The rank of a CM elliptic curve and a recurrence formula

TL;DR

This paper investigates the rank of a specific family of elliptic curves over the rationals, providing a recurrence-based criterion for when the rank equals 2, assuming the Birch and Swinnerton-Dyer conjecture.

Contribution

It introduces a recurrence formula to determine the rank of the elliptic curve $E_p$ when the BSD conjecture holds, linking algebraic properties to polynomial constants.

Findings

Rank of $E_p$ is 0 or 2 depending on $p$ modulo 16.

A recurrence formula characterizes the polynomial associated with the curve.

Necessary and sufficient condition for rank 2 under BSD conjecture.

Abstract

Let $p$ be a prime number and $E_{p}$ denote the elliptic curve $y^2=x^3+px$. It is known that for $p$ which is congruent to $1, 9$ modulo $16$, the rank of $E_{p}$ over $\mathbb{Q}$ is equal to $0, 2$. Under the condition that the Birch and Swinnerton-Dyer conjecture is true, we give a necessary and sufficient condition that the rank is $2$ in terms of the constant term of some polynomial that is defined by a recurrence formula.