We formulate and prove the weight part of Serre’s conjecture for three-dimensional mod $p$ Galois representations under a genericity condition when the field is unramified at $p$. This removes the assumption made previously that the representation be tamely ramified at $p$. We also prove a version of Breuil’s lattice conjecture and a mod $p$ multiplicity one result for the cohomology of $U(3)$-arithmetic manifolds. The key input is a study of the geometry of the Emerton–Gee stacks using the local models we introduced previously (2023).

中文翻译：

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三维广泛分支伽罗瓦表示的 Serre 权重

当场在 $p$ 处无分支时，我们在泛性条件下制定并证明了三维模 $p$ 伽罗瓦表示的塞尔猜想的权重部分。这消除了之前所做的假设，即表示在 $p$ 处被驯服地分支。我们还证明了布勒伊格猜想的一个版本以及 $U(3)$ 算术流形上同调的模 $p$ 重数结果。关键输入是使用我们之前介绍的局部模型（2023）对 Emerton-Gee 电堆的几何形状进行研究。