Continuity and finiteness of the radius of convergence of a p-adic differential equation via potential theory

TL;DR

This paper investigates the properties of the radius of convergence for p-adic differential equations on Berkovich curves, demonstrating its continuity and factorization using potential theory under certain conditions.

Contribution

The authors provide a shorter proof of the continuity and factorization of the radius of convergence, leveraging potential theory on Berkovich curves, under specific assumptions.

Findings

Radius of convergence is continuous on Berkovich curves.

Radius factorizes through a retraction to a finite graph.

Potential theory simplifies proofs of known properties.

Abstract

We study the radius of convergence of a differential equation on a smooth Berkovich curve over a non-archimedean complete valued field of characteristic 0. Several properties of this function are known: F. Baldassarri proved that it is continuous and the authors showed that it factorizes by the retraction through a locally finite graph. Here, assuming that the curve has no boundary or that the differential equation is overconvergent, we provide a shorter proof of both results by using potential theory on Berkovich curves.