A generalized PGL(2) Petersson/Bruggeman-Kuznetsov formula for analytic applications
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| Format: | Preprint |
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2024
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| _version_ | 1866908787006242816 |
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| author | Hu, Yueke Petrow, Ian Young, Matthew P. |
| author_facet | Hu, Yueke Petrow, Ian Young, Matthew P. |
| contents | We develop generalized Petersson/Bruggeman-Kuznetsov (PBK) formulas for specified local components at non-archimedean places. In fact, we introduce two hypotheses on non-archimedean test function pairs $f \leftrightarrow π(f)$, called geometric and spectral hypotheses, under which one obtains `nice' PBK formulas by the adelic relative trace function approach. Then, given a supercuspidal representation $σ$ of ${\rm PGL}_2(\mathbb{Q}_p)$, we study extensively the case that $π(f)$ is a projection onto the line of the newform if $π$ is isomorphc to $σ$ or its unramified quadratic twist, and $π(f) = 0$ otherwise. As a first application, we prove an optimal large sieve inequality for families of automorphic representations that arise in our framework. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_05672 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A generalized PGL(2) Petersson/Bruggeman-Kuznetsov formula for analytic applications Hu, Yueke Petrow, Ian Young, Matthew P. Number Theory 11F12, 11F30, 11F70, 11F72, 11F85 We develop generalized Petersson/Bruggeman-Kuznetsov (PBK) formulas for specified local components at non-archimedean places. In fact, we introduce two hypotheses on non-archimedean test function pairs $f \leftrightarrow π(f)$, called geometric and spectral hypotheses, under which one obtains `nice' PBK formulas by the adelic relative trace function approach. Then, given a supercuspidal representation $σ$ of ${\rm PGL}_2(\mathbb{Q}_p)$, we study extensively the case that $π(f)$ is a projection onto the line of the newform if $π$ is isomorphc to $σ$ or its unramified quadratic twist, and $π(f) = 0$ otherwise. As a first application, we prove an optimal large sieve inequality for families of automorphic representations that arise in our framework. |
| title | A generalized PGL(2) Petersson/Bruggeman-Kuznetsov formula for analytic applications |
| topic | Number Theory 11F12, 11F30, 11F70, 11F72, 11F85 |
| url | https://arxiv.org/abs/2411.05672 |