Modular forms for the even unimodular lattice of signature (2,10)

More than forty years ago, Igusa proved in [Ig] a “Fundamental Lemma”, according to which, theta constants give an explicit generically injective map from some modular varieties related to the symplectic group Sp(n,R) into a projective space. The embedded varieties satisfy some quartic relations, the so called Riemann’s relations. We expect that similar results hold for orthogonal instead of symplectic groups, where one should use typical “orthogonal constructions” instead of theta series. Some years ago, Borcherds described in [Bo1] two methods for constructing modular forms on modular varieties related to the orthogonal group O(2, n). They are the so called Borcherds’ additive and multiplicative lifting. The multiplicative lifting has been used by Borcherds himself and other authors to construct modular forms with known vanishing locus and interesting properties. The additive lifting has been used to construct explicit maps from some modular varieties related to O(2, 4), O(2, 6), O(2, 8), O(2, 10) and also for some unitary groups as U(1, 4) and U(1, 5). cf. [FH], [AF], [Fr2], [Fr3], [FS], [Ko1], [Ko2]. In this paper we try a more systematic treatment in certain level 2 cases: We are mainly interested in even lattices L of signature (2, n), such that the discriminant L′/L is a vector space over the field of two elements and such the the induced quadratic form q : L′/L → F2 is of even type. This means that the dimension 2m of L′/L is even and that q is of the form