Perfectoid Shimura varieties and the Calegari–Emerton conjectures

Journal article, 2023

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, generalizing previous work of Scholze.