Applications of patching to quadratic forms and central simple algebras

TL;DR

This paper applies patching techniques to quadratic forms and central simple algebras over function fields, reestablishing and extending results on the u-invariant of p-adic function fields using local-global principles.

Contribution

It introduces a patching approach to study quadratic forms and central simple algebras, generalizing recent results on the u-invariant for p-adic function fields.

Findings

Reproved and generalized the u-invariant result for p-adic function fields

Established a local-global principle for homogeneous spaces of rational algebraic groups

Demonstrated the effectiveness of patching in algebraic and arithmetic geometry

Abstract

This paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over henselian valued fields. In particular, we use a patching approach to reprove and generalize a recent result of Parimala and Suresh on the u-invariant of p-adic function fields, for p odd. The strategy relies on a local-global principle for homogeneous spaces for rational algebraic groups, combined with local computations.