Analytic and arithmetic applications of half integral weight automorphic forms

In this thesis we explore both analytic and arithmetic applications of half integral weight modular forms. In the first chapter we are motivated by a conjecture of Hoffstein (2011) that asserts that the L-series attached to a half integral weight modular form satisfies a Lindelof hypothesis. Using spectral and diophantine techniques, the first chapter establishes a twisted second moment of L-series attached to such forms with a power saving error term. The second chapter of this thesis uses the spectral theory of half integral weight forms to establish a power saving in a formula of Andrews for the coefficients of Ramanujan's famous third order mock theta function. We also prove a conjecture of Andrews (1966) relating to the convergence of said formula. The third chapter uses spectral techniques and half integral weight Hecke theory to get bounds on sums of half integral weight Kloosterman sums. This is inspired by Sarnak and Tsimerman's treatment of weight zero Kloosterman sums in the context of the Linnik--Selberg conjecture. Each chapter is completely self contained and independent of the others. They were originally written as separate manuscripts for publication.

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

2020

Author dc:creator

• Dunn, Alexander

Contributors dc:contributor

• Zaharescu , Alexandru
• Ahlgren , Scott
• Ford, Kevin
• Allen , Patrick

• L function, moment, partition, Kloosterman sums, Maass forms

Statement dc:rights

• Copyright 2020 Alexander Dunn

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/108094

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/108094