Elliptic curves with \(\mathbb Q({\mathcal E}[3])=\mathbb Q(\zeta_3)\) and counterexamples to local-global divisibility by 9
type: Item
instance of: scholarly article
title: Elliptic curves with \(\mathbb Q({\mathcal E}[3])=\mathbb Q(\zeta_3)\) and counterexamples to local-global divisibility by 9
zbMATH Open document ID: 1216.11064
author: Laura Paladino
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2011-01-03T00:00:00Z
full work available at URL: https://eudml.org/doc/116392
full work available at URL: http://jtnb.cedram.org/item
review text: The authors constructs a family \({\mathcal F}\) of elliptic curves defined over \({\mathbb Q}\), depending of two rational parameters, with the following property: an elliptic curve belongs to \({\mathcal F}\) if and only if the field of definition ot its 3-torsion is exactly \({\mathbb Q}(\zeta_3)\). A subfamily of \({\mathcal F}\), depending on just one rational parameter \(k\), gives counterexamples to the local-global divisibility problem by 9, in analogy with the examples given in [\textit{R. Dvornicich} and \textit{U. Zannier}, ``An analogue for elliptic curves of the Grunwald--Wang example'', C. R. Math., Acad. Sci. Paris 338, No. 1, 47--50 (2004; Zbl 1035.14007)] for the local-global divisibility by~4.
reviewed by: Roberto Dvornicich
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14G05
zbMATH DE Number: 5831866
zbMATH Keywords: local-global divisibility problem
zbMATH Keywords: elliptic curves
MaRDI profile type: Publication
OpenAlex ID: W2010162425
Wikidata QID: Q124979296
cites work: Q5566638
cites work: Local-global divisibility of rational points in some commutative algebraic groups
cites work: An analogue for elliptic curves of the Grunwald--Wang example.
cites work: On a local-global principle for the divisibility of a rational point by a positive integer
cites work: Principal Homogeneous Spaces Over Abelian Varieties
cites work: Ein allgemeines Existenztheorem für algebraische Zahlkörper.
cites work: Rational isogenies of prime degree. (With an appendix by D. Goldfeld)
cites work: The field generated by the points of small prime order on an elliptic curve
cites work: Sur la nature non-cyclotomique des points d'ordre fini des courbes elliptiques. (On the noncyclotomic nature of finite-order points of elliptic curves). With an appendix by E. Kowalski and P. Michel
cites work: Local-global divisibility by 4 in elliptic curves defined over \(\mathbb Q\)
cites work: Corps engendré par les points de 13-torsion des courbes elliptiques
cites work: Q4003835
cites work: Q4875376
cites work: Q3710637
cites work: Q4004187
cites work: Q4757239
cites work: A counter-example to Grunwald's theorem
cites work: On Grunwald's theorem
cites work: Non-analytic class field theory and Grünwald's theorem
cites work: Power residues on Abelian varieties
DOI: 10.5802/JTNB.708
description: scientific article; zbMATH DE number 5831866