Elliptic curves with complex multiplication and abelian division fields

Abstract

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