  In this paper, we present an application of mirror symmetry to arithmetic
geometry. The main result is the computation of the period of a mixed Hodge
structure, which lends evidence to its expected motivic origin. More precisely,
given a mirror pair $(M,W)$ of Calabi-Yau threefolds, the prepotential of the
complexified Kahler moduli space of $M$ admits an expansion with a constant
term that is frequently of the form $$-3\, \chi (M) \,\zeta(3)/(2 \pi i)^3+r,$$
where $r \in \mathbb{Q}$ and $\chi(M)$ is the Euler characteristic of $M$. We
focus on the mirror pairs for which the deformation space of the mirror
threefold $W$ forms part of a one-parameter algebraic family $W_{\varphi}$
defined over $\mathbb{Q}$ and the large complex structure limit is a rational
point. Assuming a version of the mirror conjecture, we compute the limit mixed
Hodge structure on $H^3(W_{\varphi})$ at the large complex structure limit. It
turns out to have a direct summand expressible as an extension of
$\mathbb{Q}(-3)$ by $\mathbb{Q}(0)$ whose isomorphism class can be computed in
terms of the prepotential of $M$, and hence, involves $\zeta(3)$. By way of
Ayoub's works on the motivic nearby cycle functor, this reveals in precise form
a connection between mirror symmetry and a variant of the Hodge conjecture for
mixed Tate motives.
