Diophantine Geometry

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 MordellWeil Theorem. - C. 2 The Kernel of Reduction Modulo p. - C. 3 Appendix: Finiteness Theorems in Algebraic Number Theory. - C. 4 Appendix: The Selmer and TateShafarevich 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?(zw) for Nonvanishing Sections. - E. 6 Constructing Sections of Small Height I: Applying RiemannRoch. - E. 7 Constructing Sections of Small Height II: Applying Siegel's Lemma. - E. 8 Lower Bound for h?(zw) at Admissible Version I. - E. 9 Eisenstein's Estimate for the Derivatives of an Algebraic Function. - E. 10 Lower Bound for h?(zw) 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. 2 Discreteness of Algebraic Points. - F. 3 Height Bounds and Height Conjectures. - F. 4 The Search for Effectivity. - F. 5 Geometry Governs Arithmetic. - Exercises. - References. - List of Notation. Language: English