On the Cohomology of Shimura Varieties with Torsion Coefficients

Explore the latest developments in the cohomology of Shimura varieties with torsion coefficients in this comprehensive lecture by Ana Caraiani from Imperial College London. Delve into the geometry of the Hodge-Tate period morphism, including Mingjia Zhang's recent generalization of Igusa varieties to Igusa stacks. Compare the original approach to computing cohomology with torsion coefficients, developed by Caraiani and Peter Scholze using the trace formula, with newer methods by Teruhisa Koshikawa, Linus Hamann, and Si Ying Lee that rely on advanced local results. Discover how combining these approaches leads to a new instance of local-global compatibility. This 1-hour and 12-minute talk, presented at the Institut des Hautes Etudes Scientifiques (IHES), offers a thorough survey of recent results in this field of mathematics.