For a polarized compleax Abelian surface A we study the function NA(t) counting the number of elliptic curves in A with degree bounded by t. We describe elliptic curves as solutions of an explicit Diophantine equation, and we show that computing the number of solutions is reduced to the classical problem in Number Theory of counting lattice points lying on an explicit bounded subset of Euclidean space. We obtain in this way some asymptotic estimate for the counting function.
