Crystalline Cohomology and Semi-Stable Reduction of Shimura Varieties in the p-adic Langlands Program

We investigate the arithmetic properties of the cohomology of Shimura varieties admitting semi-stable reduction at a prime p. By invoking the theory of log-crystalline cohomology and the Hyodo-Kato isomorphism, we construct a canonical comparison isomorphism between the p-adic étale cohomology of the generic fiber and the log-crystalline cohomology of the special fiber equipped with its Frobenius and Monodromy operators. We analyze the spectral sequence associated with the weight filtration on the log-crystalline cohomology, establishing that the Galois representations arising in the cohomology of these varieties are semi-stable in the sense of Fontaine. Furthermore, we articulate the implications of this result for the p-adic Langlands correspondence, specifically regarding the local-global compatibility of the Hecke algebra action on the associated admissible filtered (, N)-modules.