Minjia Zhang (Princeton University): Igusa stacks and cohomology of Shimura varieties

Shimura varieties have been playing an important role in the Langlands program. Recently, following a conjecture of Peter Scholze, we have constructed certain p-adic geometric objects, which we call ​”Igusa stacks” (in the PEL type case), which encode prime-to-p information of Shimura varieties. Together with certain flag varieties (encoding information of Shimura varieties at p), they give p-adic uniformization of Shimura varieties. In this talk, I will explain construction of Igusa stack for Hodge type Shimura varieties, which leads to new proofs of Mantovan’s product formula, Eichler-Shimura relations, and vanishing type results about torsion cohomology of Shimura varieties. This is based on joint work with Patrick Daniels, Daniel Kim and Pol van Hoften.

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