We prove the Kloosterman-Spectral sum formula for PSL 2 (Z[i])\PSL 2 (C), and apply it to derive an explicit spectral expansion for the fourth power moment of the Dedekind zeta function of the Gaussian number field. Our sum formula, Theorem 13.1, allows the extension of the spectral theory of Kloosterman sums to all algebraic number fields. 2 −δ+it g(t) Γ(δ − it) ∞ 0 r 2(s−γ)−δ−it−1 (2.16) × ∞ 0 y δ−it−1 e − 1 2 y(r+1/r) I |q| (y)dy dr dt = (−1) q ∞ −∞ 2 1−δ+it g(t) Γ(δ − it) × ∞ 0 y δ−it−1 K 2(s−γ)−δ−it (y)I |q| (y)dy dt,

We construct explicit isomorphisms between spaces of Maass wave forms and cohomology groups for discrete cofinite groups Γ ⊂ PSL 2 (R).In the case that Γ is the modular group PSL 2 (Z) this gives a cohomological framework for the results in Period functions for Maass wave forms. I, of J. Lewis and D. Zagier in Ann. Math. 153 (2001), 191-258, where a bijection was given between cuspidal Maass forms and period functions.We introduce the concepts of mixed parabolic cohomology group and semi-analytic vectors in principal series representation. This enables us to describe cohomology groups isomorphic to spaces of Maass cusp forms, spaces spanned by residues of Eisenstein series, and spaces of all Γ-invariant eigenfunctions of the Laplace operator.For spaces of Maass cusp forms we also describe isomorphisms to parabolic cohomology groups with smooth coefficients and standard cohomology groups with distribution coefficients. We use the latter correspondence to relate the Petersson scalar product to the cup product in cohomology.

.(s)2sds Res=a -6,,,(2ni)-t(4~zlm[) -1 ~ f(s)sinTzs 2sds. Res=0The functions Coo(S ) and c01.,l(s ) are coefficients occurring in the Fourier series expansion of the Eisenstein series; the function c,.,(s) is a coefficient in the Fourier series expansion of a Poincar+ series.The theorem is applied to obtain some asymptotic results concerning the Fourier coefficients ?~.. Under additional conditions on the function f the formula in the theorem is modified in such a way that the Fourier coefficients of holomorphic cusp forms appear, Key words: Cusp forms -Fourier coefficients -Modular groupThe main object of this paper is to derive a formula containing the Fourier coefficients of cusp forms with weight zero for the full modular group. This formula is similar to the Selberg trace formula (see [4/~tppendix], [1,3]), as it relates a certain sum over the spectrum of the Laplace operator to a number of integrals. The difference is that it is not obtained by computing in two ways the trace of a certain operator, but by doing the same thing for a double-sided Fourier coefficient of the kernel of the operator.The integrals in the formula contain Fourier coefficients of Eisenstein and Poincar6 series, which are better known than the Fourier coefficients of cusp forms.

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