Modular forms: a classical and computational introduction
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
London
Imperial College Press
2015
|
| Ausgabe: | 2. ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | XII, 239 S. graph. Darst. |
| ISBN: | 9781783265459 1783265450 |
Internformat
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| 245 | 1 | 0 | |a Modular forms |b a classical and computational introduction |c L. J. P. Kilford |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a London |b Imperial College Press |c 2015 | |
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Datensatz im Suchindex
| _version_ | 1804153060157554688 |
|---|---|
| adam_text | Titel: Modular forms
Autor: Kilford, L. J. P
Jahr: 2015
Contents
Acknowledgements for the Second Edition 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....................29
2.7 Eisenstein series for congruence subgroups......... 32
2.8 Derivatives of modular forms................................35
2.8.1 Quasi-modular forms................ 38
2.9 Exercises ........................... 38
3. Results on Finite-Dimensionality 43
3.1 Spaces of modular forms are finite-dimensional............43
3.2 Explicit formulae for the dimensions of spaces of modular
forms ..........................................................48
3.2.1 Formulae for the full modular group................48
3.2.2 Formulae for congruence subgroups................51
3.3 The Sturm bound ............................................54
3.4 Exercises ......................................................58
4. The Arithmetic of Modular Forms 59
4.1 Hecke operators................................................60
4.1.1 Motivation for the Hecke operators ................60
4.1.2 Hecke operators for Mfc(SL2(Z))....................61
4.1.3 Hecke operators for congruence subgroups .... 65
4.2 Bases of eigenforms............................................71
4.2.1 The Petersson scalar product........................72
4.2.2 The Hecke operators are Hermitian................77
4.2.3 Integral bases ........................................82
4.3 Oldforms and newforms......................................83
4.3.1 Multiplicity one for newforms ......................87
4.4 Exercises ......................................................90
5. Applications of Modular Forms 95
5.1 Modular functions............................................96
5.2 ^-products and ^-quotients ..................................100
5.3 The arithmetic of the j-invariant............................105
5.3.1 The j-invariant and the Monster group............108
5.3.2 Ramanujan s Constant ..............................109
5.4 Applications of the modular function A(z)..................110
5.4.1 Computing digits of ir using A(z) ..................111
5.4.2 Proving Picard s Theorem ..........................113
5.5 Identities of series and products......................114
5.6 The Ramanujan-Petersson Conjecture......................115
5.7 Elliptic curves and modular forms ..........................118
5.7.1 Format s Last Theorem..............................121
5.8 Theta functions and their applications......................122
5.8.1 Representations of n by a quadratic form in an
even number of variables............................123
5.8.2 Representations of n by a quadratic form in an odd
number of variables..................................130
5.8.3 The Shimura correspondence........................133
5.9 CM modular forms............................................135
5.10 Lacunary modular forms......................................137
5.11 Exercises ......................................................140
6. Modular Forms in Characteristic p 145
6.1 Classical treatment............................................145
6.1.1 The structure of the ring of mod p forms..........146
6.1.2 The 0 operator on mod p modular forms..........152
6.1.3 Hecke operators and Hecke eigenforms ............153
6.2 Galois representations attached to mod p modular forms . 154
6.3 Katz modular forms..........................................158
6.4 The Sturm bound in characteristic p........................160
6.5 Computations with mod p modular forms..................161
6.6 Exercises ......................................................163
7. Computing with Modular Forms 165
7.1 Historical introduction to computations in number theory 165
7.2 Magma........................................................169
7.2.1 Magma philosophy..................................172
7.2.2 Magma programming................................173
7.3 Sage ..........................................................175
7.3.1 Sage philosophy......................................177
7.3.2 SAGE programming..................................177
7.3.3 The Sage interface..................................178
7.3.4 Sage graphics........................................179
7.4 Pari and other systems......................................180
7.4.1 Pari..................................................180
7.4.2 Other systems and solutions........................181
7.5 Discussion of computation....................................182
7.5.1 Computation today..................................182
7.5.2 Expected running times..............................184
7.5.3 Using computation effectively ......................185
7.5.4 The limits of computation..........................186
7.5.5 Explicit examples of limitations....................188
7.5.6 Guy s law of small numbers ........................189
7.5.7 How hard is it to calculate Fourier coefficients of
modular forms?......................................191
7.6 Exercises ......................................................192
7.6.1 Magma ..............................................192
7.6.2 SAGE..................................................194
7.6.3 Pari..................................................195
7.6.4 Maple................................................195
8. The Future of Modular Forms? 197
8.1 Disappearance?................................................197
8.2 The Status Quo................................................198
8.3 Generalizations................................................198
8.3.1 Siegel modular forms................................198
8.3.2 Hilbert modular forms ..............................200
8.3.3 Generalized modular forms..........................201
8.3.4 Almost modular forms ..............................202
8.4 Exercises ......................................................203
9. Modular Forms Projects 205
Appendix A Magma Code for Classical Modular Forms 209
Appendix B Sage Code for Classical Modular Forms 211
Appendix C Hints and Answers to Selected Exercises 213
Bibliography 219
List of Symbols 231
Index 235
|
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| spelling | Kilford, L. J. P. Verfasser (DE-588)1047868490 aut Modular forms a classical and computational introduction L. J. P. Kilford 2. ed. London Imperial College Press 2015 XII, 239 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Forms, Modular Algebraic spaces Modulform (DE-588)4128299-1 gnd rswk-swf 1\p (DE-588)1071861417 Konferenzschrift gnd-content Modulform (DE-588)4128299-1 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027839173&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Kilford, L. J. P. Modular forms a classical and computational introduction 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 |
| title_auth | Modular forms a classical and computational introduction |
| title_exact_search | Modular forms a classical and computational introduction |
| title_full | Modular forms a classical and computational introduction L. J. P. Kilford |
| title_fullStr | Modular forms a classical and computational introduction L. J. P. Kilford |
| title_full_unstemmed | Modular forms a classical and computational introduction L. J. P. Kilford |
| title_short | Modular forms |
| title_sort | modular forms a classical and computational introduction |
| title_sub | a classical and computational introduction |
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| topic_facet | Forms, Modular Algebraic spaces Modulform Konferenzschrift |
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