{
  "id": "0710.4677",
  "version": "v1",
  "published": "2007-10-25T08:58:50.000Z",
  "updated": "2007-10-25T08:58:50.000Z",
  "title": "Congruences between modular forms and related modules",
  "authors": [
    "Miriam Ciavarella"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We fix $\\ell$ a prime and let $M$ be an integer such that $\\ell\\not|M$; let $f\\in S_2(\\Gamma_1(M\\ell^2))$ be a newform supercuspidal of fixed type related to the nebentypus, at $\\ell$ and special at a finite set of primes. Let $\\TT^\\psi$ be the local quaternionic Hecke algebra associated to $f$. The algebra $\\TT^\\psi$ acts on a module $\\mathcal M^\\psi_f$ coming from the cohomology of a Shimura curve. Applying the Taylor-Wiles criterion and a recent Savitt's theorem, $\\TT^\\psi$ is the universal deformation ring of a global Galois deformation problem associated to $\\orho_f$. Moreover $\\mathcal M^\\psi_f$ is free of rank 2 over $\\TT^\\psi$. If $f$ occurs at minimal level, by a generalization of a Conrad, Diamond and Taylor's result and by the classical Ihara's lemma, we prove a theorem of raising the level and a result about congruence ideals. The extension of this results to the non minimal case is an open problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-10-25T08:58:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F80"
    ],
    "keywords": [
      "modular forms",
      "related modules",
      "congruence",
      "global galois deformation problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0710.4677C"
    }
  }
}