1 June 2016 From pro-p Iwahori–Hecke modules to (φ,Γ)-modules, I
Elmar Grosse-Klönne
Duke Math. J. 165(8): 1529-1595 (1 June 2016). DOI: 10.1215/00127094-3450101


Let o be the ring of integers in a finite extension K of Qp, and let k be its residue field. Let G be a split reductive group over Qp, and let T be a maximal split torus in G. Let H(G,I0) be the pro-p Iwahori–Hecke o-algebra. Given a semi-infinite reduced chamber gallery (alcove walk) C() in the T-stable apartment, a period ϕN(T) of C() of length r, and a homomorphism τ:Zp×T compatible with ϕ, we construct a functor from the category Modfin(H(G,I0)) of finite-length H(G,I0)-modules to étale (φr,Γ)-modules over Fontaine’s ring OE. If G=GLd+1(Qp), then there are essentially two choices of (C(),ϕ,τ) with r=1, both leading to a functor from Modfin(H(G,I0)) to étale (φ,Γ)-modules and hence to GalQp-representations. Both induce a bijection between the set of absolutely simple supersingular H(G,I0)ok-modules of dimension d+1 and the set of irreducible representations of GalQp over k of dimension d+1. We also compute these functors on modular reductions of tamely ramified locally unitary principal series representations of G over K. For d=1, we recover Colmez’s functor (when restricted to o-torsion GL2(Qp)-representations generated by their pro-p Iwahori invariants).


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Elmar Grosse-Klönne. "From pro-p Iwahori–Hecke modules to (φ,Γ)-modules, I." Duke Math. J. 165 (8) 1529 - 1595, 1 June 2016. https://doi.org/10.1215/00127094-3450101


Received: 14 February 2014; Revised: 17 July 2015; Published: 1 June 2016
First available in Project Euclid: 25 February 2016

zbMATH: 1364.11103
MathSciNet: MR3504178
Digital Object Identifier: 10.1215/00127094-3450101

Primary: 11F85
Secondary: 11F70 , 11F80

Keywords: Galois representation , Iwahori–Hecke algebra , Mod $p$ representation , modular local Langlands program , supersingular module

Rights: Copyright © 2016 Duke University Press

Vol.165 • No. 8 • 1 June 2016
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