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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.04448 |
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Table of Contents:
- Let $ψ$ 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 $\int_{\mathbb{X}}ψ(z)E(z,1/2+it)^{3} dμz = \mathcal{O}_ψ(t^{-1/3+\varepsilon})$. In off-diagonal case we prove $\frac{1}{2\log t}\int_{\mathbb{X}}ψ(z)|E(z,1/2+it)|^{2}g(z)dμz = o(1)$ as long as $\min\{t , t_{g}\} \rightarrow \infty$. Finally we show $\int_{\mathbb{X}}ψ(z)f^{2}(z)g(z)dμz = o(1)$ in the range $|t_{f} - t_{g}| \leq t_{f}^{2/3-\varepsilon}$ where $f,g$ are two Hecke-Maass cusp forms.