Heegner points at Eisenstein primes and twists of elliptic curves

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 (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 (Mordell curves). To prove these results, we establish a general criterion for the non-triviality of the $p$-adic logarithm of Heegner points at an Eisenstein prime $p$, in terms of the relative $p$-class numbers of certain number fields and then apply this criterion to the special case $p=3$. As a by-product, we also prove the 3-part of the Birch and Swinnerton-Dyer conjecture for many elliptic curves of $j$-invariant 0.

include statements for abelian varieties of GL(2)-type; the proofs remain the same

References in corpus (1)

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

Cited by in corpus (2)

• Many cubic surfaces contain rational points
• A New $p$-adic Maass-Shimura operator and Supersingular Rankin-Selberg $p$-adic $L$-functions