Cohomological Vanishing Theorems for Non-Compact Shimura Varieties via p-adic Hodge Theory

We establish new cohomological vanishing theorems for non-compact Shimura varieties of PEL type with coefficients in torsion automorphic sheaves. By leveraging the geometry of the Faltings-Chai toroidal compactification and the infinite-level perfectoid structures introduced by Scholze, we analyze the boundary contributions to the cohomology. Specifically, we utilize the p-adic Hodge-Tate period map to transport the vanishing problem to the flag variety, employing a p-adic analogue of the Bernstein-Gelfand-Gelfand (BGG) resolution. Our main result provides a sharp bound on the cohomological degree in terms of the weight of the coefficient system and the toroidal boundary stratification, extending the classical vanishing results of Borel-Wallach and the algebraic results of Lan-Suh to the p-adic torsion setting. We further discuss applications to the Calegari-Emerton conjectures on completed cohomology.