A cohomological obstruction to weak approximation for homogeneous spaces

TL;DR

This paper investigates the Brauer-Manin obstruction to weak approximation on homogeneous spaces over number fields, providing a cohomological framework to understand when weak approximation fails.

Contribution

It introduces a cohomological method to compute the Brauer-Manin obstruction for homogeneous spaces G/H over number fields, extending previous approaches.

Findings

Cohomological formulas for the Brauer-Manin obstruction

Explicit computation of obstructions in terms of Galois cohomology

Application to homogeneous spaces with non-connected stabilizers

Abstract

Let X be a homogeneous space, X = G/H, where G is a connected linear algebraic group over a number field k, and H is a k-subgroup of G (not necessarily connected). Let S be a finite set of places of k. We compute the Brauer-Manin obstruction to weak approximation for X in S in terms of Galois cohomology.