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. Front Matter....Pages i-xv Some Historical Notes....Pages 1-8 Finiteness Theorems for Abelian Varieties over Number Fields....Pages 9-26 Group Schemes, Formal Groups, and p -Divisible Groups....Pages 29-78 Abelian Varieties over C....Pages 79-101 Abelian Varieties....Pages 103-150 The Theory of Height Functions....Pages 151-166 Jacobian Varieties....Pages 167-212 Néron Models....Pages 213-230 Siegel Moduli Schemes and Their Compactifications over C....Pages 231-251 Heights and Elliptic Curves....Pages 253-265 Lipman’s Proof of Resolution of Singularities for Surfaces....Pages 267-287 An Introduction to Arakelov Intersection Theory....Pages 289-307 Minimal Models for Curves over Dedekind Rings....Pages 309-326 Local Heights on Curves....Pages 327-339 A Higher Dimensional Mordell Conjecture....Pages 341-353 Erratum to: Erratum....Pages 354-354

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. This revised second printing now includes a comprehensive index.