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

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