The spectral decomposition of shifted convolution sums

Abstract

Let π1, π2 be cuspidal automorphic representations of PGL2(R) of conductor 1 and Hecke eigenvalues λπ1,2(n), and let h>0 be an integer. For any smooth compactly supported weight functions W1,2:R×→C and any Y>0, a spectral decomposition of the shifted convolution sum ∑m±n=hλπ1(|m|)λπ2(|n|)|mn|W1(mY)W2(nY) is obtained. As an application, a spectral decomposition of the Dirichlet series ∑m,n≥1m−n=hλπ1(m)λπ2(n)(m+n)s(mnm+n)100 is proved for Rs>1/2 with polynomial growth on vertical lines in the s-aspect and uniformity in the h-aspect