  We study zeta-functions for a one parameter family of quintic threefolds
defined over finite fields and for their mirror manifolds and comment on their
structure. The zeta-function for the quintic family involves factors that
correspond to a certain pair of genus 4 Riemann curves. The appearance of these
factors is intriguing since we have been unable to `see' these curves in the
geometry of the quintic. Having these zeta-functions to hand we are led to
comment on their form in the light of mirror symmetry. That some residue of
mirror symmetry survives into the zeta-functions is suggested by an application
of the Weil conjectures to Calabi-Yau threefolds: the zeta-functions are
rational functions and the degrees of the numerators and denominators are
exchanged between the zeta-functions for the manifold and its mirror. It is
clear nevertheless that the zeta-function, as classically defined, makes an
essential distinction between Kahler parameters and the coefficients of the
defining polynomial. It is an interesting question whether there is a `quantum
modification' of the zeta-function that restores the symmetry between the
Kahler and complex structure parameters. We note that the zeta-function seems
to manifest an arithmetic analogue of the large complex structure limit which
involves 5-adic expansion.
