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On the cohomology of simple Shimura varieties with non quasi-split local groups

Jingren Chi, Thomas J. Haines

TL;DR

The paper develops a global–local framework to study the bad reduction of simple Shimura varieties associated with unitary similitude groups, extending Scholze’s approach to non-quasi-split local groups. It establishes a vanishing property for twisted orbital integrals of Scholze-type test functions and proves a robust pseudostabilization base-change transfer for these functions, connected to the stable Bernstein center. By introducing companion groups and a generalized Kottwitz-triple method, the authors relate local harmonic analysis to global automorphic and Galois data, yielding a precise description of the cohomology and semisimple local L-factors of these Shimura varieties. The main results express trace formulas and Lefschetz numbers in terms of stable-endoscopic-central distributions and stable base-change packets, providing a broad extension of previous works to broader bad-reduction scenarios. Overall, the work deepens the understanding of how Shimura varieties with non-quasi-split local groups behave under bad reduction and clarifies the spectral side of their trace formulas via stable Bernstein-center transfers and generalized Kottwitz-triples.

Abstract

We study the Scholze test functions for bad reduction of simple Shimura varieties at a prime where the underlying local group is any inner form of a product of Weil restrictions of general linear groups. Using global methods, we prove that these test functions satisfy a vanishing property of their twisted orbital integrals, and we prove that the pseudostabilization base changes of such functions exist (even though the local group need not be quasi-split) and can be expressed in terms of explicit distributions in the stable Bernstein center. We then deduce applications to the stable trace formula and local Hasse-Weil zeta functions for these Shimura varieties.

On the cohomology of simple Shimura varieties with non quasi-split local groups

TL;DR

The paper develops a global–local framework to study the bad reduction of simple Shimura varieties associated with unitary similitude groups, extending Scholze’s approach to non-quasi-split local groups. It establishes a vanishing property for twisted orbital integrals of Scholze-type test functions and proves a robust pseudostabilization base-change transfer for these functions, connected to the stable Bernstein center. By introducing companion groups and a generalized Kottwitz-triple method, the authors relate local harmonic analysis to global automorphic and Galois data, yielding a precise description of the cohomology and semisimple local L-factors of these Shimura varieties. The main results express trace formulas and Lefschetz numbers in terms of stable-endoscopic-central distributions and stable base-change packets, providing a broad extension of previous works to broader bad-reduction scenarios. Overall, the work deepens the understanding of how Shimura varieties with non-quasi-split local groups behave under bad reduction and clarifies the spectral side of their trace formulas via stable Bernstein-center transfers and generalized Kottwitz-triples.

Abstract

We study the Scholze test functions for bad reduction of simple Shimura varieties at a prime where the underlying local group is any inner form of a product of Weil restrictions of general linear groups. Using global methods, we prove that these test functions satisfy a vanishing property of their twisted orbital integrals, and we prove that the pseudostabilization base changes of such functions exist (even though the local group need not be quasi-split) and can be expressed in terms of explicit distributions in the stable Bernstein center. We then deduce applications to the stable trace formula and local Hasse-Weil zeta functions for these Shimura varieties.
Paper Structure (59 sections, 45 theorems, 160 equations)

This paper contains 59 sections, 45 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Strategy of proof
  4. Organization of the article
  5. Notations
  6. Preparations in local harmonic analysis
  7. Transfer of twisted orbital integrals
  8. Reduction to matching at regular elements
  9. A reinterpretation of stable base change transfer
  10. Local stable base change
  11. Remark on $\sigma$-stability
  12. Proof of part (2)
  13. Remark on a relative version of (1)
  14. Reduction to a totally ramified Weil restriction
  15. Analysis of $F/\mathbb{Q}_p$ totally ramified case
  16. ...and 44 more sections

Key Result

Theorem 1.1.1

There is an identity in the Grothendieck group of $\mathbb{G}(\mathbb{A}_f^p)\times \mathcal{G}(\mathbb{Z}_p)\times W_{\mathbb{E}_\mathfrak{p}}$ representations

Theorems & Definitions (101)

  • Theorem 1.1.1
  • Theorem 1.1.2
  • Theorem 1.1.3
  • Lemma 2.1.1
  • proof
  • Definition 2.1.2
  • Remark 2.1.3
  • Remark 2.1.4
  • Definition 2.1.5
  • Proposition 2.2.1
  • ...and 91 more