# Siegel modular forms of degree three and invariants of ternary quartics

## TL;DR

This paper characterizes the structure of the graded ring of Siegel modular forms of degree 3, identifying generators and linking them to invariants of ternary quartics, thus advancing understanding of modular forms and algebraic invariants.

## Contribution

It explicitly determines the generators of the graded ring of degree 3 Siegel modular forms and establishes a correspondence with ternary quartic invariants.

## Key findings

- The graded ring is generated by 19 modular forms.
- A homogeneous system of parameters with 7 specific forms is identified.
- A complete dictionary between invariants and generators is provided.

## Abstract

We determine the structure of the graded ring of Siegel modular forms of degree 3. It is generated by 19 modular forms, among which we identify a homogeneous system of parameters with 7 forms of weights 4, 12, 12, 14, 18, 20 and 30. We also give a complete dictionary between the Dixmier-Ohno invariants of ternary quartics and the above generators.