Riemann-Hurwitz formula for finite morphisms of $p$-adic curves

TL;DR

This paper establishes a Riemann-Hurwitz formula for finite morphisms of $p$-adic curves using De Rham cohomology and valuation polygons, offering new insights into the relationship between their Euler characteristics.

Contribution

It introduces a Riemann-Hurwitz formula for $p$-adic curves over characteristic zero fields, utilizing $p$-adic Runge's theorem and valuation polygons, and extends the perspective to positive characteristic cases.

Findings

Proves a Riemann-Hurwitz formula for $p$-adic Berkovich curves.

Uses $p$-adic Runge's theorem and valuation polygons as main tools.

Provides a new viewpoint on the classical formula in positive characteristic.

Abstract

Given a finite morphism $\varphi:Y\to X$ of quasi-smooth Berkovich curves over a complete, algebraically closed field $k$ of characteristic $0$, we prove a Riemann-Hurwitz formula relating their Euler-Poincar\'e characteristics (calculated using De Rham cohomology of their overconvergent structure sheaf). The main tools are $p$-adic Runge's theorem together with valuation polygons of analytic functions. Using the results obtained, we provide another point of view on Riemann-Hurwitz formula for finite morphisms of curves over algebraically closed fields of positive characteristic.