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arXiv · 1909.12219

Diagrams in the mod $p$ cohomology of Shimura curves

Andrea Dotto·Daniel Le
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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 $\bar{r}$ that is sufficiently generic at $p$, we prove that the diagram occurring in the corresponding Hecke eigenspace of 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 $p$.

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Andrea Dotto, Daniel Le. 2019-09-26. Diagrams in the mod $p$ cohomology of Shimura curves. https://doi.org/10.1112/s0010437x21007375

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