Moduli of plane quartics, Gopel invariants and Borcherds products

TL;DR

This paper constructs a 15-dimensional space of automorphic forms that provides a birational embedding of the moduli space of plane quartics with level 2 structure into projective space, linking Gopel invariants and Borcherds products.

Contribution

It introduces a new automorphic form space that embeds the moduli space of plane quartics with level 2 structure, connecting classical invariants with modern automorphic forms.

Findings

Automorphic forms realize the moduli space embedding.

The embedding matches Coble's Gopel invariants.

Links between automorphic forms and classical invariants are established.

Abstract

It is known that the moduli space of plane quartic curves is birational to an arithmetic quotient of a 6-dimensional complex ball. In this paper, we shall show that there exists a 15-dimensional space of meromorphic automorphic forms on the complex ball which gives a birational embedding of the moduli space of plane quartics with level 2 structure into 14-dimensional projective space. This map coincides with the one given by Coble by using Gopel invariants.