#### Document Type

Dissertation

#### Date of Degree

Spring 2016

#### Degree Name

PhD (Doctor of Philosophy)

#### Degree In

Mathematics

#### First Advisor

Yangbo Ye

#### Abstract

Let * ƒ* be a Maass cusp form for Γ

_{0}(

*N)*with Fourier coefficients λ

_{ƒ}(n) and Laplace eigenvalue ¼+

*k*

^{2}. For real α≠0 and β>0 consider the sum:

^{∑}_{n}λ_{ƒ}(n)*e*(α*n*^{β})Φ(^{n}/X), where Φ is a smooth function of compact support. We prove bounds on the second spectral moment of this sum, with the eigenvalue tending toward infinity. When the eigenvalue is sufficiently large we obtain an average bound for this sum in terms of *X*. The method is adopted from proofs of subconvexity bounds for Rankin-Selberg *L*-functions for *GL*(2)×*GL*(2). It contains in particular the Kuznetsov trace formula and an asymptotic expansion of a well-known oscillatory integral with an enlarged range of *K*^{ε}≤*L*≤*K*^{1-ε}. The same bounds can be proved in the same way for holomorphic cusp forms.

Furthermore, we prove similar bounds for

^{∑}_{n}λ* _{f}*(n)λ

_{g}(n)

*e*(α

*n*

^{β})Φ(

^{n}/X),

where *g* is a holomorphic cusp form. As a corollary, we obtain a subconvexity bound for the *L*-function *L(s, f *×*g)*. This bound has the significant property of breaking convexity even for the trivial bound toward the Generalized Ramanujan Conjecture.

#### Public Abstract

Automorphic forms are complex-valued functions which satisfy a number of interesting properties, and are central objects of study in number theory. One way to learn more about these functions is to study their *Fourier coefficients*, which constitute a sequence of complex numbers associated to each automorphic form. *Resonance sums* are a means of investigating the oscillatory behavior of this sequence. For a fixed form, the corresponding resonance sum has been studied extensively, and the information given is with respect to the number of terms in the sum.

For a Maass form (resp. holomorphic form), which is a type of automorphic form, another important number associated to it is its *eigenvalue* (or *level*). Instead of fixing one such form, our approach is to consider the resonance sum for a family of automorphic forms. In this way we allow both the number of terms and its eigenvalue (or level) to vary, thereby gaining insight on the behavior of these forms in a new aspect.

#### Keywords

publicabstract

#### Pages

v, 69

#### Bibliography

66-69

#### Copyright

Copyright 2016 Nathan Salazar