Congruences between Heegner points and quadratic twists of elliptic curves

Abstract

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 $\gg X/{\log }^{5/6}X$, 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 $E$.

minor revision

References in corpus (1)

• A majority of elliptic curves over $\mathbb{Q}$ satisfy the Birch and Swinnerton-Dyer conjecture

Cited by in corpus (2)

• Comparing anticyclotomic Selmer groups of positive coranks for congruent modular forms -- Part II
• A New $p$-adic Maass-Shimura operator and Supersingular Rankin-Selberg $p$-adic $L$-functions