Diagrams in the mod p cohomology of Shimura curves
Diagrams in the mod p cohomology of Shimura curves
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DOI：
10.1112/s0010437x21007375
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发表时间：
2019-09
影响因子：
1.8
通讯作者：
Andrea Dotto;Daniel Le
中科院分区：
文献类型：
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作者：
Andrea Dotto;Daniel Le
摘要：我们证明了$\mathrm {GL}_2(\mathbf {Q}_{p^f})$的mod $p$ Langlands程序的局部全局兼容结果。即，给定一条Shimura曲线的mod $p$上同调中出现的一个全局残量表示$\bar {r}$，该曲线在$p$处是充分泛型的，并且满足Taylor-Wiles假设，我们证明了在mod $p$完全上同调的相应Hecke特征空间中出现的图是由$\bar {r}$对$p$处分解群的限制决定的。如果这些限制是半简单的，我们证明了由Breuil附在图上的$(\varphi ,\Gamma )$ -模，在Fontaine等价下，给出了$\bar {r}$的限制对偶到$p$处分解群的张量归纳。
Abstract We prove a local–global compatibility result in the mod $p$ Langlands program for $\mathrm {GL}_2(\mathbf {Q}_{p^f})$. Namely, given a global residual representation $\bar {r}$ appearing in the mod $p$ cohomology of a Shimura curve that is sufficiently generic at $p$ and satisfies a Taylor–Wiles hypothesis, we prove that the diagram occurring in the corresponding Hecke eigenspace of mod $p$ completed cohomology is determined by the restrictions of $\bar {r}$ to decomposition groups at $p$. If these restrictions are moreover semisimple, we show that the $(\varphi ,\Gamma )$-modules attached to this diagram by Breuil give, under Fontaine's equivalence, the tensor inductions of the duals of the restrictions of $\bar {r}$ to decomposition groups at $p$.