Vanishing Theorems for p-adic Étale Cohomology on Perfectoid Shimura Varieties

We establish vanishing theorems for the p-adic étale cohomology of Shimura varieties of Hodge type with torsion coefficients. By passing to the inverse limit over all levels at p, we work within the framework of perfectoid spaces, specifically utilizing the perfectoid Shimura variety SKp constructed by Scholze. We analyze the geometry of the Hodge-Tate period map HT: SKp FG,, proving that its affineness implies the vanishing of cohomology groups Hiét(SK, F) for degrees i exceeding the complex dimension of the Shimura variety. Our approach integrates the primitive comparison theorem in the pro-étale topology with the geometry of the flag variety FG,. These results confirm conjectures of Calegari and Emerton regarding the sparseness of torsion in the cohomology of locally symmetric spaces associated with reductive groups.