The Second Moment of $\mathrm{GL}_3 \times \mathrm{GL}_2$ $L$-functions at Special Points
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915549712220160 |
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| author | Qi, Zhi |
| author_facet | Qi, Zhi |
| contents | Let $ϕ$ be a fixed Hecke--Maass form for $\mathrm{SL}_3 (\mathbb{Z})$ and $u_j $ traverse an orthonormal basis of Hecke--Maass forms for $\mathrm{SL}_2 (\mathbb{Z}) $. Let $1/4+t_j^2$ be the Laplace eigenvalue of $u_j $. In this paper, we prove the mean Lindelöf hypothesis for the second moment of $ L (1/2+it_j, ϕ\times u_j) $ on $ T < t_j \leqslant T + \sqrt{T} $. Previously, this was proven by Young on $ t_j \leqslant T$. Our approach is more direct as we do not apply the Poisson summation formula to detect the `Eisenstein--Kloosterman' cancellation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_11191 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Second Moment of $\mathrm{GL}_3 \times \mathrm{GL}_2$ $L$-functions at Special Points Qi, Zhi Number Theory Let $ϕ$ be a fixed Hecke--Maass form for $\mathrm{SL}_3 (\mathbb{Z})$ and $u_j $ traverse an orthonormal basis of Hecke--Maass forms for $\mathrm{SL}_2 (\mathbb{Z}) $. Let $1/4+t_j^2$ be the Laplace eigenvalue of $u_j $. In this paper, we prove the mean Lindelöf hypothesis for the second moment of $ L (1/2+it_j, ϕ\times u_j) $ on $ T < t_j \leqslant T + \sqrt{T} $. Previously, this was proven by Young on $ t_j \leqslant T$. Our approach is more direct as we do not apply the Poisson summation formula to detect the `Eisenstein--Kloosterman' cancellation. |
| title | The Second Moment of $\mathrm{GL}_3 \times \mathrm{GL}_2$ $L$-functions at Special Points |
| topic | Number Theory |
| url | https://arxiv.org/abs/2510.11191 |