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


Chol Park ( Korea Institute for Advanced Study, South Korea )
2018-02-06  15:00 - 15:50
Room 101, Mathematics Research Center Building (ori. New Math. Bldg.)

Let K be a finite extension of Qp . It is believed that one can attach a smooth Fp -representation of GLn (K ) (or a packet of such representations) to a continuous Galois representation Gal(Qp /K ) → GLn (Fp ) in a natural way, that is called modp Langlands program for GLn (K ).

This conjecture is known only for GL2 (Qp ) : one of the main difficulties is that there is no classification of such smooth representations of GLn (K ) unless K = Qp and n = 2.

However, for a given continuous Galois representation ρ0 : Gal(Qp /Qp ) → GLn (Fp ), one can define a smooth Fp -representation Π0 of GLn (Qp ) 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 ρ0 for modp Langlands correspondence in the spirit of Emerton.

The structure of Π0 is very mysterious as a representation of GLn (Qp ), but it is conjectured that Π0 determine ρ0 .

In this talk, we discuss that Π0 determines ρ0 , provided that ρ0 is ordinary and generic. More precisely, we prove that the tamely ramified part of ρ0 is determined by the Serre weights a ttached to ρ0 , and the wildly ramified part of ρ0 is obtained in terms of refined Hecke actions on Π0 .

This is a joint work with Zicheng Qian.