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Laura Paladino
Elliptic curves with ${\mathbb{Q}}({\mathcal{E}}[3])= {\mathbb{Q}}(\zeta _3)$ and counterexamples to local-global divisibility by 9
Journal de théorie des nombres de Bordeaux, 22 no. 1 (2010), p. 139-160, doi: 10.5802/jtnb.708
Article PDF | Reviews MR 2675877 | Zbl 1216.11064

Résumé - Abstract

We give a family ${\mathcal{F}}_{h,\beta }$ of elliptic curves, depending on two nonzero rational parameters $\beta $ and $h$, such that the following statement holds: let $\mathcal{E}$ be an elliptic curve and let ${\mathcal{E}}[3]$ be its 3-torsion subgroup. This group verifies ${{\mathbb{Q}}}({\mathcal{E}}[3])={{\mathbb{Q}}}(\zeta _3)$ if and only if $\mathcal{E}$ belongs to ${\mathcal{F}}_{h,\beta }$.

Furthermore, we consider the problem of the local-global divisibility by 9 for points of elliptic curves. The number 9 is one of the few exceptional powers of primes, for which an answer to the local-global divisibility is unknown in the case of such algebraic groups. In this paper, we give a negative one. We show some curves of the family ${\mathcal{F}}_{h,\beta }$, with points locally divisible by 9 almost everywhere, but not globally, over a number field of degree at most 2 over ${\mathbb{Q}}({\zeta }_3)$.

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