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 ≥ 3. If Γ ⊂ 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 Sk(Γ) 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.