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Trace formula and functional equation

Chung-Hang Kwan, Wing Hong Leung

TL;DR

This work develops a beyond-endoscopic approach to analytic continuation and the functional equation for standard L-functions attached to holomorphic cusp forms with level and nebentypus. By combining Petersson's trace formula with a careful analytic–arithmetic reciprocity (including Hankel inversion and Poisson summation), the authors transform spectral sums into dual arithmetic sums, then return them to spectral form via a reverse Petersson step. The main contributions include an averaged Voronoi-type identity for primitive nebentypus, a precise functional-equation formula under suitable assumptions, and parallel results for trivial nebentypus (square-free level) and the $D=1$ case, all proven without relying on integral representations of L-functions. The results illuminate how root numbers and conductors influence the functional equation and demonstrate the viability of a fully trace-formula–driven route to L-function analytic continuation, with potential implications for automorphic transfer and beyond-endoscopic investigations.

Abstract

We present a "beyond-endoscopic" treatment of the functional equation for the standard $L$-function of a holomorphic cusp form with level and nebentypus. We use Petersson's formula and methods from Venkatesh's thesis and "spectral reciprocity".

Trace formula and functional equation

TL;DR

This work develops a beyond-endoscopic approach to analytic continuation and the functional equation for standard L-functions attached to holomorphic cusp forms with level and nebentypus. By combining Petersson's trace formula with a careful analytic–arithmetic reciprocity (including Hankel inversion and Poisson summation), the authors transform spectral sums into dual arithmetic sums, then return them to spectral form via a reverse Petersson step. The main contributions include an averaged Voronoi-type identity for primitive nebentypus, a precise functional-equation formula under suitable assumptions, and parallel results for trivial nebentypus (square-free level) and the case, all proven without relying on integral representations of L-functions. The results illuminate how root numbers and conductors influence the functional equation and demonstrate the viability of a fully trace-formula–driven route to L-function analytic continuation, with potential implications for automorphic transfer and beyond-endoscopic investigations.

Abstract

We present a "beyond-endoscopic" treatment of the functional equation for the standard -function of a holomorphic cusp form with level and nebentypus. We use Petersson's formula and methods from Venkatesh's thesis and "spectral reciprocity".

Paper Structure

This paper contains 32 sections, 14 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Beyond Endoscopy
  3. Set-up
  4. Main results
  5. Primitive nebentypus
  6. Trivial nebentypus
  7. Sketch of argument
  8. Local ingredients
  9. Global ingredients
  10. Trace formula
  11. Comments on the assumptions
  12. A related work
  13. Acknowledgement
  14. Preliminaries
  15. Integral transforms and Bessel functions
  16. ...and 17 more sections

Key Result

Theorem 1.2

Let $D>1$, $k\ge 4$ and $\ell\ge 1$ are integers, and $\chi\, (\bmod\, D)$ is a primitive character satisfying $\chi(-1)=(-1)^k$. Write $\ell= \ell_{0}\ell'$, where $\ell_{0}:=(\ell, D^{\infty})$ and $(\ell', D)=1$. The notations $\ell_{0}$, $\ell'$ will be used across this text with the same meanin for any $g\in C_{c}^{\infty}(0, \infty)$ and orthogonal basis $\mathcal{B}_{k}(D, \chi)$ of $S_{k}

Theorems & Definitions (19)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4: Hankel inversion formula
  • Lemma 2.5
  • Lemma 2.6
  • ...and 9 more