Finiteness of Free Algebras of Modular Forms on Unitary Groups
Yota Maeda, Kazuma Ohara
TL;DR
The paper addresses when the graded algebra $M_*(\Gamma)$ of modular forms for arithmetic subgroups of unitary groups $\mathrm{U}(1,n)$ is freely generated. It develops a unitary extension of Prasad–Vinberg volume techniques, deriving an explicit covolume formula for stabilizers of Hermitian lattices and using Hirzebruch–Mumford volumes to formulate a non-freeness criterion. The main results show that for imaginary quadratic fields with odd discriminant, $M_*(\Gamma)$ is not free for $n>99$ (with the $E=\mathbb{Q}(\sqrt{-3})$ exception requiring $n>154$), and that only finitely many lattices can admit freeness; a finiteness result for reflective modular forms follows as a byproduct. The methods are applied to questions about moduli spaces, including cubic threefolds, illustrating how these volume techniques constrain the birational geometry of ball quotients and illuminate the rarity of free modular form algebras in the unitary setting.
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 $\operatorname{O}^+(2,n)$ or the unitary group $\operatorname{U}(1,n)$. 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 $\mathbb{Q}(\sqrt{-3})$, where we require $n > 154$), the graded algebra $M_*(Γ)$ is never free for any arithmetic subgroup $Γ<\operatorname{U}(1,n)$, 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.