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By D. I. Wallace and Communicated Andrew M. Odlyzko


Abstract. There are two types of Eisenstein series associated to BL(3,Z). This paper deals with those which are built out of cuspidal Maass waveforms for SL(2, Z). We compute the inner product of two of them over a truncated fundamental region and then compute the rate of divergence as the truncation parameter tends to infinity. The solution of this problem is of use in computing the details of the trace formula for SL(3, Z). The Selberg trace formula for a discrete group T acting on SL(3,R)/ SO(3,R) is of interest to number theorists because it is one of the first examples of rank> 1 to be worked out in detail. Arthur [1] has obtained results for SL( «,R) showing that relevant operators are indeed trace class. His methods do not provide a version which allows one to obtain truly analytical results, however. A detailed computation of the contribution to the trace formula of orbital integrals and Eisenstein series would enable one, for example, to compute the rate of growth of eigenvalues associated to the discrete specrum of the Laplace-Beltrami operator, as in Hejhal [2]. A more ambitious application would be computing the rate of growth of class numbers of cubic number fields as compared with the size of their regulators, as in Sarnak [5]. Both of these applications arise primarily when T = SL(3, Z), which is the case considered in this paper. The principal difficulty in computing with the trace formula arises out of the combined contribution of the various types of Eisenstein series and the orbital integrals over elements in various parabolic subgroups. Both types of objects are badly divergent in their own right, but together they contribute a finite amount to the formula. The general shape of the formula is 1) Trkf = f £/(z lyz)-J2 f h{<p,X)E{i,4>,X)E{i,<p,X)dX dz, J ^ y€T i4J

Year: 2013
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