Igusa Stacks and the Cohomology of Shimura Varieties

2024/08/02 by Patrick Daniels, Pol van Hoften, Daniels, Patrick +5 · 2 citations
Mathematics · #11G18 #14G35 #14G45 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2408.01348

openalex publication_date 2024/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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.

Cited by

• Igusa Stack for some expectional Shimura Varieties
• On the non-generic part of cohomology of compact unitary Shimura varieties of signature (1,n)

Related

• Intersection Cohomology of Igusa Stacks
• Torsion Vanishing for Some Shimura Varieties
• On the generic part of the cohomology of Shimura varieties of abelian type
• Categorical local Langlands and torsion classes of some Shimura varieties
• Igusa Stack for some expectional Shimura Varieties