On Dwork's p-adic formal congruences theorem and hypergeometric mirror maps

TL;DR

This paper extends Dwork's p-adic formal congruences theorem to a broader class, enabling new p-adic congruences for hypergeometric series and establishing integrality properties of mirror map coefficients.

Contribution

It generalizes Dwork's theorem and provides explicit formulas for Eisenstein constants, advancing the understanding of hypergeometric series and mirror map integrality.

Findings

Proved a broad generalization of Dwork's p-adic formal congruences theorem.

Derived p-adic congruences for hypergeometric series with rational parameters for all primes.

Established integrality of Taylor coefficients of mirror maps in a new, comprehensive manner.

Abstract

Using Dwork's theory, we prove a broad generalisation of his famous p-adic formal congruences theorem. This enables us to prove certain p-adic congruences for the generalized hypergeometric series with rational parameters; in particular, they hold for any prime number p and not only for almost all primes. Along the way, using Christol's functions, we provide an explicit formula for the "Eisenstein constant" of any globally bounded hypergeometric series with rational parameters. As an application of these results, we obtain an arithmetic statement of a new type concerning the integrality of Taylor coefficients of the associated mirror maps. It essentially contains all the similar univariate integrality results in the litterature.