Growth of Odd Torsion Over Imaginary Quadratic Fields of Class Number 1

TL;DR

This paper investigates how the odd-order torsion subgroup of elliptic curves over certain imaginary quadratic fields can grow when extending the base field to quadratic extensions, identifying possible torsion groups.

Contribution

It classifies the possible odd-order torsion groups of elliptic curves over quadratic extensions of specific imaginary quadratic fields with class number 1.

Findings

Identifies all possible odd torsion groups over quadratic extensions

Provides explicit classifications for non-cylotomic imaginary quadratic fields

Enhances understanding of torsion growth in elliptic curves over quadratic fields

Abstract

Let $K$ be a non-cylotomic imaginary quadratic field of class number 1 and $E/K$ is an elliptic curve with $E(K)[2]\simeq \mathbb{Z}_1.$ We determine the odd-order torsion groups that can arise as $E(L)_{\text{tor}}$ where $L$ is a quadratic extension of $K.$