On central critical values of the degree four L-functions for GSp(4) : the fundamental lemma
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Bibliographic Information
On central critical values of the degree four L-functions for GSp(4) : the fundamental lemma
（Memoirs of the American Mathematical Society, no. 782）
American Mathematical Society, 2003
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Includes bibliographical references (p. 138-139)
"July 2003, volume 164, number 782 (fourth of 5 numbers)"
Description and Table of Contents
Description
In this paper we prove two equalities of local Kloosterman integrals on $\mathrm{GSp}\left(4\right)$, the group of $4$ by $4$ symplectic similitude matrices. One is an equality between the Novodvorsky orbital integral and the Bessel orbital integral and the other one is an equality between the Bessel orbital integral and the quadratic orbital integral. We conjecture that both of Jacquet's relative trace formulas for the central critical values of the $L$-functions for $\mathrm{g1}\left(2\right)$ in [{J1}] and [{J2}], where Jacquet has given another proof of Waldspurger's result [{W2}], generalize to the ones for the central critical values of the degree four spinor $L$-functions for $\mathrm{GSp}\left(4\right)$.We believe that our approach will lead us to a proof and also a precise formulation of a conjecture of Bocherer [{B}] and its generalization. Support for this conjecture may be found in the important paper of Bocherer and Schulze-Pillot [{BSP}]. Also a numerical evidence has been recently given by Kohnen and Kuss [{KK}]. Our results serve as the fundamental lemmas for our conjectural relative trace formulas for the main relevant double cosets.
Table of Contents
Statement of results Gauss sum, Kloosterman sum and Salie sum Matrix argument Kloosterman sums Evaluation of the Novodvorsky orbital integral Evaluation of the Bessel orbital integral Evaluation of the quadratic orbital integral Bibliography.
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