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Pré-Publication, Document De Travail Année : 2022

Moduli of Fontaine--Laffaille representations and a mod-$p$ local-global compatibility result

Résumé

Let $F/F^+$ be a CM field and let $\widetilde{v}$ be a finite unramified place of $F$ above the prime $p$. Let $\overline{r}: \mathrm{Gal}(\overline{\mathbb{Q}}/F)\rightarrow \mathrm{GL}_n(\overline{\mathbb{F}}_p)$ be a continuous representation which we assume to be modular for a unitary group over $F^+$ which is compact at all real places. We prove, under Taylor--Wiles hypotheses, that the smooth $\mathrm{GL}_n(F_{\widetilde{v}})$-action on the corresponding Hecke isotypical part of the mod-$p$ cohomology with infinite level above $\widetilde{v}|_{F^+}$ determines $\overline{r}|_{\mathrm{Gal}(\overline{\mathbb{Q}}_p/F_{\widetilde{v}})}$, when this latter restriction is Fontaine--Laffaille and has a suitably generic semisimplification.
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Dates et versions

hal-03840811 , version 1 (06-11-2022)

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Daniel Le, Bao Viet Le Hung, Stefano Morra, Chol Park, Zicheng Qian. Moduli of Fontaine--Laffaille representations and a mod-$p$ local-global compatibility result. 2022. ⟨hal-03840811⟩
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