Igusa Stacks and the Cohomology of Shimura Varieties

Patrick Daniels, Pol van Hoften, Dongryul Kim, Mingjia Zhang

Published 2024-08-02Version 1

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.