p-adic Differential Equations

This book is a detailed study of $P$-adic differential equations,building the theory from first principles and highlighting analogies with classical theory. It includes original results on $P$-adic geometry,crystalline cohomology,Hodge theory,perfectoid spaces,and algorithms for L-functions. The updated edition contains five new chapters on convergence of solutions and global index theory.

Format: Hardback
Length: 420 pages
Publication date: 09 June 2022
Publisher: Cambridge University Press

In its second edition, this comprehensive volume offers a detailed exploration of $P$-adic differential equations. Assuming a foundational understanding of number theory at the graduate level, the text constructs the theory from scratch, progressing to the forefront of current research, while highlighting analogies and connections with the classical theory of ordinary differential equations. The author includes numerous original findings that play a pivotal 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 revised edition incorporates five new chapters that revisit the theory of convergence of solutions of $P$-adic differential equations from a broader perspective. These chapters introduce the Berkovich analytification of the projective line, define convergence polygons as functions on the projective line, and derive a global index theorem in terms of the Laplacian of the convergence polygon.

Introduction:
$P$-adic differential equations have gained significant attention in recent years due to their applications in various fields, including number theory, algebraic geometry, and mathematical physics. These equations arise in the study of complex numbers and their arithmetic properties, and they exhibit unique features that distinguish them from their classical counterparts. In this volume, we aim to provide a comprehensive and up-to-date treatment of $P$-adic differential equations.

Scope and Coverage:
Our goal is to cover the fundamental aspects of $P$-adic differential equations, including their definitions, properties, and solutions. We will begin by reviewing the basic theory of $P$-adic numbers and their arithmetic operations, and then proceed to introduce the concept of $P$-adic differential equations. We will discuss the main results and techniques related to the study of these equations, including the theory of convergence, the existence and uniqueness of solutions, and the theory of regularity.

Original Contributions:
One of the key features of this volume is the inclusion of numerous original results. These results play a crucial role in the study of $P$-adic geometry, crystalline cohomology, $P$-adic Hodge theory, perfectoid spaces, and algorithms for L-functions of arithmetic varieties. We will highlight these contributions and discuss their implications and applications.

Updated Edition:
To ensure that the content remains relevant and up-to-date, we have included five new chapters in this updated edition. These chapters revisit the theory of convergence of solutions of $P$-adic differential equations from a more global viewpoint. We introduce the Berkovich analytification of the projective line, define convergence polygons as functions on the projective line, and derive a global index theorem in terms of the Laplacian of the convergence polygon.

Conclusion:
$P$-adic differential equations offer a rich field of study with numerous exciting developments and applications. This volume provides a comprehensive and accessible introduction to the theory of $P$-adic differential equations, with a focus on both the fundamental aspects and the latest research trends. We hope that it will be valuable to students, researchers, and practitioners in the field.

ISBN-13: 9781009123341
Edition number: 2 Revised edition