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E-grāmata: Neron Models and Base Change

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  • Formāts: PDF+DRM
  • Sērija : Lecture Notes in Mathematics 2156
  • Izdošanas datums: 02-Mar-2016
  • Izdevniecība: Springer International Publishing AG
  • Valoda: eng
  • ISBN-13: 9783319266381
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Neron Models and Base ChangePalielināt
  • Formāts: PDF+DRM
  • Sērija : Lecture Notes in Mathematics 2156
  • Izdošanas datums: 02-Mar-2016
  • Izdevniecība: Springer International Publishing AG
  • Valoda: eng
  • ISBN-13: 9783319266381
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Presentingthe first systematic treatment of the behavior of Néron models under ramifiedbase change, this book can be read as an introduction to various subtleinvariants and constructions related to Néron models of semi-abelian varieties,motivated by concrete research problems and complemented with explicitexamples. Néron models of abelian andsemi-abelian varieties have become an indispensable tool in algebraic andarithmetic geometry since Néron introduced them in his seminal 1964 paper.Applications range from the theory of heights in Diophantine geometry to Hodgetheory.We focus specifically on Néron component groups, Edixhoven"s filtrationand the base change conductor of Chai and Yu, and we study these invariantsusing various techniques such as models of curves, sheaves on Grothendiecksites and non-archimedean uniformization. We then apply our results to thestudy of motivic zeta functions of abelian varieties. The final chaptercontains a list of challenging open

questions. This book is aimed towardsresearchers with a background in algebraic and arithmetic geometry.

