Igusa Stacks and the Cohomology of Shimura Varieties II

TL;DR

This paper constructs Igusa stacks for all abelian type Shimura varieties and demonstrates their implications for cohomology, confirming the compatibility of Fargues--Scholze and classical local Langlands correspondences for certain groups.

Contribution

It introduces Igusa stacks for all abelian type Shimura varieties and proves the agreement of Fargues--Scholze and classical local Langlands correspondences for classical groups.

Findings

Igusa stacks constructed for all abelian type Shimura varieties

Fargues--Scholze local Langlands matches classical correspondences for groups of type A, B, D

Extends previous work by Hamann, Bertoloni Meli--Hamann--Nguyen, and Peng

Abstract

We construct Igusa stacks for all Shimura varieties of abelian type and derive consequences for the cohomology of these Shimura varieties. As an application, we prove that the Fargues--Scholze local Langlands correspondence agrees with the semi-simplification of the local Langlands correspondences constructed by Arthur, Mok and others, for all classical groups of type $A$, $B$ and $D$; this extends work of Hamann, Bertoloni Meli--Hamann--Nguyen and Peng.