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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

AuthorTable of Contents

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

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