Effective divisors on projectivized Hodge bundles and modular forms

TL;DR

This paper develops a method to construct vector-valued modular forms on moduli spaces of curves and abelian varieties using effective divisors in projectivized Hodge bundles, providing explicit examples for genus 2 and 3.

Contribution

It introduces a novel approach to generate modular forms via effective divisors on projectivized Hodge bundles, extending to hyperelliptic cases and calculating divisor classes.

Findings

Constructed basic modular forms for genus 2 and 3

Extended line bundles to compactifications for hyperelliptic curves

Calculated divisor classes in dual projectivized Hodge bundles

Abstract

We construct vector-valued modular forms on moduli spaces of curves and abelian varieties using effective divisors in projectivized Hodge bundles over moduli of curves. Cycle relations tell us the weight of these modular forms. In particular we construct basic modular forms for genus $2$ and $3$. We also discuss modular forms on the moduli of hyperelliptic curves. In that case the relative canonical bundle is a pull back of a line bundle on a ${\mathbb P}^1$-bundle over the moduli of hyperelliptic curves and we extend that line bundle to a compactification so that its push down is (close to) the Hodge bundle and use this to construct modular forms. In an appendix we use our method to calculate divisor classes in the dual projectivized $k$-Hodge bundle determined by Gheorghita-Tarasca and by Korotkin-Sauvaget-Zograf.