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The classification of free algebras of orthogonal modular forms

Part of: Discontinuous groups and automorphic forms Metric geometry

Published online by Cambridge University Press:  06 August 2021

Haowu Wang
Haowu Wang*
Affiliation:
Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111Bonn, Germanyhaowu.wangmath@gmail.com
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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 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.

Keywords

modular forms for orthogonal groupssymmetric domains of type IVweighted projective spacesreflection groups

MSC classification

Primary: 11F55: Other groups and their modular and automorphic forms (several variables) 51F15: Reflection groups, reflection geometries 32N15: Automorphic functions in symmetric domains
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 9 , September 2021 , pp. 2026 - 2045
Copyright
© 2021 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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