Modular forms: a classical and computational introduction ; ["based on notes for lectures given at the Mathematical Institute at the University of Oxford ... 2004-2007"]
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
London
Imperial College Press
2008
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 224 S. graph. Darst. 24 cm |
| ISBN: | 1848162138 9781848162136 |
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MARC
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| 245 | 1 | 0 | |a Modular forms |b a classical and computational introduction ; ["based on notes for lectures given at the Mathematical Institute at the University of Oxford ... 2004-2007"] |c L. J. P. Kilford |
| 264 | 1 | |a London |b Imperial College Press |c 2008 | |
| 300 | |a XII, 224 S. |b graph. Darst. |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 650 | 4 | |a Forms, Modular | |
| 650 | 4 | |a Algebraic spaces | |
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Record in the Search Index
| _version_ | 1804139174218956800 |
|---|---|
| adam_text | Contents
Acknowledgements
vii
Introduction
1
1.
Historical overview
5
1.1
18th Century
—
a prologue
................. 5
1.2
19th century
—
the classical period
............. 6
1.3
Early 20th century
—
arithmetic applications
....... 7
1.4
Later 20th century
—
the link to elliptic curves
...... 8
1.5
The 21st century
—
the
Langlands
Program
........ 9
2.
Introduction to modular forms
11
2.1
Modular forms for SL2(Z)
.................. 11
2.2 Eisenstein
series for the full modular group
........ 15
2.3
Computing Fourier expansions of
Eisenstein
series
.... 17
2.4
Congruence subgroups
.................... 21
2.5
Fundamental domains
.................... 25
2.6
Modular forms for congruence subgroups
.......... 28
2.7 Eisenstein
series for congruence subgroups
......... 32
2.8
Derivatives of modular forms
................ 35
2.8.1
Quasi-modular forms
................ 37
2.9
Exercises
........................... 38
3.
Results on finite-dimensionality
41
3.1
Spaces of modular forms are finite-dimensional
...... 41
3.2
Explicit formulae for the dimensions of spaces of modular
forms
............................. 46
χ
Modular
forms: a classical and computational introduction
3.2.1
Formulae for the
fuli
modular group
........ 46
3.2.2
Formulae for congruence subgroups
........ 49
3.3
The Sturm bound
...................... 52
3.4
Exercises
........................... 55
4.
The arithmetic of modular forms
57
4.1 Hecke
operators
........................ 58
4.1.1
Motivation for the
Hecke
operators
........ 58
4.1.2 Hecke
operators for Mfc(SL2(Z))
.......... 59
4.1.3 Hecke
operators for congruence subgroups
.... 63
4.2
Bases of eigenforms
...................... 69
4.2.1
The
Petersson
scalar product
............ 69
4.2.2
The
Hecke
operators are Hermitian
........ 75
4.2.3
Integral bases
.................... 79
4.3
Oldforms and newforms
................... 80
4.3.1
Multiplicity one for newforms
........... 85
4.4
Exercises
........................... 88
5.
Applications of modular forms
93
5.1
Modular functions
...................... 94
5.2
^-products and 77-quotients
................. 98
5.3
The arithmetic of the j-invariant
.............. 103
5.3.1
The j-invariant and the Monster group
...... 106
5.3.2
Ramanujan s Constant
.............. 107
5.4
Applications of the modular function
λ(ζ)
......... 108
5.4.1
Computing digits of
π
using
λ(ζ)
......... 109
5.4.2
Proving Picard s Theorem
.............
Ill
5.5
Identities of series and products
............... 112
5.6
The Ramanujan-Petersson Conjecture
........... 113
5.7
Elliptic curves and modular forms
............. 116
5.7.1
Fermat s Last Theorem
............... 119
5.8
Theta functions and their applications
........... 120
5.8.1
Representations of
η
by a quadratic form in an
even number of variables
.............. 121
5.8.2
Representations of
η
by a quadratic form in an odd
number of variables
................. 128
5.8.3
The Shimura correspondence
............ 131
5.9
CM modular forms
...................... 133
Contents xi
5.10 Lacunary
modular
forms
................... 135
5.11
Exercises
........................... 138
6.
