Local-global principles for torsors over arithmetic curves

TL;DR

This paper investigates local-global principles for torsors over arithmetic curves, explicitly computes obstructions, and establishes conditions for these principles to hold, with applications to quadratic forms and central simple algebras.

Contribution

It provides explicit calculations of the Tate-Shafarevich group for certain algebraic groups and develops a Mayer-Vietoris sequence approach for Galois cohomology in this context.

Findings

The Tate-Shafarevich group is finite for groups with rational components.

Necessary and sufficient conditions for local-global principles are established.

New applications to quadratic forms and central simple algebras are demonstrated.

Abstract

We consider local-global principles for torsors under linear algebraic groups, over function fields of curves over complete discretely valued fields. The obstruction to such a principle is a version of the Tate-Shafarevich group; and for groups with rational components, we compute it explicitly and show that it is finite. This yields necessary and sufficient conditions for local-global principles to hold. Our results rely on first obtaining a Mayer-Vietoris sequence for Galois cohomology and then showing that torsors can be patched. We also give new applications to quadratic forms and central simple algebras.