Log-Growth Filtration and Frobenius Slope Filtration of $F$-Isocrystals at the Generic and Special Points

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 $\varphi$-$\nabla$-modules, both at the generic point and at the special point. We prove that a bounded $\varphi$-$\nabla$-module at the generic point is a direct sum of pure $\varphi$-$\nabla$-modules. By this splitting of Frobenius slope filtration for bounded modules we will introduce a filtration for $\varphi$-$\nabla$-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 $\varphi$-$\nabla$-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 \cite{CT09}. Using PBQ filtrations for $\varphi$-$\nabla$-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.

2010 Mathematics Subject Classification: Primary 12H25; Secondary 11S80.

Keywords and Phrases: $p$-adic differential equations, logarithmic growth, Frobenius slopes, Newton polygon.

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