The Hirzebruch-Mumford covolume of some hermitian lattices

TL;DR

This paper computes the Hirzebruch-Mumford covolume of certain hermitian lattices associated with special unitary groups over imaginary quadratic fields, extending known results and providing new calculations for specific lattice groups.

Contribution

It provides the first detailed computation of the Hirzebruch-Mumford volume for the lattice M, complementing existing results for lattice L, and clarifies the volume formulas for these hermitian lattice groups.

Findings

Computed the covolume for $bB^n/\Gamma$ using Zeltinger's result.

Derived the covolume for $bB^n/\Gamma'$ for the first time.

Extended the understanding of volumes of hermitian lattice quotients.

Abstract

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.