Factorial property of a ring of automorphic forms

Abstract:

A ring of automorphic forms is shown to be factorial under some conditions on the domain and on the Picard group. As an application, we show that any divisor on the moduli space ${\mathfrak {M}_g}$ of curves of genus $g \geqslant 3$ is defined by a single element, and that the Satake compactification of ${\mathfrak {M}_g}$ is written as a projective spectrum of a factorial graded ring. We find a single element which defines the closure of ${\mathfrak {M}\prime _4}$ in ${\mathfrak {M}_4}$ where ${\mathfrak {M}\prime _4}$ is the moduli of curves of genus four whose canonical curves are exhibited as complete intersections of quadric cones and of cubics in ${\mathbb {P}^3}$.