Torsion of elliptic curves over cubic fields

TL;DR

This paper investigates the possible torsion groups of elliptic curves over cubic fields, identifying minimal discriminant fields for certain groups and analyzing the infinitude of such curves.

Contribution

It provides new results on torsion groups over specific cubic fields and characterizes the existence of infinitely many elliptic curves with given torsion over these fields.

Findings

Identified minimal discriminant fields for certain torsion groups.

Determined the existence of infinitely many elliptic curves with specific torsion groups over cubic fields.

Classified torsion groups that appear over cubic fields with particular Galois groups.

Abstract

Although it is not known which groups can appear as torsion groups of elliptic curves over cubic number fields, it is known which groups can appear for infinitely many non-isomorphic curves. We denote the set of these groups as $S$. In this paper we deal with three problems concerning the torsion of elliptic curves over cubic fields. First, we study the possible torsion groups of elliptic curves that appear over the field with smallest absolute value of its discriminant and having Galois group $S_3$ and over the field with smallest absolute value of its discriminant and having Galois group $\Z/3\Z$. Secondly, for all except two groups $G\in S$, we find the field $K$ with smallest absolute value of its discriminant such that there exists an elliptic curve over $K$ having $G$ as torsion. Finally, for every $G\in S$ and every cubic field $K$ we determine whether there exists infinitely many non-isomorphic elliptic curves with torsion $G$.