Let $F$ be a totally real field in which $p$ is unramified. Let $\overline{r}:G_{F}\rightarrow \text{GL}_{2}(\overline{\mathbf{F}}_{p})$ be a modular Galois representation that satisfies the Taylor–Wiles hypotheses and is tamely ramified and generic at a place $v$ above $p$ . Let $\mathfrak{m}$ be the corresponding Hecke eigensystem. We describe the $\mathfrak{m}$ -torsion in the $\text{mod}\,p$ cohomology of Shimura curves with full congruence level at $v$ as a $\text{GL}_{2}(k_{v})$ -representation. In particular, it only depends on $\overline{r}|_{I_{F_{v}}}$ and its Jordan–Hölder factors appear with multiplicity one. The main ingredients are a description of the submodule structure for generic $\text{GL}_{2}(\mathbf{F}_{q})$ -projective envelopes and the multiplicity one results of Emerton, Gee and Savitt [Lattices in the cohomology of Shimura curves, Invent. Math. 200(1) (2015), 1–96].

令$F$是一个完全真实的字段，其中$p$是未分枝的。令$\overline{r}:G_{F}\rightarrow \text{GL}_{2}(\overline{\mathbf{F}}_{p})$是满足 Taylor-Wiles 的模伽罗瓦表示假设并且在$p$上方$ v$的地方被温和地分枝和通用。令$\mathfrak{m}$为对应的 Hecke 特征系统。我们将在$v$处具有完全同余水平的 Shimura 曲线的$\text{mod}\,p$上同调中的$\mathfrak{m}$ -torsion描述为$\text{GL}_{2}(k_ {v})$ - 表示。特别是，它只依赖于$\overline{r}|_{I_{F_{v}}}$ 并且它的 Jordan-Hölder 因子以多重 1 出现。主要成分是对通用$\text{GL}_{2}(\mathbf{F}_{q})$ -射影包络的子模块结构的描述以及Emerton、Gee 和 Savitt [Lattices] 的多重性结果在志村曲线的上同调中，发明。数学。200 (1) (2015), 1-96]。