Fundamentals of Diophantine Geometry

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Springer Science & Business Media, Aug 29, 1983 - Mathematics - 370 pages
Diophantine problems represent some of the strongest aesthetic attractions to algebraic geometry. They consist in giving criteria for the existence of solutions of algebraic equations in rings and fields, and eventually for the number of such solutions. The fundamental ring of interest is the ring of ordinary integers Z, and the fundamental field of interest is the field Q of rational numbers. One discovers rapidly that to have all the technical freedom needed in handling general problems, one must consider rings and fields of finite type over the integers and rationals. Furthermore, one is led to consider also finite fields, p-adic fields (including the real and complex numbers) as representing a localization of the problems under consideration. We shall deal with global problems, all of which will be of a qualitative nature. On the one hand we have curves defined over say the rational numbers. Ifthe curve is affine one may ask for its points in Z, and thanks to Siegel, one can classify all curves which have infinitely many integral points. This problem is treated in Chapter VII. One may ask also for those which have infinitely many rational points, and for this, there is only Mordell's conjecture that if the genus is :;;; 2, then there is only a finite number of rational points.
 

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Contents

III
1
IV
5
V
9
VI
12
VII
18
VIII
19
IX
21
X
24
XLIX
183
L
188
LI
189
LII
192
LIII
194
LIV
196
LV
197
LVI
200

XI
29
XII
32
XIII
41
XIV
44
XV
50
XVI
54
XVII
62
XVIII
66
XIX
70
XX
76
XXI
83
XXII
87
XXIII
90
XXIV
91
XXV
95
XXVI
99
XXVII
106
XXVIII
110
XXIX
113
XXX
120
XXXI
124
XXXII
134
XXXIII
138
XXXIV
139
XXXV
144
XXXVI
145
XXXVII
146
XXXVIII
149
XXXIX
153
XL
158
XLI
163
XLII
165
XLIII
170
XLIV
171
XLV
173
XLVI
175
XLVII
178
XLVIII
181
LVII
205
LVIII
212
LIX
225
LX
226
LXI
229
LXII
233
LXIII
236
LXIV
239
LXV
242
LXVI
247
LXIX
252
LXX
258
LXXI
263
LXXII
266
LXXIII
271
LXXIV
276
LXXV
283
LXXVI
286
LXXVII
290
LXXVIII
296
LXXIX
297
LXXX
303
LXXXI
307
LXXXII
314
LXXXIII
320
LXXXIV
324
LXXXV
327
LXXXVI
332
LXXXVII
334
LXXXVIII
339
LXXXIX
341
XC
344
XCI
347
XCII
349
XCIII
355
XCIV
359
XCV
367
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