### Abstract

In [15] and [6] all possible torsion groups of elliptic curves E with integral j-invariant over quadratic and pure cubic number fields K are determined. Moreover, with the exception of the torsion groups of isomorphism types ℤ/2ℤ, ℤ/3ℤ and ℤ/2ℤ x ℤ/2ℤ, all elliptic curves E and all basic quadratic and pure cubic fields K such that E over K has one of these torsion groups were computed. The present paper is aimed at solving the corresponding problem for general cubic number fields K. In the general cubic case, the above groups ℤ/2ℤ, ℤ/3ℤ and ℤ/2ℤ x ℤ/2ℤ and, in addition, the groups ℤ/4ℤ, ℤ/5ℤ occur as torsion groups of infinitely many curves E with integral j-invariant over infinitely many cubic fields K. For all the other possible torsion groups, the (finitely many) elliptic curves with integral j over the (finitely many) cubic fields K are calculated here. Of course, the results obtained in [6] for pure cubic fields and in [24] for cyclic cubic fields are regained by our algorithms. However, compared with [15] and [6], a solution of the torsion group problem in the much more involved general cubic case requires some essentially new methods. In fact we shall use Gröbner basis techniques and elimination theory to settle the general case.

Original language | English |
---|---|

Pages (from-to) | 353-413 |

Number of pages | 61 |

Journal | International Journal of Algebra and Computation |

Volume | 7 |

Issue number | 3 |

Publication status | Published - Jun 1997 |

### Fingerprint

### Keywords

- Cubic number field
- Elimination
- Elliptic curve
- Gröbner basis
- Norm equation
- Parametrization
- Reduction theory
- Torsion group

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*International Journal of Algebra and Computation*,

*7*(3), 353-413.

**Torsion groups of elliptic curves with integral j-invariant over general cubic number fields.** / Pethő, A.; Weis, Thomas; Zimmer, Horst G.

Research output: Contribution to journal › Article

*International Journal of Algebra and Computation*, vol. 7, no. 3, pp. 353-413.

}

TY - JOUR

T1 - Torsion groups of elliptic curves with integral j-invariant over general cubic number fields

AU - Pethő, A.

AU - Weis, Thomas

AU - Zimmer, Horst G.

PY - 1997/6

Y1 - 1997/6

N2 - In [15] and [6] all possible torsion groups of elliptic curves E with integral j-invariant over quadratic and pure cubic number fields K are determined. Moreover, with the exception of the torsion groups of isomorphism types ℤ/2ℤ, ℤ/3ℤ and ℤ/2ℤ x ℤ/2ℤ, all elliptic curves E and all basic quadratic and pure cubic fields K such that E over K has one of these torsion groups were computed. The present paper is aimed at solving the corresponding problem for general cubic number fields K. In the general cubic case, the above groups ℤ/2ℤ, ℤ/3ℤ and ℤ/2ℤ x ℤ/2ℤ and, in addition, the groups ℤ/4ℤ, ℤ/5ℤ occur as torsion groups of infinitely many curves E with integral j-invariant over infinitely many cubic fields K. For all the other possible torsion groups, the (finitely many) elliptic curves with integral j over the (finitely many) cubic fields K are calculated here. Of course, the results obtained in [6] for pure cubic fields and in [24] for cyclic cubic fields are regained by our algorithms. However, compared with [15] and [6], a solution of the torsion group problem in the much more involved general cubic case requires some essentially new methods. In fact we shall use Gröbner basis techniques and elimination theory to settle the general case.

AB - In [15] and [6] all possible torsion groups of elliptic curves E with integral j-invariant over quadratic and pure cubic number fields K are determined. Moreover, with the exception of the torsion groups of isomorphism types ℤ/2ℤ, ℤ/3ℤ and ℤ/2ℤ x ℤ/2ℤ, all elliptic curves E and all basic quadratic and pure cubic fields K such that E over K has one of these torsion groups were computed. The present paper is aimed at solving the corresponding problem for general cubic number fields K. In the general cubic case, the above groups ℤ/2ℤ, ℤ/3ℤ and ℤ/2ℤ x ℤ/2ℤ and, in addition, the groups ℤ/4ℤ, ℤ/5ℤ occur as torsion groups of infinitely many curves E with integral j-invariant over infinitely many cubic fields K. For all the other possible torsion groups, the (finitely many) elliptic curves with integral j over the (finitely many) cubic fields K are calculated here. Of course, the results obtained in [6] for pure cubic fields and in [24] for cyclic cubic fields are regained by our algorithms. However, compared with [15] and [6], a solution of the torsion group problem in the much more involved general cubic case requires some essentially new methods. In fact we shall use Gröbner basis techniques and elimination theory to settle the general case.

KW - Cubic number field

KW - Elimination

KW - Elliptic curve

KW - Gröbner basis

KW - Norm equation

KW - Parametrization

KW - Reduction theory

KW - Torsion group

UR - http://www.scopus.com/inward/record.url?scp=0031491615&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031491615&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0031491615

VL - 7

SP - 353

EP - 413

JO - International Journal of Algebra and Computation

JF - International Journal of Algebra and Computation

SN - 0218-1967

IS - 3

ER -