# Modular forms for the $A_{1}$-tower

## TL;DR

This paper explores the structure of modular forms related to the $A_1$-tower, connecting classical Siegel forms with orthogonal groups and providing a new framework for understanding their graded rings.

## Contribution

It introduces a novel framework for the $A_1$-tower of modular forms, utilizing three different construction methods to produce simple coordinates for their graded rings.

## Key findings

- Constructed a unified framework for the $A_1$-tower
- Developed three types of modular form constructions
- Produced simplified coordinates for graded rings

## Abstract

In the 1960's Igusa determined the graded ring of Siegel modular forms of genus two. He used theta series to construct $\chi_{5}$, the cusp form of lowest weight for the group $\operatorname{Sp}(2;Z)$. In 2010 Gritsenko found three towers of orthogonal type modular forms which are connected with certain series of root lattices. In this setting Siegel modular forms can be identifed with the orthogonal group of signature $(2,3)$ for the lattice $A_{1}$ and Igusa's form $\chi_{5}$ appears as the roof of this tower. We use this interpretation to construct a framework for this tower which uses three different types of constructions for modular forms. It turns out that our method produces simple coordinates for the graded rings of modular forms.