On the Spectrum of the Metaplectic Group

The subject of this thesis is the theory of nonholomorphic modular forms of non-integral weight, and its applications to arithmetical functions involving Dedekind sums and Kloosterman sums. As was discovered by Andre Weil, automorphic forms of non-integral weight correspond to invariant funtions on Metaplectic groups. We thus give an explicit description of Meptaplectic groups corresponding to rational weight automorphic forms and explain this correspondence. We also describe the spectral decomposition of automorphic forms, and use this to find the spectral decomposition of a class of automorphic forms: the Poincare series. The applications center around the fact that for congruence subgroups of SL(2,Z), the Dedekind a function can be used to define multiplier systems of arbitrary weight, and these involve the Dedekind sum. We can use the general theory to bound sums of Kloosterman sums which involve the above multiplier systems, and therefore Dedekind sums. From this follows results about the distribution of values of the Dedekind sum.