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

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*Masaaki Furusawa; Joseph A. Shalika*

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.

#### Table of Contents

# Table of Contents

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

- Contents vii8 free
- Abstract viii9 free
- Introduction ix10 free
- Chapter 1. Statement of Results 112 free
- Chapter 2. Gauss Sum, Kloosterman Sum and Salié Sum 1425
- Chapter 3. Matrix Argument Kloosterman Sums 2637
- Chapter 4. Evaluation of the Novodvorsky Orbital Integral 8192
- Chapter 5. Evaluation of the Bessel Orbital Integral 104115
- Chapter 6. Evaluation of the Quadratic Orbital Integral 116127
- Bibliography 138149