{
  "id": "1311.1460",
  "version": "v3",
  "published": "2013-11-06T17:49:30.000Z",
  "updated": "2014-11-14T22:28:33.000Z",
  "title": "On spaces of modular forms spanned by eta-quotients",
  "authors": [
    "Jeremy Rouse",
    "John J. Webb"
  ],
  "comment": "23 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "An eta-quotient of level $N$ is a modular form of the shape $f(z) = \\prod_{\\delta | N} \\eta(\\delta z)^{r_{\\delta}}$. We study the problem of determining levels $N$ for which the graded ring of holomorphic modular forms for $\\Gamma_{0}(N)$ is generated by (holomorphic, respectively weakly holomorphic) eta-quotients of level $N$. In addition, we prove that if $f(z)$ is a holomorphic modular form that is non-vanishing on the upper half plane and has integer Fourier coefficients at infinity, then $f(z)$ is an integer multiple of an eta-quotient. Finally, we use our results to determine the structure of the cuspidal subgroup of $J_{0}(2^{k})(\\mathbb{Q})$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-11-23T01:29:27.000Z",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-11-14T22:28:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F20",
      "11F30",
      "11G18"
    ],
    "keywords": [
      "modular forms",
      "eta-quotient",
      "holomorphic modular form",
      "integer fourier coefficients",
      "upper half plane"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.1460R"
    }
  }
}