Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The second moment of $\mathrm{GL}(2)\times \mathrm{GL}(2)$ Rankin-Selberg $L$-functions in the level aspect

Jakub Dobrowolski

TL;DR

This work analyzes the second moment of GL(2)×GL(2) Rankin–Selberg L-values in the level (conductor) aspect over a number field F, with a fixed unramified π0. The authors implement a period–spectral method, deploying a regularised spectral decomposition and a shifted inner product identity to separate the analysis into a spectral side and a period side, each further decomposed into degenerate and regularised contributions. The main result is an asymptotic expansion with a cubic-log main term and a power-saving error term, which becomes square-root cancellation under the Generalised Ramanujan Conjecture for GL(2). The techniques hinge on precise local zeta integrals, spectral weights J_q, and the careful treatment of the degenerate term, yielding explicit constants and polynomial-in-log main-term structure that generalise prior Q and archimedean cases to arbitrary number fields and levels. The work thus advances understanding of level-aspect moments for Rankin–Selberg L-functions and provides a robust framework for conductor-variation analyses in the automorphic setting.

Abstract

We prove an asymptotic formula with a power-saving error term for a specific weighted second moment of $\mathrm{GL}(2)\times \mathrm{GL}(2)$ Rankin-Selberg $L$-function, $L(1/2,π\otimes π_0)$ over any number field $F$ where $π$ runs over representations with the non-archimedean conductor dividing an ideal which tends to infinity and $π_0$ is a fixed cuspidal representation unramified everywhere. The error term shows the square root cancellation under the assumption of the Generalised Ramanujan Conjecture.

The second moment of $\mathrm{GL}(2)\times \mathrm{GL}(2)$ Rankin-Selberg $L$-functions in the level aspect

TL;DR

This work analyzes the second moment of GL(2)×GL(2) Rankin–Selberg L-values in the level (conductor) aspect over a number field F, with a fixed unramified π0. The authors implement a period–spectral method, deploying a regularised spectral decomposition and a shifted inner product identity to separate the analysis into a spectral side and a period side, each further decomposed into degenerate and regularised contributions. The main result is an asymptotic expansion with a cubic-log main term and a power-saving error term, which becomes square-root cancellation under the Generalised Ramanujan Conjecture for GL(2). The techniques hinge on precise local zeta integrals, spectral weights J_q, and the careful treatment of the degenerate term, yielding explicit constants and polynomial-in-log main-term structure that generalise prior Q and archimedean cases to arbitrary number fields and levels. The work thus advances understanding of level-aspect moments for Rankin–Selberg L-functions and provides a robust framework for conductor-variation analyses in the automorphic setting.

Abstract

We prove an asymptotic formula with a power-saving error term for a specific weighted second moment of Rankin-Selberg -function, over any number field where runs over representations with the non-archimedean conductor dividing an ideal which tends to infinity and is a fixed cuspidal representation unramified everywhere. The error term shows the square root cancellation under the assumption of the Generalised Ramanujan Conjecture.
Paper Structure (27 sections, 11 theorems, 182 equations)

This paper contains 27 sections, 11 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Introduction to the problem
  3. Overview of the proof
  4. Acknowledgments
  5. Notation and Preliminaries
  6. Notations
  7. Matrix groups
  8. Automorphic representations
  9. Eisenstein series
  10. Whittaker model
  11. Rankin--Selberg L--function
  12. Regularised spectral decomposition
  13. Spectral weight
  14. Main Theorem
  15. Proof of the Main Theorem
  16. ...and 12 more sections

Key Result

Theorem 1

Let $\pi_0$ be a fixed cuspidal representation for $\mathrm{PGL}_2(F)$ that is unramified at all places. One has the following expansion of a weighted second moment over the conductor family eq: conductor family where $c_F$ is a constant depending only on $F$ and $P(\mathfrak{q})$ is a degree 2 polynomial in $\log N(\mathfrak{q})$ with coefficients bounded by $(\log\log N(\mathfrak{q}))^4$, and $\

Theorems & Definitions (27)

  • Theorem
  • Theorem 1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • proof : Proof of Theorem \ref{['Theorem']}
  • Proposition 4.1
  • Lemma 4.2
  • proof
  • Remark 4.1
  • ...and 17 more