{
  "id": "2202.09219",
  "version": "v1",
  "published": "2022-02-18T14:37:37.000Z",
  "updated": "2022-02-18T14:37:37.000Z",
  "title": "$\\mathbb{Q}$-curves and the Lebesgue-Nagell equation",
  "authors": [
    "Michael A. Bennett",
    "Philippe Michaud-Jacobs",
    "Samir Siksek"
  ],
  "comment": "13 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, we consider the equation \\[ x^2 - q^{2k+1} = y^n, \\qquad q \\nmid x, \\quad 2 \\mid y, \\] for integers $x,q,k,y$ and $n$, with $k \\geq 0$ and $n \\geq 3$. We extend work of the first and third-named authors by finding all solutions in the cases $q= 41$ and $q = 97$. We do this by constructing a Frey-Hellegouarch $\\mathbb{Q}$-curve defined over the real quadratic field $K=\\mathbb{Q}(\\sqrt{q})$, and using the modular method with multi-Frey techniques.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-18T14:37:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D41",
      "11D61",
      "11F80",
      "11G05"
    ],
    "keywords": [
      "lebesgue-nagell equation",
      "real quadratic field",
      "extend work",
      "modular method",
      "multi-frey techniques"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}