Faltings height and N\'eron-Tate height of a theta divisor

TL;DR

This paper establishes a formula linking the Faltings height of a principally polarized abelian variety over algebraic numbers with the Néron--Tate height of a symmetric theta divisor, incorporating tropical geometry insights.

Contribution

It completes previous formulas by explicitly relating these heights and expressing local non-archimedean terms via tropical moments.

Findings

Derived a comprehensive height relation formula.

Connected non-archimedean terms to tropical geometry.

Extended prior results by Bost, Hindry, Autissier, Wagener.

Abstract

We prove a formula, which, given a principally polarized abelian variety $(A,\lambda)$ over the field of algebraic numbers, relates the stable Faltings height of $A$ with the N\'eron--Tate height of a symmetric theta divisor on $A$. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener. The local non-archimedean terms in our formula can be expressed as the tropical moments of the tropicalizations of $(A,\lambda)$.