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Spectral Reciprocity: A Fourier--Analytic Approach

Liyang Yang

TL;DR

The paper develops a robust Fourier-analytic framework to establish spectral reciprocity between GL3 and GL2 spectra over number fields, extending Motohashi-type and Blomer–Khan-type recurrences to general automorphic representations (including non-cuspidal GL3). It provides explicit weight transforms in analytic newvector and spherical settings, and derives powerful arithmetic consequences such as first moments for GL3×GL2 L-functions, an explicit twisted fourth moment for GL2 L-functions, sharp fifth-moment bounds, subconvexity for triple-product L-functions, and simultaneous nonvanishing results. The methodology blends restricted Fourier expansions, additive and multiplicative Poisson summation, Rankin–Selberg integrals, and local spectral analysis, with careful meromorphic continuation to access values at the central point. By treating the cuspidal and non-cuspidal GL3 spectra uniformly and making weight transforms explicit, the work broadens the reach of spectral reciprocity to number fields and opens avenues for refined moment estimates and nonvanishing results across general arithmetic settings.

Abstract

We develop a Fourier--analytic framework for establishing spectral reciprocity formulas linking $\mathrm{GL}_3$ and $\mathrm{GL}_2$ automorphic spectra over number fields. The method applies uniformly to cuspidal and non-cuspidal $\mathrm{GL}_3$ representations and treats Motohashi-type and Blomer--Khan-type reciprocities in a parallel manner, revealing intrinsic connections between them and extending each to new settings. We also obtain explicit weight transforms in the analytic newvector and spherical cases. Applications include first-moment estimates for $\mathrm{GL}_3\times\mathrm{GL}_2$ $L$-functions over number fields, an explicit twisted fourth moment for $\mathrm{GL}_2$ $L$-functions over totally real fields, a sharp upper bound for the fifth moment, subconvexity for triple product $L$-functions, and new simultaneous nonvanishing results.

Spectral Reciprocity: A Fourier--Analytic Approach

TL;DR

The paper develops a robust Fourier-analytic framework to establish spectral reciprocity between GL3 and GL2 spectra over number fields, extending Motohashi-type and Blomer–Khan-type recurrences to general automorphic representations (including non-cuspidal GL3). It provides explicit weight transforms in analytic newvector and spherical settings, and derives powerful arithmetic consequences such as first moments for GL3×GL2 L-functions, an explicit twisted fourth moment for GL2 L-functions, sharp fifth-moment bounds, subconvexity for triple-product L-functions, and simultaneous nonvanishing results. The methodology blends restricted Fourier expansions, additive and multiplicative Poisson summation, Rankin–Selberg integrals, and local spectral analysis, with careful meromorphic continuation to access values at the central point. By treating the cuspidal and non-cuspidal GL3 spectra uniformly and making weight transforms explicit, the work broadens the reach of spectral reciprocity to number fields and opens avenues for refined moment estimates and nonvanishing results across general arithmetic settings.

Abstract

We develop a Fourier--analytic framework for establishing spectral reciprocity formulas linking and automorphic spectra over number fields. The method applies uniformly to cuspidal and non-cuspidal representations and treats Motohashi-type and Blomer--Khan-type reciprocities in a parallel manner, revealing intrinsic connections between them and extending each to new settings. We also obtain explicit weight transforms in the analytic newvector and spherical cases. Applications include first-moment estimates for -functions over number fields, an explicit twisted fourth moment for -functions over totally real fields, a sharp upper bound for the fifth moment, subconvexity for triple product -functions, and new simultaneous nonvanishing results.

Paper Structure

This paper contains 151 sections, 105 theorems, 942 equations.

Table of Contents

  1. Introduction
  2. The Spectral Reciprocity Formulas
  3. Spectral Reciprocity of Type I
  4. Spectral Reciprocity of Type II
  5. Applications of the Spectral Reciprocity Formulas
  6. The First Moment of $\mathrm{GL}_3\times\mathrm{GL}_2$ $L$-functions
  7. Explicit Twisted $4$-th Moment of $\mathrm{GL}_2$ $L$-functions
  8. The $5$-th Moment Estimates
  9. Subconvexity for Triple Product $L$-functions
  10. Simultaneous Nonvanishing
  11. Reciprocity Mechanisms via Partial Fourier Expansion
  12. Type I: Partial Fourier Expansion
  13. Type II: Partial Additive Poisson Summation
  14. Type III: Partial Multiplicative Poisson Summation
  15. Outline of the Paper
  16. ...and 136 more sections

Key Result

Theorem A

Let $|\mathop{\mathrm{Re}}\nolimits(s)|<1/2$. Then the following identity of meromorphic functions holds:

Theorems & Definitions (198)

  • Theorem A
  • Theorem B
  • Theorem C
  • Remark 1.1
  • Remark 1.2
  • Theorem D
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • ...and 188 more