Goldfeld’s Conjecture and Congruences Between Heegner Points

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.

The average size of the 3‐isogeny Selmer groups of elliptic curves y2=x3+k

“…Finally, in recent work, Kriz and Li [31] prove that (over Q) at least 10% of the curves y 2 = x 3 + k have rank 0 (respectively, 1). Their p-adic methods are completely different from ours, and while their rank 0 and 1 proportions are lower, their rank 1 results over Q are unconditional.…”

“…The finiteness of the number of integral points on such curves was first proven by Mordell , which is why these curves are sometimes called Mordell curves . Another application of our parameterization and methods is the existence of rational points on cubic surfaces: Browning has recently used our results, along with , to prove that a positive proportion of cubic surfaces of the form have a rational point.…”

Towards Hilbert’s tenth problem for rings of integers through Iwasawa theory and Heegner points

“…Hence log 2,ω E (σ(π E (P N,K ))) remains the same up to sign. • The proof in [KL19] also shows that the quantity in (5.3) is a 2-adic integer. So, the condition actually is about indivisibility by 2.…”

“…The theory of [KL19] allows one to achieve rank E [p] (Q) = 0 for many primes p under suitable conditions. For our applications it would be of great interest to extend this theory in order to control the two relevant cubic twists simultaneously.…”

“…The first condition is achieved by a study of the variation of cyclotomic Iwasawa invariants under base-change, while the second one is achieved by means of a recent result by Kriz and Li [KL19] on congruences of Heegner points in the context of the celebrated Goldfeld's conjecture. Both methods -Iwasawa theory and Heegner points-impose a series of technical conditions on the admissible elliptic curves E, and a critical aspect of the proof of Theorem 1.2 is to make sure that there exist elliptic curves that are admissible for both methods simultaneously.…”