La droite de Berkovich sur Z

TL;DR

This paper explores the Berkovich line over the ring of integers of a number field, revealing its topological and algebraic properties, and applying these findings to arithmetic power series and related problems.

Contribution

It introduces the Berkovich line over integers of a number field, analyzing its properties and applications to arithmetic power series and Galois problems.

Findings

The Berkovich line over integers has favorable topological and algebraic properties.

Examples of Stein spaces within the Berkovich line are identified.

Applications include results on zeroes, poles, and the inverse Galois problem.

Abstract

We study here the Berkovich line over the ring of integers of a number field. It is a natural object which contains complex and non-Archimedean analytic spaces associated to each place. We prove that this line satisfies good topological and algebraic properties and exhibit a few examples of Stein spaces that lie in it. We derive applications to the study of convergent arithmetic power series: choice of zeroes and poles, noetherianity of global rings and inverse Galois problem. Typical examples of such power series are given by analytic functions on the open complex unit disk whose Taylor development in 0 has integer coefficients.