Local deformation rings and a Breuil-M\'{e}zard conjecture when l\neq p

TL;DR

This paper computes deformation rings of two-dimensional mod l Galois representations with fixed inertial type over finite extensions of Q_p, establishing an analogue of the Breuil-Mézard conjecture relating these rings to mod l reductions of GL_2 representations.

Contribution

It introduces explicit calculations of deformation rings for mod l Galois representations and proves an analogue of the Breuil-Mézard conjecture in this setting.

Findings

Deformation rings are explicitly computed for the given Galois representations.

An analogue of the Breuil-Mézard conjecture is established for odd primes l and p.

The special fibers of deformation rings relate to mod l reductions of irreducible GL_2 representations.

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).