  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.
