Zeros of Rankin-Selberg $L$-functions in families
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866910396688891904 |
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| author | Humphries, Peter Thorner, Jesse |
| author_facet | Humphries, Peter Thorner, Jesse |
| contents | Let $\mathfrak{F}_n$ be the set of all cuspidal automorphic representations $π$ of $\mathrm{GL}_n$ with unitary central character over a number field $F$. We prove the first unconditional zero density estimate for the set $\mathcal{S}=\{L(s,π\timesπ')\colonπ\in\mathfrak{F}_n\}$ of Rankin-Selberg $L$-functions, where $π'\in\mathfrak{F}_{n'}$ is fixed. We use this density estimate to establish (i) a hybrid-aspect subconvexity bound at $s=\frac{1}{2}$ for almost all $L(s,π\timesπ')\in \mathcal{S}$, (ii) a strong on-average form of effective multiplicity one for almost all $π\in\mathfrak{F}_n$, and (iii) a positive level of distribution for $L(s,π\times\tildeπ)$, in the sense of Bombieri-Vinogradov, for each $π\in\mathfrak{F}_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_05634 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Zeros of Rankin-Selberg $L$-functions in families Humphries, Peter Thorner, Jesse Number Theory Let $\mathfrak{F}_n$ be the set of all cuspidal automorphic representations $π$ of $\mathrm{GL}_n$ with unitary central character over a number field $F$. We prove the first unconditional zero density estimate for the set $\mathcal{S}=\{L(s,π\timesπ')\colonπ\in\mathfrak{F}_n\}$ of Rankin-Selberg $L$-functions, where $π'\in\mathfrak{F}_{n'}$ is fixed. We use this density estimate to establish (i) a hybrid-aspect subconvexity bound at $s=\frac{1}{2}$ for almost all $L(s,π\timesπ')\in \mathcal{S}$, (ii) a strong on-average form of effective multiplicity one for almost all $π\in\mathfrak{F}_n$, and (iii) a positive level of distribution for $L(s,π\times\tildeπ)$, in the sense of Bombieri-Vinogradov, for each $π\in\mathfrak{F}_n$. |
| title | Zeros of Rankin-Selberg $L$-functions in families |
| topic | Number Theory |
| url | https://arxiv.org/abs/2103.05634 |