Theta Liftings of Maass Forms

In 1993, Katok and Sarnak introduced a formula which describes periods of Maass cusp forms of weight 0 in terms of Fourier coefficients of Maass cusp forms of weight 1/2. Such periods are given by either the sum of Maass cusp forms evaluated at Heegner points or the sum over cycle integrals of Maass cusp forms associated to certain geodesics. This formula has since been generalized to periods twisted by a genus character and to Maass forms of even weight for Γ0(N) with N being odd. It can be applied to prove the equidistribution of geometric invariants in the modular curve. In this thesis, we give a version of the Katok-Sarnak formula which describes twisted periods of Maass cusp forms of even weight for the congruence subgroup Γ0(N) for square-free N in terms of Fourier coefficients of vector valued Maass cusp forms of half-integral weight associated to a lattice of signature (2, 1) for the metaplectic group Mp2(Z). The proof of this formula is obtained as follows: We associate to every even Maass cusp form which is a normalized newform, a vector valued half integral weight Maass cusp form, which is given by the Siegel and Millson theta lift. The Fourier coefficients of both lifts can be expressed in terms of periods of the input function. We express the Siegel and Millson theta lift by a basis of half-integral vector valued Maass cusp forms and once more compute its Fourier coefficients, which involves the computation of the Shimura and Millson-Shimura lift – the adjoint theta lifts to the Siegel and Millson lift. The generalized vector valued Katok-Sarnak formula is derived by comparing the results of both computations.