Modular forms in characteristic
ρ
143
6.1
Classical treatment
...................... 143
6.1.1
The structure of the ring of mod
ρ
forms
..... 144
6.1.2
The
θ
operator on mod
ρ
modular forms
..... 150
6.1.3 Hecke
operators and
Hecke eigenforms ...... 151
6.2
Galois representations attached to mod
ρ
modular forms
. 152
6.3
Katz modular forms
..................... 156
6.4
The Sturm bound in characteristic
ρ
............ 158
6.5
Computations with mod
ρ
modular forms
......... 159
6.6
Exercises
........................... 161
7.
Computing with modular forms
163
7.1
Historical introduction to computations in number theory
163
7.2
Magma
............................ 167
7.2.1
Magma philosophy
................. 170
7.2.2
Magma programming
................ 171
7.3
Sage
............................. 173
7.3.1
Sage philosophy
................... 175
7.3.2
Sage programming
................. 175
7.3.3
The Sage interface
................. 176
7.3.4
Sage graphics
.................... 177
7.4 Pari
and other systems
................... 177
7.4.1 Pari........................ . 177
7.4.2
Other systems and solutions
............ 179
7.5
Discussion of computation
.................. 180
7.5.1
Computation today
................. 180
7.5.2
Expected running times
............... 182
7.5.3
Using computation effectively
........... 183
7.5.4
The limits of computation
............. 184
7.5.5
Guy s law of small numbers
............ 187
7.5.6
How hard is it to calculate Fourier coefficients of
modular forms?
................... 189
7.6
Exercises
........................... 189
7.6.1
Magma
....................... 190
7.6.2
Sage
......................... 191
xii
Modular
forms: a classical and computational introduction
7.6.3
Pari
......................... 193
7.6.4
Maple
........................ 193
Appendix A Magma code for classical modular forms
195
Appendix
В
Sage code for classical modular forms
197
Appendix
С
Hints and answers to selected exercises
199
Bibliography
205
List of Symbols
217
Index
221
|
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| illustrated | Illustrated |
| indexdate | 2024-07-09T21:39:53Z |
| institution | BVB |
| isbn | 1848162138 9781848162136 |
| language | English |
| lccn | 2008301117 |
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| physical | XII, 224 S. graph. Darst. 24 cm |
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| spelling | Kilford, L. J. P. Verfasser (DE-588)1047868490 aut Modular forms a classical and computational introduction ; ["based on notes for lectures given at the Mathematical Institute at the University of Oxford ... 2004-2007"] L. J. P. Kilford London Imperial College Press 2008 XII, 224 S. graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Forms, Modular Algebraic spaces Modulform (DE-588)4128299-1 gnd rswk-swf (DE-588)1071861417 Konferenzschrift gnd-content Modulform (DE-588)4128299-1 s DE-604 Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017592458&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kilford, L. J. P. Modular forms a classical and computational introduction ; ["based on notes for lectures given at the Mathematical Institute at the University of Oxford ... 2004-2007"] Forms, Modular Algebraic spaces Modulform (DE-588)4128299-1 gnd |
| subject_GND | (DE-588)4128299-1 (DE-588)1071861417 |
| title | Modular forms a classical and computational introduction ; ["based on notes for lectures given at the Mathematical Institute at the University of Oxford ... 2004-2007"] |
| title_auth | Modular forms a classical and computational introduction ; ["based on notes for lectures given at the Mathematical Institute at the University of Oxford ... 2004-2007"] |
| title_exact_search | Modular forms a classical and computational introduction ; ["based on notes for lectures given at the Mathematical Institute at the University of Oxford ... 2004-2007"] |
| title_full | Modular forms a classical and computational introduction ; ["based on notes for lectures given at the Mathematical Institute at the University of Oxford ... 2004-2007"] L. J. P. Kilford |
| title_fullStr | Modular forms a classical and computational introduction ; ["based on notes for lectures given at the Mathematical Institute at the University of Oxford ... 2004-2007"] L. J. P. Kilford |
| title_full_unstemmed | Modular forms a classical and computational introduction ; ["based on notes for lectures given at the Mathematical Institute at the University of Oxford ... 2004-2007"] L. J. P. Kilford |
| title_short | Modular forms |
| title_sort | modular forms a classical and computational introduction based on notes for lectures given at the mathematical institute at the university of oxford 2004 2007 |
| title_sub | a classical and computational introduction ; ["based on notes for lectures given at the Mathematical Institute at the University of Oxford ... 2004-2007"] |
| topic | Forms, Modular Algebraic spaces Modulform (DE-588)4128299-1 gnd |
| topic_facet | Forms, Modular Algebraic spaces Modulform Konferenzschrift |
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