Description:

Title: Perfectoidness via Sen Theory and Applications to Shimura Varieties

Abstract: There is a longstanding question about the perfectoidness of Shimura varieties at infinite level, which would lead to a profound connection between étale cohomology and coherent cohomology. By developing p-adic Hodge theory for general valuation rings, we establish “stalkwise perfectoidness” for Shimura varieties at infinite level, which is sufficient for such a connection. As an application, we prove that the integral completed cohomology groups vanish in higher degrees, verifying a conjecture of Calegari-Emerton for arbitrary Shimura varieties. This is my recent work arXiv.2407.14488.