Elliptic Curves with CM and Class Field Theory - MathOverflow 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 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>