Torsion p-adic Galois representations and a conjecture of Fontaine

Abstract

Let p be a prime, K a finite extension of Q p and T a finite free Z p -representation of Gal ( K ¯ / K ) . We prove that T ⊗ Z p Q p is semi-stable (resp. crystalline) with Hodge–Tate weights in { 0 , … , r } if and only if, for all n, T / p n T is torsion semi-stable (resp. crystalline) with Hodge–Tate weights in { 0 , … , r } .