Weak approximation for homogeneous spaces: reduction to the case with finite stabilizer

TL;DR

This paper investigates whether the Brauer-Manin obstruction is the only barrier to weak approximation on homogeneous spaces, reducing the problem to cases with finite stabilizers and providing some unconditional results.

Contribution

It reduces the general problem to the case with finite stabilizer and offers unconditional results based on existing knowledge.

Findings

Reduction of the weak approximation problem to finite stabilizer cases

Unconditional results derived from known finite stabilizer cases

Clarification of the role of the Brauer-Manin obstruction

Abstract

We reduce the question about whether the Brauer-Manin obstruction to weak approximation for homogeneous spaces is the only obstruction to the "simpler" question of the particular case of homogeneous spaces of $\mathrm{SL}_n$ with finite stabilizer. We also give unconditional results based on the few known results on finite stabilizers.