{
  "id": "0907.2578",
  "version": "v2",
  "published": "2009-07-15T13:51:14.000Z",
  "updated": "2009-11-17T10:08:52.000Z",
  "title": "On the integrality of the Taylor coefficients of mirror maps, II",
  "authors": [
    "Christian Krattenthaler",
    "Tanguy Rivoal"
  ],
  "comment": "27 pages, AmS-LaTeX. This is the second part of an originally larger paper (arXiv:0709.1432) of the same title. The first part is arXiv:0907.2577. This final version is to appear in Commun. Number Theory Phys",
  "journal": "Commun. Number Theory Phys. 3 (2009), 555-591",
  "categories": [
    "math.NT",
    "hep-th",
    "math.AG"
  ],
  "abstract": "We continue our study begun in \"On the integrality of the Taylor coefficients of mirror maps\" (arXiv:0907.2577) of the fine integrality properties of the Taylor coefficients of the series ${\\bf q}(z)=z\\exp({\\bf G}(z)/{\\bf F}(z))$, where ${\\bf F}(z)$ and ${\\bf G}(z)+\\log(z) {\\bf F}(z)$ are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at $z=0$. More precisely, we address the question of finding the largest integer $v$ such that the Taylor coefficients of $(z ^{-1}{\\bf q}(z))^{1/v}$ are still integers. In particular, we determine the Dwork-Kontsevich sequence $(u_N)_{N\\ge1}$, where $u_N$ is the largest integer such that $q(z)^{1/u_N}$ is a series with integer coefficients, where $q(z)=\\exp(G(z)/F(z))$, $F(z)=\\sum_{m=0}^{\\infty} (Nm)! z^m/m!^N$ and $G(z)=\\sum_{m=1}^{\\infty} (H_{Nm}-H_m)(Nm)! z^m/m!^N$, with $H_n$ denoting the $n$-th harmonic number, conditional on the conjecture that there are no prime number $p$ and integer $N$ such that the $p$-adic valuation of $H_N-1$ is strictly greater than 3.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-11-17T10:08:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11S80",
      "11J99",
      "14J32",
      "33C20"
    ],
    "keywords": [
      "taylor coefficients",
      "mirror maps",
      "largest integer",
      "th harmonic number",
      "maximal unipotent monodromy"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.4310/CNTP.2009.v3.n3.a5"
    },
    "note": {
      "typesetting": "LaTeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 825839,
      "adsabs": "2009arXiv0907.2578K"
    }
  }
}