Torsion of rational elliptic curves over different types of cubic fields

TL;DR

This paper classifies the possible torsion groups of rational elliptic curves over various types of cubic fields, identifying when these groups can occur infinitely often and providing explicit examples.

Contribution

It determines which torsion groups occur over different cubic field types and whether these occurrences are finite or infinite, including explicit constructions.

Findings

Classification of torsion groups over cyclic cubic fields

Identification of torsion groups over non-Galois totally real cubic fields

Explicit examples of elliptic curves with specified torsion over cubic fields

Abstract

Let $E$ be an elliptic curve defined over $\Q$, and let $G$ be the torsion group $E(K)_{tors}$ for some cubic field $K$ which does not occur over $\Q$. In this paper, we determine over which types of cubic number fields (cyclic cubic, non-Galois totally real cubic, complex cubic or pure cubic) $G$ can occur, and if so, whether it can occur infinitely often or not. Moreover, if it occurs, we provide elliptic curves $E/\Q$ together with cubic fields $K$ so that $G= E(K)_{tors}$.