Wild coverings of Berkovich curves

Abstract

This paper is an extended version of the author’s talk given at the conference ”Non-Archimedean analytic geometry: theory and practice” held in August 2015 at Papeete, and I wish to thank the organizers. It gives a brief overview of results and methods of works [CTT14] and [Tem14] on the structure of finite morphisms between Berkovich curves. The structure of tame morphisms between smooth Berkovich curves is pretty well known and it is completely controlled by the simultaneous semistable reduction theorem, see, for example, [ABBR13]. The structure of wild morphisms was for a long time terra incognita, though one should mention some special results recently obtained by Faber in [Fab13a] and [Fab13b]. In this project we obtain a relatively complete description of the combinatorial structure of an arbitrary finite morphism f : Y → X between smooth Berkovich curves, and it is divided into two parts.