  We consider certain families of Calabi-Yau orbifolds and their mirror
partners constructed from Fermat hypersurfaces in weighted projective 4-spaces.
Our focus is the topological mirror symmetry. There are at least three known
ingredients to describe the topological mirror symmetry, namely, integral
vertices in reflexive polytopes, monomials in graded polynomial rings (with
some group actions), and periods (and Picard-Fuchs differential equations).
  In this paper we will introduce Fermat motives associated to these Calabi-Yau
orbifolds and then use them to give motivic interpretation of the topological
mirror symmetry phenomenon between mirror pairs of Calabi-Yau orbifolds. We
establish, at the Fermat (the Landau-Ginzburg) point in the moduli space, the
one-to-one correspondence between the monomial classes and Fermat motives. This
is done by computing the number of ${\bf F}_q$-rational points on our
Calabi-Yau orbifolds over ${\bf F}_q$ in two different ways: Weil's algebraic
number theoretic method involving Jacobi (Gauss) sums, and Dwork's $p$-adic
analytic method involving Dwork characters and Gauss sums. We will discuss
specific examples in detail.
