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

Abstract

Given a finite morphism $\phi :Y\to X$ of quasi-smooth Berkovich curves over a complete, algebraically closed field $k$ of characteristic $0$, we prove a Riemann-Hurwitz formula relating their Euler-Poincaré 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.

arXiv commentThis is second, completed version, 32 pages; Accepted in Mathematische Zeitschrift