[Paper Review] Refined global Gross-Prasad conjecture on special Bessel periods and Boecherer's conjecture

Masaaki Furusawa, Kazuki Morimoto|arXiv (Cornell University)|Nov 17, 2016

Advanced Algebra and Geometry42 references3 citations

TL;DR

This paper proves an explicit Ichino-Ikeda-type formula for the central value $ L(1/2, \pi)L(1/2, \pi \times \chi_E) $ in terms of special Bessel periods for $ \mathrm{SO}(2n+1) \times \mathrm{SO}(2) $, under temperedness and discrete series conditions. The result confirms Liu's refined Gross-Prasad conjecture in this case and establishes Böcherer's conjecture for degree-two Siegel cusp forms as a corollary.

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 $\mathrm{SO}\left(2n+1 ight) imes\mathrm{SO}\left(2 ight)$. Recall that a Bessel period for $\mathrm{SO}\left(2n+1 ight) imes\mathrm{SO}\left(2 ight)$ is called special when the representation of $\mathrm{SO}\left(2 ight)$ 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 $L\left(1/2,π ight)L\left(1/2,π imesχ_E ight)$, 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.

Motivation & Objective

• To establish an explicit formula for the central value $ L(1/2, \pi)L(1/2, \pi \times \chi_E) $ in terms of special Bessel periods for $ \mathrm{SO}(2n+1) \times \mathrm{SO}(2) $, as conjectured by Yifeng Liu.
• To extend the refined Gross-Prasad conjecture to the case of special Bessel periods on $ \mathrm{SO}(2n+1) \times \mathrm{SO}(2) $, where the $ \mathrm{SO}(2) $-representation is trivial.
• To prove Böcherer's conjecture on central critical values of spinor $ L $-functions for holomorphic Siegel cusp forms of degree two that are Hecke eigenforms.
• To establish the non-vanishing of the central $ L $-value if and only if the special Bessel period does not vanish identically, under tempered and discrete series conditions.

Proposed method

• Utilizes the global theta correspondence and the descent method to relate automorphic representations on $ \mathrm{SO}(2n+1) $ to generic representations on $ \widetilde{\mathrm{Sp}}_n $.
• Applies the local Langlands correspondence and the theory of $ L $-parameters to compare local components of automorphic representations.
• Employs the Howe duality and the uniqueness of generic elements in tempered $ L $-packets to deduce local isomorphisms $ \pi_v \simeq \pi'_v $ from genericity of $ \theta(\pi_v, \psi_v) $.
• Uses the non-vanishing of the Bessel period to deduce non-vanishing of the central $ L $-value via the global descent and weak lift to $ \mathrm{GL}_{2n} $.
• Applies the work of Dickson, Pitale, Saha, and Schmidt to link the result to Siegel modular forms of degree two.
• Relies on the assumption that Conjecture 9.5.4 in Arthur [3] holds for groups in $ \mathcal{G}_n $, ensuring weak lifts and multiplicity-one for $ \pi $.

Experimental results

Research questions

• RQ1 Does the Ichino-Ikeda type formula for the central $ L $-value $ L(1/2, \pi)L(1/2, \pi \times \chi_E) $ hold in the case of special Bessel periods for $ \mathrm{SO}(2n+1) \times \mathrm{SO}(2) $?
• RQ2 Can the refined global Gross-Prasad conjecture for special Bessel periods be proven under the assumption that $ \pi $ is tempered and its archimedean components are discrete series?
• RQ3 Does the non-vanishing of the special Bessel period imply the non-vanishing of the central $ L $-value $ L(1/2, \pi)L(1/2, \pi \times \chi_E) $?
• RQ4 Can Böcherer's conjecture on central critical values of spinor $ L $-functions for degree-two Siegel cusp forms be deduced from this result?
• RQ5 Under what conditions does the global theta lift $ \theta(\pi, \psi) $ being $ \psi_\lambda $-generic imply that $ \pi $ has a special Bessel model of type $ E $?

Key findings

• The paper proves the Ichino-Ikeda type formula for $ L(1/2, \pi)L(1/2, \pi \times \chi_E) $ in terms of the special Bessel period for $ \mathrm{SO}(2n+1) \times \mathrm{SO}(2) $, under the assumption that $ \pi $ is tempered and its archimedean components are discrete series.
• It establishes that the non-vanishing of the special Bessel period implies the non-vanishing of the central $ L $-value $ L(1/2, \pi)L(1/2, \pi \times \chi_E) $, confirming the refined Gross-Prasad conjecture in this case.
• The proof shows that if the Bessel period vanishes identically, then the central $ L $-value must also vanish, under the assumption of Arthur's conjecture on weak lifts.
• The authors deduce a proof of Böcherer's conjecture on the central critical values of spinor $ L $-functions for holomorphic Siegel cusp forms of degree two that are Hecke eigenforms.
• The result relies on the global descent method and the theta correspondence to construct a nearly equivalent generic automorphic representation $ \pi^\circ $, enabling the use of known results on $ L $-values.
• The proof establishes that $ \pi \simeq \pi' $ globally by showing $ \pi_v \simeq \pi'_v $ at all places $ v $, using local $ L $-parameters and the uniqueness of generic representations in tempered $ L $-packets.