On shifted convolution of half-integral weight cusp forms
type: Item
instance of: scholarly article
title: On shifted convolution of half-integral weight cusp forms
zbMATH Open document ID: 1229.11073
author: Wenzhi Luo
published in: Science China. Mathematics
publication date: 2011-02-25T00:00:00Z
review text: Let \[ f(z)=\sum_{n\geq 1}a_f(n)n^{\frac{l-1}{2}}e^{2\pi nz}\in S_l(\Gamma_0(1)) \] and \[ g(z)=\sum_{n\geq 1}a_g(n)n^{\frac{k-1}{2}}e^{2\pi nz}\in S_k(\Gamma_0(4N)) \] be two cusp forms, where \(l\in{\mathbb N}\) and \(k\geq \frac 52\) is a half-integer, \(\phi(x)\) a smooth function with support in \([\frac{X}{2},\frac{5X}{2}]\) satisfying \(\phi^{(p)}(x)\ll (\frac{X}{P})^{-p}\) for all integers \(p\geq 0\), where \(1\leq P\leq X\). The author proves that for a fixed integer \(b>0\) \[ \sum_{n\geq 1}a_g(n+b)a_f(n)\phi(n)\ll_{f,g,b,\epsilon}X^{\frac 34 +\epsilon} P^{\frac 32}. \] If \[ h(z)=y^{\frac 12}\sum_{n\in{\mathbb Z}}\rho_h(n) K_{it}(2\pi|n|y)e^{2\pi inx} \] is a Maass cusp form for \(SL(2,{\mathbb Z})\) with the Laplacian eigenvalue \( \lambda=\frac 14+t^2,t>0\) then \[ \sum_{n\geq 1}a_g(n+b)\rho_h(n)\phi(n)\ll_{f,h,b,\epsilon}X^{\frac 34 +\epsilon} P^{\frac 32}. \] It is expected that the bounds would hold with \(X^{\frac 12+\epsilon}\) in place of \(X^{\frac 34+\epsilon}\), which can be proved in the case \(b=0\).
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11F37
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F30
zbMATH DE Number: 5857686
zbMATH Keywords: shifted convolution
zbMATH Keywords: half-integral weight cusp forms
zbMATH Keywords: Poincaré series
zbMATH Keywords: Poisson-Voronoi summation formulas
MaRDI profile type: Publication
full work available at URL: s11425-010-4038-z
OpenAlex ID: W2032980210
cites work: Sums of Hecke Eigenvalues over Values of Quadratic Polynomials
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cites work: Q4184932
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cites work: On exponential sums involving the Fourier coefficients of Maass wave forms.
cites work: Q5517939
DOI: 10.1007/S11425-010-4038-Z
description: scientific article; zbMATH DE number 5857686