Modular forms of weight 8 for Γg(1, 2)

We complete the program indicated by the Ansatz of D'Hoker and Phong in genus 4 by proving the uniqueness of the restriction to Jacobians of the weight 8 Siegel cusp forms satisfying the Ansatz. We prove dim[Γ4(1, 2), 8]0 = 2 and dim[Γ4(1, 2), 8] = 7. In each genus, we classify the linear relations among the self-dual lattices of rank 16. We extend the program to genus 5 by constructing the unique linear combination of theta series that satisfies the Ansatz. © Springer-Verlag 2009.