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

Fontaine, Jean-Marc

Ann. Math. (2) 115, 529-577 (1982).

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

Keywords:

Barsotti-Tate ring; p-adic étale cohomology groups; de Rham cohomology; Hodge-Tate module

Citations:

Zbl 0429.14016; Zbl 0157.27601