{
  "id": "2412.14693",
  "version": "v1",
  "published": "2024-12-19T09:53:09.000Z",
  "updated": "2024-12-19T09:53:09.000Z",
  "title": "Rational points in a family of conics over $\\mathbb{F}_2(t)$",
  "authors": [
    "Daniel Loughran",
    "Judith Ortmann"
  ],
  "comment": "18 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Serre famously showed that almost all plane conics over $\\mathbb{Q}$ have no rational point. We investigate versions of this over global function fields, focusing on a specific family of conics over $\\mathbb{F}_2(t)$ which illustrates new behaviour. We obtain an asymptotic formula using harmonic analysis, which requires a new Tauberian theorem over function fields for Dirichlet series with branch point singularities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-19T09:53:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G05",
      "14F22"
    ],
    "keywords": [
      "rational point",
      "global function fields",
      "branch point singularities",
      "plane conics",
      "asymptotic formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}