On the 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 subgroup orders.

Contribution

It proves that specific cyclic groups of orders 169, 143, 91, 65, 77, and 55 do not occur as torsion subgroups over cubic fields, advancing understanding of torsion structures.

Findings

Certain cyclic torsion groups are impossible over cubic fields

Non-existence of torsion subgroups of orders 169, 143, 91, 65, 77, 55

Provides new constraints on elliptic curve torsion over cubic fields

Abstract

Let $E$ be an elliptic defined over a number field $K$. Then its Mordell-Weil group $E(K)$ is finitely generated: $E(K)\cong E(K)_{tor}\times\mathbb{Z}^r$. In this paper, we discuss the cyclic torsion subgroup of elliptic curves over cubic number fields. For $N=169,143,91,65,77$ or $55$, 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 $K$.