{
  "id": "1909.09622",
  "version": "v1",
  "published": "2019-09-20T17:14:00.000Z",
  "updated": "2019-09-20T17:14:00.000Z",
  "title": "On the distribution of the Eichler-Shimura map and the zeroes of period polynomials",
  "authors": [
    "Asbjorn Christian Nordentoft"
  ],
  "comment": "16 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper we determine the limiting distribution of the Eichler-Shimura map or equivalently the limiting joint distribution of the coefficients of the period polynomials associated to a fixed cusp form (as the element of the group varies). The limiting distribution is shown to be a product of two independent distributions, one of which is connected to the additive twist of the cuspidal $L$-function. Furthermore we determine the asymptotic behavior of the zeroes of the period polynomials in the same limit. We use the method of moments and the main ingredients in the proofs are additive twists of $L$-functions and bounds for both individual and sums of Kloosterman sums.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-20T17:14:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "period polynomials",
      "eichler-shimura map",
      "limiting distribution",
      "additive twist",
      "main ingredients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}