Dissertation
Square moments and bounds for resonance sums of two cusp forms
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2023
DOI: 10.25820/etd.007128
Abstract
Let f and g be holomorphic cusp forms for the modular group SL2(Z) of weight k1 and k2 with Fourier coefficients λf (n) and λg(n), respectively. For real $\alpha\neq0$ and $0<\beta\leq1$, consider a smooth resonance sum $S_X(f,g;\alpha,\beta)$ $$ S_X(f, g;\alpha,\beta)
:= \sum_n \lambda_f(n) \lambda_g(n) e(\alpha n^{\beta}) \phi\Big(\frac nX\Big).$$ Following Iwaniec, Luo, and Sarnak, a general form of Hypothesis S predicts a bound $S_X(f, g;\alpha,\beta)\ll X^{1/2+\varepsilon}$.
In this thesis, double square moments of $S_X(f,g;\alpha,\beta)$ over both $f$ and $g$ are nontrivially bounded when their weights $k_1$ and $k_2$ tend to infinity together. By allowing both $f$ and $g$ to move, these double moments are indeed square moments associated with automorphic forms for $GL(4)$. By taking out a small exceptional set of $f$ and $g$, bounds for individual $S_X(f,g;\alpha,\beta)$ shall then be proved. These individual bounds break the resonance barrier of $X^{5/8}$ for $1/6<\beta<1$ and achieve a square-root cancellation for $1/3<\beta<1$ for almost all $f$ and $g$ as an evidence for Hypothesis S.
Details
- Title: Subtitle
- Square moments and bounds for resonance sums of two cusp forms
- Creators
- Praneel Samanta
- Contributors
- Yangbo Ye (Advisor)Sergii Bezuglyi (Committee Member)Frauke Bleher (Committee Member)Tim Gillespie (Committee Member)Palle Jorgensen (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Spring 2023
- Publisher
- University of Iowa
- DOI
- 10.25820/etd.007128
- Number of pages
- vi, 58 pages
- Copyright
- Copyright 2023 Praneel Samanta
- Language
- English
- Date submitted
- 04/24/2023
- Date approved
- 04/26/2023
- Description bibliographic
- Includes bibliographical references (pages 52-58).
- Public Abstract (ETD)
- Modern number theory studies certain non-Euclidean waveforms called automorphic forms. Their behavior can be characterized by the oscillation of their Fourier coefficients, a sequence of associated complex numbers. These Fourier coefficients can be used to define a certain class of functions called L-functions, whose zeros are known to encode information about prime numbers. For specific automorphic forms associated with 4 X 4 matrices, this oscillation is studied by testing it against fractional exponential functions. Fixing one of these two vibrating systems, we may control the second to obtain oscillatory information about the first. Non-trivial bounds are proved in this thesis when the automorphic forms are allowed to change. Improvements of these results would lead to a deeper understanding of zeros of the related L-functions.
- Academic Unit
- Mathematics
- Record Identifier
- 9984437258002771
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