Finiteness of Free Algebras of Modular Forms on Unitary Groups

TL;DR

This paper proves that for high-dimensional unitary groups over imaginary quadratic fields, the algebra of modular forms is rarely free, confirming a conjecture and establishing finiteness results for certain lattices and modular forms.

Contribution

It demonstrates the finiteness of free algebras of modular forms for unitary groups in high dimensions and over specific fields, extending previous orthogonal group results.

Findings

Finitely many Hermitian lattices admit free modular form algebras in high dimensions.

For n > 99 (except over ()), the algebra is never free.

Established a volume formula generalizing Prasad's for special unitary groups.

Abstract

Classical results on the classification of reflections in an arithmetic subgroup $\Gamma$ imply that if the graded algebra of modular forms $M_*(\Gamma)$ is freely generated, then $\Gamma$ 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 $M_*(\Gamma)$ is never free for any arithmetic subgroup $\Gamma<\operatorname{U}(1,n)$, 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.