Joint cubic moment of Eisenstein series and Hecke-Maass cusp forms
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912600387747840 |
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| author | Guo, Chengliang |
| author_facet | Guo, Chengliang |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04448 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Joint cubic moment of Eisenstein series and Hecke-Maass cusp forms Guo, Chengliang Number Theory 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. |
| title | Joint cubic moment of Eisenstein series and Hecke-Maass cusp forms |
| topic | Number Theory |
| url | https://arxiv.org/abs/2410.04448 |