# On central critical values of the degree four -functions for GSp(4): the fundamental lemma

### About this Title

**Masaaki Furusawa** and **Joseph A. Shalika**

Publication: Memoirs of the American Mathematical Society

Publication Year
2003: Volume 164, Number 782

ISBNs: 978-0-8218-3328-5 (print); 978-1-4704-0380-5 (online)

DOI: http://dx.doi.org/10.1090/memo/0782

MathSciNet review: 1983033

MSC (2000): Primary 11F67; Secondary 11F70, 11F72, 22E55

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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.

Readership

Graduate students and research mathematicians interested in
number theory.

### Table of Contents

**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