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In this paper we study algebras of modular forms on unitary groups ofsignature $(n,1)$. We give a necessary and sufficient condition for an algebraof unitary modular forms to be free in terms of the modular Jacobian. As acorollary we obtain a criterion that guarantees in many cases that, if $L$ isan even lattice with complex multiplication and the ring of modular forms forits orthogonal group is a polynomial algebra, then the ring of modular formsfor its unitary group is also a polynomial algebra. We prove that a number ofrings of unitary modular forms are freely generated by applying these criteriato 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 examplesof finite-covolume reflection groups acting on complex hyperbolic space.

Free algebras of modular forms on ball quotients