TABLE OF CONTENTS

LIST OF NOTATIONS • • V

INTRODUCTION •• •• 1

CHAPTER I CONJUGACY CLASSES OF Sp(2, Z) • • • • 7

1.1 Introduction • • • • 7

1.2 Representatives of conjugacy classes of finite

order elements 8

1.3 Conjugacy classes of finite order elements in

Sp(2,Z) •• , • 11

1.4 Conjugacy classes of Ti 14

1.5 Conjugacy classes of r0°° 17

CHAPTER II DIMENSION FORMULA FOR THE VECTOR SPACE OF CUSP

FORMS OF DEGREE TWO WITH RESPECT TO Sp(2,Z) ••• 21

2 .1 Introduction 21

2.2 Contributions from elliptic conjugacy classes •• 22

2.3 Contributions from conjugacy classes of elements

having a one-dimensional set of fixed points (I) 23

2.4 Contributions from conjugacy classes of elements

having a one-dimensional set of fixed points (H) 30

2.5 Contributions from conjugacy classes of elements

having a two-dimensional set of fixed points ••• 37

2.6 Contributions from conjugacy classes of unipotent

elements * • • • 43

2.7 A dimension formula for the vector space of cusp

forms with respect to Sp(2,Z) •• 44

CHAPTER III REPRESENTATIVES OF CONJUGACY CLASSES OF ELEMENTS

OF Sp(3,Z) IN Sp(3,R) 46

3 .1 Introduction 46

3.2 Conjugacy classes of torsion elements in

Sp(3,Z) •••• 47

3.3 A classification of conjugacy classes of

Sp(3,Z) •••» 51

3.4 Selberg's trace formula and its modification ••• 56

3.5 Conjugacy classes with zero contribution (I) ••• 62

3.6 Conjugacy classes with zero contribution (IE) ••• 66

CHAPTER IV CONTRIBUTIONS FROM CONJUGACY CLASSES IN

All A1 U A2 U A0 ** * 81

4 .1 Introduction » 81

4.2 Contributions from elliptic conjugacy classes •• 82

4.3 Contributions from conjugacy classes of elements

having a one-dimensional set of fixed points ••• 86

4.4 Contributions from conjugacy classes of elements

having a one-dimensional set of fixed points ••• 97

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