Semi-simplified modulo p of semi-stable representations: an algorithmic approach

The aim of this paper is to present an algorithm the complexity of which is polynomial to compute the semi-simplified modulo $p$ of a semi-stable $\Q_p$-representation of the absolute Galois group of a $p$-adic field (\emph{i.e.} a finite extension of $\Q_p$). In order to do so, we use abundantly the $p$-adic Hodge theory and, in particular, the Breuil-Kisin modules theory.