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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 Q_p . It is believed that one can attach a smooth F_p -representation of GL_n (K ) (or a packet of such representations) to a continuous Galois representation Gal(Q_p /K ) → GL_n (F_p ) in a natural way, that is called modp Langlands program for GL_n (K ).

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

However, for a given continuous Galois representation ρ₀ : Gal(Q_p /Q_p ) → GL_n (F_p ), one can define a smooth F_p -representation Π₀ of GL_n (Q_p ) 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 (Q_p ), 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.