ON A RAMIFICATION BOUND OF THE MOD p REPRESENTATIONS OF THE p-ADIC ÉTALE COHOMOLOGY GROUPS OVER A LOCAL FIELD

For a rational prime p > 2, let 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 an integer satisfying r < p − 1. In this paper, we prove the upper numbering ramification group G (j) K for j > 1+ e(1+ r/(p− 1)) acts trivially on the mod p representation associated to the p-adic étale cohomology group H ét(XK̄ , Qp) for a proper smooth scheme XK over K with semi-stable reduction.