{
  "id": "2210.13045",
  "version": "v1",
  "published": "2022-10-24T08:58:00.000Z",
  "updated": "2022-10-24T08:58:00.000Z",
  "title": "Quadratic forms and Genus Theory : a link with 2-descent and an application to non-trivial specializations of ideal classes",
  "authors": [
    "William Dallaporta"
  ],
  "comment": "27 pages, 1 figure. Submitted",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Genus Theory is a classical feature of integral binary quadratic forms. Using the author's generalization of the well-known correspondence between quadratic form classes and ideal classes of quadratic algebras, we extend it to the case when quadratic forms are twisted and have coefficients in any PID $R$. When ${R = \\mathbb{K}[X]}$, we show that the Genus Theory map is the quadratic form version of the $2$-descent map on a certain hyperelliptic curve. As an application, we make a contribution to a question of Agboola and Pappas regarding a specialization problem of divisor classes on hyperelliptic curves. Under suitable assumptions, we prove that the set of non-trivial specializations has density $1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-24T08:58:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E16",
      "14H25",
      "14H40",
      "11R45"
    ],
    "keywords": [
      "non-trivial specializations",
      "ideal classes",
      "application",
      "hyperelliptic curve",
      "integral binary quadratic forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}