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

Abstract

Given a finite morphism $$\varphi :Y\rightarrow X$$ of quasi-smooth Berkovich curves over a complete, non-archimedean, nontrivially valued algebraically closed field k of characteristic 0, we prove a Riemann–Hurwitz formula relating their Euler–Poincare 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.