Quadratic forms and elliptic curves
Suggested by the antique congruent number problem for right triangles, the author obtained, using arbitrary triangles, a family of infinitely many elliptic curves each of it provided with a non-torsion point [see \textit{T. Ono}, Proc. Japan Acad., Ser. A 70, No. 4, 106-108 and No. 7, 223-225 (1994; respectively Zbl 0824.14028 and Zbl 0824.14029), and also \textit{S. Kwon}, ibid. 72, No. 6, 118-120 (1996; Zbl 0865.14016)]. Now the author pursues the same theme, replacing triangles by quadratic forms. Let \((V,q)\) be a quadratic space over a field \(k\) of characteristic \(\neq 2\), and \(w=(u,v)\) a pair of independent and nonisotropic vectors in \(V\). In this first part, the author associates an elliptic curve \(E_w: Y^2=X^3+ A_w X^2+B_w X\) to some special constants \(A_w,B_w\in k\) defined by \(w\). If \(k=\mathbb{Q}\), \(E_w\) is provided with a non-torsion point \(P_w\) under some simple conditions on \(w\) in a quadratic extension of \(\mathbb{Q}\). To achieve that these points are rational, the values of \(q\) must be squares. The author renames such \(q\)'s as ``Hopf maps''. [For part II and III see the two following reviews].
scientific article; zbMATH DE number 993535