Supersingular Hecke modules as Galois representations

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 $\mathcal{\mathscr{H}}$ 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 $\mathcal{\mathscr{H}}$-modules to the category of $Gal\left(\bar{F}∕F\right)$-representations over $k$. More generally, for a certain $k$-algebra ${\mathcal{\mathscr{H}}}^{♯}$ surjecting onto $\mathcal{\mathscr{H}}$ we define the notion of $♯$-supersingular modules and construct an exact and fully faithful functor from the category of $♯$-supersingular ${\mathcal{\mathscr{H}}}^{♯}$-modules to the category of $Gal\left(\bar{F}∕F\right)$-representations over $k$.