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Title:Arithmetic of Maass forms of half-integral weight
Author(s):Andersen, Nickolas Robert
Director of Research:Ahlgren, Scott
Doctoral Committee Chair(s):Berndt, Bruce
Doctoral Committee Member(s):Ford, Kevin; Luu, Martin
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(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.
Issue Date:2016-06-15
Rights Information:Copyright 2016 Nickolas Andersen
Date Available in IDEALS:2016-11-10
Date Deposited:2016-08

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