This book presents a graduate student-level introduction to the classical theory of modular forms and computations involving modular forms, including modular functions and the theory of Hecke operators. It also includes applications of modular forms to such diverse subjects as the theory of quadratic forms, the proof of Fermat s last theorem and the approximation of pi . It provides a balanced overview of both the theoretical and computational sides of the subject, allowing a variety of courses to be taught from it.
Contents: Historical Overview; Introduction to Modular Forms; Results on Finite-Dimensionality; The Arithmetic of Modular Forms; Applications of Modular Forms; Modular Forms in Characteristic p ; Computing with Modular Forms; Appendices: ; MAGMA Code for Classical Modular Forms; SAGE Code for Classical Modular Forms; Hints and Answers to Selected Exercises.
Author(s): Lloyd Kilford
Edition: illustrated edition
Publisher: World Scientific Pub Co (
Year: 2008
Language: English
Pages: 237
Introduction......Page 10
Acknowledgements......Page 8
Possible courses......Page 14
An overview of this book1......Page 16
1.1 18th Century — a prologue......Page 18
1.2 19th century — the classical period......Page 19
1.3 Early 20th century — arithmetic applications......Page 20
1.4 Later 20th century — the link to elliptic curves......Page 21
1.5 The 21st century — the Langlands Program......Page 22
2.1 Modular forms for SL2(Z)......Page 24
2.2 Eisenstein series for the full modular group......Page 28
2.3 Computing Fourier expansions of Eisenstein series......Page 30
2.4 Congruence subgroups......Page 34
2.5 Fundamental domains......Page 38
2.6 Modular forms for congruence subgroups......Page 41
2.7 Eisenstein series for congruence subgroups......Page 45
2.8 Derivatives of modular forms......Page 48
2.8.1 Quasi-modular forms......Page 50
2.9 Exercises......Page 51
3.1 Spaces of modular forms are finite-dimensional......Page 54
3.2.1 Formulae for the full modular group......Page 59
3.2.2 Formulae for congruence subgroups......Page 62
3.3 The Sturm bound......Page 65
3.4 Exercises......Page 68
4. The arithmetic of modular forms......Page 70
4.1.1 Motivation for the Hecke operators......Page 71
4.1.2 Hecke operators for Mk(SL2(Z))......Page 72
4.1.3 Hecke operators for congruence subgroups......Page 76
4.2.1 The Petersson scalar product......Page 82
4.2.2 The Hecke operators are Hermitian......Page 88
4.2.3 Integral bases......Page 92
4.3 Oldforms and newforms......Page 93
4.3.1 Multiplicity one for newforms......Page 98
4.4 Exercises......Page 101
5. Applications of modular forms......Page 106
5.1 Modular functions......Page 107
5.2 η-products and η-quotients......Page 111
5.3 The arithmetic of the j-invariant......Page 116
5.3.1 The j-invariant and the Monster group......Page 119
5.3.2 “Ramanujan’s Constant”......Page 120
5.4 Applications of the modular function λ(z)......Page 121
5.4.1 Computing digits of π using λ(z)......Page 122
5.4.2 Proving Picard’s Theorem......Page 124
5.5 Identities of series and products......Page 125
5.6 The Ramanujan-Petersson Conjecture......Page 126
5.7 Elliptic curves and modular forms......Page 129
5.7.1 Fermat’s Last Theorem......Page 132
5.8 Theta functions and their applications......Page 133
5.8.1 Representations of n by a quadratic form in an even number of variables......Page 134
5.8.2 Representations of n by a quadratic form in an odd number of variables......Page 141
5.8.3 The Shimura correspondence......Page 144
5.9 CM modular forms......Page 146
5.10 Lacunary modular forms......Page 148
5.11 Exercises......Page 151
6.1 Classical treatment......Page 156
6.1.1 The structure of the ring of mod p forms......Page 157
6.1.2 The θ operator on mod p modular forms......Page 163
6.1.3 Hecke operators and Hecke eigenforms......Page 164
6.2 Galois representations attached to mod p modular forms......Page 165
6.3 Katz modular forms......Page 169
6.4 The Sturm bound in characteristic p......Page 171
6.5 Computations with mod p modular forms......Page 172
6.6 Exercises......Page 174
7.1 Historical introduction to computations in number theory......Page 176
7.2 Magma......Page 180
7.2.1 Magma philosophy......Page 183
7.2.2 Magma programming......Page 184
7.3 Sage......Page 186
7.3.2 Sage programming......Page 188
7.3.3 The Sage interface......Page 189
7.4.1 Pari......Page 190
7.4.2 Other systems and solutions......Page 192
7.5.1 Computation today......Page 193
7.5.2 Expected running times......Page 195
7.5.3 Using computation effectively......Page 196
7.5.4 The limits of computation......Page 197
7.5.4.1 Explicit examples of limitations......Page 199
7.5.5 Guy’s law of small numbers......Page 200
7.6 Exercises......Page 202
7.6.1 Magma......Page 203
7.6.2 Sage......Page 204
7.6.4 Maple......Page 206
Appendix A Magma code for classical modular forms......Page 208
Appendix B Sage code for classical modular forms......Page 210
Appendix C Hints and answers to selected exercises......Page 212
Bibliography......Page 218
List of Symbols......Page 230
Index......Page 234
