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 , arise naturally in investigating the surjectivity of Borcherds' singular theta lift , and their Fourier coefficients seem to encode interesting arithmetic information. 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. / Les formes de Maass faiblement harmoniques ont recemment trouve des applications dans plusieurs domaines des mathématiques. Elles fournissent un cadre pour la théorie des ``Mock Modular forms" de Ramanujan, surviennent naturellement dans l'etude de la surjectivite de la correspondance de Borcherds et leurs coefficients de Fourier semblent donner des informations arithmetiques sur les derivees des tordues quadratiques de certaines fonctions L associees aux formes modulaires. Jusqu'a present, les formes de Maass faiblement harmoniques ont uniquement ete etudiees en tant qu' objets analytiques sur les nombres complexes. L'objectif de cette these est de les decrire dans un cadre algebrique plus conceptuel, et de jeter les bases d'une theorie p-adique des formes de Maass faiblement harmoniques, par analogie avec le point de vue geometrique de Katz sur la theorie des formes modulaires p-adiques. Cette these traite uniquement du cas des formes de Maass faiblement harmoniques de poids 0.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:QMM.92409 |
| Date | January 2010 |
| Creators | Candelori, Luca |
| Contributors | Henri Darmon (Supervisor) |
| Publisher | McGill University |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Format | application/pdf |
| Coverage | Master of Science (Department of Mathematics and Statistics) |
| Rights | All items in eScholarship@McGill are protected by copyright with all rights reserved unless otherwise indicated. |
| Relation | Electronically-submitted theses. |
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