{
  "id": "math/0608372",
  "version": "v2",
  "published": "2006-08-15T04:42:01.000Z",
  "updated": "2007-07-10T08:47:30.000Z",
  "title": "Period polynomials and explicit formulas for Hecke operators on Γ_0(2)",
  "authors": [
    "Shinji Fukuhara",
    "Yifan Yang"
  ],
  "comment": "AMS-LaTeX, 30 pages, final version, to appear on the Mathematical Proceedings of the Cambridge Philosophical Society",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let S_{w+2}(\\Gamma_0(N)) be the vector space of cusp forms of weight w+2 on the congruence subgroup \\Gamma_0(N). We first determine explicit formulas for period polynomials of elements in S_{w+2}(\\Gamma_0(N)) by means of Bernoulli polynomials. When N=2, from these explicit formulas we obtain new bases for S_{w+2}(\\Gamma_0(2)), and extend the Eichler-Shimura-Manin isomorphism theorem to \\Gamma_0(2). This implies that there are natural correspondences between the spaces of cusp forms on \\Gamma_0(2) and the spaces of period polynomials. Based on these results, we will find explicit form of Hecke operators on S_{w+2}(\\Gamma_0(2)). As an application of our main theorems, we will also give an affirmative answer to a speculation of Imamo\\=glu and Kohnen on a basis of S_{w+2}(\\Gamma_0(2)).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-07-10T08:47:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F25",
      "11F11",
      "11F67"
    ],
    "keywords": [
      "period polynomials",
      "hecke operators",
      "cusp forms",
      "first determine explicit formulas",
      "eichler-shimura-manin isomorphism theorem"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......8372F"
    }
  }
}