On the $\Gamma$-equivalence of binary quadratic forms

Abstract

For a congruence subgroup $\Gamma$, we define the notion of $\Gamma$-equivalence on binary quadratic forms which is the same as proper equivalence if $\Gamma ={\mathrm{SL}}_2(\mathbb{Z})$. We develop a theory on $\Gamma$-equivalence such as the finiteness of $\Gamma$-reduced forms, the isomorphism between ${\Gamma }_0(N)$-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.

submitted for publication