I Some Historical Notes.- 1. The Theorems of Mordell and Mordell-Weil.- 2. Siegel's Theorem About Integral Points.- 3. The Proof of the Mordell Conjecture for Function Fields, by Manin and Grauert.- 4. The New Ideas of Parshin and Arakelov, Relating the Conjectures of Mordell and Shafarevich.- 5. The Work of Szpiro, Extending This to Positive Characteristic.- 6. The Theorem of Tate About Endomorphisms of Abelian Varieties over Finite Fields.- 7. The Work of Zarhin.- Bibliographic Remarks.- II Finiteness Theorems for Abelian Varieties over Number Fields.- 1. Introduction.- 2. Semiabelian Varieties.- 3. Heights.- 4. Isogenies.- 5. Endomorphisms.- 6. Finiteness Theorems.- References.- Erratum.- III Group Schemes, Formal Groups, and p-Divisible Groups.- 1. Introduction.- 2. Group Schemes, Generalities.- 3. Finite Group Schemes.- 4. Commutative Finite Group Schemes.- 5. Formal Groups.- 6. p-Divisible Groups.- 7. Applications of Groups of Type (p, p,..., p) to p-Divisible Groups.- References.- IV Abelian Varieties over ?.- 0. Introduction.- 1. Complex Tori.- 2. Isogenies of Complex Tori.- 3. Abelian Varieties.- 4. The Neron-Severi Group and the Picard Group.- 5. Polarizations and Polarized Abelian Manifolds.- 6. The Space of Principally Polarized Abelian Manifolds.- References.- V Abelian Varieties.- 1. Definitions.- 2. Rigidity.- 3. Rational Maps into Abelian Varieties.- 4. Review of the Cohomology of Schemes.- 5. The Seesaw Principle.- 6. The Theorems of the Cube and the Square.- 7. Abelian Varieties Are Projective.- 8. Isogenies.- 9. The Dual Abelian Variety: Definition.- 10. The Dual Abelian Variety: Construction.- 11. The Dual Exact Sequence.- 12. Endomorphisms.- 13. Polarizations and the Cohomology of Invertible Sheaves.- 14. A Finiteness Theorem.- 15. The Etale Cohomology of an Abelian Variety.- 16. Pairings.- 17. The Rosati Involution.- 18. Two More Finiteness Theorems.- 19. The Zeta Function of an Abelian Variety.- 20. Abelian Schemes.- References.- VI The Theory of Height Functions.- The Classical Theory of Heights.- 1. Absolute Values.- 2. Height on Projective Space.- 3. Heights on Projective Varieties.- 4. Heights on Abelian Varieties.- 5. The Mordell-Weil Theorem.- Heights and Metrized Line Bundles.- 6. Metrized Line Bundles on Spec (R).- 7. Metrized Line Bundles on Varieties.- 8. Distance Functions and Logarithmic Singularities.- References.- VII Jacobian Varieties.- 1. Definitions.- 2. The Canonical Maps from C to its Jacobian Variety.- 3. The Symmetric Powers of a Curve.- 4. The Construction of the Jacobian Variety.- 5. The Canonical Maps from the Symmetric Powers of C to its Jacobian Variety.- 6. The Jacobian Variety as Albanese Variety; Autoduality.- 7. Weil's Construction of the Jacobian Variety.- 8. Generalizations.- 9. Obtaining Coverings of a Curve from its Jacobian; Application to Mordell's Conjecture.- 10. Abelian Varieties Are Quotients of Jacobian Varieties.- 11. The Zeta Function of a Curve.- 12. Torelli's Theorem: Statement and Applications.- 13. Torelli's Theorem: The Proof.- Bibliographic Notes for Abelian Varieties and Jacobian Varieties.- References.- VIII Neron Models.- 1. Properties of the Neron Model, and Examples.- 2. Weil's Construction: Proof.- 3. Existence of the Neron Model: R Strictly Local.- 4. Projective Embedding.- 5. Appendix: Prime Divisors.- References.- IX Siegel Moduli Schemes and Their Compactifications over ?.- 0. Notations and Conventions.- 1. The Moduli Functors and Their Coarse Moduli Schemes.- 2. Transcendental Uniformization of the Moduli Spaces.- 3. The Satake Compactification.- 4. Toroidal Compactification.- 5. Modular Heights.- References.- X Heights and Elliptic Curves.- 1. The Height of an Elliptic Curve.- 2. An Estimate for the Height.- 3. Weil Curves.- 4. A Relation with the Canonical Height.- References.- XI Lipman's Proof of Resolution of Singularities for Surfaces.- 1. Introduction.- 2. Proper Intersection Numbers and the Vanishing Theorem.- 3. Step 1: Reduction to Rational Singularities.- 4. Basic Properties of Rational Singularities.- 5. Step 2: Blowing Up the Dualizing Sheaf.- 6. Step 3: Resolution of Rational Double Points.- References.- XII An Introduction to Arakelov Intersection Theory.- 1. Definition of the Arakelov Intersection Pairing.- 2. Metrized Line Bundles.- 3. Volume Forms.- 4. The Riemann-Roch Theorem and the Adjunction Formula.- 5. The Hodge Index Theorem.- References.- XIII Minimal Models for Curves over Dedekind Rings.- 1. Statement of the Minimal Models Theorem.- 2. Factorization Theorem.- 3. Statement of the Castelnuovo Criterion.- 4. Intersection Theory and Proper and Total Transforms.- 5. Exceptional Curves.- 5A. Intersection Properties.- 5B. Prime Divisors Satisfying the Castelnuovo Criterion.- 6. Proof of the Castelnuovo Criterion.- 7. Proof of the Minimal Models Theorem.- References.- XIV Local Heights on Curves.- 1. Definitions and Notations.- 2. Neron's Local Height Pairing.- 3. Construction of the Local Height Pairing.- 4. The Canonical Height.- 5. Local Heights for Divisors with Common Support.- 6. Local Heights for Divisors of Arbitrary Degree.- 7. Local Heights on Curves of Genus Zero.- 8. Local Heights on Elliptic Curves.- 9. Green's Functions on the Upper Half-plane.- 10. Local Heights on Mumford Curves.- References.- XV A Higher Dimensional Mordell Conjecture.- 1. A Brief Introduction to Nevanlinna Theory.- 2. Correspondence with Number Theory.- 3. Higher Dimensional Nevanlinna Theory.- 4. Consequences of the Conjecture.- 5. Comparison with Faltings' Proof.- References.
