**Memoirs of the American Mathematical Society**

2003;
139 pp;
Softcover

MSC: Primary 11;
Secondary 22

Print ISBN: 978-0-8218-3328-5

Product Code: MEMO/164/782

List Price: $65.00

Individual Member Price: $39.00

**Electronic ISBN: 978-1-4704-0380-5
Product Code: MEMO/164/782.E**

List Price: $65.00

Individual Member Price: $39.00

# 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 [

#### 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

#### Readership

Graduate students and research mathematicians interested in number theory.