Octavic theta series

In aprevious paper [ we considered the even unimodular lattice of signature (2, 10). It can be realized as direct sum of the negative of the lattice E8 and two hyperbolic planes. We considered the orthogonal group O(L). A certain subgroup of index two O+(L) acts biholomorphically on a ten dimensional tube domain H10. A speial form f(2z) belongs to the principal congruence subgroup of level two O+(L)[2]. The O+(L)-orbit of f(2z) spans a 715-dimensional space which is the direct sum of Cf(z) and a 714-dimensional irreducible space. The 715-dimensional space defines an everywhere regular birational embedding of the associated modular variety into aprojective space of dimension 714 714. The image is contained in a certain system of quadrics. If one is very optimistic, one may conjecture that the ring of modular forms of weight divisible by 4 is generated by the 715-dimensional space and that the quadratic relations are the defining ones. We have been asked by one of the authors whether Φ is related to theta series. In this paper we give an affirmative answer. We will construct a modular embedding of H10 into the Siegel half plane H16 of degree 16. This means that every substitution of O+(L) extends to a Siegel modular substitution. Even more, we will show that it extends to a substitution of the theta group Γ16,ϑ which is the group Γ16[1, 2] in Igusa’s notation.