I will report on my work in progress done jointly with Frits Beukers. In 1990s V. Batyrev used Dwork modules to describe the mixed Hodge structure on the middle cohomology of affine hypersurfaces in algebraic tori. Dwork modules are crystals where the Frobenius map and connection can be described explicitly. We use these crystals to show several p-adic congruences for the coefficients of powers of a Laurent polynomial. The congruence mentioned in the title involves the L-function of the toric exponential sums and yields p-adic unit-root formulas.