Congruences between Heegner Points and Quadratic Twists of Elliptic Curves
نویسندگان
چکیده
We establish a congruence formula between p-adic logarithms of Heegner points for two elliptic curves with the same mod p Galois representation. As a first application, we use the congruence formula when p = 2 to explicitly construct many quadratic twists of analytic rank zero (resp. one) for a wide class of elliptic curves E. We show that the number of twists of E up to twisting discriminant X of analytic rank zero (resp. one) is X/ log X, improving the current best general bound towards Goldfeld’s conjecture due to Ono–Skinner (resp. Perelli–Pomykala). We also prove the 2-part of the Birch and Swinnerton-Dyer conjecture for many rank zero and rank one twists of E, which was only recently established for specific CM elliptic curves E.
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