From the reviews: "This book gives a thorough introduction to several theories that are fundamental to research on modular forms. Most of the material, despite its importance, had previously been unavailable in textbook form. Complete and readable proofs are given... In conclusion, this book is a welcome addition to the literature for the growing number of students and mathematicians in other fields who want to understand the recent developments in the theory of modular forms."

Mathematical Reviews#

"This book will certainly be indispensable to all those wishing to get an up-to-date initiation to the theory of modular forms."

Publicationes Mathematicae#

Inhalt

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.