On the independence of Heegner points associated to distinct quadratic imaginary fields

Let E/Q be an elliptic curve with a fixed modular parametrization F : X_0(N) --> E and let P_1,...,P_r be Heegner points on E attached to the rings of integers of distinct quadratic imaginary field k_1,...,k_r. We prove that if the odd parts of the class numbers of k_1,...,k_r are larger than a constant C=C(E,F) depending only on E and F, then the points P_1,...,P_r are independent in E/(torsion). We also discuss a possible application to the elliptic curve discrete logarithm problem.