Spectral reciprocity for $\mathrm{GL}(n)$ and simultaneous non-vanishing of central $L$-values
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866910005605695488 |
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| author | Jana, Subhajit Nunes, Ramon |
| author_facet | Jana, Subhajit Nunes, Ramon |
| contents | Let $F$ be a totally real number field and $n\ge 3$. Let $Π$ and $π$ be cuspidal automorphic representations for $\mathrm{PGL}_{n+1}(F)$ and $\mathrm{PGL}_{n-1}(F)$, respectively, that are unramified and tempered at all finite places. We prove simultaneous non-vanishing of the Rankin--Selberg $L$-values $L(1/2,Π\otimes\widetildeσ)$ and $L(1/2,σ\otimes\widetildeπ)$ for certain sequences of $σ$ varying over cuspidal automorphic representations for $\mathrm{PGL}_n(F)$ with conductor tending to infinity in the level aspect and bearing certain local conditions. Along the way, we also prove a reciprocity formula for the average of the product of Rankin--Selberg $L$-functions $L(1/2,Π\otimes\widetildeσ)L(1/2,σ\otimes\widetildeπ)$ over a conductor aspect family of $σ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_02297 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Spectral reciprocity for $\mathrm{GL}(n)$ and simultaneous non-vanishing of central $L$-values Jana, Subhajit Nunes, Ramon Number Theory 11F55, 11F72 Let $F$ be a totally real number field and $n\ge 3$. Let $Π$ and $π$ be cuspidal automorphic representations for $\mathrm{PGL}_{n+1}(F)$ and $\mathrm{PGL}_{n-1}(F)$, respectively, that are unramified and tempered at all finite places. We prove simultaneous non-vanishing of the Rankin--Selberg $L$-values $L(1/2,Π\otimes\widetildeσ)$ and $L(1/2,σ\otimes\widetildeπ)$ for certain sequences of $σ$ varying over cuspidal automorphic representations for $\mathrm{PGL}_n(F)$ with conductor tending to infinity in the level aspect and bearing certain local conditions. Along the way, we also prove a reciprocity formula for the average of the product of Rankin--Selberg $L$-functions $L(1/2,Π\otimes\widetildeσ)L(1/2,σ\otimes\widetildeπ)$ over a conductor aspect family of $σ$. |
| title | Spectral reciprocity for $\mathrm{GL}(n)$ and simultaneous non-vanishing of central $L$-values |
| topic | Number Theory 11F55, 11F72 |
| url | https://arxiv.org/abs/2111.02297 |