Berkovich curves and Schottky uniformization

TL;DR

This paper provides an accessible introduction to Berkovich geometry and its application to Schottky uniformization, emphasizing explicit proofs and the analytic construction of Mumford curves without assuming algebraically closed fields.

Contribution

It offers a detailed, self-contained exposition of Berkovich curves and Schottky uniformization, including new proofs and insights into non-Archimedean analytic geometry.

Findings

Classification of points on Berkovich affine line

Analytic construction of Mumford curves

Proof that Mumford curves are quotients by Schottky groups

Abstract

This text is an exposition of non-Archimedean curves and Schottky uniformization from the point of view of Berkovich geometry. It consists of two parts, the first one of an introductory nature, and the second one more advanced. The first part is meant to be an introduction to the theory of Berkovich spaces focused on the case of the affine line. We define the Berkovich affine line and present its main properties, with many details: classification of points, path-connectedness, metric structure, variation of rational functions, etc. Contrary to many other introductory texts, we do not assume that the base field is algebraically closed. The second part is devoted to 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 curves 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.