A criterion to rule out torsion groups for elliptic curves over number fields

TL;DR

This paper introduces a new criterion to determine which finite abelian groups cannot be torsion subgroups of elliptic curves over certain number fields, and applies it to classify torsion groups over cubic and quartic fields.

Contribution

The paper develops a novel criterion for ruling out specific torsion groups over number fields and uses it to classify all torsion groups over particular cubic and quartic fields.

Findings

Eliminated certain groups as torsion groups over cubic and quartic fields.

Provided complete lists of torsion groups over specific cubic and quartic fields.

Established a new method for analyzing torsion subgroups over number fields.

Abstract

We present a criterion for proving that certain groups of the form $\mathbb Z/m\mathbb Z\oplus\mathbb Z/n\mathbb Z$ do not occur as the torsion subgroup of any elliptic curve over suitable (families of) number fields. We apply this criterion to eliminate certain groups as torsion groups of elliptic curves over cubic and quartic fields. We also use this criterion to give the list of all torsion groups of elliptic curves occurring over a specific cubic field and over a specific quartic field.