  We study mirror symmetric pairs of Calabi--Yau manifolds over finite fields.
In particular we compute the number of rational points of the manifolds as a
function of the complex structure parameters. The data of the number of
rational points of a Calabi--Yau $X/\mathbb{F}_q$ can be encoded in a
generating function known as the congruent zeta function. The Weil Conjectures
(proved in the 1970s) show that for smooth varieties, these functions take a
very interesting form in terms of the Betti numbers of the variety. This has
interesting implications for mirror symmetry, as mirror symmetry exchanges the
odd and even Betti numbers. Here the zeta functions for a one-parameter family
of K3 surfaces, $\mathbb{P}_3[4]$, and a two-parameter family of octics in
weighted projective space, $\mathbb{P}_4{}^{(1, 1, 2, 2, 2)} [8]$, are
computed. The form of the zeta function at points in the moduli space of
complex structures where the manifold is singular (where the Weil conjectures
apart from rationality are not applicable), is investigated. The zeta function
appears to be sensitive to monomial and non-monomial deformations of complex
structure (or equivalently on the mirror side, toric and non-toric divisors).
Various conjectures about the form of the zeta function for mirror symmetric
pairs are made in light of the results of this calculation. Connections with
$L$-functions associated to both elliptic and Siegel modular forms are
suggested.
