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The étale Brauer-Manin obstruction for classifying stacks

Ajneet Dhillon, Nicole Lemire, Jonathan Martin, Yidi Wang

TL;DR

This work extends local-global principles to classifying stacks by developing an étale Brauer-Manin obstruction for stacks and proving it is the only obstruction to strong approximation for $BG$ when $G$ is linear. The authors formalize torsors and Galois twists in a 2-categorical setting, establish that the étale Brauer-Manin locus is a closed subset of adelic points, and prove dense rational points in this locus off a finite set of places via twists of the connected component $G^{ullet}$ and inner forms. The results connect stack-theoretic torsors with adelic obstructions, offering potential applications to inverse Galois theory and height-based counting on classifying stacks. This framework refines previous Br-Obstruction results for connected groups and provides a robust method to analyze arithmetic of stacks with symmetries.

Abstract

We study the strong approximation for classifying stacks $BG$, where $G$ is a linear algebraic group over a number field $k$. More specifically, we prove that the étale Brauer-Manin obstruction is the only obstruction to strong approximation for $BG$. To prove the result, we formulate the theory of torsors and Galois twists for algebraic stacks.

The étale Brauer-Manin obstruction for classifying stacks

TL;DR

This work extends local-global principles to classifying stacks by developing an étale Brauer-Manin obstruction for stacks and proving it is the only obstruction to strong approximation for when is linear. The authors formalize torsors and Galois twists in a 2-categorical setting, establish that the étale Brauer-Manin locus is a closed subset of adelic points, and prove dense rational points in this locus off a finite set of places via twists of the connected component and inner forms. The results connect stack-theoretic torsors with adelic obstructions, offering potential applications to inverse Galois theory and height-based counting on classifying stacks. This framework refines previous Br-Obstruction results for connected groups and provides a robust method to analyze arithmetic of stacks with symmetries.

Abstract

We study the strong approximation for classifying stacks , where is a linear algebraic group over a number field . More specifically, we prove that the étale Brauer-Manin obstruction is the only obstruction to strong approximation for . To prove the result, we formulate the theory of torsors and Galois twists for algebraic stacks.

Paper Structure

This paper contains 10 sections, 18 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Torsors over algebraic stacks
  3. Galois Twists
  4. The basic example
  5. The étale Brauer-Manin Obstruction
  6. A review of topologizing the adelic points of a stack
  7. The étale Brauer-Manin locus
  8. The étale Brauer-Manin obstruction
  9. The main result
  10. Right group actions on stacks and groupoids

Key Result

Theorem 1.1

Let $G$ be a linear algebraic group over a number field $k$. Let $S$ be a finite collection of places of $k$ containing all the infinite places of $k$.

Theorems & Definitions (49)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 39 more