Weak approximation for homogeneous spaces over some two-dimensional geometric global fields

TL;DR

This paper investigates obstructions to weak approximation for homogeneous spaces over certain two-dimensional geometric global fields, demonstrating the limitations of the Brauer-Manin obstruction and proposing a torsor-based alternative.

Contribution

It shows the Brauer-Manin obstruction suffices for linear groups but not for homogeneous spaces, and introduces a torsor-based obstruction to explain failures.

Findings

Brauer-Manin obstruction works for linear groups over these fields.

Brauer-Manin obstruction is insufficient for homogeneous spaces.

A torsor-based obstruction explains failures of weak approximation.

Abstract

In this article, we study obstructions to weak approximation for connected linear groups and homogeneous spaces with connected or abelian stabilizers over finite extensions of $\mathbb C((x,y))$ or function fields of curves over $\mathbb C((t))$. We show that for connected linear groups, the usual Brauer-Manin obstruction works as in the case of tori. However, this Brauer-Manin obstruction is not enough for homogeneous spaces, as shown by the examples we give. We then construct an obstruction using torsors under quasi-trivial tori that explains the failure of weak-approximation.