This marvellous and highly original book fills a significant gap in the extensive literature on classical modular forms. This is not just yet another introductory text to this theory, though it could certainly be used as such in conjunction with more traditional treatments.

Its novelty lies in its computational emphasis throughout: Stein not only defines what modular forms are, but shows in illuminating detail how one can compute everything about them in practice. This is illustrated throughout the book with examples from his own (entirely free) software package SAGE, which really bring the subject to life while not detracting in any way from its theoretical beauty.

The author is the leading expert in computations with modular forms, and what he says on this subject is all tried and tested and based on his extensive experience.

PREFACE
This is a graduate-level textbook about algorithms for computing with modular forms. It is nontraditional in that the primary focus is not on underlying theory; instead, it answers the question “how do you use a computer to explicitly compute spaces of modular forms?”

This book emerged from notes for a course the author taught at Harvard University in 2004, a course at UC San Diego in 2005, and a course at the University of Washington in 2006.

The author has spent years trying to find good practical ways to compute with classical modular forms for congruence subgroups of SL2(Z) and has implemented most of these algorithms several times, first in C++ [Ste99b], then in MAGMA [BCP97], and as part of the free open source computer algebra system SAGE (see [Ste06]). Much of this work has involved turning formulas and constructions buried in obscure research papers into precise computational recipes then testing these and eliminating inaccuracies.

The author is aware of no other textbooks on computing with modular forms, the closest work being Cremona’s book [Cre97a], which is about computing with elliptic curves, and Cohen’s book [Coh93] about algebraic number theory. ...

Visit Modular Forms: A Computational Approach Download Page

You can download Modular Forms: A Computational Approach in PDF format.

William A. Stein
(with an appendix by Paul E. Gunnells)
Department of Mathematics, University of Washington
Department of Mathematics and Statistics, University of Massachusetts

Contents
Preface xi
Chapter 1. Modular Forms 1
§1.1. Basic Definitions 1
§1.2. Modular Forms of Level 1 3
§1.3. Modular Forms of Any Level 4
§1.4. Remarks on Congruence Subgroups 7
§1.5. Applications of Modular Forms 9
§1.6. Exercises 11
Chapter 2. Modular Forms of Level 1 13
§2.1. Examples of Modular Forms of Level 1 13
§2.2. Structure Theorem for Level 1 Modular Forms 17
§2.3. The Miller Basis 20
§2.4. Hecke Operators 22
§2.5. Computing Hecke Operators 26
§2.6. Fast Computation of Fourier Coefficients 29
§2.7. Fast Computation of Bernoulli Numbers 29
§2.8. Exercises 33
Chapter 3. Modular Forms of Weight 2 35
§3.1. Hecke Operators 36
§3.2. Modular Symbols 39
§3.3. Computing with Modular Symbols 41
§3.4. Hecke Operators 47
§3.5. Computing the Boundary Map 51
§3.6. Computing a Basis for S2(¡0(N)) 53
§3.7. Computing S2(¡0(N)) Using Eigenvectors 58
§3.8. Exercises 60
Chapter 4. Dirichlet Characters 63
§4.1. The Definition 64
§4.2. Representing Dirichlet Characters 64
§4.3. Evaluation of Dirichlet Characters 67
§4.4. Conductors of Dirichlet Characters 70
§4.5. The Kronecker Symbol 72
§4.6. Restriction, Extension, and Galois Orbits 75
§4.7. Alternative Representations of Characters 77
§4.8. Dirichlet Characters in SAGE 78
§4.9. Exercises 81
Chapter 5. Eisenstein Series and Bernoulli Numbers 83
§5.1. The Eisenstein Subspace 83
§5.2. Generalized Bernoulli Numbers 83
§5.3. Explicit Basis for the Eisenstein Subspace 88
§5.4. Exercises 90
Chapter 6. Dimension Formulas 91
§6.1. Modular Forms for ¡0(N) 92
§6.2. Modular Forms for ¡1(N) 95
§6.3. Modular Forms with Character 98
§6.4. Exercises 102
Chapter 7. Linear Algebra 103
§7.1. Echelon Forms of Matrices 103
§7.2. Rational Reconstruction 105
§7.3. Echelon Forms over Q 107
§7.4. Echelon Forms via Matrix Multiplication 110
§7.5. Decomposing Spaces under the Action of Matrix 114
§7.6. Exercises 119
Chapter 8. General Modular Symbols 121
Contents ix
§8.1. Modular Symbols 122
§8.2. Manin Symbols 124
§8.3. Hecke Operators 128
§8.4. Cuspidal Modular Symbols 133
§8.5. Pairing Modular Symbols and Modular Forms 137
§8.6. Degeneracy Maps 142
§8.7. Explicitly Computing Mk(¡0(N)) 144
§8.8. Explicit Examples 147
§8.9. Refined Algorithm for the Presentation 154
§8.10. Applications 155
§8.11. Exercises 156
Chapter 9. Computing with Newforms 159
§9.1. Dirichlet Character Decomposition 159
§9.2. Atkin-Lehner-Li Theory 161
§9.3. Computing Cusp Forms 165
§9.4. Congruences between Newforms 170
§9.5. Exercises 176
Chapter 10. Computing Periods 177
§10.1. The Period Map 178
§10.2. Abelian Varieties Attached to Newforms 178
§10.3. Extended Modular Symbols 179
§10.4. Approximating Period Integrals 180
§10.5. Speeding Convergence Using Atkin-Lehner 183
§10.6. Computing the Period Mapping 185
§10.7. All Elliptic Curves of Given Conductor 187
§10.8. Exercises 190
Chapter 11. Solutions to Selected Exercises 191
§11.1. Chapter 1 191
§11.2. Chapter 2 193
§11.3. Chapter 3 194
§11.4. Chapter 4 196
§11.5. Chapter 5 197
§11.6. Chapter 6 197
§11.7. Chapter 7 198
§11.8. Chapter 8 199
§11.9. Chapter 9 201
§11.10. Chapter 10 201
Appendix A. Computing in Higher Rank 203
§A.1. Introduction 203
§A.2. Automorphic Forms and Arithmetic Groups 205
§A.3. Combinatorial Models for Group Cohomology 213
§A.4. Hecke Operators and Modular Symbols 225
§A.5. Other Cohomology Groups 232
§A.6. Complements and Open Problems 244
Bibliography 253
Index 265