Theta Nullvalues of Supersingular Abelian varieties

Let $\eta$ be a polarization with connected kernel on a superspecial abelian variety $E^g$. We give a sufficient criterion which allows the computation of the theta nullvalues of any quotient of $E^g$ by a maximal isotropic subgroup scheme of $\ker (\eta )$ effectively. This criterion is satisfied in many situations studied by Li and Oort. We used our method to implement an algorithm that constructs supersingular curves of genus 3.