The $\ell$-modular local Langlands correspondence and local factors

Abstract

Let $F$ be a non-archimedean local field of residual characteristic $p$, $\ell \ne p$ be a prime number, and ${\mathrm{W}}_F$ the Weil group of $F$. We classify the indecomposable ${\mathrm{W}}_F$-semisimple Deligne $\overline{{\mathbb{F}}_{\ell }}$-representations in terms of the irreducible $\overline{{\mathbb{F}}_{\ell }}$-representations of ${\mathrm{W}}_F$, and extend constructions of Artin-Deligne local factors to this setting. Finally, we define a variant of the $\ell$-modular local Langlands correspondence which satisfies a preservation of local factors statement for generic representations.