p-adic differential equations
By: Kiran S. Kedlaya.
Series: Cambridge studies in advanced mathematics -- 125
Publisher: Cambridge University Press
Published: 2010
Place of publication: Cambridge; New York
Language: eng
ISBN-13: 9780521768795
Subjects: p-adic analysis; Differential equations
Description: "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"--

"Although the very existence of a highly developed theory of p-adic ordinary differential equations is not entirely well known even within number theory, the subject is actually almost 50 years old. Here are circumstances, past and present, in which it arises"--
Notes: Includes bibliographical references and index.
Table of contents: title: Machine generated contents note: 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.