Berkovich Curves and Schottky Uniformization I: The Berkovich Affine Line

Abstract

AbstractThis is the first part of a survey on the theory of non-Archimedean curves and Schottky uniformization from the point of view of Berkovich geometry. This text is of an introductory nature and aims at giving a general idea of the theory of Berkovich spaces by focusing 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.