Supersingular Hecke modules as Galois representations

Abstract

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

Slight rearrangements and improvements