Sch\'emas en groupes et poids de Diamond-Serre

TL;DR

This paper corrects a key proposition in Gee's work on Galois representations, comparing group scheme extensions with crystalline representations using Breuil's theory, enhancing understanding of their weight structures.

Contribution

It provides a corrected proof of a proposition relating Galois representations from group schemes with descent data to crystalline representations with specified Hodge-Tate weights.

Findings

Corrected the statement and proof of Proposition 3.3.1 in Gee's preprint.

Established a comparison between Galois representations from group schemes and crystalline representations.

Utilized Breuil's theory to achieve the comparison.

Abstract

This note is a correction of (statement and proof of) proposition 3.3.1 of Toby Gee's preprint intitled *On the weights of mod p Hilbert modular forms*. The aim is to compare Galois representations arising from extensions of some group schemes (over the ring of integers of a p-adic field) endowed with a descent data, and extensions of some crystalline representations with given Hodge-Tate weights. The main tool of the proof is the theory of Breuil.