Introduction to Modular Forms

I. Classical Theory.- I. Modular Forms.- § 1. The Modular Group.- § 2. Modular Forms.- § 3. The Modular Function j.- § 4. Estimates for Cusp Forms.- § 5. The Mellin Transform.- II. Hecke Operators.- § 1. Definitions and Basic Relations.- § 2. Euler Products.- III. Petersson Scalar Product.- § 1. The Riemann Surface ?\?.- § 2. Congruence Subgroups.- § 3. Differential Forms and Modular Forms.- § 4. The Petersson Scalar Product.- Appendix by D. Zagier. The Eichler-Selberg Trace Formula on SL2(Z).- II. Periods of Cusp Forms.- IV. Modular Symbols.- § 1. Basic Properties.- § 2. The Manin-Drinfeld Theorem.- § 3. Hecke Operators and Distributions.- V. Coefficients and Periods of Cusp Forms on SL2(Z).- § 1. The Periods and Their Integral Relations.- § 2. The Manin Relations.- § 3. Action of the Hecke Operators on the Periods.- § 4. The Homogeneity Theorem.- VI. The Eichler-Shimura Isomorphism on SL2(Z).- § 1. The Polynomial Representation.- § 2. The Shimura Product on Differential Forms.- § 3. The Image of the Period Mapping.- § 4. Computation of Dimensions.- § 5. The Map into Cohomology.- III. Modular Forms for Congruence Subgroups.- VII. Higher Levels.- § 1. The Modular Set and Modular Forms.- § 2. Hecke Operators.- § 3. Hecke Operators on q-Expansions.- § 4. The Matrix Operation.- § 5. Petersson Product.- § 6. The Involution.- VIII. Atkin-Lehner Theory.- § 1. Changing Levels.- § 2. Characterization of Primitive Forms.- § 3. The Structure Theorem.- § 4. Proof of the Main Theorem.- IX. The Dedekind Formalism.- § 1. The Transformation Formalism.- § 2. Evaluation of the Dedekind Symbol.- IV. Congruence Properties and Galois Representations.- X. Congruences and Reduction mod p.- § 1. Kummer Congruences.- § 2. Von Staudt Congruences.- § 3. q-Expansions.- § 4. Modular Forms over Z[1/2, 1/3].- § 5. Derivatives of Modular Forms.- § 6. Reduction mod p.- § 7. Modular Forms mod p, p?5.- § 8. The Operation of ? on M?.- XI. Galois Representations.- § 1. Simplicity.- § 2. Subgroups of GL2.- § 3. Applications to Congruences of the Trace of Frobenius.- Appendix by Walter Feit. Exceptional Subgroups of GL2.- V. p-Adic Distributions.- XII. General Distributions.- § 1. Definitions.- § 2. Averaging Operators.- § 3. The Iwasawa Algebra.- § 4. Weierstrass Preparation Theorem.- § 5. Modules over Zp[[T]].- XIII. Bernoulli Numbers and Polynomials.- § 1. Bernoulli Numbers and Polynomials.- § 2. The Integral Distribution.- § 3. L-Functions and Bernoulli Numbers.- XIV. The Complex L-Functions.- § 1. The Hurwitz Zeta Function.- § 2. Functional Equation.- XV. The Hecke-Eisenstein and Klein Forms.- § 1. Forms of Weight 1.- § 2. The Klein Forms.- § 3. Forms of Weight 2.