On elliptic curves of bounded degree in a polarized Abelian variety

Abstract

For a polarized complex Abelian variety $A$ we study the function $N_A(t)$ counting the number of elliptic curves in $A$ with degree bounded by $t$. This extends our previous work in dimension two. We describe the collection of elliptic curves in the product $A = S \times F$ of an Abelian variety and an elliptic curve by means of an explicit parametrization, and in terms of the parametrization we express the degrees of elliptic curves relative to a split polarization. When this is applied to the self product $A = E^k$ of an elliptic curve, it turns out that an asymptotic estimate of the counting function $N_A(t)$ can be obtained from an asymptotic study of the degree form on the group of endomorphisms of the elliptic curve.