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

We generalise Dworks 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 $q(mathbf z)=z_iexp(G(mathbf z)/F(mathbf z))$ 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.