Period differential equations for the families of $K3$ surfaces with $2$ parameters derived from the reflexive polytopes

TL;DR

This paper investigates the period mappings of K3 surface families from reflexive polytopes, deriving differential equations and monodromy groups, and links one equation to the Hilbert modular orbifold for ().

Contribution

It derives explicit period differential equations and monodromy groups for K3 families from reflexive polytopes, connecting them to modular forms.

Findings

Determined lattice structures of K3 surface families.

Derived period differential equations.

Connected one differential equation to Hilbert modular orbifold.

Abstract

In this paper, we study the period mappings for the families of $K3$ surfaces derived from the $3$-dimensional $5$-verticed reflexive polytopes. We determine the lattice structures, the period differential equations and the projective monodromy groups. Moreover, we show that one of our period differential equations coincides with the unifomizing differential equation of the Hilbert modular orbifold for the field $\mathbb{Q}(\sqrt{5})$.