Quotients of $L$-functions: degrees $n$ and $n-2$
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| Format: | Preprint |
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2024
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| _version_ | 1866913275837415424 |
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| author | Raghunathan, Ravi |
| author_facet | Raghunathan, Ravi |
| contents | If $L(s,π)$ and $L(s,ρ)$ are the Dirichlet series attached to cuspidal automorphic representations $π$ and $ρ$ of ${\rm GL}_n({\mathbb A}_{\mathbb Q})$ and ${\rm GL}_{n-2}({\mathbb A}_{\mathbb Q})$ respectively, we show that $F_2(s)=L(s,π)/L(s,ρ)$ has infinitely many poles.
We also establish analogous results for Artin $L$-functions and other $L$-functions not yet proven to be automorphic. Using the classification theorems of \cite{Ragh20} and \cite{BaRa20}, we show that cuspidal $L$-functions of ${\rm GL}_3({\mathbb A}_{\mathbb Q})$ are primitive in ${\mathfrak G}$, a monoid that contains both the Selberg class ${\mathcal{S}}$ and $L(s,σ)$ for all unitary cuspidal automorphic representations $σ$ of ${\rm GL}_n({\mathbb A}_{\mathbb Q})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_13895 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quotients of $L$-functions: degrees $n$ and $n-2$ Raghunathan, Ravi Number Theory 11F66, 11M41, 11F70 If $L(s,π)$ and $L(s,ρ)$ are the Dirichlet series attached to cuspidal automorphic representations $π$ and $ρ$ of ${\rm GL}_n({\mathbb A}_{\mathbb Q})$ and ${\rm GL}_{n-2}({\mathbb A}_{\mathbb Q})$ respectively, we show that $F_2(s)=L(s,π)/L(s,ρ)$ has infinitely many poles. We also establish analogous results for Artin $L$-functions and other $L$-functions not yet proven to be automorphic. Using the classification theorems of \cite{Ragh20} and \cite{BaRa20}, we show that cuspidal $L$-functions of ${\rm GL}_3({\mathbb A}_{\mathbb Q})$ are primitive in ${\mathfrak G}$, a monoid that contains both the Selberg class ${\mathcal{S}}$ and $L(s,σ)$ for all unitary cuspidal automorphic representations $σ$ of ${\rm GL}_n({\mathbb A}_{\mathbb Q})$. |
| title | Quotients of $L$-functions: degrees $n$ and $n-2$ |
| topic | Number Theory 11F66, 11M41, 11F70 |
| url | https://arxiv.org/abs/2403.13895 |