Automatic meromorphy in non-archimedean geometry

TL;DR

This paper proves that certain analytic functions on non-archimedean spaces are meromorphic, and shows that invertible functions on analytifications of schemes are algebraic, bridging analytic and algebraic geometry in this setting.

Contribution

It establishes a new criterion for meromorphy in non-archimedean geometry and links invertible analytic functions to algebraic functions on schemes.

Findings

Analytic functions with dense Zariski-open zero-loci are meromorphic.

Invertible analytic functions on analytifications are algebraic.

Provides a bridge between non-archimedean analytic and algebraic geometry.

Abstract

In this text we prove that if X is a reduced non-archimedean analytic space and f is a analytic function on a dense Zariski-open subspace of X whose zero-locus is closed in X, then f is a meromorphic function on X. As a corollary, we deduce that every invertible analytic function on the analytification of a reduced scheme of finite type over an affinoid algebra is algebraic.