Weierstrass and Kleinian mock modular forms and related constructions

When Bruinier and Funke introduced the notion of harmonic Maaß forms, they already showed that for any given cusp form g, there is a harmonic Maaß form F whose shadow is g. There are various theoretical ways to construct these forms, e.g. via Poincaré series or via holomorphic projection. An alternative way, which works specifically for newforms of weight 2 with rational coefficients, which is based on very classical concepts and works quite efficiently in practice, is through Weierstrass mock modular forms. In my talk, I will discuss this construction and various generalizations of it, relaxing the requirement that the coefficients of the given newform be rational. This is joint work with Claudia Alfes-Neumann. If time permits, I shall also discuss a generalization of the construction to higher weights, which is joint work with Claudia Alfes-Neumann, Jens Funke, and Eugenia Rosu.