{
  "id": "1208.3946",
  "version": "v3",
  "published": "2012-08-20T08:50:11.000Z",
  "updated": "2014-09-25T09:00:16.000Z",
  "title": "Automorphy of Symm^5(GL(2)) and base change",
  "authors": [
    "Luis V. Dieulefait"
  ],
  "comment": "55 pages. Appendices A and B not included (cf. ArXiv preprints 1208.4128, 1209.5105, respect.). We have improved the exposition following the referee's suggestions. Some diagrams showing the congruences involved have been added at some steps for the reader's convenience. The proof of base change in Section 5 has been simplified",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove that for any Hecke eigenform f of level 1 and arbitrary weight there is a self-dual cuspidal automorphic form $\\pi$ of $GL_6(\\Q)$ corresponding to $\\Symm^5 (f)$, i.e., such that the system of Galois representations attached to $\\pi$ agrees with the 5-th symmetric power of the one attached to f. We also improve the base change result that we obtained in a previous work: for any newform f, and any totally real number field F (no extra assumptions on f or F), we prove the existence of base change relative to the extension $F/\\Q$. Finally, we combine the previous results to deduce that base change also holds for $\\Symm^5(f)$: for any Hecke eigenform f of level 1 and any totally real number field F, the automorphic form corresponding to $\\Symm^5 (f)$ can be base changed to F.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-09-23T19:27:28.000Z",
      "abstract": "We prove that for any Hecke eigenform f of level 1 and arbitrary weight there is a self-dual cuspidal automorphic form $\\pi$ of $GL_6(\\Q)$ corresponding to $\\Symm^5 (f)$, i.e., such that the system of Galois representations attached to $\\pi$ agrees with the 5-th symmetric power of the one attached to f. Assuming a slight strengthening of the Automorphic Lifting Theorems that we use in the proof of this result, we note that the same conclusion applies also to any newform without CM of level prime to 30. We also improve the base change result that we obtained in a previous work: for any newform f, and any totally real number field F (no extra assumptions on f or F), we prove the existence of base change relative to the extension $F/\\Q$. Finally, we combine the previous results to deduce that base change also holds for $\\Symm^5(f)$: for any Hecke eigenform f of level 1 and any totally real number field F, the automorphic form corresponding to $\\Symm^5 (f)$ can be base changed to F (again, this can be conditionally extended to non-CM newforms of level prime to 30).",
      "comment": "a new Appendix has been incorporated: \"Automorphy lifting for small $\\ell$\", written jointly with T. Gee. This is referred to as Appendix B, and will be uploaded separately",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-09-25T09:00:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "totally real number field",
      "self-dual cuspidal automorphic form",
      "automorphy",
      "hecke eigenform",
      "level prime"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 55,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.3946D"
    }
  }
}