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

Yueke Hu, Ian Petrow, Matthew P. Young

TL;DR

The paper develops generalized Petersson/Bruggeman/Kuznetsov (PBK) formulas at finite places for PGL2 by employing the adelic relative trace formula and introducing geometric and spectral hypotheses on non-archimedean test-function pairs. It proves a refined PBK identity (theorem theoGeomSpec) that isolates spectral contributions from representations with prescribed local components, and expresses the diagonal term via local Plancherel data, while the geometric side is governed by generalized Kloosterman sums $H(m,n;c)$ with controlled moduli and conductors. The authors provide several explicit local constructions (e.g., supercuspidal and principal-series test functions) that realize the narrow spectral support predicted by the hypotheses, and they demonstrate powerful applications including a harmonically-weighted Weyl-Selberg law, optimal large sieve inequalities, and moments bounds for L-functions. The framework yields precise arithmetic information about the Kloosterman sums and spectral weights, enabling equidistribution statements and robust analytic applications in the non-archimedean setting with potential for broad generalization. The results thus connect local representation-theoretic data with global automorphic phenomena, offering a versatile tool for analytic number theory on GL2/QQ.

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 $f \leftrightarrow π(f)$, 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.

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

TL;DR

The paper develops generalized Petersson/Bruggeman/Kuznetsov (PBK) formulas at finite places for PGL2 by employing the adelic relative trace formula and introducing geometric and spectral hypotheses on non-archimedean test-function pairs. It proves a refined PBK identity (theorem theoGeomSpec) that isolates spectral contributions from representations with prescribed local components, and expresses the diagonal term via local Plancherel data, while the geometric side is governed by generalized Kloosterman sums with controlled moduli and conductors. The authors provide several explicit local constructions (e.g., supercuspidal and principal-series test functions) that realize the narrow spectral support predicted by the hypotheses, and they demonstrate powerful applications including a harmonically-weighted Weyl-Selberg law, optimal large sieve inequalities, and moments bounds for L-functions. The framework yields precise arithmetic information about the Kloosterman sums and spectral weights, enabling equidistribution statements and robust analytic applications in the non-archimedean setting with potential for broad generalization. The results thus connect local representation-theoretic data with global automorphic phenomena, offering a versatile tool for analytic number theory on GL2/QQ.

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 , called geometric and spectral hypotheses, under which one obtains `nice' PBK formulas by the adelic relative trace function approach. Then, given a supercuspidal representation of , we study extensively the case that is a projection onto the line of the newform if is isomorphc to or its unramified quadratic twist, and otherwise. As a first application, we prove an optimal large sieve inequality for families of automorphic representations that arise in our framework.

Paper Structure

This paper contains 84 sections, 82 theorems, 533 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Statement of generalized Petersson/Bruggeman/Kuznetsov formula
  4. Relations between parameters
  5. Level versus ramification
  6. Level versus conductors of representations
  7. Level versus geometric conductor
  8. Conductors of representations versus $f(1)$
  9. Level versus $f(1)$
  10. Diagonal weight versus $f(1)$
  11. Weighted Weyl-Selberg Law and equidistribution
  12. Large sieve inequality
  13. Moments of $L$-functions
  14. Acknowledgements
  15. Notation, conventions, and measure normalizations
  16. ...and 69 more sections

Key Result

Theorem 1.7

Let $f \in \mathcal{H}_{\rm fin}$ be a pure tensor satisfing the geometric and spectral assumptions. For all $m_1,m_2 \in \mathbb{Z}$ with $m_1m_2 >0$ and $(m_1m_2,N)=1$ we have where $(\text{ cts. })$ is a similar continuous spectrum term that we give explicitly in ctsdots.

Theorems & Definitions (184)

  • Definition 1.1: Family cut out by $f$
  • Definition 1.2
  • Definition 1.3: Admissible moduli
  • Definition 1.5: Newform projector
  • Definition 1.6: Local family
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Lemma 1.10
  • Corollary 1.11: Harmonically-weighted Weyl-Selberg Law
  • ...and 174 more