Pseudo-absolute values: foundations

TL;DR

This paper introduces pseudo-absolute values, a generalization of absolute values on fields, and explores their topological and analytic properties, connecting them to Arakelov theory and Zariski-Riemann spaces.

Contribution

It defines pseudo-absolute values, studies their topological space, and develops local and global analytic spaces over pseudo-valued fields, extending classical concepts.

Findings

The space of pseudo-absolute values is compact and Hausdorff.

Pseudo-absolute values include pathological cases relevant to Diophantine approximation.

Analytic spaces over pseudo-valued fields generalize Zariski-Riemann spaces.

Abstract

In this article, we introduce pseudo-absolute values, which generalise usual absolute values. Roughly speaking, a pseudo-absolute value on a field $K$ is a map $|\cdot| : K \to [0,+\infty]$ satisfying axioms similar to those of usual absolute values. This notion allows to include "pathological" absolute values one can encounter trying to incorporate the analogy between Diophantine approximation and Nevanlinna theory in an Arakelov theoretic framework. It turns out that the space of all pseudo-absolute values can be endowed with a compact Hausdorff topology in a similar way as the Berkovich analytic spectrum of a Banach ring. Moreover, we introduce both local and global notions of analytic spaces over pseudo-valued fields and interpret them as analytic counterparts to Zariski-Riemann spaces.