## On certain types of $$p$$-adic representations of the Galois group of a local field; construction of a Barsott-Tate ring. (Sur certains types de représentations $$p$$-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate.)(French)Zbl 0544.14016

The paper under review is devoted to $$p$$-adic representations $$G\to \text{Aut}(V)$$ of the Galois group $$G=\text{Gal}(\bar K/K)$$ of local field $$K$$ of characteristic 0 with perfect residue field of positive characteristic $$p$$. Here $$V$$ is a finite-dimensional vector space over the field $${\mathbb{Q}}_ p$$ of $$p$$-adic numbers. The author gives a construction of the Barsotti-Tate ring announced in his previous paper [Astérisque 65, 3-80 (1979; Zbl 0429.14016)]. He also constructs a complete discrete valuation field $$B_{DR}$$ enjoying the following properties. $$B_{DR}$$ contains $$K$$ and the Galois group acts on $$B_{DR}$$. The residue field of $$B_{DR}$$ coincides with the completion $$C$$ of $$K$$. The valuation on $$B_{DR}$$ defines a filtration and the corresponding graded ring is the Barsotti-Tate ring. For any $$p$$-adic representation $$V$$ $$\underline D_{DR}(V)=(B_{DR}\otimes_{{\mathbb{Q}}_ p}V)^ G$$ is a finite- dimensional filtered vector space over $$K$$ and its dimension does not exceed the dimension of $$V$$; if the equality holds then $$V$$ is a Hodge-Tate module [J. Tate, Proc. Conf. local fields, NUFFIC Summer School Driebergen 1966, 158-183 (1967; Zbl 0157.27601)]. The most important examples of $$p$$-adic representations arised from $$p$$-adic étale cohomology groups $$V=H^ i_{et}(X)=H^ i_{et}(X\times \bar K,{\mathbb{Q}}_ p)$$ of smooth projective varieties $$X$$ over $$K$$. - Conjecture. The filtered $$K$$-space $$\underline{D}_{DR}(H^ i_{et}(X))$$ is canonically isomorphic to the de Rham cohomology group $$H^ i_{DR}(X)$$ with Hodge filtration. (Since dimensions of étale and de Rham cohomology groups coincide, $$H^ i_{et}(X)$$ ought to be a Hodge-Tate module.) - This conjecture is a deep refinement of the famous conjecture of Tate concerning existence of $$p$$-adic Hodge decomposition J. Tate, ibid.).
Reviewer: Yu.G.Zarhin

### MSC:

 14G99 Arithmetic problems in algebraic geometry; Diophantine geometry 14G15 Finite ground fields in algebraic geometry 14F40 de Rham cohomology and algebraic geometry 14G20 Local ground fields in algebraic geometry 11S25 Galois cohomology 14L05 Formal groups, $$p$$-divisible groups

### Citations:

Zbl 0429.14016; Zbl 0157.27601
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