The classification of free algebras of orthogonal modular forms

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 $\mathrm{\Gamma }$ is a subgroup of the integral orthogonal group of $M$ containing the discriminant kernel. It is proved that there are exactly 26 groups $\mathrm{\Gamma }$ such that the space of modular forms for $\mathrm{\Gamma }$ is a free algebra. Using the sufficient condition, we recover some well-known results.