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

Abstract

We consider the principal congruence subgroup Γ [ 2 ] \Gamma [2] of level two inside the orthogonal group of an even and unimodular lattice of signature ( 2 , 10 ) (2,10) . Using Borcherds’ additive lifting construction, we construct a 715 715 -dimensional space of singular modular forms. This space is the direct sum of the one-dimensional trivial and a 714 714 -dimensional irreducible representation of the finite group O ⁡ ( F 2 12 ) \operatorname {O}(\mathbb {F}^{12}_2) (even type). It generates an algebra whose normalization is the ring of all modular forms. We define a certain ideal of quadratic relations. This system appears as a special member of a whole system of ideals of quadratic relations. At least some of them have geometric meaning. For example, we work out the relation to Kondo’s approach to the modular variety of Enriques surfaces.