Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on a classical relative trace formula

Zhining Wei

TL;DR

This note presents a classical derivation of a relative trace formula for holomorphic cusp forms on $\mathrm{SL}_2(\mathbb{Z})$, yielding a closed formula for the eigenvalue-weighted second moment of central $L$-values. It decomposes the trace into spectral and geometric sides, with explicit expressions for singular and regular orbital integrals and a regularization that handles divergences, ultimately matching Kuznetsov’s preprint (and Iwaniec–Sarnak). The main result expresses the weighted second moment as $M_2(n;s_1,s_2)+E(n;s_1,s_2)$, with a detailed breakdown of terms from the geometric side and their analytic continuations, and a corollary at $(s_1,s_2)=(0,0)$. The approach combines classical kernel methods with hypergeometric representations and divisor arithmetic to produce explicit, computable terms, reinforcing connections to the Kuznetsov framework and related trace-formula techniques.

Abstract

In this note, we derive a relative trace formula (RTF) using classical methods. We obtain a closed formula for the second moment of the central values of holomorphic cusp forms, a result originally established in Kuznetsov's preprint.

A note on a classical relative trace formula

TL;DR

This note presents a classical derivation of a relative trace formula for holomorphic cusp forms on , yielding a closed formula for the eigenvalue-weighted second moment of central -values. It decomposes the trace into spectral and geometric sides, with explicit expressions for singular and regular orbital integrals and a regularization that handles divergences, ultimately matching Kuznetsov’s preprint (and Iwaniec–Sarnak). The main result expresses the weighted second moment as , with a detailed breakdown of terms from the geometric side and their analytic continuations, and a corollary at . The approach combines classical kernel methods with hypergeometric representations and divisor arithmetic to produce explicit, computable terms, reinforcing connections to the Kuznetsov framework and related trace-formula techniques.

Abstract

In this note, we derive a relative trace formula (RTF) using classical methods. We obtain a closed formula for the second moment of the central values of holomorphic cusp forms, a result originally established in Kuznetsov's preprint.

Paper Structure

This paper contains 19 sections, 10 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The spectral side
  4. The geometric side
  5. The complex logarithm and complex power functions
  6. Integral representations for hypergeometric functions
  7. The divisor functions
  8. Some possible extensions
  9. The singular orbital integrals
  10. The small cell orbital integrals
  11. The dual orbital integrals
  12. The singular orbital integrals: a summary
  13. The regular orbital integral
  14. The calculation of $J_{\operatorname{Reg}}^1(\textbf{s},n)$
  15. The calculation of $J_{\operatorname{Reg}}^3(\textbf{s},n)$
  16. ...and 4 more sections

Key Result

Theorem A

(Kuznetsov, preprint $1994$) Assume that $\textbf{s}=(s_1,s_2)$ is in the region: and $s_1-s_2$ is not an integer. Then where and Here: Moreover, $(s_1,s_2)=(0,0)$ is a removable singularity.

Theorems & Definitions (23)

  • Theorem A
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 13 more