| Harmonic weak Maass forms are instances of real analytic modular forms which have recently found applications in several areas of mathematics. They provide a framework for Ramanujan's theory of mock modular forms ([Ono08]), arise naturally in investigating the surjectivity of Borcherds' singular theta lift ([BF04]), and their Fourier coefficients seem to encode interesting arithmetic information ([BO]). Until now, harmonic weak Maass forms have been studied solely as complex analytic objects. The aim of this thesis is to recast their definition in more conceptual, algebro-geometric terms, and to lay the foundations of a p-adic theory of harmonic weak Maass forms analogous to the theory of p-adic modular forms formulated by Katz in the classical context. This thesis only discusses harmonic weak Maass forms of weight 0. The treatment of more general integral weights requires no essentially new idea but involves further notational complexities which may obscure the main features of our approach. This more general theory is presented in the article [CD], to which this thesis may serve as a motivated introduction. |
