Invitation to the Mathematics of Fermat-Wiles:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English French |
| Veröffentlicht: |
San Diego [u.a.]
Academic Press
2002
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XI, 381 S.: graph. Darst. |
| ISBN: | 0123392519 |
Internformat
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Datensatz im Suchindex
| _version_ | 1804128904102805504 |
|---|---|
| adam_text | INVITATION TO THE MATHEMATICS OF FERMAT-WILES YVES HELLEGOUARCH
UNIVERSITY OF CAEN, FRANCE THIS WORK HAS BEEN PUBLISHED WITH THE HELP OF
THE FRENCH MINISTERE DE LA CULTURE - CENTRE NATIONAL DU LIVRE ACADEMIC
PRESS AN IMPRINT OF ELSEVIER AMSTERDAM * BOSTON * HEIDELBERG * LONDON *
NEW YORK * OXFORD PARIS * SAN DIEGO * SAN FRANCISCO * SINGAPORE * SYDNEY
* TOKYO CONTENTS FOREWORD VIII 1 PATHS 1 1.1 DIOPHANTUS AND HIS A
RITHMETICA 2 1.2 TRANSLATIONS OF DIOPHANTUS 2 1.3 FERMAT 3 1.4 INFINITE
DESCENT 4 1.5 FERMAT S THEOREM IN DEGREE 4 7 1.6 THE THEOREM OF TWO
SQUARES 9 1.6.1 A MODERN PROOF 10 1.6.2 FERMAT-STYLE PROOF OF THE
CRUCIAL THEOREM 12 1.6.3 REPRESENTATIONS AS SUMS OF TWO SQUARES 13 1.7
EULER-STYLE PROOF OF FERMAT S LAST THEOREM FOR N = 3 16 1.8 KUMMER, 1847
18 1.8.1 THE RING OF INTEGERS OF Q() 18 1.8.2, A LEMMA OF KUMMER ON THE
UNITS OF Z[F ] 23 1.8.3 THE IDEALS OF Z[] 25 1.8.4 RUMMER S PROOF
(1847) 26 1.8.5 REGULAR PRIMES 31 1.9 THE CURRENT APPROACH ; 33
EXERCISES AND PROBLEMS 35 2 ELLIPTIC FUNCTIONS 68 2.1 ELLIPTIC INTEGRALS
68 2.2 THE DISCOVERY OF ELLIPTIC FUNCTIONS IN 1718 71 2.3 EULER S
CONTRIBUTION (1753) _ _ 75 2.4 ELLIPTIC FUNCTIONS: STRUCTURE THEOREMS 77
2.5 WEIERSTRASS-STYLE ELLIPTIC FUNCTIONS 80 2.6 EISENSTEIN SERIES 85 2.7
THE WEIERSTRASS CUBIC 87 2.8 ABEL S THEOREM 89 VI CONTENTS 2.9
LOXODROMIC FUNCTIONS 92 2.10 THE FUNCTION P 95 2.11 COMPUTATION OF THE
DISCRIMINANT 97 2.12 RELATION TO ELLIPTIC FUNCTIONS 99 EXERCISES AND
PROBLEMS 101 NUMBERS AND GROUPS 118 3.1 ABSOLUTE VALUES ON Q 118 3.2
COMPLETION OF A FIELD EQUIPPED WITH AN ABSOLUTE VALUE 123 3.3 THE FIELD
OF P-ADIC NUMBERS 127 3.4 ALGEBRAIC CLOSURE OF A FIELD 131 3.5
GENERALITIES ON THE LINEAR REPRESENTATIONS OF GROUPS 134 3.6 GALOIS
EXTENSIONS 140 3.6.1 THE GALOIS CORRESPONDENCE 141 3.6.2 QUESTIONS OF
DIMENSION 143 3.6.3 STABILITY 146 3.6.4 CONCLUSIONS 146 3.7 RESOLUTION
OF ALGEBRAIC EQUATIONS 149 3.7.1 SOME GENERAL PRINCIPLES 149 3.7.2
RESOLUTION OF THE EQUATION OF DEGREE THREE 152 EXERCISES AND PROBLEMS
155 ELLIPTIC CURVES 172 4.1 CUBICS AND ELLIPTIC CURVES 172 4.2 BEZOUT S
THEOREM 179 4.3 NINE-POINT THEOREM 183 4.4 GROUP LAWS ON AN ELLIPTIC
CURVE 185 4.5 REDUCTION MODULO P 18 9 4.6 - W-DIVISION POINTS OF AN
ELLIPTIC CURVE 192 4.6.1 2-DIVISION POINTS 192 4.6.2 3-DIVISION POINTS
193 4.6.3 N-DIVISION POINTS OF AN ELLIPTIC CURVE DEFINED OVER Q 194 4.7
A MOST INTERESTING GALOIS REPRESENTATION 195 4.8 RING OF ENDOMORPHISMS
OF AN ELLIPTIC CURVE 197 4.9 ELLIPTIC CURVES OVER A FINITE FIELD 202
4.10 TORSION ON AN ELLIPTIC CURVE DEFINED OVER Q 205 4.11 MORDELL-WEIL
THEOREM 211 4.12 BACK TO THE DEFINITION OF ELLIPTIC CURVES 211 4.13
FORMULAE 215 4.14 MINIMAL WEIERSTRASS EQUATIONS (OVER Z) 218 4.15
HASSE-WEIL L-FUNCTIONS 223 4.15.1 RIEMANN ZETA FUNCTION 223 4.15.2 ARTIN
ZETA FUNCTION 224 4.15.3 HASSE-WEIL L-FUNCTION 226 EXERCISES AND
PROBLEMS 228 CONTENTS VII 5 MODULAR FORMS 255 5.1 BRIEF HISTORICAL
OVERVIEW 255 5.2 THE THETA FUNCTIONS 260 5.3 MODULAR FORMS FOR THE
MODULAR GROUP 5L 2 (Z)/{/,-/) 274 5.3.1 MODULAR PROPERTIES OF THE
EISENSTEIN SERIES 274 5.3.2 THE MODULAR GROUP 280 5.3.3 DEFINITION OF
MODULAR FORMS AND FUNCTIONS 287 5.4 THE SPACE OF MODULAR FORMS OF WEIGHT
