Polygones de Hodge, de Newton et de l'inertie mod\'er\'ee des repr\'esentations semi-stables

TL;DR

This paper introduces a new polygon associated with semi-stable p-adic Galois representations, showing it lies between the Hodge and Newton polygons under certain conditions, and explores an example related to crystalline representations.

Contribution

It defines a third polygon from the semi-simplification mod p of semi-stable representations and proves its position relative to existing polygons under specific conditions.

Findings

The new polygon lies above the Hodge polygon with the same endpoint.

The position of the new polygon is established under conditions on Hodge-Tate weights.

An example related to crystalline representations is analyzed.

Abstract

Let k be a perfect field, and K be a totally ramified extension of K_0 = Frac W(k) of degree e. To a semi-stable p-adic representation of G_K (the absolute Galois group of K), one can classicaly associate two polygons : the Hodge polygon et the Newton polygon. It is well known that the former lies below the latter, and that they have same endpoints. In this note, we introduce a third polygon gotten from the semi-simplification of the representation mod p, and, under some conditions on Hodge-Tate weights, we prove that it lies above the Hodge polygon again with same endpoint. We finally examine one exemple associated to a crystalline representation.