Torsion subgroups of elliptic curves over quintic and sextic number fields

TL;DR

This paper determines the possible torsion subgroups of elliptic curves over degree 5 and 6 number fields, extending known classifications for lower degrees.

Contribution

It explicitly characterizes the sets of torsion subgroups that occur infinitely often over quintic and sextic fields, filling a gap in the classification.

Findings

Determined (5) and (6) sets of torsion subgroups

Extended classification of torsion subgroups to degree 5 and 6 fields

Provided a complete list of groups occurring infinitely often

Abstract

Let $\Phi^\infty(d)$ denote the set of finite abelian groups that occur infinitely often as the torsion subgroup of an elliptic curve over a number field of degree $d$. The sets $\Phi^\infty(d)$ are known for $d\le 4$. In this article we determine $\Phi^\infty(5)$ and $\Phi^\infty(6)$.