{
  "id": "1209.6293",
  "version": "v2",
  "published": "2012-09-27T17:27:17.000Z",
  "updated": "2017-07-16T01:20:42.000Z",
  "title": "Modularity Lifting Theorems beyond the Taylor-Wiles Method. II",
  "authors": [
    "Frank Calegari",
    "David Geraghty"
  ],
  "comment": "This contents of this paper have been entirely merged with part I (arXiv:1207.4224), and so this paper is being withdrawn. The results remain unchanged, but one should reference the merged version of the paper which is the one to appear",
  "categories": [
    "math.NT"
  ],
  "abstract": "In a previous paper [CG], we showed how one could generalize Taylor-Wiles modularity lifting theorems [Wil95, TW95] to contexts beyond those in which the automorphic forms in question arose from the middle degree cohomology of Shimura varieties, in particular, to contexts in which the relevant automorphic forms contributed to cohomology in exactly two degrees. In this sequel, we extend our method to the general case in which Galois representations are expected to occur in cohomology, contingent on the (as yet unproven) existence of certain Galois representations with the expected properties. As an application, we prove the following result (conditional on the conjectures mentioned above): If E is an elliptic curve over an arbitrary number field, then E is potentially modular, and the Sato-Tate conjecture holds for E.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-09-27T17:27:17.000Z",
      "abstract": "In a previous paper [CG], we showed how one could generalize Taylor-Wiles modularity lifting theorems [Wil95, TW95] to contexts beyond those in which the automorphic forms in question arose from the middle degree cohomology of Shimura varieties; in particular, to contexts in which the relevant automorphic forms contributed to cohomology in exactly two degrees. In this sequel, we extend our method to the general case in which Galois representations are expected to occur in cohomology, contingent on the (as yet unproven) existence of certain Galois representations with the expected properties. As an application, we prove the following result (conditional on the conjectures mentioned above): If E is an elliptic curve over an arbitrary number field, then E is potentially modular, and the Sato-Tate conjecture holds for E.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2017-07-16T01:20:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "taylor-wiles method",
      "galois representations",
      "arbitrary number field",
      "middle degree cohomology",
      "generalize taylor-wiles modularity lifting theorems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.6293C"
    }
  }
}