{
  "id": "2010.06297",
  "version": "v1",
  "published": "2020-10-13T11:26:37.000Z",
  "updated": "2020-10-13T11:26:37.000Z",
  "title": "Arithmetic properties of Fourier coefficients of meromorphic modular forms",
  "authors": [
    "Steffen Löbrich",
    "Markus Schwagenscheidt"
  ],
  "comment": "20 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We investigate integrality and divisibility properties of Fourier coefficients of meromorphic modular forms of weight $2k$ associated to positive definite integral binary quadratic forms. For example, we show that if there are no non-trivial cusp forms of weight $2k$, then the $n$-th coefficients of these meromorphic modular forms are divisible by $n^{k-1}$ for every natural number $n$. Moreover, we prove that their coefficients are non-vanishing and have either constant or alternating signs. Finally, we obtain a relation between the Fourier coefficients of meromorphic modular forms, the coefficients of the $j$-function, and the partition function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-13T11:26:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F30",
      "11F33",
      "11F37",
      "11F25",
      "11F27"
    ],
    "keywords": [
      "meromorphic modular forms",
      "fourier coefficients",
      "arithmetic properties",
      "definite integral binary quadratic forms",
      "positive definite integral binary quadratic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}