Néron Models [electronic resource] by Siegfried Bosch, Werner Lütkebohmert, Michel Raynaud.

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 al...

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Bibliographic Details
Uniform Title:Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 2197-5655 ; 21
Main Authors: Bosch, Siegfried (Author)
Lütkebohmert, Werner (Author)
Raynaud, Michel (Author)
Corporate Author: SpringerLink (Online service)
Language:English
Published: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1990.
Edition:1st ed. 1990.
Series:Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 21
Subjects:
Algebraic geometry.
Online Access:
Springer Book Archive - Mathematics: 1990 (Springer Link)
ProQuest Ebook Central - Academic Complete: 1990 (Ebook Central @ Proquest)
Format: Electronic eBook
Contents:
  • 1. What Is a Néron Model?
  • 1.1 Integral Points
  • 1.2 Néron Models
  • 1.3 The Local Case: Main Existence Theorem
  • 1.4 The Global Case: Abelian Varieties
  • 1.5 Elliptic Curves
  • 1.6 Néron’s Original Article
  • 2. Some Background Material from Algebraic Geometry
  • 2.1 Differential Forms
  • 2.2 Smoothness
  • 2.3 Henselian Rings
  • 2.4 Flatness
  • 2.5 S-Rational Maps
  • 3. The Smoothening Process
  • 3.1 Statement of the Theorem
  • 3.2 Dilatation
  • 3.3 Néron’s Measure for the Defect of Smoothness
  • 3.4 Proof of the Theorem
  • 3.5 Weak Néron Models
  • 3.6 Algebraic Approximation of Formal Points
  • 4. Construction of Birational Group Laws
  • 4.1 Group Schemes
  • 4.2 Invariant Differential Forms
  • 4.3 R-Extensions of K-Group Laws
  • 4.4 Rational Maps into Group Schemes
  • 5. From Birational Group Laws to Group Schemes
  • 5.1 Statement of the Theorem
  • 5.2 Strict Birational Group Laws
  • 5.3 Proof of the Theorem for a Strictly Henselian Base
  • 6. Descent
  • 6.1 The General Problem
  • 6.2 Some Standard Examples of Descent
  • 6.3 The Theorem of the Square
  • 6.4 The Quasi-Projectivity of Torsors
  • 6.5 The Descent of Torsors
  • 6.6 Applications to Birational Group Laws
  • 6.7 An Example of Non-Effective Descent
  • 7. Properties of Néron Models
  • 7.1 A Criterion
  • 7.2 Base Change and Descent
  • 7.3 Isogenies
  • 7.4 Semi-Abelian Reduction
  • 7.5 Exactness Properties
  • 7.6 Weil Restriction
  • 8. The Picard Functor
  • 8.1 Basics on the Relative Picard Functor
  • 8.2 Representability by a Scheme
  • 8.3 Representability by an Algebraic Space
  • 8.4 Properties
  • 9. Jacobians of Relative Curves
  • 9.1 The Degree of Divisors
  • 9.2 The Structure of Jacobians
  • 9.3 Construction via Birational Group Laws
  • 9.4 Construction via Algebraic Spaces
  • 9.5 Picard Functor and Néron Models of Jacobians
  • 9.6 The Group of Connected Components of a Néron Model
  • 9.7 Rational Singularities
  • 10. Néron Models of Not Necessarily Proper Algebraic Groups
  • 10.1 Generalities
  • 10.2 The Local Case
  • 10.3 The Global Case.