For a $\nabla$-module M over the ring $K[[x]}_0$ of bounded functions over a p-adic local field K we define the notion of special and generic log-growth filtrations oil the base of the power series development of the solutions and horizontal sections. Moreover, if M also admits a Frobenius structure then it is endowed with generic and special Frobenius slope filtrations. We will show that in the case of M a $\phi-\nabla$-module of rank 2, the Frobenius polygon for M and the log-growth polygon for its dual, $M^\star$, coincide, this is proved by showing explicit relationships between the filtrations. This will lead us to formulate some conjectural links between the behaviours of the filtrations arising from the log-growth and Frobenius structures of the differential module. This coincidence between the two polygons was only known for the hypergeometric cases by Dwork
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 under tensor products. So, the overholonomicity gives a p-adic cohomology stable under Grothendieck's cohomological operations.
We discuss Tate-type problems for F-isocrystals, that is, the full faithfulness of the natural restriction functors between categories of overconvergent F-isocrystals on schemes of positive characteristic. We prove it in the cases of unit-root F-isocrystals. Using this result, we prove that an overconvergent unit-root F-isocrystal has a finite monodromy.
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