Morphisms of Berkovich curves and the different function

TL;DR

This paper introduces a new invariant called the different function for morphisms of Berkovich curves, enabling detailed analysis of ramification and skeletons in non-Archimedean geometry.

Contribution

It defines the different function $\delta_f$ for Berkovich curve morphisms, proving its properties and applications in describing ramification loci and skeletons.

Findings

$\delta_f$ is a piecewise monomial function.

$\delta_f$ satisfies a balancing condition at type 2 points.

Complete description of ramification locus when degree equals residue characteristic $p$.

Abstract

Given a generically \'etale morphism $f\colon Y\to X$ of quasi-smooth Berkovich curves, we define a different function $\delta_f\colon Y\to[0,1]$ that measures the wildness of the topological ramification locus of $f$. This provides a new invariant for studying $f$, which cannot be obtained by the usual reduction techniques. We prove that $\delta_f$ is a piecewise monomial function satisfying a balancing condition at type 2 points analogous to the classical Riemann-Hurwitz formula, and show that $\delta_f$ can be used to explicitly construct the simultaneous skeletons of $X$ and $Y$. As an application, we use our results to completely describe the topological ramification locus of $f$ when its degree equals to the residue characteristic $p$.