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

TL;DR

This paper establishes bounds on the ramification groups acting trivially on torsion semi-stable Galois representations over local fields, linking ramification theory with p-adic Hodge theory.

Contribution

It provides explicit upper bounds for ramification groups acting trivially on torsion semi-stable representations with specified Hodge-Tate weights over local fields.

Findings

Derived explicit formulas for ramification bounds u(K,r,n).

Proved triviality of ramification groups above these bounds.

Connected ramification bounds with semi-stable Galois representations.

Abstract

For a rational prime p, let k be a perfect field of characteristic p, K be a finite totally ramified extension of Frac(W(k)) of degree e and r be a non-negative integer satisfying r<p-1. In this article, we prove the upper numbering ramification group G^(j) for j>u(K,r,n) acts trivially on the p^n-torsion semi-stable G_K-representations with the 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 r>1.