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

Summary · qwen2.5:32b

The article extends the theta divisor over the moduli space of abelian varieties, showing that its pure weight 2 extension differs from the Zariski closure by a tropicalization of the Riemann theta function; this work applies Moret-Bailly's "key formula" to derive a universal formula for the Néron-Tate height of a point.

Suggested post angle

Discussing the extension of theta divisors in abelian varieties and its applications to arithmetic, including a universal formula for the N'eron--Tate height of a point.

Excerpt

arXiv:2602.22162v2 Announce Type: replace 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.