Moments of $L$-functions via a relative trace formula
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866914483626049536 |
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| author | Jana, Subhajit Nunes, Ramon |
| author_facet | Jana, Subhajit Nunes, Ramon |
| contents | We prove an asymptotic formula for the second moment of the $\mathrm{GL}(n)\times\mathrm{GL}(n+1)$ Rankin--Selberg central $L$-values $L(1/2,Π\otimesπ)$, where $π$ is a fixed cuspidal representation of $\mathrm{GL}(n)$ that is tempered and unramified at every place, while $Π$ varies over a family of automorphic representations of $\mathrm{PGL}(n+1)$ ordered by (archimedean or non-archimedean) conductor. As another application of our method, we prove the existence of infinitely many cuspidal representations $Π$ of $\mathrm{PGL}(n+1)$ such that $L(1/2,Π\otimesπ_1)$ and $L(1/2,Π\otimesπ_2)$ do not vanish simultaneously where $π_1$ and $π_2$ are cuspidal representations of $\mathrm{GL}(n)$ that are unramified and tempered at every place and have trivial central characters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_06461 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Moments of $L$-functions via a relative trace formula Jana, Subhajit Nunes, Ramon Number Theory 11F41, 11F70, 11F72 We prove an asymptotic formula for the second moment of the $\mathrm{GL}(n)\times\mathrm{GL}(n+1)$ Rankin--Selberg central $L$-values $L(1/2,Π\otimesπ)$, where $π$ is a fixed cuspidal representation of $\mathrm{GL}(n)$ that is tempered and unramified at every place, while $Π$ varies over a family of automorphic representations of $\mathrm{PGL}(n+1)$ ordered by (archimedean or non-archimedean) conductor. As another application of our method, we prove the existence of infinitely many cuspidal representations $Π$ of $\mathrm{PGL}(n+1)$ such that $L(1/2,Π\otimesπ_1)$ and $L(1/2,Π\otimesπ_2)$ do not vanish simultaneously where $π_1$ and $π_2$ are cuspidal representations of $\mathrm{GL}(n)$ that are unramified and tempered at every place and have trivial central characters. |
| title | Moments of $L$-functions via a relative trace formula |
| topic | Number Theory 11F41, 11F70, 11F72 |
| url | https://arxiv.org/abs/2309.06461 |