Patching over Analytic Fibers and the Local-Global Principle

TL;DR

This paper develops a patching technique for higher-dimensional Berkovich analytic curves, establishing local-global principles over certain function fields by proving algebraicity of germs of meromorphic functions.

Contribution

It introduces a new patching method applicable to fibers of Berkovich curves and proves local-global principles in this non-archimedean setting.

Findings

Patching technique applicable around specific fibers of Berkovich curves

Local-global principles established over overconvergent meromorphic functions

Germs of meromorphic functions shown to be algebraic

Abstract

As a starting point for higher-dimensional patching in the Berkovich setting, we show that this technique is applicable around certain fibers of a relative Berkovich analytic curve. As a consequence, we prove a local-global principle over the field of overconvergent meromorphic functions on said fibers. By showing that these germs of meromorphic functions are algebraic, we also obtain local-global principles over function fields of algebraic curves defined over a class of (not necessarily complete) ultrametric fields.