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On central critical values of the degree four \(L\)-functions for \({\text {GSp}}\left( 4\right) \): a simple trace formula

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Abstract

We establish a simple relative trace formula for \({\text {GSp}}(4)\) and inner forms with respect to Bessel subgroups to obtain a certain Bessel identity. From such an identity, one can hope to prove a formula relating central values of degree four spinor \(L\)-functions to squares of Bessel periods as conjectured by Böcherer. Under some local assumptions, we obtain nonvanishing results, i.e., a global Gross–Prasad conjecture for \((\text {SO}(5),\text {SO}(2))\).

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Acknowledgments

We would like to thank Yiannis Sakellaridis and Wei Zhang for helpful discussions about trace formulas. We also thank Wee Teck Gan, Hervé Jacquet, Alan Roche, Ralf Schmidt and Shuichiro Takeda for answering some questions related to our project. Finally we thank the referee for several suggestions.

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Authors and Affiliations

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

    Masaaki Furusawa

  2. Department of Mathematics, University of Oklahoma, Norman, OK , 73019-0315, USA

    Kimball Martin

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  1. Masaaki Furusawa
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  2. Kimball Martin
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Correspondence to Kimball Martin.

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To the memory of Hiroshi Saito

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Furusawa, M., Martin, K. On central critical values of the degree four \(L\)-functions for \({\text {GSp}}\left( 4\right) \): a simple trace formula. Math. Z. 277, 149–180 (2014). https://doi.org/10.1007/s00209-013-1248-4

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  • DOI: https://doi.org/10.1007/s00209-013-1248-4

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