The Grunwald problem and approximation properties for homogeneous spaces

TL;DR

This paper investigates the Grunwald problem and weak approximation for homogeneous spaces, providing positive results for certain groups and counterexamples at bad places, highlighting new transcendental obstructions.

Contribution

It establishes affirmative solutions for iterated semidirect products with abelian kernels away from bad places and introduces the first examples of transcendental Brauer-Manin obstructions.

Findings

Positive results for iterated semidirect products with abelian kernels

Counterexamples at bad places demonstrating obstructions

First examples of transcendental Brauer-Manin obstructions

Abstract

Given a group $G$ and a number field $K$, the Grunwald problem asks whether given field extensions of completions of $K$ at finitely many places can be approximated by a single field extension of $K$ with Galois group G. This can be viewed as the case of constant groups $G$ in the more general problem of determining for which $K$-groups $G$ the variety $\mathrm{SL}_n/G$ has weak approximation. We show that away from an explicit set of bad places both problems have an affirmative answer for iterated semidirect products with abelian kernel. Furthermore, we give counterexamples to both assertions at bad places. These turn out to be the first examples of transcendental Brauer-Manin obstructions to weak approximation for homogeneous spaces.