Multiplicity one for wildly ramified representations
Multiplicity one for wildly ramified representations
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DOI：
10.2140/ant.2019.13.1807
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发表时间：
2017-08
影响因子：
1.3
通讯作者：
Daniel Le
中科院分区：
文献类型：
--
作者：
Daniel Le
设$F$是一个完全实数域，其中$p$是无分支的。设$\overline{r}：G_F\right tarrow\mathm{GL}_2(\overline{\mathbb{F}}_p)$是满足Taylor-Wiles假设的模Galois表示，并且在$v$上$p$处是一般的。设$\mathfrak{m}$是相应的Hecke本征系。则全同余水平为$v$的Shimura曲线的mod$p$上同调中的$mathfrak{m}$-挠与由Breuil和Pav{S}k={u}Nas构造的$\mathfrak{m}$-表示$D_0(上线{r}|_{G_{F_v})$重合.特别地，它只依赖于局部表示{r}|{G_{F_v}}，而且它的Jordan-H‘老因子是以1的重数出现的.这是作者与Morra和Schraen的工作的基础和推广，不依赖于Hu-Wang，证明了当另外假设$overline{r}|_{G_{F_v}}是温和分枝的情况下的这些结果.主要的新工具是利用多类型Tanly潜在Barsotti-Tate变形环及其交理论来计算积分射影包络的Taylor-Wiles面片模.
Let $F$ be a totally real field in which $p$ is unramified. Let $\overline{r}: G_F \rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a modular Galois representation which satisfies the Taylor-Wiles hypotheses and is generic at a place $v$ above $p$. Let $\mathfrak{m}$ be the corresponding Hecke eigensystem. Then the $\mathfrak{m}$-torsion in the mod $p$ cohomology of Shimura curves with full congruence level at $v$ coincides with the $\mathrm{GL}_2(k_v)$-representation $D_0(\overline{r}|_{G_{F_v}})$ constructed by Breuil and Pa\v{s}k\={u}nas. In particular, it depends only on the local representation $\overline{r}|_{G_{F_v}}$, and its Jordan-H\"older factors appear with multiplicity one. This builds on and extends work of the author with Morra and Schraen and independently of Hu-Wang, which proved these results when $\overline{r}|_{G_{F_v}}$ was additionally assumed to be tamely ramified. The main new tool is a method for computing Taylor-Wiles patched modules of integral projective envelopes using multitype tamely potentially Barsotti-Tate deformation rings and their intersection theory.