The Siegel modular forms of genus 2 with the simplest divisor

TL;DR

This paper identifies exactly eight genus 2 Siegel modular forms with simple divisors, extending classical forms and answering a question from a 2007 conference, highlighting their unique properties.

Contribution

It precisely classifies all such modular forms with simple divisors, expanding understanding of their structure and relation to classical and previously constructed forms.

Findings

Exactly eight such modular forms exist.

These forms generalize classical Igusa and Gritsenko-Nikulin forms.

The forms vanish only along the diagonal of the Siegel upper half-plane.

Abstract

We prove that there exist exactly eight Siegel modular forms with respect to the congruence subgroups of Hecke type of the paramodular groups of genus two vanishing precisely along the diagonal of the Siegel upper half-plane. This is a solution of a question formulated during the conference "Black holes, Black Rings and Modular Forms" (ENS, Paris, August 2007). These modular forms generalize the classical Igusa form and the forms constructed by Gritsenko and Nikulin in 1998.