cohomology on the Borel-Serre boundary. As a key input to this induction, we prove a new perfectoidness result for towers of minimally compactified Shimura varieties of pre-abelian type, generalizing previous work of Scholze.