Binary quadratic forms and ray class groups

Abstract

AbstractLetKbe an imaginary quadratic field different from$\open{Q}(\sqrt {-1})$and$\open{Q}(\sqrt {-3})$. For a positive integerN, letKNbe the ray class field ofKmodulo$N {\cal O}_K$. By using the congruence subgroup ± Γ1(N) of SL2(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(KN/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(KNab/K) in terms of these extended form class groups for whichKNabis the maximal abelian extension ofKunramified outside prime ideals dividing$N{\cal O}_K$.