Cohomological Vanishing and p-adic Etale Cohomology of Perfectoid Shimura Varieties at Infinite Level

This monograph investigates the asymptotic cohomological behavior of Shimura varieties at infinite level within the framework of perfectoid geometry. We establish that the inverse limit of the tower of Shimura varieties associated to a Shimura datum (G, X) of Hodge type admits a unique perfectoid space structure S. Utilizing the Hodge-Tate period map HT: S FG,, we analyze the p-adic étale cohomology of these spaces. Our central result demonstrates a vanishing theorem for torsion cohomology classes in degrees above the middle dimension, confirming specific instances of the Calegari-Emerton conjectures. We further explore the implications for completed cohomology, providing a geometric interpretation of the "sparse" distribution of Hecke eigenvalues in the torsion setting. The methods rely heavily on the tilting equivalence, the geometry of diamonds, and the spectral sequence associated with the Hodge-Tate fibration.