The Hirzebruch-Mumford volume for the orthogonal group and applications

  • V. Gritsenko

    Université Lille 1 Institut für UFR de Mathématiques Algebraische Geometrie F-59655 Villeneuve d'Ascq, Leibniz Universität Hannover Cedex D-30060 Hannover France Germany
  • K. Hulek

    Université Lille 1 Institut für UFR de Mathématiques Algebraische Geometrie F-59655 Villeneuve d'Ascq, Leibniz Universität Hannover Cedex D-30060 Hannover France Germany
  • G.K. Sankaran

    Department of Mathematical Sciences University of Bath Bath BA2 7AY England
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Abstract

In this paper we derive an explicit formula for the Hirzebruch-Mumford volume of an indefinite lattice LL of rank 3\ge 3. If Γ\Orth(L)\Gamma \subset \Orth(L) is an arithmetic subgroup and LL has signature (2,n)(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)S_k(\Gamma) of cusp forms of weight kk, as kk goes to infinity. We compute this in a number of examples, which are important for geometric applications.

Cite this article

V. Gritsenko, K. Hulek, G.K. Sankaran, The Hirzebruch-Mumford volume for the orthogonal group and applications. Doc. Math. 12 (2007), pp. 215–241

DOI 10.4171/DM/224