K FOR SL 2 (Z) 289 5.5 THE FIFTH OPERATION OF ARITHMETIC 294 5.6 THE
PETERSSON HERMITIAN PRODUCT 297 5.7 HECKE FORMS 299 5.7.1 HECKE
OPERATORS FOR SL 2 (Z) 300 5.8 HECKE S THEORY 304 5.8.1 THE MELLIN
TRANSFORM 306 5.8.2 FUNCTIONAL EQUATIONS FOR THE FUNCTIONS L(F, S) 307
5.9 WILES THEOREM 308 EXERCISES AND PROBLEMS 313 6 NEW PARADIGMS, NEW
ENIGMAS 325 6.1 A SECOND DEFINITION OF THE RING 1 P OF P-ADIC INTEGERS
326 6.2 THE TATE MODULE T T (E) 328 6.3 A MARVELLOUS RESULT 330 6.4 TATE
LOXODROMIC FUNCTIONS 331 6.5 CURVES AJB , C 332 6.5.1 REDUCTION OF
CERTAIN CURVES ^BC 333 6.5.2 PROPERTY OF THE FIELD K P ASSOCIATED TO E
AP ^ TCP 335 6.5.3 SUMMARY OF THE PROPERTIES OF A PTP C P 335 6.6 THE
SERRE CONJECTURES 336 6.7 MAZUR-RIBET S THEOREM 339 6.7.1 MAZUR-RIBET S
THEOREM 340 6.7.2 OTHER APPLICATIONS 341 6.8 SZPIRO S CONJECTURE AND THE
ABC CONJECTURE 343 6.8.1 SZPIRO S CONJECTURE , 343 6.8.2 ABC CONJECTURE
344 6.8.3 CONSEQUENCES 344 EXERCISES AND PROBLEMS 348 APPENDIX: THE
ORIGIN OF THE ELLIPTIC APPROACH TO FERMAT S LAST THEOREM 359
BIBLIOGRAPHY X 371 INDEX 375
|
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| indexdate | 2024-07-09T18:56:39Z |
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| isbn | 0123392519 |
| language | English French |
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| physical | XI, 381 S.: graph. Darst. |
| publishDate | 2002 |
| publishDateSearch | 2002 |
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| publisher | Academic Press |
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| spelling | Hellegouarch, Yves Verfasser aut Invitation aux Mathématiques de Fermat-Wiles Invitation to the Mathematics of Fermat-Wiles Yves Hellegouarch San Diego [u.a.] Academic Press 2002 XI, 381 S.: graph. Darst. txt rdacontent n rdamedia nc rdacarrier Geschichte 1650-1995 gnd rswk-swf Curves, Elliptic Fermat's last theorem Forms, Modular Geschichte (DE-588)4020517-4 gnd rswk-swf Fermatsche Vermutung (DE-588)4154012-8 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Mathematik (DE-588)4037944-9 s Geschichte 1650-1995 z 1\p DE-604 Fermatsche Vermutung (DE-588)4154012-8 s Geschichte (DE-588)4020517-4 s 2\p DE-604 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009617298&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Hellegouarch, Yves Invitation to the Mathematics of Fermat-Wiles Curves, Elliptic Fermat's last theorem Forms, Modular Geschichte (DE-588)4020517-4 gnd Fermatsche Vermutung (DE-588)4154012-8 gnd Mathematik (DE-588)4037944-9 gnd |
| subject_GND | (DE-588)4020517-4 (DE-588)4154012-8 (DE-588)4037944-9 |
| title | Invitation to the Mathematics of Fermat-Wiles |
| title_alt | Invitation aux Mathématiques de Fermat-Wiles |
| title_auth | Invitation to the Mathematics of Fermat-Wiles |
| title_exact_search | Invitation to the Mathematics of Fermat-Wiles |
| title_full | Invitation to the Mathematics of Fermat-Wiles Yves Hellegouarch |
| title_fullStr | Invitation to the Mathematics of Fermat-Wiles Yves Hellegouarch |
| title_full_unstemmed | Invitation to the Mathematics of Fermat-Wiles Yves Hellegouarch |
| title_short | Invitation to the Mathematics of Fermat-Wiles |
| title_sort | invitation to the mathematics of fermat wiles |
| topic | Curves, Elliptic Fermat's last theorem Forms, Modular Geschichte (DE-588)4020517-4 gnd Fermatsche Vermutung (DE-588)4154012-8 gnd Mathematik (DE-588)4037944-9 gnd |
| topic_facet | Curves, Elliptic Fermat's last theorem Forms, Modular Geschichte Fermatsche Vermutung Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009617298&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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