  Mirror symmetry suggests unexpected relationships between arithmetic
properties of distinct families of algebraic varieties. For example, Wan and
others have shown that for some mirror pairs, the number of rational points
over a finite field matches modulo the order of the field. In this paper, we
obtain a similar result for certain mirror pairs of toric hypersurfaces. We use
recent results by Huang, Lian, Yau and Yu describing the relationship between
the Picard-Fuchs equations of these varieties and their Hasse--Witt matrices,
which encapsulate information about the number of points. The result allows us
to compute the number of points modulo the order of the field explicitly. We
illustrate this by computing K3 surface examples related to hypergeometric
functions.
