Dwork crystals and related congruences

In the talk I will describe a realization of the p-adic cohomology of an affine toric hypersurface which originates in Dwork's work and give an explicit description of the unit-root subcrystal based on certain congruences for the coeficient of powers of a Laurent polynomial. This is joint work with Frits Beukers. The formulas for the Frobenius and connection on the unit-root subcrystal that we give were conjectured in my preprint ``Higher Hasse--Witt matrices'' (arXiv:1605.06440).
A few days ago a different proof was given by An Huang, Bong Lian, Shing-Tung Yau and Chenglong Yu in their preprint ''Hasse--Witt matrices, unit roots and period integrals'' (arXiv:1801.01189). I will comment on the difference of the two approaches.