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Refined global Gross-Prasad conjecture on special Bessel periods and Boecherer's conjecture

Masaaki Furusawa, Kazuki Morimoto

Abstract

In this paper we pursue the refined global Gross-Prasad conjecture for Bessel periods formulated by Yifeng Liu in the case of special Bessel periods for $\mathrm{SO}\left(2n+1\right)\times\mathrm{SO}\left(2\right)$. Recall that a Bessel period for $\mathrm{SO}\left(2n+1\right)\times\mathrm{SO}\left(2\right)$ is called special when the representation of $\mathrm{SO}\left(2\right)$ is trivial. Let $π$ be an irreducible cuspidal tempered automorphic representation of a special orthogonal group of an odd dimensional quadratic space over a totally real number field $F$ whose local component $π_v$ at any archimedean place $v$ of $F$ is a discrete series representation. Let $E$ be a quadratic extension of $F$ and suppose that the special Bessel period corresponding to $E$ does not vanish identically on $π$. Then we prove the Ichino-Ikeda type explicit formula conjectured by Liu for the central value $L\left(1/2,π\right)L\left(1/2,π\timesχ_E\right)$, where $χ_E$ denotes the quadratic character corresponding to $E$. Our result yields a proof of Boecherer's conecture on holomorphic Siegel cusp forms of degree two which are Hecke eigenforms.

Refined global Gross-Prasad conjecture on special Bessel periods and Boecherer's conjecture

Abstract

In this paper we pursue the refined global Gross-Prasad conjecture for Bessel periods formulated by Yifeng Liu in the case of special Bessel periods for . Recall that a Bessel period for is called special when the representation of is trivial. Let be an irreducible cuspidal tempered automorphic representation of a special orthogonal group of an odd dimensional quadratic space over a totally real number field whose local component at any archimedean place of is a discrete series representation. Let be a quadratic extension of and suppose that the special Bessel period corresponding to does not vanish identically on . Then we prove the Ichino-Ikeda type explicit formula conjectured by Liu for the central value , where denotes the quadratic character corresponding to . Our result yields a proof of Boecherer's conecture on holomorphic Siegel cusp forms of degree two which are Hecke eigenforms.

Paper Structure

This paper contains 32 sections, 16 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Special Bessel periods
  4. Refined Gan-Gross-Prasad conjecture by Liu in our case
  5. Local integral
  6. Normalization of local integrals
  7. Refined global Gross-Prasad conjecture on $B_{\lambda, \psi}$
  8. Main Theorem
  9. Böcherer's conjecture
  10. Organization of the paper
  11. Reduction to a local equality
  12. Set up
  13. Rallis inner product formula
  14. Local doubling zeta integrals
  15. Rallis inner product formula
  16. ...and 17 more sections

Key Result

Theorem 1

Let $F$ be a totally real number field and $\pi=\otimes_v\,\pi_v$ an irreducible cuspidal tempered automorphic representation of $G\left(\mathbb{A}\right)$ for $G\in\mathcal{G}$ such that $\pi_v$ is a discrete series representation at any archimedean place $v$ of $F$. Suppose that the special Bessel Here $L\left(s,\pi,\mathrm{Ad}\right)$ is defined as $L\left(s,\pi, \mathrm{Ad}\right)=\prod_v L\le

Theorems & Definitions (50)

  • Definition 1
  • Definition 2
  • Remark 1
  • Definition 3
  • Definition 4
  • Remark 2
  • Definition 5
  • Remark 3
  • Conjecture
  • Remark 4
  • ...and 40 more