Integration of oneforms on padic analytic spaces


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Among the many differences between classical and padic objects, those related to differential equations occupy a special place. For example, a closed padic analytic oneform defined on a simplyconnected 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 oneforms on certain smooth padic analytic curves in a bigger class of functions. Since then, there have been several attempts to generalize his ideas to smooth padic analytic spaces of higher dimension, but the spaces considered were invariably associated with algebraic varieties.This book aims to show that every smooth padic 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 oneforms 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 oneform 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 padic 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 nonArchimedean analytic geometry, number theory, and algebraic geometry.