Patching over Berkovich Curves and Quadratic Forms

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.