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

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 SymðSym8 ðVÞ l Sym12 ðVÞÞ with respect to the action of SLðVÞ by adding a Borcherds product of weight 132 with one relation of weight 264, where V is a 2-dimensional C-vector space. The proof is based on the study of the moduli space of elliptic K3 surfaces with a section.