Inverse period mappings of $K3$ surfaces and a construction of modular forms for a lattice with the Kneser conditions

Abstract

We explicitly construct modular forms on a $4$-dimensional bounded symmetric domain of type $IV$ based on the variation of the Hodge structures of $K3$ surfaces. We study the ring of our modular forms. Because of the Kneser conditions of the transcendental lattice of our family of $K3$ surfaces, our modular group has a good arithmetic property. Also, our results can be regarded as natural extensions of classical Siegel modular forms from the viewpoint of $K3$ surfaces.

arXiv comment23 pages, 1 figure, Section 3 is modified, new references are added, minor misprints are corrected

References in corpus (6)

• The Igusa modular forms and ``the simplest'' Lorentzian Kac--Moody algebras
• K3 surfaces and equations for Hilbert modular surfaces
• Six line configurations and string dualities
• The ring of modular forms for the even unimodular lattice of signature (2,10)
• A theta expression of the Hilbert modular functions for $\sqrt{5}$ via the periods of $K3$ surfaces
• Icosahedral invariants and a construction of class fields via periods of $K3$ surfaces

Cited by in corpus (3)

• Geometric interpretation of Hermitian modular forms via Burkhardt invariants
• Elliptic fibrations on toric $K3$ hypersurfaces and mirror symmetry derived from Fano polytopes
• Sequence of families of lattice polarized $K3$ surfaces, modular forms and degrees of complex reflection groups