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arXiv · 2308.00668

Elliptic curves with complex multiplication and abelian division fields

Asimina S. Hamakiotes·Alvaro Lozano-Robledo
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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.

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Asimina S. Hamakiotes, Alvaro Lozano-Robledo. 2023-08-01. Elliptic curves with complex multiplication and abelian division fields. https://arxiv.org/abs/2308.00668

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