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Igusa Stacks and the Cohomology of Shimura Varieties II

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

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.

Igusa Stacks and the Cohomology of Shimura Varieties II

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 , and ; this extends work of Hamann, Bertoloni Meli--Hamann--Nguyen and Peng.

Paper Structure

This paper contains 48 sections, 75 theorems, 281 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Main geometric results
  4. Compatibility of local Langlands correspondences
  5. Cohomological results
  6. The proof of Theorem \ref{['Thm:IntroIgusaGeneric']}
  7. Organization of the paper
  8. Notation and conventions
  9. Acknowledgements
  10. Preliminaries
  11. Igusa stacks
  12. Locally profinite groups
  13. An integral conjecture
  14. Igusa stacks for tori
  15. Construction of the Igusa stack
  16. ...and 33 more sections

Key Result

Theorem 1

There is an open immersion of v-stacks $\mathrm{Igs}_{K^p}^{\circ}{(\mathsf{G}, \mathsf{X})} \hookrightarrow \mathrm{Igs}_{K^p}{(\mathsf{G}, \mathsf{X})}$ on $\operatorname{Perf}$ sitting in a commutative diagram \begin{tikzcd} \mathbf{Sh}_{K^p}\gx^{\circ,\diamondsuit} \arrow{d}{\IgsQuot^{\c

Theorems & Definitions (191)

  • Theorem 1: Theorem \ref{['Thm:MainThmIgusa']}, \ref{['Thm:FiniteLevelIgusa']}
  • Theorem 2
  • Remark 1.3.2
  • Corollary 3: Theorem \ref{['Thm:ActualEvenOrthogonalLanglands']}
  • Remark 1.3.3
  • Remark 1.3.6
  • Theorem 4: Theorem \ref{['Thm:Plectic']}
  • Remark 1.4.1
  • Corollary 5: Corollary \ref{['Cor:PartialEichlerShimura']}
  • Remark 1.4.2
  • ...and 181 more