AbstractsMathematics

In this thesis, we show that the Fourier coefficients of certain half-integral weight harmonic Maass forms are given as ``twisted traces'' of CM values of integral weight harmonic Maass forms. These results generalize work of Zagier, Bruinier, Funke, and Ono on traces of CM values of harmonic Maass forms of weight 0 and -2. We utilize two theta lifts: one of them is a generalization of the Kudla-Millson theta lift considered by Bruinier, Funke, and Ono and the other one is defined using a theta kernel recently studied by Hövel. Both of the lifts have interesting applications. For instance, we show that the vanishing of the central derivative of the Hasse-Weil zeta function of an elliptic curve over the rational numbers is encoded by the Fourier coefficients of a harmonic Maass form arising from the Weierstrass zeta-function of the elliptic curve.