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Néron Models Softcover reprint of the original 1st ed. 1990 [Mīkstie vāki]

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Néron Models Softcover reprint of the original 1st ed. 1990Palielināt
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Permanent link: https://www.kriso.lv/db/9783642080739.html
Keywords:
Néron models were invented by A. Néron in the early 1960s in order to study the integral structure of abelian varieties over number fields. Since then, arithmeticians and algebraic geometers have applied the theory of Néron models with great success. Quite recently, new developments in arithmetic algebraic geometry have prompted a desire to understand more about Néron models, and even to go back to the basics of their construction. The authors have taken this as their incentive to present a comprehensive treatment of Néron models. This volume of the renowned "Ergebnisse" series provides a detailed demonstration of the construction of Néron models from the point of view of Grothendieck's algebraic geometry. In the second part of the book the relationship between Néron models and the relative Picard functor in the case of Jacobian varieties is explained. The authors helpfully remind the reader of some important standard techniques of algebraic geometry. A special chapter surveys the theory of the Picard functor.

Papildus informācija

Springer Book Archives
Introduction 1(5)
Chapter 1 What Is a Neron Model?
6(25)
1.1 Integral Points
6(6)
1.2 Neron Models
12(4)
1.3 The Local Case: Main Existence Theorem
16(2)
1.4 The Global Case: Abelian Varieties
18(2)
1.5 Elliptic Curves
20(7)
1.6 Neron's Original Article
27(4)
Chapter 2 Some Background Material from Algebraic Geometry
31(29)
2.1 Differential Forms
31(3)
2.2 Smoothness
34(10)
2.3 Henselian Rings
44(7)
2.4 Flatness
51(4)
2.5 S-Rational Maps
55(5)
Chapter 3 The Smoothening Process
60(34)
3.1 Statement of the Theorem
60(2)
3.2 Dilatation
62(2)
3.3 Neron's Measure for the Defect of Smoothness
64(7)
3.4 Proof of the Theorem
71(2)
3.5 Weak Neron Models
73(4)
3.6 Algebraic Approximation of Formal Points
77(17)
Chapter 4 Construction of Birational Group Laws
94(18)
4.1 Group Schemes
94(5)
4.2 Invariant Differential Forms
99(4)
4.3 R-Extensions of K-Group Laws
103(6)
4.4 Rational Maps into Group Schemes
109(3)
Chapter 5 From Birational Group Laws to Group Schemes
112(17)
5.1 Statement of the Theorem
112(2)
5.2 Strict Birational Group Laws
114(5)
5.3 Proof of the Theorem for a Strictly Henselian Base
119(10)
Chapter 6 Descent
129(43)
6.1 The General Problem
129(9)
6.2 Some Standard Examples of Descent
138(10)
6.3 The Theorem of the Square
148(4)
6.4 The Quasi-Projectivity of Torsors
152(4)
6.5 The Descent of Torsors
156(6)
6.6 Applications to Birational Group Laws
162(4)
6.7 An Example of Non-Effective Descent
166(6)
Chapter 7 Properties of Neron Models
172(27)
7.1 A Criterion
172(4)
7.2 Base Change and Descent
176(2)
7.3 Isogenies
178(3)
7.4 Semi-Abelian Reduction
181(3)
7.5 Exactness Properties
184(7)
7.6 Weil Restriction
191(8)
Chapter 8 The Picard Functor
199(37)
8.1 Basics on the Relative Picard Functor
199(10)
8.2 Representability by a Scheme
209(14)
8.3 Representability by an Algebraic Space
223(8)
8.4 Properties
231(5)
Chapter 9 Jacobians of Relative Curves
236(53)
9.1 The Degree of Divisors
236(7)
9.2 The Structure of Jacobians
243(8)
9.3 Construction via Birational Group Laws
251(8)
9.4 Construction via Algebraic Spaces
259(5)
9.5 Picard Functor and Neron Models of Jacobians
264(9)
9.6 The Group of Connected Components of a Neron Model
273(13)
9.7 Rational Singularities
286(3)
Chapter 10 Neron Models of Not Necessarily Proper Algebraic Groups
289(28)
10.1 Generalities
289(7)
10.2 The Local Case
296(13)
10.3 The Global Case
309(8)
Bibliography 317(5)
Subject Index 322