Siegfried Bosch, Werner Lütkebohmert, Michel Raynaud

Néron Models

• Gebundenes Buch

Produktbeschreibung

N ron models were invented by A. N ron 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 N ron models with great success. Quite recently, new developments in arithmetic algebraic geometry have prompted a desire to understand more about N ron 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 N ron models. This volume of the renowned "Ergebnisse" series provides a detailed demonstration of the construction of N ron models from the point of view of Grothendieck's algebraic geometry. In the second part of the book the relationship between N ron 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.

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Produktdetails

• Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
• Verlag: Springer / Springer Berlin Heidelberg / Springer, Berlin
• Artikelnr. des Verlages: 978-3-540-50587-7
• 1990
• Seitenzahl: 344
• Erscheinungstermin: 12. April 1990
• Englisch
• Abmessung: 260mm x 183mm x 25mm
• Gewicht: 815g
• ISBN-13: 9783540505877
• ISBN-10: 3540505873
• Artikelnr.: 23885869

Inhaltsangabe

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 ofConnected 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.