Perfectoid Spaces

Description

Preface Contents About the Editors On ψ-Lattices in Modular (,Γ)-Modules 1 Multivariable Modular étale (,Γ)-Modules 2 The lattices D and D inside D 3 Examples References The Relative (de-)Perfectoidification Functor and Motivic p-Adic Cohomologies 1 Introduction 2 Definitions and Main Properties of Adic Motives 3 De-Perfectoidification 4 Classic De Rham-Like Cohomologies via Motives 5 New De Rham-Like Cohomologies via Motives 6 A Betti-Like Cohomology via Motives References Diagrams and Mod p Representations of p-adic Groups 1 Introduction 1.1 Notation 2 Diagrams and the Existence Theorem 3 Diagrams Attached to Galois Representations 3.1 Diamond Diagrams 3.2 The Map δ 3.3 Irreducible Admissible Supercuspidal Representations 3.4 Extra Characters 4 Infinite-dimensional Diagrams and Non-admissible Representations References A Short Review on Local Shtukas and Divisible Local Anderson Modules 1 Introduction 2 Local Shtukas 3 Finite Shtukas 4 Divisible Local Anderson Modules 5 Cohomology Realizations of Local Shtukas 6 Crystalline Representations over Function Fields References An Introduction to p-Adic Hodge Theory 1 Local Fields: Preliminaries 1.1 Non-Archimedean Fields 1.2 Local Fields 1.3 Ramification Filtration 1.4 Norms and Traces 1.5 Witt Vectors 1.6 Non-abelian Cohomology 2 Galois Groups of Local Fields 2.1 Unramified and Tamely Ramified Extensions 2.2 Local Class Field Theory 2.3 The Absolute Galois Group of a Local Field 3 Zp-Extensions 3.1 The Different in Zp-Extensions 3.2 The Normalized Trace 3.3 Application to Continuous Cohomology 4 Deeply Ramified Extensions 4.1 Deeply Ramified Extensions 4.2 Almost étale Extensions 4.3 Continuous Cohomology of GK 5 From Characteristic 0 to Characteristic p and Vice Versa I: Perfectoid Fields 5.1 Perfectoid Fields 5.2 Tilting 5.3 The Ring Ainf(E) 5.4 The Tilting Equivalence 6 From Characteristic 0 to Characteristic p and Vice Versa II: The Field of Norms 6.1 Arithmetically Profinite Extensions 6.2 The Field of Norms 6.3 Functorial Properties 6.4 Comparison with the Tilting Equivalence 7 ell-Adic Representations 7.1 Preliminaries 7.2 ell-Adic Representations of Local Fields (ellneqp) 8 Classification of p-Adic Representations 8.1 The Case of Characteristic p 8.2 The Case of Characteristic 0 9 B-Admissible Representations 9.1 General Approach 9.2 First Examples 10 Tate–Sen Theory 10.1 Hodge–Tate Representations 10.2 Sen's Theory 11 Rings of p-Adic Periods 11.1 The Field BdR 11.2 The Rings Bcris and Bmax 11.3 The Ring Bst 12 Filtered (,N)-Modules 12.1 Filtered Vector Spaces 12.2 -Modules 12.3 Slope Filtration 12.4 Filtered (,N)-Modules 13 The Hierarchy of p-Adic Representations 13.1 de Rham Representations 13.2 Crystalline and Semi-Stable Representations 13.3 The Hierarchy of p-Adic Representations 13.4 Comparison Theorems 14 p-Divisible Groups 14.1 Formal Groups 14.2 p-Divisible Groups 14.3 Classification of p-Divisible Groups 14.4 p-Adic Integration on Formal Groups 15 Formal Complex Multiplication 15.1 Lubin–Tate Theory 15.2 Hodge–Tate Decomposition for Lubin–Tate Formal Groups 15.3 Formal Complex Multiplication for p-Divisible Groups 16 The Exponential Map 16.1 The Group of Points of a Formal Group 16.2 The Universal Covering 16.3 Application to Galois Cohomology 16.4 The Bloch–Kato Exponential Map 16.5 Hilbert Symbols for Formal Groups 17 The Weak Admissibility: The Case of Dimension One 17.1 Formal Groups of Dimension One 17.2 Geometric Interpretation of (Bcris+)h=p References Perfectoid Spaces: An Annotated Bibliography 1 Perfectoid Fields 2 Tilting, Untilting, and Witt Vectors 3 Perfectoid Rings and Spaces 4 Fargues–Fontaine Curves 5 Vector Bundles on Fargues–Fontaine Curves References The Fargues-Fontaine Curve and p-Adic Hodge Theory 1 Introduction 1.1 The Fargues-Fontaine Curve 1.2 Applications 2 Construction of XFF 2.1 Tilting and the Space |Y| 2.2 Rings of Functions on |YF| 2.3 The Schematic Fargues-Fontaine Curve and Its Main Properties 2.4 Construction of Curves (ch7FaspsFo, Chap. 5) 2.5 Vector Bundles on Curves 2.6 Conclusion of the Proof of Theorem 42 3 Vector Bundles on XF 3.1 Harder-Narasimhan Categories 3.2 Classification of Vector Bundles—Statement of the Results 3.3 The Curves XF,Eh for hge1 3.4 Construction of Vector Bundles on XF 4 An Application to Galois Representations: Weakly Admissible Equals Admissible 4.1 Fontaine's Formalism of B-Admissible Galois Representations 4.2 Weak Admissibility Implies Admissibility 5 The Classification Theorem 5.1 Preparations 5.2 An Abstract Classification Theorem 5.3 Some Constructions Related to p-Divisible Groups 5.4 The Crystal of a p-Divisible Group 5.5 The Grothendieck-Messing Period Map 5.6 The Hodge-Tate Decomposition and the HT Period Map 6 Conclusion of the Classification Theorem 6.1 What We Have to Prove and a Reduction Theorem 6.2 Modification of Vector Bundles Associated with p-Divisible Groups 6.3 Conclusion of the Proof 7 Appendix: Weierstrass Division in Ainf 7.1 Notation 7.2 Weierstrass Division in Wn(mathcalOF) 7.3 Weierstrass Division in W(mathcalOF) References Simplicial Galois Deformation Functors 1 Introduction 2 Classical and Simplicial Galois Deformation Functors 2.1 Classical Deformations 2.2 Simplicial Reformulation of Classical Deformations 2.3 Simplicial Reformulation of Classical Framed Deformations 2.4 Derived Deformation Functors 3 Pseudo-Deformation Functors 3.1 Classical Pseudo-Characters and Functors on FFS 3.2 Classical Pseudo-Characters and Simplicial Objects 3.3 Derived Deformations of Pseudo-Characters 4 (Co)tangent Complexes and Pro-Representability 4.1 Dold-Kan Correspondence 4.2 (Co)tangent Complexes of Simplicial Commutative Rings 4.3 Tangent Complexes of Formally Cohesive Functors and Lurie's Criterion 4.4 Pro-Representability of Derived Deformation Functors 4.5 Relative Derived Deformations and Relative Tangent Complexes 5 Application to the Galatius-Venkatesh Homomorphism References