Let $K$ be a non-trivially valued complete non-Archimedean field. Given an
algebraic group over $K$ such that every regular function is constant, every
rigid-analytic function on it is shown to be constant. In particular, an
algebraic group whose analytification is Stein (in Kiehl's sense) is
necessarily affine--a remarkable difference between the complex and the
non-Archimedean worlds.Comment: 49 pages, comments welcome

Maculan, Marco

English

arXiv

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.Comment: The paper has been split into two. This is the second par

Rigid-analytic functions on the universal vector extension

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