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Mixed fourth moments of automorphic forms and the shifted moments of $L$-functions

Chengliang Guo

TL;DR

This work analyzes the mixed fourth moments of automorphic forms, specifically the interaction between Hecke--Maass cusp forms and Eisenstein series of type $(2,2)$, under GRH and GRC. The authors derive an explicit asymptotic for the mixed moment $\langle \phi^2, |E_T|^2\rangle$, separating main, continuous, and discrete spectral contributions via regularized Plancherel and Watson-type formulas, while also introducing a truncated Eisenstein framework to reveal an independent random-wave behavior on compact subsets. The centerpiece is a shifted-moments analysis for $L$-functions, implemented through Soundararajan’s method, to bound shifted central values and their density, which in turn yields the needed control on the spectral sums. The results lead to a precise nonequidistribution statement on the full fundamental domain, a smooth-averaged equidistribution result for truncated Eisenstein series, and conjectural joint-value-distribution statements for automorphic forms, with concrete implications for Rankin--Selberg $L$-functions and broader quantum chaos contexts.

Abstract

In this article, we study the mixed fourth moments of Hecke--Maass cusp forms and Eisenstein series with type $(2, 2)$. Under the assumptions of the Generalized Riemann Hypothesis (GRH) and the Generalized Ramanujan Conjecture (GRC), we establish asymptotic formulas for these moments. Our results give an interesting non-equidistribution phenomenon over the full fundamental domain. In fact, this independent equidistribution should be true in a compact set. We further investigate this behaviour by examining a truncated version involving truncated Eisenstein series. Additionally, we propose a conjecture on the joint value distribution of Eisenstein series. The proofs are based on the bounds of the shifted mixed moments of $L$-functions.

Mixed fourth moments of automorphic forms and the shifted moments of $L$-functions

TL;DR

This work analyzes the mixed fourth moments of automorphic forms, specifically the interaction between Hecke--Maass cusp forms and Eisenstein series of type , under GRH and GRC. The authors derive an explicit asymptotic for the mixed moment , separating main, continuous, and discrete spectral contributions via regularized Plancherel and Watson-type formulas, while also introducing a truncated Eisenstein framework to reveal an independent random-wave behavior on compact subsets. The centerpiece is a shifted-moments analysis for -functions, implemented through Soundararajan’s method, to bound shifted central values and their density, which in turn yields the needed control on the spectral sums. The results lead to a precise nonequidistribution statement on the full fundamental domain, a smooth-averaged equidistribution result for truncated Eisenstein series, and conjectural joint-value-distribution statements for automorphic forms, with concrete implications for Rankin--Selberg -functions and broader quantum chaos contexts.

Abstract

In this article, we study the mixed fourth moments of Hecke--Maass cusp forms and Eisenstein series with type . Under the assumptions of the Generalized Riemann Hypothesis (GRH) and the Generalized Ramanujan Conjecture (GRC), we establish asymptotic formulas for these moments. Our results give an interesting non-equidistribution phenomenon over the full fundamental domain. In fact, this independent equidistribution should be true in a compact set. We further investigate this behaviour by examining a truncated version involving truncated Eisenstein series. Additionally, we propose a conjecture on the joint value distribution of Eisenstein series. The proofs are based on the bounds of the shifted mixed moments of -functions.
Paper Structure (31 sections, 27 theorems, 290 equations)

This paper contains 31 sections, 27 theorems, 290 equations.

Table of Contents

  1. Introduction
  2. Mixed fourth moments problem of automorphic forms
  3. Conjecture on joint value distribution of automorphic forms
  4. Applications to the moments of Rankin--Selberg $L$-functions
  5. Fractional mixed moments problem of $L$-functions
  6. Ideas of proofs
  7. Plan for this paper
  8. Preliminaries
  9. Automorphic forms
  10. Stirling's formula
  11. Rankin--Selberg theory and Watson's formula
  12. Regularized inner product and regularized Plancherel formula
  13. The proof of Theorem \ref{['Nonequidistribution-theorem']}
  14. The main term
  15. The contribution of continuous spectrum
  16. ...and 16 more sections

Key Result

Theorem 1.1

Assume GRH and GRC. Let $\phi$ be a Hecke--Maass form with spectral parameter $t_{\phi}$. For $T \geq 1$ we get as $\min\{t_{\phi}, T\} \rightarrow \infty$.

Theorems & Definitions (52)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Conjecture 1.8
  • Conjecture 1.9
  • Conjecture 1.10
  • ...and 42 more