Integration of one-forms on p-adic analytic spaces / Vladimir G. Berkovich.

Format:

Book

Author/Creator:

Berkovich, Vladimir G.

Series:

Annals of Mathematics Studies ; v. 162

Annals of mathematics studies ; no. 162

Annals of Mathematics Studies ; 162

Language:

English

Subjects (All):

p-adic analysis.

Physical Description:

1 online resource (163 p.)

Edition:

Course Book

Place of Publication:

Princeton, N.J. : Princeton University Press, 2007.

Language Note:

English

Summary:

Among the many differences between classical and p-adic objects, those related to differential equations occupy a special place. For example, a closed p-adic analytic one-form defined on a simply-connected domain does not necessarily have a primitive in the class of analytic functions. In the early 1980s, Robert Coleman discovered a way to construct primitives of analytic one-forms on certain smooth p-adic analytic curves in a bigger class of functions. Since then, there have been several attempts to generalize his ideas to smooth p-adic analytic spaces of higher dimension, but the spaces considered were invariably associated with algebraic varieties. This book aims to show that every smooth p-adic analytic space is provided with a sheaf of functions that includes all analytic ones and satisfies a uniqueness property. It also contains local primitives of all closed one-forms with coefficients in the sheaf that, in the case considered by Coleman, coincide with those he constructed. In consequence, one constructs a parallel transport of local solutions of a unipotent differential equation and an integral of a closed one-form along a path so that both depend nontrivially on the homotopy class of the path. Both the author's previous results on geometric properties of smooth p-adic analytic spaces and the theory of isocrystals are further developed in this book, which is aimed at graduate students and mathematicians working in the areas of non-Archimedean analytic geometry, number theory, and algebraic geometry.

Contents:

Frontmatter

Contents

Introduction

1. Naive Analytic Functions and Formulation of the Main Result

2. Étale Neighborhoods of a Point in a Smooth Analytic Space

3. Properties of Strictly Poly-stable and Marked Formal Schemes

4. Properties of the Sheaves Ω1.dx/dOX

5. Isocrystals

6. F-isocrystals

7. Construction of the Sheaves SλX

8. Properties of the sheaves SλX

9. Integration and Parallel Transport along a Path

References

Index of Notation

Index of Terminology

Notes:

Description based upon print version of record.

Includes bibliographical references and indexes.

Description based on online resource; title from PDF title page (publisher's Web site, viewed 08. Jul 2019)

ISBN:

9781400837151

1400837154

9781299133334

1299133339

9780691127415

0691127417

OCLC:

845250381