{
  "id": "1306.5662",
  "version": "v3",
  "published": "2013-06-24T16:04:08.000Z",
  "updated": "2014-07-13T23:59:40.000Z",
  "title": "Modular-type functions attached to Calabi-Yau varieties: integrality properties",
  "authors": [
    "Hossein Movasati",
    "Khosro Monsef Shokri"
  ],
  "comment": "This paper will appear as an appendix in a monograph of the first author",
  "categories": [
    "math.NT",
    "hep-th",
    "math.GR"
  ],
  "abstract": "We study the integrality properties of the coefficients of the mirror map attached to the generalized hypergeometric function $_{n}F_{n-1}$ with rational parameters and with a maximal unipotent monodromy. We present a conjecture on the $p$-integrality of the mirror map which can be verified experimentally. We prove its consequence on the $N$-integrality of the mirror map for the particular cases $1\\leq n\\leq 4$. For $n=2$ we obtain the Takeuchi's classification of arithmetic triangle groups with a cusp, and for $n=4$ we prove that $14$ examples of hypergeometric Calabi-Yau equations are the full classification of hypergeometric mirror maps with integral coefficients. As a by-product we get the integrality of the corresponding algebra of modular-type functions. These are natural generalizations of the algebra of classical modular and quasi-modular forms in the case $n=2$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-07-13T23:59:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "integrality properties",
      "modular-type functions",
      "calabi-yau varieties",
      "hypergeometric mirror maps",
      "maximal unipotent monodromy"
    ],
    "tags": [
      "monograph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1239687,
      "adsabs": "2013arXiv1306.5662M"
    }
  }
}