The local lattice conjecture for $U(3)$ arithmetic manifolds

Let $\rho$ be the Galois representation attached to an automorphic representation of a unitary group of semisimple rank $2$, and let $\tau_v\subset\pi_v$ be the $K$-type corresponding to $\operatorname{WD}(\rho|_{D_v})$. Let $V_v$ be the algebraic representation corresponding to the Hodge–Tate weights of $\rho|_{D_v}$. Local lattice conjecture. The lattice structure on $\tau_v\otimes V_v$ given by completed cohomology depends only on $\rho|_{D_v}$. This is the $\mathrm{GL}_3$ analogue of Breuil's $\mathrm{GL}_2$ conjecture and proposes a local-global compatibility statement for the integral structure supplied by completed cohomology. The paper establishes related results under Taylor–Wiles hypotheses, but the conjectural dependence solely on the local Galois representation is not stated as resolved here.