CONSTRUCTION OF P-ADIC SEMI-STABLE REPRES ENTATIONS

The aim of this work is to generalize to the semi-stable setting the F onraine-Laffaille crystalline theory. Let k be a perfect field of cara cteristic p > 0, W the Witt vectors in k, K-0 = Fr(W) and S = W<u> the p-adic completion of the P.D. polynomial algebra. We define a categor y of S-modules with p-torsion and show it is abelian and has the same simple objects as Fontaine-Laffaille's <(MF)under bar>(f.p-2)(tvr) cat egory. We define an exact and fully faithfull functor from this catego ry to the category of p-adic representations of Gal((K) over bar(0)/K- 0) of finite length. We define ''strongly divisible'' free S-modules a nd show how one can build p-adic semi-stable representations with them , using the previous torsion theory. By finding strongly divisible S-m odules in dimension 2, we build all the dimension 2 p-adic semi-stable representations with differences in Hodge-Tate weights not exceeding p - 2. (C) Elsevier, Paris.