Jutila's circle method and \(\mathrm{GL}(2) \times \mathrm{GL}(2)\) shifted convolution sums
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.
scientific article; zbMATH DE number 7595075
scientific article