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Spectral Moment Formulae for $\hbox{GL}(3)\times \hbox{GL}(2)$ $\hbox{L}$-functions II: The Eisenstein Case

Chung-Hang Kwan

TL;DR

This work proves an exact Motohashi-type reciprocity for GL(3)×GL(2) L-functions in the Eisenstein case by exploiting GL(3) period integrals and unipotent Fourier–Whittaker structures. It introduces a shifted cubic moment \\mathfrak{M}_{\boldsymbol{\alpha}}^{(3)}(s; H) and a corresponding integral transform \\mathcal{F}_{\boldsymbol{\alpha}}H, providing a unified, intrinsic description of the full main-term structure that matches the CFKRS moment conjectures. The authors perform a detailed spectral analysis, including Eisenstein contributions, and develop three new Mellin–Barnes identities to capture the complete set of main terms for both the cubic and fourth moments, with analytic continuation and polar-term analysis ensuring a complete CFKRS-compatible identity. The results illuminate the arithmetic sources and symmetries behind moment conjectures and offer a robust automorphic framework that could extend to other high-rank reciprocity phenomena and mixed moments.

Abstract

This work is the second in a series, following Part I (Algebra Number Theory 18.10 (2024)) and preceding Part III (Math. Ann. 391.1 (2025)). We continue our investigation of spectral moments of $\hbox{GL}(3)\times \hbox{GL}(2)$ $\hbox{L}$-functions from the perspective of period integrals. Using an identity between two distinct periods for the $\hbox{GL}(3)$ Eisenstein series, we establish an exact Motohashi-type identity linking the shifted cubic moment of $\hbox{GL}(2)$ $\hbox{L}$-functions to the shifted fourth moment of $\hbox{GL}(1)$ $\hbox{L}$-functions. In addition, we offer a novel, intrinsic and automorphic account for the sources and symmetries of the full set of main terms for both moments, in agreement with the CFKRS Moment Conjectures (Proc. Lond. Math. Soc.(3) 91 (2005)).

Spectral Moment Formulae for $\hbox{GL}(3)\times \hbox{GL}(2)$ $\hbox{L}$-functions II: The Eisenstein Case

TL;DR

This work proves an exact Motohashi-type reciprocity for GL(3)×GL(2) L-functions in the Eisenstein case by exploiting GL(3) period integrals and unipotent Fourier–Whittaker structures. It introduces a shifted cubic moment \\mathfrak{M}_{\boldsymbol{\alpha}}^{(3)}(s; H) and a corresponding integral transform \\mathcal{F}_{\boldsymbol{\alpha}}H, providing a unified, intrinsic description of the full main-term structure that matches the CFKRS moment conjectures. The authors perform a detailed spectral analysis, including Eisenstein contributions, and develop three new Mellin–Barnes identities to capture the complete set of main terms for both the cubic and fourth moments, with analytic continuation and polar-term analysis ensuring a complete CFKRS-compatible identity. The results illuminate the arithmetic sources and symmetries behind moment conjectures and offer a robust automorphic framework that could extend to other high-rank reciprocity phenomena and mixed moments.

Abstract

This work is the second in a series, following Part I (Algebra Number Theory 18.10 (2024)) and preceding Part III (Math. Ann. 391.1 (2025)). We continue our investigation of spectral moments of -functions from the perspective of period integrals. Using an identity between two distinct periods for the Eisenstein series, we establish an exact Motohashi-type identity linking the shifted cubic moment of -functions to the shifted fourth moment of -functions. In addition, we offer a novel, intrinsic and automorphic account for the sources and symmetries of the full set of main terms for both moments, in agreement with the CFKRS Moment Conjectures (Proc. Lond. Math. Soc.(3) 91 (2005)).
Paper Structure (43 sections, 35 theorems, 168 equations)

This paper contains 43 sections, 35 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Moment conjectures
  4. Our main results
  5. The reciprocity formula
  6. Main terms
  7. Integral transform
  8. Sketch, road map and discussion of our argument
  9. Additional Remarks
  10. Outline
  11. Notations and conventions
  12. The parameters (Convention \ref{['con: smallshift']})
  13. Test functions (Regularity \ref{['regassuCeta']})
  14. Groups
  15. Additive characters
  16. ...and 28 more sections

Key Result

Theorem 1.3

Let $H\in \mathcal{C}_{\eta}$. On the vertical strip $1/4 + \epsilon_{0} < \mathop{\mathrm{Re}}\nolimits s < 3/4$, the following formula holds: where

Theorems & Definitions (77)

  • Theorem 1.3
  • Theorem 1.4: Propositions \ref{['1swaprop']}, \ref{['Eissimpl']}, & \ref{['secMTcom']}
  • Theorem 1.5: Corollary \ref{['cor: hypergtrans']}
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 67 more