Journal article
Double square moments and bounds for resonance sums of cusp forms
Journal of number theory, Vol.246, pp.166-188
05/2023
DOI: 10.1016/j.jnt.2022.11.012
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 α≠0 and 0<β≤1, consider a smooth resonance sum SX(f,g;α,β) of λf(n)λg(n) against e(αnβ) over X≤n≤2X. Double square moments of SX(f,g;α,β) over both f and g are nontrivially bounded when their weights k1 and k2 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 SX(f,g;α,β) shall then be proved. These individual bounds break the resonance barrier of X5/8 for 1/6<β<1 and achieve a square-root cancellation for 1/3<β<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 Gillespie - Saint Ambrose UniversityPraneel Samanta - Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USAYangbo Ye - Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA
- Resource Type
- Journal article
- Publication Details
- Journal of number theory, Vol.246, pp.166-188
- Publisher
- Elsevier Inc
- DOI
- 10.1016/j.jnt.2022.11.012
- ISSN
- 0022-314X
- eISSN
- 1096-1658
- Language
- English
- Electronic publication date
- 12/2022
- Date published
- 05/2023
- Academic Unit
- Mathematics
- Record Identifier
- 9984355056202771
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