{
  "id": "1506.02860",
  "version": "v1",
  "published": "2015-06-09T10:56:49.000Z",
  "updated": "2015-06-09T10:56:49.000Z",
  "title": "On the generalized Fermat equation $x^{2\\ell}+y^{2m}=z^p$ for $3 \\le p \\le 13$",
  "authors": [
    "Samuele Anni",
    "Samir Siksek"
  ],
  "comment": "20 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We show that the generalized Fermat equation $x^{2\\ell}+y^{2 m}=z^p$ has no non-trivial primitive solutions for primes $\\ell$, $m \\ge 5$, and $3 \\le p \\le 13$. This is achieved by relating a putative solution to a Frey curve over a real subfield of the $p$-th cyclotomic field, and studying its mod $\\ell$ representation using modularity and level lowering. Along the way we prove the following modularity theorem. Let $K$ be a real abelian field of odd class number in which $5$ is unramified. Let $S_5$ be the set of places of $K$ above $5$. Suppose for every non-empty proper subset $S \\subset S_5$ there is a totally positive unit $u \\in \\mathcal{O}_K$ such that $\\prod_{\\mathfrak{q} \\in S} \\mathrm{Norm}_{\\mathbb{F}_\\mathfrak{q}/\\mathbb{F}_5}(u \\bmod{\\mathfrak{q}}) \\ne \\overline{1}$. Then every semistable elliptic curve over $K$ is modular.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-09T10:56:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D41",
      "11F80",
      "11G05",
      "11F41"
    ],
    "keywords": [
      "generalized fermat equation",
      "non-empty proper subset",
      "odd class number",
      "th cyclotomic field",
      "real abelian field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150602860A"
    }
  }
}