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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 f, then $K(j(E))$ is the ring class field of $K$ with conductor f, but I'm wondering if anything more can be said.

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 There is an isogeny from $E$ to a regular CM curve naturally defined by the subgroup of the fundamental group which has full CM, so the isogeny is defined over $\mathbb Q$. It has a cyclic kernel of degree $k$, so for $(m,k)=1$ it induces an isomorphism on $m$-torsion points and so preserves the CM theory for those torsion points. – Will Sawin Aug 30 at 16:52
Beyond that, you can use the induced map on Tate modules to determine the Galois representation of $E$ and so the action on torsion points. – Will Sawin Aug 30 at 16:56
I don't think this characterization of $K(j(E))$ is correct. I believe that it is a subfield of that ray class field. Instead of modding out by principal ideals with a generator $1$ mod $k$, I think you should mod out by ideals with a generator integral mod $k$, that is, ideals with a generator in the order. The reason is because you can define a CM curve by an order using a $\mathbb Z/k$-subgroup of a CM curve by the full ring, and the defining equation for a subgroup doesn't get you the full ray class field, but the subfield I described. – Will Sawin Aug 30 at 18:00
Thanks for your comments! About what I said concerning K(j(E))--this is coming specifically from the first page of Kwon's "Degree of Isogenies of Elliptic Curves with Complex Multiplication." Other places (for example, Silverman's The Arithmetic of Elliptic Curves, page 427) at least say that [K(j(E)):K] is equal to the class number of the order by which E has CM. Am I misunderstanding something? There could be some assumptions I've missed, or perhaps I've misinterpreted. – abourdon Aug 30 at 20:06
As for your first response... I don't think I'm quite following. Are you saying that if I start with an elliptic curve E with CM by a non-maximal order, it's isogenous (with the isogeny defined over Q) to a curve with CM by the maximal order? Is this new curve no longer an elliptic curve? And would we know anything about the degree k? – abourdon Aug 30 at 20:25

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