Let $K$ be an imaginary quadratic field with Hilbert class field $H$, and let $E$ be an elliptic curve defined over $H$ with complex multiplication by the ring of integers $O\_K$ of $K$. It is known that for an integral ideal $\mathfrak{m}$ of $O\_K$, $K(j(E),h(E[\mathfrak{m}]))$ is the ray class field of K modulo $\mathfrak{m}$, where $h$ is the Weber function for $E/H$. (This is stated, for example, on page 135 of Silverman's *Advanced Topics in the Arithmetic of Elliptic Curves*.)


My question is this: what if $E$ has CM by an arbitrary order? Can any generalization of this statement be made? I've read that if $E$ has CM by an order of conductor $\mathfrak{f}$, then $K(j(E))$ is the ring class field of $K$ with conductor $\mathfrak{f}$, but I'm wondering if anything more can be said.



Elliptic Curves with CM and Class Field Theory
In fact Shimura handled the case of an abelian variety $A$ with complex multiplication by an order $O$ inside the maximal order $O\_K$ of the CM field $K$.
A very good modern reference is the following article by Marco Streng:
[An explicit version of Shimura's reciprocity law for Siegel modular functions.](http://pub.math.leidenuniv.nl/~strengtc/reciprocity.pdf)
I also recommend his PhD thesis.
Both are available here:
<http://pub.math.leidenuniv.nl/~strengtc/research.html>


Now for the statement (see Theorem 2.2 of the paper above):
let $\tau$ be the element of the Siegel space that corresponds to the abelian variety $A$ above. The Siegel modular functions of level $N$ with $q$-expansion in $\mathbb{Q}(\zeta\_N)$ evaluated at $\tau$ generate an abelian extension $H(N)$ of the reflex field $K^r$ (for the type norm $\Phi$ associated to $A$).


This abelian extension is associated by class field theory to the class group
$I\_K(NF)/H\_{\Phi,O}(N)$ where


* $F$ is the conductor of $O$
* $I\_K(NF)$ are the fractional ideals of $K$ prime to $NF$
* $H\_{\Phi,O}(N)$ are the ideals $\mathfrak{a} \in I\_K(NF)$ such that $\exists \mu \in K$ with $N\_{\Phi^r,O}(\mathfrak{a})=\mu O$, $\mu \overline{\mu}=N(\mathfrak{a})\in \mathbb{Q}$ and $\mu \equiv 1 {\bmod^\times} NO$.
(Here $N\_{\Phi^r,O}$ is the type norm from the reflex field $K^r$ to $K$.)


Now specializing to elliptic curves, we get that


* If $E$ has CM by $O\_K$, then $K(j(E),h(E[]))$ is the *ray class field* associated to the class group $I\_K(m)/\{ \mu O\_K \mid \mu \equiv 1 {\bmod^\times} m\}$
* If $E$ has CM by $O$ where the conductor of $O$ is $F \in \mathbb{Z}$, then $K(j(E))$ is the *ring class field* of $O$, meaning the extension
associated to the class group $I\_K(F)/\{ \text{principal ideals of $O$ primes to $F$}\}$. And finally, $K(j(E),h(E[]))$ will correspond to 
$I\_K(mF)/\{ \mu O \mid \text{$\mu$ is prime to $F$ and}\ \mu \equiv 1 {\bmod^\times} mO\}$.
(At least we have the inclusion, but I am pretty sure that in the elliptic curve case, the evaluation of Weber functions at the points of $m$-torsion over the $j$-invariant give the field generated by level-$m$ modular forms evaluated at $\tau$.) So in this case we have something intermediate between a ring class field and a ray class field. [I have not checked carefully but it should be the compositium of the ring class field of conductor $F$ and the ray class field of modulus $m$ when $m$ is prime to $F$].


