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

TL;DR

This paper constructs a sequence of lattice polarized K3 surface families linked to exceptional complex reflection groups, revealing new modular forms and their relations to group invariants through period mappings.

Contribution

It introduces a novel sequence of K3 surface families associated with complex reflection groups and explores their modular forms and invariants, especially focusing on the Shepherd-Todd group No.34.

Findings

Derived modular forms from period mappings of K3 families

Established relations between modular forms and reflection group invariants

Analyzed arithmetic and geometric properties of the associated lattices

Abstract

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.