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Abstract We prove a local–global compatibility result in the mod $$p$$ Langlands program for $$\mathrm {GL}_2(\mathbf {Q}_{p^f})$$ . Namely, given a global residual representation $$\bar {r}$$ appearing in the mod $$p$$ cohomology of a Shimura curve that is sufficiently generic at $$p$$ and satisfies a Taylor–Wiles hypothesis, we prove that the diagram occurring in the corresponding Hecke eigenspace of mod $$p$$ completed cohomology is determined by the restrictions of $$\bar {r}$$ to decomposition groups at $$p$$ . If these restrictions are moreover semisimple, we show that the $$(\varphi ,\Gamma )$$ -modules attached to this diagram by Breuil give, under Fontaine's equivalence, the tensor inductions of the duals of the restrictions of $$\bar {r}$$ to decomposition groups at $$p$$ . 

Let $$F$$ be a totally real field in which $$p$$ is unramified. Let $$\overline{r}:G_{F}\rightarrow \text{GL}_{2}(\overline{\mathbf{F}}_{p})$$ be a modular Galois representation that satisfies the Taylor–Wiles hypotheses and is tamely ramified and generic at a place $$v$$ above $$p$$ . Let $$\mathfrak{m}$$ be the corresponding Hecke eigensystem. We describe the $$\mathfrak{m}$$ -torsion in the $$\text{mod}\,p$$ cohomology of Shimura curves with full congruence level at $$v$$ as a $$\text{GL}_{2}(k_{v})$$ -representation. In particular, it only depends on $$\overline{r}|_{I_{F_{v}}}$$ and its Jordan–Hölder factors appear with multiplicity one. The main ingredients are a description of the submodule structure for generic $$\text{GL}_{2}(\mathbf{F}_{q})$$ -projective envelopes and the multiplicity one results of Emerton, Gee and Savitt [Lattices in the cohomology of Shimura curves, Invent. Math.   200 (1) (2015), 1–96]. 

We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a $U(3)$ -arithmetic manifold is purely local, that is, only depends on the Galois representation at places above $$p$$ . This is a generalization to $$\text{GL}_{3}$$ of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil–Mézard conjecture for (tamely) potentially crystalline deformation rings with Hodge–Tate weights $(2,1,0)$ as well as the Serre weight conjectures of Herzig [‘The weight in a Serre-type conjecture for tame $$n$$ -dimensional Galois representations’, Duke Math. J.   149 (1) (2009), 37–116] over an unramified field extending the results of Le et al. [‘Potentially crystalline deformation 3985 rings and Serre weight conjectures: shapes and shadows’, Invent. Math.   212 (1) (2018), 1–107]. We also prove results in modular representation theory about lattices in Deligne–Lusztig representations for the group $$\text{GL}_{3}(\mathbb{F}_{q})$$ .