Preprint
Double square moments and bounds for resonance sums of cusp forms
ArXiv.org
09/08/2022
Abstract
Let $f$ and $g$ be holomorphic cusp forms for the modular group $SL_2(\mathbb
Z)$ of weight $k_1$ and $k_2$ with Fourier coefficients $\lambda_f(n)$ and
$\lambda_g(n)$, respectively. For real $\alpha\neq0$ and $0<\beta\leq1$,
consider a smooth resonance sum $S_X(f,g;\alpha,\beta)$ of
$\lambda_f(n)\lambda_g(n)$ against $e(\alpha n^\beta)$ over $X\leq n\leq2X$.
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
$S_X(f,g;\alpha,\beta)$ will then be proved. These individual bounds break the
resonance barrier of $X^\frac58$ for $\frac16<\beta<1$ and achieve a
square-root cancellation for $\frac13<\beta<1$ for almost all $f$ and $g$ as an
evidence for Hypothesis S for cusp forms over integers. The methods used in
this study include Petersson's formula, Poisson's summation formula, and
stationary phase integrals.
Details
- Title: Subtitle
- Double square moments and bounds for resonance sums of cusp forms
- Creators
- Tim GillespiePraneel SamantaYangbo Ye
- Resource Type
- Preprint
- Publication Details
- ArXiv.org
- ISSN
- 2331-8422
- Language
- English
- Date posted
- 09/08/2022
- Academic Unit
- Mathematics
- Record Identifier
- 9984296000202771
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