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F-iscocrystals and p-adic representations. (English) Zbl 0639.14011
Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, part 2, Proc. Symp. Pure Math. 46, 111-138 (1987).
[For the entire collection see Zbl 0626.00011.]
This paper discusses the relation between p-adic representations of the fundamental group of a variety in characteristic $$p>0$$ and unit-root F- isocrystals. First it is shown that the category of K-representations of the fundamental group (where K is a p-adic field) is equivalent to the category of convergent unit-root F-isocrystals (it is now undestood that the technical condition of convergence is necessary to impose on isocrystals in order to get the right notion) through the obvious functors. Then the author discusses to which condition on the p-adic representation the condition of overconvergence corresponds. Overconvergent F-isocrystals seem to correspond to $$\ell$$-adic representations and unit-root overconvergent F-isocrystals should be pure. In accordance with this philosophy it is proved that on a curve representations of the fundamental group with finite monodromy at all points at infinity give rise to overconvergent F-isocrystals and it is shown that the converse is true for rank 1 F-isocrystals. The proofs are too technical to enable a discussion of them here.
Reviewer: T.Ekedahl

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14F35 Homotopy theory and fundamental groups in algebraic geometry
Zbl 0626.00011