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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.04022 |
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Table of 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.