  This paper intends to give a mathematical explanation for results on the
zeta-function of some families of varieties recently obtained in the context of
Mirror Symmetry. In doing so, we obtain concrete and explicit examples for some
results recently used in algorithms to count points on smooth hypersurfaces in
P^n. In particular, we extend the monomial-motive correspondence of Kadir and
Yui and we give explicit solutions to the p-adic Picard-Fuchs equation
associated with monomial deformations of Fermat hypersurfaces. As a by-product
we obtain Poincare Duality for the Rigid cohomology of certain singular affine
varieties.
