A new geometric description for Igusa's modular form $(azy)_5$

TL;DR

This paper introduces a novel geometric perspective to describe Igusa's modular form $(azy)_5$, which is fundamental in the structure of genus 2 modular forms, offering a potentially simpler understanding.

Contribution

It provides a new geometric description of $(azy)_5$, replacing the complex algebraic expression with a more intuitive geometric approach.

Findings

New geometric interpretation of $(azy)_5$

Simplifies understanding of genus 2 modular forms

Potential for easier computation and analysis

Abstract

The modular form $(azy)_5$ notably appears in one of Igusa's classic structure theorems as a generator of the ring of full modular forms in genus 2, being exhibited by means of a complicated algebraic expression. In this work a different description for this modular form is provided by resorting to a peculiar geometrical approach.