This material has been published in "Théories galoisiennes et arithmétiques des équations différentielles," L. Di Vizio and T. Rivoal (eds.), Séminaires et Congrès, Soc. Math. France, vol. 27, Paris, 2011, pp. 279-307, the only definitive repository of the content that has been certified and accepted after peer review. Copyright and all rights therein are retained by the Société Mathématiques de France. This material may not be copied or reposted without explicit permission.

Christian Krattenthaler and Tanguy Rivoal

Multivariate p-adic formal congruences and integrality of Taylor coefficients of mirror maps

(27 pages)

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 z=(z1,z2,...,zd), we show that the Taylor coefficients of the multi-variable series q(z)=ziexp(G(z)/F(z)) are integers, where F(z) and G(z)+log(zi) F(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" 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.

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