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

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

2008-04-18

arxiv.org/abs/0804.3049v3

in: "Th\'eories galoisiennes et arithm\'etiques des \'equations diff\'erentielles," L. Di Vizio and T. Rivoal (eds.), S\'emina

Mathematics

Number Theory

28 pages, AmS-LaTeX; title changed; the original manuscript was restructured

Scientific paper

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.

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