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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.14741 |
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| _version_ | 1866911510420258816 |
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| author | Hu, Yueke Petrow, Ian Young, Matthew P. |
| author_facet | Hu, Yueke Petrow, Ian Young, Matthew P. |
| contents | We prove Lindelöf-on-average upper bounds on the cubic moment of central values of $L$-functions over certain families of $\operatorname{PGL}_2/\mathbb{Q}$ automorphic representations $π$ given by specifying the local representation $π_p$ of $π$ at finitely many primes. Such bounds were previously known in the case that $π_p$ belongs to the principal series or is a ramified quadratic twist of the Steinberg representation; here we handle the supercuspidal case. Crucially, we use new Petersson/Bruggeman-Kuznetsov forumulas for supercuspidal local component families recently developed by the authors. As corollaries, we derive Weyl-strength subconvex bounds for central values of $\operatorname{PGL}_2$ $L$-functions in the square-full aspect, and in the depth aspect, or in a hybrid of these two situations. A special case of our results is the Weyl-subconvex bound for all cusp forms of level $p^2$. Previously, such a bound was only known for forms that are twists from level $p$, which cover roughly half of the level $p^2$ forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_14741 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The cubic moment of $L$-functions for specified local component families Hu, Yueke Petrow, Ian Young, Matthew P. Number Theory We prove Lindelöf-on-average upper bounds on the cubic moment of central values of $L$-functions over certain families of $\operatorname{PGL}_2/\mathbb{Q}$ automorphic representations $π$ given by specifying the local representation $π_p$ of $π$ at finitely many primes. Such bounds were previously known in the case that $π_p$ belongs to the principal series or is a ramified quadratic twist of the Steinberg representation; here we handle the supercuspidal case. Crucially, we use new Petersson/Bruggeman-Kuznetsov forumulas for supercuspidal local component families recently developed by the authors. As corollaries, we derive Weyl-strength subconvex bounds for central values of $\operatorname{PGL}_2$ $L$-functions in the square-full aspect, and in the depth aspect, or in a hybrid of these two situations. A special case of our results is the Weyl-subconvex bound for all cusp forms of level $p^2$. Previously, such a bound was only known for forms that are twists from level $p$, which cover roughly half of the level $p^2$ forms. |
| title | The cubic moment of $L$-functions for specified local component families |
| topic | Number Theory |
| url | https://arxiv.org/abs/2506.14741 |