Morphisms of Berkovich curves and the different function

Abstract

Given a generically étale morphism $f:Y\to X$ of quasi-smooth Berkovich curves, we define a different function ${\delta }_f:Y\to [0,1]$ 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 $p$.

Final version, 49 pages, to appear in Adv.Math