{
  "id": "0709.1432",
  "version": "v3",
  "published": "2007-09-10T16:16:12.000Z",
  "updated": "2009-07-15T13:52:49.000Z",
  "title": "On the integrality of the Taylor coefficients of mirror maps",
  "authors": [
    "Christian Krattenthaler",
    "Tanguy Rivoal"
  ],
  "comment": "AmS-LaTeX; 54 pages. This paper was cut in two separate papers, arXiv:0907.2577 and arXiv:0907.2578, and is thereby superseded by these two",
  "categories": [
    "math.NT",
    "hep-th",
    "math.AG"
  ],
  "abstract": "We show that the Taylor coefficients of the series ${\\bf q}(z)=z\\exp({\\bf G}(z)/{\\bf F}(z))$ are integers, 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 $z=0$. We also address the question of finding the largest integer $u$ such that the Taylor coefficients of $(z ^{-1}{\\bf q}(z))^{1/u}$ are still integers. As consequences, we are able to prove numerous integrality results for the Taylor coefficients of mirror maps of Calabi-Yau complete intersections in weighted projective spaces, which improve and refine previous results by Lian and Yau, and by Zudilin. In particular, we prove the general ``integrality'' conjecture of Zudilin about these mirror maps. A further outcome of the present study is the determination of 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 $q(z)=\\exp(F(z)/G(z))$, $F(z)=\\sum_{m=0} ^{\\infty} (Nm)! z^m/m!^N$ 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": "v3",
      "updated": "2009-07-15T13:52:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11S80",
      "11J99",
      "14J32",
      "33C20"
    ],
    "keywords": [
      "taylor coefficients",
      "mirror maps",
      "integrality",
      "largest integer",
      "maximal unipotent monodromy"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 54,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0709.1432K"
    }
  }
}