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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2501.04022 |
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| _version_ | 1866917887021678592 |
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| author | Miao, Xinchen |
| author_facet | Miao, Xinchen |
| contents | Let $F$ be a number field with adele ring $\mathbb{A}_F$, $π_1, π_2$ be two unitary cuspidal automorphic representations of $\mathrm{PGL}_2(\mathbb{A}_F)$ with finite analytic conductor. We study the twisted first moment of the triple product $L$-function $L(\frac{1}{2}, π\otimes π_1 \otimes π_2)$ and the Hecke eigenvalues $λ_π(\mathfrak{l})$, where $π$ is a unitary automorphic representation of $\mathrm{PGL}_2(\mathbb{A}_F)$ and $\mathfrak{l}$ is an integral ideal coprimes with the finite analytic conductor $C(π\otimes π_1 \otimes π_2)$. The estimation becomes a reciprocity formula between different moments of $L$-functions. Combining with the ideas and estimations established in [HMN23] and [MV10], we study the subconvexity problem for the triple product $L$-function in the level aspect and give a new explicit hybrid subconvexity bound for $L(\frac{1}{2}, π\otimes π_1 \otimes π_2)$, allowing joint ramifications and conductor dropping range. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04022 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral Reciprocity and Hybrid Subconvexity Bound for triple product $L$-functions Miao, Xinchen Number Theory Let $F$ be a number field with adele ring $\mathbb{A}_F$, $π_1, π_2$ be two unitary cuspidal automorphic representations of $\mathrm{PGL}_2(\mathbb{A}_F)$ with finite analytic conductor. We study the twisted first moment of the triple product $L$-function $L(\frac{1}{2}, π\otimes π_1 \otimes π_2)$ and the Hecke eigenvalues $λ_π(\mathfrak{l})$, where $π$ is a unitary automorphic representation of $\mathrm{PGL}_2(\mathbb{A}_F)$ and $\mathfrak{l}$ is an integral ideal coprimes with the finite analytic conductor $C(π\otimes π_1 \otimes π_2)$. The estimation becomes a reciprocity formula between different moments of $L$-functions. Combining with the ideas and estimations established in [HMN23] and [MV10], we study the subconvexity problem for the triple product $L$-function in the level aspect and give a new explicit hybrid subconvexity bound for $L(\frac{1}{2}, π\otimes π_1 \otimes π_2)$, allowing joint ramifications and conductor dropping range. |
| title | Spectral Reciprocity and Hybrid Subconvexity Bound for triple product $L$-functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2501.04022 |