Supersingular representations of GL_2(Q_p) and (phi,Gamma)-modules

TL;DR

This paper provides a direct proof linking two-dimensional irreducible Galois representations over F_p^bar to supersingular representations of GL_2(Q_p) via Colmez's functor, clarifying their correspondence.

Contribution

It offers a direct proof of the relationship between Galois representations and supersingular representations of GL_2(Q_p) using Colmez's functor.

Findings

Colmez's functor applied to irreducible Galois representations yields supersingular representations

Establishes a direct link between Galois representations and supersingular representations

Clarifies the structure of supersingular representations in the p-adic setting

Abstract

The purpose of this note is to give a direct proof of the fact that if one applies Colmez' functor to a two dimensional irreducible F_p^bar-representation of Gal(Q_p^bar/Q_p), one gets the restriction to the Borel subgroup of GL_2(Q_p) of a supersingular representation.