Binary quadratic forms and ray class groups

Abstract

Abstract Let K be an imaginary quadratic field different from $\open{Q}(\sqrt {-1})$ and $\open{Q}(\sqrt {-3})$ . For a positive integer N , let K N be the ray class field of K modulo $N {\cal O}_K$ . By using the congruence subgroup ± Γ 1 ( N ) of SL 2 (ℤ), we construct an extended form class group whose operation is basically the Dirichlet composition, and explicitly show that this group is isomorphic to the Galois group Gal( K N / K ). We also present an algorithm to find all distinct form classes and show how to multiply two form classes. As an application, we describe Gal( K N ab / K ) in terms of these extended form class groups for which K N ab is the maximal abelian extension of K unramified outside prime ideals dividing $N{\cal O}_K$ .