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

Shotton, Jack

doi:10.2140/ant.2016.10.1437

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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

DOI: 10.2140/ant.2016.10.1437

Abstract

We compute the deformation rings of two dimensional mod $l$ representations of $Gal (\bar{F} ∕ F)$ 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 ${GL}_{2} (O_{F})$ .

Keywords

Galois representations, deformation rings, local Langlands, Breuil–Mézard

Mathematical Subject Classification 2010

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

Vol. 10, No. 7, 2016