# P-adic Differential Equations Kedlaya Cambridge University Press 9780521768795

EAN 9780521768795, ISBN10: 0521768799

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.

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.