Acknowledgment
Some Standard Notation
CHAPTER 1 Absolute Values
1.Definitions， dependence and independence
2.Completions
3.Unramified extensions
4.Finite extensions
CHAPTER 2 Proper Sets of Absolute Values.
Divisors and Units
1.Proper sets of absolute values
2.Number fields
3.Divisors on varieties
4.Divisors on schemes
5.Mr-divisors and divisor classes
6.Ideal classes and units in number fields
7.Relative units and divisor classes
8.The Chevalley-Weil theorem
CHAPTER 3 Heights
1.Definitions
2.Gauss' lemma
3.Heights in function fields
4.Heights on abelian groups
5.Counting points of bounded height
CHAPTER 4
Geometric Properties of Heights
1.Functorial properties
2.Heights and linear systems
3.Ample linear systems
4.Projections on curves
5.Heights associated with divisor classes
CHAPTER 5 Heights on Abelian Varieties
1.Some linear and quasi-linear algebra
2.Quadraticity ofendomorphisms on divisor classes
3.Quadraticity of the height
4.Heights and Poincare divisors
5.Jacobian varieties and curves
6.Definiteness properties Over number fields
7.Non-degenerate heights and Euclidean spaces
8.Mumford's theorem
CHAPTER 6 The Mordell-Weil Theorem
1.Kummer theory
2.The weak Mordell-Weil theorem
3.The infinite descent
4.Reduction steps
5.Points of bounded height
6.Theorem of the base
CHAPTER 7 The Thue-Siegel-Roth Theorem
1.Statement of the theorem
2.Reduction to simultaneous approximations
3.Basic steps of the proof
4.A combinatorial lemma
5.Proof of Proposition 3.1
6.Wronskians
7.Factorization of a polynomial
8.The index
9.Proof of Proposition 3.2
10.A geometric formulation of Roth's theorem
CHAPTER 8 Siegel's Theorem and Integral Points
1.Height of integral points
2.Finiteness theorems
3.The curve ax + by = 1
4.The Thue-Siegel curve
5.Curves of genus 0
6.Torsion points on curves
7.Division points on curves
8.Non-abelian Kummer theory
CHAPTER 9 Hilbert's Irreducibility Theorem
1.Irreducibility and integral points
2.Irreducibility Over the rational numbers
3.Reduction steps
4.Function fields
5.Abstract definition of Hilbert sets
6.Applications to commutative group varieties
CHAPTER 10 Well Functions and Neron Divisors
I.Bounded sets and functions
2.Neron divisors and Well functions
3.Positive divisors
4.The associated height function
CHAPTER 11 Neron Functions on Abelian Varieties
i.Existence of Neron functions
2.Translation properties of Neron functions
3.Neron functions on varieties
4.Reciprocity laws
5.Neron functions as intersection multiplicities
6.The Neron symbol and group extensions
CHAPTER 12 Algebraic Families of Neron Functions
1.Variation of Neron functions in an algebraic family
2.Silverman's height and specialization theorems
3.Neron heights as intersection multiplicities
4.Fibral divisors
5.The height determined by a section ： Tate's theorem
CHAPTER 13 Neron Functions Over the Complex Numbers
1.The Neron function of an abelian variety
2.The scalar product of differentials of first kind
3.The canonical 2-form and the Riemann theta function
4.The divisor of the Riemann theta function
5.Green， Neron， and theta functions
6.The law of interchange of argument and parameter
7.Differentials of third kind and Green's function
Appendix
Review of S.Lang's Diophantine Geometry， by L.J.Mordell
Review of L.J.Mordell's Diophantine Equations， by S.Lang
Bibliography
Index
