Elliptic curves with ( [ 3 ] ) = ( ζ 3 ) and counterexamples to local-global divisibility by 9

Laura Paladino[1]

  • [1] Dipartimento di Matematica Università di Pisa Largo Bruno Pontecorvo, 5 56126 Pisa, Italy

Journal de Théorie des Nombres de Bordeaux (2010)

  • Volume: 22, Issue: 1, page 139-160
  • ISSN: 1246-7405

Abstract

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We give a family h , β of elliptic curves, depending on two nonzero rational parameters β and h , such that the following statement holds: let be an elliptic curve and let [ 3 ] be its 3-torsion subgroup. This group verifies ( [ 3 ] ) = ( ζ 3 ) if and only if belongs to h , β .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 h , β , with points locally divisible by 9 almost everywhere, but not globally, over a number field of degree at most 2 over ( ζ 3 ) .

How to cite

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Paladino, Laura. "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 22.1 (2010): 139-160. <http://eudml.org/doc/116392>.

@article{Paladino2010,
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 $\{\mathbb\{Q\}\}(\{\zeta \}_3)$.},
affiliation = {Dipartimento di Matematica Università di Pisa Largo Bruno Pontecorvo, 5 56126 Pisa, Italy},
author = {Paladino, Laura},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {local-global divisibility problem; elliptic curves},
language = {eng},
number = {1},
pages = {139-160},
publisher = {Université Bordeaux 1},
title = {Elliptic curves with $\{\mathbb\{Q\}\}(\{\mathcal\{E\}\}[3])= \{\mathbb\{Q\}\}(\zeta _3)$ and counterexamples to local-global divisibility by 9},
url = {http://eudml.org/doc/116392},
volume = {22},
year = {2010},
}

TY - JOUR
AU - Paladino, Laura
TI - Elliptic curves with ${\mathbb{Q}}({\mathcal{E}}[3])= {\mathbb{Q}}(\zeta _3)$ and counterexamples to local-global divisibility by 9
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2010
PB - Université Bordeaux 1
VL - 22
IS - 1
SP - 139
EP - 160
AB - 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 ${\mathbb{Q}}({\zeta }_3)$.
LA - eng
KW - local-global divisibility problem; elliptic curves
UR - http://eudml.org/doc/116392
ER -

References

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  1. E. Artin, J. Tate, Class field theory. Benjamin, Reading, MA, 1967. Zbl0176.33504MR223335
  2. R. Dvornicich, U. Zannier, Local-global divisibility of rational points in some commutative algebraic groups. Bull. Soc. Math. France, 129 (2001), 317–338. Zbl0987.14016MR1881198
  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. Zbl1035.14007MR2038083
  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. Zbl1115.14011MR2303515
  5. S. Lang, J. TatePrincipal homogeneous spaces over abelian varieties. American J. Math., no. 80 (1958), 659–684. Zbl0097.36203MR106226
  6. W. Grunwald, Ein allgemeines Existenztheorem für algebraische Zahlkörper. Journ. f.d. reine u. angewandte Math., 169 (1933), 103–107. Zbl0006.25204
  7. B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld. Invent Math., 44 (1978), no. 2, 129–162. Zbl0386.14009MR482230
  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. Zbl1027.11041MR1857596
  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. Zbl1020.11041MR1861089
  10. L. Paladino, Local-global divisibility by 4 in elliptic curves defined over . Annali di Matematica Pura e Applicata, DOI 10.1007/s10231-009-0098-5. Zbl1208.11074
  11. M. Rebolledo, Corps engendré par les points de 13-torsion des courbes elliptiques. Acta Arith., no. 109 (2003), no. 3, 219–230. Zbl1049.11058MR1980258
  12. J.-P. Serre, Topics in galois Theory. Jones and barlett, Boston, 1992. Zbl0746.12001MR1162313
  13. G. Shimura, Introduction to the arithmetic theory of automorphic functions. Princeton University Press, 1994. Zbl0872.11023MR1291394
  14. J. H. Silverman, The arithmatic of elliptic curves. Springer, 1986. Zbl0585.14026
  15. J. H. Silverman, J. Tate, Rational points on elliptic curves. Springer, 1992. Zbl0752.14034MR1171452
  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. Zbl0032.10802MR26992
  18. Sh. Wang, On Grunwald’s theorem. Annals of Math., no. 51 (1950), 471–484. Zbl0036.15802MR33801
  19. G. Whaples, Non-analytic class field theory and Grunwald’s theorem . Duke Math. J., no. 9 (1942), 455–473. Zbl0063.08226MR7010
  20. S. Wong, Power residues on abelian variety. Manuscripta Math., no. 102 (2000), 129–137. Zbl1025.11019MR1771232

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