K3 Surfaces and Orthogonal Modular Forms

TL;DR

This paper explicitly constructs generators for the ring of modular forms related to certain K3 surface moduli spaces, revealing their structure through elliptic fibrations and Weierstrass equations.

Contribution

It provides explicit generators for the modular form ring associated with specific K3 surface moduli, a novel explicit description in the field.

Findings

Generators are explicitly described as coefficients in Weierstrass equations.

The work applies to K3 surfaces with automorphism group (Z/2Z)^2 and Picard rank ≥13.

The modular forms are linked to elliptic fibrations of the K3 surfaces.

Abstract

We determine explicit generators for the ring of modular forms associated with the moduli spaces of K3 surfaces with automorphism group $(\mathbb{Z}/2\mathbb{Z})^2$ and of Picard rank 13 and higher. The K3 surfaces in question carry a canonical Jacobian elliptic fibration and the modular form generators appear as coefficients in the Weierstrass-type equations describing these fibrations.