Pentes en cohomologie rigide et F -isocristaux unipotents

Chiarellotto, Bruno; Stum, Bernard Le

doi:10.1007/s002290050212

Cited by 25 publications

(20 citation statements)

References 9 publications

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“…By a theorem of Chiarellotto and Le Stum ( [8], Proposition 2.4.1), this implies that the comparison morphism y in De®nition 1.8 is an isomorphism. The only thing preventing the object Log n from splitting is the connection.…”

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confidence: 81%

Rigid syntomic cohomology and p-adic polylogarithms

Bannai

2000

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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The main purpose of this paper is to construct the p-adic realization of the classical polylogarithm following the method of Beilinson and Deligne as explained by Huber and Wildeshaus. A simplicial construction of the p-adic polylogarithm was previously obtained by Somekawa. In this paper, we will give a sheaf theoretic interpretation of this construction. In particular, we will give an interpretation of the p-adic polylogarithm as an object in the p-adic analogue of the category of variation of mixed Hodge structures. We will also calculate the restriction of this object to torsion points, and will prove a result which is compatible with the results of Gros-Kurihara, Gros and Somekawa. IntroductionIn the paper [23], M. Somekawa constructed the p-adic polylogarithm in the syntomic cohomology of a simplicial scheme over the projective line minus three points. This is a p-adic analogue of the elements in absolute Hodge and l-adic cohomology de®ned by Beilinson [2] and is the image by the syntomic regulator of the motivic polylogarithm in motivic cohomology. The purpose of this paper is to develop a theory of p-adic mixed sheaves which is su½cient to give a sheaf theoretic interpretation of the construction by Somekawa. In particular, as in the classical case, the p-adic polylogarithm has an interpretation as an object in the p-adic analogue of the category of variation of mixed Hodge structures on the projective line minus three points. In Theorem 2, which is our main result, we will explicitly determine the shape of the p-adic polylogarithm.As a corollary, we will show that the specialization of this object to the roots of unity can be expressed by the p-adic polylogarithm function de®ned by Coleman. This result is compatible with the result of Somekawa, and is compatible with the results of Gros-Kurihara [14] and Gros [15] on the calculation of the image of the Beilinson element with respect to the syntomic regulator. This gives evidence that the p-adic formalism which we develop in this paper is potentially a powerful tool in the calculation of the image by the syntomic regulator of elements in motivic cohomology.

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confidence: 81%

Rigid syntomic cohomology and p-adic polylogarithms

Bannai

2000

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…The upshot of the previous section is that we now have an affine group scheme π rig 1 (X/S, p) over the Tannakian category F-Isoc † (S/K) whose fibre (ignoring Frobenius structures) over any closed point s is the usual rigid fundamental group π rig 1 (X s , p s ) as defined by Chiarellotto and le Stum in [CLS99a]. In Chapter II of [Chi98], Chiarellotto defines a Frobenius isomorphism F * : π rig 1 (X s , p s ) ∼ → π rig 1 (X s , p s ), by using the fact that Frobenius pullback induces an automorphism of the category N Isoc † (X s /K).…”

Section: Extension To Proper Curves Frobenius Structuresmentioning

confidence: 99%

Relative fundamental groups and rational points

Lazda¹

2015

Rend. Sem. Mat. Univ. Padova

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ABSTRACT. In this paper we define a relative rigid fundamental group, which associates to a section p of a smooth and proper morphism f : X → S in characteristic p, with dim S = 1, a Hopf algebra in the ind-category of overconvergent F-isocrystals on S. We prove a base change property, which says that the fibres of this object are the Hopf algebras of the rigid fundamental groups of the fibres of f . We explain how to use this theory to define period maps as Kim does for varieties over number fields, and show in certain cases that the targets of these maps can be interpreted as varieties. CONTENTS

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“…This includes work of Chiarellotto and Le Stum [5], Kim and Hain [19,20], Shiho [35,36], and Vologodsky [42] among others. Thus the main new contribution of this paper is to extend some of the above work to a theory with non-unipotent coefficients, though we also obtain some new results about rational homotopy theory.…”

Section: Acknowledgementsmentioning

confidence: 97%

F-isocrystals and homotopy types

Olsson

2007

Journal of Pure and Applied Algebra

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We study a positive characteristic analogue of the nonabelian Hodge structure constructed by Katzarkov, Pantev, and Toen on the homotopy type of a complex algebraic variety. Given a proper smooth scheme X over a perfect field of characteristic p and a Tannakian category C of isocrystals on X , we construct an object X C in a suitable homotopy category of simplicial presheaves whose category of local systems is equivalent to C in a manner compatible with cohomology. We then study F-isocrystal structure on these simplicial presheaves. As applications of the theory, we prove a p-adic analogue of a result of Hain on relative Malcev completions, a generalization to the level of homotopy types of a theorem of Katz relating p-adicétale local systems and Fisocrystals, as well as a p-adic version of the formality theorem in homotopy theory. We have also included a new proof based on reduction modulo p of the formality theorem for complex algebraic varieties.

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Pentes en cohomologie rigide et F -isocristaux unipotents

Cited by 25 publications

References 9 publications

Rigid syntomic cohomology and p-adic polylogarithms

Rigid syntomic cohomology and p-adic polylogarithms

Relative fundamental groups and rational points

F-isocrystals and homotopy types

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