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Vendredi 3 octobre 2008


Recent Progress in Arithmetic D-modules theory

on Friday 3rd October 2008, at Institut de Recherche Mathematique Avancée, Strasbourg.

The aim of this conference is the exposition of recent progress on arithmetic D-modules theory. We will focus in particular on the work of Caro and Tsuzuki, which, combined with the semi-stable theorem of Kedlaya, proves that the category of overholonomic arithmetic D-modules with Frobenius, contructed by Caro, is stable by the 6 Grothendieck cohomological operations.

Organizers: C.Noot-Huyghe and A.Marmora

  • Vendredi 3 octobre 2008

  • 08:00 - 08:30

    Coffee and reception of participants
  • 08:30 - 09:00

    Pierre Berthelot, Univ Rennes 1

    An introduction to finiteness conditions in arithmetic D-module theory

    In this introductory talk, we will review the main initeness conditions introduced in the theory of arithmetic D-modules, and we will discuss their stability properties under the basic cohomological operations.
  • 09:05 - 10:05

    Kiran Kedlaya, MIT

    Semistable reduction for overconvergent F-isocrystals

    Let k be a field of characteristic p > 0, and let X be a k-variety. The alterations theorem of de Jong states that we can find a proper generically finite morphism from some smooth variety Y to X, such that Y admits a smooth compactification overline{Y} with boundary divisor having strict normal crossings. Now let E be an overconvergent F-isocrystal on X. In its simplest form, the semistable reduction problem (conjecture of Shiho) is to prove that we can choose Y and overline{Y} as above so that the pullback of E to Y admits a logarithmic extension to overline{Y} with nilpotent residues on the boundary. One can also state a refined conjecture applying to partially overconvergent F-isocrystals. We will give a detailed account of our proof of the conjecture of shiho. We start by recallying the Zariski-Nagata purity theorem for overconvergent F-isocrystals and logarithmic extensions. We then formulate the problem of local semistable reduction, which allows us to work in a neighborhood of a single valuation on the function field of an irreducible X. We reduce this problem to the case where the valuation has height 1 and residual transcendence degree 0, then set up an induction on a certain numerical invariant (the corank). The base case is that of an Abhyankar valuation; using the strong form of local uniformization available for such varieties, we can proceed much as the one-dimensional case (i.e., the conjecture of Crew). For the induction, we fiber X in curves over a variety X_0 of dimension one lower, and view our given valuation as a point in a Berkovich affine line (over the completion of the function field of X_0). We then make an intricate calculation (using our quantitative refinement of Christol-Mebkhout decomposition theory) to how that we can reduce from our given valuation to another point in this Berkovich line corresponding to a valuation of lower corank.
  • 10:20 - 11:20

    Nobuo Tsuzuki, Tohoku Univ.

    On the overholonomicity of overconvergent F-isocrystals on smooth varieties I. Comparison between log-rigid and rigid cohomologies

    This is the first part of the joint work with D.Caro. The goal is to prove that an overconvergent F-isocrystal on a smooth variety is verholonomic arithmetic D-modules. In this talk we discuss a relative version of comparison between log-rigid cohomology and rigid cohomology with oefficients in log-isocrystals. This is one of the keys to prove our main result which will be explained in the talk of D. Caro. This type of comparison between log-rigid cohomology and rigid cohomology was first studied by Baldassarri-Chiarellotto and by many others. We assume that an O_X^dagger-coherent module with connection is locally free only in the rigid analytic side. Hence it might be locally projective in the formal scheme side in general. By this reason we can not apply Christol's transfer theorem, generalized by Baldassarri-Chiarellotto, directly. We will explain the technical part of our proof.
  • 11:20 - 11:45

    Coffee break.
  • 11:45 - 12:45

    Daniel Caro, Univ. Caen

    On the overholonomicity of overconvergent F-isocrystals on smooth varieties II

    This is the second part of the joint work with N. Tsuzuki. The goal is to prove that an overconvergent F-isocrystal on a smooth variety is an overholonomic arithmetic D-modules. In this talk we will give some explanations on this result by using the relative version of comparison between log-rigid cohomology and rigid cohomology with coefficients in log-isocrystals explained by N. Tsuzuki in the previous talk. We will give some applications of this result in theory of arithmetic D-modules of Berthelot.
  • 12:45 - 14:30

  • 14:30 - 15:30

    Richard Crew, Florida Univ.

    Rings of p-adic differential operators on tubes

  • 15:45 - 16:45

    Takeshi Tsuji, Tokyo Univ.

    Nearby cycles and D-modules of log schemes in characteristic p>0

    For a semi-stable scheme over a complete discrete valuation ring of mixed characteristic (0,p), I define nearby cycles as a D-module and study its properties. P. Berthelot recently proved that the cohomology of the nearby cycles coincides with Hyodo-Kato cohomology (log crystalline cohomology). Combining his result, we obtain a new construction of a weight spectral sequence, which was constructed by A. Mokrane using de Rham-Witt complex.