Fundamentals of diophantine geometry
著者
書誌事項
Fundamentals of diophantine geometry
Springer-Verlag, 1983
- : New York
- : Berlin
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注記
"An earlier version of this book, Diophantine geometry, was published by Wiley-Interscience"--T.p. verso
Bibliography: p. [359]-365
Includes index
内容説明・目次
内容説明
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.
目次
1 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 Neron Divisors.- 11 Neron Functions on Abelian Varieties.- 12 Algebraic Families of Neron Functions.- 13 Neron 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.
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