Regularized Theta Lifts of Harmonic Maass Forms

In this thesis we study regularized theta lifts between various spaces of harmonic Maass forms and their applications. The work consists of three main parts. In the first part we investigate the so-called Millson theta lift, which maps harmonic Maass forms of weight -2k (with k a non-negative integer) for congruence subgroups of the modular group to vector valued harmonic Maass forms of weight 1/2-k. We show that the Fourier coefficients of the lift of a harmonic Maass form F are given by traces of CM values and cycle integrals of non-holomorphic modular forms arising from F by application of certain differential operators, and that the Millson lift is related to the classical Shintani theta lift via the xi-operator. This part is based on joint work with Claudia Alfes-Neumann. The second part discusses some new applications of the Millson and the Kudla-Millson theta lifts. First we construct completions of two of Ramanujan's mock theta functions using the Millson lift of a suitable weakly holomorphic modular function F and use this to derive formulas for the coefficients of the mock theta functions in terms of traces of CM values of F. Further, we use the Millson and the Kudla-Millson theta lifts to obtain xi-preimages of unary theta functions of weight 3/2 and 1/2 whose holomorphic parts have rational Fourier coefficients. We also use these preimages to compute the Petersson inner products of harmonic Maass forms of weight 1/2 and 3/2 with unary theta series, and thereby obtain formulas and rationality results for the Weyl vectors of Borcherds products at the cusps. This part is based on joint work with Jan Hendrik Bruinier. In the third part we extend Borcherds' regularized theta lift in signature (1,2) to the full space of harmonic Maass forms of weight 1/2, i.e., those forms whose non-holomorphic part is allowed to grow linearly exponentially at infinity. We obtain real analytic modular functions with logarithmic singularities at CM points and new types of singularities along geodesics in the upper half-plane. Further, we use the theta lift to construct modular integrals of weight 2 with rational period functions, whose coefficients are given by linear combinations of Fourier coefficients of harmonic Maass forms of weight 1/2