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It is known that the moduli space of plane quartic curves is birational to an arithmetic quotient of a six-dimensional complex ball [Kondō, S. “A complex hyperbolic structure of the moduli space of curves of genus three.”  Journal für die Reine und Angewandte Mathematik  525 (2000): 219–32]. 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. This map coincides with the one given by Coble [“Algebraic geometry and theta functions.”  Algebraic geometry and theta functions . American Mathematical Society Colloquium Publication 10. Providence, RI: American Mathematical Society, 1929 (3rd ed., 1969)] by using Göpel invariants.

Moduli of Plane Quartics, Göpel Invariants and Borcherds Products