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

Abstract

Let $F/F^{+}$ be a CM field and let $\tilde{v}$ be a finite unramified place of $F$ above the prime $p$. Let $\overline{r}:\mathrm{Gal}(\overline{\mathbb{Q}}/F)\to {\mathrm{GL}}_n({\overline{\mathbb{F}}}_p)$ 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 ${\mathrm{GL}}_n(F_{\tilde{v}})$-action on the corresponding Hecke isotypical part of the mod-$p$ cohomology with infinite level above $\tilde{v}{|}_{F^{+}}$ determines $\overline{r}{|}_{\mathrm{Gal}({\overline{\mathbb{Q}}}_p/F_{\tilde{v}})}$, when this latter restriction is Fontaine--Laffaille and has a suitably generic semisimplification.

Updated the references in the bibliography