Sum formula for SL2 over imaginary quadratic number fields

Abstract

The subject of this thesis is generalization of the classical sum formula of Bruggeman and Kuznetsov to the upper half-space H3. The derivation of the preliminary sum formula involves computation of the inner product of two specially chosen Poincar´e series in two different ways: the spectral description where the Fourier coefficients CV of cusp forms on H3 appear and the geometric description where the sums of Kloosterman sums appear. An extension of the classical Lebedev transformation will be the building block for the Poincar´e series used. Our main result, the spectral sum formula, is obtained by extending the class of test functions in its preliminary version using the method of Miatello and Wallach. As an application, we obtain weighted density results concerning the cuspidal automorphic representations in L2(SL2(O)\SL2(C)) with eigenvalue not exceeding X and prescribed spectral parameter pV . A one-sided inversion of the Bessel transformation occurring in the spectral sum formula allows us to prove the sum formula in a reversed form, so called Kloosterman sum formula.