Arithmetic of Maass forms of half-integral weight

We investigate the arithmetic properties of coefficients of Maass forms in three contexts. First, we discuss connections to invariants of real and imaginary quadratic fields, expanding on the work of Zagier and Duke-Imamoglu-Toth. Next, we examine the deep relationship between sums of Kloosterman sums and Maass cusp forms, motivated by work of Kuznetsov and Sarnak-Tsimerman, among others. Finally, we focus on the classical mock theta functions of Ramanujan, and give a simple proof of the mock theta conjectures using the modern theory of harmonic Maass forms, especially work of Zwegers and Bringmann-Ono, together with the theory of vector-valued modular forms.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2016

Author dc:creator

• Andersen, Nickolas Robert

Contributors dc:contributor

• Ahlgren, Scott
• Berndt, Bruce
• Ford, Kevin
• Luu, Martin

• number theory
• modular forms

Statement dc:rights

• Copyright 2016 Nickolas Andersen

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/92714

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/92714