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

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 . In this paper, we prove the upper numbering ramification group G K ( j ) for j > u ( K , r , n ) acts trivially on the p n -torsion semi-stable G K -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 .