| Description |
1 online resource (vi, 137 pages) |
| Series |
London Mathematical Society student texts ; 24 |
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London Mathematical Society student texts ; 24.
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| Contents |
1 Curves of genus 0. Introduction 3 -- 2 p-adic numbers 6 -- 3 Local-global principle for conics 13 -- 4 Geometry of numbers 17 -- 5 Local-global principle. Conclusion of proof 20 -- 6 Cubic curves 23 -- 7 Non-singular cubics. The group law 27 -- 8 Elliptic curves. Canonical form 32 -- 9 Degenerate laws 39 -- 10 Reduction 42 -- 11 P-adic case 46 -- 12 Global torsion 50 -- 13 Finite basis theorem. Strategy and comments 54 -- 14 A 2-isogeny 58 -- 15 Weak finite basis theory 66 -- 16 Remedial mathematics. Resultants 75 -- 17 Heights. Finite basis Theorem 78 -- 18 Local-global for genus 1 85 -- 19 Elements of Galois cohomology 89 -- 20 Construction of the jacobian 92 -- 21 Some abstract nonsense 98 -- 22 Principal homogeneous spaces and Galois cohomology 104 -- 23 Tate-Shafarevich group 108 -- 24 Endomorphism group 114 -- 25 Points over finite fields 118 -- 26 Factorizing using elliptic curves 124 |
| Summary |
The study of (special cases of) elliptic curves goes back to Diophantos and Fermat, and today it is still one of the liveliest centres of research in number theory. This book, which is addressed to beginning graduate students, introduces basic theory from a contemporary viewpoint but with an eye to the historical background. The central portion deals with curves over the rationals: the Mordell-Weil finite basis theorem, points of finite order (Nagell-Lutz) etc. The treatment is structured by the local-global standpoint and culminates in the description of the Tate-Shafarevich group as the obstruction to a Hasse principle. In an introductory section the Hasse principle for conics is discussed. The book closes with sections on the theory over finite fields (the 'Riemann hypothesis for function fields') and recently developed uses of elliptic curves for factoring large integers. Prerequisites are kept to a minimum; an acquaintance with the fundamentals of Galois theory is assumed, but no knowledge either of algebraic number theory or algebraic geometry is needed. The p-adic numbers are introduced from scratch, as is the little that is needed on Galois cohomology. Many examples and exercises are included for the reader. For those new to elliptic curves, whether they are graduate students or specialists from other fields, this will be a fine introductory text |
| Bibliography |
Includes bibliographical references (page 135) and index |
| Notes |
English |
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Print version record |
| Subject |
Curves, Elliptic.
|
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MATHEMATICS -- Geometry -- Algebraic.
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Curves, Elliptic
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Diophantische Gleichung
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Elliptische Kurve
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Elliptische functies.
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Courbes elliptiques.
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Elliptiske kurver.
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| Form |
Electronic book
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| ISBN |
9781107088290 |
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1107088291 |
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9781139172530 |
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1139172530 |
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1316086992 |
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9781316086995 |
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1107091322 |
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9781107091320 |
|
1107100313 |
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9781107100312 |
|
