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

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.