Goldfeld's conjecture and congruences between heegner points

Abstract

© The Author(s) 2019. Given an elliptic curve E over Q, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (respectively 1). We show that this conjecture holds whenever E has a rational 3-isogeny. We also prove the analogous result for the sextic twists of j-invariant 0 curves. For a more general elliptic curve E, we show that the number of quadratic twists of E up to twisting discriminant X of analytic rank 0 (respectively 1) is ≫ X= log5/6 X, improving the current best general bound toward Goldfeld's conjecture due to Ono-Skinner (respectively Perelli-Pomykala). To prove these results, we establish a congruence formula between p-adic logarithms of Heegner points and apply it in the special cases p = 3 and p = 2 to construct the desired twists explicitly. As a by-product, we also prove the corresponding p-part of the Birch and Swinnerton-Dyer conjecture for these explicit twists.