Sequence of families of lattice polarized $K3$ surfaces, modular forms and degrees of complex reflection groups

We introduce a sequence of families of lattice polarized $K3$ surfaces. This sequence is closely related to complex reflection groups of exceptional type. Namely, we obtain modular forms coming from the inverse correspondences of the period mappings attached to our sequence. We study a non-trivial relation between our modular forms and invariants of complex reflection groups. Especially, we consider a family concerned with the Shepherd-Todd group of No.34 based on arithmetic properties of lattices and algebro-geometric properties of the period mappings.