Integration of One-forms on P-adic Analytic Spaces.

AM-162

Vladimir G. Berkovich

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English

Princeton University Pres
12 February 2007

Mathematics & Sciences; Differential calculus & equations; Analytic geometry

Series: Annals of Mathematics Studies

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.

By: Vladimir G. Berkovich
Imprint: Princeton University Pres
Country of Publication: United States
Volume: No. 162
Dimensions: Height: 235mm, Width: 152mm, Spine: 12mm
Weight: 28g
ISBN: 9780691128627
ISBN 10: 0691128626
Series: Annals of Mathematics Studies
Pages: 168
Publication Date: 12 February 2007
Audience: Professional and scholarly , College/higher education , Undergraduate , Primary
Format: Paperback
Publisher's Status: Active

Vladimir G. Berkovich is Matthew B. Rosenhaus Professor of Mathematics at the Weizmann Institute of Science in Rehovot, Israel. He is the author of Spectral Theory and Analytic Geometry over Non-Archimedean Fields .