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Register a new userGhosts and congruences for $p^s$-appoximations of hypergeometric periods
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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 hypergeometric and KZ equations, solutions which come as coefficients of master polynomials and whose coefficients are integers. As an application we show that the simplest example of a $p$-adic KZ connection has an invariant line subbundle while its complex analog has no nontrivial subbundles due to the irreducibility of the monodromy group.
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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 $p^s$ of Knizhnik--Zamolodchikov (KZ) equations, solutions which come as coefficients of master polynomials and whose coefficients are integers. As an application we show that the $p$-adic KZ connection associated with the family of hyperelliptic curves $y^2=(t-z_1)dots (t-z_{2g+1})$ has an invariant subbundle of rank $g$. Notice that the corresponding complex KZ connection has no nontrivial subbundles due to the irreducibility of its monodromy representation.
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 provide several new $q$-congruences for truncated basic hypergeometric series, mostly of arbitrary order. Our results include congruences modulo the square or the cube of a cyclotomic polynomial, and in some instances, parametric generalizations thereof. These are established by a variety of techniques including polynomial argument, creative microscoping (a method recently introduced by the first author in collaboration with Zudilin), Andrews multiseries generalization of the Watson transformation, and induction. We also give a number of related conjectures including congruences modulo the fourth power of a cyclotomic polynomial.
We provide several new $q$-congruences for truncated basic hypergeometric series with the base being an even power of $q$. Our results mainly concern congruences modulo the square or the cube of a cyclotomic polynomial and complement corresponding ones of an earlier paper containing $q$-congruences for truncated basic hypergeometric series with the base being an odd power of $q$. We also give a number of related conjectures including $q$-congruences modulo the fifth power of a cyclotomic polynomial and a congruence for a truncated ordinary hypergeometric series modulo the seventh power of a prime greater than 3.
The sequence $A(n)_{n geq 0}$ of Apery numbers can be interpolated to $mathbb{C}$ by an entire function. We give a formula for the Taylor coefficients of this function, centered at the origin, as a $mathbb{Z}$-linear combination of multiple zeta values. We then show that for integers $n$ whose base-$p$ digits belong to a certain set, $A(n)$ satisfies a Lucas congruence modulo $p^2$.
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