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LMSST : 24 Lectures on Elliptic Curves - J. W. S. Cassels

LMSST

24 Lectures on Elliptic Curves

By: J. W. S. Cassels, J. Bruce (Editor)

Paperback

Published: 20th January 1992
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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.

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

Introductionp. 1
Curves of genus 0. Introductionp. 3
p-adic numbersp. 6
The local-global principle for conicsp. 13
Geometry of numbersp. 17
Local-global principle. Conclusion of proofp. 20
Cubic curvesp. 23
Non-singular cubics. The group lawp. 27
Elliptic curves. Canonical formp. 32
Degenerate lawsp. 39
Reductionp. 42
The p-adic casep. 46
Global torsionp. 50
Finite basis theorem. Strategy and commentsp. 54
A 2-isogenyp. 58
The weak finite basis theoryp. 66
Remedial mathematics. Resultantsp. 75
Heights. Finite basis Theoremp. 78
Local-global for genus 1p. 85
Elements of Galois cohomologyp. 89
Construction of the jacobianp. 92
Some abstract nonsensep. 98
Principal homogeneous spaces and Galois cohomologyp. 104
The Tate-Shafarevich groupp. 108
The endomorphism groupp. 114
Points over finite fieldsp. 118
Factorizing using elliptic curvesp. 124
Formularyp. 130
Further Readingp. 135
Indexp. 136
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521425308
ISBN-10: 0521425301
Series: London Mathematical Society Student Texts
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 144
Published: 20th January 1992
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 22.86 x 15.88  x 0.64
Weight (kg): 0.22