Approximation theorems for classifying stacks over number fields

TL;DR

This paper establishes approximation theorems for algebraic stacks over number fields, demonstrating strong approximation with Brauer-Manin obstruction for classifying stacks of connected linear algebraic groups, and addressing torsor approximation problems.

Contribution

It proves strong approximation with Brauer-Manin obstruction for classifying stacks of connected linear algebraic groups over number fields, providing concrete criteria for torsor approximation.

Findings

Proves strong approximation with Brauer-Manin obstruction for BG

Provides criteria for approximating local G-torsors by global torsors

Answers specific questions on torsor approximation over number fields

Abstract

Approximation theorems for algebraic stacks over a number field $k$ are studied in this article. For G a connected linear algebraic group over a number field we prove strong approximation with Brauer-Manin obstruction for the classifying stack $BG$. This result answers a very concrete question, given $G$-torsors $P_v$ over $k_v$, where $v$ ranges over a finite number of places, when can you approximate the $P_v$ by a $G$-torsor $P$ defined over $k$.