Supersingular Hecke modules as Galois representations

TL;DR

This paper constructs a functor linking supersingular modules over a Hecke algebra associated with GL_{d+1}(F) to Galois representations, revealing deep connections between algebraic and Galois-theoretic structures.

Contribution

It introduces a new exact and fully faithful functor from supersingular Hecke modules to Galois representations, extending to a broader class of modules via a surjective algebra.

Findings

Establishment of a functor from supersingular Hecke modules to Galois representations

Extension of the functor to a broader class of modules using a surjective algebra

Deepening the understanding of the relationship between Hecke modules and 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 ${\rm GL}_{d+1}(F)$ for some $d\ge1$. We construct an exact and fully faithful functor from the category of supersingular ${\mathcal H}$-modules to the category of ${\rm Gal}(\overline{F}/F)$-representations over $k$. More generally, for a certain $k$-algebra ${\mathcal H}^{\sharp}$ surjecting onto ${\mathcal H}$ we define the notion of $\sharp$-supersingular modules and construct an exact and fully faithful functor from the category of $\sharp$-supersingular ${\mathcal H}^{\sharp}$-modules to the category of ${\rm Gal}(\overline{F}/F)$-representations over $k$.