Admissible unitary completions of locally $Q_p$-rational representations of $GL_2(F)$

TL;DR

This paper constructs admissible unitary completions of certain $GL_2(F)$ representations over $L$-vector spaces and applies these results to lift 2-dimensional mod $p$ Galois representations to crystabelline ones with specified Hodge-Tate weights.

Contribution

It introduces a method to construct admissible unitary completions of $GL_2(F)$ representations and applies existing results to lift mod $p$ Galois representations in the case $F=Q_p$.

Findings

Constructed admissible unitary completions for certain $GL_2(F)$ representations.

Obtained results on lifting mod $p$ Galois representations to crystabelline representations.

Extended known lifting results to more general local fields.

Abstract

Let $F$ be a finite extension of $Q_p$, $p>2$. We construct admissible unitary completions of certain representations of $GL_2(F)$ on $L$-vector spaces, where $L$ is a finite extension of $F$. When $F=Q_p$ using the results of Berger, Breuil and Colmez we obtain some results about lifting 2-dimensional mod $p$ representations of the absolute Galois group of $Q_p$ to crystabelline representations with given Hodge-Tate weights.