Spectral square moments of a resonance sum for Maass forms

Abstract

Let fbe a Maass cusp form for {\Gamma }_{\mathrm{0}}(N) with Fourier coefficients {\lambda }_f (n)and Laplace eigenvalue {\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}+k^{\mathrm{2}}.For real \alpha \ne \mathrm{0} and β>0,consider the sum S_X(f; \alpha , \beta ) ={\displaystyle {\sum }_n{\lambda }_f(n)e(\alpha n^{\beta })\varphi (n/X)},where φis a smooth function of compact support. We prove bounds for the second spectral moment of S_X(f; \alpha ,\hspace{0.17em}\beta ),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 fhas its eigenvalue beyond X^{\frac{\mathrm{1}}{\mathrm{2}}+\epsilon },the standard resonance main term for S_X(f;\pm 2\sqrt{q},\mathrm{ 1}/2), q\in {\mathbb{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^{\epsilon }\le L\le K^{1-\epsilon }. The same bounds can be proved in a similar way for holomorphic cusp forms.

Key words： Cusp form; Maass form; Fourier coefficient of cusp form; Kuznetsov trace formula; resonance sum; first derivative test; weighted stationary phase

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