On a ramification bound of torsion semi-stable representations over a local field

Let p be a rational prime, k be a perfect field of characteristic p, W=W(k) be the ring of Witt vectors, K be a finite totally ramified extension of Frac(W) of degree e and r be a non-negative integer satisfying r<p−1. In this paper, we prove the upper numbering ramification group for j>u(K,r,n) acts trivially on the pn-torsion semi-stable GK-representations with Hodge-Tate weights in {0,…,r}, where u(K,0,n)=0, u(K,1,n)=1+e(n+1/(p−1)) and u(K,r,n)=1−p−n+e(n+r/(p−1)) for 1<r<p−1.