Elliptic curves, modular forms, and their L-functions:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
2011
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| Series: | Student mathematical library
Volume 58 : IAS/Park City mathematical subseries |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xiv, 195 Seiten Illustrationen, Diagramme |
| ISBN: | 9780821852422 |
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MARC
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| 245 | 1 | 0 | |a Elliptic curves, modular forms, and their L-functions |c Álvaro Lozano-Robledo |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c 2011 | |
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Record in the Search Index
| _version_ | 1804145615309897729 |
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| adam_text | Titel: Elliptic curves, modular forms, and their L-functions
Autor: Lozano-Robledo, Álvaro
Jahr: 2011
Contents
Preface xi
Chapter 1. Introduction 1
§1.1. Elliptic curves 1
§1.2. Modular forms 7
§1.3. L-functions 11
§1.4. Exercises 15
Chapter 2. Elliptic curves 17
§2.1. Why elliptic curves? 17
§2.2. Definition 20
§2.3. Integral points 23
§2.4. The group structure on E(Q) 24
§2.5. The torsion subgroup 32
§2.6. Elliptic curves over finite fields 35
§2.7. The rank and the free part of E(Q) 43
§2.8. Linear independence of rational points 46
§2.9. Descent and the weak Mordeil-Weil theorem 49
§2.10. Homogeneous Spaces 59
§2.11. Selmer and Sha 66
vii
Contents
§2.12. Exercises 69
Chapter 3. Modular curves 77
§3.1. Elliptic curves over C 77
§3.2. Functions on lattices and elliptic functions 82
§3.3. Elliptic curves and the upper half-plane 84
§3.4. The modular curve X(l) 87
§3.5. Congruence subgroups 90
§3.6. Modular curves 91
§3.7. Exercises 94
Chapter 4. Modular forms 99
§4.1. Modular forms for the modular group 99
§4.2. Modular forms for congruence subgroups 105
§4.3. The Petersson inner product HO
§4.4. Hecke Operators acting on cusp forms Hl
§4.5. Exercises 118
Chapter 5. L-functions 123
§5.1. The L-function of an elliptic curve 123
§5.2. The Birch and Swinnerton-Dyer conjecture 127
§5.3. The L-function of a modular (cusp) form 135
§5.4. The Taniyama-Shimura-Weil conjecture 137
§5.5. Fermat s last theorem 140
§5.6. Looking back and looking forward 142
§5.7. Exercises 143
Appendix A. PARI/GP and Sage 147
§A.l. Elliptic curves 147
§A.2. Modular forms l54
§A.3. L-functions I56
§A.4. Other Sage commands 158
Appendix B. Complex analysis l59
Contents ix
§B.l. Complex numbers 159
§B.2. Analytic functions 160
§B.3. Meromorphic functions 163
§B.4. The complex exponential function 165
§B.5. Theorems in complex analysis 166
§B.6. Quotients of the complex plane 168
§B.7. Exercises 169
Appendix C. Projective space 171
§C.l. The projective line 171
§C2. The projective plane 173
§C3. Over an arbitrary field 174
§C4. Curves in the projective plane 175
§C5. Singular and smooth curves 176
Appendix D. The p-adic numbers 179
§D.l. Hensel s lemma 181
§D.2. Exercises 182
Appendix E. Parametrization of torsion structures 185
Bibliography 189
Index 193
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| id | DE-604.BV037331538 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T23:22:15Z |
| institution | BVB |
| isbn | 9780821852422 |
| language | English |
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| physical | xiv, 195 Seiten Illustrationen, Diagramme |
| publishDate | 2011 |
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| publisher | American Mathematical Society |
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| series | Student mathematical library |
| series2 | Student mathematical library IAS/Park City mathematical subseries |
| spelling | Lozano-Robledo, Álvaro 1978- Verfasser (DE-588)143876678 aut Elliptic curves, modular forms, and their L-functions Álvaro Lozano-Robledo Providence, Rhode Island American Mathematical Society 2011 xiv, 195 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Student mathematical library Volume 58 IAS/Park City mathematical subseries Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-1641-6 Student mathematical library Volume 58 : IAS/Park City mathematical subseries (DE-604)BV013184751 58 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022485442&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lozano-Robledo, Álvaro 1978- Elliptic curves, modular forms, and their L-functions Student mathematical library Algebraische Geometrie (DE-588)4001161-6 gnd |
| subject_GND | (DE-588)4001161-6 |
| title | Elliptic curves, modular forms, and their L-functions |
| title_auth | Elliptic curves, modular forms, and their L-functions |
| title_exact_search | Elliptic curves, modular forms, and their L-functions |
| title_full | Elliptic curves, modular forms, and their L-functions Álvaro Lozano-Robledo |
| title_fullStr | Elliptic curves, modular forms, and their L-functions Álvaro Lozano-Robledo |
| title_full_unstemmed | Elliptic curves, modular forms, and their L-functions Álvaro Lozano-Robledo |
| title_short | Elliptic curves, modular forms, and their L-functions |
| title_sort | elliptic curves modular forms and their l functions |
| topic | Algebraische Geometrie (DE-588)4001161-6 gnd |
| topic_facet | Algebraische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022485442&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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