On some extensions of Gauss’ work and applications

Let K be an imaginary quadratic field of discriminant dK{d}_{K} with ring of integers OK{{\mathcal{O}}}_{K}, and let τK{\tau }_{K} be an element of the complex upper half plane so that OK=[τK,1]{{\mathcal{O}}}_{K}={[}{\tau }_{K},1]. For a positive integer N, let QN(dK){{\mathcal{Q}}}_{N}({d}_{K}) be the set of primitive positive definite binary quadratic forms of discriminant dK{d}_{K} with leading coefficients relatively prime to N. Then, with any congruence subgroup Γ{\mathrm{\Gamma}} of SL2(Z){\text{SL}}_{2}({\mathbb{Z}}) one can define an equivalence relation ∼Γ{\sim }_{{\mathrm{\Gamma}}} on QN(dK){{\mathcal{Q}}}_{N}({d}_{K}). Let ℱΓ,ℚ{ {\mathcal F} }_{{\mathrm{\Gamma}},{\mathbb{Q}}} denote the field of meromorphic modular functions for Γ{\mathrm{\Gamma}} with rational Fourier coefficients. We show that the set of equivalence classes QN(dK)/∼Γ{{\mathcal{Q}}}_{N}({d}_{K})/{\sim }_{{\mathrm{\Gamma}}} can be equipped with a group structure isomorphic to Gal(KℱΓ,ℚ(τK)/K)\text{Gal}(K{ {\mathcal F} }_{{\mathrm{\Gamma}},{\mathbb{Q}}}({\tau }_{K})/K) for some Γ{\mathrm{\Gamma}}, which generalizes the classical theory of form class groups.