Refined global Gross–Prasad conjecture on special Bessel periods and Böcherer’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 \mathrm{SO}(2n+1)\times\mathrm{SO}(2) . Recall that a Bessel period for \mathrm{SO}(2n+1)\times\mathrm{SO}(2) is called special when the representation of \mathrm{SO} (2) is trivial. Let \pi 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 \pi_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 \pi . Then we prove the Ichino–Ikeda type explicit formula conjectured by Liu for the central value L (1/2, \pi) L (1/2, \pi\times\chi_E ) , where \chi_E denotes the quadratic character corresponding to E . Our result yields a proof of Böcherer’s conjecture on holomorphic Siegel cusp forms of degree two which are Hecke eigenforms.

On Fundamental Fourier Coefficients of Siegel Cusp Forms of Degree 2

“…Now assume that the refined Gan-Gross-Prasad conjecture (see [15,Conjecture 1.12] and [40, (1.1)]) for the pair (φ,Λ) holds true. In fact, this conjecture for Λ = 1 K is now known, thanks to work of Furusawa and Morimoto [18] (which, combined with [15], completes the proof of Böcherer's conjecture), who have also recently announced the proof for general Λ. Then [15,Theorem 1.13] implies that under some mild assumptions,…”

A symplectic restriction problem

“…However Siegel cusp forms are not globally generic, albeit that they are equivalent to a generic form everywhere but the archimedean place. Nevertheless, their Petersson norm may be expressed in terms of its adjoint L-function via Böcherer's conjecture (now a theorem due to the breakthrough result of Furusawa and Morimoto [29] alongside the explicit formulation in [23]).…”

Arithmetic and analytical aspects of Siegel modular forms

“…In addition, the weights w F,N are related to central values of L-functions. This remarkable conjecture is due to Böcherer and was recently proven in [20,Theorem 2 & Remark 6]. Let S p2q k pN q new,T denote the space of newforms orthogonal to Saito-Kurokawa lifts.…”

“…For π F,p " χ ¸σ St GSpp2q , the attached Langlands L-parameter is pρ, N q with ρ : The proof of Böcherer's conjecture in [17] and [20] is obtained via local computations. In the introduction, we already stated relations for newforms; now we present similar results for members of the oldspace basis constructed in the previous section.…”

“…If φ p is not spherical, then it is of type VIb and we apply the results from [17]. If φ p is spherical, then by construction of δr,s,t it is one the three vectors from (20). We have…”