Complete classification of the torsion structures of rational elliptic curves over quintic number fields

TL;DR

This paper provides a complete classification of the torsion structures of rational elliptic curves over quintic number fields, identifying all possible torsion subgroup configurations and their occurrences.

Contribution

It offers a comprehensive classification of torsion structures over quintic fields and determines the uniqueness of such extensions for rational elliptic curves.

Findings

Classified all possible torsion groups over quintic fields.

Proved that at most one quintic extension changes the torsion subgroup.

Identified conditions under which torsion structures can grow over quintic fields.

Abstract

We classify the possible torsion structures of rational elliptic curves over quintic number fields. In addition, let E be an elliptic curve defined over Q and let G = E(Q)_tors be the associated torsion subgroup. We study, for a given G, which possible groups G \subseteq H could appear such that H=E(K)_tors, for [K:Q]=5. In particular, we prove that at most there is a quintic number field K such that E(Q)_tors\neq E(K)_tors.