{
  "id": "1311.0577",
  "version": "v3",
  "published": "2013-11-04T03:38:37.000Z",
  "updated": "2014-08-04T02:37:42.000Z",
  "title": "Computations of Galois Representations Associated to Modular Forms",
  "authors": [
    "Peng Tian"
  ],
  "comment": "This paper has been published in Acta Arithmetica",
  "journal": "Acta Arith. 164 (2014), 399-411",
  "doi": "10.4064/aa164-4-5",
  "categories": [
    "math.NT"
  ],
  "abstract": "We propose an improved algorithm for computing mod $\\ell$ Galois representations associated to a cusp form $f$ of level one. The proposed method allows us to explicitly compute the case with $\\ell=29$ and $f$ of weight $k=16$, and the cases with $\\ell=31$ and $f$ of weight $k=12,20, 22$. All the results are rigorously proved to be correct. As an example, we will compute the values modulo $31$ of Ramanujan's tau function at some huge primes up to a sign. Also we will give an improved higher bound on Lehmer's conjecture for Ramanujan's tau function.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-08-04T02:37:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "galois representations",
      "modular forms",
      "ramanujans tau function",
      "computations",
      "values modulo"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.0577T"
    }
  }
}