Modular Forms of weight 8 for $\Gamma_g(1,2)$

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 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.