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S=T for Shimura Varieties and Moduli Spaces of p-adic Shtukas

Zhiyou Wu

TL;DR

This work proves the $S=T$ conjecture for Shimura varieties of Hodge type by leveraging Scholze’s diamonds and v-stacks together with the Fargues–Scholze geometric Satake correspondence, establishing a local–global compatibility that identifies excursion operators with Hecke operators on shtuka moduli spaces. The authors define excursion operators via creation, partial Frobenius, and annihilation correspondences on moduli of shtukas and their Witt-vector variants, and prove $S=T$ first for moduli spaces of $p$-adic shtukas and then for Shimura varieties via integral period maps. A key outcome is the Eichler–Shimura relation in this context, derived by transporting the $S=T$ identity from shtukas to Shimura varieties through period morphisms. This framework deepens the connection between geometric Langlands methods in mixed and equal characteristic and provides a robust pathway to local-global compatibility in the Langlands program for Shimura data.

Abstract

We prove the $S=T$ conjecture proposed by Xiao--Zhu in \cite{2017arXiv170705700X}, making use of Scholze's theory of diamonds and v-stacks and Fargues--Scholze's geometric Satake equivalence. Following \cite{2018arXiv180205299X}, we deduce the Eichler--Shimura relation for Shimura varieties of Hodge type.

S=T for Shimura Varieties and Moduli Spaces of p-adic Shtukas

TL;DR

This work proves the conjecture for Shimura varieties of Hodge type by leveraging Scholze’s diamonds and v-stacks together with the Fargues–Scholze geometric Satake correspondence, establishing a local–global compatibility that identifies excursion operators with Hecke operators on shtuka moduli spaces. The authors define excursion operators via creation, partial Frobenius, and annihilation correspondences on moduli of shtukas and their Witt-vector variants, and prove first for moduli spaces of -adic shtukas and then for Shimura varieties via integral period maps. A key outcome is the Eichler–Shimura relation in this context, derived by transporting the identity from shtukas to Shimura varieties through period morphisms. This framework deepens the connection between geometric Langlands methods in mixed and equal characteristic and provides a robust pathway to local-global compatibility in the Langlands program for Shimura data.

Abstract

We prove the conjecture proposed by Xiao--Zhu in \cite{2017arXiv170705700X}, making use of Scholze's theory of diamonds and v-stacks and Fargues--Scholze's geometric Satake equivalence. Following \cite{2018arXiv180205299X}, we deduce the Eichler--Shimura relation for Shimura varieties of Hodge type.

Paper Structure

This paper contains 22 sections, 40 theorems, 386 equations.

Table of Contents

  1. Introduction
  2. Basic Definitions
  3. Hecke stacks
  4. Moduli spaces of $p$-adic shtukas
  5. Moduli spaces of Witt vector shtukas and Witt vector Hecke stacks
  6. The Geometric Satake
  7. The Hecke Operators
  8. The excursion operators
  9. The creation correspondences
  10. The generic fiber
  11. The special fiber
  12. The annihilation correspondences
  13. The generic fiber
  14. The special fiber
  15. The partial Frobenius
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $X_K$ be a Shimura variety of Hodge type with respect to a reductive group $G$ and level $K$. Let $p$ be a prime over which $G$ is unramified, and $K$ is hyperspecial at $p$. Let $l$ be a prime different from $p$, then we have an identity as endormorphisms of $H^i_c(\mathfrak{X}_{K,\overline{\mathbb{F}_p}},\overline{\mathbb{Q}}_\ell)$, where $\mathfrak{X}_K$ is the smooth integral model of $X

Theorems & Definitions (111)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Definition 2.9
  • ...and 101 more