SHIFTED CONVOLUTION SUMS OF FOURIER COEFFICIENTS OF CUSP FORMS

Abstract

Let g be a holomorphic Hecke eigenform for Gamma(0)(N) of weight l, or a Maass eigenform for Gamma(0)(N) with Laplace eigenvalue 1/4 + l(2). Let lambda(g)(n) be the nth Fourier coefficient of g. A shifted convolution sum of lambda(g)(n) is a sum of the form Sigma(n) lambda(g)(n)lambda(g)(n + h)w(n), where h is a nonzero integer, and w a nice weight function. These shifted convolution sums play a crucial role in analytic number theory, and in particular, in subconvexity bound problems of automorphic L-functions. This article will survey historical developments and recent progress on estimation and analytic continuation of a type of shifted convolution sums. The techniques to be used include spectral decomposition using Poincare series, a special choice of an orthonormal basis of Hecke eigenforms, a classical result of Good and its generalization by Krotz and Stanton, and a spectral large sieve. The shifted convolution sum will be meromorphically continued to Rs > -1/2, passing through all poles from Laplace eigenvalues.