Patching over Analytic Fibers and the Local-Global Principle

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.