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The weight two and opposite sign cases for the Fourier relative trace formulas

Matteo Di Scipio

Abstract

We provide an adelic relative trace formula proof to the Petersson/Bruggeman-Kuznetsov (PBK) formulas, specifically in the holomorphic case for $κ=2$ and the non-holomorphic case for $m_1m_2<0$. Given two sets of hypothesis on the non archimedean test function $f$, called the geometric and spectral assumptions, this approach allows us to obtain refined PBK formulas.

The weight two and opposite sign cases for the Fourier relative trace formulas

Abstract

We provide an adelic relative trace formula proof to the Petersson/Bruggeman-Kuznetsov (PBK) formulas, specifically in the holomorphic case for and the non-holomorphic case for . Given two sets of hypothesis on the non archimedean test function , called the geometric and spectral assumptions, this approach allows us to obtain refined PBK formulas.
Paper Structure (25 sections, 23 theorems, 161 equations)

This paper contains 25 sections, 23 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Statement of main results
  4. Parity equidistribution
  5. Notation and conventions
  6. Acknowledgments
  7. Preliminary results
  8. Local representation theory
  9. Global representation theory
  10. Whittaker/Fourier coefficients
  11. Distribution results for spectral parameters
  12. Kloosterman sums and local weights
  13. An overview of the relative trace approach
  14. The pre-trace formula
  15. The BK formula in the same signs case
  16. ...and 10 more sections

Key Result

Theorem 1.2.1

Suppose $f=\bigotimes_p f_p \in \mathcal{H}$ is non-zero, and that for each $p$ that $f_p$ is supported inside the subgroup of matrices $g \in G(\mathbb{Q}_p)$ with $v_p(\det g)\in 2 \mathbb{Z}$. For $m_1,m_2 \in \tfrac{1}{N}\mathbb{Z}$ with $m_1m_2<0$ we have that as absolute convergent sums/integrals. Here

Theorems & Definitions (44)

  • Theorem 1.2.1: Unrefined BK formula
  • Theorem 1.2.2: Refined BK formula
  • Theorem 1.2.3: Petersson formula
  • Remark 1.2.4
  • Corollary 1.3.1: Parity equidistribution
  • Proposition 2.3.1
  • proof
  • Proposition 2.4.1: Weak Weyl's law
  • proof
  • Lemma 2.4.2
  • ...and 34 more