Joint IAS/Princeton University Number Theory Seminar

Given an elliptic curve $E$ over $\mathbb{Q}$, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (resp. 1). We show 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 (resp. 1) is $\gg\frac{X}{\log^{5/6}X}$, improving the current best general bound towards Goldfeld's conjecture due to Ono-Skinner (resp. Perelli-Pomykala). We prove these results by establishing a congruence formula between $p$-adic logarithms of Heegner points based on Coleman's integration. This is joint work with Daniel Kriz.