Local heights of toric varieties over non-archimedean fields

TL;DR

This paper extends the theory of local heights on toric varieties to non-archimedean fields with rank 1 absolute values, generalizing previous results and applying them to global heights in a new setting.

Contribution

It generalizes local height formulas for toric varieties to arbitrary non-archimedean absolute values of rank 1, including new applications to global heights.

Findings

Generalized induction formula for non-archimedean absolute values.

Proved toric local height formula in the new setting.

Applied results to Moriwaki's global heights in a toric fibration.

Abstract

We generalize results about local heights previously proved in the case of discrete absolute values to arbitrary non-archimedean absolute values of rank 1. First, this is done for the induction formula of Chambert-Loir and Thuillier. Then we prove the formula of Burgos--Philippon--Sombra for the toric local height of a proper normal toric variety in this more general setting. We apply the corresponding formula for Moriwaki's global heights over a finitely generated field to a fibration which is generically toric. We illustrate the last result in a natural example where non-discrete non-archimedean absolute values really matter.