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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.