Integration of One-forms on P-adic Analytic Spaces. (AM-162)

Integration of One-forms on P-adic Analytic Spaces. (AM-162)

Vladimir G. Berkovich
Copyright Date: 2007
Pages: 168
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    Integration of One-forms on P-adic Analytic Spaces. (AM-162)
    Book Description:

    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.

    eISBN: 978-1-4008-3715-1
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-vi)
  3. Introduction
    (pp. 1-6)

    One of the basic facts of complex analysis is the exactness of the de Rham complex ofsheavesof analytic differential forms on a smooth complex analytic space. In its turn, its proof is based on the fact that every point of such a space has an open neighborhood isomorphic to an open polydisc, which reduces the verification of the exactness to the classical Poincaré lemma. The latter states that the de Rham complex ofspacesof analytic differential forms on an open polydisc is exact. Its proof actually works over any non-Archimedean field$ k $of characteristic zero as well,...

  4. Chapter One Naive Analytic Functions and Formulation of the Main Result
    (pp. 7-22)

    After recalling some notions and notation, we give a precise definition of the sheaves of naive analytic functions$ \mathfrak{N}_X^K $. We then recall the definition of a$ {\mathcal{D}_X} $-module, introduce a related notion of a$ D_X $-module, and establish a simple relation between the de Rham complexes of a$ D_X $-module and those of its pullback under a so-called discoid morphism$ Y \to X $. In §1.4, we introduce the logarithmic function$ {\text{Lo}}{{\text{g}}^\lambda }(T) \in {\mathfrak{N}^{K,1}}({{\mathbf{G}}_{\text{m}}}) $and a filtered$ D_X $-module$ {L^\lambda }(X) $which is generated over$ \mathcal{O}(X) $by the logarithms$ {\text{Lo}}{{\text{g}}^\lambda }(f) $of invertible analytic functions on$ X $. Furthermore, given a so-called semi-annular morphism$Y \rightarrow X$...

  5. Chapter Two Étale Neighborhoods of a Point in a Smooth Analytic Space
    (pp. 23-38)

    In this section we describe a fundamental system of étale neighborhoods of a point$ x $in a smooth$ k $-analytic space$ X $. If$ s (x) = \text{dim}(X) $, we use a result of J. de Jong [deJ3] to show that there is a morphism$ \varphi: \mathfrak{X}_{\eta} \to X $from the generic fiber of a so-called proper marked formal scheme$ \mathfrak{X} $over the ring of integers$ k'^{\circ} $of a finite extension$ k' $of$ k $such that$ \varphi $is étale in an open neighborhood of the generic point of$ \mathfrak{X} $which is a unique preimage of$ x $in$ \mathfrak{X}_{\eta} $. (This morphism$ \varphi $is in fact étale...

  6. Chapter Three Properties of Strictly Poly-stable and Marked Formal Schemes
    (pp. 39-54)

    In this section we study properties of a proper marked formal scheme$ \mathfrak{X} $over$ k^{\circ} $and its generic fiber$ \mathfrak{X}_{\eta} $. For example, the reduction map$ \pi : \mathfrak{X}_{\eta} \mapsto \mathfrak{X}_{s} $is surjective and the preimage$ \pi^{-1}(\mathbf{x}) $of a closed point$ \mathbf{x}\in \mathfrak{X}_{s} $with$ \tilde{k}(\mathbf{x}) $separable over$ \tilde{k} $is a semi-annular space. In particular, if the characteristic of$ k $is zero and the field$ \tilde{k} $is perfect, any closed one-form on$ \mathfrak{X}_{ \eta } $whose restriction to the residue class$ \pi^{-1}(\mathbf{x}) $of every closed point$ \mathbf{x}\in \mathfrak{X}_{s} $is in$ \Omega_{L^{\lambda}}^{1}(\pi^{-1}(\mathbf{x})) $has a primitive in the class of functions whose restrictions to every residue class$\pi^{-1}(\mathbf{x})$are in$L^{\lambda}(\pi^{-1}(\mathbf{x}))$...

  7. Chapter Four Properties of the Sheaves $ \Omega _{X}^{1,\text{cl}}/d\mathcal{O}_{X} $
    (pp. 55-70)

    In this section we study the quotient sheaf$ \Omega _{X}^{1,\text{cl}}/d\mathcal{O}_{X} $, which measures nonexactness of the de Rham complex of a smooth$ k $-analytic space$ X $(at$ \Omega_{X}^{1} $). The study is based on the following property of analytic spaces: any two points from the subset$ X_{0} $of a connected closed analytic space$ X $can be connected by smooth analytic curves. This property allows one to reduce certain problems to the one-dimensional case. It is used in the proof of the main result in §7 and of the following facts here. Let$ \mathcal{O}_{X}^{1} $be the subsheaf of$ \mathcal{O}_{X}^{\ast} $consisting of the...

