On the cyclic torsion of elliptic curves over cubic number fields (III)

TL;DR

This paper proves that certain cyclic groups of order 39 cannot appear as torsion subgroups of elliptic curves over cubic number fields, advancing understanding of torsion structures in this setting.

Contribution

It establishes the non-existence of cyclic torsion subgroups of order 39 over cubic number fields, completing part of a series on torsion classifications.

Findings

No elliptic curve over a cubic number field has torsion subgroup of order 39.

Supports the classification of possible torsion subgroups over cubic fields.

Contributes to the broader understanding of torsion structures in elliptic curves.

Abstract

This is the third part of a series of papers discussing the cyclic torsion subgroup of elliptic curves over cubic number fields. For $N=39$, we show that $\mathbb{Z}/N\mathbb{Z}$ is not a subgroup of $E(K)_{tor}$ for any elliptic curve $E$ over a cubic number field $K$.