The ring of modular forms for the even unimodular lattice of signature (2,18)

TL;DR

This paper describes the structure of the ring of modular forms for a specific lattice, linking it to invariant theory and Borcherds products, and studies related moduli spaces of elliptic K3 surfaces.

Contribution

It establishes a new description of the modular forms ring for the (2,18) lattice using invariant theory and Borcherds products, connecting to moduli spaces of elliptic K3 surfaces.

Findings

Ring of modular forms is generated by invariants and Borcherds product.

Identifies a relation of weight 264 among generators.

Connects modular forms to the geometry of elliptic K3 surfaces.

Abstract

We show that the ring of modular forms with characters for the even unimodular lattice of signature (2,18) is obtained from the invariant ring of $\mathrm{Sym}(\mathrm{Sym}^8(V) \oplus \mathrm{Sym}^{12}(V))$ with respect to the action of $\mathrm{SL}(V)$ by adding a Borcherds product of weight 132 with one relation of weight 264, where $V$ is a 2-dimensional $\mathbb{C}$-vector space. The proof is based on the study of the moduli space of elliptic K3 surfaces with a section.