Spectral square moments of a resonance sum for Maass forms.

Let f be a Maass cusp form for Γ( N) with Fourier coefficients λ( n) and Laplace eigenvalue $$\frac{1} {4} + k^2 $$ . For real α ≠ 0 and β > 0, consider the sum S ( f; α, β) = ∑ λ( n) e( αn ) ϕ( n/ X), where ϕ is a smooth function of compact support. We prove bounds for the second spectral moment of S ( f; α, β), with the eigenvalue tending towards infinity. When the eigenvalue is sufficiently large, we obtain an average bound for this sum in terms of X. This implies that if f has its eigenvalue beyond $$X^{\tfrac{1} {2} + \varepsilon } $$ , the standard resonance main term for S ( f; $$ \pm 2\sqrt q $$ , 1/2), q ∈ ℤ, cannot appear in general. The method is adopted from proofs of subconvexity bounds for Rankin-Selberg L-functions for GL(2) × GL(2). It contains in particular a proof of an asymptotic expansion of a well-known oscillatory integral with an enlarged range of K ⩽ L ⩽ K . The same bounds can be proved in a similar way for holomorphic cusp forms. [ABSTRACT FROM AUTHOR]