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

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

43 pages, 6 figures. This paper was published in March 2012. But, on Proposition 4.1 and Theorem 4.1, there exist some typos. Here, these errors are corrected. On 5 September 2016, typos in page 28 are corrected. arXiv admin note: substantial text overlap with arXiv:1009.5725, arXiv:1001.5312, arXiv:1012.0156

References in corpus (1)

• Computing Differential Equations for Integrals Associated to Smooth Fano Polytopes

Cited by in corpus (6)

• A theta expression of the Hilbert modular functions for $\sqrt{5}$ via the periods of $K3$ surfaces
• Mirror symmetry and K3 surfaces
• Modular surfaces associated with toric K3 hypersurfaces
• Icosahedral invariants and Shimura curves
• Elliptic fibrations on toric $K3$ hypersurfaces and mirror symmetry derived from Fano polytopes
• Computer-assisted proofs for finding the monodromy of Picard-Fuchs differential equations for a family of K3 toric hypersurfaces