Torsion of elliptic curves over cyclic cubic fields

TL;DR

This paper classifies all possible torsion groups of elliptic curves over various types of cubic fields, including cyclic, non-cyclic totally real, and complex cubic fields.

Contribution

It provides a complete classification of torsion groups of elliptic curves over different cubic field types, extending prior knowledge in the field.

Findings

All torsion groups over cyclic cubic fields identified

Torsion groups over non-cyclic totally real cubic fields characterized

Torsion groups over complex cubic fields determined

Abstract

We determine all the possible torsion groups of elliptic curves over cyclic cubic fields, over non-cyclic totally real cubic fields and over complex cubic fields.