{
  "id": "1712.04140",
  "version": "v1",
  "published": "2017-12-12T06:26:51.000Z",
  "updated": "2017-12-12T06:26:51.000Z",
  "title": "Binary quadratic forms and ray class groups",
  "authors": [
    "Ick Sun Eum",
    "Ja Kyung Koo",
    "Dong Hwa Shin"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $K$ be an imaginary quadratic field different from $\\mathbb{Q}(\\sqrt{-1})$ and $\\mathbb{Q}(\\sqrt{-3})$. For a positive integer $N$, let $K_\\mathfrak{n}$ be the ray class field of $K$ modulo $\\mathfrak{n}=N\\mathcal{O}_K$. By using the congruence subgroup $\\pm\\Gamma_1(N)$, we construct an extended form class group whose operation is basically the Dirichlet composition, and explicitly show that this group is isomorphic to the Galois group $\\mathrm{Gal}(K_\\mathfrak{n}/K)$. We also present algorithms to find all form classes and show how to multiply two form classes. As an application, we describe $\\mathrm{Gal}(K_\\mathfrak{n}^\\mathrm{ab}/K)$ in terms of these extended form class groups for which $K_\\mathfrak{n}^\\mathrm{ab}$ is the maximal abelian extension of $K$ unramified outside prime ideals dividing $\\mathfrak{n}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-12T06:26:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R29",
      "11E16",
      "11G15",
      "11R37"
    ],
    "keywords": [
      "binary quadratic forms",
      "ray class groups",
      "extended form class group",
      "outside prime ideals dividing",
      "ray class field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}