Jutila's circle method and \(\mathrm{GL}(2) \times \mathrm{GL}(2)\) shifted convolution sums
type: Item
instance of: scholarly article
title: Jutila's circle method and \(\mathrm{GL}(2) \times \mathrm{GL}(2)\) shifted convolution sums
author: Guangwei Hu
author: Guangshi Lü
published in: Journal of Number Theory
publication date: 2022-09-30T00:00:00Z
review text: In this work, the authors considered the problem of Jutila's circle method and \(\mathrm{GL}(2)\times\mathrm{GL}(2)\) shifted convolution sums. Let \(A=(a_{ij})\) be a symmetric \(l\times l\) positive definite integral matrix such that \(a_{ii}\equiv 0\pmod 2\) for \(1\leq i\leq l\), and let \(D=\det A\) denote the determinant of \(A\). Then the quadratic form \(Q(x)\) associated to \(A\) is defined by \[ Q(x)=\frac{1}{2}\sum_{i}a_{ii}x_{i}^{2}+\sum_{i<j}a_{ij}x_{i}x_{j}. \] The corresponding theta function of \(Q\) is given by \[ \Theta (z,Q)=\sum_{n=0}^{\infty }r(n,Q)e(nz), \] where \(z\in \mathbb{H},e(x)=e^{2\pi ix}\) is an additive character, and \( r(n,Q)\) denotes the number of representations of \(n\) by the quadratic form \(Q\). Let \(\lambda _{f}(n)\) be the normalized Fourier coefficients of a Hecke-Maass or Hecke holomorphic cusp form \(f\) for congruence group \(\Gamma _{0}(N)\) with level \(N\) and nebentypus \(\chi _{N}\). Let \(Q(x)\) be a positive definite integral quadratic form as above and let \(r(n,Q)\) denote the number of representations of \(n\) by the quadratic form \(Q\). In that paper, the authors apply Jutila's circle method to derive a sharp bound for the shifted convolution sum of Fourier coefficients \(\lambda _{f}(n)\) and \(r(n,Q)\). They generalize and improve previous results without the Ramanujan conjecture.
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11E76
Mathematics Subject Classification ID: 11P55
zbMATH DE Number: 7595075
zbMATH Keywords: Jutila's circle method
zbMATH Keywords: shifted convolution sums
zbMATH Keywords: Fourier coefficients
reviewed by: Ahmet Tekcan
MaRDI profile type: Publication
full work available at URL: j.jnt.2022.05.008
OpenAlex ID: W4283586580
Wikidata QID: Q114156467
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DOI: 10.1016/J.JNT.2022.05.008
description: scientific article; zbMATH DE number 7595075