# On the small cyclic torsion of elliptic curves over cubic number fields

## TL;DR

This paper investigates the possible cyclic torsion subgroups of elliptic curves over cubic number fields, establishing non-existence results for certain torsion orders based on Merel's uniform boundedness theorem.

## Contribution

It provides new non-existence results for specific cyclic torsion subgroups of elliptic curves over cubic fields, extending the classification of torsion structures.

## Key findings

- $ ext{Z}/49 ext{Z}$, $ ext{Z}/40 ext{Z}$, $ ext{Z}/25 ext{Z}$, and $ ext{Z}/22 ext{Z}$ are not subgroups of $E(K)_{tor}$ over any cubic field $K$.
- The results rely on bounds derived from Merel's theorem for elliptic curves over number fields.
- The paper advances the understanding of torsion subgroup possibilities over cubic number fields.

## Abstract

Merel's result on the strong uniform boundedness conjecture made it meaningful to classify the torsion part of the Mordell-Weil groups of all elliptic curves defined over number fields of fixed degree $d$. In this paper, we discuss the cyclic torsion subgroup of elliptic curves over cubic number fields. For $N=49,40,25$ or $22$, 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 $K$.