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

I
1
II
3
III
6
IV
13
V
17
VI
20
VII
23
VIII
27
XVII
75
XVIII
78
XIX
85
XX
89
XXI
92
XXII
98
XXIII
104
XXIV
108

IX
32
X
39
XI
42
XII
46
XIII
50
XIV
54
XV
58
XVI
66
XXV
112
XXVI
118
XXVII
124
XXVIII
130
XXIX
135
XXX
136
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