LMSST: 24 Lectures on Elliptic Curves

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Cambridge University Press, Nov 21, 1991 - Mathematics - 137 pages
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The study of special cases of elliptic curves goes back to Diophantos and Fermat, and today it is still one of the liveliest centers of research in number theory. This book, 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-Wei 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. Many examples and exercises are included for the reader, and those new to elliptic curves, whether they are graduate students or specialists from other fields, will find this a valuable introduction.
 

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Contents

Introduction
1
Curves of genus 0 Introduction
3
padic numbers
6
The localglobal principle for conies
13
Geometry of numbers
17
Localglobal principle Conclusion of proof
20
Cubic curves
23
Nonsingular cubics The group law
27
Remedial mathematics Resultants
75
Heights Finite Basis Theorem
78
Localglobal for genus 1
85
Elements of Galois cohomology
89
Construction of the jacobian
92
Some abstract nonsense
98
Principal homogeneous spaces and Galois cohomology
104
The TateShafarevich group
108

Elliptic curves Canonical Form
32
Degenerate laws
39
Reduction
42
The padic case
46
Global torsion
50
Finite Basis Theorem Strategy and comments
54
A 2isogeny
58
The weak finite basis theory
66
The endomorphism group
112
Points over finite fields
118
Factorizing using elliptic curves
124
Formulary
130
Further Reading
135
Index
136
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