$$L$$ functions associated to overconvergent $$F$$-isocrystals. II: Unit roots and poles. (Fonctions $$L$$ associées aux $$F$$-isocristaux surconvergents. II: Zéros et pôles unités.)(French)Zbl 0911.14011

[For part I of this paper see Math. Ann., 296, No. 3, 557-576 (1993; Zbl 0789.14015).]
It is given a construction of the Artin-Schreier exact sequence on the syntomic site of a scheme, not necessarily smooth and proper. Under the last conditions the construction was done by J. M. Fontaine and W. Messing [in: Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata 1985, Contemp. Math. 67, 179-207 (1987; Zbl 0632.14016)]. N. Katz has conjectured that $$p$$-adic unit roots and poles of the $$L$$-functions associated to a representation of the fundamental group of a variety over $$\mathbb{F}_q$$ can be expressed in terms of $$p$$-adic étale cohomologies. The authors prove this conjecture as well as many interesting corollaries.

MSC:

 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14F35 Homotopy theory and fundamental groups in algebraic geometry 14F30 $$p$$-adic cohomology, crystalline cohomology

Citations:

Zbl 0632.14016; Zbl 0789.14015
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