On some lattice computations related to moduli problems

The method used in [6] to prove that most moduli spaces of K3 surfaces are of general type leads to a combinatorial problem about the possible number of roots orthogonal to a vector of given length in E8. A similar problem arises for E7 in [8]. Both were solved partly by computer methods. We use an improved computation and find one further case, omitted from [6]: the moduli space ℒ2d of K3 surfaces with polarisation of degree 2d is also of general type for d = 52. We also apply this method to some related problems. In Appendix A, V. Gritsenko shows how to arrive at the case d = 52 and some others directly.