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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.
We show that the Taylor coefficients of the series ${bf q}(z)=zexp({bf G}(z)/{bf F}(z))$ are integers, where ${bf F}(z)$ and ${bf G}(z)+log(z) {bf F}(z)$ are specific solutions of certain hypergeometric differential equations with maximal unipotent m
We prove general Dwork-type congruences for Hasse--Witt matrices attached to tuples of Laurent polynomials. We apply this result to establishing arithmetic and $p$-adic analytic properties of functions originating from polynomial solutions modulo $p^
We prove general Dwork-type congruences for constant terms attached to tuples of Laurent polynomials. We apply this result to establishing arithmetic and $p$-adic analytic properties of functions originating from polynomial solutions modulo $p^s$ of
We propose higher-order generalizations of Jacobsthals $p$-adic approximation for binomial coefficients. Our results imply explicit formulae for linear combinations of binomial coefficients $binom{ip}{p}$ ($i=1,2,dots$) that are divisible by arbitrarily large powers of prime $p$.
The explicit formulas of operations, in particular addition and multiplication, of $p $-adic integers are presented. As applications of the results, at first the explicit formulas of operations of Witt vectors with coefficients in $mathbb{F}_{2}$ are