#### Vol. 14, No. 1, 2020

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

##### Keywords
pro-$p$ Iwahori Hecke algebra, supersingular module, Galois representation, $(\varphi, \Gamma)$-module
Primary: 11F85