most recent 30 from http://mathoverflow.net 2013-05-18T21:09:59Z http://mathoverflow.net/feeds/question/105942 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105942/elliptic-curves-with-cm-and-class-field-theory abourdon 2012-08-30T13:06:59Z 2012-08-30T13:06:59Z

<p>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 <em>Advanced Topics in the Arithmetic of Elliptic Curves</em>.)</p> <p>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.</p>