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
$\text{GL}_{3}(\mathbb{F}_{q})$
.
On deformation rings of residually reducible Galois representations and R = T theorems
