Arithmetic Geometry

This volume is the result of a (mainly) instructional conference on arithmetic geometry, held from July 30 through August 10, 1984 at the University of Connecticut in Storrs. This volume contains expanded versions of almost all the instructional lectures given during the conference. In addition to these expository lectures, this volume contains a translation into English of Falt­ ings' seminal paper which provided the inspiration for the conference. We thank Professor Faltings for his permission to publish the translation and Edward Shipz who did the translation. We thank all the people who spoke at the Storrs conference, both for helping to make it a successful meeting and enabling us to publish this volume. We would especially like to thank David Rohrlich, who delivered the lectures on height functions (Chapter VI) when the second editor was unavoidably detained. In addition to the editors, Michael Artin and John Tate served on the organizing committee for the conference and much of the success of the conference was due to them-our thanks go to them for their assistance. Finally, the conference was only made possible through generous grants from the Vaughn Foundation and the National Science Foundation.

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.