Finiteness of Free Algebras of Modular Forms on Unitary Groups

2025/05/19 by Yota Maeda, Maeda, Yota, Kazuma Ohara +1 · 1 citation
Mathematics · #11E39 #11F55 #11G18 #20G25 #22E40 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2505.13698

openalex publication_date 2025/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Classical results on the classification of reflections in an arithmetic subgroup Γ imply that if the graded algebra of modular forms M_*(Γ) is freely generated, then Γ must be an arithmetic subgroup of either the orthogonal group O⁺(2,n) or the unitary group U(1,n). Vinberg and Schwarzman showed that in the orthogonal case, if n>10, then it is never free. In this paper, we investigate the remaining unitary case and prove that, up to scaling, there are only finitely many isometry classes of Hermitian lattices of signature (1, n) with n > 2 over imaginary quadratic fields with odd discriminant that admit a free algebra of modular forms. In particular, when n>99 (except over ℚ(√(-3)), where we require n > 154), the graded algebra M_*(Γ) is never free for any arithmetic subgroup Γ<U(1,n), thereby partially confirming a conjecture by Wang and Williams. As a byproduct, we also establish a finiteness result for reflective modular forms. In the course of this proof, we derive a formula for the covolume of an arithmetic subgroup of a special unitary group, presented as the stabiliser of a Hermitian lattice, which generalises Prasad's volume formula for principal arithmetic subgroups in the case of special unitary groups.

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• The Kodaira dimension of even-dimensional ball quotients

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