TABLE OF CONTENTS

Preface vii

Notation and Terminology ix

Introduction 1

Chapter I. MULTIPLE GAMMA FUNCTION AND ITS GENERALIZA-

TIONS 13

§1. Basic integral representations 13

§2. Shintani's formulas 17

§3. The second derivative of £(s, A, x) at s = 0 24

§4. An asymptotic expansion of the multiple gamma function 34

Exercises 41

Chapter II. THE STARK-SHINTANI CONJECTURE 43

§1. Stark's general conjecture 43

§2. Transition to a more precise conjecture 48

§3. Shintani's formulas for the partial zeta function 52

§4. An example 54

Exercises 58

Chapter III. ABSOLUTE CM-PERIODS 61

§1. Shimura's period symbol PK 61

§2. The case of abelian fields 65

§3. Conjectures on absolute CM-periods 74

§4. Numerical examples 94

§5. Further investigations on the invariant X(c) 108

§6. Numerical examples (continued) 126

Exercises 133

Chapter IV. EXPLICIT CONE DECOMPOSITIONS AND APPLICA-

TIONS 135

§1. A special decomposition of a higher dimensional cube 135

§2. Topological preparations 139

§3. A sufficient condition for a cone decomposition 144

§4. Examples 147

§5. Explicit cone decompositions for index finite subgroups 149

§6. Applications 161

Exercises 166