FIELD Mathematics October 10 (Thu), 2019 16:00-17:00 1423 ¹ÚÃ¶ Choi, Junhwa UNIST [GS_M_NT] Fontaine--Laffaille modules and their mod $p$ local-global compatibility For a given mod $p$ Galois representation $\overline{\rho}$, one can define a mod $p$ automoprhic representation $\overline{\Pi}$ by a certain space of mod $p$ algebraic automoprhic forms on a unitary group. We wish that $\overline{\Pi}$ corresponds to $\overline{\rho}$ for a mod $p$ Langlands correspondence, but the structure of $\overline{\Pi}$ is quite mysterious as a representation. In this talk, we will discuss our approach to the conjecture, mod $p$ local-global compatibility which states that $\overline{\Pi}$ determines $\overline{\rho}$, in the case that $\overline{\rho}$ is Fontaine--Laffaille. This is a joint work with Daniel Le, Bao Le Hung, Stefano Morra, and Zicheng Qian.