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

Attila Pethö, Thomas Weis, Horst G. Zimmer

Research output: Contribution to journalArticle

10 Citations (Scopus)

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 languageEnglish
Pages (from-to)353-413
Number of pages61
JournalInternational Journal of Algebra and Computation
Volume7
Issue number3
DOIs
Publication statusPublished - Jun 1997

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

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