Determination of $\textrm{GL}(3)$ cusp forms by central values of quadratic twisted $L$-functions
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866908468741406720 |
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| author | Hua, Shenghao Huang, Bingrong |
| author_facet | Hua, Shenghao Huang, Bingrong |
| contents | Let $ϕ$ and $ϕ'$ be two $\textrm{GL}(3)$ Hecke--Maass cusp forms. In this paper, we prove that $ϕ=ϕ'\textrm{ or }\widetilde{ϕ'}$ if there exists a nonzero constant $κ$ such that $$L(\frac{1}{2},ϕ\otimes χ_{8d})=κL(\frac{1}{2},ϕ'\otimes χ_{8d})$$ for all positive odd square-free positive $d$. Here $\widetilde{ϕ'}$ is dual form of $ϕ'$ and $χ_{8d}$ is the quadratic character $(\frac{8d}{\cdot})$. To prove this, we obtain asymptotic formulas for twisted first moment of central values of quadratic twisted $L$-functions on $\textrm{GL}(3)$, which will have many other applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_00473 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Determination of $\textrm{GL}(3)$ cusp forms by central values of quadratic twisted $L$-functions Hua, Shenghao Huang, Bingrong Number Theory Let $ϕ$ and $ϕ'$ be two $\textrm{GL}(3)$ Hecke--Maass cusp forms. In this paper, we prove that $ϕ=ϕ'\textrm{ or }\widetilde{ϕ'}$ if there exists a nonzero constant $κ$ such that $$L(\frac{1}{2},ϕ\otimes χ_{8d})=κL(\frac{1}{2},ϕ'\otimes χ_{8d})$$ for all positive odd square-free positive $d$. Here $\widetilde{ϕ'}$ is dual form of $ϕ'$ and $χ_{8d}$ is the quadratic character $(\frac{8d}{\cdot})$. To prove this, we obtain asymptotic formulas for twisted first moment of central values of quadratic twisted $L$-functions on $\textrm{GL}(3)$, which will have many other applications. |
| title | Determination of $\textrm{GL}(3)$ cusp forms by central values of quadratic twisted $L$-functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2201.00473 |