Quadratic forms and elliptic curves
type: Item
instance of: scholarly article
title: Quadratic forms and elliptic curves
zbMATH Open document ID: 0877.14020
author: Takashi Ono
published in: Proceedings of the Japan Academy. Series A
publication date: 1997-12-10T00:00:00Z
review text: 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].
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11E25
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14G05
zbMATH DE Number: 993535
zbMATH Keywords: rational points
zbMATH Keywords: quadratic space
zbMATH Keywords: elliptic curve
reviewed by: Bernd Richter
MaRDI profile type: Publication
full work available at URL: pjaa.72.157
OpenAlex ID: W4234111938
cites work: Q3346401
cites work: Elliptic curves related with triangles
cites work: On the hopf fibration over Z
cites work: Triangles and elliptic curves
cites work: Q4867187
cites work: Q4004187
cites work: A classical Diophantine problem and modular forms of weight \(3/2\)
DOI: 10.3792/PJAA.72.157
Recommended article: Quadratic forms, elliptic curves and integer sequences
Recommended article: Quadratic forms and elliptic curves. II
Recommended article: Quadratic forms and elliptic curves. IV
Recommended article: Quadratic forms and elliptic curves. III
Recommended article: Quadratic algebras related to elliptic curves
Recommended article: On Elliptic Curves and Binary Quartic Forms
Recommended article: Q4777372
Recommended article: Q4146817
Recommended article: Q5686095
Recommended article: Q4264382
description: scientific article; zbMATH DE number 993535