Patching over analytic fibers and the local–global principle

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, thus generalizing the results of Mehmeti(Compos Math 155:2399–2438, 2019).