Duke Mathematical Journal

From pro-p Iwahori–Hecke modules to (φ,Γ)-modules, I

Elmar Grosse-Klönne

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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).

Article information

Duke Math. J., Volume 165, Number 8 (2016), 1529-1595.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F85: $p$-adic theory, local fields [See also 14G20, 22E50]
Secondary: 11F80: Galois representations 11F70: Representation-theoretic methods; automorphic representations over local and global fields

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


Grosse-Klönne, Elmar. From pro- $p$ Iwahori–Hecke modules to $(\varphi,\Gamma)$ -modules, I. Duke Math. J. 165 (2016), no. 8, 1529--1595. doi:10.1215/00127094-3450101. https://projecteuclid.org/euclid.dmj/1456412785

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