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

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.