# Metric uniformization of morphisms of Berkovich curves via $p$-adic differential equations

## TL;DR

This paper establishes a connection between the metric uniformization of morphisms of Berkovich curves and $p$-adic differential equations, characterizing when a morphism is radial based on control of differential equations and preimage counts.

## Contribution

It proves that a skeleton radializes a finite étale morphism of Berkovich curves if and only if it controls the pushforward of the associated $p$-adic differential equation, providing a new criterion for radiality.

## Key findings

- $ ext{Gamma}_f$ radializes $f$ iff it controls the pushforward of the differential equation.
- For finite étale morphisms of open unit discs, $f$ is radial iff preimage counts depend only on radius.
- Preimage counts are constant for points with the same radius in the open unit disc case.

## Abstract

We consider a finite \'etale morphism $f:Y \to X$ of quasi-smooth Berkovich curves over a complete nonarchimedean non-trivially valued field $k$, assumed algebraically closed and of characteristic 0, and a skeleton $\Gamma_f=(\Gamma_Y,\Gamma_X)$ of the morphism $f$. We prove that $\Gamma_f$ radializes $f$ if and only if $\Gamma_X$ controls the pushforward of the constant $p$-adic differential equation $f_*(\mathcal{O}_Y,d_Y)$. Furthermore, when $f$ is a finite \'etale morphism of open unit discs, we prove that $f$ is radial if and only if the number of preimages of a point $x\in X$, counted without multiplicity, only depends on the radius of the point $x$.