# Concomitants of Ternary Quartics and Vector-valued Siegel and Teichm\"uller Modular Forms of Genus Three

## TL;DR

This paper develops a representation-theoretic approach to construct all vector-valued Siegel and Teichmüller modular forms of genus three, linking algebraic geometry, modular forms, and cohomology.

## Contribution

It introduces a method using ternary quartics to explicitly construct and analyze genus three modular forms and their geometric properties.

## Key findings

- Constructed all genus three vector-valued modular forms via ternary quartics.
- Established the relation between concomitant vanishing orders and modular form vanishing on loci.
- Connected Teichmüller cusp forms with the cohomology of symplectic local systems.

## Abstract

We show how one can use the representation theory of ternary quartics to construct all vector-valued Siegel modular forms and Teichm\"uller modular forms of degree 3. The relation between the order of vanishing of a concomitant on the locus of double conics and the order of vanishing of the corresponding modular form on the hyperelliptic locus plays an important role. We also determine the connection between Teichm\"uller cusp forms on \overline{M}_g and the middle cohomology of symplectic local systems on M_g. In genus 3, we make this explicit in a large number of cases.