Non-cyclic Torsion of Elliptic Curves Over Imaginary Quadratic Fields of Class Number 1

TL;DR

This paper classifies possible torsion groups of elliptic curves over quadratic extensions of certain imaginary quadratic fields with class number 1, focusing on non-cyclic torsion structures.

Contribution

It determines all torsion groups that can occur for elliptic curves over quadratic extensions of specified imaginary quadratic fields with class number 1.

Findings

Classification of torsion groups over quadratic extensions

Identification of non-cyclic torsion structures

Explicit description of torsion possibilities

Abstract

Let $K$ be a non-cylotomic imaginary quadratic field of class number 1 and $E/K$ is an elliptic curve with $E(K)[2]\simeq \mathbb{Z}/2\mathbb{Z} \oplus\mathbb{Z}/2\mathbb{Z}.$ In this article, we determine the torsion groups that can arise as $E(L)_{\text{tor}}$ where $L$ is a quadratic extension of $K.$