On some crystalline representations of $GL_2(Q_p)$

TL;DR

This paper demonstrates that certain locally algebraic representations of GL_2(Q_p) have a universal unitary completion that is non-zero, irreducible, admissible, and corresponds to a specific 2-dimensional crystalline Galois representation.

Contribution

It establishes a link between locally algebraic representations of GL_2(Q_p) and 2-dimensional crystalline Galois representations with non-semisimple Frobenius, expanding the p-adic Langlands correspondence.

Findings

Universal unitary completion is non-zero and irreducible.

Corresponds to a 2-dimensional crystalline representation.

Applicable for p > 2.

Abstract

We show that the universal unitary completion of certain locally algebraic representation of $G:=\GL_2(\Qp)$ with $p>2$ is non-zero, topologically irreducible, admissible and corresponds to a 2-dimensional crystalline representation with non-semisimple Frobenius via the $p$-adic Langlands correspondence for $G$.