#### DOI

10.17077/etd.2gc5yu28

#### Document Type

Dissertation

#### Date of Degree

Spring 2017

#### Access Restrictions

.

#### Degree Name

PhD (Doctor of Philosophy)

#### Degree In

Mathematics

#### First Advisor

Ye, Yangbo

#### First Committee Member

Kutzko, Philip

#### Second Committee Member

Krishnamurthy, Muthu

#### Third Committee Member

Strohmer, Gerhard

#### Fourth Committee Member

McKee, Mark

#### Abstract

The purpose of this study is to investigate the oscillatory behavior of the fractional exponential sum weighted by certain automorphic forms for GL(2) x GL(3) case. Automorphic forms are complex-values functions defined on some topological groups which satisfy a number of applicable properties. One nice property that all automorphic forms admit is the existence of Fourier series expansions, which allows us to study the properties of automorphic forms by investigating their corresponding Fourier coefficients. The Maass forms is one family of the classical automorphic forms, which is the major focus of this study.

Let f be a fixed Maass form for SL(3, Z) with Fourier coefficients Af(m, n). Also, let {gj} be an orthonormal basis of the space of the Maass cusp form for SL(2, Z) with corresponding Laplacian eigenvalues 1/4+kj^2, kj>0. For real α be nonzero and β>0, we considered the asymptotics for the sum in the following form Sx(f x gj, α, β) = ∑Af(m, n)λgj(n)e(αn^β)φ(n/X) where φ is a smooth function with compactly support, λgj(n) denotes the nth Fourier coefficient of gj, and X is a real parameter that tends to infinity. Also, e(x) = exp(2πix) throughout this thesis.

We proved a bound of the weighted average sum of Sx(f x gj, α, β) over all Laplacian eigenvalues, which is better than the trivial bound obtained by the classical Rankin-Selberg method. In this case, we allowed the form varies so that we can obtain information about their oscillatory behaviors in a different aspect. Similar to the proofs of the subconvexity bounds for Rankin-Selberg L-functions for GL(2) x GL(3) case, the method we used in this study includes several sophisticated techniques such as weighted first and second derivative test, Kuznetsov trace formula, and Voronoi summation formula.

#### Public Abstract

During the last half-century, the theory of automorphic forms has become a major focus in the development of the modern number theory. Automorphic forms are functions from some topological groups to the complex plane, which have many applications to different aspects in Mathematics. Because automorphic forms have Fourier expansions. We can study its properties by studying the corresponding Fourier coefficients. Taking the weighted sums of these Fourier coefficients against various exponential functions will case a rise of resonance. We call this type of sum as a resonance sum. Resonance is a physical phenomenon that occurs between two interactive vibrating systems. Fixing one of these two vibrating systems, we may control the second one to detect the resonance frequencies of the first system, and thus obtain its oscillation spectrum. The most classical example of this is the Fourier series expansion of a periodic function, which is the resonance sum for *GL _{1}* case. This study is to learn the property of a resonance sum in a higher dimensional space.

#### Pages

v, 45 pages

#### Bibliography

Includes bibliographical references (pages 43-45).

#### Copyright

Copyright © 2017 Huan Qin

#### Recommended Citation

Qin, Huan. "Averages of fractional exponential sums weighted by Maass forms." PhD (Doctor of Philosophy) thesis, University of Iowa, 2017.

https://doi.org/10.17077/etd.2gc5yu28