Igusa Stacks and the Cohomology of Shimura Varieties

Abstract

We construct functorial Igusa stacks for all Hodge-type Shimura varieties, proving a conjecture of Scholze and extending earlier results of the fourth-named author for PEL-type Shimura varieties. Using the Igusa stack, we construct a sheaf on ${\mathrm{Bun}}_G$ that controls the cohomology of the corresponding Shimura variety. We use this sheaf and the spectral action of Fargues-Scholze to prove a compatibility between the cohomology of Shimura varieties of Hodge type and the semisimple local Langlands correspondence of Fargues-Scholze, generalizing the Eichler-Shimura relation of Blasius-Rogawski to arbitrary level at $p$. When the given Shimura variety is proper, we show moreover that the sheaf is perverse, which allows us to prove new torsion vanishing results for the cohomology of Shimura varieties.

We give a different proof of equidimensionality of central leaves without reference to local model diagrams; as a result Theorems V & VI have been strengthened. Some missing Tate twists have been added, and other minor revisions have been made throughout. 125 pages, comments welcome!