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

TL;DR

This paper introduces a polynomial-time algorithm for computing the semi-simplified modulo p of semi-stable p-adic Galois representations, leveraging p-adic Hodge theory and Breuil-Kisin modules.

Contribution

It provides the first efficient algorithm for this computation, integrating p-adic Hodge theory with Breuil-Kisin modules.

Findings

Algorithm runs in polynomial time

Effective computation of semi-simplified modulo p representations

Bridges p-adic Hodge theory with computational methods

Abstract

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.