Canonical local heights and Berkovich skeleta

TL;DR

This paper explores canonical local heights on abelian varieties over non-archimedean fields using Berkovich spaces, extending Tate's formulas to higher dimensions and refining Néron's classical results.

Contribution

It provides a new perspective on canonical local heights via Berkovich spaces and extends Tate's explicit formulas to higher-dimensional abelian varieties.

Findings

Refined Néron's classical relation between heights and intersection multiplicities.

Extended Tate's explicit formulas to higher-dimensional cases.

Connected Berkovich analytic spaces with height theory.

Abstract

We discuss canonical local heights on abelian varieties over non-archimedean fields from the point of view of Berkovich analytic spaces. Our main result is a refinement of N\'eron's classical result relating canonical local heights with intersection multiplicities on the N\'eron model. We also revisit Tate's explicit formulas for N\'eron's canonical local heights on elliptic curves. Our results can be viewed as extensions of Tate's formulas to higher dimensions.