JournalsjemsVol. 23, No. 4pp. 1295–1331

Refined global Gross–Prasad conjecture on special Bessel periods and Böcherer’s conjecture

  • Masaaki Furusawa

    Osaka City University, Japan
  • Kazuki Morimoto

    Kobe University, Japan
Refined global Gross–Prasad conjecture on special Bessel periods and Böcherer’s conjecture cover
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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)\mathrm{SO}(2n+1)\times\mathrm{SO}(2). Recall that a Bessel period for SO(2n+1)×SO(2)\mathrm{SO}(2n+1)\times\mathrm{SO}(2) is called special when the representation of SO(2)\mathrm{SO} (2) is trivial. Let π\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 FF whose local component πv\pi_v at any archimedean place vv of FF is a discrete series representation. Let EE be a quadratic extension of FF and suppose that the special Bessel period corresponding to EE does not vanish identically on π\pi. Then we prove the Ichino–Ikeda type explicit formula conjectured by Liu for the central value L(1/2,π)L(1/2,π×χE)L (1/2, \pi) L (1/2, \pi\times\chi_E ), where χE\chi_E denotes the quadratic character corresponding to EE. Our result yields a proof of Böcherer’s conjecture on holomorphic Siegel cusp forms of degree two which are Hecke eigenforms.

Cite this article

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. 1295–1331

DOI 10.4171/JEMS/1034