{
  "id": "1606.06535",
  "version": "v1",
  "published": "2016-06-21T12:28:47.000Z",
  "updated": "2016-06-21T12:28:47.000Z",
  "title": "On lifting and modularity of reducible residual Galois representations over imaginary quadratic fields",
  "authors": [
    "Tobias Berger",
    "Krzysztof Klosin"
  ],
  "comment": "29 pages, this is a pre-copyedited, author-produced PDF of an article published in Int. Math. Res. Not. following peer review. The version of record is available online at: http://imrn.oxfordjournals.org/content/2015/20/10525",
  "journal": "Int. Math. Res. Not. Volume 2015, Issue 20, 10525-10562",
  "doi": "10.1093/imrn/rnu266",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper we study deformations of mod $p$ Galois representations $\\tau$ (over an imaginary quadratic field $F$) of dimension $2$ whose semi-simplification is the direct sum of two characters $\\tau_1$ and $\\tau_2$. As opposed to our previous work we do not impose any restrictions on the dimension of the crystalline Selmer group $H^1_{\\Sigma}(F, {\\rm Hom}(\\tau_2, \\tau_1)) \\subset {\\rm Ext}^1(\\tau_2, \\tau_1)$. We establish that there exists a basis $\\mathcal{B}$ of $H^1_{\\Sigma}(F, {\\rm Hom}(\\tau_2, \\tau_1))$ arising from automorphic representations over $F$ (Theorem 8.1). Assuming among other things that the elements of $\\mathcal{B}$ admit only finitely many crystalline characteristic 0 deformations we prove a modularity lifting theorem asserting that if $\\tau$ itself is modular then so is its every crystalline characteristic zero deformation (Theorems 8.2 and 8.5).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-21T12:28:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F80",
      "11F55"
    ],
    "keywords": [
      "imaginary quadratic field",
      "reducible residual galois representations",
      "modularity",
      "crystalline characteristic zero deformation",
      "crystalline selmer group"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}