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

TL;DR

This paper introduces a generalized notion of equivalence for binary quadratic forms relative to congruence subgroups, extending classical theory and exploring applications to integer representations with congruence conditions.

Contribution

It develops a comprehensive theory of $\Gamma$-equivalence for binary quadratic forms, including finiteness, class group isomorphisms, and representation results under congruence conditions.

Findings

Finiteness of $\Gamma$-reduced forms established

Isomorphism between $\Gamma_0(N)$-class group and ideal class group shown

Representation of integers with congruence conditions analyzed

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.