ON DWORK’S p-ADIC FORMAL CONGRUENCES THEOREM AND HYPERGEOMETRIC MIRROR MAPS

Abstract

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 literature. 1