On the ramification of non-abelian Galois coverings of degree $p^3$

TL;DR

This paper explores the refined Swan conductor for certain l-adic sheaves over schemes in characteristic p, providing explicit formulas and proving integrality of the ramification measure.

Contribution

It offers an explicit expression for the refined Swan conductor in specific cases and demonstrates its integrality, advancing understanding of ramification in non-abelian Galois coverings.

Findings

Explicit formula for rsw in certain situations

Proof of integrality of the refined Swan conductor

Enhanced understanding of ramification in non-abelian coverings

Abstract

The refined Swan conductor is defined by K.\ Kato \cite{KK2}, and generalized by T.\ Saito \cite{wild}. In this part, we consider some smooth $l$-adic \'{e}tale sheaves of rank $p$ such that we can be define the $rsw$ following T.\ Saito, on some smooth dense open subscheme $U$ of a smooth separated scheme X of finite type over a perfect fields $\kappa$ of characteristic $p>0$. We give an explicit expression of $rsw(\mathcal{F})$ in some situation. As a consequence, we show that it is integral.