Rigid analytic Stein algebraic groups are affine

TL;DR

This paper proves that algebraic groups over non-Archimedean fields are affine if their associated rigid analytic spaces are Stein, establishing a link between algebraic and analytic properties in non-Archimedean geometry.

Contribution

It demonstrates that rigid analytic Stein algebraic groups over non-Archimedean fields are necessarily affine, connecting algebraic and analytic structures in this setting.

Findings

Algebraic groups with constant regular functions have constant rigid analytic functions.

An algebraic group over a non-Archimedean field is affine iff its associated rigid analytic space is Stein.

The statement over complex numbers does not hold, highlighting differences in non-Archimedean geometry.

Abstract

Let $K$ be a complete non-trivially valued non-Archimedean field. Given an algebraic group over $K$ on which every regular function is constant, any rigid analytic function is shown to be constant too. It follows that an algebraic group over $K$ is affine if and only if the associated $K$-analytic space is Stein; that is, rigid analytic embeddings of it in an affine space may always be chosen to be given by algebraic functions. Arguably curiously, the corresponding statement over the complex numbers is false.