Local deformation rings for $\operatorname{GL}_2$ and a Breuil–Mézard conjecture when $l \neq p$

Abstract

We compute the deformation rings of two dimensional mod $l$ representations of $Gal\left(\overline{F}∕F\right)$ with fixed inertial type for $l$ an odd prime, $p$ a prime distinct from $l$, and $F∕{\mathbb{Q}}_p$ a finite extension. We show that in this setting an analogue of the Breuil–Mézard conjecture holds, relating the special fibres of these deformation rings to the mod $l$ reduction of certain irreducible representations of ${GL}_2\left({\mathcal{O}}_F\right)$.