Hyperelliptic Schottky Problem and Stable Modular Forms

TL;DR

This paper investigates the structure of stable modular forms related to hyperelliptic Jacobians, showing that differences of theta series vanish on this locus and are generated by such differences, with geometric and algebraic insights.

Contribution

It demonstrates that the ideal of stable modular forms vanishing on the hyperelliptic locus is generated by differences of theta series, linking geometric boundary results to algebraic form generation.

Findings

Differences of theta series vanish on hyperelliptic Jacobian locus.

The ideal of stable modular forms vanishing on the locus is generated by theta series differences.

Boundary geometry of the Satake compactification plays a key role.

Abstract

It is well known that, fixed an even, unimodular, positive definite quadratic form, one can construct a modular form in each genus; this form is called the theta series associated to the quadratic form. Varying the quadratic form, one obtains the ring of stable modular forms. We show that the differences of theta series associated to specific pairs of quadratic forms vanish on the locus of hyperelliptic Jacobians in each genus. In our examples, the quadratic forms have rank 24, 32 and 48. The proof relies on a geometric result about the boundary of the Satake compactification of the hyperelliptic locus. We also study the monoid formed by the moduli space of all principally polarised abelian varieties, the operation being the product of abelian varieties. We use this construction to show that the ideal of stable modular forms vanishing on the hyperelliptic locus in each genus is generated by differences of theta series.