Diamond diagrams and multivariable $(\varphi,\mathcal{O}_K^{\times})$-modules
Yitong Wang
TL;DR
This work analyzes mod $p$ representations of GL onde{2}(K) arising from Shimura curves and their associated local Galois representations. By computing explicit constants in the Diamond diagram for non-semisimple residual representations, the authors establish a local–global compatibility: the étale $(\varphi,\mathcal{O}_K^{\times})$-module $D_A(\pi)$ is explicitly determined by the restriction $\bar r_v$ to the decomposition group at $p$, via the functor $D_A^{\otimes}$. The approach combines a detailed study of Iwahori invariants, Kisin modules for tamely potentially Barsotti–Tate deformations, and a careful analysis of $S$-operators to derive concrete scalars controlling the diagram. The main result, $D_A(\pi) \cong D_A^{\otimes}(\bar r_v(1))$, generalizes prior semisimple cases and achieves a concrete realization of the mod $p$ Langlands correspondence in this setting, under Taylor–Wiles and genericity hypotheses at $p$. These computations enable explicit descriptions of the associated Galois–modular correspondence and contribute to a deeper understanding of $p$-adic and mod $p$ Langlands program for GL onde{2}(K).
Abstract
Let $p$ be a prime number and $K$ a finite unramified extension of $\mathbb{Q}_p$. Let $π$ 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 $π$, generalizing the results of Dotto-Le. As a result, we prove that the associated étale $(\varphi,\mathcal{O}_K^{\times})$-module $D_A(π)$ 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.