Patching over Berkovich Curves and Quadratic Forms

TL;DR

This paper extends patching techniques to Berkovich curves, establishing a local-global principle for function fields of analytic curves and applying it to quadratic forms and the u-invariant.

Contribution

It generalizes existing patching methods to Berkovich geometry, providing new local-global principles and applications to quadratic forms over analytic curves.

Findings

Established a local-global principle for function fields of Berkovich curves.

Applied patching to derive results on quadratic forms and the u-invariant.

Extended previous patching results to a broader geometric setting.

Abstract

We extend field patching to the setting of Berkovich analytic geometry and use it to prove a local-global principle over function fields of analytic curves. We apply this result to quadratic forms, and combine it with sufficient conditions for local isotropy over a Berkovich curve to obtain applications on the u-invariant. The patching method we adapt was introduced by Harbater and Hartmann, and further developed by these two authors and Krashen. This paper generalizes their results on the local-global principle and quadratic forms.