A period differential equation for a family of $K3$ surfaces and the Hilbert modular orbifold for the field $\mathbb{Q}(\sqrt{5})$

Abstract

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 $\mathbb{Q}(\sqrt{5})$.