Crystalline Cohomology and the Weight-Monodromy Conjecture for Singular Shimura Varieties

This monograph establishes a rigorous framework for the Weight-Monodromy Conjecture in the context of singular Shimura varieties exhibiting bad reduction of parahoric type. We utilize the theory of log-crystalline cohomology and Hyodo-Kato theory to construct the requisite weight and monodromy filtrations on the cohomology of the special fiber. By invoking the Rapoport-Zink weight spectral sequence and the theory of perfectoid spaces developed by Scholze, we prove the coincidence of these filtrations for a class of Shimura varieties that admit p-adic uniformization by Drinfeld upper half spaces. Furthermore, we analyze the spectral properties of the Frobenius operator on the associated (, N)-modules, demonstrating that the purity of the weight filtration is preserved under the transition to singular loci via the local model theory. The results provide a crucial step towards the full resolution of the Langlands-Rapoport conjecture for varieties with non-smooth integral models.