p-adic Hodge Theoretic Aspects of Crystalline Cohomology in Shimura Varieties of PEL Type

This monograph establishes a rigorous framework for the integral comparison theorems between the p-adic étale cohomology and the crystalline cohomology of the special fiber of Shimura varieties of PEL type with hyperspecial level structure at p. We leverage the theory of integral canonical models constructed by Kisin and Lan to define the geometric objects over the ring of integers of a p-adic local field. By invoking the integral p-adic Hodge theory developed by Bhatt, Morrow, and Scholze, we analyze the torsion in the cohomology groups, providing a spectral sequence relating the logarithmic crystalline cohomology of the special fiber to the p-adic étale cohomology of the generic fiber. Furthermore, we investigate the degeneration of the Hodge-Tate spectral sequence in this context, extending results of Faltings and Chai to the setting of torsion coefficients. The results have significant implications for the modularity of torsion classes in the cohomology of locally symmetric spaces associated with reductive groups of type A and C.