On the torsion of rational elliptic curves over quartic fields

TL;DR

This paper investigates the possible torsion subgroup structures of rational elliptic curves over quartic fields, identifying which torsion groups can occur infinitely often and characterizing their occurrence.

Contribution

It classifies the torsion groups of rational elliptic curves over quartic fields that appear infinitely often, extending understanding of torsion phenomena in elliptic curves.

Findings

Identifies torsion groups occurring infinitely often over quartic fields.

Provides criteria for possible torsion subgroup structures.

Classifies the torsion structures for elliptic curves over quartic extensions.

Abstract

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 <= H could appear such that H=E(K)_tors, for [K:Q]=4 and H is one of the possible torsion structures that occur infinitely often as torsion structures of elliptic curves defined over quartic number fields.