The Hirzebruch-Mumford Volume for the Orthogonal Group and Applications

In this paper we derive an explicit formula for the Hirzebruch-Mumford volume of an indefinite lattice $L$ of rank $\ge 3$. If $\Gamma \subset \Orth(L)$ 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.

2000 Mathematics Subject Classification: 11F55, 32N15, 14G35

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