Generators and integer points on the elliptic curve $y^{2}=x^{3}-nx$

Yasutsugu Fujita ¹   ; Nobuhiro Terai ²

¹ College of Industrial Technology Nihon University 2-11-1 Shin-ei Narashino, Chiba 275-8576, Japan
² Division of Information System Design Ashikaga Institute of Technology 268-1 Omae Ashikaga, Tochigi 326-8558, Japan

Acta Arithmetica, Tome 160 (2013) no. 4, pp. 333-348

Résumé

Let $E$ be an elliptic curve over the rationals ${\mathbb {Q}}$ given by $y^2=x^3-nx$ with a positive integer $n$. We consider first the case where $n=N^2$ for a square-free integer $N$. Then we show that if the Mordell–Weil group $E({\mathbb {Q}})$ has rank one, there exist at most 17 integer points on $E$. Moreover, we show that for some parameterized $N$ a certain point $P$ can be in a system of generators for $E( {\mathbb {Q}})$, and we determine the integer points in the group generated by the point $P$ and the torsion points. Secondly, we consider the case where $n=s^4+t^4$ for distinct positive integers $s$ and $t$. We then show that if $n$ is fourth-power-free, the points $P_1=(-t^2,s^2t)$ and $P_2=(-s^2,st^2)$ can be in a system of generators for $E( {\mathbb {Q}})$. Furthermore, we prove that if $n$ is square-free, then there exist at most nine integer points in the group $\varGamma $ generated by the points $P_1$, $P_2$ and the torsion point $(0,0)$. In particular, in case $n=s^4+1$ the group $\varGamma $ has exactly seven integer points.

Détail

DOI : 10.4064/aa160-4-3

Keywords: elliptic curve rationals mathbb given nx positive integer nbsp consider first where square free integer nbsp mordell weil group mathbb has rank there exist integer points nbsp moreover parameterized certain point system generators mathbb determine integer points group generated point torsion points secondly consider where distinct positive integers nbsp fourth power free points t s system generators mathbb furthermore prove square free there exist nine integer points group vargamma generated points torsion point particular group vargamma has exactly seven integer points