On Torsion Subgroups of Elliptic Curves over Quartic, Quintic and Sextic Number Fields

TL;DR

This paper investigates which torsion subgroups of elliptic curves can occur over specific degree 4, 5, and 6 number fields, providing criteria for finiteness, minimal discriminants, and rank bounds for such torsion groups.

Contribution

It offers explicit criteria to determine whether a torsion subgroup appears finitely or infinitely often over a given degree d number field, advancing understanding of elliptic curve torsion structures.

Findings

Criteria for finiteness of torsion subgroup occurrences

Identification of minimal discriminant fields for given torsion groups

Examples of degree d fields with bounded Mordell-Weil rank

Abstract

The list of all groups that can appear as torsion subgroups of elliptic curves over number fields of degree $d$, $d=4,5,6$, is not completely determined. However, the list of groups $\Phi^{\infty}(d)$, $d=4,5,6$, that can be realized as torsion subgroups for infinitely many non-isomorphic elliptic curves over these fields are known. We address the question of which torsion subgroups can arise over a given number field of degree $d$. In fact, given $G\in\Phi^{\infty}(d)$ and a number field $K$ of degree $d$, we give explicit criteria telling whether $G$ is realized finitely or infinitely often over $K$. We also give results on the field with the smallest absolute value of its discriminant such that there exists an elliptic curve with torsion $G$. Finally, we give examples of number fields $K$ of degree $d$, $d=4,5,6$, over which the Mordell-Weil rank of elliptic curves with prescribed torsion is bounded from above.