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Catalogue Information
Field name
Details
Dewey Class
516.35
Title
Néron Models ([EBook]) / by Siegfried Bosch, Werner Lütkebohmert, Michel Raynaud.
Author
Bosch, Siegfried. , 1944-
Added Personal Name
Lütkebohmert, Werner
Raynaud, Michel
Other name(s)
SpringerLink (Online service)
Publication
Berlin, Heidelberg : Springer , 1990.
Physical Details
X, 328 pages : online resource.
Series
Ergebnisse der Mathematik und ihrer Grenzgebiete, A Series of Modern Surveys in Mathematics
0071-1136 ; ; 21
ISBN
9783642514388
Summary Note
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.:
Contents note
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.
System details note
Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
Internet Site
http://dx.doi.org/10.1007/978-3-642-51438-8
Links to Related Works
Subject References:
Abelian varieties
.
Algebraic Geometry
.
Geometry, Algebraic
.
Néron models
.
Authors:
author
.
Bosch, Siegfried. 1944-
.
Lütkebohmert, Werner
.
Raynaud, Michel
.
Corporate Authors:
SpringerLink (Online service)
.
Series:
Ergebnisse der Mathematik und ihrer Grenzgebiete, A Series of Modern Surveys in Mathematics
.
Classification:
516.35
.
.
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