Joint cubic moment of Eisenstein series and Hecke-Maass cusp forms

Abstract

Let $\psi$ be a smooth compactly supported function on $\mathbb{X}=SL(2,\mathbb{Z})\backslash \mathbb{H}$. In this paper, we are interested in the joint cubic moments of automorphic forms when the spectral parameters go to infinity. We show that the diagonal case for Eisenstein series ${\int }_{\mathbb{X}}\psi (z)E(z,1/2+it{)}^{3}d\mu z={\mathcal{O}}_{\psi }(t^{-1/3+\epsilon })$. In off-diagonal case we prove $\frac{1}{2\log t}{\int }_{\mathbb{X}}\psi (z)|E(z,1/2+it){|}^{2}g(z)d\mu z=o(1)$ as long as $\min \{t,t_g\}\to \infty$. Finally we show ${\int }_{\mathbb{X}}\psi (z)f^2(z)g(z)d\mu z=o(1)$ in the range $|t_f-t_g|\le t_f^{2/3-\epsilon }$ where $f,g$ are two Hecke-Maass cusp forms.

arXiv comment29 pages, completely rewrite and polish. All results are unconditional. Final version, to appear in Journal of Number theory

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• The cubic moment of Hecke--Maass cusp forms and moments of $L$-functions