Elliptic curves with complex multiplication and abelian division fields

TL;DR

This paper classifies elliptic curves with complex multiplication over imaginary quadratic fields where their division fields are abelian extensions, focusing on cases where these fields match cyclotomic extensions.

Contribution

It provides a complete classification of elliptic curves with CM for which division fields are abelian and coincide with cyclotomic fields, advancing understanding of their Galois properties.

Findings

Classification of N and E for abelian division fields

Identification of cases where division fields equal cyclotomic fields

Insights into Galois representations of CM elliptic curves

Abstract

Let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f\geq 1$. Let $E$ be an elliptic curve with CM by $\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j_{K,f})$, where $j_{K,f}=j(E)$. In this article, we classify the values of $N\geq 2$ and the elliptic curves $E$ such that (i) the division field $\mathbb{Q}(j_{K,f},E[N])$ is an abelian extension of $\mathbb{Q}(j_{K,f})$, and (ii) the $N$-division field coincides with the $N$-th cyclotomic extension of the base field.