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arXiv · 2411.05672

A generalized PGL(2) Petersson/Bruggeman-Kuznetsov formula for analytic applications

Yueke Hu·Ian Petrow·Matthew P. Young
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Abstract

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 $f \leftrightarrow \pi(f)$, called geometric and spectral hypotheses, under which one obtains `nice' PBK formulas by the adelic relative trace function approach. Then, given a supercuspidal representation $\sigma$ of ${\rm PGL}_2(\mathbb{Q}_p)$, we study extensively the case that $\pi(f)$ is a projection onto the line of the newform if $\pi$ is isomorphc to $\sigma$ or its unramified quadratic twist, and $\pi(f) = 0$ otherwise. As a first application, we prove an optimal large sieve inequality for families of automorphic representations that arise in our framework.

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Yueke Hu, Ian Petrow, Matthew P. Young. 2024-11-08. A generalized PGL(2) Petersson/Bruggeman-Kuznetsov formula for analytic applications. https://arxiv.org/abs/2411.05672

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