Averages of shifted convolution sums for GL(3)×GL(2)

Abstract

Let Af(1,n) be the normalized Fourier coefficients of a GL(3) Hecke–Maass cusp form f and let ag(n) be the normalized Fourier coefficients of a GL(2) cusp form g. Let λ(n) be either Af(1,n) or the triple divisor function d3(n). It is proved that for any ϵ>0, any integer r≥1 and r5/2X1/4+7δ/2≤H≤X with δ>0,1H∑h≥1W(hH)∑n≥1λ(n)ag(rn+h)V(nX)≪X1−δ+ϵ, where V and W are smooth compactly supported functions, and the implied constants depend only on the associated forms and ϵ.