Local-global principles for homogeneous spaces of reductive groups over global function fields

TL;DR

This paper proves that for homogeneous spaces of reductive groups over global function fields, the Brauer-Manin obstruction is the only barrier to the Hasse principle and approximation properties, using advanced duality methods.

Contribution

It establishes the sufficiency of Brauer-Manin obstructions for these spaces over global function fields, extending previous results to positive characteristic cases.

Findings

Brauer-Manin obstructions are the only obstructions to the Hasse principle.

The methods involve abelianization and arithmetic duality theorems.

Results apply to homogeneous spaces with reductive stabilizers over global function fields.

Abstract

Let $K$ be a global field of positive characteristic. We prove that the Brauer-Manin obstructions to the Hasse principle, to weak approximation and to strong approximation are the only ones for homogeneous spaces of reductive groups with reductive stabilizers. The methods involve abelianization techniques and arithmetic duality theorems for complexes of tori over K.