Cohomology of Perfectoid Shimura Varieties and Vanishing Cycles in the Local Langlands Program

This monograph explores the intersection of p-adic Hodge theory, the cohomology of Shimura varieties, and the local Langlands correspondence. We analyze the geometry of Shimura varieties at infinite level, demonstrating their perfectoid nature as established by Scholze. Central to this study is the Hodge-Tate period map HT, which relates the perfectoid Shimura variety to the flag variety. We investigate the vanishing cycles formalism in this context, specifically focusing on the cohomology of the special fiber and its relation to the generic fiber via the nearby cycles functor R. Furthermore, we articulate the implications of these geometric structures for the local Langlands program, leveraging the Fargues-Fontaine curve and the geometrization conjecture of Fargues and Scholze. We provide a rigorous treatment of torsion in the cohomology of Shimura varieties and its role in constructing Galois representations associated with mod p automorphic forms.