When $A$ is a CM abelian variety of dimension $1$ (i.e., an elliptic curve), then we have a result that if it has CM by a maximal order then it has a model over a number field $F$ where $F$ is the Hilbert class field of the CM field if $A$ has CM by the maximal order and is otherwise a ray class field. In particular, if $A$ has CM by a non-maximal order then it has a "strictly larger" field of definition. (Experts, please excuse some sloppiness here.)

Is the analogous result true in general? I.e., is it true that if we take two CM abelian varieties $A$ and $A^\prime$ with CM by $(K,\Phi)$ but such that $A$ has CM by the maximal order $\mathcal{O}_K\subseteq K$ and $A^\prime$ has CM by a non-maximal order $\mathcal{O}\subseteq K$ that for the analogous model of $A^\prime$ over $F^\prime$ and $A$ over $F$ we must have that $[F^\prime:F]>1$?

  • 2
    $\begingroup$ For an elliptic curve $E/F$ with CM endomorphisms defined over $F$ : there is some $ n$ such that it has CM by $\mathbb{Z}+n\mathcal{O}_K$, let $H = \{ P \in E, \forall a \in n \mathcal{O}_K, [a] P = 0\}$ then $E/H$ has CM by $\mathcal{O}_K$. The isogeny $E \to E/H$ is defined over $F$, so $F$ contains the field where curves with CM by $\mathcal{O}_K$ are defined. $\endgroup$ – reuns Jan 29 at 0:51

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.