Let p be a prime number and K a finite unramified extension of $\mathbb{Q}_p$. Let $\pi$ be an admissible smooth mod p representation of $\operatorname{GL}_2(K)$ 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 $\pi$, generalizing the results of Dotto and Le. As a result, we prove that the associated étale $(\varphi,\mathcal{O}_K^{\times})$-module $D_A(\pi)$ defined by Breuil, Herzig, Hu, Morra and Schraen is explicitly determined by the restriction of $\overline{r}$ to the decomposition group at p, generalizing the results of Breuil et al. and the author.

We present a new construction of the $p$ -adic local Langlands correspondence for $\operatorname{GL}_{2}(\mathbb{Q}_{p})$ via the patching method of Taylor–Wiles and Kisin. This construction sheds light on the relationship between the various other approaches to both the local and the global aspects of the $p$ -adic Langlands program; in particular, it gives a new proof of many cases of the second author’s local–global compatibility theorem and relaxes a hypothesis on the local mod $p$ representation in that theorem.

We compute the image of any choice of complex conjugation on the Galois representations associated to regular algebraic cuspidal automorphic representations and to torsion classes in the cohomology of locally symmetric spaces for $\text{GL}_{n}$ over a totally real field $F$ .