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arXiv · 1711.00230

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

Bumkyu Cho
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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.

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Bumkyu Cho. 2017-11-01. On the $\Gamma$-equivalence of binary quadratic forms. https://arxiv.org/abs/1711.00230

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