Elliptic Curves: Diophantine Analysis

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Springer Science & Business Media, Nov 1, 1978 - Mathematics - 264 pages
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It is possible to write endlessly on elliptic curves. (This is not a threat.) We deal here with diophantine problems, and we lay the foundations, especially for the theory of integral points. We review briefly the analytic theory of the Weierstrass function, and then deal with the arithmetic aspects of the addition formula, over complete fields and over number fields, giving rise to the theory of the height and its quadraticity. We apply this to integral points, covering the inequalities of diophantine approximation both on the multiplicative group and on the elliptic curve directly. Thus the book splits naturally in two parts. The first part deals with the ordinary arithmetic of the elliptic curve: The transcendental parametrization, the p-adic parametrization, points of finite order and the group of rational points, and the reduction of certain diophantine problems by the theory of heights to diophantine inequalities involving logarithms. The second part deals with the proofs of selected inequalities, at least strong enough to obtain the finiteness of integral points.
  

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Contents

II
3
III
6
IV
10
V
13
VI
17
VII
19
VIII
23
IX
26
XLII
148
XLIII
151
XLIV
155
XLV
159
XLVII
162
XLVIII
164
XLIX
166
L
169

X
33
XII
37
XIII
43
XIV
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XV
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XVI
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XVII
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XVIII
62
XIX
68
XX
73
XXI
77
XXIII
84
XXIV
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XXV
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XXVI
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XXVII
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XXVIII
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XXX
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XXXI
107
XXXII
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XXXIII
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XXXIV
128
XXXV
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XXXVI
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XXXVII
140
XXXVIII
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XXXIX
144
XL
146
XLI
147
LI
173
LII
176
LIII
181
LV
184
LVI
186
LVII
188
LVIII
190
LIX
192
LX
193
LXII
197
LXIII
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LXIV
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LXV
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LXVI
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LXVII
210
LXVIII
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LXIX
218
LXX
221
LXXI
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LXXII
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LXXIII
232
LXXIV
234
LXXV
235
LXXVI
238
LXXVII
241
LXXVIII
246
LXXIX
253
LXXX
260
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About the author (1978)

Lang, Yale University, New Haven, CT.