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

Abstract

Abstract. We construct infinite families of elliptic curves with given torsion group structures over quartic number fields. In a 2006 paper, the first two authors and Park determined all of the group structures which occur infinitely often as the torsion of elliptic curves over quartic number fields. Our result presents explicit examples of their theoretical result. This paper also presents an efficient way of finding such families of elliptic curves with prescribed torsion group structures over quadratic or quartic number fields.

JKL-ECM: an implementation of ECM using Hessian curves

“…Here we will also state another theorem of [15], which allows the generation of curves with Z /4 Z ⊕ Z /8 Z torsion over the given quartic number field.…”

“…We wish to investigate the use of Hessian curves, and a particular subclass consisting of curves arising in families described by Jeon, Kim and Lee (JKL) [15], in ECM. These curves have a large torsion subgroup over a quartic number field, which yields an improvement to ECM as outlined in [5, § 6], for curves over Q, and [9, § 5] for curves over quartic number fields (also referred to in [11]).…”

Gonality of the modular curveX1(N)

“…The values of N for which the curve X 1 (N ) has infinitely many places of degree d over Q are known for d = 1 (see [13]: N =1...10,12), d = 2 (see [9]: N =1...16,18), d = 3 (see [7]: N =1...16,18,20) and d = 4 (see [8]: N =1...18,20...22,24). In this section, we extend this to d 8.…”

Infinite families of elliptic curves over Dihedral quartic number fields

“…There has been some development on characterization of groups which appear as torsion groups of elliptic curves over rational number fields [9], quadratic number fields [5,6,11], cubic number fields [2,4,10], and quartic number fields [1,3]. In particular, for the quartic case, in [3] they determined which ✩ The first author was supported by Basic Science Research Program through the National Research Foundation of Korea groups occur infinitely often as torsion groups E(K ) tors when K varies over all quartic number fields and E varies over all elliptic curves over K .…”