Geodesic
Log-growth filtration and Frobenius slope filtration of $F$-isocrystals at the generic and special points
Documenta mathematica, Tome 16 (2011), pp. 33-69.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 citeCT09. 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.
Classification : 12H25, 11S80
Keywords: $p$-adic differential equations, logarithmic growth, Frobenius slopes, Newton polygon 
@article{DOCMA_2011__16__a32,
     author = {Chiarellotto, B. and Tsuzuki, N.},
     title = {Log-growth filtration and {Frobenius} slope filtration of $F$-isocrystals at the generic and special points},
     journal = {Documenta mathematica},
     pages = {33--69},
     publisher = {mathdoc},
     volume = {16},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2011__16__a32/}
}
TY  - JOUR
AU  - Chiarellotto, B.
AU  - Tsuzuki, N.
TI  - Log-growth filtration and Frobenius slope filtration of $F$-isocrystals at the generic and special points
JO  - Documenta mathematica
PY  - 2011
SP  - 33
EP  - 69
VL  - 16
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DOCMA_2011__16__a32/
LA  - en
ID  - DOCMA_2011__16__a32
ER  - 
%0 Journal Article
%A Chiarellotto, B.
%A Tsuzuki, N.
%T Log-growth filtration and Frobenius slope filtration of $F$-isocrystals at the generic and special points
%J Documenta mathematica
%D 2011
%P 33-69
%V 16
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DOCMA_2011__16__a32/
%G en
%F DOCMA_2011__16__a32
Chiarellotto, B.; Tsuzuki, N. Log-growth filtration and Frobenius slope filtration of $F$-isocrystals at the generic and special points. Documenta mathematica, Tome 16 (2011), pp. 33-69. http://geodesic.mathdoc.fr/item/DOCMA_2011__16__a32/