Prime twists of elliptic curves

TL;DR

This paper establishes a criterion based on 2-adic logarithms of Heegner points to determine the analytic rank of prime twists of certain elliptic curves, contributing to understanding their rank distribution.

Contribution

It introduces a new criterion for the analytic rank of prime twists of elliptic curves with specific torsion, linking it to 2-adic logarithms of Heegner points.

Findings

Criteria for prime twists to have rank 0 or 1

New cases supporting Silverman's conjecture

Application to rank distribution of elliptic curve twists

Abstract

For certain elliptic curves $E/\mathbb{Q}$ with $E(\mathbb{Q})[2]=\mathbb{Z}/2 \mathbb{Z}$, we prove a criterion for prime twists of $E$ to have analytic rank 0 or 1, based on a mod 4 congruence of 2-adic logarithms of Heegner points. As an application, we prove new cases of Silverman's conjecture that there exists a positive proposition of prime twists of $E$ of rank zero (resp. positive rank).