## 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 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.

Original language | English |
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Pages (from-to) | 33-69 |

Number of pages | 37 |

Journal | Documenta Mathematica |

Volume | 16 |

Issue number | 1 |

Publication status | Published - 2011 Jun 1 |

## Keywords

- Frobenius slopes
- Logarithmic growth
- Newton polygon
- p-adic differential equations

## ASJC Scopus subject areas

- Mathematics(all)