Hasse principle for hermitian spaces over semi-global fields

TL;DR

This paper proves a local-global principle for hermitian spaces over certain function fields, extending known results and applying patching techniques to establish isotropy over odd degree extensions.

Contribution

It confirms the conjecture for groups related to hermitian forms over central simple algebras with involutions, using patching methods.

Findings

Proves the local-global principle for hermitian spaces in specified cases.

Establishes a Springer-type theorem for hermitian forms over odd degree extensions.

Extends the understanding of isotropy of hermitian forms over function fields.

Abstract

In a recent paper, Colliot-Th\'el\`ene, Parimala and Suresh conjectured that a local-global principle holds for projective homogeneous spaces of connected linear algebraic groups over function fields of p-adic curves. In this paper, we show that the conjecture is true for a linear algebraic group whose almost simple factors of its semisimple part are isogenous to unitary groups or special unitary groups of hermitian or skew-hermitian spaces over central simple algebras with involutions. The proof implements patching techniques of Harbater, Hartmann and Krashen. As an application, we obtain a Springer-type theorem for isotropy of hermitian forms over odd degree extensions of function fields of p-adic curves.