Current Trends in Arithmetical Algebraic Geometry

✍ Scribed by Kenneth A. Ribet

• Publisher: American Mathematical Society
• Year: 1987
• Tongue: English
• Leaves: 294
• Series: Contemporary Mathematics 67
• Category: Library

✦ Synopsis

Mark Sepanski's Algebra is a readable introduction to the delightful world of modern algebra. Beginning with concrete examples from the study of integers and modular arithmetic, the text steadily familiarizes the reader with greater levels of abstraction as it moves through the study of groups, rings, and fields. The book is equipped with over 750 exercises suitable for many levels of student ability. There are standard problems, as well as challenging exercises, that introduce students to topics not normally covered in a first course. Difficult problems are broken into manageable subproblems and come equipped with hints when needed. Appropriate for both self-study and the classroom, the material is efficiently arranged so that milestones such as the Sylow theorems and Galois theory can be reached in one semester.

✦ Table of Contents

Title......Page 1
Copyright......Page 2
CONTENTS......Page 3
INTRODUCTION......Page 4
Introduction......Page 6
§1 Height pairing in geometric case......Page 8
§2. Local indexes over non-archimedean places......Page 10
§3. Local indexes over C and S......Page 15
§4. Height pairing over number fields......Page 17
§5 Some conjectures and problems......Page 21
References......Page 28
SURVEY OF DRINFEL'D MODULES......Page 30
Fixed notations throughout the article......Page 32
§1. ENDOMORPHISMS OF THE ADDITIVE GROUP......Page 35
§2. DEFINITION OF DRINFEL'D MODULE OVER A FIELD......Page 37
§3. DIVISION POINTS......Page 39
§4. ISOGENIES......Page 42
§5. DRINFEL'D MODULES OVER A BASE SCHEME......Page 45
§6. LEVEL STRUCTURE AND THE MODULI SPACE......Page 46
§1. EXPONENTIAL FUNCTION ASSOCIATED TO A LATTICE......Page 49
§2. CHARACTERIZATION OF DRINFEL'n MODULES OVER Coo......Page 50
§3. DISCRETE MODULES IN A VECTOR SPACE OVER A LOCAL FIELD......Page 53
§4. MODULI SPACES AS HOMOGENEOUS SPACES (LOCAL THEORY)......Page 55
§5. MODULI SPACES AS HOMOGENEOUS SPACES (ADELle THEO~Y)......Page 58
§l. NORMS ON VECTOR SPACES OVER A LOCAL FIELD......Page 63
§2. THE BUILDING FOR PGL(V) OVER A LOCAL FIELD......Page 66
§3. METRIC ON THE BUILDING......Page 68
§4. THE MAPPING FROM THE p-ADIC SYMMETRIC SPACE TO THE BUILDING......Page 69
§s. FILTRATION OF THE I-DIMENSIONAL p-ADIC SYMMETRIC SPACE......Page 71
§l. GENERALITIES ON THE COHOMOLOGY OF RIGID ANALYTIC SPACES......Page 76
§2. COHOMOLOGY OF rl (CcJ......Page 78
§3. A SIMPLE TOPOLOGICAL MODEL......Page 80
§4. COHOMOLOGY OF THE MODULI SPACE WITH FIXED LEVEL STRUCTURE......Page 82
§l. COCLOSED l-COCHAINS AND THE SPECIAL REPRESENTATION......Page 84
§2. LIMIT COHOMOLOGY AND AUTOMORPHIC FORMS......Page 87
§3. PROPERTIES OF THE CORRESPONDENCE 'IT f--+ a(n)......Page 89
§4. LOCAL LANGLANDS' ·CONJECTURE IN CHARACTERISTIC P......Page 91
BIBLIOGRAPHY......Page 93
Ie determinant de la cohomologie......Page 97
1.1 Rappels: determinants ([10] ) .......Page 100
1.Z Rappels: torsion analytigue ([14] ) •......Page 101
1.5 Calculer la torsion analytigue.......Page 102
2.1 Incarner une integrale de classes de Chern.......Page 104
