Elliptic curves with complex multiplication and abelian division fields

Abstract

Let $K$ be an imaginary quadratic field, and let ${\mathcal{O}}_{K,f}$ be an order in $K$ of conductor $f\ge 1$. 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 $N\ge 2$ 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.