Elliptic curves of bounded degree in a polarized Abelian variety

摘要

For a polarized complex Abelian variety A, of dimension g>1, we study thefunction N_A(t) counting the number of elliptic curves in A with degree boundedby 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, andwe show that computing the number of solutions is reduced to the classicaltopic in Number Theory of counting points of the lattice Z^n lying on anexplicit bounded subset of R^n. We obtain, for Abelian varieties of smalldimension, some upper bounds for the counting function.