## Abstract

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 $r<p-1$. Let $V$ be a semi-stable $p$-adic
$G_K$-representation with Hodge-Tate weights in $\{0,\dots,r\}$. In
this paper, we prove the upper numbering ramification group
$G_{K}^{(j)}$ for $j>u(K,r,n)$ acts trivially on the mod $p^n$
representations associated to $V$, where $u(K,0,n)=0$,
$u(K,1,n)=1+e(n+1/(p-1))$ and
$u(K,r,n)=1-p^{-n}e(K(\zeta_p)/K)^{-1}+e(n+r/(p-1))$ for $r>1$.