Pointwise perfectoidness of Shimura varieties at infinite level

A longstanding question in the theory of Shimura varieties concerns their perfectoidness at infinite level—a property that would reveal deep connections between étale and coherent cohomology. In this talk, we establish a criterion for perfectoidness via Sen theory, building on a new development of p-adic Hodge theory for general valuation rings that extends Tate’s foundational work on discrete valuation rings. We further provide a conceptual explanation, based on the p-adic Simpson correspondence after Abbes and Gros, for why Shimura varieties satisfy this criterion, at least in the case of modular curves. For general Shimura varieties, it follows through additional technical arguments due to Pan and Rodríguez Camargo. This yields the “pointwise perfectoidness” of Shimura varieties at infinite level, which suffices to establish the desired connection between different cohomologies. As an application, we show that integral completed cohomology groups vanish in higher degrees, thereby confirming a conjecture of Calegari and Emerton for arbitrary Shimura varieties.