02871nam a22003135i 4500978-1-4757-1810-2cr nn 008mamaa130611s1983 xxu: s :::: 0:eng d9781475718102ENGUSQA440-699516Online resource: SpringerLang, Serge.1927-2005.Fundamentals of Diophantine Geometry[EBook]by Serge Lang.New York, NYSpringer1983.XVIII, 370 pagesonline resource.textonline resource1 Absolute Values -- 2 Proper Sets of Absolute Values. Divisors and Units -- 3 Heights -- 4 Geometric Properties of Heights -- 5 Heights on Abelian Varieties -- 6 The Mordell-Weil Theorem -- 7 The Thue-Siegel-Roth Theorem -- 8 Siegel’s Theorem and Integral Points -- 9 Hilbert’s Irreducibility Theorem -- 10 Weil Functions and Néron Divisors -- 11 Néron Functions on Abelian Varieties -- 12 Algebraic Families of Néron Functions -- 13 Néron Functions Over the Complex Numbers -- Review of S. Lang’s Diophantine Geometry, by L. J. Mordell -- Review of L. J. Mordell’s Diophantine Equations, by S. Lang.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.Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)Mathematics.Geometry.Mathematics.Geometry.SpringerLink (Online service)http://dx.doi.org/10.1007/978-1-4757-1810-2