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

Preprint Series # 917
Hattori, Shin On a ramification bound of semi-stable torsion representations over a local field. (2008);

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## 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 $r>1$.

Item Type: Preprint 30 ramification, Galois representation, integral p-adic Hodge theory 11-xx NUMBER THEORY 1868