Triangles and elliptic curves. II
type: Item
instance of: scholarly article
title: Triangles and elliptic curves. II
zbMATH Open document ID: 0824.14029
author: Takashi Ono
published in: Proceedings of the Japan Academy. Series A
publication date: 1995-11-13T00:00:00Z
review text: This is a continuation of part I [ibid. No. 4, 106-108 (1994; see the preceding review)]. In part I, to each parameter \(t = (a,b,c) \in k^ 3\), \(k\) a field of characteristic \(\neq 2\), there was associated an elliptic plane curve \(E_ t = E\) over \(k\) with \(k\)-rational point \(\pi_ t = \pi\) on it. Now, an elliptic space curve \(C\) in a fibre of the map \(t \mapsto E_ t\) is considered, such that the map \(t \mapsto \pi_ t\) is an isogeny: \(C \to E = E_ t\), \(t \in C\). The Ansatz is the observation that the elliptic curve \(y^ 2 - x(x + m^ 2 - l^ 2) (x + n^ 2 - l^ 2)\) \((l,m,n\) in any field extension \(k'\) of \(k)\) has the \(k'\)-rational point \(x_ 0 = l^ 2\), \(y_ 0 = lmn\). Taking \(l = c/2\), \(m = (a + b)/2\), \(n = (a - b)/2\) we arrive at part I.
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 723974
zbMATH Keywords: Mordell-Weil rank of elliptic curves
reviewed by: Bernd Richter
MaRDI profile type: Publication
full work available at URL: pjaa.70.223
OpenAlex ID: W1984337211
cites work: Triangles and elliptic curves
cites work: Q4867187
cites work: Q3710637
DOI: 10.3792/PJAA.70.223
description: scientific article; zbMATH DE number 723974