Do there exist elliptic curves over $H_K$ having everywhere good reduction and CM by $\mathcal{O}_K$?

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 $$H$$)?

By Fontaine's Il n'y a pas de variété abélienne sur Z, corollary of Théorème B, there do not exist such curves for $$K=\mathbb{Q}(\sqrt{-1})$$ or $$\mathbb{Q}(\sqrt{-3})$$.

• I am nothing close to a specialist but there is a paper Elliptic curves with everywhere good reduction May 26 at 17:17
• There are clearly no examples with $H_K = K$, since in that case the Groessencharacter attached to $E$ would be a Groessencharacter of conductor 1 and infinity-type $(1, 0)$, which is impossible. May 26 at 19:40
• I have a feeling there might be one for $K = \mathbf{Q}(\sqrt{-21})$. For this field, $H_K$ has class number 1 and all units of $H_K$ get sent to 1 by the norm map to $K$. So there is a conductor 1 Groessencharacter of $H_K$ which sends $\mathfrak{a}$ to $\sigma_1(\operatorname{N}_{H_K / K} \alpha)$ where $\sigma_1$ is your favourite embedding of $K$ into $\mathbf{C}$ and $\alpha$ is any generator of $\mathfrak{a}$. This has the correct infinity-type to be the GC of a CM elliptic curve. May 27 at 10:28

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 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 $$d < 100$$).