zbMATH — the first resource for mathematics

A note on p-adic étale cohomology in the semi-stable reduction case. (English) Zbl 0619.14013
This paper studies p-adic étale cohomology of varieties over a p-adic field (i.e. a complete discrete valuation field of mixed characteristics (0,p)) under the assumption that the varieties have semi-stable reduction. This assumption is rather general because every variety is conjectured to admit a semi-stable model after enlarging the base field. Results of S. Bloch and K. Kato [Publ. Math., Inst. Hautes Étud. Sci. 63, 107-152 (1986; Zbl 0613.14017)] are generalized to the semi-stable reduction case (they considered the good reduction case): The structure of the sheaf of p-adic vanishing cycles are described in terms of the ”modified differential modules”. (As the special fiber may be singular, the usual differential modules do not work well.) And a global result is obtained under the ”ordinary” assumption. Namely the p-adic étale cohomology groups of varieties with ”ordinary semi-stable reduction” has a filtration whose subquotients are unramified representations of the absolute Galois group of the base field with some Tate twist.

14F30 \(p\)-adic cohomology, crystalline cohomology
14C99 Cycles and subschemes
Full Text: DOI EuDML
[1] [BK] Bloch, S., Kato, K.:p-adic etale cohomology. Publ. Math. IHES63, 107-152 (1986) · Zbl 0613.14017
[2] [D] Deligne, P.: Equations diff?rentielles ? points singuliers r?guliers (Lect. Notes Math., vol. 163) Berlin Heidelberg New York: Springer 1970
[3] [F] Faltings G.:p-adic Hodge theory (Preprint 1985)
[4] [FM] Fontaine, J.-M., Messing, W.:p-adic periods andp-adic etale cohomology. Contemporary Math.67, 179-209 (1987)
[5] [IR] Illusie, L., Raynaud, M.: Les suites spectrales associ?es au complexe de de Rham-Witt. Publ. Math. IHES57, 73-212 (1983) · Zbl 0538.14012
[6] [K] Kurihara, A.: Construction ofp-adic unit balls and the Hirzebruch proportionality. Am. J. Math.102, 565-648 (1980) · Zbl 0498.14011 · doi:10.2307/2374116
[7] [Me] Messing, W.: The crystals associated to Barsotti-Tate groups: with applications to abelian schemes (Lect. Notes Math., vol. 264) Berlin Heidelberg New York: Springer 1972 · Zbl 0243.14013
[8] [Mum] Mumford, D.: An analytic construction of degenerating curves over complete local rings. Compos. Math.24, 129-174 (1972) · Zbl 0228.14011
[9] [Mus] Mustafin, G.A.: Nonarchimedean uniformization. Math. USSR Sbornik34, 187-214 (1978) · Zbl 0411.14006 · doi:10.1070/SM1978v034n02ABEH001156
[10] [S] Serre, J.-P.: Abelianl-adic representations and elliptic curves. New York Amsterdam: Benjamin 1968
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.