p -adic Differential Equations

Over the last 50 years the theory of p-adic differential equations has grown into an active area of research in its own right, and has important applications to number theory and to computer science. This book, the first comprehensive and unified introduction to the subject, improves and simplifies existing results as well as including original material. Based on a course given by the author at MIT, this modern treatment is accessible to graduate students and researchers. Exercises are included at the end of each chapter to help the reader review the material, and the author also provides detailed references to the literature to aid further study.

Spis treści:
Preface
Introductory remarks
Part I. Tools of p-adic Analysis
1. Norms on algebraic structures
2. Newton polygons
3. Ramification theory
4. Matrix analysis
Part II. Differential Algebra
5. Formalism of differential algebra
6. Metric properties of differential modules
7. Regular singularities
Part III. p-adic Differential Equations on Discs and Annuli
8. Rings of functions on discs and annuli
9. Radius and generic radius of convergence
10. Frobenius pullback and pushforward
11. Variation of generic and subsidiary radii
12. Decomposition by subsidiary radii
13. p-adic exponents
Part IV. Difference Algebra and Frobenius Modules
14. Formalism of difference algebra
15. Frobenius modules
16. Frobenius modules over the Robba ring
Part V. Frobenius Structures
17. Frobenius structures on differential modules
18. Effective convergence bounds
19. Galois representations and differential modules
20. The p-adic local monodromy theorem, Statement
21. The p-adic local monodromy theorem, Proof
Part VI. Areas of Application
22. Picard-Fuchs modules
23. Rigid cohomology
24. p-adic Hodge theory
References
Index of notation
Index.