The classification of free algebras of orthogonal modular forms

Abstract

We prove a necessary and sufficient condition for the graded algebra of automorphic forms on a symmetric domain of type IV to be free. From the necessary condition, we derive a classification result. Let $M$ be an even lattice of signature $(2,n)$ splitting two hyperbolic planes. Suppose $\Gamma$ is a subgroup of the integral orthogonal group of $M$ containing the discriminant kernel. It is proved that there are exactly 26 groups $\Gamma$ such that the space of modular forms for $\Gamma$ is a free algebra. Using the sufficient condition, we recover some well-known results.

Revised version, to appear in Compositio Mathematica

References in corpus (2)

• On some free algebras of orthogonal modular forms
• The classification of 2-reflective modular forms

Cited by in corpus (4)

• Projective spaces as orthogonal modular varieties
• Simple lattices and free algebras of modular forms
• Maximal Discrete Subgroups of $S{O}^{+}(2,n+2)$
• Siegel modular forms of degree two and level five