This book originated in the idea that open problems act as crystallization points in mathematical research. Mathematical books usually deal with fully developed theories. But here we present work at an earlier stagewhen challenging questions can give new directions to mathematical research. In mathematics, significant progress is often made by looking at the underlying structures of open problems and discovering new directions that are developed to find solutions. In that process, the search for finding the "true" nature of the problem at hand is the impetus for our thoughts. It is only much later, in retrospect, that we see the "flow of mathematics"from problem to theory and new insight. This is the gist of the present volume.
The origin of this volume lies in a collection of nineteen problems presented in 1995 to the participants of the conference Arithmetic and Geometry of Abelian Varieties.
Author(s): [various contributors], Frans Oort (Universiteit Utrecht) (editor)
Series: Advanced Lectures in Mathematics 46
Publisher: International Press of Boston
Year: 2019
Language: English
Commentary: Improvements with respect to [MD5]E149203AFAB35E7EC39A17A37EFC6253: added OCR by adobe ACROBAT, pagenated and added bookmarks
Pages: 345
Introduction
Contents
Chapter 1. Is a Finite Locally Free Group Scheme Killed by Its Order?
1 Introduction
2 Finite group schen1es over fields
3 Commutative group schemes
4 Finite group schemes over Artin rings
References
Chapter 2. Lifting of Curves with Automorphisms
1 Introduction
1.1 Riemai1n's existence theorem
1.2 Lifting of covers of curves
1.3 A remark on this exposition
1.4 Notation and conventions
2 Global results
2.1 Tame covers
2.2 Wild covers
2.3 Oort groups and the Oo1't conjecture
3 The local-global principle
3.1 Preliminaries arid etale lifting
3.2 A patching result
3.3 Proof of Theorem 3.1
4 The local lifting problein and its obstructions
4.1 The (local) KGB obstruction
4.2 The ( differential) Hurwitz tree obstruction
5 Surnniary of present local lifting problem results
5.1 Local Oort groups
5.2 Weak local Oort groups
6 Lifting techniques and examples
6.1 Birational lifts and the different criterion
6.2 Explicit lifts
6.3 Sekiguchi-Suwa theory
6.4 Hurwitz trees
6.5 Successive approximation
6.6 The ''Mumford method''
7 Approach using deforination theory
7.1 Setup
7.2 The local-global principle via deformation theory
7.3 Examples of local miniversal deformation rings
8 Open probleins
8.1 Existence of local lifts
8.2 Rings of definition
8.3 Moduli/deformations/geometry of local lifts
8.4 Non-algebraically closed residue fields
A Some algebraic preliminaries
A.1 Homological Algebra
A.2 Complete local rings
A.3 Ramification theory
References
Chapter 3. The Andre-Oort Conjecture
Introduction
1 The Manin-Muinford conjecture
2 The Andre-Oort conjecture
3 Special subvarieties are linear in Serre-Tate canonical coordinates
4 The Andre-Oort conjecture: recent results
5 Generalizations: Unlikely intersections
6 Open problems and questions
References
Chapter 4. Special Subvarieties in the Torelli Locus
1 Introduction
2 An expectation
3 Known counterexatnples
4 Weyl CM fields
5 Positive characteristic
6 Jacobians in Inixed characteristic
7 Appendix: CM abelian varieties
8 Appendix: special subvarieties
9 Appendix: notations
10 Open probleins and questions
References
Chapter 5. Moduli of Abelian Varieties
0 Introduction
1 p-divisible groups
2 Stratifications and foliations
3 Newton polygon strata
4 Foliations
5 The EO stratification
6 Irreducibility
7 CM lifting
8 Generalized Serre-Tate coordinates
9 Some historical remarks
10 Sonie Questions
References
Chapter 6. Current Results on Newton Polygons of Curves
1 Introduction
2 Supersingular curves
3 Newton polygons and Ekedahl-Oort types of curves
4 Notation and background
5 Stratifications of moduli spaces
6 An inductive result about Newton polygon and Ekedahl-Oort strata
7 Open problems
References
Chapter 7. Sustained p-divisible Groups: A Foliation Retraced
1 What is a sustained p-divisible group
2 Stabilized Horn schemes for truncations of p-divisible groups
3 Descent of sustained p-divisible groups, torsors for stabilized Isom schemes and the slope filtration
4 Pointwise criterion for sustained p-divisible groups
5 Deforniation of sustained p-divisible groups and local structure of central leaves
References
Chapter 8. The Hecke Orbit Conjecture: A Survey and Outlook
1 Introduction
2 Hecke syrnnietry on Inodular varieties of PEL type
3 Sustained p-divisible groups and central leaves
4 Local structure of leaves
5 Action of the local stabilizer subgroup
6 Local rigidity for subvarieties of leaves
7 Monodroniy of Hecke invariant subvarieties
8 The Hecke orbit conjecture for A_g
9 Open questions
References
Appendix 1. Some Questions in Algebraic Geometry
References
Appendix 2. Automorphisms of Curves-2005 Collection
References
Appendix 3. Questions in Arithmetic Algebraic Geometry
References
An afterthought: When do we use the word conjecture?
