The Hirzebruch-Mumford volume for the orthogonal group and applications

Abstract

In this paper we derive an explicit formula for the Hirzebruch-Mumford volume of an indefinite lattice $L$ of rank $\ge 3$. If $\Gamma \subset \mathrm{O}(L)$ is an arithmetic subgroup and $L$ has signature $(2,n)$, then an application of Hirzebruch-Mumford proportionality allows us to determine the leading term of the growth of the dimension of the spaces $S_k(\Gamma )$ of cusp forms of weight $k$, as $k$ goes to infinity. We compute this in a number of examples, which are important for geometric applications.