Talks## MOD <i>p</i> local-global compatibility for GL<i><sub>n</i></sub> (Q<i><sub>p</i></sub> ) in the ordinary case

**43**

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2018-02-06

15:00 - 15:50

101 , Mathematics Research Center Building (ori. New Math. Bldg.)

Let *K* be a finite extension of Q* _{p}* . It is believed that one can attach a smooth F

*-representation of GL*

_{p}*(*

_{n}*K*) (or a packet of such representations) to a continuous Galois representation Gal(Q

*/*

_{p}*K*) → GL

*(F*

_{n}*) in a natural way, that is called modp Langlands program for GL*

_{p}*(*

_{n}*K*).

This conjecture is known only for GL

*(Q*

_{2}*) : one of the main difficulties is that there is no classification of such smooth representations of GL*

_{p}*(*

_{n}*K*) unless

*K*= Q

*and*

_{p}*n*= 2.

However, for a given continuous Galois representation ρ

_{0}: Gal(Q

*/Q*

_{p}*) → GL*

_{p}*(F*

_{n}*), one can define a smooth F*

_{p}*-representation Π*

_{p}_{0}of GL

*(Q*

_{n}*) by a space of mod p automorphic forms on a compact unitary group, which is believed to be a candidate on the automorphic side corresponding to ρ*

_{p}_{0}for modp Langlands correspondence in the spirit of Emerton.

The structure of Π

_{0}is very mysterious as a representation of GLn (Q

*), but it is conjectured that Π*

_{p}_{0}determine ρ

_{0}.

In this talk, we discuss that Π

_{0}determines ρ

_{0}, provided that ρ

_{0}is ordinary and generic. More precisely, we prove that the tamely ramified part of ρ

_{0}is determined by the Serre weights a ttached to ρ

_{0}, and the wildly ramified part of ρ

_{0}is obtained in terms of refined Hecke actions on Π

_{0}.

This is a joint work with Zicheng Qian.