Metric uniformization of morphisms of Berkovich curves via p-adic differential equations

Abstract

We consider a finite rig-etale morphism f: Y → X of quasi-smooth Berkovich curves over a complete algebraically closed valued field extension k of ℚp and a skeleton Γf = (ΓY, ΓX) of the morphism f. We prove that Γf radializes f if and only if ΓX controls the pushforward of the constant p-adic differential equation $${f_*}\left( {{{\cal O}_Y},{d_Y}} \right)$$ . Furthermore, when f is a finite morphism of open unit discs and k is of arbitrary characteristic, we prove that f is radial if and only if the number of preimages of a point x ∈ X, counted without multiplicity, only depends on the radius of the point x.