LEADER 05228nam 22006255 450 001 9910972120703321 005 20250818100421.0 010 $a9783642514388 010 $a3642514383 024 7 $a10.1007/978-3-642-51438-8 035 $a(CKB)3400000000103668 035 $a(SSID)ssj0001297766 035 $a(PQKBManifestationID)11861127 035 $a(PQKBTitleCode)TC0001297766 035 $a(PQKBWorkID)11229586 035 $a(PQKB)11386519 035 $a(DE-He213)978-3-642-51438-8 035 $a(MiAaPQ)EBC3089336 035 $a(PPN)238008290 035 $a(EXLCZ)993400000000103668 100 $a20121227d1990 u| 0 101 0 $aeng 135 $aurnn#008mamaa 181 $ctxt 182 $cc 183 $acr 200 10$aNéron Models /$fby Siegfried Bosch, Werner Lütkebohmert, Michel Raynaud 205 $a1st ed. 1990. 210 1$aBerlin, Heidelberg :$cSpringer Berlin Heidelberg :$cImprint: Springer,$d1990. 215 $a1 online resource (X, 328 p.) 225 1 $aErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics,$x2197-5655 ;$v21 300 $aBibliographic Level Mode of Issuance: Monograph 311 08$a9783540505877 311 08$a3540505873 311 08$a9783642080739 311 08$a3642080731 320 $aIncludes bibliographical references and index. 327 $a1. 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. 330 $aNéron models were invented by A. Néron in the early 1960s in order to study the integral structure of abelian varieties over number fields. Since then, arithmeticians and algebraic geometers have applied the theory of Néron models with great success. Quite recently, new developments in arithmetic algebraic geometry have prompted a desire to understand more about Néron models, and even to go back to the basics of their construction. The authors have taken this as their incentive to present a comprehensive treatment of Néron models. This volume of the renowned "Ergebnisse" series provides a detailed demonstration of the construction of Néron models from the point of view of Grothendieck's algebraic geometry. In the second part of the book the relationship between Néron models and the relative Picard functor in the case of Jacobian varieties is explained. The authors helpfully remind the reader of some important standard techniques of algebraic geometry. A special chapter surveys the theory of the Picard functor. 410 0$aErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics,$x2197-5655 ;$v21 606 $aGeometry, Algebraic 606 $aAlgebraic Geometry 615 0$aGeometry, Algebraic. 615 14$aAlgebraic Geometry. 676 $a516.35 700 $aBosch$b Siegfried$4aut$4http://id.loc.gov/vocabulary/relators/aut$041946 702 $aLu?tkebohmert$b Werner$4aut$4http://id.loc.gov/vocabulary/relators/aut 702 $aRaynaud$b Michel$4aut$4http://id.loc.gov/vocabulary/relators/aut 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910972120703321 996 $aNeron models$9382528 997 $aUNINA