MULTIVARIATE p-ADIC FORMAL CONGRUENCES AND INTEGRALITY OF TAYLOR COEFFICIENTS OF MIRROR MAPS

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) = zi exp(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” [arχiv: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.