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.

42 pages. Proved a local-global principle with respect to completions and generalized the one obtained in the previous version. Added a section showing that Harbater, Hartmann and Krashen's local-global principle can be obtained as a consequence of ours and that the converse is true as well under certain conditions. Final version. To appear in Compositio Math