Heegner Points and the Arithmetic of Elliptic Curves over Ring Class Extensions

To Appear in Journal of Number Theory

by Robert Bradshaw and William Stein

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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 a small issue in the literature regarding generalizing the Gross-Zagier formula, and offer a conjecturally correct formula.