The defect of weak approximation for homogeneous spaces. II

TL;DR

This paper investigates the failure of weak approximation on homogeneous spaces of linear algebraic groups over number fields, providing criteria based on toric quotients and establishing conditions for real and weak approximation properties.

Contribution

It introduces a method to compute the defect of weak approximation using quasi-trivial groups and toric quotients, extending previous results and identifying new conditions for approximation properties.

Findings

The defect of weak approximation can be expressed via the toric quotient T of the stabilizer.

If T splits over a metacyclic extension, the homogeneous space satisfies weak approximation.

Homogeneous spaces with connected stabilizers have the real approximation property regardless of T.

Abstract

Let X be a homogeneous space of a connected linear algebraic group G' over a number field k, containing a k-point x. Assume that the stabilizer of x in G' is connected. Using the notion of a quasi-trivial group, recently introduced by Colliot-Th\'el\`ene, we can represent X in the form X=G/H, where G is a quasi-trivial k-group and H is a connected k-subgroup of G. Let S be a finite set of places of k. Applying results of [B2], we compute the defect of weak approximation for X with respect to S in terms of the biggest toric quotient T of H. In particular, we show that if T splits over a metacyclic extension of k, then X has the weak approximation property. We show also that any homogeneous space X with connected stabilizer (without assumptions on T) has the real approximation property.