Elliptic curves with \(\mathbb Q({\mathcal E}[3])=\mathbb Q(\zeta_3)\) and counterexamples to local-global divisibility by 9. (English) Zbl 1216.11064

Paladino, Laura

J. Théor. Nombres Bordx. 22, No. 1, 139-160 (2010).

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 [R. Dvornicich and 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.

Reviewer: Roberto Dvornicich (Pisa)

MSC:

11G05	Elliptic curves over global fields
14G05	Rational points

Keywords:

local-global divisibility problem; elliptic curves

Citations:

Zbl 1035.14007

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