Congruences between Heegner points and quadratic twists of elliptic curves

TL;DR

This paper proves a congruence formula linking p-adic logarithms of Heegner points for elliptic curves with the same mod p Galois representation, enabling explicit construction of quadratic twists with specific ranks and advancing the understanding of the Birch and Swinnerton-Dyer conjecture.

Contribution

It introduces a congruence formula for Heegner points on elliptic curves sharing the same mod p Galois representation and applies it to construct many quadratic twists with rank zero or one.

Findings

Constructs many quadratic twists with rank zero or one.

Improves bounds towards Goldfeld's conjecture.

Proves the 2-part of the Birch and Swinnerton-Dyer conjecture for many twists.

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 use the congruence formula when $p=2$ to 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 $\gg X/\log^{5/6}X$, 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 many rank zero and rank one twists of $E$, which was only recently established for specific CM elliptic curves $E$.