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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.

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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.