Torsion groups of elliptic curves over quadratic cyclotomic fields in elementary abelian 2-extensions

TL;DR

This paper investigates the structure of torsion subgroups of elliptic curves over quadratic cyclotomic fields within their maximal elementary abelian 2-extensions, providing insights into their algebraic properties.

Contribution

It characterizes the torsion groups of elliptic curves over quadratic cyclotomic fields in elementary abelian 2-extensions, a novel analysis in this specific algebraic setting.

Findings

Classification of possible torsion subgroups

Identification of constraints on torsion points

Extension of known results to new field settings

Abstract

Let K denote the quadratic field $\mathbb{Q}(\sqrt{d})$ where d=$-1$ or $-3$. Let E be an elliptic curve defined over K. In this paper, we analyze the torsion subgroups of E in the maximal elementary abelian $2$-extension of $K$.