  We study Calabi-Yau manifolds defined over finite fields. These manifolds
have parameters, which now also take values in the field and we compute the
number of rational points of the manifold as a function of the parameters. The
intriguing result is that it is possible to give explicit expressions for the
number of rational points in terms of the periods of the holomorphic
three-form. We show also, for a one parameter family of quintic threefolds,
that the number of rational points of the manifold is closely related to as the
number of rational points of the mirror manifold. Our interest is primarily
with Calabi-Yau threefolds however we consider also the interesting case of
elliptic curves and even the case of a quadric in CP_1 which is a zero
dimensional Calabi-Yau manifold. This zero dimensional manifold has trivial
dependence on the parameter over C but a not trivial arithmetic structure.
