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Heights in Diophantine Geometry

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ISBN-10: 0521846153

ISBN-13: 9780521846158

Edition: 2006

Authors: Enrico Bombieri, Walter Gubler, Bela Bollobas, William Fulton, Anatole Katok

List price: $141.99
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Description:

Diophantine geometry has been studied by number theorists for thousands of years, since the time of Pythagoras, and has continued to be a rich area of ideas such as Fermat's Last Theorem, and most recently the ABC conjecture. This monograph is a bridge between the classical theory and modern approach via arithmetic geometry. The authors provide a clear path through the subject for graduate students and researchers. They have re-examined many results and much of the literature, and provide a thorough account of several topics at a level not seen before in book form. The treatment is largely self-contained, with proofs given in full detail.
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Book details

List price: $141.99
Copyright year: 2006
Publisher: Cambridge University Press
Publication date: 1/19/2006
Binding: Hardcover
Pages: 668
Size: 6.25" wide x 9.00" long x 1.25" tall
Weight: 2.288
Language: English

Dr Walter Gubler is a Lecturer in Mathematics at the University of Dortmund.

Preface
Terminology
Heights
Introduction
Absolute values
Finite-dimensional extensions
The product formula
Heights in projective and affine space
Heights of polynomials
Lower bounds for norms of products of polynomials
Bibliographical notes
Weil heights
Introduction
Local heights
Global heights
Weil heights
Explicit bounds for Weil heights
Bounded subsets
Metrized line bundles and local heights
Heights on Grassmannians
Siegel's lemma
Bibliographical notes
Linear tori
Introduction
Subgroups and lattices
Subvarieties and maximal subgroups
Bibliographical notes
Small points
Introduction
Zhang's theorem
The equidistribution theorem
Dobrowolski's theorem
Remarks on the Northcott property
Remarks on the Bogomolov property
Bibliographical notes
The unit equation
Introduction
The number of solutions of the unit equation
Applications
Effective methods
Bibliographical notes
Roth's theorem
Introduction
Roth's theorem
Preliminary lemmas
Proof of Roth's theorem
Further results
Bibliographical notes
The subspace theorem
Introduction
The subspace theorem
Applications
The generalized unit equation
Proof of the subspace theorem
Further results: the product theorem
The absolute subspace theorem and the Faltings-Wustholz theorem
Bibliographical notes
Abelian varieties
Introduction
Group varieties
Elliptic curves
The Picard variety
The theorem of the square and the dual abelian variety
The theorem of the cube
The isogeny multiplication by n
Characterization of odd elements in the Picard group
Decomposition into simple abelian varieties
Curves and Jacobians
Bibliographical notes
Neron-Tate heights
Introduction
Neron-Tate heights
The associated bilinear form
Neron-Tate heights on Jacobians
The Neron symbol
Hilbert's irreducibility theorem
Bibliographical notes
The Mordell-Weil theorem
Introduction
The weak Mordell-Weil theorem for elliptic curves
The Chevalley-Weil theorem
The weak Mordell-Weil theorem for abelian varieties
Kummer theory and Galois cohomology
The Mordell-Weil theorem
Bibliographical notes
Faltings's theorem
Introduction
The Vojta divisor
Mumford's method and an upper bound for the height
The local Eisenstein theorem
Power series, norms, and the local Eisenstein theorem
A lower bound for the height
Construction of a Vojta divisor of small height
Application of Roth's lemma
Proof of Faltings's theorem
Some further developments
Bibliographical notes
The abc-conjecture
Introduction
The abc-conjecture
Belyi's theorem
Examples
Equivalent conjectures
The generalized Fermat equation
Bibliographical notes
Nevanlinna theory
Introduction
Nevanlinna theory in one variable
Variations on a theme: the Ahlfors-Shimizu characteristic
Holomorphic curves in Nevanlinna theory
Bibliographical notes
The Vojta conjectures
Introduction
The Vojta dictionary
Vojta's conjectures
A general abc-conjecture
The abc-theorem for function fields
Bibliographical notes
Algebraic geometry
Introduction
Affine varieties
Topology and sheaves
Varieties
Vector bundles
Projective varieties
Smooth varieties
Divisors
Intersection theory of divisors
Cohomology of sheaves
Rational maps
Properties of morphisms
Curves and surfaces
Connexion to complex manifolds
Ramification
Discriminants
Unramified field extensions
Unramified morphisms
The ramification divisor
Geometry of numbers
Adeles
Minkowski's second theorem
Cube slicing
References
Glossary of notation
Index