F-isocrystals on open varieties. Results and conjectures. (English) Zbl 0736.14004

The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. II, Prog. Math. 87, 219-248 (1990).
[For the entire collection see Zbl 0717.00009.]
P. Deligne proved the Weil conjectures in the context of \(\ell\)- adic étale cohomology [the signum \([De]\) refers to his paper ‘La conjecture de Weil. II’, Publ. Math., Inst. Hautes Étud. Sci. 52, 137- 252 (1980; Zbl 0456.14014) which, however, mistakingly fails in the list of references]. The present paper contains analogous results with \(\mathbb{Q}_ p\)-crystalline cohomology, for instance how the weights of a mixed convergent \(F\)-isocrystal (analogue of \(\mathbb{Q}_ \ell\)-adic étale sheaf) carry over to its cohomology. In order to treat open varieties, one has to consider differentials with logarithmic poles at infinity, etc., and to this end logarithmic analogues of notions of commutative algebra are given, from the notion of a log-étale map (reminding the very beginning of SGA) to that of logarithmic crystalline cohomology. Results include the crystalline Lefschetz trace formula and unipotency of crystalline monodromy.


14F30 \(p\)-adic cohomology, crystalline cohomology
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)