Abstract
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 (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$\gg X/\log ^{5/6}X$, 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.
Highlights
One can ask the following natural question: how is ran(E) distributed when E varies in families? The simplest (1-parameter) family is given by the quadratic twist family of a given curve E
We work directly with the p-adic incarnation of the L-values – the p-adic logarithm of Heegner points – and we prove the following key congruence formula
The exterior differential is given on q-expansions by d = θ(dq/q) where θ is the Atkin–Serre operator on p-adic modular forms acting via q(d/dq) on q-expansions
Summary
− p), and Theorem 1.20 specializes to provide a mod p congruence between the p-adic logarithm of the Heegner point on E and certain Bernoulli numbers, which can be viewed as a generalization of Rubin’s formula from the rank-0 case to the rank-1 case. (2) When N is a prime different from p, Mazur, in his seminal paper [47], proved a congruence formula at an Eisenstein prime above p, between the algebraic part of L(J0(N ), χ , 1) and a quantity involving generalized Bernoulli numbers attached to χ, for certain odd Dirichlet characters χ This was later generalized by Vatsal [83] for more general N and used to prove weak Goldfeld for r = 0 for infinitely many elliptic curves.
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