## Algebra & Number Theory

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

Jack Shotton

#### Abstract

We compute the deformation rings of two dimensional mod $l$ representations of $Gal(F¯∕F)$ with fixed inertial type for $l$ an odd prime, $p$ a prime distinct from $l$, and $F∕ℚp$ 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 $GL2(OF)$.

#### Article information

Source
Algebra Number Theory, Volume 10, Number 7 (2016), 1437-1475.

Dates
Revised: 14 April 2016
Accepted: 18 July 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ant/1510842566

Digital Object Identifier
doi:10.2140/ant.2016.10.1437

Mathematical Reviews number (MathSciNet)
MR3554238

Zentralblatt MATH identifier
06633202

#### Citation

Shotton, Jack. Local deformation rings for $\operatorname{GL}_2$ and a Breuil–Mézard conjecture when $l \neq p$. Algebra Number Theory 10 (2016), no. 7, 1437--1475. doi:10.2140/ant.2016.10.1437. https://projecteuclid.org/euclid.ant/1510842566

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