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

## TL;DR

This paper constructs explicit modular forms related to K3 surfaces using their Hodge structures, explores their algebraic properties, and extends classical Siegel modular forms within this geometric framework.

## Contribution

It introduces a new method to construct modular forms for K3 surfaces with Kneser lattice conditions, expanding the understanding of their arithmetic and geometric properties.

## Key findings

- Explicit construction of modular forms on a 4-dimensional domain
- Analysis of the ring of these modular forms
- Extension of classical Siegel modular forms to K3 surfaces

## 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.