Torsion of rational elliptic curves over quartic Galois number fields

TL;DR

This paper classifies the possible torsion subgroup structures of rational elliptic curves when extended over quartic Galois number fields, advancing understanding of elliptic curve torsion phenomena in this setting.

Contribution

It provides a comprehensive classification of torsion subgroups of rational elliptic curves over quartic Galois fields, filling a gap in the existing literature.

Findings

Complete list of torsion subgroup types over quartic Galois fields

Identification of new torsion structures not seen over smaller fields

Extension of known classifications to degree four Galois extensions

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $K$ be a number field of degree four that is Galois over $\mathbb{Q}$. The goal of this article is to classify the different isomorphism types of $E(K)_{\text{tors}}$.