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

TL;DR

This paper investigates the asymptotic behavior of joint cubic moments of Eisenstein series and Hecke-Maass cusp forms as their spectral parameters tend to infinity, revealing decay rates and off-diagonal correlations.

Contribution

It provides new bounds and asymptotic results for joint moments of automorphic forms in the high spectral parameter limit.

Findings

Diagonal cubic moment of Eisenstein series decays as t^{-1/3+ε}

Off-diagonal correlations tend to zero as spectral parameters grow

Joint integrals of cusp forms show negligible contribution in specified ranges

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 $\int_{\mathbb{X}}\psi(z)E(z,1/2+it)^{3} d\mu z = \mathcal{O}_{\psi}(t^{-1/3+\varepsilon})$. 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 $\min\{t , t_{g}\} \rightarrow \infty$. Finally we show $\int_{\mathbb{X}}\psi(z)f^{2}(z)g(z)d\mu z = o(1)$ in the range $|t_{f} - t_{g}| \leq t_{f}^{2/3-\varepsilon}$ where $f,g$ are two Hecke-Maass cusp forms.