{
  "id": "1609.04458",
  "version": "v1",
  "published": "2016-09-14T22:13:53.000Z",
  "updated": "2016-09-14T22:13:53.000Z",
  "title": "On the asymptotic Fermat's Last Theorem over number fields",
  "authors": [
    "Mehment Haluk Şengün",
    "Samir Siksek"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Assuming two deep but standard conjectures from the Langlands Programme, we prove that the asymptotic Fermat's Last Theorem holds for imaginary quadratic fields $\\mathbb{Q}(\\sqrt{-d})$ with $-d \\equiv 2$, $3 \\pmod{4}$. For a general number field $K$, again assuming standard conjectures, we give a criterion based on the solutions to a certain $S$-unit equation, which if satisfied implies the asymptotic Fermat's Last Theorem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-14T22:13:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D41"
    ],
    "keywords": [
      "asymptotic fermats",
      "general number field",
      "imaginary quadratic fields",
      "langlands programme",
      "theorem holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}