Local deformation rings for GL2 and a Breuil–Mézard conjecture when l≠p

Abstract

We compute the deformation rings of two dimensional mod l representations of Gal(Fbar/F) with fixed inertial type, for l an odd prime, p a prime distinct from p and F/Q_p a finite extension. We show that in this setting (when p is also odd) an analogue of the Breuil-M\'{e}zard conjecture holds, relating the special fibres of these deformation rings to the mod l reduction of certain irreducible representations of GL_2(O_F).