Arithmetic of Maass forms of half-integral weight

number theory;modular forms

Andersen, Nickolas Robert

关键词: number theory;    modular forms;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/92714/ANDERSEN-DISSERTATION-2016.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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.