Related papers: Unipotency and semistability of overconvergent F-c…

Building on our previous papers (math.AG/0102173, math.AG/0105244, math.AG/0106192) we prove that every overconvergent F-isocrystal over k((t)) is quasi-unipotent (in the sense of Crew), for k a field of positive characteristic.

R. Crew conjectured that every overconvergent F-isocrystal over k((t)) (k a field of positive characteristic) is quasi-unipotent (equivalently, potentially semistable), and so has ``generic'' and ``special'' Newton polygons. It is easy to…

We complete our proof that given an overconvergent F-isocrystal on a variety over a field of positive characteristic, one can pull back along a suitable generically finite cover to obtain an isocrystal which extends, with logarithmic…

We introduce a valuation-theoretic approach to the problem of semistable reduction (i.e., existence of logarithmic extensions on suitable covers) of overconvergent isocrystals with Frobenius structure. The key tool is the quasicompactness…

Let X be a smooth variety over a field of positive characteristic, and let E be an overconvergent isocrystal on X. We establish a criterion for the existence of a "canonical logarithmic extension" of E to a good compactification of X. In…

Given a smooth affine curve X over a field k of positive characteristic, and an overconvergent F-isocrystal on X, we prove after replacing k by a finite purely inseparable extension, there exists a finite separable cover of X, the pullback…

Tsuzuki has conjectured that for crystals with Frobenius and connection over a local field k((t)), the embedding of the category of overconvergent crystals into the category of convergent crystals is fully faithful. We prove Tsuzuki's…

We show that a semisimple overconvergent "absolutely unit-root" F-isocrystal on a geometrically connected smooth variety over a finite field becomes constant over a finite covering.

In this article we prove Crew's parabolicity conjecture of $F$-isocrystals. For this purpose, we introduce and study the notion of $\dagger$-hull of a sub-$F$-isocrystal. On the way, we prove a new Lefschetz theorem for overconvergent…

In the proof of Crew's parabolicity conjecture, we established a key property concerning the slopes of $\dagger$-hulls of $F$-isocrystals, extending a result of Tsuzuki. This article presents an alternative proof of this theorem for a…

Let $\mathcal{V}$ be a mixed characteristic complete discrete valuation ring, $k$ its residual field, $\mathcal{P}$ a proper smooth formal scheme over $\mathcal{V}$, $P$ its special fiber, $T$ a divisor of $P$, $U:=P\setminus T$, $Y$ a…

We resolve the local semistable reduction problem for overconvergent F-isocrystals at monomial valuations (Abhyankar valuations of height 1 and residue transcendence degree 0). We first introduce a higher-dimensional analogue of the generic…

In the framework of Berthelot's theory of arithmetic $\mathcal{D}$-modules, we introduce the notion of arithmetic $\mathcal{D}$-modules having potentially-unipotent monodromy. For example, from Kedlaya's semistable reduction theorem,…

Let $X$ be a smooth scheme over a finite field. It is conjectured that a convergent $F$-isocrystal on $X$ is overconvergent if its restriction to every curve contained in $X$ is overconvergent. Using the theory of \'etale and crystalline…

We show a Lefschetz theorem for irreducible overconvergent $F$-isocrystals on smooth varieties defined over a finite field. We derive several consequences from it.

We prove that cohomological descent holds for finitely presented crystals on the overconvergent site with respect to proper or fppf hypercovers.

Let X be a smooth variety over a field of characteristic p>0. We prove that the forgetful functor from the category of overconvergent F-isocrystals on X to the category of convergent F-isocrystals is fully faithful. The argument uses the…

We prove the overholonomicity of overconvergent $F$-isocrystals over smooth varieties. This implies that the notions of overholonomicity and devissability in overconvergent $F$-isocrystals are equivalent. Then the overholonomicity is stable…

We prove the semistable reduction theorem for $\mathcal{E}^{\dag}_K$-valued and $K$-valued overconvergent $F$-isocrystals over $k((t))$-varieties which were introduced by Lazda and P\'{a}l. As an application, we prove the finite…

In this paper, we prove the generic overconvergence of relative rigid cohomology with coefficient, by using the semistable reduction conjecture for overconvergent $F$-isocrystals (which is recently shown by Kedlaya).