Integration of one-forms on p-adic analytic spaces
著者
書誌事項
Integration of one-forms on p-adic analytic spaces
（Annals of mathematics studies, no. 162）
Princeton University Press, c2007
- : hard
- : pbk
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注記
Includes bibliographical references (p. [149]-151) and index
内容説明・目次
- 巻冊次
-
: hard ISBN 9780691127415
内容説明
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.
- 巻冊次
-
: pbk ISBN 9780691128627
内容説明
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.
目次
*Frontmatter, pg. i*Contents, pg. v*Introduction, pg. 1*1. Naive Analytic Functions and Formulation of the Main Result, pg. 7*2. Etale Neighborhoods of a Point in a Smooth Analytic Space, pg. 23*3. Properties of Strictly Poly-stable and Marked Formal Schemes, pg. 39*4. Properties of the Sheaves OMEGA1.dx/dOX, pg. 55*5. Isocrystals, pg. 71*6. F-isocrystals, pg. 87*7. Construction of the Sheaves SlambdaX, pg. 95*8. Properties of the sheaves SlambdaX, pg. 113*9. Integration and Parallel Transport along a Path, pg. 131*References, pg. 149*Index of Notation, pg. 153*Index of Terminology, pg. 155
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