2.2 Incarner la composante de degre 2 de Riemarm-Roch......Page 106
2.3 Metriser......Page 107
3.2 Soit V un espace vectoriel de dimension finie......Page 112
3.4 Soit f un fibre vectoriel sur X......Page 113
3.5 Dans la fin de ce paragraphe, X est une courbe propre et lisse.......Page 114
3.6 Soient E et F deux diviseurs eoncentres sur la fibre speciale......Page 115
3.7 Un diviseur canpactifie......Page 116
4. Objets virtuels.......Page 117
5. Fibres metrigues virtuels.......Page 130
6. Integrale d'un produit de premieres classes de Chern.......Page 145
7. <L z M) et cohomologie.......Page 150
8. Integrale d 'un produit de premieres classes de Chern. Cas general (esguisse).......Page 156
9. ICZet isomorphisme de Riemann-Roch......Page 159
10. Metriser IC2......Page 175
11. I.e theoreme.......Page 178
Bibliographie......Page 180
Introduction......Page 182
1. Crystalline Representations......Page 183
2. Crystalline Representations and Elale Cohomology......Page 186
3. "Arithmetical" Applications......Page 187
4. Geometric Applications......Page 188
1. The Syntomic Site, Crystalline Cohomology and the cartier Isomorphism......Page 190
2. Crystalline and de Rham Cohomology......Page 193
1. The Sheaves S~and Crystalline Cohomology......Page 198
2. An Indication of the Proof of the Lemma of 1.6......Page 200
3. The Sheaves Sh......Page 201
4. The Syntomic Etsle Site......Page 202
5. Construction of the Sheaves.,g,~......Page 204
6. Construction of p- Adic Etale Cohomology......Page 206
BIBLIOGRAPHY......Page 208
AN INTRODUCTION TO HIGHER DIMENSIONAL ARAKELOV THEORY......Page 211
1. CLASSICAL INTERSECTION THEORY.......Page 212
2 • ARITHMETIC VARIETIES......Page 215
3. RINGS OF ALGEBRAIC INTEgERS.......Page 216
4. GREEN'S CURRENTS......Page 218
5. THE CHOW GROUPS OF AN ARAKELOV VARIETY......Page 220
6. THE INTERSECTION PAIRING......Page 222
7. VECTOR BUNDLES AND CHARACTERISTIC CLASSES......Page 225
BIBLIOGRAPHY......Page 229
0. INTRODUCTION......Page 231
1. HYPERBOLICITY AND NOGUCHI'S THEOREM......Page 232
2. ON CARTAN'S THEOREM......Page 239
BIBLIOGRAPHY......Page 247
0. Introduction......Page 249
1. Cubic exponential sums and Wu......Page 250
2. The Shimura Correspondence......Page 253
3. The L-function......Page 257
4. A theorem of Serre on Galois representations We describe here......Page 258
REFERENCES......Page 262
1. Les conjectures sur les formes de poids 2......Page 264
2. Conjectures generales sur les representations......Page 266
1. Wotl Heights......Page 270
2. l1elrlzed Line Bundles......Page 271
3. The Degree of a r1etrlzed Line Bundle......Page 272
4. The Modular Height of an Abelian Veriety......Page 273
5. Canonical Heights on Abelian Varieties......Page 274
7. lower Bounds for the Canonical Height......Page 275
8. Speclaltzatlon Theorems......Page 277
BIBLIOGRAPHY......Page 278
PRESENTATION DE LA THEORIE D'ARAKELOV......Page 280
I. DEGRE, RIEMANN-ROCH, DUALITE POUR LES FIBRES INVERSIBLES HERJ.\1ITIENS SUR UNANNEAU D'ENTIERS ALGEBRIQUES......Page 281
I I. LA THEoRIE DES INrERSECTIONS D'ARAKELOV SUR LES SURFACES ARlruMETIQUESC Metriques pennises).......Page 283
III. FORMULE D'ADJONCTION ET 'I'HEoREME DE RIfMANN-ROrn......Page 286
BIBLIOGRAPHIE......Page 294