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# memo_has_moved_text();On central critical values of the degree four $L$-functions for $\mathrm {GSp}(4)$: The fundamental lemma. III

Masaaki Furusawa, Department of Mathematics, Graduate School of Science, Osaka City University, Sugimoto 3–3–138, Sumiyoshi, Osaka 558–8585, Japan, Kimball Martin, Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019-0315, USA and Joseph A. Shalika, Department of Mathematics, The Johns Hopkins University, 3400 North Charles Street, Baltimore, Maryland 21218–2689, USA

Publication: Memoirs of the American Mathematical Society
Publication Year: 2013; Volume 225, Number 1057
ISBNs: 978-0-8218-8742-4 (print); 978-1-4704-1057-5 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-2013-00675-2
Published electronically: February 11, 2013
Keywords:Relative trace formula, fundamental lemma, Kloosterman integral
MSC: Primary 11F67; Secondary 11F46

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Chapters

• Preface
• Chapter 1. Introduction
• Chapter 2. Reduction Formulas
• Chapter 3. Anisotropic Bessel Orbital Integral
• Chapter 4. Split Bessel and Novodvorsky Orbital Integrals
• Chapter 5. Rankin-Selberg Orbital Integral

### Abstract

Some time ago, the first and third authors proposed two relative trace formulas to prove generalizations of Böcherer's conjecture on the central critical values of the degree four $L$-functions for $\mathrm{GSp}(4)$, and proved the relevant fundamental lemmas. Recently, the first and second authors proposed an alternative third relative trace formula to approach the same problem and proved the relevant fundamental lemma. In this paper the authors extend the latter fundamental lemma and the first of the former fundamental lemmas to the full Hecke algebra. The fundamental lemma is an equality of two local relative orbital integrals. In order to show that they are equal, the authors compute them explicitly for certain bases of the Hecke algebra and deduce the matching.