  In this article we study the period map for a family of $K3$ surfaces which
is given by the anticanonial divisor of a toric variety. We determine the
period differential equation and its monodromy group. Moreover we show the
exact relation between our period differential equation and the unifomizing
differential equation of the Hilbert modular orbifold for the field
$\mathbb{Q}(\sqrt{5})$.
