Introduction to Modular Forms

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