On some crystalline representations of GL$_{\mathsf 2}(\mathbb{Q}_{\mathsf p})$

Abstract

We show that the universal unitary completion of certain locally algebraic representation of $G:={GL}_2\left({\mathbb{Q}}_p\right)$ with $p>2$ is nonzero, topologically irreducible, admissible and corresponds to a $2$-dimensional crystalline representation with nonsemisimple Frobenius via the $p$-adic Langlands correspondence for $G$.