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

Abstract

We consider a finite étale morphism $f:Y\to X$ of quasi-smooth Berkovich curves over a complete nonarchimedean non-trivially valued field $k$, assumed algebraically closed and of characteristic 0, and a skeleton ${\Gamma }_f=({\Gamma }_Y,{\Gamma }_X)$ of the morphism $f$. We prove that ${\Gamma }_f$ radializes $f$ if and only if ${\Gamma }_X$ controls the pushforward of the constant $p$-adic differential equation $f_{\ast }({\mathcal{O}}_Y,d_Y)$. Furthermore, when $f$ is a finite étale morphism of open unit discs, we prove that $f$ is radial if and only if the number of preimages of a point $x\in X$, counted without multiplicity, only depends on the radius of the point $x$.

33 pages; Comments are welcome