Free algebras of modular forms on ball quotients - Wang, Haowu; Williams, Brandon (Research in the mathematical sciences, 2025)
In this paper, we study algebras of modular forms on unitary groups of signature (n, 1). We give a necessary and sufficient condition for an algebra of unitary modular forms to be free in terms of the modular Jacobian. As a corollary, we obtain a criterion that guarantees in many cases that, if L is an even lattice with complex multiplication and the ring of modular forms for its orthogonal group is a polynomial algebra, then the ring of modular forms for its unitary group is also a polynomial algebra. We prove that a number of rings of unitary modular forms are freely generated by applying these criteria to Hermitian lattices over the rings of integers of $$\mathbb {Q}(\sqrt{d})$$for $$d=-1,-2,-3$$. As a byproduct, our modular groups provide many explicit examples of finite-covolume reflection groups acting on complex hyperbolic space.