{
  "id": "1711.00230",
  "version": "v1",
  "published": "2017-11-01T07:22:41.000Z",
  "updated": "2017-11-01T07:22:41.000Z",
  "title": "On the $Γ$-equivalence of binary quadratic forms",
  "authors": [
    "Bumkyu Cho"
  ],
  "comment": "submitted for publication",
  "categories": [
    "math.NT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-01T07:22:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E16",
      "11E25",
      "11R29"
    ],
    "keywords": [
      "binary quadratic forms",
      "ideal class group",
      "form class group",
      "representation",
      "proper equivalence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}