Integration of One-forms on P-adic Analytic Spaces. (AM-162) / / Vladimir G. Berkovich.

• Superior document: Title is part of eBook package: De Gruyter Princeton Annals of Mathematics eBook-Package 1940-2020
• VerfasserIn: Berkovich, Vladimir G.,
• Place / Publishing House: Princeton, NJ : : Princeton University Press, , [2006]; ©2007
• Year of Publication: 2006
• Edition: Course Book
• Language: English
• Series: Annals of Mathematics Studies ; 162
• Physical Description: 1 online resource (168 p.) :; 14 line illus.

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.

Table of 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