Open Problems in Arithmetic Algebraic Geometry

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?