  8. Chapter Five Isocrystals
    (pp. 71-86)

    In this section we study various objects related to a wide germ of a strictly$ k $-affinoid space, i.e., a germ of an analytic space ($ X, Y $) in which$ Y $is a strictly affinoid domain in the interior of a separated analytic space$ X $, and consider a related notion of a wide germ of a formal scheme ($ X, \mathfrak{Y} $) (if$ \mathfrak{Y} $is affine, it gives rise to a wide germ of a strictly$ k $-affinoid germ ($ X, \mathfrak{Y}_{\eta} $)). First of all, we show that the correspondence$ (X, Y) \mapsto B = \mathcal{O} (X, Y) $gives rise to an anti-equivalence between the category of such germs and...

  9. Chapter Six $ F $-isocrystals
    (pp. 87-94)

    Beginning with this section the ground field$ k $is assumed to be a closed subfield of$ \mathbf{C}_{p} $. We apply constructions of the previous section to a wide germ of a smooth affine formal scheme ($ X, \mathfrak{Z} $) which is a lifting of a similar germ defined over a finite extension of$ \mathbf{Q}_{p} $. First of all, we consider the notions of a Frobenius lifting on the associated germ ($ X, \mathfrak{Z}_{\eta} $) and of a Frobenius structure on isocrystals over$ B=\mathcal{O}(X,\mathfrak{Z}_{\eta}) $($ F $-isocrystals). We provide the unipotent isocrystals$ E^{i}(X,\mathfrak{Z})=E_{B}^{i} $with such a structure and, using a result of B. Chiarellotto [Chi], show...

  10. Chapter Seven Construction of the Sheaves $ \mathcal{S}_{X}^{\lambda} $
    (pp. 95-112)

    The construction of the sheaves$ \mathcal{S}_{X}^{\lambda, n} $is carried out by double induction on$ m = \text{dim}(X) $and the number$ n $. Assume that the sheaves$ \mathcal{S}_{X}^{\lambda, i} $with all required properties exist for all pairs ($ i,X $) with$ 0 \leq i \leq n $, or$ i = n + 1 $and$ \text{dim}(X) \leq m $. In order to construct the sheaves$ \mathcal{S}_{X}^{\lambda, n+1 } $for$ X $of dimension$ m + 1 $, we have to construct a primitive of a closed one-form$ \omega $with coefficients in$ \mathcal{S}_{X}^{\lambda, n} $in an open neighborhood of every point of$ X $. For this we make additional induction hypotheses that describe the form of a function$f \in \mathcal{S}^{\lambda, i} (X)$in an étale neighborhood of...

  11. Chapter Eight Properties of the Sheaves $ \mathcal{S}_{X}^{\lambda} $
    (pp. 113-130)

    In this section we refine information on the sheaves$ \mathcal{S}_{X}^{\lambda} $using the fact that they exist. First of all, we show that any connected wide germ with good reduction ($ X, Y $) can be provided with a unique filtered$ D_{(X,Y)} $-subalgebra$ \mathcal{E}^{\lambda} (X, Y) \subset \mathcal{S}^{\lambda}(X, Y) $with properties (a)–(d) and (f) of Coleman’s algebras mentioned in the introduction, and we relate it to the isocrystal$ E^{\lambda}(X, \mathfrak{Y}) $considered in §5.5 and §7.1. We also show that any proper marked formal scheme$ \mathfrak{X} $over$ k^{\circ} $can be provided with a unique filtered$ \mathcal{O}({\mathfrak{X}_\eta }) $-subalgebra$ \mathcal{E}^{\lambda}(\mathfrak{X}) \subset \mathcal{S}^{\lambda}({\mathfrak{X}_\eta }) $which is a filtered$ D_{{\mathfrak{X}_\eta }} $-algebra that possesses similar properties and...

  12. Chapter Nine Integration and Parallel Transport along a Path
    (pp. 131-148)

    For a$ k $-analytic space$ X $, we set$ \bar X = X\hat \otimes {\hat k^a} $. At the beginning of this section we construct, for every smooth$ k $-analytic space$ X $with$ {H_1}(\bar X,{\mathbf{Q}})\tilde \to {H_1}(X,{\mathbf{Q}}) $, every closed one-form$ \omega \in \Omega _{{\mathcal{S}^{\lambda ,n}}}^1(X) $and every path$ \gamma :[0,1] \to X $with ends in$ X(k) $, an integral$ {\int _\gamma }\omega \in {K^{n + 1}} $. This integral possesses all of the natural properties and, in particular, it only depends on the homotopy class of$ \gamma $. That this dependence is nontrivial is shown in §9.2. Furthermore, for a$ {\mathcal{D}_X} $-module$ \mathcal{F} $, let$ {\mathcal{F}_{{\mathcal{S}^\lambda }}} $denote the$ {\mathcal{D}_X} $-module$ \mathcal{F}{ \otimes _{{\mathcal{O}_X}}}\mathcal{S}_X^\lambda $. We prove that a locally unipotent$\mathcal{D}_{X}$-module$\mathcal{F}$...

  13. References
    (pp. 149-152)
  14. Index of Notations
    (pp. 153-154)
  15. Index of Terminology
    (pp. 155-156)