The \'{e}tale Brauer-Manin obstruction for classifying stacks

TL;DR

This paper proves that the étale Brauer-Manin obstruction is the sole barrier to strong approximation for classifying stacks of linear algebraic groups over number fields.

Contribution

It introduces a new framework of torsors and Galois twists for algebraic stacks to establish this result.

Findings

Étale Brauer-Manin obstruction is the only obstruction for strong approximation on BG.

Develops a theory of torsors and Galois twists for algebraic stacks.

Provides a new perspective on strong approximation in the context of algebraic stacks.

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 \'etale 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.