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 being 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.

Siegel modular forms of degree two and level five

“…, χ ) and is, therefore, a constant denoted c by Lemma 3.1. We will now prove the claim by an argument which appeared essentially in [12]. Suppose that M !…”

“…Since det(σ v ) = −1, the chain rule implies that the above Jacobian vanishes on all mirrors of 2-reflections. Conversely, the main theorem of [12], and its generalization to meromorphic modular forms with constrained poles, implies that Theorem 3. 6 The ring M !…”

“…We will show that M ! * ( O(L)) is freely generated using a generalization of the modular Jacobian approach of [12,Theorem 5.1]. We briefly introduce the main objects of this approach.…”

“…(2) 0,1 (5)) of meromorphic Siegel modular forms on (2) 0,1 (5) with singularities supported on H. Using a generalization of the modular Jacobian approach of [12], we prove in Theorem 3.6 that M ! * (…”

On the classification of reflective modular forms

“…Reflective modular forms have many significant applications. They are useful to classify and construct various algebraic objects, such as generalized Kac-Moody algebras [4,2,21,22,18,31], hyperbolic reflection groups [8,21] and free algebras of modular forms [37]. They also identify the geometric type of moduli spaces and modular varieties [6,20,19,14,17,27].…”