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Character sum, reciprocity and Voronoi formula

Chung-Hang Kwan, Wing Hong Leung

TL;DR

This work constructs a novel four-variable character-sum reciprocity identity that functions as a twisted, non-archimedean analogue of Weber’s integral. By integrating Petersson trace formulas, Poisson summation, and additive reciprocity with a spectral framework inspired by Venkatesh, the authors obtain a new, spectral proof of the twisted Hecke $L$-function functional equation and the Voronoi formula for holomorphic cusp forms. The approach hinges on a detailed local analysis of character sums and a careful cancellation that enables a backward-Petersson maneuver, culminating in an explicit Voronoi-type transformation with arithmetic and archimedean data intertwined. The results advance beyond-endoscopy methods and illuminate how nontrivial reciprocity in character sums can drive global automorphic identities, with analytic continuation and polynomial growth established for the associated Dirichlet-polynomial averages. An accompanying corollary extends these ideas to the full functional equation landscape for twisted $L$-functions in this setting.

Abstract

We prove a novel four-variable character sum identity which serves as a twisted, non-archimedean counterpart to Weber's integrals for Bessel functions. Using this identity and ideas from Venkatesh's thesis, we present a new, spectral proof of the Voronoi formula for classical modular forms.

Character sum, reciprocity and Voronoi formula

TL;DR

This work constructs a novel four-variable character-sum reciprocity identity that functions as a twisted, non-archimedean analogue of Weber’s integral. By integrating Petersson trace formulas, Poisson summation, and additive reciprocity with a spectral framework inspired by Venkatesh, the authors obtain a new, spectral proof of the twisted Hecke -function functional equation and the Voronoi formula for holomorphic cusp forms. The approach hinges on a detailed local analysis of character sums and a careful cancellation that enables a backward-Petersson maneuver, culminating in an explicit Voronoi-type transformation with arithmetic and archimedean data intertwined. The results advance beyond-endoscopy methods and illuminate how nontrivial reciprocity in character sums can drive global automorphic identities, with analytic continuation and polynomial growth established for the associated Dirichlet-polynomial averages. An accompanying corollary extends these ideas to the full functional equation landscape for twisted -functions in this setting.

Abstract

We prove a novel four-variable character sum identity which serves as a twisted, non-archimedean counterpart to Weber's integrals for Bessel functions. Using this identity and ideas from Venkatesh's thesis, we present a new, spectral proof of the Voronoi formula for classical modular forms.

Paper Structure

This paper contains 25 sections, 15 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Character sums and modular forms
  3. Global results
  4. A road map for Theorem \ref{['ourmainthmtwist']}
  5. Two dualities
  6. Multiplicative (analytic--arithmetic) cancellation
  7. Three-fold reciprocity and local analysis
  8. Additive cancellations and back to the global aspect
  9. Remarks on analytic continuation
  10. Acknowledgement
  11. Preliminaries
  12. Setting the stage: Petersson--Poisson--Reciprocity
  13. Step 1: Petersson trace formula
  14. Step 2: Poisson summation
  15. Step 3: Additive reciprocity.
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $q, \ell \ge 1$, $a, b, u, v$ be integers such that $ab|q^\infty$ and $(uv,q)=1$. The notation $c\mid q^{\infty}$ refers to $c\mid q^{k}$ for some $k\ge 1$. Let $\chi\, (\bmod\, q)$ be a Dirichlet character. Define Then we have

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 1.5
  • Lemma 2.1
  • Lemma 2.2
  • ...and 14 more