The unramified Brauer group of homogeneous spaces with finite stabilizer

TL;DR

This paper provides formulas for the unramified Brauer group of certain homogeneous spaces and applies them to analyze the Brauer-Manin obstruction to rational points over number fields.

Contribution

It introduces explicit formulas for the unramified Brauer group of homogeneous spaces with finite stabilizer and applies these to study rational points and obstructions over number fields.

Findings

Brauer-Manin pairing is constant outside a finite set of places.

Formulas enable computation of the unramified Brauer group for broad classes of homogeneous spaces.

Application to the Hasse principle and weak approximation over number fields.

Abstract

We give formulas for calculating the unramified Brauer group of a homogeneous space $X$ of a semisimple simply connected group $G$ with finite geometric stabilizer $\bar F$ over a wide family of fields of characteristic 0. When $k$ is a number field, we use these formulas in order to study the Brauer-Manin obstruction to the Hasse principle and weak approximation. We prove in particular that the Brauer-Manin pairing is constant on $X(k_v)$ for every $v$ outside from an explicit finite set of non archimedean places of $k$.