Supersingular Hecke modules as Galois representations

Grosse-Klönne, Elmar

doi:10.2140/ant.2020.14.67

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Supersingular Hecke modules as Galois representations

Elmar Grosse-Klönne

Vol. 14 (2020), No. 1, 67–118

DOI: 10.2140/ant.2020.14.67

Abstract

Let $F$ be a local field of mixed characteristic $(0, p)$ , let $k$ be a finite extension of its residue field, let $ℋ$ be the pro- $p$ -Iwahori Hecke $k$ -algebra attached to ${GL}_{d + 1} (F)$ for some $d \geq 1$ . We construct an exact and fully faithful functor from the category of supersingular $ℋ$ -modules to the category of $Gal (\bar{F} ∕ F)$ -representations over $k$ . More generally, for a certain $k$ -algebra $ℋ^{♯}$ surjecting onto $ℋ$ we define the notion of $♯$ -supersingular modules and construct an exact and fully faithful functor from the category of $♯$ -supersingular $ℋ^{♯}$ -modules to the category of $Gal (\bar{F} ∕ F)$ -representations over $k$ .

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Keywords

pro-$p$ Iwahori Hecke algebra, supersingular module, Galois representation, $(\varphi, \Gamma)$-module

Mathematical Subject Classification 2010

Primary: 11F85

Milestones

Received: 10 December 2018

Revised: 6 May 2019

Accepted: 1 September 2019

Published: 15 March 2020

Authors

Elmar Grosse-Klönne
Mathematisch-Naturwissenschaftliche Fakultät, Institut für Mathematik Humboldt-Universität zu Berlin Germany

Vol. 14, No. 1, 2020