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

Abstract

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 (see [Invent. Math. 182 (2010), no. 3, 513–584]) and the authors showed that it factorizes by the retraction through a locally finite graph (see [`The convergence Newton polygon of a p-adic differential equation I: Affinoid domains of the Berkovich affine line', preprint 2012] and [`The convergence Newton polygon of a p-adic differential equation II: Continuity and finiteness on Berkovich curves', preprint 2012]). 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.