Elliptic curves with \(\mathbb Q({\mathcal E}[3])=\mathbb Q(\zeta_3)\) and counterexamples to local-global divisibility by 9
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.
scientific article; zbMATH DE number 5831866
scientific article