On the family of elliptic curves $y^2 = x^3 − m^2x + p^2$

Proceedings of the Indian Academy of Sciences. Mathematical sciences

ABHISHEK JUYAL^1¹ 

[1] Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad 211 004, India^1

关键词: Elliptic curve;    rank;    torsion subgroup;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

In this paper, we study the torsion subgroup and rank of elliptic curves for the subfamilies of $E_{m,p} : y^{2} = x^{3} − m^{2}x + p^{2}$, where $m$ is a positive integer and $p$ is a prime. We prove that for any prime $p$, the torsion subgroup of $E_{m,p}(\mathbb{Q})$ is trivial for both the cases $\{m \geq 1, m \nequiv 0 (mod 3)\}$ and $\{m \geq 1, m \equiv 0 (mod 3)$, with $gcd(m, p) = 1\}$. We also show that given any odd prime $p$ and for any positive integer $m$ with $m \nequiv 0 (mod 3)$ and $m \equiv 2 (mod 32)$, the lower bound for the rank of $E_{m,p}(\mathbb{Q})$ is 2. Finally, we find curves of rank 9 in this family.

【 授权许可】

CC BY