Exceptional zeros of $\mathrm{GL}_3\times\mathrm{GL}_3$ Rankin-Selberg $L$-functions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915714848260096 |
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| author | Thorner, Jesse |
| author_facet | Thorner, Jesse |
| contents | Let $χ$ be an idele class character over a number field $F$, and let $π,π'$ be any two cuspidal automorphic representations of $\mathrm{GL}_2(\mathbb{A}_F)$. We prove that the Rankin-Selberg $L$-function $L(s,\mathrm{Sym}^2(π)\times(\mathrm{Sym}^2 (π')\otimesχ))$ has a "standard" zero-free region with no exceptional Landau-Siegel zero except possibly when it is divisible by the $L$-function of a real idele class character. In particular, no such zero exists if $π$ is non-dihedral and $π'$ is not a twist of $π$. Until now, this was only known when $π=π'$, $π$ is self-dual, and $χ$ is trivial. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_09984 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exceptional zeros of $\mathrm{GL}_3\times\mathrm{GL}_3$ Rankin-Selberg $L$-functions Thorner, Jesse Number Theory Let $χ$ be an idele class character over a number field $F$, and let $π,π'$ be any two cuspidal automorphic representations of $\mathrm{GL}_2(\mathbb{A}_F)$. We prove that the Rankin-Selberg $L$-function $L(s,\mathrm{Sym}^2(π)\times(\mathrm{Sym}^2 (π')\otimesχ))$ has a "standard" zero-free region with no exceptional Landau-Siegel zero except possibly when it is divisible by the $L$-function of a real idele class character. In particular, no such zero exists if $π$ is non-dihedral and $π'$ is not a twist of $π$. Until now, this was only known when $π=π'$, $π$ is self-dual, and $χ$ is trivial. |
| title | Exceptional zeros of $\mathrm{GL}_3\times\mathrm{GL}_3$ Rankin-Selberg $L$-functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2508.09984 |