Torsion groups of Mordell curves over number fields of higher degree

TL;DR

This paper classifies all possible torsion subgroups of Mordell curves over certain higher degree number fields, specifically those with degrees 2p and 3p where p ≥ 5 is prime.

Contribution

It provides a complete classification of torsion subgroups for Mordell curves over number fields of degrees 2p and 3p, extending previous results over smaller fields.

Findings

Identifies all possible torsion subgroups over specified fields.

Extends classification to higher degree number fields.

Provides explicit descriptions of torsion structures.

Abstract

Mordell curves over a number field $K$ are elliptic curves of the form $ y^2 = x^3 + c$, where $c \in K \setminus \{ 0 \}$. Let $p \geq 5$ be a prime number, $K$ a number field such that $[K:\mathbb{Q}] \in \{ 2p, 3p \}$ and let $E$ be a Mordell curve defined over $K$. We classify all the possible torsion subgroups $E(K)_{\text{tors}}$ for all Mordell curves $E$ defined over $\mathbb{Q}$ when $[K: \mathbb{Q}] \in \{2p, 3p \}$.