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G-companions on algebraic stacks and applications to canonical $\ell$-adic local systems on Shimura stacks

Min Shi

TL;DR

The work extends Drinfeld’s companion theorem from schemes to smooth Artin stacks of finite presentation, enabling compatibility of canonical $\,\ell$-adic local systems on Shimura stacks across primes $\,\ell$. It develops a stack-compatible companion theory for reductive groups via pro-semisimple completions and a lambda-semiring framework, and proves a main theorem for semisimple $G$ with reductions to DM-stacks and schemes. The paper then applies these results to Shimura stacks by constructing integral models, good compactifications, and spreading-out techniques, establishing independence of $\,\ell$ in adjoint and reductive settings and providing tame specialization maps. These results broaden Langlands-type compatibility from schemes to stacks and deepen the arithmetic understanding of Shimura-stack local systems with non-neat levels.

Abstract

Cases of Deligne's companion conjecture for normal schemes over finite fields have been proven by L. Lafforgue, Drinfeld, and Zheng in recent years: L. Lafforgue proved the conjecture for curves, Drinfeld proved the conjecture for all smooth schemes and later also for representations valued in a reductive group, and Zheng proved Deligne's conjecture for smooth Artin stacks. In this paper, we extend Drinfeld's theorem for general reductive groups to smooth Artin stacks of finite presentation and apply the result to the study of compatibility of the canonical $\ell$-adic local systems on Shimura stacks.

G-companions on algebraic stacks and applications to canonical $\ell$-adic local systems on Shimura stacks

TL;DR

The work extends Drinfeld’s companion theorem from schemes to smooth Artin stacks of finite presentation, enabling compatibility of canonical -adic local systems on Shimura stacks across primes . It develops a stack-compatible companion theory for reductive groups via pro-semisimple completions and a lambda-semiring framework, and proves a main theorem for semisimple with reductions to DM-stacks and schemes. The paper then applies these results to Shimura stacks by constructing integral models, good compactifications, and spreading-out techniques, establishing independence of in adjoint and reductive settings and providing tame specialization maps. These results broaden Langlands-type compatibility from schemes to stacks and deepen the arithmetic understanding of Shimura-stack local systems with non-neat levels.

Abstract

Cases of Deligne's companion conjecture for normal schemes over finite fields have been proven by L. Lafforgue, Drinfeld, and Zheng in recent years: L. Lafforgue proved the conjecture for curves, Drinfeld proved the conjecture for all smooth schemes and later also for representations valued in a reductive group, and Zheng proved Deligne's conjecture for smooth Artin stacks. In this paper, we extend Drinfeld's theorem for general reductive groups to smooth Artin stacks of finite presentation and apply the result to the study of compatibility of the canonical -adic local systems on Shimura stacks.

Paper Structure

This paper contains 23 sections, 42 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Backgrounds and Notations
  4. Zheng's generalization
  5. Drinfeld's generalization
  6. Main theorem for semisimple $G$
  7. Reduce to dense open subschemes
  8. Reduction from DM stack to schemes
  9. Reduction from the algebraic stacks case to DM stacks
  10. Main theorem for reductive $G$
  11. Application
  12. the Adjoint Case
  13. Definition of the Shimura stacks and Construction of the Canonical $l$-adic local system on Shimura stacks
  14. Definition of the Shimura stacks
  15. Construction of the Canonical $l$-adic local system on Shimura stacks
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\mathcal{X}$ be a smooth algebraic stack of finite presentation over $\mathbb{F}_q$. Let $\lambda$ and $\lambda'$ be non-Archimedean places of $\overline{\mathbb{Q}}$ not dividing $p$. Let $G$ be a reductive group (not necessarily connected) over $\mathbb{Q}$ and let $\Pi$ be the $\Acute{e}$tal

Theorems & Definitions (98)

  • Theorem 1.1: c.f. theorem \ref{['theorem 4.1']}
  • Theorem 1.2: c.f. theorem \ref{['theorem 1.26']}
  • Theorem 3.1: drinfeld2018conjecture
  • Definition 3.2: LMB
  • Definition 3.3: Zheng_2018
  • Theorem 3.4: Zheng_2018
  • Theorem 3.5: drinfeld2018prosemisimple
  • Remark 3.6
  • Definition 3.7
  • Remark 3.8: Interpretation of Drinfeld's theorem
  • ...and 88 more