p-adic Differential Equations

Kiran S. Kedlaya

• Publisher: Cambridge University Press
• Year: 2022
• Print ISBN: 9781009123341
• eBook ISBN: 9781009275651
• Edition: 2
• Topic: Mathematics
• Subtopic: Number Theory
• Index: Mathematics

Now in its second edition, this volume provides a uniquely detailed study of $P$-adic differential equations. Assuming only a graduate-level background in number theory, the text builds the theory from first principles all the way to the frontiers of current research, highlighting analogies and links with the classical theory of ordinary differential equations. The author includes many original results which play a key role in the study of $P$-adic geometry, crystalline cohomology, $P$-adic Hodge theory, perfectoid spaces, and algorithms for L-functions of arithmetic varieties. This updated edition contains five new chapters, which revisit the theory of convergence of solutions of $P$-adic differential equations from a more global viewpoint, introducing the Berkovich analytification of the projective line, defining convergence polygons as functions on the projective line, and deriving a global index theorem in terms of the Laplacian of the convergence polygon.

Table of contents

• Cover
• Half-title
• Series information
• Title page
• Copyright information
• Contents
• Preface
• Acknowledgments
• 0 Introductory remarks
• Part I Tools of p-adic Analysis
• Part II Differential Algebra
• Part III p-adic Differential Equations on Discs and Annuli
• Part IV Difference Algebra and Frobenius Modules
• Part V Frobenius Structures
• Part VI The p-adic local monodromy theorem
• Part VII Global theory
• APPENDICES AND BACK MATTER
• References
• Index of notation
• Subject index