On some modular representations of the Borel subgroup of GL_2(Q_p)

TL;DR

This paper explicitly computes certain modular representations of the Borel subgroup of GL_2(Q_p) associated with Galois representations, linking Colmez's recipe to Breuil's correspondence for irreducible 2-dimensional cases.

Contribution

It provides explicit calculations of Omega(W) and establishes its connection to supersingular representations in Breuil's correspondence for irreducible 2-dimensional W.

Findings

Omega(W) explicitly computed for given W

Omega(W) matches supersingular representations in Breuil's correspondence

Results deepen understanding of modular representations of Borel subgroups

Abstract

Colmez has given a recipe to associate a smooth modular representation Omega(W) of the Borel subgroup of GL_2(Q_p) to a F_p^bar-representation W of Gal(Qp^bar/Qp) by using Fontaine's theory of (phi,Gamma)-modules. We compute Omega(W) explicitly and we prove that if W is irreducible and dim(W)=2, then Omega(W) is the restriction to the Borel subgroup of GL_2(Q_p) of the supersingular representation associated to W in Breuil's correspondence.