On the family of elliptic curves $y^2=x^3-m^2x + (pqr)^2$

TL;DR

The article analyzes the elliptic-curve family $E_m: y^2 = x^3 - m^2 x + (pqr)^2$ with distinct odd primes $p,q,r$ under $m \not\equiv 0 \\pmod{3}$ and $m \equiv 2 \\pmod{2^{k}}$ ($k \ge 5$), proving that $E_m(\\mathbb{Q})_{tors} = {\\mathcal{O}}$ and that $\operatorname{rank} E_m(\\mathbb{Q}) \ge 2$. The approach combines Mordell-Weil theory, reduction modulo primes with good reduction to constrain torsion via the discriminant $\\Delta(E_m) = 16[4m^6 - 27(pqr)^4]$, and explicit descent-style arguments showing two independent rational points $A_m=(0,pqr)$ and $B_m=(m,pqr)$ yield rank at least $2$. The paper also provides a concrete example with rank $2$ (e.g., $m=2$, $p=3$, $q=7$, $r=11$) and notes computational evidence suggesting some curves in the family may have rank $\ge 3$, raising questions about subfamilies with higher rank. These results contribute to understanding how parameter choices influence torsion and rank in elliptic curves of this form.

Abstract

In this article, we consider a family of elliptic curves defined by $E_{m}: y^2= x^3 -m^2 x + (pqr)^2 $ where $m $ is a positive integer and $p, q, ~\text{and}~ r$ are distinct odd primes and study the torsion as well the rank of $E_{m}(\mathbb{Q})$. More specifically, we proved that if $m \not \equiv 0 \pmod{3}, m \not \equiv 0 \pmod{4} ~\text{and}~ m \equiv 2 \pmod {2^{k}}$ where $k \geq 5$, then the torsion subgroup of $E_{m}(\mathbb{Q})$ is trivial and lower bound of the $\mathbb{Q}$ rank of this family of elliptic curves is $2$.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Theorem 3.1
• Remark 3.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3