{
  "abstract": "Let $D$ be a negative discriminant, and let $K=\\mathbb{Q}(\\sqrt{D})$. Let $\\mathcal{Q}(D)$ denote the set of primitive positive definite binary quadratic forms over $\\mathbb{Z}$ of discriminant $D$. We introduce the set of adelic framed forms \\begin{equation*} \\widehat{\\mathcal{Q}}(D)= \\left\\{(Q,\\,γ)\\in \\mathcal{Q}(D)\\times\\mathrm{SL}_2(\\widehat{\\mathbb{Z}})~|~ Q\\left(γ\\begin{bmatrix}1\\\\0\\end{bmatrix}\\right)\\in \\widehat{\\mathbb{Z}}^\\times\\right\\} \\end{equation*} and its orbit space $\\widehat{C}(D)$ under the natural action of $\\mathrm{SL}_2(\\mathbb{Z})$. We define an explicit adelic analogue of the Gauss-Dirichlet composition law on $\\widehat{C}(D)$ and endow $\\widehat{C}(D)$ with the quotient topology induced by the subspace topology on $\\widehat{\\mathcal{Q}}(D)$ inherited from the product topology on $\\mathcal{Q}(D)\\times\\mathrm{SL}_2(\\widehat{\\mathbb{Z}})$, where $\\mathcal{Q}(D)$ is discrete and $\\mathrm{SL}_2(\\widehat{\\mathbb{Z}})$ has its profinite topology. We then prove that there is an isomorphism of topological groups \\begin{equation*} \\widehat{C}(D)\\simeq\\mathrm{Gal}\\left(K^\\mathrm{ab}(\\mathfrak{t}^{1/\\infty})/K(\\mathfrak{t})\\right), \\end{equation*} where the Galois group is endowed with the Krull topology, $\\mathfrak{t}$ is a positive transcendental real number, and $\\mathfrak{t}^{1/\\infty}=\\{\\sqrt[N]{\\mathfrak{t}}~|~N\\geq1\\}$. Moreover, we identify an explicitly defined subgroup of $\\widehat{C}(D)$ with $\\mathrm{Gal}(K^\\mathrm{ab}/K)$ and describe the corresponding Galois action on special values of modular functions. In this way, classical Gauss composition, finite-level form class groups, and Shimura reciprocity are brought together within a single adelic framework. Finally, we show that the abstract group structure of $\\widehat{C}(D)$ uniquely determines the imaginary quadratic field $K$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.04873",
  "author_total": 3,
  "authors": [
    {
      "name": "Ja Kyung Koo"
    },
    {
      "name": "Dong Hwa Shin"
    },
    {
      "name": "Dong Sung Yoon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/adsyrqz2cttda.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.04873",
  "fulltext_url": null,
  "id": "adsyrqz2cttda",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.04873",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7196966119",
  "pdf_url": "https://arxiv.org/pdf/2608.04873",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/adsyrqz2cttda.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2608.04873",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.04873"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Adelic framed form class groups and explicit class field theory",
  "title_text": "Adelic framed form class groups and explicit class field theory",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/adsyrqz2cttda",
  "venue": "arXiv",
  "versions": []
}