Berkovich Curves and Schottky Uniformization II: Analytic Uniformization of Mumford Curves

Abstract

AbstractThis is the second part of a survey on the theory of non-Archimedean curves and Schottky uniformization from the point of view of Berkovich geometry. It is more advanced than the first part and covers the theory of Mumford curves and Schottky uniformization. We start by briefly reviewing the theory of Berkovich curves, then introduce Mumford curves in a purely analytic way (without using formal geometry). We define Schottky groups acting on the Berkovich projective line, highlighting how geometry and group theory come together to prove that the quotient by the action of a Schottky group is an analytic Mumford curve. Finally, we present an analytic proof of Schottky uniformization, showing that any analytic Mumford curve can be described as a quotient of this kind. The guiding principle of our exposition is to stress notions and fully prove results in the theory of non-Archimedean curves that, to our knowledge, are not fully treated in other texts.