  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.
