Ramification groups of Galois extensions over local fields of positive characteristic with Galois group isomorphic to the group of unitriangular matrices

Abstract

We study the ramification groups of finite Galois extensions $L/K$ of a complete discrete valuation field $K$ of equal characteristic $p>0$ with perfect residue field and Galois group isomorphic to the group of unitriangular matrices $U{T}_n({\mathbb{F}}_p)$ over ${\mathbb{F}}_p$. We show that the upper ramification breaks can be expressed as a linear function of the valuation of the entries of a matrix directly constructed from the coefficients of a defining equation of the extension. This allows us to compute the ramification groups without using any elements of $L-K$.