# Fundamentals of Diophantine Geometry

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 Copyright

### References to this book

 Introduction to Complex Hyperbolic SpacesSerge LangLimited preview - 1987
 LMSST: 24 Lectures on Elliptic CurvesLimited preview - 1991
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