Igusa's map on modular forms vanishing on the hyperelliptic locus

TL;DR

This paper extends Igusa's map to modular forms vanishing on the hyperelliptic locus, computes their derivatives using Thomae's formula, and illustrates the approach with examples in genera 3 and 4.

Contribution

It introduces an extension of Igusa's map for hyperelliptic-vanishing modular forms and computes their derivatives via Thomae's formula, providing new tools for studying these forms.

Findings

Derived explicit derivatives of hyperelliptic-vanishing modular forms.

Applied the method to genera 3 and 4 examples.

Connected derivatives to vector-valued modular forms.

Abstract

We extend Igusa's map $\rho$ to modular forms which vanish on the hyperelliptic locus of the Siegel upper half-plane. The lowest non-vanishing derivatives of such modular forms are computed with the help of the general Thomae formula, they serve as vector-valued modular forms. This approach is illustrated by examples in genera $3$ and $4$.