Some Bounds for ramification of $p^n$-torsion semi-stable representations

Abstract

Let p be an odd prime, K a finite extension of Q_p, G=Gal(\bar K/K) the Galois group and e=e(K/Q_p) the ramification index. Suppose T is a p^n torsion representation such that T is isomorphic to a quotient of two G-stable Z_p-lattices in a semi-stable representation with Hodge-Tate weights in {0,...,r}. We prove that there exists a constant μexplicitly depending on n, e and r such that the upper numbering ramification group G^{(μ)} acts on T trivially.