#### Vol. 10, No. 7, 2016

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

### Jack Shotton

Vol. 10 (2016), No. 7, 1437–1475
##### 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 $F∕{ℚ}_{p}$ 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 ${GL}_{2}\left({\mathsc{O}}_{F}\right)$.

##### Keywords
Galois representations, deformation rings, local Langlands, Breuil–Mézard
Primary: 11S37
##### Milestones
Received: 28 April 2015
Revised: 14 April 2016
Accepted: 18 July 2016
Published: 27 September 2016
##### Authors
 Jack Shotton University of Chicago 5442 S Ellis Ave Chicago, IL 60615 United States