Analytic functions on tubes of non-Archimedean analytic spaces

TL;DR

This paper provides an explicit description of bounded analytic functions on tubes of non-Archimedean analytic spaces, extending known results and analyzing their connectedness using formal-analytic function relations.

Contribution

It offers a new explicit characterization of bounded analytic functions on tubes of special formal schemes in non-Archimedean geometry, generalizing previous results by Bosch.

Findings

Explicit description of bounded analytic functions on tubes

Generalization of connectedness results for these tubes

Application of formal-analytic function relations in non-Archimedean spaces

Abstract

Let $k$ be a discretely valued non-Archimedean field. We give an explicit description of analytic functions whose norm is bounded by a given real number $r$ on tubes of reduced $k$-analytic spaces associated to special formal schemes (those include $k$-affinoid spaces as well as open polydiscs). As an application we study the connectedness of these tubes. This generalizes (in the discretely valued case) a result of Siegfried Bosch. We use as a main tool a result of A.J. de Jong relating formal and analytic functions on special formal schemes and a generalization of de Jong's result which is proved in the joint appendix with Christian Kappen.