Chao Li ( Columbia )
Congruence between Heegner points and applications to 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 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^{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 these twists of E. If time permits, we will discuss further applications to quadratic twists families with a 3-isogeny, the j-invariant 0 family, the Rubin--Silverberg families, etc. This is joint work with Daniel Kriz.