\(L\) 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.


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
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