For $K$ a number field, denote by $\mathcal{O}\_K$ its ring of integers and by $H\_K$ its Hilbert class field.


For which imaginary quadratic field $K$ does there exist an elliptic curve $E$, defined over $H$, with complex multiplication by $\mathcal{O}\_K$ having everywhere good reduction (on $H$)?



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> By Fontaine's [Il n'y a pas de variété abélienne sur Z](https://link.springer.com/article/10.1007/BF01388584), corollary of Théorème B, there do not exist such curves for $K=\mathbb{Q}(\sqrt{-1})$ or $\mathbb{Q}(\sqrt{-3})$.
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Do there exist elliptic curves over $H_K$ having everywhere good reduction and CM by $\mathcal{O}_K$?
Here is an example.


Let $K = \mathbf{Q}(\sqrt{-21})$. Then the class group of $K$ is $C\_2 \times C\_2$ and its Hilbert class field is $H\_K = \mathbf{Q}(\sqrt{-1}, \sqrt{3}, \sqrt{7})$. In particular, $H\_K$ is a CM-field and its maximal totally real subfield $H\_K^+$ is $\mathbf{Q}(\sqrt{3}, \sqrt{7})$.


The LMFDB database has an elliptic curve [4.4.7056.1-1.1.a1](https://beta.lmfdb.org/EllipticCurve/4.4.7056.1/1.1/a/1) over $H\_K^+$ which has everywhere good reduction and has CM by $\mathcal{O}\_K$. Base-extending this from $H\_K^+$ to $H\_K$ gives the example you seek.


Similar examples should exist whenever $H\_K$ has class number 1 and all units of $H\_K$ are in the kernel of the norm map to $K$. (The class number condition may well not be needed, but the condition on the units certainly is). I have no idea if there are infinitely many such fields $K$, but there definitely some! This happens for $\mathbf{Q}(\sqrt{-d})$ for $d = 21, 33, 42, 57, 66, 77, 93$ (and no others for $d < 100$).


