Arithmetic GeometryThis 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. |
Contents
5 The Work of Szpiro Extending This to Positive Characteristic | 6 |
2 Semiabelian Varieties | 23 |
Group Schemes Formal Groups and pDivisible Groups 29 | 29 |
2 Group Schemes Generalities | 30 |
3 Finite Group Schemes | 37 |
4 Commutative Finite Group Schemes | 45 |
5 Formal Groups | 56 |
6 pDivisible Groups | 60 |
7 Weils Construction of the Jacobian Variety | 189 |
8 Generalizations | 192 |
9 Obtaining Coverings of a Curve from its Jacobian Application to Mordells Conjecture | 195 |
10 Abelian Varieties Are Quotients of Jacobian Varieties | 198 |
11 The Zeta Function of a Curve | 200 |
Statement and Applications | 202 |
The Proof | 204 |
Bibliographic Notes for Abelian Varieties and Jacobian Varieties | 208 |
7 Applications of Groups of Type p p p to pDivisible Groups | 76 |
References | 78 |
CHAPTER IV | 79 |
2 Isogenies of Complex Tori | 81 |
3 Abelian Varieties | 83 |
4 The NéronSeveri Group and the Picard Group | 92 |
5 Polarizations and Polarized Abelian Manifolds | 95 |
6 The Space of Principally Polarized Abelian Manifolds | 97 |
References | 100 |
CHAPTER V | 103 |
1 Definitions | 104 |
3 Rational Maps into Abelian Varieties | 105 |
4 Review of the Cohomology of Schemes | 108 |
5 The Seesaw Principle | 109 |
6 The Theorems of the Cube and the Square | 110 |
7 Abelian Varieties Are Projective | 112 |
8 Isogenies | 114 |
Definition | 117 |
Construction | 119 |
11 The Dual Exact Sequence | 120 |
12 Endomorphisms | 121 |
13 Polarizations and the Cohomology of Invertible Sheaves | 126 |
14 A Finiteness Theorem | 127 |
15 The Étale Cohomology of an Abelian Variety | 128 |
16 Pairings | 131 |
17 The Rosati Involution | 137 |
18 Two More Finiteness Theorems | 140 |
19 The Zeta Function of an Abelian Variety | 143 |
20 Abelian Schemes | 145 |
References | 150 |
CHAPTER VI | 151 |
3 Heights | 153 |
4 Heights on Abelian Varieties | 156 |
5 The MordellWeil Theorem | 158 |
Heights and Metrized Line Bundles | 161 |
8 Distance Functions and Logarithmic Singularities | 163 |
References | 166 |
CHAPTER VII | 167 |
2 The Canonical Maps from C to its Jacobian Variety | 171 |
3 The Symmetric Powers of a Curve | 174 |
4 The Construction of the Jacobian Variety | 179 |
5 The Canonical Maps from the Symmetric Powers of C to its Jacobian Variety | 182 |
6 The Jacobian Variety as Albanese Variety Autoduality | 185 |
4 Isogenies | 210 |
References | 211 |
CHAPTER VIII | 213 |
1 Properties of the Néron Model and Examples | 214 |
Proof | 221 |
R Strictly Local | 223 |
4 Projective Embedding | 227 |
2 Transcendental Uniformization of the Moduli Spaces | 235 |
4 Toroidal Compactification | 243 |
CHAPTER X | 253 |
3 Weil Curves | 260 |
6 Finiteness Theorems | 264 |
CHAPTER XI | 267 |
Reduction to Rational Singularities | 274 |
Blowing Up the Dualizing Sheaf | 281 |
CHAPTER XII | 289 |
4 The RiemannRoch Theorem and the Adjunction Formula | 299 |
CHAPTER XIII | 309 |
3 Statement of the Castelnuovo Criterion | 314 |
4 Intersection Theory and Proper and Total Transforms | 315 |
5 Exceptional Curves | 317 |
5B Prime Divisors Satisfying the Castelnuovo Criterion | 319 |
6 Proof of the Castelnuovo Criterion | 321 |
7 Proof of the Minimal Models Theorem | 323 |
References | 325 |
CHAPTER XIV | 327 |
2 Nérons Local Height Pairing | 328 |
3 Construction of the Local Height Pairing | 329 |
4 The Canonical Height | 331 |
5 Local Heights for Divisors with Common Support | 332 |
6 Local Heights for Divisors of Arbitrary Degree | 333 |
7 Local Heights on Curves of Genus Zero | 334 |
8 Local Heights on Elliptic Curves | 335 |
9 Greens Functions on the Upper HalfPlane | 336 |
10 Local Heights on Mumford Curves | 337 |
References | 339 |
CHAPTER XV | 341 |
2 Correspondence with Number Theory | 344 |
3 Higher Dimensional Nevanlinna Theory | 347 |
4 Consequences of the Conjecture | 349 |
5 Comparison with Faltings Proof | 352 |
References | 353 |
Copyright | |