MicrosoftInternetExplorer4Introduction.- Preliminaries.- Models of curves and theNeron component series of a Jacobian.- Component groups andnon-archimedean uniformization.- The base change conductor and Edixhoven"s ltration.-The base change conductor and the Artin conductor.- Motivic zeta functions ofsemi-abelian varieties.- Cohomological interpretation of the motivic zeta
Part I About This Book
1 Content of This Book
3(6)
2 Introduction
9(12)
2.1 Motivation and Background
9(7)
2.1.1 Abelian Varieties in Number Theory and Geometry
9(1)
2.1.2 Degenerating Families of Curves
10(1)
2.1.3 Neron Models
10(1)
2.1.4 Semi-Abelian Reduction
11(1)
2.1.5 Behaviour Under Base Change
12(1)
2.1.6 Jacobians, Stable Curves and Semi-Abelian Reduction
13(1)
2.1.7 Example: Elliptic Curves
13(2)
2.1.8 Motivic Zeta Functions
15(1)
2.2 Aim of This Book
16(5)
2.2.1 Semi-Abelian Varieties and Wildly Ramified Jacobians
16(1)
2.2.2 A Guiding Principle
17(1)
2.2.3 Notation
18(3)
3 Preliminaries
21(18)
3.1 Galois Theory of K
21(2)
3.1.1 The Artin Conductor
21(2)
3.1.2 Isolating the Wild Part of the Conductor
23(1)
3.2 Subtori of Algebraic Groups
23(4)
3.2.1 Maximal Subtori
23(1)
3.2.2 Basic Properties of the Reductive Rank
24(3)
3.3 Neron Models
27(7)
3.3.1 The Neron Model and the Component Group
27(2)
3.3.2 The Toric Rank
29(1)
3.3.3 Neron Models and Base Change
30(1)
3.3.4 Example: The Neron lft-Model of a Split Algebraic Torus
30(1)
3.3.5 The Neron Component Series
31(1)
3.3.6 Semi-Abelian Reduction
32(1)
3.3.7 Non-Archimedean Uniformization
33(1)
3.4 Models of Curves
34(5)
3.4.1 Sncd-Models and Combinatorial Data
34(1)
3.4.2 A Theorem of Winters
35(1)
3.4.3 Neron Models of Jacobians
35(1)
3.4.4 Semi-Stable Reduction
36(3)
Part II Neron Component Groups of Semi-Abelian Varieties
4 Models of Curves and the Neron Component Series of a Jacobian
39(20)
4.1 Sncd-Models and Tame Base Change
39(4)
4.1.1 Base Change and Normalization
39(1)
4.1.2 Local Computations
40(2)
4.1.3 Minimal Desingularization
42(1)
4.2 The Characteristic Polynomial and the Stabilization Index
43(7)
4.2.1 The Characteristic Polynomial
43(2)
4.2.2 The Stabilization Index
45(4)
4.2.3 Applications to sncd-Models and Base Change
49(1)
4.3 The Neron Component Series of a Jacobian
50(3)
4.3.1 Rationality of the Component Series
51(2)
4.4 Appendix: Locally Toric Rings
53(6)
4.4.1 Resolution of Locally Toric Singularities
53(2)
4.4.2 Tame Cyclic Quotient Singularities
55(4)
5 Component Groups and Non-Archimedean Uniformization
59(30)
5.1 Component Groups of Smooth Sheaves
59(11)
5.1.1 The Work of Bosch and Xarles
59(2)
5.1.2 Identity Component and Component Group of a Smooth Sheaf
61(3)
5.1.3 Some Basic Properties of the Component Group
64(4)
5.1.4 The Trace Map
68(2)
5.2 The Index of a Semi-Abelian K-Variety
70(1)
5.2.1 Definition of the Index
70(1)
5.2.2 Example: The Index of a K-Torus
70(1)
5.3 Component Groups and Base Change
71(18)
5.3.1 Uniformization of Semi-Abelian Varieties
71(1)
5.3.2 Bounded Rigid Varieties and Torsors Under Analytic Tori
72(2)
5.3.3 Behaviour of the Component Group Under Base Change
74(11)
5.3.4 The Component Series of a Semi-Abelian Variety
85(4)
Part III Chai and Yu's Base Change Conductor and Edixhoven's Filtration
6 The Base Change Conductor and Edixhoven's Filtration
89(18)
6.1 Basic Definitions
89(8)
6.1.1 The Conductor of a Morphism of Modules
89(1)
6.1.2 The Base Change Conductor of a Semi-Abelian Variety
90(1)
6.1.3 Jumps and Edixhoven's Filtration
91(6)
6.2 Computing the Base Change Conductor
97(5)
6.2.1 Invariant Differential Forms
97(1)
6.2.2 Elliptic Curves
98(3)
6.2.3 Behaviour Under Non-Archimedean Uniformization
101(1)
6.3 Jumps of Jacobians
102(5)
6.3.1 Dependence on Reduction Data
102(5)
7 The Base Change Conductor and the Artin Conductor
107(12)
7.1 Some Comparison Results
107(2)
7.1.1 Algebraic Tori
107(1)
7.1.2 Saito's Discriminant-Conductor Formula
108(1)
7.2 Elliptic Curves
109(4)
7.2.1 The Potential Degree of Degeneration
109(1)
7.2.2 A Formula for the Base Change Conductor
110(3)
7.3 Genus Two Curves
113(6)
7.3.1 Hyperelliptic Equations
113(1)
7.3.2 Minimal Equations
114(1)
7.3.3 Comparison of the Base Change Conductor and the Minimal Discriminant
115(4)
Part IV Applications to Motivic Zeta Functions
8 Motivic Zeta Functions of Semi-Abelian Varieties
119(10)
8.1 The Motivic Zeta Function
119(2)
8.1.1 Definition
119(1)
8.1.2 Decomposing the Identity Component
120(1)
8.2 Motivic Zeta Functions of Jacobians
121(1)
8.2.1 Behaviour of the Identity Component
121(1)
8.2.2 Behaviour of the Order Function
122(1)
8.3 Rationality and Poles
122(7)
8.3.1 Rationality of the Zeta Function
122(2)
8.3.2 Poles and Monodromy
124(1)
8.3.3 Prym Varieties
124(5)
9 Cohomological Interpretation of the Motivic Zeta Function
129(14)
9.1 The Trace Formula for Semi-Abelian Varieties
129(8)
9.1.1 The Rational Volume
129(2)
9.1.2 The Trace Formula and the Number of Neron Components
131(5)
9.1.3 Cohomological Interpretation of the Motivic Zeta Function
136(1)
9.2 The Trace Formula for Jacobians
137(6)
9.2.1 The Monodromy Zeta Function
137(2)
9.2.2 The Trace Formula for Jacobians
139(4)
Part V Some Open Problems
10 Some Open Problems
143(6)
10.1 The Stabilization Index
143(2)
10.2 The Characteristic Polynomial
145(1)
10.3 The Motivic Zeta Function and the Monodromy Conjecture
145(1)
10.4 Base Change Conductor for Jacobians
146(1)
10.5 Component Groups of Jacobians
147(2)
References 149
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