Hjem

A The Geometry of Curves and Abelian Varieties. - A.
1 Algebraic Varieties. - A.
2 Divisors. - A.
3 Linear Systems. - A.
4 Algebraic Curves. - A.
5 Abelian Varieties over C. - A.
6 Jacobians over C. - A.
7 Abelian Varieties over Arbitrary Fields. - A.
8 Jacobians over Arbitrary Fields. - A.
9 Schemes. - B Height Functions. - B.
1 Absolute Values. - B.
2 Heights on Projective Space. - B.
3 Heights on Varieties. - B.
4 Canonical Height Functions. - B.
5 Canonical Heights on Abelian Varieties. - B.
6 Counting Rational Points on Varieties. - B.
7 Heights and Polynomials. - B.
8 Local Height Functions. - B.
9 Canonical Local Heights on Abelian Varieties. - B.
10 Introduction to Arakelov Theory. - Exercises. - C Rational Points on Abelian Varieties. - C.
1 The Weak Mordell--Weil Theorem. - C.
2 The Kernel of Reduction Modulo p. - C.
3 Appendix: Finiteness Theorems in Algebraic Number Theory. - C.
4 Appendix: The Selmer and Tate--Shafarevich Groups. - C.
5 Appendix: Galois Cohomology and Homogeneous Spaces. - Exercises. - D Diophantine Approximation and Integral Points on Curves. - D.
1 Two Elementary Results on Diophantine Approximation. - D.
2 Roth's Theorem. - D.
3 Preliminary Results. - D.
4 Construction of the Auxiliary Polynomial. - D.
5 The
Index Is Large. - D.
6 The
Index Is Small (Roth's Lemma). - D.
7 Completion of the Proof of Roth's Theorem. - D.
8 Application: The Unit Equation U + V =
1. - D.
9 Application: Integer Points on Curves. - Exercises. - E Rational Points on Curves of Genus at Least
2. - E.
I Vojta's Geometric Inequality and Faltings' Theorem. - E.
2 Pinning Down Some Height Functions. - E.
3 An Outline of the Proof of Vojta's Inequality. - E.
4 An Upper Bound for h?(z, w). - E.
5 A Lower Bound for h?(z,w) for Nonvanishing Sections. - E.
6 Constructing Sections of Small Height
I: Applying Riemann--Roch. - E.
7 Constructing Sections of Small Height
II: Applying Siegel's Lemma. - E.
8 Lower Bound for h?(z,w) at Admissible Version
I. - E.
9 Eisenstein's Estimate for the Derivatives of an Algebraic Function. - E.
10 Lower Bound for h?(z,w) at Admissible: Version
II. - E.
11 A Nonvanishing Derivative of Small Order. - E.
12 Completion of the Proof of Vojta's Inequality. - Exercises. - F Further Results and Open Problems. - F.
1 Curves and Abelian Varieties. - F.
1.
1 Rational Points on Subvarieties of Abelian Varieties. - F.
1.
2 Application to Points of Bounded Degree on Curves. - F.
2 Discreteness of Algebraic Points. - F.
2.
1 Bogomolov's Conjecture. - F.
2.
2 The Height of a Variety. - F.
3 Height Bounds and Height Conjectures. - F.
4 The Search for Effectivity. - F.
4.
1 Effective Computation of the Mordell--Weil Group A(k). - F.
4.
2 Effective Computation of Rational Points on Curves. - F.
4.
3 Quantitative Bounds for Rational Points. - F.
5 Geometry Governs Arithmetic. - F.
5.
1 Kodaira Dimension. - F.
5.
2 The Bombieri-Lang Conjecture. - F.
5.
3 Vojta's Conjecture. - F.
5.
4 Varieties Whose Rational Points Are Dense. - Exercises. - References. - List of Notation.