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

Abstract

Given a finite morphism ϕ : Y → 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-Poincaré characteristics (calculated using De Rham cohomology of their overconvergent structure sheaf). The main tools are padic 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.

Mittag-Leffler problems on Berkovich curves

“…In[3] the field k is assumed to be of mixed characteristic, but the proof carries on to the case of equal characteristic 0…”