Director of Research (if dissertation) or Advisor (if thesis)
Ahlgren, Scott
Doctoral Committee Chair(s)
Berndt, Bruce
Committee Member(s)
Ford, Kevin
Luu, Martin
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
number theory
modular forms
Abstract
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.
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