Diamond diagrams and multivariable $(\ddot{I}†,{\mathcal{O}}_K^{\times })$-modules

Abstract

Let $p$ be a prime number and $K$ a finite unramified extension of ${\mathbb{Q}}_p$. Let $\ddot{I}€$ be an admissible smooth mod $p$ representation of ${\mathrm{GL}}_2(K)$ occurring in some Hecke eigenspaces of the mod $p$ cohomology and $\overline{r}$ be its underlying global two-dimensional Galois representation. When $\overline{r}$ satisfies some Taylor-Wiles hypotheses and is sufficiently generic at $p$, we compute explicitly certain constants appearing in the diagram associated to $\ddot{I}€$, generalizing the results of Dotto-Le. As a result, we prove that the associated étale $(\ddot{I}†,{\mathcal{O}}_K^{\times })$-module $D_A(\ddot{I}€)$ defined by Breuil-Herzig-Hu-Morra-Schraen is explicitly determined by the restriction of $\overline{r}$ to the decomposition group at $p$, generalizing the results of Breuil-Herzig-Hu-Morra-Schraen and the author.

34 pages