Elliptic curves on abelian varieties

TL;DR

This paper characterizes all elliptic curves on a principally polarized abelian variety using intersection numbers in its Néron-Severi group, providing a new geometric criterion.

Contribution

It introduces a novel intersection-theoretic characterization of elliptic curves on abelian varieties, linking divisor classes to elliptic curve embeddings.

Findings

Elliptic curves are characterized by specific intersection numbers.

The method applies to principally polarized abelian varieties.

Provides a geometric criterion for elliptic curves on abelian varieties.

Abstract

Given a principally polarized abelian variety $(A,\Theta)$, we give a characterization of all elliptic curves that lie on $A$ in terms of intersection numbers of divisor classes in its N\'eron-Severi group.