On the rank of the multivariable $(\varphi,\mathcal{O}_K^{\times})$-modules associated to mod $p$ representations of $\operatorname{GL}_2(K)$

Abstract

Let $p$ be a prime number, $K$ a finite unramified extension of $\mathbb{Q}_p$ and $\mathbb{F}$ a finite extension of $\mathbb{F}_p$. For $\pi$ an admissible smooth representation of $\operatorname{GL}_2(K)$ over $\mathbb{F}$ satisfying certain multiplicity-one properties, we compute the rank of the associated \'etale $(\varphi,\mathcal{O}_K^{\times})$-module $D_A(\pi)$ defined by Breuil-Herzig-Hu-Morra-Schraen, extending their results.