Analytic and arithmetic applications of half integral weight automorphic forms

Dunn, Alexander

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Description

Title

Analytic and arithmetic applications of half integral weight automorphic forms

Author(s)

Dunn, Alexander

Issue Date

2020-03-31

Director of Research (if dissertation) or Advisor (if thesis)

• Zaharescu , Alexandru
• Ahlgren , Scott

Doctoral Committee Chair(s)

Ford, Kevin

Committee Member(s)

Allen , Patrick

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

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

Abstract

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.

Graduation Semester

2020-05

Type of Resource

Thesis

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Copyright and License Information

Copyright 2020 Alexander Dunn

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