| |
| |
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 | |