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On Central Critical Values of the Degree Four $L$-functions for $\mathrm{GSp}(4)$: The Fundamental Lemma

Masaaki Furusawa Osaka City University, Osaka, Japan
Joseph A. Shalika Johns Hopkins University, Baltimore, MD
Available Formats:
Electronic ISBN: 978-1-4704-0380-5
Product Code: MEMO/164/782.E
List Price: $65.00 MAA Member Price:$58.50
AMS Member Price: $39.00 Click above image for expanded view On Central Critical Values of the Degree Four$L$-functions for$\mathrm{GSp}(4)$: The Fundamental Lemma Masaaki Furusawa Osaka City University, Osaka, Japan Joseph A. Shalika Johns Hopkins University, Baltimore, MD Available Formats:  Electronic ISBN: 978-1-4704-0380-5 Product Code: MEMO/164/782.E  List Price:$65.00 MAA Member Price: $58.50 AMS Member Price:$39.00
• Book Details

Memoirs of the American Mathematical Society
Volume: 1642003; 139 pp
MSC: Primary 11; Secondary 22;

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 $\mathrm{gl}\left(2\right)$ 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 Böcherer [{B}] and its generalization. Support for this conjecture may be found in the important paper of Böcherer 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.

Graduate students and research mathematicians interested in number theory.

• Chapters
• 1. Statement of results
• 2. Gauss sum, Kloosterman sum and Salié sum
• 3. Matrix argument Kloosterman sums
• 4. Evaluation of the Novodvorsky orbital integral
• 5. Evaluation of the Bessel orbital integral
• 6. Evaluation of the quadratic orbital integral
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Volume: 1642003; 139 pp
MSC: Primary 11; Secondary 22;

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 $\mathrm{gl}\left(2\right)$ 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 Böcherer [{B}] and its generalization. Support for this conjecture may be found in the important paper of Böcherer 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.