The convergence Newton polygon of a p-adic differential equation II: Continuity and finiteness on Berkovich curves

TL;DR

This paper investigates how the convergence Newton polygon of a p-adic differential equation varies along Berkovich curves, demonstrating that its slopes form continuous functions that factor through a finite subgraph.

Contribution

It extends previous work by proving the continuity and finiteness properties of the convergence Newton polygon's slopes on Berkovich curves.

Findings

Slopes of the convergence Newton polygon are continuous functions.

These functions factor through a locally finite subgraph.

The results generalize properties known on affinoid domains to entire Berkovich curves.

Abstract

We study the variation of the convergence Newton polygon of a differential equation along a smooth Berkovich curve over a non-archimedean complete valued field of characteristic 0. Relying on work of the second author who investigated its properties on affinoid domains of the affine line, we prove that its slopes give rise to continuous functions that factorize by the retraction through a locally finite subgraph of the curve.