Integration of one-forms on p-adic analytic spaces
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.
14 pages matching parallel transport in this book
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Naive Analytic Functions and Formulation of the Main Result
F tale Neighborhoods of a Point in a Smooth Analytic Space
9 other sections not shown
affinoid subdomain analytic curve analytic functions assume basic curve Ber2 Ber7 bijection canonical morphism center at zero closed one-form closed point x e closed subfield coincides connected construction contained Corollary defined denote dim(X element embedding etale morphism etale neighborhood etale topology exact sequence exists F-isocrystals filtered algebra finite extension Frobenius lifting functor Furthermore geometric point given gives rise homomorphism homotopy implies injective irreducible component isocrystal isomorphism k-analytic latter Lemma logarithm marked formal scheme marked neighborhood module non-Archimedean field nontrivial Notice open affine subscheme open annulus open disc open neighborhood open subset parallel transport polynomial preimage primitive Proof proper marked formal Proposition Px-module quotient required fact follows resp semi-annular sheaves shrink smooth analytic space smooth morphism strictly affinoid strictly poly-stable subspace suffices to show surjective Theorem trivial true unipotent unique valuation Vx,x wide germ Zariski