Volume of truncated fundamental domains

Abstract

For T > 0, by applying Arthur's analytic truncation AT to the constant function 1 on G(F)\G(A)1, we obtain a characteristic function of a compact subset i(T) of G(F)\G(A)1. The aim of this paper is to give a general formula for the volume of a(T), which is an exponential polynomial function of T. Consequently, by letting T -* oo, we obtain a new way to evaluate the volume of fundamental domain for G(F)\G(A)1. While a(T) seems to be quite arbitrary, its volume admits beautiful structures: (i) Surprisingly enough, the coefficients of T are special values of zeta functions. (ii) The volume is an alternating sum associated to standard parabolic subgroups. ((ii) is suggested by geometry after all, a(T) is obtained from G(F)\G(A)1 by truncating off cuspidal regions which are known to be parametrized by standard parabolic subgroups.) Besides all basic properties of Arthur's analytic truncation, our method is based on a fundamental formula of Jacquet-Lapid-Rogawski on an integration of truncated Eisenstein series associated to cusp forms. Their formula, an advanced version of the Rankin-Selberg method, is obtained using the so-called Bernstein principle via regularized integration over cones.