Cohomology of Perfectoid Shimura Varieties and the p-adic Local Langlands Correspondence

This monograph explores the intersection of perfectoid geometry and the p-adic Local Langlands program, focusing on the cohomology of Shimura varieties at infinite level. We analyze the geometric structure of perfectoid Shimura varieties, denoted as SKp, arising as inverse limits of finite-level Shimura varieties with full level structure at p. Central to our study is the Hodge-Tate period map HT: SKp FG, which fibers the perfectoid space over the flag variety. We investigate the vanishing theorems for torsion in the cohomology of these varieties, utilizing the geometry of the Hodge-Tate period map and the tilting equivalence. Furthermore, we articulate the realization of the p-adic Local Langlands correspondence within the completed cohomology groups H"0362Hi(SKp, Zp), establishing a categorical link between the spectral action of the stack of L-parameters on the Fargues-Fontaine curve and the Hecke eigenspaces of the cohomology. The results synthesize recent breakthroughs by Scholze, Fargues, and Caraiani, providing a unified framework for understanding the p-adic arithmetic of automorphic forms.