The Hirzebruch-Mumford covolume of some hermitian lattices

Let $L=diag(1,1,\ldots,1,-1)$ and $M=diag(1,1,\ldots,1,-2)$ be the lattices of signature $(n,1)$. We consider the groups $\Gamma=SU(L,\mathcal{O}_K)$ and $\Gamma'=SU(M,\mathcal{O}_K)$ for an imaginary quadratic field $K=\mathbb{Q}(\sqrt{-d})$ of discriminant $D$ and it's ring of integers $\mathcal{O}_K$, $d$ odd and square free. We compute the Hirzebruch-Mumford volume of the factor spaces $\mathbb{B}^n/\Gamma$ and $\mathbb{B}^n/\Gamma'$. The result for the factor space $\mathbb{B}^n/\Gamma$ is due to Zeltinger, but as we're using it to prove the result for $\mathbb{B}^n/\Gamma'$ and it is hard to find his article, we prove the first result here as well

Nonfreeness of some algebras of hermitian modular forms

We study the algebras of hermitian automorphic forms for the lattice $L_n=diag(1,1,\ldots,1,-1)$ and for the field $K=\mathbb{Q}(\sqrt{-d})$ such that $p=2$ is unramified and the ring of integers $\mathcal{O}_K$ is a p.i.d. We prove that for $d>7$ these algebras can't be free. When $d=7$ and $d=3$ we give an estimate for the dimension of the symmetric spaces for which these algebras might be free. We also compare our results with the known results for $d=3$