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Construction of Harmonic Maass Forms in Small Weight

Rohloff, Marc

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Abstract

This thesis deals with various problems regarding automorphic forms of small weight. We study the continuation of Poincaré and Eisenstein series, as well as more abstract construction principles for harmonic Maass forms. Further, we show by using Riemann-Roch that the automorphic forms we constructed provide us with a basis for the weakly harmonic Maass forms. The difficulties encountered are solved by the introduction of a new type of vector space for square integrable automorphic forms, the Petersson-Sobolev spaces, which are defined in analogy to functional analytic Sobolev spaces. We show that these spaces provide us with a notion of invertibility of the Laplace operator as well as regularity theorems for the Petersson-Sobolev spaces, which are similar to the Sobolev imbedding theorems of functional analysis. The properties of the Petersson-Sobolev spaces then provide us with the tools required to solve the problems studied.

Translation of abstract (German)

Diese Arbeit behandelt verschiedene Probleme in der Theorie automorpher Formen, unter anderen betrachten wir die Fortsetzung von Poincaré- und Eisensteinreihen und weitere Konstruktionsmethoden harmonischer Maassformen. Mit Hilfe von Riemann-Roch zeigen wir, dass die so konstruierten automorphen Formen ein Erzeugendensystem des Raumes der schwach harmonischen Maassformen bilden. Hierbei ergeben sich einige Schwierigkeiten die wir durch Einführung einer neuen Art von Vektorräumen für quadratintegrierbare automorphe Formen lösen, der sogenannten Petersson-Sobolev Räume, in Analogie zur Definition der funktionalanalytischen Sobolevräume. Wir zeigen dann, dass diese Räume es uns erlauben, den Laplaceoperator zu invertieren und uns zusätzlich mit Regularitätsaussagen, ähnlich der Sobolevschen Einbettungssätze, ausstatten die wir dann als Werkzeuge nutzen um die analysierten Probleme zu lösen.

Item Type: Dissertation
Supervisor: Kohnen, Prof. Dr. Winfried
Date of thesis defense: 19 May 2017
Date Deposited: 01 Jun 2017 06:35
Date: 2017
Faculties / Institutes: The Faculty of Mathematics and Computer Science > Dean's Office of The Faculty of Mathematics and Computer Science
The Faculty of Mathematics and Computer Science > Department of Mathematics
Subjects: 510 Mathematics
Controlled Keywords: Modulform
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