Elliptic curves with $\mathbb{Q}\left(ℰ\left[3\right]\right)=\mathbb{Q}\left({\zeta }_3\right)$ and counterexamples to local-global divisibility by 9

Paladino, Laura

Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 139-160.

Nous donnons une famille ${ℱ}_{h,\beta }$ de courbes elliptiques, dépendant de deux paramètres rationnels non nuls $\beta$ et $h$, telle que nous avons la propriété suivante : soit $ℰ$ une courbe elliptique et soit $ℰ\left[3\right]$ son sous-groupe de 3-torsion. On a que $\mathbb{Q}\left(ℰ\left[3\right]\right)=\mathbb{Q}\left({\zeta }_3\right)$ si et seulement si $ℰ$ est une courbe de la famille ${ℱ}_{h,\beta }$.

De plus, nous considérons le problème de la divisibilité locale-globale par 9 pour les points d’une courbe elliptique. Le nombre 9 est une des rares puissances d’un nombre premier pour laquelle on ne connait pas la réponse à la divisibilité locale-globale dans le cas de tels groupes algébriques. Dans ce papier nous donnons une réponse négative. Nous exhibons des courbes de la famille ${ℱ}_{h,\beta }$, avec des points qui sont localement divisibles par 9 presque partout, mais qui ne sont pas globalement divisibles par 9, sur un corps de nombres de degré au plus 2 sur $\mathbb{Q}\left({\zeta }_3\right)$.

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

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 ${ℱ}_{h,\beta }$, with points locally divisible by 9 almost everywhere, but not globally, over a number field of degree at most 2 over $\mathbb{Q}\left({\zeta }_3\right)$.

[1] E. Artin, J. Tate, Class field theory. Benjamin, Reading, MA, 1967. | MR | Zbl

[2] R. Dvornicich, U. Zannier, Local-global divisibility of rational points in some commutative algebraic groups. Bull. Soc. Math. France, 129 (2001), 317–338. | Numdam | MR | Zbl

[3] R. Dvornicich, U. Zannier, An analogue for elliptic curves of the Grunwald-Wang example. C. R. Acad. Sci. Paris, Ser. I 338 (2004), 47–50. | MR | Zbl

[4] R. Dvornicich, U. Zannier, On local-global principle for the divisibility of a rational point by a positive integer. Bull. Lon. Math. Soc., no. 39 (2007), 27–34. | MR | Zbl

[5] S. Lang, J. Tate Principal homogeneous spaces over abelian varieties. American J. Math., no. 80 (1958), 659–684. | MR | Zbl

[6] W. Grunwald, Ein allgemeines Existenztheorem für algebraische Zahlkörper. Journ. f.d. reine u. angewandte Math., 169 (1933), 103–107. | Zbl

[7] B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld. Invent Math., 44 (1978), no. 2, 129–162. | MR | Zbl

[8] L. Merel, W. Stein, The field generated by the points of small prime order on an elliptic curve. Math. Res. Notices, no. 20 (2001), 1075–1082. | MR | Zbl

[9] L. Merel, Sur la nature non-cyclotomique des points d’ordre fini des courbes elliptiques. (French) [On the noncyclotomic nature of finite-order points of elliptic curves] With an appendix by E. Kowalski and P. Michel. Duke Math. J. 110 (2001), no. 1, 81–119. | MR | Zbl

[10] L. Paladino, Local-global divisibility by 4 in elliptic curves defined over $\mathbb{Q}$. Annali di Matematica Pura e Applicata, DOI 10.1007/s10231-009-0098-5.

[11] M. Rebolledo, Corps engendré par les points de 13-torsion des courbes elliptiques. Acta Arith., no. 109 (2003), no. 3, 219–230. | MR | Zbl

[12] J.-P. Serre, Topics in galois Theory. Jones and barlett, Boston, 1992. | MR | Zbl

[13] G. Shimura, Introduction to the arithmetic theory of automorphic functions. Princeton University Press, 1994. | MR | Zbl

[14] J. H. Silverman, The arithmatic of elliptic curves. Springer, 1986. | Zbl

[15] J. H. Silverman, J. Tate, Rational points on elliptic curves. Springer, 1992. | MR | Zbl

[16] E. Trost, Zur theorie des Potenzreste. Nieuw Archief voor Wiskunde, no. 18 (2) (1948), 58–61.

[17] Sh. Wang, A counter example to Grunwald’s theorem. Annals of Math., no. 49 (1948), 1008–1009. | MR | Zbl

[18] Sh. Wang, On Grunwald’s theorem. Annals of Math., no. 51 (1950), 471–484. | MR | Zbl

[19] G. Whaples, Non-analytic class field theory and Grunwald’s theorem . Duke Math. J., no. 9 (1942), 455–473. | MR | Zbl

[20] S. Wong, Power residues on abelian variety. Manuscripta Math., no. 102 (2000), 129–137. | MR | Zbl