Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on Poisson summation for GL(2)

Tian An Wong

TL;DR

This work revisits the cancellation of the trivial representation in the GL(2) Arthur–Selberg trace formula using adelic Poisson summation, providing two parallel adelic frameworks (Poisson après Langlands and Poisson après Matz) that reexpress the elliptic contribution to expose the trivial term. It develops the necessary measure theory, completion techniques, and convergence analyses to extend the results from compactly supported test functions to the broader class of functions used in the r-trace formula, and connects these constructions to L-functions through archimedean basic functions. The main contributions include a rigorous adelic reformulation that yields the cancellation of the trivial representation, a convergence-controlled dual sum, and a meaningful interpretation in terms of L-functions and Ramanujan-type bounds. Together, these results advance the Beyond Endoscopy program for GL(2) by solidifying adelic methods for dealing with non-tempered contributions and by providing a robust template for incorporating L-functions and archimedean data into trace formulas.

Abstract

Using analytic number theory techniques, Altuğ showed that the contribution of the trivial representation to the Arthur-Selberg trace formula for GL(2) over $\Q$ could be cancelled by applying a modified Poisson summation formula to the regular elliptic contribution. Drawing on recent works, we re-examine these methods from an adelic perspective.

A note on Poisson summation for GL(2)

TL;DR

This work revisits the cancellation of the trivial representation in the GL(2) Arthur–Selberg trace formula using adelic Poisson summation, providing two parallel adelic frameworks (Poisson après Langlands and Poisson après Matz) that reexpress the elliptic contribution to expose the trivial term. It develops the necessary measure theory, completion techniques, and convergence analyses to extend the results from compactly supported test functions to the broader class of functions used in the r-trace formula, and connects these constructions to L-functions through archimedean basic functions. The main contributions include a rigorous adelic reformulation that yields the cancellation of the trivial representation, a convergence-controlled dual sum, and a meaningful interpretation in terms of L-functions and Ramanujan-type bounds. Together, these results advance the Beyond Endoscopy program for GL(2) by solidifying adelic methods for dealing with non-tempered contributions and by providing a robust template for incorporating L-functions and archimedean data into trace formulas.

Abstract

Using analytic number theory techniques, Altuğ showed that the contribution of the trivial representation to the Arthur-Selberg trace formula for GL(2) over could be cancelled by applying a modified Poisson summation formula to the regular elliptic contribution. Drawing on recent works, we re-examine these methods from an adelic perspective.

Paper Structure

This paper contains 19 sections, 19 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Poisson après Langlands
  4. Relation to the $r$-trace formula
  5. Poisson après Matz
  6. Application to $L$-functions
  7. Poisson après Langlands
  8. Measures
  9. Trivial representation
  10. The regular elliptic contribution
  11. The problem of completion
  12. Convergence of the dual sum
  13. Poisson summation
  14. Completion of main theorem
  15. Poisson après Matz
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $f \in {\mathscr C}(G)$. Assume that $\theta_{f_v}$ is continuous on $F_v$ when $F_v$ is nonarchimedean with residual characteristic $2$. Then the elliptic part of the trace formula for ${\mathrm{GL}}(2)$ can be expressed as

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 25 more