Generators for the elliptic curve \(y^2=x^3-nx\) of rank at least three
Let \(n\) be a positive integer and consider the curve \(E_n: y^2=x^3-nx\). It is well known that \(E_n(\mathbb{Q})_{\mathrm{tors}}\) (the rational torsion points of \(E_n\,\)) is isomorphic to \(\mathbb{Z}/2\mathbb{Z}\) or to \((\mathbb{Z}/2\mathbb{Z})^2\,\), and this paper provides some generators for the torsion free part of \(E_n(\mathbb{Q})\) for an infinite family of these curves. Let \(n=st\) with positive non square integers \(s\) and \(t\) (\(n\) non square and 4-th power free as well) and assume there exist positive integers \(\alpha\), \(\beta\), \(\gamma\), an odd \(m_1\) and an even \(m_2\) such that \[ t-s=\alpha^2\;\;,\;\;m_1^4s-t=\beta^2\;\;\text{and}\;\;t-m_2^4s=\varepsilon \gamma^2 \] (with \(\varepsilon = \pm 1\)). The main result of the paper shows that the previous conditions describe infinite families of elliptic curves (for example for \(\varepsilon=1\) and \((m_1,m_2)=(3,2)\) or for \(\varepsilon=-1\) and \((m_1,m_2)=(5,2)\,\)) and that the points \[ G_1=(-s,s\alpha)\;\;,\;\;G_2=(m_1^2s,m_1s\beta)\;\;{\text{and}}\;\;G_3=(-\varepsilon m_2^2s,m_2s\gamma) \] are part of a basis for \(E_n(\mathbb{Q})\) (at least for very large \(n \geq \max\{m_1,m_2\}^{60}\,\)). After checking that the given points are independent modulo \(E_n(\mathbb{Q})_{\mathrm{tors}}\,\), the author uses computations of heights of linear combinations of the \(G_i\) to show that no such combination is in \(3E_n(\mathbb{Q})\) if \(n\) is large enough. Then a result of \textit{S. Siksek} [Rocky Mt. J. Math. 25, No. 4, 1501--1538 (1995; Zbl 0852.11028)], bounding the index \(\nu\) of \(\mathbb{Z}G_1+\mathbb{Z}G_2+\mathbb{Z}G_3\) in a rank 3 lattice \(\mathbb{Z}P_1+\mathbb{Z}P_2+\mathbb{Z}P_3\,\), yields \(\nu<5\) and then the theorem. For small \(n\) and the triples \((\varepsilon,m_1,m_2)\in \{(1,3,2),(-1,5,2)\}\), Siksek's theorem only provides \(\nu<11\) and the remaining part of the proof (namely checking that no linear combination of the \(G_i\) is in \(\mu E(\mathbb{Q})\) for \(5\leq \mu \leq 10\)) is carried out via computations using MAGMA.
scientific article; zbMATH DE number 6139694
scientific article