Nron Models [Hardcover]

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.1. What Is a N?ron Model?.- 1.1 Integral Points.- 1.2 N?ron Models.- 1.3 The Local Case: Main Existence Theorem.- 1.4 The Global Case: Abelian Varieties.- 1.5 Elliptic Curves.- 1.6 N?rons 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 N?rons Measure for the Defect of Smoothness.- 3.4 Proof of the Theorem.- 3.5 Weak N?ron 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.- 6.2lÃ.