Modular Forms

✍ Scribed by Toshitsune Miyake

• Publisher: Springer
• Year: 2006
• Tongue: English
• Leaves: 345
• Series: Springer monographs in mathematics
• Edition: Corrected
• Category: Library

✦ Synopsis

This book is a translation of the earlier book written by Koji Doi and the author, who revised it substantially for this English edition. It offers the basic knowledge of elliptic modular forms necessary to understand recent developments in number theory. It also treats the unit groups of quaternion algebras, rarely dealt with in books; and in the last chapter, Eisenstein series with parameter are discussed following the recent work of Shimura.

✦ Table of Contents

Contents......Page 7
Notation and Terminology......Page 9
§1.1. The Group of Automorphisms of the Upper Half Plane......Page 10
§1.2. Actions of Groups......Page 13
§1.3. Classification of Linear Fractional Transformations......Page 16
§1.4. The Invariant Metric and Measure on H......Page 19
§1.5. Fuchsian Groups......Page 26
§1.6. Fundamental Domains......Page 29
§1.7. Quotient Spaces Γ \ H......Page 33
§1.8. The Structure of Γ \ H as a Riemann Surface......Page 37
§1.9. Fuchsian Groups of the First Kind......Page 40
§2.1. Automorphic Forms......Page 46
§2.2. Differentials on Compact Riemann Surfaces......Page 54
§2.3. Automorphic Forms and Differentials......Page 57
§2.4. The Measure of Γ \ H*......Page 62
§2.5. Dimensions of gsub(k) and jsub(k)......Page 66
§2.6. Poincaré Series and Eisenstein Series......Page 70
§2.7. Hecke Algebras......Page 78
§2.8. Hecke Operators on the Space of Automorphic Forms......Page 83
§3.1. Dirichlet Characters......Page 88
§3.2. The Riemann Zeta-Function......Page 93
§3.3. Hecke L-Functions......Page 99
§4.1. SLsub(2)......Page 105
§4.2. Congruence Modular Groups......Page 112
§4.3. Modular Forms and Dirichlet Series......Page 123
§4.4. Δ(z) and η(z)......Page 138
§4.5. Hecke Algebras of Modular Groups......Page 140
§4.6. Primitive Forms......Page 162
§4.7. Dirichlet L-Functions and Modular Forms......Page 184
§4.8. L-Functions of Quadratic Fields and Cusp Forms......Page 191
§4.9. Theta Functions......Page 194
§5.1. Algebras over Q and Adelization......Page 204
§5.2. Quaternion Algebras......Page 207
§5.3. Hecke Algebras of Unit Groups of Quaternion Algebras......Page 219
§6.1. Spaces of Functions on H......Page 228
§6.2. The Projection of L[sup(p)]sub(k) onto H[sup(p)]sub(k)......Page 234
§6.3. Function Spaces Consisting of Automorphic Forms......Page 237
§6.4. Traces of Hecke Operators (Calculation of Integrals)......Page 240
§6.5. Traces of Hecke Operators (Algebraic Calculation)......Page 252
§6.6. Local Conjugacy Classes......Page 257
§6.7. Class Numbers of Orders of Q[α]......Page 265
§6.8. An Explicit Formula for tr(T(n))......Page 268
§7.1. Eisenstein Series of Weight k ≥ 3......Page 277
§7.2. Analytic Continuation of Eisenstein Series......Page 283
Numerical Tables......Page 303
References......Page 323
List of Symbols......Page 325
E......Page 341
P......Page 342
Z......Page 343