Exceptional zeros of Rankin-Selberg $L$-functions and joint Sato-Tate distributions
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866908697883574272 |
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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 non-dihedral twist-inequivalent cuspidal automorphic representations of $\mathrm{GL}_2(\mathbb{A}_F)$. We prove that if $m,n\geq 0$ are integers, $m+n\geq 1$, $F$ is totally real, $χ$ corresponds with a ray class character, and $π,π'$ correspond with primitive non-CM holomorphic Hilbert cusp forms, then the Rankin--Selberg $L$-function $L(s,\mathrm{Sym}^m(π)\times(\mathrm{Sym}^n(π')\otimesχ))$ has a standard zero-free region with no exceptional Landau--Siegel zero. This is new even for $F=\mathbb{Q}$. As an application, we establish the strongest known unconditional effective rates of convergence in the Sato--Tate distribution for $π$ and the joint Sato--Tate distribution for $π$ and $π'$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06482 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exceptional zeros of Rankin-Selberg $L$-functions and joint Sato-Tate distributions Thorner, Jesse Number Theory Let $χ$ be an idele class character over a number field $F$, and let $π,π'$ be non-dihedral twist-inequivalent cuspidal automorphic representations of $\mathrm{GL}_2(\mathbb{A}_F)$. We prove that if $m,n\geq 0$ are integers, $m+n\geq 1$, $F$ is totally real, $χ$ corresponds with a ray class character, and $π,π'$ correspond with primitive non-CM holomorphic Hilbert cusp forms, then the Rankin--Selberg $L$-function $L(s,\mathrm{Sym}^m(π)\times(\mathrm{Sym}^n(π')\otimesχ))$ has a standard zero-free region with no exceptional Landau--Siegel zero. This is new even for $F=\mathbb{Q}$. As an application, we establish the strongest known unconditional effective rates of convergence in the Sato--Tate distribution for $π$ and the joint Sato--Tate distribution for $π$ and $π'$. |
| title | Exceptional zeros of Rankin-Selberg $L$-functions and joint Sato-Tate distributions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2404.06482 |