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Expository notes on Spectral Reciprocity with Explicit Transform

Haonan Gu

TL;DR

Expository notes develop a modular framework to analyze smoothed GL3 spectral averages of $L(1/2,\varphi\times u_j)$ over SL2 Maass cusp forms by combining the Kuznetsov trace formula, the GL3 Voronoi summation, and $t$-aspect second-moment bounds. The main result isolates a diagonal contribution $L(1,\varphi)\mathcal{H}_0[h]$ and proves non-diagonal and Eisenstein terms are bounded by $T^{5/4+\varepsilon}$ and $T^{3/2+\varepsilon}$, respectively, with potential improvements from stronger $t$-aspect moments; the argument also develops spectral-window tools, including Kuznetsov-admissible weights and near-flat windows. The work provides a transparent, step-by-step approach to spectral reciprocity in GL3 with explicit transforms, and it discusses generalizations to higher level, twists, and mollified moments. Collectively, the paper clarifies how diagonal, off-diagonal, and continuous-spectrum components interact in smoothed GL3 averages and sets a foundation for sharper moment bounds via improved $L$-functions technology.

Abstract

We assemble three basic analytic inputs -- the Kuznetsov trace formula on $\mathrm{SL}_2(\mathbb Z)$ with explicit continuous spectrum, the $\mathrm{GL}_3$ Voronoi formula, and $t$-aspect second-moment bounds for $L(1/2+it,\varphi)$ -- into a single framework for a smoothed $\mathrm{GL}_3$ spectral average. For a fixed Hecke-Maass cusp form $\varphi$ on $\mathrm{SL}_3(\mathbb Z)$, we evaluate a weight-$0$ spectral average of $L(1/2,\varphi\times f_j)$ over the $\mathrm{GL}_2$ Maass spectrum. In the Kuznetsov normalization where the diagonal transform has density $t\tanh(πt)$, the diagonal contributes exactly $2\mathcal H_0[h]$; the off-diagonal and the continuous spectrum are bounded with power savings consistent with the currently best unconditional second-moment bounds in the $\mathrm{GL}_3$ $t$-aspect. The argument is organized into a sequence of steps: normalizations and approximate functional equations, insertion into Kuznetsov, Voronoi summation on $\mathrm{GL}_3$, a bilinear estimate for the off-diagonal, evaluation of the diagonal and the Eisenstein contribution, moment refinements and parameter optimization, and finally plateau-smooth spectral windows and standard generalizations.

Expository notes on Spectral Reciprocity with Explicit Transform

TL;DR

Expository notes develop a modular framework to analyze smoothed GL3 spectral averages of over SL2 Maass cusp forms by combining the Kuznetsov trace formula, the GL3 Voronoi summation, and -aspect second-moment bounds. The main result isolates a diagonal contribution and proves non-diagonal and Eisenstein terms are bounded by and , respectively, with potential improvements from stronger -aspect moments; the argument also develops spectral-window tools, including Kuznetsov-admissible weights and near-flat windows. The work provides a transparent, step-by-step approach to spectral reciprocity in GL3 with explicit transforms, and it discusses generalizations to higher level, twists, and mollified moments. Collectively, the paper clarifies how diagonal, off-diagonal, and continuous-spectrum components interact in smoothed GL3 averages and sets a foundation for sharper moment bounds via improved -functions technology.

Abstract

We assemble three basic analytic inputs -- the Kuznetsov trace formula on with explicit continuous spectrum, the Voronoi formula, and -aspect second-moment bounds for -- into a single framework for a smoothed spectral average. For a fixed Hecke-Maass cusp form on , we evaluate a weight- spectral average of over the Maass spectrum. In the Kuznetsov normalization where the diagonal transform has density , the diagonal contributes exactly ; the off-diagonal and the continuous spectrum are bounded with power savings consistent with the currently best unconditional second-moment bounds in the -aspect. The argument is organized into a sequence of steps: normalizations and approximate functional equations, insertion into Kuznetsov, Voronoi summation on , a bilinear estimate for the off-diagonal, evaluation of the diagonal and the Eisenstein contribution, moment refinements and parameter optimization, and finally plateau-smooth spectral windows and standard generalizations.

Paper Structure

This paper contains 29 sections, 3 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Main theorem
  3. Kuznetsov diagonal transform
  4. Kuznetsov weights (discrete and continuous)
  5. Admissible test functions and spectral windows
  6. The spectral average
  7. Overview of the argument
  8. Normalizations and approximate functional equations
  9. Archimedean factors and completed $L$--functions
  10. A Mellin--Barnes kernel
  11. Approximate functional equation for $L(1/2,\varphi\times u_j)$
  12. Approximate functional equation for $L(1/2+it,\varphi)$
  13. Kuznetsov and the first spectral decomposition
  14. Insertion of the approximate functional equation
  15. Kuznetsov formula (even spectrum)
  16. ...and 14 more sections

Key Result

Theorem 2.2

Let $\varphi$ be a fixed self--dual Hecke--Maass cusp form on $\mathrm{SL}_3(\mathbb{Z})$. With $\mathcal{H}_0$ as in eq:H0-def and with $h$ as in eq:h-def, one has Here:

Theorems & Definitions (6)

  • Definition 2.1: Kuznetsov--admissible test functions
  • Theorem 2.2: Spectral evaluation with unconditional error terms
  • Remark 2.3: Effect of improved $t$--aspect moments
  • Lemma 4.1: Uniform derivative bounds
  • Proposition 7.1: Off--diagonal bound
  • Remark 7.2