On mod p local-global compatibility for GLn(Qp)


Salle Kampé de Fériet
Zicheng Qian
Université Paris-Sud
Jeudi, 22 Novembre, 2018 - 11:00 - 12:00

We fix a prime $p$, a sufficiently large finite field $\mathbf{k}$ with characteristic $p$ and a number field $F$ such that its completion at certain finite place $v$ is $F_v=\mathbf{Q}_p$. We consider a $n$-dimensional continuous representation $\overline{r}: \mathrm{Gal}(\overline{F}/F)\rightarrow\mathrm{GL}_n(\mathbf{k})$ as well as $\overline{\rho}:=\overline{r}|_{\mathrm{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)}$. There is a standard method to start with $\overline{r}$ and construct a smooth admissible $k$-representation $\Pi(\overline{r})$ of $\mathrm{GL}_n(\mathbf{Q}_p)$. In this talk, we are going to quickly recall $\Pi(\overline{r})$ and then sketch how to prove that $\Pi(\overline{r})$ determines $\overline{\rho}$ when $\overline{\rho}$ is Fontaine-Laffaille with a sufficiently generic semisimplification, generalizing the previous results including Park-Qian, Le-Morra-Park and Herzig-Le-Morra. This result is widely expected due to the philosophy of mod $p$ Langlands correspondence. This result is a combination of several results by Bao Le Hung, Brandon Levin, Daniel Le, Stefano Morra, Chol Park and myself.