On the independence of Heegner points on elliptic curves

Given an elliptic curve $E$ defined over $Q$ with a fixed modular parametrisation, we can associate Heegner points to quadratic imaginary fields on $E$. So, given a set of quadratic imaginary fields how to know whether the Heegner points associated to them are linearly independent or not? Rosen and Silverman has find a criteria on the fields for the independence of Heegner points in the case of non-CM elliptic curves. We will show how to extend their result to CM elliptic curves. If time permits we will look at independence of more general Heegner points, i.e Heegner points associated to the orders of quadratic imaginary fields, Heegner points arising from Shimura curve parametrisations and Darmon's Heegner points. We will also talk about the motivation for solving these kind of independence problems.