Pure extension of the theta divisor over the moduli space of abelian varieties

TL;DR

This paper explores the extension of the theta divisor over the moduli space of abelian varieties, comparing different methods and their tropicalizations, with implications for arithmetic geometry and height formulas.

Contribution

It introduces a novel comparison between Zariski closure and pure weight 2 extensions of the theta divisor using tropicalization, and extends key formulas in arithmetic geometry.

Findings

The two extension methods differ by a tropicalization of the Riemann theta function.

A generalized key formula for the pure weight 2 extension is established.

Applications include a universal formula for the Néron--Tate height of points.

Abstract

A theta divisor on the universal principally polarised abelian variety can be extended to a compactification either by taking the Zariski closure, or by taking the unique extension which is pure of weight 2. For the latter, following ideas of Yuan and Zhang, we need to pass to the category of adelic- or b-divisors. We show that the two choices of extension differ by a tropicalisation of the Riemann theta function. We prove an extension of Moret-Bailly's ''key formula'' that features the pure weight 2 extension of the theta divisor, and discuss various arithmetic applications, including a ''universal'' formula for the N\'eron--Tate height of a point. A key technical input is the systematic use of the theory of logarithmic abelian varieties due to Kajiwara, Kato, and Nakayama.