Log-growth filtration and Frobenius slope filtration of $F$-isocrystals at the generic and special points

Abstract

We study, locally on a curve of characteristic $p>0$, the relation between the log-growth filtration and the Frobenius slope filtration for $F$-isocrystals, which we will indicate as $\phi -\nabla$-modules, both at the generic point and at the special point. We prove that a bounded $\phi -\nabla$-module at the generic point is a direct sum of pure $\phi -\nabla$-modules. By this splitting of Frobenius slope filtration for bounded modules we will introduce a filtration for $\phi -\nabla$-modules (PBQ filtration). We solve our conjectures of comparison of the log-growth filtration and the Frobenius slope filtration at the special point for particular $\phi -\nabla$-modules (HPBQ modules). Moreover we prove the analogous comparison conjecture for PBQ modules at the generic point. These comparison conjectures were stated in our previous work citeCT09. Using PBQ filtrations for $\phi -\nabla$-modules, we conclude that our conjecture of comparison of the log-growth filtration and the Frobenius slope filtration at the special point implies Dwork's conjecture, that is, the special log-growth polygon is above the generic log-growth polygon including the coincidence of both end points.