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KAIST 수리과학과 세미나 



Mod p localglobal compatibility for GL_n(Q_p) in the ordinary 

Let K be a finite extension of Qp. It is believed that one can attach a smooth Fprepresentation 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 mod p 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 Fprepresentation 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 mod p Langlands correspondence in the spirit of Emerton. The structure of Π0 is very mysterious as a representation of GLn(Qp), and it is not known that ρ0 and Π0 determine each other. 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 attached to ρ0 , and the wildly ramified part of ρ0 is obtained in terms of refined Hecke actions on Π0. The talk is based on a joint work with Zicheng Qian. 







