{
  "id": "0804.3049",
  "version": "v3",
  "published": "2008-04-18T15:57:15.000Z",
  "updated": "2010-04-12T14:46:54.000Z",
  "title": "Multivariate $p$-adic formal congruences and integrality of Taylor coefficients of mirror maps",
  "authors": [
    "Christian Krattenthaler",
    "Tanguy Rivoal"
  ],
  "comment": "28 pages, AmS-LaTeX; title changed; the original manuscript was restructured",
  "journal": "in: \"Th\\'eories galoisiennes et arithm\\'etiques des \\'equations diff\\'erentielles,\" L. Di Vizio and T. Rivoal (eds.), S\\'eminaires et Congr\\`es, Soc. Math. France, vol. 27, Paris, 2011, pp. 279-307",
  "categories": [
    "math.NT",
    "hep-th",
    "math.AG"
  ],
  "abstract": "We generalise Dwork's theory of $p$-adic formal congruences from the univariate to a multi-variate setting. We apply our results to prove integrality assertions on the Taylor coefficients of (multi-variable) mirror maps. More precisely, with $\\mathbf z=(z_1,z_2,...,z_d)$, we show that the Taylor coefficients of the multi-variable series $q(\\mathbf z)=z_i\\exp(G(\\mathbf z)/F(\\mathbf z))$ are integers, where $F(\\mathbf z)$ and $G(\\mathbf z)+\\log(z_i) F(\\mathbf z)$, $i=1,2,...,d$, are specific solutions of certain GKZ systems. This result implies the integrality of the Taylor coefficients of numerous families of multi-variable mirror maps of Calabi-Yau complete intersections in weighted projective spaces, as well as of many one-variable mirror maps in the \"Tables of Calabi-Yau equations\" [arXiv:math/0507430] of Almkvist, van Enckevort, van Straten and Zudilin. In particular, our results prove a conjecture of Batyrev and van Straten in [Comm. Math. Phys. 168 (1995), 493-533] on the integrality of the Taylor coefficients of canonical coordinates for a large family of such coordinates in several variables.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-04-12T14:46:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11S80",
      "11J99",
      "14J32",
      "33C70"
    ],
    "keywords": [
      "taylor coefficients",
      "adic formal congruences",
      "multivariate",
      "van straten",
      "calabi-yau complete intersections"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 851443,
      "adsabs": "2008arXiv0804.3049K"
    }
  }
}