Serge Lang
Introduction
to Modular Forms
With 9 Figures
Springer
Table of Contents
Part I. Classical Theory
Chapter I. Modular Forms
§ 1.
§ 2.
§3.
§ 4.
§ 5.
The Modular Group
Modular Forms
The Modular Functiony
Estimates for Cusp Forms
The Meilin Transform
3
3
5
12
12
14
Chapter II. Hecke Operators
16
§ 1. Definitions and Basic Relations
§ 2. Euler Products
16
21
Chapter III. Petersson Scalar Product
24
§ 1.
§ 2.
§ 3.
§4.
24
29
32
35
The Riemann Surface T\§*
Congruence Subgroups
Differential Forms and Modular Forms
The Petersson Scalar Product
Appendix by D. Zagier. The Eichler-Selberg Trace Formula on SL2(Z)
. . 44
Part II. Periods of Cusp Forms
Chapter IV. Modular Symbols
57
§ 1. Basic Properties
§2. The Manin-Drinfeld Theorem
§ 3. Hecke Operators and Distributions
57
61
65
Chapter V. Coefficients and Periods of Cusp Forms on 5L2(Z)
68
§ 1. The Periods and Their Integral Relations
§ 2. The Manin Relations
69
73
Vlll
Table of Contents
§3. Action ofthe Hecke Operators onthe Periods
§ 4. The Homogeneity Theorem
76
81
Chapter VI. The Eichler-Shimura Isomorphism on SL2(Z)
84
§ 1. The Polynomial Representation
§ 2. The Shimura Product on Differential Forms
§3. The Image ofthe Period Mapping
§ 4. Computation of Dimensions
§ 5. The Map into Cohomology
85
88
89
93
96
Part III. Modular Forms for Congruence Subgroups
Chapter VII. Higher Levels.
101
§ 1.
§ 2.
§ 3.
§ 4.
§5.
§6.
101
105
108
111
112
114
The Modular Set and Modular Forms
Hecke Operators
Hecke Operators on <?-Expansions
The Matrix Operation
Petersson Product
The Involution
Chapter VIII. Atkin-Lehner Theory
118
§1.
§2.
§ 3.
§4.
118
122
123
126
Changing Levels
Characterization of Primitive Forms
The Structure Theorem
Proof ofthe Main Theorem
Chapter IX. The Dedekind Formalism
138
§ 1. The Transformation Formalism
§2. Evaluation of the Dedekind Symbol
138
142
Part IV. Congruence Properties and Galois Representations
Chapter X. Congruences and Reductiori mod p
151
§1.
§2.
§ 3.
§ 4.
§5.
§6.
§7.
§8.
151
153
154
156
159
162
164
169
Kummer Congruences
Von Staudt Congruences
^-Expansions
Modular Forms over Z[£, | ]
Derivatives of Modular Forms
Reductionmodp
Modular Forms modp,p>5
The Operation o f ö o n M
Tableof Contents
ix
Chapter XI. Galois Representations
176
§ 1. Simplicity
§2. SubgroupsofGL2
§3. Applications to Congruences of the Trace of Frobenius
177
180
187
Appendix by Walter Feit. Exceptional Subgroups of GL2
198
Part V. p-Adic Distributions
Chapter XII. General Distributions
207
§ 1.
§ 2.
§ 3.
§ 4.
§ 5.
207
210
217
219
221
Definitions
Averaging Operators
The Iwasawa Algebra
Weierstrass Preparation Theorem
Modules over Zp[[7]]
Chapter XIII. Bernoulli Numbers and Polynomials
228
§ 1. Bernoulli Numbers and Polynomials
§ 2. The Integral Distribution
§ 3. L-Functions and Bernoulli Numbers
228
233
236
Chapter XIV. The Complex L-Functions
240
§1. The Hurwitz Zeta Function
§ 2. Functional Equation
240
244
Chapter XV. The Hecke-Eisenstein and Klein Forms
247
§1. Forms ofWeight 1
§ 2. The Klein Forms
§3. Forms ofWeight 2
247
251
252
Bibliography . . .
255
Subject Index
260