TY - JOUR

T1 - Log-growth filtration and frobenius slope filtration of F-isocrystals at the generic and special points

AU - Chiarellotto, Bruno

AU - Tsuzuki, Nobuo

PY - 2011

Y1 - 2011

N2 - 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 φ-∇-modules, both at the generic point and at the special point. We prove that a bounded φ-∇-module at the generic point is a direct sum of pure φ-∇-modules. By this splitting of Frobenius slope filtration for bounded modules we will introduce a filtration for φ-∇-modules (PBQ filtration). We solve our conjectures of comparison of the log-growth filtra-tion and the Frobenius slope filtration at the special point for particular φ-∇-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 [CT09]. Using PBQ filtrations for φ-∇-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.

AB - 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 φ-∇-modules, both at the generic point and at the special point. We prove that a bounded φ-∇-module at the generic point is a direct sum of pure φ-∇-modules. By this splitting of Frobenius slope filtration for bounded modules we will introduce a filtration for φ-∇-modules (PBQ filtration). We solve our conjectures of comparison of the log-growth filtra-tion and the Frobenius slope filtration at the special point for particular φ-∇-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 [CT09]. Using PBQ filtrations for φ-∇-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.

KW - Frobenius slopes

KW - Logarithmic growth

KW - Newton polygon

KW - p-adic differential equations

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M3 - Article

AN - SCOPUS:79957615278

SN - 1431-0635

VL - 16

SP - 33

EP - 69

JO - Documenta Mathematica

JF - Documenta Mathematica

IS - 1

ER -