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LMSST

24 Lectures on Elliptic Curves

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Paperback | 20 January 1992

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144 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 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.

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'... an excellent introduction ... written with humour.' Monatshefte fur Mathematik

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Introduction	p. 1
Curves of genus 0. Introduction	p. 3
p-adic numbers	p. 6
The local-global principle for conics	p. 13
Geometry of numbers	p. 17
Local-global principle. Conclusion of proof	p. 20
Cubic curves	p. 23
Non-singular cubics. The group law	p. 27
Elliptic curves. Canonical form	p. 32
Degenerate laws	p. 39
Reduction	p. 42
The p-adic case	p. 46
Global torsion	p. 50
Finite basis theorem. Strategy and comments	p. 54
A 2-isogeny	p. 58
The weak finite basis theory	p. 66
Remedial mathematics. Resultants	p. 75
Heights. Finite basis Theorem	p. 78
Local-global for genus 1	p. 85
Elements of Galois cohomology	p. 89
Construction of the jacobian	p. 92
Some abstract nonsense	p. 98
Principal homogeneous spaces and Galois cohomology	p. 104
The Tate-Shafarevich group	p. 108
The endomorphism group	p. 114
Points over finite fields	p. 118
Factorizing using elliptic curves	p. 124
Formulary	p. 130
Further Reading	p. 135
Index	p. 136
Table of Contents provided by Syndetics. All Rights Reserved.

LMSST

24 Lectures on Elliptic Curves

At a Glance

Paperback

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

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