p-adic di􀀋erential equations - Version 7 - Jan 08 - Table of contents

Oct 22, 2026

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Comprehensive reference on p-adic differential equations, organized into an introduction, tools of p-adic analysis, differential algebra, and applications on discs and annuli. Establishes foundational concepts such as absolute values, valuations, Newton polygons, and matrix analysis, then develops formalism for differential algebra and studies metric properties. Progresses through regular singularities, radii of convergence, Frobenius pullback/pushforward, decomposition theorems, and p-adic exponents, culminating in difference algebra and Frobenius structures.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages202

Reading Time8 h 29 min

AccessFree to download

Preface

About the book

Structure of the book

Acknowledgments

Chapter 0 . Introductory remarks

Why p-adic di􀀋erential equations?

Zeta functions of varieties

Zeta functions and p-adic di􀀋erential equations

A word of caution

Notes

Exercises

Part 1 . Tools of p-adic analysis

Chapter 1 . Absolute values

Chapter 2 . Newton polygons

Chapter 3 . Matrix analysis

Part 2 . Di􀀋erential algebra

Chapter 4 . Formalism of di􀀋erential algebra

Chapter 5 . Metric properties of di􀀋erential modules

Chapter 6 . Regular singularities

Part 3 . p-adic di􀀋erential equations on discs and annuli

Chapter 7 . Rings of functions on discs and annuli

Chapter 8 . Radius and generic radius of convergence

Chapter 9 . Frobenius pullback and pushforward

Chapter 10 . Variation of generic and subsidiary radii

Chapter 11 . Decomposition by subsidiary radii

Chapter 12 . p-adic exponents

Part 4 . Di􀀋erence algebra and Frobenius structures

Chapter 13 . Formalism of di􀀋erence algebra

Chapter 14 . Frobenius modules

Chapter 15 . Frobenius structures on di􀀋erential modules

Chapter 16 . E􀀋ective convergence bounds

Chapter 17 . Quasiconstant di􀀋erential modules

Chapter 18 . The p-adic local monodromy theorem

Part 5 . Areas of application

Chapter 19 . Picard-Fuchs modules

p-adic di􀀋erential equations (version of 7 Jan 08) Kiran S. Kedlaya Contents Preface ix 1. About the book ix 2. Structure of the book x 3. Acknowledgments x Chapter 0 . Introductory remarks 1 1. Why p-adic di􀀋erential equations? 1 2. Zeta functions of varieties 2 3. Zeta functions and p-adic di􀀋erential equations 3 4. A word of caution 5 Notes 6 Exercises 6 Part 1 . Tools of p-adic analysis 7 Chapter 1 . Absolute values 9 1. Absolute values on abelian groups 9 2. Valuations and nonarchimedean absolute values 10 3. Norms on modules 11 4. Examples of nonarchimedean absolute values 12 5. Unrami􀀌ed extensions 14 6. Tamely rami􀀌ed extensions 15 7. Spherical completeness 16 Notes 17 Exercises 17 Chapter 2 . Newton polygons 19 1. Gauss norms and Newton polygons 19 2. Slope factorizations and a master factorization theorem 20 3. Applications to nonarchimedean 􀀌eld theory 22 Exercises 23 Chapter 3 . Matrix analysis 25 1. Singular values and eigenvalues (archimedean case) 25 2. Perturbations (archimedean case) 27 3. Singular values and eigenvalues (nonarchimedean case) 29 4. Perturbations (nonarchimedean case) 32 5. Horn’s inequalities 34 Notes 35 Exercises 36 Part 2 . Di􀀋erential algebra 37 Chapter 4 . Formalism of di􀀋erential algebra 39 1. Di􀀋erential rings and di􀀋erential modules 39 2. Di􀀋erential modules and di􀀋erential systems 39 3. Operations on di􀀋erential modules 40 4. Cyclic vectors 41 5. Di􀀋erential polynomials 42 6. Di􀀋erential equations 43 7. Cyclic vectors: a mixed blessing 44 8. Taylor series 45 Notes 45 Exercises 46 Chapter 5 . Metric properties of di􀀋erential modules 47 1. Spectral norms of linear operators 47 2. Spectral norms of di􀀋erential operators 48 3. A coordinate-free approach 51 4. Newton polygons for twisted polynomials 51 5. Twisted polynomials and spectral norms 52 6. The visible decomposition theorem 53 7. Matrices and the visible spectrum 54 Notes 57 Exercises 57 Chapter 6 . Regular singularities 59 1. Irregularity 59 2. Exponents in the complex analytic setting 60 3. Formal solutions of regular di􀀋erential equations 61 4. Index and irregularity 63 Notes 63 Exercises 63 Part 3 . p-adic di􀀋erential equations on discs and annuli 65 Chapter 7 . Rings of functions on discs and annuli 67 1. Power series on closed discs and annuli 67 2. Gauss norms and Newton polygons 67 3. Factorization results 69 4. Open discs and annuli 70 Notes 70 Exercises 71 Chapter 8 . Radius and generic radius of convergence 73 1. Di􀀋erential modules on rings and annuli 73 2. Radius of convergence on a disc 73 3. Generic radius of convergence 74 4. Some examples in rank 1 76 5. Transfer theorems 76 6. Geometric interpretation 78 7. Subsidiary radii 79 Notes 79 Exercises 80 Chapter 9 . Frobenius pullback and pushforward 81 1. Why Frobenius? 81 2. p-th roots 81 3. Moving along Frobenius 83 4. Frobenius antecedents 84 5. Frobenius descendants and subsidiary radii 85 6. Decomposition by spectral norm 86 7. Integrality, or lack thereof 87 8. O􀀋-centered Frobenius descendants 88 Notes 89 Exercises 89 Chapter 10 . Variation of generic and subsidiary radii 91 1. Harmonicity of the valuation function 91 2. Variation of Newton polygons 91 3. Variation of subsidiary radii: statements 93 4. Convexity for the generic radius 94 5. Finding lattices 95 6. Measuring small radii 96 7. Larger radii 96 8. Monotonicity 98 9. Radius versus generic radius 99 10. Subsidiary radii as radii of convergence 100 Notes 101 Exercises 101 Chapter 11 . Decomposition by subsidiary radii 103 1. Metrical detection of units 103 2. Decomposition over a closed disc 104 3. Decomposition over a closed annulus 105 4. Decomposition over an open disc or annulus 106 5. Modules solvable at a boundary 107 Notes 108 Exercises 108 Chapter 12 . p-adic expo

FAQs

What topics does the book cover before developing p-adic differential equations?

It begins with introductory remarks and then develops tools of p-adic analysis, including absolute values, Newton polygons, and matrix analysis. These provide the foundational language for later chapters.

How does the book treat regular singularities in the p-adic setting?

Regular singularities are addressed in a dedicated section, discussing irregularity, exponents in the complex analytic setting, formal solutions, and the relationship between index and irregularity.

What role do Frobenius structures and monodromy play?

Later parts focus on difference algebra and Frobenius structures, then establish effective convergence bounds and study quasiconstant modules. The p-adic local monodromy theorem is presented with multiple approaches and applications.

Oct 22, 2026

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