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 SO (2n + 1) × SO (2). Recall that a Bessel period for SO (2n + 1) × SO (2) is called special when the representation of SO ( 2) 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 πv 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 (1/2, π) L (1/2, π × χ E ), where χ 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.

Explicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree level

“…Above, f, f is the Petersson norm of f normalized as in (74), L f (s, π f , Ad) is the finite part of the adjoint (degree 10) L-function for π f , and L f (s, π f ) is the finite part of the spinor (degree 4) L-function for π f . Indeed, combining the results of this paper with recent work of Furusawa and Morimoto [13], we get the following theorem.…”

Moments of Spinor L-Functions and Symplectic Kloosterman Sums

“…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 [8,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 [7] and [8] 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.…”

Spectral average of central values of automorphic L-functions for holomorphic cusp forms on SO_0(m,2) II

“…The refined Gan-Gross-Prasad conjecture posed by ) in the co-dimension 1 case was extended by Liu ([24]) in higher codimensional case; the conjecture predicts the quantity |a f F (ξ)| 2 , the norm-square of the Bessel period on SO(m+2)×SO(m−1), should be related to the central value of the convolution L-function of F and f . An important case of the conjecture on the special Bessel period on SO(m+2)×SO (2) for an odd m has been proved by Furusawa-Morimoto [9] recently. For the Siegel modular case, Liu's conjecture is further refined by [7].…”

“…We also remark that when m is odd the point s = 1/2 is critical in the sense that both Γ L (s) and Γ L (1 − s) are regular at s = 1/2. Let B + l (♮) denote the set of F ∈ B + l such that π F is tempered, i.e., the local representations π F,v is tempered for all places, where π F ∼ = v π F,v is the cuspidal representation of G(A) generated by F , and set 1 There seems to be a good reason to expect that the following assertions are true ( [8], [9], [33], [22], [21], [41]):…”