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

TL;DR

This paper develops a generalized Petersson/Bruggeman-Kuznetsov formula for non-archimedean local components, introducing hypotheses that enable adelic trace formulas and applications like large sieve inequalities.

Contribution

It introduces geometric and spectral hypotheses for non-archimedean test functions, enabling new PBK formulas and automorphic representation analysis.

Findings

Established generalized PBK formulas under new hypotheses

Analyzed projections onto newform lines for supercuspidal representations

Proved an optimal large sieve inequality for specific automorphic families

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.