Unramified Brauer groups of homogeneous spaces with finite stabilisers and the Grunwald Problem

TL;DR

This paper develops an algorithm to compute the unramified Brauer group of certain homogeneous spaces, enabling effective analysis of the Brauer-Manin obstruction and solving the Grunwald problem for specific groups.

Contribution

It introduces a new algorithm for calculating unramified Brauer groups of homogeneous spaces with finite stabilisers over fields of characteristic zero.

Findings

Algorithm effectively computes the unramified Brauer group.

Application to Brauer-Manin obstruction and weak approximation.

Guarantees effectivity of the Grunwald problem for supersolvable groups.

Abstract

We provide an algorithm for calculating the unramified Brauer group of a homogeneous space $X$ of a semi-simple simply connected group $H$ with finite geometric stabiliser over any field of characteristic 0. When $k$ is a number field, we use the obtained description of the unramified Brauer group in order to study the Brauer-Manin obstruction to weak approximation on $X$. In particular, we provide an algorithm to compute the Brauer-Manin obstruction on $X$, which guarantees effectivity of the Grunwald problem for supersolvable groups thanks to previous work of Harpaz and Wittenberg.