On the $Γ$-equivalence of binary quadratic forms

Abstract

For a congruence subgroup $Γ$, we define the notion of $Γ$-equivalence on binary quadratic forms which is the same as proper equivalence if $Γ= \mathrm{SL}_2(\mathbb Z)$. We develop a theory on $Γ$-equivalence such as the finiteness of $Γ$-reduced forms, the isomorphism between $Γ_0(N)$-form class group and the ideal class group, $N$-representation of integers, and $N$-genus of binary quadratic forms. As an application, we deal with representations of integers by binary quadratic forms under certain congruence condition on variables.