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

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 ${\mathrm{GL}}_2(K)$ over $\mathbb{F}$ satisfying certain multiplicity-one properties, we compute the rank of the associated étale $(\phi ,{\mathcal{O}}_K^{\times })$-module $D_A(\pi )$ defined by Breuil-Herzig-Hu-Morra-Schraen, extending their results.

61 pages, 4 appendices