Determine all relations among theta constants on A4

Determine the complete set of algebraic relations among theta constants on the moduli space A4 of principally polarized abelian fourfolds, i.e., characterize all polynomial identities between (even) theta constants that hold identically on A4 and describe the ideal of relations among them.

Background

In genus 4, Jacobians admit explicit characterizations by equations in theta constants (e.g., Igusa’s relation), but theta constants themselves satisfy many intrinsic relations even on A4. Understanding these relations is essential for distinguishing identities that characterize the Jacobian locus from identities valid on all abelian fourfolds.

The authors note that, despite classical and modern progress, the global structure of the ideal of relations among theta constants on A4 is not fully known. Identifying this ideal would clarify which theta-constant identities are special to Jacobians and which are universal constraints on A4.