Nonlinear twists of Rankin-Selberg coefficients in the eigenvalue aspect
Abstract
Let g be a fixed primitive Hecke--Maass cusp form, and let f vary with spectral parameter kf=K, at a fixed level and with trivial nebentypus. We study smooth sums of λf(n)λg(n)e(αnβ) for fixed real α≠0. The crucial input is the smooth spectral second moment of Lau, Liu, and Ye, for which the window length can be taken to be L=K1/3+a for arbitrarily small fixed a>0. We show that, for every κ>0, there is β0>0 such that the sum is O(K2/3+κX1/2) for 1≤ X≤ K2 and 0<β≤β0. We give a detailed Baire category argument which makes the dependence on the smooth weight uniform through one finite derivative norm. A second argument keeps the phase explicit, applies Voronoi summation and shifted convolution estimates, and gives the numerical bound O(K423/224+(5/7)β+ε) at X=K2 for 0<β<5/32.
Related
- A Rankin-Selberg Integral for SU2,2
- Archimedean Modular Symbols for Rankin-Selberg Convolutions of GLn(ℂ)×GLn(ℂ) and GLn(ℂ)×GLn+1(ℂ)
- Rankin--Selberg integrals of opposite conductor--one newforms
- Regularized spectral expansion of a Rankin--Selberg period and moments
- Weyl Bounds for GL2× GL2 Rankin--Selberg L-functions via Short Spectral Moments
BibTeX
@misc{ye2026nonlinear,
title = {{Nonlinear twists of Rankin-Selberg coefficients in the eigenvalue aspect}},
author = {Yangbo Ye},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15979},
eprint = {2610.15979},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15979},
}
RIS
TY - UNPB
TI - Nonlinear twists of Rankin-Selberg coefficients in the eigenvalue aspect
AU - Yangbo Ye
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15979
UR - https://arxiv.org/abs/2610.15979
AB - Let g be a fixed primitive Hecke--Maass cusp form, and let f vary with spectral parameter k_f=K, at a fixed level and with trivial nebentypus. We study smooth sums of λ_f(n)λ_g(n)e(αn^β) for fixed real α≠0. The crucial input is the smooth spectral second moment of Lau, Liu, and Ye, for which the window length can be taken to be L=K^1/3+a for arbitrarily small fixed a>0. We show that, for every κ>0, there is β_0>0 such that the sum is O(K^2/3+κX^1/2) for 1≤ X≤ K^2 and 0<β≤β_0. We give a detailed Baire category argument which makes the dependence on the smooth weight uniform through one finite derivative norm. A second argument keeps the phase explicit, applies Voronoi summation and shifted convolution estimates, and gives the numerical bound O(K^423/224+(5/7)β+ε) at X=K^2 for 0<β<5/32.
ER -