On Mazur's conjecture
type: Item
instance of: scholarly article
title: On Mazur's conjecture
zbMATH Open document ID: 0944.14012
DOI: 10.1007/BF02308767
author: V. A. Dem'yanenko
published in: Mathematical Notes
publication date: 2000-09-18T00:00:00Z
review text: Let \(n\) be a positive integer, \(\varepsilon (n) = e^{2\pi i\over n}\), \(F\) an elliptic curve over \({\mathbb Q}(\varepsilon (n))\), and \(\{O_m ,O_m'\}\) the basis of all points of order \(m\) on \(F\); in this paper it is proved that (1) if \(O_{2^\alpha}(F)\), \(2^{\alpha -1}O_{2^\alpha}'(F)=O _{2}'(F)\in {\mathbb Q}(\varepsilon (2^t))\), where \(t\) is arbitrary, then \(\alpha \leq 3\); (2) if \(O_{3^\beta}(F)\),\(O_{3^\beta}'(F)\in {\mathbb Q}(\varepsilon (3^t))\), where \(t\) is arbitrary, then \(\beta \leq 1\). The result yields to a conjecture about the groups of \(m\)-torsion points on two elliptic curves over an algebraic field.
reviewed by: Alessandro Gimigliano
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14G05
zbMATH DE Number: 1228281
zbMATH Keywords: Mazur's conjecture
zbMATH Keywords: elliptic curves
zbMATH Keywords: algebraic fields
Wikidata QID: Q122901460
MaRDI profile type: Publication
cites work: BEHAVIOR OF THE CURVEx3+y3= 1 IN A CYCLOTOMIC Γ-EXTENSION
description: scientific article; zbMATH DE number 1228281