{
  "id": "1307.3162",
  "version": "v3",
  "published": "2013-07-11T16:22:53.000Z",
  "updated": "2014-07-16T13:27:41.000Z",
  "title": "The Asymptotic Fermat's Last Theorem for Five-Sixths of Real Quadratic Fields",
  "authors": [
    "Nuno Freitas",
    "Samir Siksek"
  ],
  "comment": "20 pages. New title. The paper is rewritten and reorganized (a second time). The proofs are substantially shorter and more efficient",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $K$ be a totally real field. By the asymptotic Fermat's Last Theorem over $K$ we mean the statement that there is a constant $B_K$ such that for prime exponents $p>B_K$ the only solutions to the Fermat equation $a^p + b^p + c^p = 0$ with $a$, $b$, $c$ in $K$ are the trivial ones satisfying $abc = 0$. With the help of modularity, level lowering and image of inertia comparisons we give an algorithmically testable criterion which if satisfied by $K$ implies the asymptotic Fermat's Last Theorem over $K$. Using techniques from analytic number theory, we show that our criterion is satisfied by $K = \\mathbb{Q}(\\sqrt{d})$ for a subset of $d$ having density $5/6$ among the squarefree positive integers. We can improve this to density 1 if we assume a standard \"Eichler-Shimura\" conjecture.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-07-16T13:27:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D41"
    ],
    "keywords": [
      "asymptotic fermats",
      "real quadratic fields",
      "five-sixths",
      "analytic number theory",
      "fermat equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.3162F"
    }
  }
}