Elliptic curves of bounded degree in a polarized Abelian variety

TL;DR

This paper investigates the enumeration of elliptic curves within polarized Abelian varieties, providing explicit descriptions and bounds for small dimensions by connecting geometric problems to classical lattice point counting in number theory.

Contribution

It offers explicit descriptions of elliptic curves as solutions to Diophantine equations and reduces the counting problem to lattice point enumeration for small dimensions.

Findings

Derived upper bounds for the number of elliptic curves in small-dimensional Abelian varieties.

Explicitly described elliptic curves as solutions to Diophantine equations.

Connected geometric counting problems to classical lattice point counting in number theory.

Abstract

For a polarized complex Abelian variety A, of dimension g>1, we study the function N_A(t) counting the number of elliptic curves in A with degree bounded by t. We describe elliptic curves as solutions of Diophantine equations which, at least for small dimensions g=2 and g=3, can actually be made explicit, and we show that computing the number of solutions is reduced to the classical topic in Number Theory of counting points of the lattice Z^n lying on an explicit bounded subset of R^n. We obtain, for Abelian varieties of small dimension, some upper bounds for the counting function.