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 $\text{GL}_{3}(\mathbb{F}_{q})$ .

Federal, state, and local laws shape the use of health information for public health purposes, such as the mandated collection of data through electronic disease reporting systems. Health professionals can leverage these data to better anticipate and plan for the needs of communities, which is seen in the use of electronic case reporting.

We construct, over any CM field, compatible systems of $l$ -adic Galois representations that appear in the cohomology of algebraic varieties and have (for all $l$ ) algebraic monodromy groups equal to the exceptional group of type $E_{6}$ .

We extend the group-theoretic construction of local models of Pappas and Zhu [Local models of Shimura varieties and a conjecture of Kottwitz, Invent. Math. 194 (2013), 147–254] to the case of groups obtained by Weil restriction along a possibly wildly ramified extension. This completes the construction of local models for all reductive groups when $p\geqslant 5$ . We show that the local models are normal with special fiber reduced and study the monodromy action on the sheaves of nearby cycles. As a consequence, we prove a conjecture of Kottwitz that the semi-simple trace of Frobenius gives a central function in the parahoric Hecke algebra. We also introduce a notion of splitting model and use this to study the inertial action in the case of an unramified group.