de Rham Representation

A de Rham representation of the absolute Galois group of a finite extension is a finite-dimensional representation whose de Rham period space has dimension equal to over . Using Fontaine's de Rham period field , this space is

The field is a filtered period field with a continuous Galois action and fixed field . Its filtration induces one on . A representation with coefficients in a finite extension of is tested after restriction of scalars.

The local de Rham condition occurs in the regular Fontaine-Mazur modularity formulation.