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On special Bessel periods and the Gross–Prasad conjecture for \(\mathrm {SO}(2n+1) \times \mathrm {SO}(2)\)

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Abstract

In this paper we investigate one instance of the global Gross–Prasad conjecture. Our main result proves that if an irreducible cuspidal automorphic representation \(\pi \) of an odd dimensional special orthogonal group, whose local component \(\pi _{w}\) at some finite place w is generic, admits the special Bessel model corresponding to a quadratic extension E over a base field F, then the central L-value \(L(1/2,\pi )L(1/2,\pi \times \chi _E)\) does not vanish. Here \(\chi _E\) denotes the quadratic character of \(\mathbb A_F^\times \) corresponding to E. As an application, we obtain the equivalence between the non-vanishing of the special Bessel period and that of the corresponding central L-value when \(\pi \) is associated to a full modular holomorphic Siegel cusp form of degree two, which is a Hecke eigenform, and E is an imaginary quadratic extension of \(\mathbb Q\).

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Acknowledgments

The authors would like to thank Hiraku Atobe and Atsushi Ichino for helpful and enlightening discussions. The authors are extremely grateful to an anonymous referee who read the earlier version of the manuscript with great care and pointed out several inaccuracies and oversights with generous suggestions as to how to rectify them. Thanks are also due to Dihua Jiang and Lei Zhang for bringing their recent significant preprint [30] to the authors’ attention.

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Author notes

  1. Kazuki Morimoto

    Present address: Department of Mathematics, Graduate School of Science, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe, 657-8501, Japan

Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science, Osaka City University, Sugimoto 3-3-138, Sumiyoshi-ku, Osaka, 558-8585, Japan

    Masaaki Furusawa

  2. Department of Mathematics, Kyoto University, Kitashirakawa, Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan

    Kazuki Morimoto

Authors
  1. Masaaki Furusawa
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Correspondence to Masaaki Furusawa.

Additional information

Communicated by Toby Gee.

The research of Masaaki Furusawa was supported in part by Grant-in-Aid for Scientific Research (C) 25400020. The research of Kazuki Morimoto was supported in part by Grant-in-Aid for JSPS fellow (26-1158), and Grant-in-Aid for Young Scientists (B) 26800021.

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Furusawa, M., Morimoto, K. On special Bessel periods and the Gross–Prasad conjecture for \(\mathrm {SO}(2n+1) \times \mathrm {SO}(2)\) . Math. Ann. 368, 561–586 (2017). https://doi.org/10.1007/s00208-016-1440-z

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