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 : ℝx → ℂ and any Y > 0, a spectral decomposition of the shifted convolution sum {equation presented} is obtained. As an application, a spectral decomposition of the Dirichlet series {equation presented} is proved for ℜs > 1/2 with polynomial growth on vertical lines in the s-aspect and uniformity in the h-aspect. ©2008.