Theta Nullvalues of Supersingular Abelian varieties

TL;DR

This paper provides a criterion for computing theta nullvalues of quotients of superspecial abelian varieties, enabling the construction of supersingular genus 3 curves through an effective algorithm.

Contribution

It introduces a new criterion for calculating theta nullvalues and implements an algorithm to construct supersingular curves of genus 3.

Findings

Criterion effectively computes theta nullvalues in many cases.

Algorithm successfully constructs supersingular genus 3 curves.

Method aligns with previous studies by Li and Oort.

Abstract

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.