000 02252nam a2200337Ia 4500
003 OSt
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020 _a9783540612230 (pbk.)
_c€ 99.99
040 _bENG
_cIISER-BPR
041 _aENG
082 _a512.73
_bLAN
_223rd
100 _aLang, Serge
_91085
222 _aMathematics
245 0 _aSurvey of diophantine geometry
250 _a1st ed.
260 _aBerlin:
_bSpringer-Verlag,
_cc1997
300 _axiii, 298p. :
_bill. ;
_c24cm
504 _aIncludes illustrations, notation, bibliographic references and subject index.
520 _aDiophantine 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.
650 _aMathematics
_92466
650 _aAlgebra
_92467
650 _aNumber theory
_92468
650 _aAnalytic number theory
_92469
650 _aDiophantine approximations
_92470
650 _aDiophantine geometry
_92471
942 _cBK
_2ddc
942 _2ddc
947 _a7331.0835
948 _a22
999 _c4225
_d4225