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 Böcherer’s conjecture on holomorphic Siegel cusp forms of degree two which are Hecke eigenforms.
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Masaaki Furusawa, Kazuki Morimoto, Refined global Gross–Prasad conjecture on special Bessel periods and Böcherer’s conjecture. J. Eur. Math. Soc. 23 (2021), no. 4, pp. 1295–1331DOI 10.4171/JEMS/1034