Local-global questions for tori over $p$-adic function fields

TL;DR

This paper investigates local-global principles for tori over p-adic function fields, establishing duality results and analyzing obstructions to rational points via Tate-Shafarevich groups and étale cohomology.

Contribution

It introduces a duality between Tate-Shafarevich groups of tori and their duals, and links failures of local-global principles to specific étale cohomology subquotients.

Findings

Established perfect duality between first and second Tate-Shafarevich groups for tori.

Linked failure of local-global principles to a subquotient of third étale cohomology.

Generalized results to principal homogeneous spaces of reductive groups with good reduction.

Abstract

We study local-global questions for Galois cohomology over the function field of a curve defined over a p-adic field (a field of cohomological dimension 3). We define Tate-Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of the base field coming from a closed point of the curve. In the case of a torus we establish a perfect duality between the first Tate-Shafarevich group of the torus and the second Tate-Shafarevich group of the dual torus. As an application, we show that the failure of the local-global principle for rational points on principal homogeneous spaces under tori is controlled by a certain subquotient of a third etale cohomology group. We also prove a generalization to principal homogeneous spaces of certain reductive group schemes in the case when the base curve has good reduction.