Vanishing of algebraic Brauer-Manin obstructions

TL;DR

This paper investigates conditions under which algebraic Brauer-Manin obstructions vanish for certain homogeneous spaces, leading to results on the Hasse principle and weak approximation over number fields.

Contribution

It establishes new criteria for the vanishing of algebraic Brauer-Manin obstructions based on the character group of the stabilizer, and applies these to linear algebraic groups.

Findings

Vanishing of algebraic Brauer-Manin obstructions under specific conditions.

Hasse principle and weak approximation hold when H is connected or abelian.

New criteria for the Hasse principle in the context of linear algebraic groups.

Abstract

Let X be a homogeneous space of a quasi-trivial k-group G, with geometric stabilizer H, over a number field k. We prove that under certain conditions on the character group of H, certain algebraic Brauer-Manin obstructions to the Hasse principle and weak approximation vanish, because the abelian groups where they take values vanish. When H is connected or abelian, these algebraic Brauer-Manin obstructions to the Hasse principle and weak approximation are the only ones, so we prove the Hasse principle and weak approximation for X under certain conditions. As an application, we obtain new sufficient conditions for the Hasse principle and weak approximation for linear algebraic groups.