Compositio Mathematica

Une application de Corps des Normes

a1 Mathématiques, Bât. 425, U.R.A. 752 du C.N.R.S., Université Paris-Sud, F-91405 Orsay cedex, France; e-mail:

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Let k be a perfect field of characteristic p > 0, K$_0$ = Frac(W(k)), π a uniformizer in K$_0$ and π$_n$ [set membership] K $_0$ (n[set membership] N) such that π$_0$ = π and π$^p$$_n$ $_ + 1$ = π$_n$. We write K$_∞$ = [cup B: union or logical sum]$_n$ $_∈N$ K$_0$ (π$_n$), H$_∞$ = Gal (K$_0$/ K$_∞$ and G = Gal(K$_0$/ K$_0$). 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 [less-than-or-eq, slant] 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.

Key Words: Mod. p Galois representation; norm field; crystalline representation..