Arithmetic Geometry

This book is the result of a conference on arithmetic geometry, held July 30 through August 10, 1984 at the University of Connecticut at Storrs, the purpose of which was to provide a coherent overview of the subject. This subject has enjoyed a resurgence in popularity due in part to Faltings' proof of Mordell's conjecture. Included are extended versions of almost all of the instructional lectures and, in addition, a translation into English of Faltings' ground-breaking paper. ARITHMETIC GEOMETRY should be of great use to students wishing to enter this field, as well as those already working in it.

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 Néron-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 Étale 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 Néron Models.- §1. Properties of the Néron Model, and Examples.- §2. Weil’s Construction: Proof.- §3. Existence of the Néron 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. Néron’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.