Heegner points and the arithmetic of elliptic curves over ring class extensions

Let E be an elliptic curve over Q and let K be a quadratic imaginary field that satisfies the Heegner hypothesis. We study the arithmetic of E over ring class extensions of K, with particular focus on the case when E has analytic rank at least 2 over Q. We also point out an issue in the literature regarding generalizing the Gross–Zagier formula, and offer a conjecturally correct formula.