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Beyond Endoscopy: Poisson Summation Formula and Kuznetsov Trace Formula on $\mathrm{GL}_2$

Zhaolin Li

Abstract

In the first part of this paper, we present a direct proof of a Poisson summation formula on the Whittaker space of $\mathrm{GL}_2$, which underlies the local Hankel transform computed by H. Jacquet. In the second part, we derive a Kuznetsov-type trace formula for incorporating non-standard test functions. This formulation reveals that the Poisson summation formula naturally yields the functional equation for the standard $L$-functions of $\mathrm{GL}_2$.

Beyond Endoscopy: Poisson Summation Formula and Kuznetsov Trace Formula on $\mathrm{GL}_2$

Abstract

In the first part of this paper, we present a direct proof of a Poisson summation formula on the Whittaker space of , which underlies the local Hankel transform computed by H. Jacquet. In the second part, we derive a Kuznetsov-type trace formula for incorporating non-standard test functions. This formulation reveals that the Poisson summation formula naturally yields the functional equation for the standard -functions of .

Paper Structure

This paper contains 21 sections, 33 theorems, 245 equations.

Table of Contents

  1. Introduction
  2. Organization of the Paper
  3. Notations and Convensions
  4. Acknowdegement
  5. Local Theory
  6. Speces of Orbital Integrals
  7. Hankel Tranforms
  8. Irregular Distributions
  9. Global Theory
  10. Orbital Integrals, Hankel Transforms, and the Poisson Summation Formula
  11. Poisson Summation Formula: An Indirect Proof
  12. Poisson Summation Formula: A Direct Proof
  13. Kuznetsov Trace Formula with Godement-Jacquet Sections
  14. The Spectrum Side of the Trace Formula
  15. The Geometric Side of the Trace Formula
  16. ...and 6 more sections

Key Result

Proposition 2.7

The restriction of $f^{\circ}$ to ${\mathrm {T}}({\mathfrak {o}}_F)$ is $(\,\mathrm{d} t)^{\frac{1}{2}}$ in the case that $\psi$ is unramified.

Theorems & Definitions (78)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Proposition 2.8
  • proof
  • ...and 68 more