Representations modulo p of the p-adic group GL(2, F)

By M. Vigneras. Compositio Mathematica, 2004

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The classification of irreducible admissible mod p representations of a p-adic GLn (2010, methodology): “The map ϕ 7 ϕ|M from (2.21) to (2.20) (resp., from (2.22) to (2.20)) is an algebra isomorphism. (compare with Vigneras [ Vig04 ])”
The image of Colmez’s Montreal functor (2010, methodology): “Moreover, Vignéras in [64, Thm.5.4] shows that I induces a bijection between irreducible objects in ModsmG,ζ(k) and ModH.”
Pro-p-Iwahori Hecke ring and supersingular -representations (2005, background): “When n = 2, the functor of Iw(1)-invariants gives a bijection [10] – from the irreducible NON supercuspidal smooth Fp-representations of GL(2, F ) (subquotients of parabolic induced representations), to the NON supersingular irreducible H Fp (2, q)-modules, – when F = Qp, from the irreducible superc”
Parabolic induction and Hecke modules in characteristic p for p-adic GLn (2010, background): “In the case where L is the diagonal torus T and M is a character of AT , the map F is an isomorphism [Schneider and Stuhler 1991; Vignéras 2004].”