Modular Forms of weight 8 for ${\Gamma }_g(1,2)$

Abstract

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 Anstaz. We prove $\dim [{\Gamma }_4(1,2),8{]}_0=2$ and $\dim [{\Gamma }_4(1,2),8]=7$. 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.

accepted for publication in Math.Ann