{
  "id": "1906.06982",
  "version": "v1",
  "published": "2019-06-17T12:15:56.000Z",
  "updated": "2019-06-17T12:15:56.000Z",
  "title": "Central limit theorems for elliptic curves and modular forms with smooth weight functions",
  "authors": [
    "Stephan Baier",
    "Neha Prabhu",
    "Kaneenika Sinha"
  ],
  "comment": "24 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "The second and third-named authors (arXiv:1705.04115) established a Central Limit Theorem for the error term in the Sato-Tate law for families of modular forms. This method was adapted to families of elliptic curves in by the first and second-named authors (arXiv:1705.09229). In this context, a Central Limit Theorem was established only under a strong hypothesis going beyond the Riemann Hypothesis. In the present paper, we consider a smoothed version of the Sato-Tate conjecture, which allows us to overcome several limitations. In particular, for the smoothed version, we are able to establish a Central Limit Theorem for much smaller families of modular forms, and we succeed in proving a theorem of this type for families of elliptic curves under the Riemann Hypothesis for $L$-functions associated to Hecke eigenforms for the full modular group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-17T12:15:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F11",
      "11F25",
      "11F41",
      "11G05",
      "11G40"
    ],
    "keywords": [
      "central limit theorem",
      "elliptic curves",
      "modular forms",
      "smooth weight functions",
      "riemann hypothesis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}