Hodge Tate Weights Research Articles

Une application de Corps des Normes

Let k be a perfect field of characteristic p > 0, K0 = Frac(W(k)), π a uniformizer in K0 and πn ∈ K 0 (n∈ N) such that π0 = π and πn+1p = πn. We write K∞ = ∪n∈N K0 (πn), H∞ = Gal (K0/ K∞ and G = Gal(K0/ K0). The main result of this paper is that the functor ‘restriction of the Galois action’ from the category of crystalline representations of G with Hodge–Tate weights in an interval of length ≤ p - 2 to the category of p-adic representations of H∞ is fully faithful and its essential image is stable by sub-object and quotient. The proof uses the comparison between two ways of building mod. p representations of H∞: one thanks to the norm field of K∞, the other thanks to some categories of ‘filtered’ modules with divided powers previously introduced by the author.