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

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 Sym(Sym(8) (V) circle plus Sym(12)(V)) with respect to the action of SL(V) 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.