Integration of one-forms on p-adic analytic spaces /
Berkovich, Vladimir G.
Integration of one-forms on p-adic analytic spaces / Vladimir G. Berkovich. - Princeton, N.J. : Princeton University Press, 2007. - 1 online resource (vi, 156 pages) - Annals of mathematics studies ; no. 162 . - Annals of mathematics studies ; no. 162. .
Includes bibliographical references and indexes.
Naive analytic functions and formulation of the main result -- Étale neighborhoods of a point in a smooth analytic space -- Properties of strictly poly-stable and marked formal schemes -- Properties of the sheaves -- Isocrystals -- F-isocrystals -- Construction of the Sheaves -- Properties of the sheaves -- Integration and parallel transport along a path.
Use copy
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.
Electronic reproduction.
[Place of publication not identified] :
HathiTrust Digital Library,
2010.
Master and use copy. Digital master created according to Benchmark for Faithful Digital Reproductions of Monographs and Serials, Version 1. Digital Library Federation, December 2002.
http://purl.oclc.org/DLF/benchrepro0212
In English.
9781400837151 (electronic bk.) 1400837154 (electronic bk.) 1299133339 9781299133334
10.1515/9781400837151 doi
22573/ctt249r54 JSTOR
GBA687868 bnb
013578338 Uk
p-adic analysis.
Analyse p-adique.
MATHEMATICS--Number Theory.
MATHEMATICS--Differential Equations--General.
p-adic analysis.
Electronic books.
Electronic books.
QA241 / .B475 2007
512.74
Integration of one-forms on p-adic analytic spaces / Vladimir G. Berkovich. - Princeton, N.J. : Princeton University Press, 2007. - 1 online resource (vi, 156 pages) - Annals of mathematics studies ; no. 162 . - Annals of mathematics studies ; no. 162. .
Includes bibliographical references and indexes.
Naive analytic functions and formulation of the main result -- Étale neighborhoods of a point in a smooth analytic space -- Properties of strictly poly-stable and marked formal schemes -- Properties of the sheaves -- Isocrystals -- F-isocrystals -- Construction of the Sheaves -- Properties of the sheaves -- Integration and parallel transport along a path.
Use copy
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.
Electronic reproduction.
[Place of publication not identified] :
HathiTrust Digital Library,
2010.
Master and use copy. Digital master created according to Benchmark for Faithful Digital Reproductions of Monographs and Serials, Version 1. Digital Library Federation, December 2002.
http://purl.oclc.org/DLF/benchrepro0212
In English.
9781400837151 (electronic bk.) 1400837154 (electronic bk.) 1299133339 9781299133334
10.1515/9781400837151 doi
22573/ctt249r54 JSTOR
GBA687868 bnb
013578338 Uk
p-adic analysis.
Analyse p-adique.
MATHEMATICS--Number Theory.
MATHEMATICS--Differential Equations--General.
p-adic analysis.
Electronic books.
Electronic books.
QA241 / .B475 2007
512.74