Torsion Groups of Elliptic Curves with Integral j-Invariant over General Cubic Number Fields

Abstract

In [15] and [16] 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×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×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 any) 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.

Computation on elliptic curves with complex multiplication

“…This may seem like a relatively short amount of time to be worried about, but for j = 0 and a number field of degree 6 it takes about a minute to find [2,3,4,6,7,9,14,19] as the list of torsion exponents. For degree 12 it takes over an hour.…”

Families of elliptic curves over quartic number fields with prescribed torsion subgroups

“…which is one of the equations X 1 (11), called the raw form of X 1 (11). By the coordinate changes s = 1 − u and r = 1 + uv, we get the following equation:…”

“…(3) over quartic number fields as Kubert did over Q. While the subject of the torsion of elliptic curves over number fields of higher order has been studied by Kamienny and Mazur [5], Merel [9], Parent [12,13], Zimmer et al [11,18] and Jeon et al [3,4], there has not been much development for finding elliptic curves with a given torsion group over number fields of higher order. Recently, the cubic number field case is treated in [2,4]; in [4], all of the group structures which occur infinitely often as the torsion of elliptic curves over cubic number fields are determined, and in [2], we construct infinite families of elliptic curves with given torsion group structures over cubic number fields.…”

Torsion des courbes elliptiques sur les corps cubiques

“…-Comme on l'a déjà remarqué en introduction, on trouve dans [23] des paramétrisations de courbes elliptiques (à j-invariant entier au-dessus de 2, 3, ou 5) ayant de la 11 ou 13-torsion sur des corps cubiques. Relevons maintenant les opérateurs de Hecke et les losanges sur cette présentation.…”