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

中文翻译：

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SERRE 权重和 BREUIL 的三维格猜想

我们在一般情况下证明，驯服类型中的晶格是由 a 的完全上同调引起的 $U(3)$ -算术流形是纯局部的，也就是说，仅取决于上面位置的伽罗瓦表示 $p$ . 这是一个概括 $\文本{GL}_{3}$ 布勒伊的格子猜想。在这个过程中，我们还用 Hodge-Tate 权重证明了几何 Breuil-Mézard 猜想对于（温和的）潜在结晶变形环 $(2,1,0)$ 以及 Herzig 的 Serre 重量猜想 ['The weight in a Serre-type conjecture for tame $n$ -维伽罗瓦表示'，杜克数学。J。 149(1) (2009), 37–116] 在一个未分支的领域上扩展了 Le 的结果等。['潜在的结晶变形 3985 环和塞尔重量猜想：形状和阴影'，发明。数学。 212(1) (2018), 1-107]。我们还证明了关于群的 Deligne-Lusztig 表示中格的模表示理论的结果 $\text{GL}_{3}(\mathbb{F}_{q})$ .